THE LOGIC OF EXPRESSION

Engaging with the challenging and controversial reading of Spinoza presented by Gilles Deleuze in Expressionism in Philosophy (1968), this book focuses on Deleuze’s redeployment of Spinozist concepts within the context of his own philosophical project of constructing a philosophy of difference as an alternative to the Hegelian dialectical philosophy.

Duffy demonstrates that a thorough understanding of Deleuze’s Spinozism is necessary in order to fully engage with Deleuze’s philosophy of difference.

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The Logic of Expression

Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze

SIMON DUFFY
University of Sydney, Australia

First published 2006 by Ashgate Publishing

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British Library Cataloguing in Publication Data
Duffy, Simon

The logic of expression: quality, quantity and intensity in Spinoza, Hegel and Deleuze. – (Ashgate new critical thinking in philosophy) 1.Spinoza, Benedictus de, 1632–1677 2.Deleuze, Gilles 3.Hegel, Georg Wilhelm Friedrich, 1770–1831 4.Expression (Philosophy) 5.Logic 6.Philosophy – History
I. Title
121.6’8

Library of Congress Cataloging-in-Publication Data
Duffy, Simon.

 The logic of expression: quality, quantity, and intensity in Spinoza, Hegel, and Deleuze / Simon Duffy.

 p. cm. – (Ashgate new critical thinking in philosophy)
Includes bibliographical references and index.
ISBN 0-7546-5618-7 (hardcover: alk. paper)

 1. Spinoza, Benedictus de, 1632–1677. 2. Expression. 3. Deleuze, Gilles. 4. Hegel, Georg Wilhelm Friedrich, 1770–1831. I. Title. II. Series.

B3999.E9D84 2006
190–dc22

2006008837

ISBN 13: 978-0-7546-5618-0 (hbk)

To Lizzy, Patrick, Oliver, Gus & Harriet …
and in memory of Stephen Delmore Foster

Acknowledgements

I would like to thank the following institutions for facilitating the research that has gone into this book: the Centre for the History of European Discourses at the University of Queensland; the Department of Philosophy at the University of Sydney; The New School Postgraduate Research Center, New York; and the Department of Philosophy at the University of Paris X-Nanterre.

In relation to the research that went in to producing the book, I’d like to thank in particular Etienne Balibar, for imparting a wider appreciation of the discipline of philosophy; Paul Patton, for his invaluable suggestions of possible directions to take in my research; Dan Smith, for sharing his expertise in the field of Deleuze studies; and Moira Gatens, both for sharing her expertise in the field of Spinoza studies, and for her support and generosity while I have been working on this project.

In the Department of Philosophy at the University of Sydney, I’d like to thank Paul Redding, for his comments on the earlier drafts of my engagement with Hegel; and, in the Department of French Studies, Ross Steele, for his generous assistance and support during my language studies.

I am indebted to Brooke, Gustavo and Lucianna Fasani for their generous support during my stay in New York while preparing the final chapters of the book; and also to Eli King and Robin Auchincloss, and to Jo Gould, James and Stephen Anspacher.

Those friends and colleagues who have been supportive, or provided commentary, and relief, at the different stages of the writing process, have been instrumental to the completion of this project. I would like to thank in particular: Dechen and Vincent Bejon, Lucianne and Ruby Brady, Ada Meinch, Rachel Ann Wood, Greg Johnston, Alex Squire, Peta Mcdowell, Nicholas van Stom, Maren Goerdel, Phil Young, Justine McGill, Yuji Sone, Nicola O’Shea, Sean Bowden, David Curtin and Sandra Field. A special mention also to my family, without whose support the undertaking of this project would not have been possible.

Parts of chapters 2, 3 and 10, and the conclusion, appear in the article ‘The Logic of Expression in Deleuze’s Expressionism in Philosophy: Spinoza. A strategy of engagement’ published in the International Journal of Philosophical Studies 12.1 (2004). Parts of chapters 2 and 3 appear in the article ‘Schizo-Math. The logic of different/ciation and the philosophy of difference’ published in Angelaki 9.3 (2004). I gratefully acknowledge the permission of the publisher of these journals, the Routledge, Taylor & Francis Group, to republish this material. An edited and reworked version of the Angelaki article appears as chapter 6 of Virtual Mathematics: the logic of difference (Clinamen, 2005), Simon Duffy (ed.), entitled ‘The Mathematics of Deleuze’s differential logic and metaphysics’. I gratefully acknowledge the permission of the publisher at Clinamen Press to republish this material. Parts of chapters 2 and 3 appear in the article ‘The differential point of view of the infinitesimal calculus in Spinoza, Leibniz and Deleuze’ published in the Journal of the British Society for Phenomenology (forthcoming, 2006). I gratefully acknowledge the permission of the editors of the JBSP to republish this material.

Introduction

Spinoza and the problem of expression

The reading of Spinoza that Gilles Deleuze presents in Expressionism in Philosophy¹ is both challenging and controversial: challenging from the point of view of the complexity with which it engages the ideas of Spinoza; and controversial from the point of view of the extent to which it serves to redeploy Spinoza within the context of Deleuze’s own philosophical project. While closely examining his reading of Spinoza, the present work focuses on the more controversial issue of Deleuze’s Spinozism, or the way in which Deleuze redeploys Spinoza, or the Spinozist concepts that he extracts from Spinoza’s philosophy, in his project of constructing a philosophy of difference. Deleuze’s Spinozism is examined in relation to both Expressionism in Philosophy and Difference and Repetition,² and to the seminars that Deleuze gave on Spinoza.³ What is proposed therefore is a Deleuzian reading of Expressionism in Philosophy that positions itself within the trajectory of the development of Deleuze’s philosophy.

Deleuze’s reading of Spinoza is explicated within the context of contemporary French Spinoza studies, particularly in relation to the work of Martial Gueroult and Pierre Macherey. However, it is in relation to Hegel’s interpretation of Spinoza and the position that Hegel assigns to Spinoza in both the dialectical progression of the history of philosophy and the development of his dialectical logic that Deleuze strategically redeploys Spinoza. The process of actualization determined by the Hegelian dialectical logic in relation to the history of philosophy is determinately linear and progressive, insofar as it is predominantly preoccupied with overcoming moments of discontinuity, an example of which would be the system of the philosophy of Spinoza, each of which is ‘at the centre of the necessity of an evolutionary process’, which determines ‘the continuation of “history”’.⁴

In ‘Spinoza et la méthode de Gueroult’, when reviewing Gueroult’s twovolume study of Spinoza,⁵ Deleuze details those elements of Gueroult’s method of engagement with Spinoza that he considers to have had a profound effect both on contemporary Spinoza studies, and also on the role of the history of philosophy in the contemporary practice of philosophy in general. He argues that ‘Gueroult renewed the history of philosophy by a structural–genetic method, which he had elaborated well before structuralism imposed itself in other domains. Such a structure is defined by an order of reasons, the reasons being differential and generative elements of the corresponding system, veritable philosophemes which only exist in their relations with each other’.⁶ ‘Gueroult’s admirable book’, he continues, ‘has a double importance, both from the point of view of the general method that it puts to work, and from the point of view of his Spinozism, which does not represent one application of this method among others but rather constitutes the most adequate, the most saturated, the most exhaustive term or object concluding the series on Descartes, Malebranche and Leibniz. This book founds the veritable scientific study of Spinozism’.⁷

This characterization of the method of Gueroult actually outlines Deleuze’s own philosophical project in Expressionism in Philosophy. It is there that Deleuze undertakes the project of renewing the history of philosophy in relation to Spinoza. The structural–genetic criteria that Deleuze deploys are determined according to a logic of expression, which is elaborated in Difference and Repetition in relation to the differential calculus as a logic of different/ciation. The ‘differential and generative elements’ of this logic exist solely in the differential relations that they have with each other. Expressionism in Philosophy is also doubly important, not only insofar as it too advocates the ‘scientific study of Spinozism’, in particular Spinoza’s relation to mathematics and the differential calculus of Leibniz, but also insofar as it is the final text of a series of works on figures in the history of philosophy: Hume, Nietzsche, Bergson,⁸ which constitutes Deleuze’s project of renewing the history of philosophy by constructing an alternative lineage in the history of philosophy. With this series of texts Deleuze proposes a new way of reading the figures of the history of philosophy, ‘neither as the possessors of a truth whose reactualization is being attempted, nor as the objects of an infinite “deconstruction”, nor as the occupants of a privileged enclave accessible only to the erudite historian, but as interlocutors’ directly implicated in ‘the construction and experimentation of new [modes] of thought’.⁹

In ‘The encounter with Spinoza’, Macherey critiques Deleuze’s reading of Spinoza by questioning whether it is ‘consistent with the original sense of the work he purports to analyze, or does it rather misrepresent Spinoza’s philosophy’.¹⁰ Macherey maintains that Deleuze attempts to ‘introduce a new version of Spinozism that [is] at variance, if not completely at odds, with the model of demonstrative rationality explicitly adopted by Spinoza himself’.¹¹ Macherey is one of the most respected of contemporary Spinoza scholars in France, whose fivevolume study of the Ethics, entitled Introduction à l’Ethique de Spinoza,¹² attempts, by extracting the persistent difficulties of Spinozism, to establish an internal coherence to the system of the text. However, it is in relation to Hegel’s interpretation of Spinoza that Macherey first distinguished himself as a Spinoza scholar of repute. Chapter 1 examines his first major work on Spinoza, Hegel ou Spinoza,¹³ which challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. Macherey argues that Hegel provides a defensive misreading of Spinoza, and that he had to ‘misread him’ in order to maintain his teleological subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can be simply sublated and subsumed within the dialectical progression of the history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism.

Deleuze also considers Spinoza’s philosophy to resist the totalizing effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. Rather than attempting to determine the elements of Deleuze already in Hegel,¹⁴ or the adequacy of the Hegelian elements in Deleuze,¹⁵ or elaborating a Deleuzian Hegel¹⁶ or a Hegelian Deleuze,¹⁷ or, on the contrary, claiming that ‘no encounter is possible’,¹⁸ an alternative approach to the determination of the relation between Deleuze and Hegel is to read their respective interpretations of Spinoza, and the role that they each assign to Spinoza in the development of their respective philosophical projects, together, alongside of each other, and also in direct relation to each other. This strategy entails examining those points of convergence between the elements of their respective interpretations of Spinoza, in order then to determine what sets them radically apart. It is in the logic of the relations established between the elements of their respective interpretations of Spinoza that this radical difference is manifested, and it is by means of the determination of this difference in logic between their respective interpretations of Spinoza that the difference in the logic of their respective philosophical projects in general is determined.

Spinoza’s role in this strategy is demonstrated in chapter 2 by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) from that which Hegel presents in the Science of Logic.¹⁹ Both Hegel and Deleuze each position the geometrical example at different stages in the early development of the differential calculus. By demonstrating the relation between ‘the differential point of view of the infinitesimal calculus’ and the differential calculus of contemporary mathematics, it is argued in chapter 3 that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic. By exploiting the implications of the differential point of view of the infinitesimal calculus in his interpretation of the physics of bodies in the second part of the Ethics, Deleuze is able to read the system of the Ethics as a whole as determined according to the logic of different/ciation. The explication of this reading strategy is what constitutes the Deleuzian reading of Expressionism in Philosophy that is undertaken in the present work. This strategy of reading the Ethics as determined according to a logic of different/ciation marks not only the originality of Deleuze’s interpretation of Spinoza, but also one of the points where Deleuze can be considered to depart from the Cartesian and Hegelian Spinoza familiar to scholars working in the field of Spinoza studies by tracing an alternative lineage in the history of philosophy between Spinoza’s ontology and the mathematics of Leibniz.

One of the recurring critiques of Deleuze’s reading of Spinoza in Expressionism in Philosophy is that the Deleuzian concepts inspired by Spinozist ideas are nowhere to be found in the text of the Ethics. Macherey conducts his critique of Expressionism in Philosophy in ‘The encounter with Spinoza’ by disputing such Deleuzian concepts, in particular the concept of ‘joyful passive affections’, which is addressed in chapter 7, and the concept of ‘expression’, which is addressed in chapter 10. In quality and quantity in the philosophy of Spinoza, Charles Ramond disputes Deleuze’s use of the concept of ‘intensive quantity’ in Expressionism in Philosophy. He argues that ‘we can … only be surprised to see Deleuze make the notion of “intensive quantity” the foundation of his interpretation of Spinoza’s theory of the essence of singular things, as if Spinoza had effectively supported such a position’. In relation to the concept of ‘intensity’ and to that of ‘intensive parts’ Ramond argues that: ‘According to Deleuze, Spinoza locates, by using the notion of “intensity”, “a long Scholastic tradition”, of which only “Scotism”, without more precision, is evoked’; and that ‘When Deleuze … declares that, in Spinoza, “modal essences are … intensive parts”, he utters an assertion strictly incomprehensible within the framework of Spinozism’.²⁰ However, these Deleuzian concepts are, more often than not, considered and criticized for themselves, independently of the context which supports their exposition because of its extremely technical character. It is by means of these concepts in Expressionism in Philosophy that Deleuze shifts the reading of the philosophy of Spinoza onto an entirely different terrain. Rather than reading the philosophy of Spinoza either solely in relation to Descartes, or simply as one stage in the dialectical progression of the history of philosophy, Deleuze reads the philosophy of Spinoza insofar as it contributes to the construction of an alternative lineage in the history of philosophy, which features the work of Duns Scotus, in particular the concept of formal distinction, the concept of the univocity of being and the concept of individuality. Deleuze does not read Spinoza as a Scotist, but rather, by reading Scotus alongside of Spinoza, he examines Spinoza’s reformulation of these Scotist concepts in order to develop those aspects of Spinoza’s philosophy that are specifically Anticartesian. The mechanism by means of which Deleuze does this is elaborated in chapter 4. This Anticartesian reading of Spinoza provides the framework for the development of a reading of Spinoza that challenges the Hegelian concept of the dialectical progression of the history of philosophy. The Deleuzian domain of engagement with Spinoza is determined therefore by deterritorializing a fairly traditional reading of Spinoza from a particularly Cartesian and Hegelian point of view, to that of a more Scotist and even Leibnizian point of view, thereby determining an alternative lineage in the history of philosophy.

Deleuze considers Spinoza’s Ethics to have given expression to the concept of individuality, whose themes can be found scattered among several other authors of the seventeenth century: for Deleuze’s purposes most notably in the work of Duns Scotus. In relation to the concept of individuality, the Hegelian categories of quantity, quality and relation, which articulate differences in the distances or dimensions of things (quantity), in their nature (quality), and in their order (relation), are considered by Deleuze to be antedated by the three different dimensions of the individual as presented by Spinoza, namely relation, power (quantity), and mode (quality). The individual is characterized as relation insofar as there is a composition of individuals in relation to one another, or amongst themselves. In chapter 2 it is argued that the infinitesimal calculus puts into play a certain type of relation for Deleuze – the differential relation – which is characteristic of the compositional relations between individuals. The characterization of the individual as power (potentia) indicates the individual’s capacity to compose new relations with other individuals. The concept of potentia, which is translated by the concept of power as force or capacity, is different to that of potestas, which is translated by power in the juridico–political sense of the term,²¹ insofar as it expresses that which an individual body can do, and which is verified by joy. Composition therefore refers not only to the characteristic relations between individuals, but also to the capacity or potential to create these kinds of relations, the mechanism of which is determined as operating according to the logic of different/ciation, and is elaborated in chapter 3. The third dimension of the individual is characterized as mode, which Deleuze considers to be expressive of the Scotist concept of an ‘intrinsic mode’. Deleuze’s characterization of the ‘intrinsic mode’ of an individual as the ‘intensive part’, or singular modal essence, of an individual, or of its corresponding finite existing mode, is elaborated in chapter 4. The mechanism by means of which intensive parts are distinguished not only from one another, but also from the extensive parts corresponding to them, is determined as operating according to the logic of different/ciation. The question of how the intensive parts of different individuals are differentiated from one another as more or less powerful is then investigated in chapter 5. Chapter 6 raises the question of whether the power, or capacity, of an individual, which Deleuze characterizes as its power to act, should be understood to be fixed or variable, and how such a power to act is implicated in the transformations of the characteristic relations determinative of an individual. Deleuze’s disputed distinction of passions into joyful passive affections and sad passive affections is elaborated in chapter 7, as is the manner by means of which joyful passive affections are pivotal in determining the transition from inadequate ideas of the imagination to adequate ideas of reason. Chapter 8 investigates in detail the mechanism by means of which joyful passive affections are implicated in the dynamic changes or transformations of the characteristic relations determinative of an individual by mapping the mathematical concept of accumulation onto the ontological concept of the accumulation of joyful passive affections, such that there are ‘correspondences without resemblance’ that ‘are of a structural–genetic nature’²² between them, that is, insofar as both are determined as operating according to the logic of different/ciation.

The distinction between the duration of an individual, or of its finite modal existence, and its eternity is investigated in chapter 9, where the thesis that occupies the fifth part of the Ethics which deals with the relation between death and the eternity of the soul according to the third kind of knowledge, and which is often mistaken for a discussion advocating the immortality of the soul, is developed. That which concerns the concept of eternity in the Ethics has interested scholars and commentators since the earliest interpretations of Spinoza, which were used either to reinforce piety by espousing one’s opposition to Spinoza’s atheism, or, on the contrary, and more rarely, to justify one’s pantheism by defending Spinozism.²³ These studies often took the form of demonstrating that the thesis on eternity in part five was incoherent with that of the preceding parts of the Ethics, notably with the principle of parallelism. This aspect of the Ethics is still widely disputed among scholars working in the field of Spinoza studies. My contention is that Deleuze’s argument from powers, which is explicated in chapter 9 according to the logic of different/ciation, demonstrates an alternative approach that allows a better comprehension of the coherence between the different parts of the Ethics. The logic of different/ciation is therefore determinative not only of the mechanism by means of which joyful passive affections operate in Deleuze’s reading of the Ethics, but it is also the logic that is determinative of Deleuze’s reading of the system of the Ethics as a whole.

The project of the present work is to develop an understanding of Deleuze’s Spinoza by providing a Deleuzian reading of Expressionism in Philosophy, which involves not only determining the mechanism of operation of the logic of different/ciation in Expressionism in Philosophy, but also involves positioning Deleuze’s Spinoza, and the logic with which it is explicated, within the context of the development of Deleuze’s wider philosophical project. Such a Deleuzian reading of Expressionism in Philosophy is therefore important for the determination of the way in which Spinoza becomes for Deleuze a significant figure in his project of tracing an alternative lineage in the history of philosophy, and serves well to situate Expressionism in Philosophy within the context of Deleuze’s project of constructing a philosophy of difference. The context of this dual Deleuzian project is explicated in chapter 10 in relation to Deleuze’s other major text from the same period, Difference and Repetition.

It is in Difference and Repetition, in relation to the Scotist concept of the univocity of being, that Deleuze specifically positions Spinoza as a significant figure in the project of tracing an alternative lineage in the history of philosophy. He argues that there are ‘three principal moments in the history of the philosophical elaboration of the univocity of being’,²⁴ which he locates respectively in the work of Scotus, Spinoza and Nietzsche. My argument in chapter 10 is that this alternative lineage in the history of philosophy only makes sense in relation to the demands of a concept of temporality that is determined according to the logic of different/ciation.

In What is Philosophy?, Deleuze claims that ‘philosophy is the discipline that involves creating concepts’.²⁵ The manner by means of which concepts are created is determined by the encounters between the concepts belonging to philosophies of the past which generate the problems of which the new concepts are the solution. The concepts of philosophies of the past, to which Deleuze refers, are determined in relation to the history of philosophy and include, for example, the Scotist concept of univocity, the Spinozist concept of Substance as immanent cause, and the Nietzschean concept of the eternal return. It is by means of the encounters between these concepts (of the past) that, on the one hand, new concepts are created, and that, on the other hand, the problem, onto which each of these concepts (of the past) is grafted, is generated. The determination of these concepts (of the past) and of the manner by means of which the encounters between them create new concepts and generate problems is explicated in chapter 10 according to the logic of different/ciation.

The concept of expression and the concept of intensity are two other concepts whose explication is pivotal in determining the manner by means of which Deleuze reads the Ethics in Expressionism in Philosophy. What is expression? And what is the relation between Spinoza’s Ethics and the concept of expression explicated by Deleuze in Expressionism in Philosophy? These questions are addressed in chapter 10 where I contend that it is in relation to the concept of expression that the logic of different/ciation is explicated in the text of Expressionism in Philosophy as a logic of expression. It is according to the logic of expression that the Spinozist concept of Substance as immanent cause and the Nietzschean concept of the eternal return are problematized in relation to one another in Difference and Repetition, thereby creating the Deleuzian concept of intensity. It is by means of the development of the problematic relation between the Deleuzian concept of intensity, which is determined according to the logic of expression, and the concept of intensity determined rather by the Hegelian dialectical logic, that the logic of expression is determined as an alternative to the Hegelian dialectical logic. Insofar as the logic of expression is determined as the logic according to which the philosophy of difference is constructed, it functions not only as an alternative to the Hegelian dialectical logic, but it is also determined as the logic according to which the philosophy of difference functions as an alternative to the dialectical philosophy determined by the Hegelian dialectical logic. Deleuze’s reading of Spinoza in Expressionism in Philosophy therefore redeploys Spinoza in relation to an alternative lineage in the history of philosophy in order to mobilize his philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectical logic.

While acknowledging some of the problematical elements of Deleuze’s challenging interpretation of Spinoza, the central focus of the work remains an examination of the manner by means of which Spinoza is mobilized by Deleuze in his project of constructing a philosophy of difference. The reading of Deleuze’s philosophy that is presented in the present work is therefore resolutely partial. Neither the implications of Deleuze’s Spinozism for understanding his other works, nor the extent to which these are influenced by his Spinozism is treated in any great detail. Indeed when certain of these texts are nevertheless referred to, it is solely in order to better determine the schema of problematization associated with the development of the logic of expression in Expressionism in Philosophy, or serves merely to suggest the implications of such a development for Deleuze’s subsequent works. As for the latter, rather than suggesting in detail the full range of potential implications, that which is suggested is solely the manner by means of which these implications are determined, that is, that they too are determinable according to the logic of expression. For example, the present work concludes by suggesting the implications of the logic of expression for determining an understanding of the relation between, what Deleuze and Guattari refer to in What is Philosophy? as, a philosophy’s plane of immanence, and the concepts created by this philosophy that are distributed over this plane.

_______________

Notes

¹ Gilles Deleuze, Expressionism in Philosophy, Spinoza, trans. M. Joughin (New York, 1992).

² Gilles Deleuze, Difference and Repetition, trans. P. Patton (New York, 1994).

³ Gilles Deleuze’s seminars on Spinoza, entitled ‘sur Spinoza’, given between 1971 and 1987 at the Université Paris VIII Vincennes and Vincennes St-Denis, have been published on the internet at URL < http://www.webdeleuze.com>.

⁴ Juliette Simont, Essai sur la quantité, la qualité, la relation chez Kant, Hegel, Deleuze. Les ‘fleurs noires’ de la logique philosophique (Paris, 1997), p. 230. Simont also writes that ‘the continuity of history is the form or the method of deciphering the actual connections of the historian’s reading, the individuation of the systems of which is the material or the discontinuous content’ (p. 231). All citations quoted from French language texts are my translations, unless otherwise indicated.

⁵ Martial Gueroult, Spinoza (t. I–II, Paris, 1968–74).

⁶ Deleuze, ‘Spinoza et la méthode générale de Gueroult’, Revue de Métaphysique et de Morale, 74 (1969), p. 426.

⁷ Ibid., p. 437.

⁸ Deleuze later returns to this series with a text on Leibniz.

⁹ Manola Antonioli, Deleuze et l’histoire de la philosophie (Paris, 1999), p. 10.

¹⁰ Pierre Macherey, ‘The encounter with Spinoza’, trans. M. Joughin, in Paul Patton (ed.), Deleuze, A Critical Reader (Oxford and Cambridge, 1996), p. 142.

¹¹ Ibid., p. 141.

¹² Pierre Macherey, Introduction à l’Ethique de Spinoza (t. I–V, Paris, 1994–98).

¹³ Pierre Macherey, Hegel ou Spinoza (Paris, 1979).

¹⁴ Pierre Verstraten recognizes in ‘the thought of Hegel … an unexpected premonition of that of Deleuze’, see ‘La question du negative chez Deleuze’, Gilles Deleuze (Paris, 1998), p. 176.

¹⁵ Is Deleuze’s Hegel perhaps too Kojèvian? See Simont, p. 276.

¹⁶ Juliette Simont actually argues that Deleuze’s Hegel is perhaps too ‘Hegelian’, ‘a terribly Hegelian Hegel’, and proposes instead another Hegel, ‘the other Hegel’, who ‘is perhaps too Deleuzian, or is only possible when read across Deleuze’ (Ibid., p. 297).

¹⁷ Catherine Malabou, in ‘Who’s afraid of Hegelian Wolves?’ (Patton, Deleuze, A Critical Reader), argues that ‘doing justice to Deleuze’s finely wrought thinking concerning affirmation implies, in my opinion, affirming Hegel’s role in it’ (p. 136).

¹⁸ Manola Antonioli, in Deleuze et l’histoire de la philosophie, argues that ‘with Hegel … no encounter is possible: the recourse to the negative and to contradiction is irreconcilable with the affirmative presuppositions of the Deleuzian philosophy, there is nothing “to put together” between Deleuze and Hegel’ (p. 80).

¹⁹ G.W.F. Hegel, Hegel’s Science of Logic, trans. A.V. Miller (London, 1969), p. 251.

²⁰ Charles Ramond, qualité et quantité dans la philosophie de Spinoza (Paris, 1995), p. 201.

²¹ Eugene Holland notes that Martial Gueroult, in his two-volume study of Spinoza, is one of the first to demonstrate the importance of the distinction between potentia and potestas in Spinoza (Holland, ‘Spinoza and Marx’, Cultural Logic, 2.1 (Fall, 1998), §29). For a thorough examination of the social and political implications of this distinction in relation to Spinoza see Antonio Negri, The Savage Anomaly: the power of Spinoza’s metaphysics and politics, trans. M. Hardt (Minneapolis, 1991).

²² Deleuze, Difference and Repetition, p. 184.

²³ For an account of the multiple conflicting readings of Spinoza’s early interpreters see Frederick C. Beiser, The fate of reason: German philosophy from Kant to Fichte (Cambridge, Massachusetts, 1987).

²⁴ Deleuze , Difference and Repetition, p. 39.

²⁵ Gilles Deleuze and Félix Guattari, What is Philosophy?, trans. H. Tomlinson and G. Burchell (New York, 1994), p. 5.

Chapter 1

Spinoza from the point of view of an idealist or a materialist dialectic

Spinoza’s metaphysical philosophy

Hegel’s most coherent understanding of Spinozism is developed in book two of the Science of Logic: The Doctrine of Essence, in which he characterizes both the limitations of Spinoza’s philosophy and the necessary steps required to overcome these limitations. Much of the distinction that Hegel makes between his own approach to philosophy, in contrast to Spinoza’s, is presented as the distinction between metaphysical philosophy and the dialectical development of philosophy. For Hegel, metaphysical philosophy seeks to assert and produce only what comes under the category of being, or substance. He argues that ‘the philosophy which adopts the standpoint of Substance and stops there is the system of Spinoza’.¹ The critical method that Hegel brings to Spinoza’s philosophy is his dialectical logic, one step in the development of which is constituted by his refutation of Spinoza’s philosophy. He outlines his method as follows: ‘The only possible refutation of Spinozism must … consist … in recognizing its standpoint as essential and necessary and then going on to raise that standpoint to the higher one through its own immanent dialectic. The relationship of substance considered simply and solely in its own intrinsic nature leads on to its opposite, to the Concept. The exposition of substance which leads on to the Concept is, therefore, the sole and genuine refutation of Spinozism’.² The question can be raised as to what exactly Hegel considers Spinozist substance to be such that it can be subsumed in the concept according to the dialectical logic? What is required in order to respond to this question is an understanding both of the difference between their respective conceptions of substance, and an understanding of how these different conceptions figure in the development of their respective philosophies. After presenting this difference, the manner by means of which Spinoza is figured by Hegel in the development of his dialectical logic will be addressed, with close attention being paid to those aspects of Hegel’s interpretation of the limits of Spinoza’s philosophy which enable this development. My argument will be that Hegel’s interpretation of Spinoza develops from a misreading of a particular passage from the correspondence of Spinoza. It is this reading which specifically allows Hegel to figure Spinoza within the dialectical logic. By offering a reading of the passage from this letter within the context of Spinoza’s Ethics, a different conception of Spinoza’s substance is presented. One which allows an interpretation of Spinoza that overcomes those limitations presented by Hegel which are necessary to figure the Spinozist system within the dialectical logic.

The Ethics commences with the idea of an absolutely unconditioned cause, or causa sui,³ which Spinoza characterizes as one substance, developed in the set of definitions and principles at the beginning of the Ethics. There is one and only one substance, which varies in an infinite number of ways. Spinoza’s concept of substance is interchangeable with his concept of both Nature and God. This substance has an absolutely infinite power of existing, that is, of producing all things that exist – including itself, causa sui – and an absolutely infinite power of thinking, hence of self-comprehension, which entails the power of comprehending all that is produced. This unique, universal and infinite substance can be understood to be an expression of being, which Spinoza organizes into substance, attributes and modes. Substance expresses itself in an infinity of attributes, each of which is expressed infinitely. Attributes are infinite forms of being which are really distinct, that is, they are irreducible to one another. Although the different attributes express the same thing insofar as they each refer to the same single substance. The essence of substance has no existence outside the attributes, therefore each attribute expresses a certain eternal and infinite essence,⁴ and the essence of each of the attributes together form the essence of an absolutely single substance. In this way the attributes constitute the essence of substance.

Substance is the cause of all things in the same sense that it is cause of itself; that is, it produces things through the very forms that constitute its own essence. It does this by producing things through the attributes. Things in general can therefore be understood to be modifications of substance through the attributes. As Spinoza says: ‘Particular things are nothing but affections of God’s attributes, or modes by which God’s attributes are expressed in a certain and determinate way’.⁵ Attributes are absolutely common to substance and to modes; modes implicate the same attributes that constitute the essence of substance, and these same attributes contain all of the essences of modes. Modes have an essence and existence of their own, but do not exist, and have no being, independently of the attributes in which they are produced.

Hegel’s refutation of Spinozism

Hegel conceives of substance as only one part of the system of his dialectical logic. At the beginning of the Science of Logic, Hegel attempts to think the primitive concept of being. But to think being is only to think it as absolutely opposed to its contrary category – nothing. The thought of pure, indeterminate being thus ‘passes over’ into the thought of nothing. And similarly, the thought of nothing ‘passes over’ into the thought of being. The difference between the primitiveness of these categories collapses; each thought ‘passes over’ into the other, its immediate opposite.⁶ The concepts of being and nothing are therefore seen not as singularly coherent and isolable concepts but as aspects of some broader thought determination – becoming. Therefore, the thought of pure being becomes the thought of pure becoming. In turn, the category of becoming can also only be thought as a contrastive determination. Hegel’s point is that thought determinations are premised upon a relation of contrastive determination, the elements of which are contradictory. The progress of thought is determined by the development of more complex categories which resolve the contradictions which emerge between thought determinations. Progressively, and by this mechanism, an increasingly complex conceptual structure is developed. This is the structure of Hegel’s dialectical logic. The attempt to specify the concept of being develops into a dialectical progression which goes beyond the positing of dichotomies, such as being and nothing, to the positing of new determinations within which these contradictory determinations are subsumed, in this case, within the thought of pure becoming.⁷

This evolving structure constitutes the topic of book one of the Science of Logic, ‘Being–logic’. Being–logic is a dialectical structure, the thought of which is developed by a contrastive determination with the structure of what Hegel calls, in book two, ‘Essence–logic’. The general determination of essence, in the essence–logic, is the negation of the starting point of being–logic, namely being – the immediately given. This negation determines being as merely posited, or as appearance, that is, that which is immediately given must be negated to determine the underlying essence. The determinations which make up the structure of being–logic and essence–logic are subsumed by a further negation into the structure of the ‘Concept–logic’, introduced in book three. In this way, the concept is the subsumption of being and essence. The definition of essence is the result of the first negation of being, which has thereby become appearance, and the Concept is the result of the second negation, or the negation of the negation. Hegel argues that ‘substance is essence insofar as it is united with being … Consequently, the Concept has substance for its immediate presupposition; what is implicit in substance is manifested in the Concept’.⁸ Therefore, the application of the dialectical logic to substance, in which being and essence are subsumed, results in the production of the Concept, which is the exposition of the process of its becoming.

The final concept of the Science of Logic is that of self-determination – what Hegel calls the ‘absolute idea’. The true content of the ‘absolute idea’ is the entire system of the Science of Logic, the development of which I have been considering so far. This idea is the pure form of the Concept. Hegel argues that the Concept ‘is the unity of the subjective and the objective idea’.⁹ What he means by this is that the logical rational structure of our thought is simultaneously the structure of the real or the ‘true’. Hegel considers the rational structure of the absolute idea, that is, the structure of the entire Science of Logic, to provide the schematic means for undertaking philosophical, as opposed to empirical, inquiries into the nature of what he calls ‘spirit’.¹⁰

Hegel considers subjectivity or self-consciousness to be identical to that which he designates as the Concept. ‘The Concept … is none other than the ‘‘I’’ or pure self-consciousness’.¹¹ The substance from which subjectivity or self-consciousness is determined is the objective cultural realm within which the individual self-conscious subject is situated. The characteristic which determines consciousness is its negativity, that is when consciousness stands in opposition to an object which it regards as other than itself. In determining itself as a conscious subject in relation to substance, an individual becomes conscious of itself both in its relation to substance, and in its distinction from substance, that is, in its relation to itself or its identity as self-conscious through a process of reciprocal negation. In this process, substance itself acquires the individual characteristics of those determined conscious subjects; in particular, the characteristic of ‘self-consciousness’. This is exemplified in the preface to the Phenomenology of Spirit, in which Hegel discusses substance as being essentially subject,¹² and where he notes that this identity ‘is expressed in the representation of the Absolute as Spirit’.¹³

Hegel commonly uses notions like ‘subject’ and even ‘self’, when referring to substance and spirit, to characterize the same internal ‘negation’ which he sees as characterizing consciousness with its separation of itself from its object. In this way, the idea of substance becoming ‘subject’ can be understood to refer to the emergence within cultural life of forms of representation which constitute the defining characteristics of that culture. These forms of representation include art, religion, politics and philosophy. Hegel refers to the manifestation of these cultural forms of representation as the development of the ‘self-consciousness’ of that culture. To view the absolute idea as spirit is to endow it with this form of self-consciousness and with a becoming mediated by human history. Hegel argues that the structural conditions, both social and political, which had been developing throughout the European continent, in which finite subjects become fully self-conscious, constitute the realm of spirit.

Omnis determinatio est negatio

In The Encyclopaedia Logic, Hegel comments on the general reception of Spinoza’s philosophy when he says that: ‘In the history of philosophy, we meet with substance as the principle of Spinoza’s philosophy. About the significance and value of this philosophy, which has been as much praised as decried, there has been from the first much misunderstanding, and much argument pro and con’.¹⁴ In order to overcome these misunderstandings, Hegel argues that Spinoza’s substance must be taken as one of the moments of the dialectical logic. Where Spinoza goes wrong is in his failure to see that his basic concept, substance, is only one moment in the determination of God as spirit. This is the point, as far as Hegel is concerned, that Spinoza never reached.

Hegel considers Spinoza’s philosophy to be the most advanced and modern form of substance–metaphysics. The common feature that Hegel ascribes to those metaphysical schemas is their appeal to an ultimate essence as the basis of being, or that which appears in the phenomenal world; that is, they are forms of metaphysics based on the idea of an immediately given substance.¹⁵ To this extent, as far as Hegel is concerned, they all fail to complete the second negation associated with the concept–logic; that which determines the subsumption of substance in the concept. In the progress of the dialectical logic, Hegel considers it important to always distinguish what is still in itself – the determinations as they are in the concept – and what is posited – substance. He argues that ‘this distinction … belongs only to the dialectical development and … is unknown to metaphysical philosophizing, which also includes the critical philosophy; the definitions of metaphysics, like its presuppositions, distinctions and conclusions, seek to assert and produce only what comes under the category of being, and that, too, of being-in-itself’.¹⁶

Throughout book two of the Science of Logic, Hegel sketches various forms of substance–metaphysics, culminating in the philosophy of Spinoza. Redding argues that, as far as Hegel is concerned, ‘metaphysical thought has to be liberated from the one-sided logic of essence with its category of substance: it must become articulated in the framework of book three of the Logic, the doctrine of the concept’.¹⁷ In this way, Hegel systematizes the various difficulties that he finds in Spinoza by attributing them to Spinoza’s one-sided view of negation, his substance–metaphysics, or his non-dialectical concept of substance.

In his refutation of Spinozism at the end of book two, Hegel argues that ‘Spinozism is a defective philosophy because in it reflection and its manifold determining is an external thinking’.¹⁸ Substance–metaphysics, as far as Hegel is concerned, implicitly restricts thought to the level of ‘external reflection’. However, before determining how Hegel comes to this conclusion in relation to Spinoza, the position that Hegel ascribes to Spinoza within the dialectical logic must be examined.

Hegel considers the absolute principle of Spinoza’s philosophy to be that ‘Determinateness is negation’. He argues that: ‘this true and simple insight establishes the absolute unity of substance. But Spinoza stops short at negation as determinateness or quality; he does not advance to a cognition of negation as absolute, that is, self-negating, negation; thus his substance does not itself contain the absolute form, and cognition of it is not an immanent cognition’.¹⁹ This socalled absolute principle of Spinoza’s philosophy is understood by Hegel to mean that ‘All determination is negation’. This ‘true and simple insight’ that Hegel ascribes to Spinoza is what establishes the absolute unity of substance. In coming to this conclusion, Hegel considers there to be only two attributes, thought and extension, which he presents in a relation of opposition – their determinateness being one’s negation of the other. As negations of each other, they have no separate existence of their own, but exist rather only as moments of substance. The attributes are determined therefore only insofar as they are the determination of substance. As such, they are posited as one, as the unity of absolute substance.

This insistence on a negative movement of determination is also at the heart of Hegel’s refutation of Spinoza. Hegel argues that the determination of the attribute is first posited by Spinoza in the mode. Therefore, the mode, which Spinoza defines as an affection of substance, should be the specific determinateness of substance. However, Hegel argues, Spinoza stops short at negation as simple determination, both at the level of the attribute, as was just elaborated, and also at the level of the mode. He argues that the negation of negation, the negation that subsumes the essence–logic within the doctrine of the concept, negates negation as simple determination. This form of negation he calls negation as absolute. The absolute negation is the negation which, in negating something, is negated itself as negation. It is therefore resolved by discovering itself as positive. In this way, the negative appears as a mediator or an intermediary which is retrospectively subordinated to the positive. Hegel contends that absolute negation is lacking in the case of Spinoza.

Hegel argues that in his exposition of the absolute, Spinoza first poses absolute substance as a whole, which is then converted into successive determinations, first with the attribute, followed then with the mode, but these two successive determinations ‘are only enumerated one after the other, without any inner sequence of development’.²⁰ As with the attributes, Hegel argues that the mode in Spinozism remains as something immediately given which is only determined according to the logic of negation as simple determination in relation to other modes. The modes are not then determined by negation as negation, there is no absolute negation of their determining negation.

What this means for Hegel is that the subsumption of the modes, or the determination of their identity in self-consciousness, by a process of reciprocal negation is not achieved by Spinozism. Therefore, neither is the establishment of a dialectical logic which would determine the subsumption of substance in the concept, and thus as the absolute idea, or spirit. As far as Hegel is concerned, it is this subsumption which constitutes the specific determinateness of being as such. Because of this, Hegel argues that Spinoza does not get beyond an essence–logic: ‘the necessity of the advance of the absolute to unessentiality is lacking … there is lacking the becoming both of identity and of its determination’.²¹

The crux of the issue here is the Hegelian conception of determination. Hegel claims that all determinations are dissolved within Spinozist substance. He says that: ‘The substance of this system is one substance, one indivisible totality; there is no determinateness that is not contained and dissolved in this absolute; and … in this necessary notion, everything … is completely reduced to a mere positedness’.²² According to Hegel, the unique and absolute being of Spinozism cannot provide a basis for determination because it involves no absolute negation between the modes, which would otherwise be the specific determinateness of substance. Determinate being must negate and subsume its other within itself in order to attain identity, or reality. This is what Hegel figures in the doctrine of the concept: substance subsumed in the absolute idea, or spirit. Without both moments of negation being remains indifferent and abstract.

Et determinatio negatio est

In order to establish Spinoza’s relation to the first negation of the dialectical logic, that is, simple determination, Hegel takes a phrase from Spinoza: ‘Omnis determinatio est negatio’. It seems that this debate ultimately turns entirely around this single phrase and its interpretation. Hegel translates this phrase as ‘all determination is negation’, that is, that all particular finite things or modes are determined insofar as they are contrasted with other things, which for Hegel is determined by a process of reciprocal negation. In a footnote to the English translation of The Encyclopaedia Logic, the translators provide a gloss for this reference. It states that ‘this tag, which Hegel loves, is a misquotation. The nearest equivalent in Spinoza’s surviving texts is in Letter L, “Figure is nothing else but determination, and determination is negation (et determinatio negatio est)”’.²³ Hegel not only changes the quotation to simplify it for his purposes, but in his interpretation he also distorts its Spinozist meaning. Rather than taking into account the context of the argument in which Spinoza introduced the correlation between negation and determination, Hegel takes a specific reference to figure and generalizes it to encompass all of being.

In book one of the Science of Logic, Hegel outlines his own conception of the relation of figure to the being of the object as follows: ‘The more stable determinatenesses … such as figure, shape, are not called properties, nor even qualities perhaps, because they are conceived as alterable, as not identical with the being of the object’.²⁴ Given this Hegelian gloss on figure, it is difficult to understand how Hegel comes to the conclusion that by figure Spinoza means all determination, as though figure is synonymous with the being of the object, when Hegel himself points out that they are not identical. Arguably, for Hegel both figure and the being of the object, although not identical, are both determined through the process of absolute negation. However, for Spinoza, the determination of figure is not the same as the determination of the being of an object. For Spinoza, the idea of figure is an abstract idea which implies a certain ignorance of the true or adequate causes of the actual existence of the object. The idea of figure is determined solely as an idea of the imagination, or an inadequate idea. For example, to consider the determination of a finite mode as nothing but negation is to consider the finite mode in itself abstractly apart from what causes it to be. Finite and determinate bodies or modes are determined negatively, that is, insofar as they oppose or negate each other, only if understood from the point of view of their reciprocal limitation – that is, insofar as one ‘finite and determinate body’ is limited by another ‘finite and determinate body’ – independently of the expressive order of nature, according to which they would agree amongst themselves as parts of a whole, or more generally as affections of substance. The finite being of the object can therefore be understood to be determined by infinite substance itself, insofar as infinite substance is cause of all finite things and expresses itself both positively and infinitely in them. To pose the finite outside of the infinite is to consider it from the abstract point of view of the imagination, which separates the whole, or substance, from the relation of its parts, or affections. So, although Spinoza states in Letter L that ‘figure is … determination’, the form of determination to which he is referring is abstract determination, and it is only this form of determination that is determination by negation, or according to which ‘determination is negation’. There is therefore no justification for the Hegelian extrapolation from the Spinozist concept that abstract determination is negation to the concept that ‘all determination is negation’.

Letter XII and ‘the problem of the infinite’

The relation of implication and involvement which ties the finite to the infinite, ‘for an adequate knowledge’,²⁵ is determined by Spinoza in Letter XII where he develops ‘the problem of the infinite’: ‘Everyone has always found the problem of the Infinite very difficult, indeed insoluble. This is because they have not distinguished between what is infinite as a consequence of its own nature, or by the force of its definition, and what has no bounds, not indeed by the force of its essence, but by the force of its cause. And also because they have not distinguished between what is called infinite because it has no limits and that whose parts we cannot explain or equate [NS: determine or express] with any number, though we know its maximum and minimum [NS, LC: or it is determined]. Finally, they have not distinguished between what we can only understand, but not imagine, and what we can also imagine’.²⁶

In order to illustrate the distinctions involved in the ‘problem of the infinite’, Spinoza proposes for our reflection an example from geometry, to which Hegel makes reference on many occasions: ‘… all the inequalities of the space between two circles, A and B, and all the variations which the matter moving in it must undergo, exceed every number. That is not inferred from the excessive size of the intervening space. For however small a portion of it we take, the inequalities of this small portion will still exceed every number. Nor is it inferred because, as happens in other cases, we do not know its maximum and minimum. For we know both in this example of ours: AB is the maximum and CD is the minimum. Instead it is inferred simply from the fact that the nature of the space between two non-concentric circles does not admit anything of the kind’.²⁷

The example that Spinoza develops (Fig. 1.1) consists of ‘two given nested non-concentric circles’,²⁸ and the two extreme orthogonal distances between them, AB and CD.²⁹

Fig. 1.1

It allows Spinoza to present the relation between the finite and the infinite as follows: ‘… some things are infinite by their nature and cannot in any way be conceived to be finite, that others [are infinite] by the force of the cause in which they inhere, though when they are conceived abstractly they can be divided into parts and regarded as finite, and that others, finally, are called infinite, or if you prefer, indefinite, because they cannot be equated with any number, though they can be conceived to be greater or lesser. For if things cannot be equated with a number, it does not follow that they must be equal. This is manifest enough from the example adduced’.³⁰

In the case of the double figure of non-concentric circles, what Spinoza suggests is that the variation of distances included between the external circle and the internal circle, not able to be determined by any number, is infinite, ‘or if you prefer, indefinite’, even though it is understood within certain limits, that is, within the two circumferences and the greatest and least orthogonal distance, which function as thresholds or limits, as in a maximum and a minimum.

Infinite actu and the ‘bad infinite’

In the Science of Logic and the History of Philosophy, when Hegel makes reference to this geometrical example he speaks only of the space interposed between the two circumferences. In the History of Philosophy he glosses Letter XII as follows: ‘The inequalities of the space <die ungleichen abstände des raumes> between AB and CD exceed every number; and yet the space which lies between is not so very great’.³¹ The attempt to determine all of the spaces would be to constitute an infinite series, an infinity of unequal spaces. Hegel, however, argues that ‘Spinoza rejects’ such a Mathematical conception of the infinite ‘which represents it as an amount or as a series which is not completed’, since, for Hegel, ‘the determination of magnitude contained in [such an unlimited series] cannot at the same time be represented as a quantum’.³² According to Hegel, any attempt to divide a continuous variation, included between a minimum and a maximum, into parts in order to reconstitute the variation by a number will result in a false infinite, since the variation, which is continuous, cannot be divided and extrapolated in this way, that is, it cannot be determined numerically. What is discovered is a magnitude that does not equal any quantum or number, and which is determined rather as exceeding all number, as expressed in statements by Spinoza such as ‘and so on to infinity’.³³ Hegel argues that this non-numerical infinite is ‘there to hand, circumscribed, affirmative, actual and present in that plane as a complete space between the two circles’.³⁴ Hegel describes this circumscribed infinite as ‘Philosophic infinity, that which is infinite actu’.³⁵ Hegel designates the relation of substance to its affections by the concept of the ‘actual infinite’, which is the realization of the cause in its effects, that is, the actual infinite is the realization in the affections, or the finite, of infinite substance, which is their cause. ‘It is an infinite which is not given in an unlimited series [as the bad infinite], but [rather] all at once: it is this [actual] infinite which is present in a limited reality, such as that included between a minimum and a maximum’.³⁶ The actual infinite ‘is not beyond, but actually present and complete’, as bounded within this space.³⁷

The affirmation by Spinoza of the existence of an actual infinite expresses the implication and envelopment of the infinite in the finite. According to Hegel, it would be a manifest contradiction to assign number to the actual infinite since this is what distinguishes it from the bad infinite. The bad infinite being the idea of the actual infinite from the point of view of the finite. The idea of the bad infinite ignores the actual infinite, or ‘mistakes it with the idea of an unlimited series which excludes the possibility of an actual infinite’.³⁸

Hegel argues that Spinoza’s concept of the actual infinite ‘might have been better expressed as “the negation of negation”’.³⁹ The finite, is primarily negated by its opposite, the infinite, in the form of the bad infinite, since it is an infinite conceived from the point of view of the finite as an unlimited series. The bad infinite remains defective since the only negation with which it is affected is the primary negation of the finite. The bad infinite itself must then be negated, that is, the primary negation of the finite which is determinative of the bad infinite must itself be negated, in order for the finite to realize itself as actually infinite. The negation of the primary negation of finitude, or the negation of the bad infinite, is therefore determinative of the actual infinite.

When Spinoza introduces the concept of the actual infinite in Letter XII from the point of view of the seventeenth-century Mathematician, contrary to Hegel’s understanding, Mathematicians are presented as being on the side of philosophers. Spinoza argues that Mathematicians ‘do not infer that … [‘things which cannot be equated with any number’] … exceed every number because of the multiplicity of their parts’, as Hegel suggests Mathematicians do. Spinoza argues that Mathematicians ‘infer’ rather that ‘the nature of [such things] cannot admit number without a manifest contradiction’.⁴⁰ With this statement Spinoza suggests that ‘no Infinite is explained in terms of the multitude of its parts; and that, if every Infinite escapes number, it is not because it contains too many parts, but because, by nature, it is not expressible by it’.⁴¹ As a consequence, the introduction of a relation of contradiction as a means of determining the concept of a non-numerical, or actual, infinite can be avoided. Rather than acknowledging the consequences of this statement, Hegel reads it solely as a statement about the distinction between two infinities, the bad or Mathematical infinite, and the actual or Philosophic infinite. According to Hegel, the Mathematicians rely on a concept of number in order to determine the concept of the infinite as the greatest of all numbers, or the indefinite or unlimited. This admission of number in the determination of the infinite enables Hegel to establish a vigorous contrast between the Mathematical infinite and the Philosophic infinite, which, according to the dialectical logic, subscribes to a relation of negation. The concept of a non-numerical, Philosophic or actual infinite can only be conceived in a relation of contradiction to a concept of the infinite that does admit number, namely the bad or Mathematical infinite. The distinction between an infinite which exceeds every number because of the multiplicity of its parts, and one whose nature does not admit number without contradiction is represented by Hegel in the relation between the bad infinite and the actual infinite, insofar as the bad infinite only manifests as contradictory to the actual infinite. When the Mathematical or bad infinite is rejected, it is rejected according to the dialectical logic, that is, it is negated. Spinoza’s supposed rejection of the Mathematical infinite then involves sublating it in the name of the Philosophic infinite, which is conceived by Hegel as a negation according to the dialectical logic.

Hegel thus interprets Spinoza’s presentation of the Mathematical infinite in Letter XII as a bad infinite which is in a relation of contradiction with the actual or Philosophic infinite, an interpretation which is contrary to Spinoza’s actual remarks. According to Hegel, the bad infinite functions only as the immediate contrary of the finite, and is abrogated in the name of the finite realized as actually infinite. ‘As soon as the cause has something else opposed to it – the effect – finitude is present; but here this something else is at the same time abrogated and it becomes once more the cause itself. The affirmative is thus negation of negation’.⁴² The infinite, when opposed to the finite, is conceived as the bad infinite, which is then sublated and subsumed in the actual infinite, that is, the finite realizes itself as actually infinite. This is how Hegel resolves the relation of the infinite to the finite from the point of view of his interpretation of Spinoza.

This explanation by Hegel of ‘the problem of the infinite’, in relation to the text of Spinoza, renders superfluous the particularities of the geometrical example as exposed by Spinoza. Macherey argues that, as far as Hegel is concerned, every finite mode ‘expresses an infinity, which it formally envelops … as its cause’⁴³ –for example, the space included between the two circles, or, to take another example put forward by Hegel, the infinity of points included in a segment of a line. This however ignores the essential fact that the two circles are non-concentric.

Hegel doesn’t distinguish between, on the one hand, the expression of the circumference of circle A as a maximum and that of circle B as a minimum, and, on the other hand, the expression of a maximum and a minimum as the greatest and smallest distances between the circumference of circle A and that of circle B. The ‘maximum and minimum’ serve Hegel simply as a means for defining the finite as a contained space or limited distance. Such an interpretation does not account for the specificity of presenting non-concentric circles in the example, since the same reasoning would hold if the distances which separate the two circumferences were all equal. It is therefore evident that ‘the two circles could not be concentric, for in this case there would be no inequalities among the segments’.⁴⁴ Macherey argues that ‘if Spinoza had only wanted … to represent the idea of a finite quantum which entails nevertheless an infinity of parts [which] exceeds all assignable number’,⁴⁵ he would not have had need of the precision that the two circles are non-concentric. Hegel therefore neglects something that is essential to the reasoning proposed by Spinoza. This simplification by Hegel renders inevitable the reduction of the infinity between the circles to a simple unlimited series represented by the idea of the bad infinite, which is contradictory to the nature or essence of the finite thing, and which when negated is abrogated in the expression of it as actually infinite.

The problem of the ‘bad infinite’

Macherey contends that, contrary to this Hegelian argument, things are not so simple. The reestablishment of a clear separation between the infinite and the finite is not sufficient to address what Macherey considers to be the tendency of the imagination to represent the infinite in the finite as the ‘bad infinite’, but merely begs the question of that tendency. Given that the relation of opposition between the finite and the infinite imposed by the dialectical logic springs from the negation of the infinite by the finite, in Appendix IX to Spinoza, Dieu, Gueroult argues that ‘far from causing their opposition to vanish by suppressing’ the expression of the infinite in favour of the expression of the finite, ‘the imagination transforms it into an irremediable conflict, for the infinite, wrongly understood’ from the point of view of the imagination, ‘subsists no less truly in things, and its negation’, again by the imagination, ‘amounts to nothing other than to confer upon it the properties of the finite. Thus, refusing to submit itself to violation, [the infinite] … affirms its presence … by causing to arise, from the depths of the alleged solutions dictated by the imagination, the fundamental absurdity which is at their source’.⁴⁶ The infinite refuses to submit itself to, that is, it negates, the absurdity of the imaginary idea of the infinite as the bad infinite, the alleged solution dictated by the imagination, or the primary negation of the infinite by the finite. Therefore the establishment of the actual infinite is dependent on the irremediable imaginary conflict.

Following on from this argument presented by Gueroult, Macherey contends that Hegel’s argument ‘therefore repeats the point of view of the imagination from which it seeks to distinguish itself’.⁴⁷ What is neglected by Hegel is ‘that essential character of the finite’, which, to return to the distinctions of the problem of the infinite presented by Spinoza in Letter XII, is ‘infinite by the force of its cause’, that which, Macherey contends, is peculiar to all the modes, whether perceived to be infinite (unlimited) or finite (limited). ‘The variation of distances included between the two non-concentric circles is also infinite, not in itself since it is limited, but as an affection of substance which expresses itself in it as the cause in its effect’,⁴⁸ that is, as the positive immanent expression of substance in its affections.

Macherey re-presents the distinctions of the ‘problem of the infinite’ as being between that which is infinite by its nature – the absolutely infinite, or substance, and the infinite in kind, or the attributes – and that which is infinite by the force of its cause – the affections of substance, or the modes. The second term of this distinction is then carried over as the first term of the next distinction: that which is infinite because it has no limits is distinguished from that which is infinite because it is not determinable numerically, that which is unlimited or indefinite, even though included between a maximum and a minimum. For Macherey, these two distinctions also characterize that between reason (which is the adequate understanding of things) and the imagination (which involves the inadequate representation of things).⁴⁹ Substance, or the absolutely infinite, can only be understood adequately and not imagined, and its affections, which are infinite by the force of their cause, can be understood adequately as infinite from the point of view of reason but only inadequately as unlimited or indefinite from the point of view of the imagination.

Macherey argues that ‘the traditional paradoxes of the infinite come from these distinctions not being respected’.⁵⁰ It is only necessary to reestablish these distinctions in order that ‘all the contradictions are not resolved, but effaced’.⁵¹ ‘For then’, Spinoza argues, ‘they would have understood clearly what kind of Infinite cannot be divided into any parts, or cannot have any parts, and what kind of Infinite can, on the other hand, be divided into parts without contradiction. They would also have understood what kind of Infinite can be conceived to be greater than another Infinite, without any contradiction, and what kind cannot be so conceived’.⁵² The presentation of these distinctions as contradictions depends only on the presentation of a problem whose terms have been inadequately distinguished and thus inadequately posed.

Rather than presenting the bad infinite and the actual infinite as contradictory, the latter being the result of the negation of the former, Macherey argues that the bad infinite, or what he refers to as the unlimited or indefinite, and the actual infinite, are one and the same thing as perceived respectively from the point of view of the imagination and from that of reason.

Macherey argues that the adequate understanding of a thing, which grasps the thing according to or after its cause, must be understood ‘affirmatively, in the sense of the absolute affirmation of the nature of the thing’.⁵³ It is an understanding which is thus not determined negatively by the negation of the bad infinite, as it is according to the Hegelian dialectical logic, but positively from the point of view of the immanent cause of a thing, which is substance in the form of its attributes. The actual infinite ‘expresses directly the immanent, and non-transitive, relation which ties substance and its affections, and which is known only by the understanding’.⁵⁴ The bad infinite or unlimited is the inadequate idea of the way in which the actual infinite is expressed in a finite mode. According to Macherey, the actual infinite, as apprehended adequately in the modes, ‘is not different to that infinity constitutive of substance, but is formally the same’.⁵⁵ The actual infinite is therefore the immanent expression in the affections, or the finite modes, of infinite substance, which is their cause. It is the adequate knowledge of this actual infinite as the immanent expression of the infinity of substance in and by a finite mode ‘which constitutes the “intellectual love of God”, or the third kind of knowledge’.⁵⁶

The adequate and the inadequate idea of the actual infinite

In the text of the example of Letter XII, the introductory sentence of which Macherey translates similarly to Hegel as ‘All the inequalities of the space interposed between two circles’,⁵⁷ the reference to ‘the space interposed’ between the two non-concentric circles actually ‘designates the set of distances’,⁵⁸ that is, all of the different distances, which separate the two circumferences, including the two extreme distances, represented by the segments AB and CD. The ‘inequalities of the space’ should then be understood to refer to ‘the set of the differences between these unequal distances’, or, what Macherey emphasizes as ‘the variation’ of these ‘differences’, which is determined by the rotation of the segments from AB towards CD, ‘in the sense of the hands of a watch’.⁵⁹ This set, which is ‘the sum of the inequalities of distance included in this … total space’,⁶⁰ is a continuous and therefore infinite variation.

Rather than being limited because included between a maximum, the circumference of circle A, and a minimum, that of circle B, and therefore being finite, as Hegel seems to indicate, the ‘inequalities of the space’, or the differences between the unequal distances, and the ‘variation’ of these differences, are only limited by the difference between the maximum distance, AB, and the minimum distance, CD.

It is this idea of the variation of differences which depends on the fact that the two circles are non-concentric, which is neglected by Hegel in his account of the geometrical example. For Macherey, the actual infinite is expressed in the example by the infinite variation of differences. The infinite variation can be understood as actually infinite from the point of view of reason, insofar as it has its cause in substance, and as indefinite or unlimited from the point of view of the imagination, insofar as it is represented as the sum of the multiplicity of its parts, since it is only from the point of view of the imagination that number can be applied to it.

When Spinoza defines the indefinite in Letter XII, the problem of various infinities, some greater than others, is addressed. He writes that ‘some things are … indefinite, because they cannot be equated with any number, though they can be conceived to be greater or lesser. For if things cannot be equated with a number, it does not follow that they must be equal. This is manifest enough from the example adduced’.⁶¹

These infinites which are greater or smaller than one another are exhibited by the geometrical example. From the point of view of the imagination, the infinite variation of differences is composed by an infinite number of different or varying differences. Borrowing the Hegelian example of the infinity of points included in a segment of a line, these differences which vary, insofar as each contains, according to its different size, a greater or lesser number of parts, would therefore contain a greater or lesser infinity. Hegel ignored the problem of greater and lesser infinities by relegating them, regardless of their relative distinction, to the same bad infinite. Gueroult argues that, as far as Hegel is concerned, ‘there cannot be unequal infinities [not because] every infinite can be expressed only by the greatest of all numbers [which cannot be more or less large] … [but rather because] it presupposes the absurdity that number is applicable to the infinite’.⁶² According to Hegel, the concept of infinities, some of which are greater than others, is therefore contradictory in the same way that the bad infinite is itself contradictory to the actual infinite.

Macherey argues that ‘the difficulties illustrated by this example are only such for the imagination which wants to represent everything by numbers, … which, in the present case … leads it to insoluble paradoxes’.⁶³ The example of the infinite variation of differences in Letter XII illustrates the difference between the inadequate idea of the infinite from the point of view of the imagination and the adequate idea of the infinite from the point of view of reason. The infinite variation or actual infinite is only inadequately or partially understood from the point of view of the imagination, according to which the infinite variation of differences is represented as unlimited or indefinite, and as composed by varying differences which are represented by the imagination as greater or smaller infinities, and this same infinite is only adequately expressed in the finite from the point of view of the third kind of knowledge. It is therefore only ‘the imagination which perceives a contradiction, … while for the understanding the notion of the continuous is perfectly clear and distinct’.⁶⁴

According to Macherey, and contrary to Hegel, Spinoza’s example does not function to represent the mediation between the different infinities that it represents, but rather it represents both infinities as existing simultaneously, their difference not being a matter of one’s negation of the other, as embodied in an opposition or contradiction, but rather a matter of a difference in the degree of understanding, that is, an epistemological difference, which can vary from an inadequate idea of the way in which the actual infinite is expressed in the finite (as unlimited or indefinite, and as greater or smaller infinities), to the adequate understanding of the expression of the actual infinite in the finite, as the substantial cause of the finite insofar as substance is the immanent cause of all finite things. The idea of infinite variation in the geometric example is therefore conceived by Macherey as inadequate both when the infinite variation of differences is represented by the imagination as unlimited or indefinite, and when the variations in the differences are represented as greater or smaller infinities, and as adequate when the infinite variation of differences is understood adequately as the actual infinite expressed in the finite, that is, infinite substance as expressed in its finite modes. Macherey can therefore conclude that ‘it seems … the error of the imagination consists in taking as infinite, in the attempt to determine it numerically, a thing which is … finite’.⁶⁵

The adequate expression of the actual infinite in the finite is, according to Macherey, the adequate knowledge of its immanent cause in substance. This form of positive determination, which does not involve negation, is excluded from the principle forged by Hegel in the formula, ‘Omnis determinatio est negatio’. When the distinctions between the different infinities introduced by Spinoza in the discussion of ‘the problem of the Infinite’ of Letter XII, are respected, Macherey argues that the contradictions involved in the relation of the finite to the infinite are not resolved, as Hegel considers them to have been by means of the dialectical logic of the negation of the negation, but, rather, they are effaced. The relation between the finite and the infinite is no longer one of contradiction. As Gueroult argues, ‘the solution to the problem of the infinite, due to [an adequate] idea of substance, puts an end to all antinomies’.⁶⁶

The adequate idea of the actual infinite, as the positive immanent expression of the absolutely infinite in the finite, is the adequate understanding of its implication in this relation, which is thereby incorporated in the finite mode as the third kind of knowledge. The inadequate idea of the actual infinite, as unlimited or indefinite, is not in a relation of negation to this positive expression of the infinite, but is rather only an imaginary representation of the actual infinite, and, as such, it betrays only a partial degree of understanding.

The point of view of an ‘external intellect’

Hegel’s conception of Spinoza’s philosophy as a substance–metaphysics which implicitly restricts thought to the level of ‘external reflection’ can now be elaborated. The argument that the attributes are merely constructs of the intellect hinges upon the interpretation of Definition 4 of the Ethics, I (EID4), in which Spinoza introduces the mind, or intellect, into his definition of ‘attribute’. Definition 4 reads as follows: ‘By attribute I understand what the intellect perceives of a substance, as constituting (tanquam constituens) its essence’.⁶⁷ Curley argues that ‘the meaning of this definition is much disputed. One important question of translation is whether tanquam should be rendered ‘as if’ or ‘as’. The former would favour those who hold the ‘subjective’ interpretation, according to which the differences between the attributes are illusory, all the attributes being identical in substance. The latter would be more congenial to those who think attributes are really distinct and not merely constructions of the intellect’.⁶⁸ This controversy turns around the question of whether the attribute, as defined by Spinoza, is ‘attributed’ to substance by an intellect as a subjective representation, or whether it should rather be considered as belonging to the objective reality of substance.⁶⁹

In book two of the Science of Logic, Hegel offers a subjectivist interpretation⁷⁰ of EID4 which reads as follows: ‘The attribute is that which the understanding thinks of God’. In regards to this proposition Hegel poses the question: ‘How does it come that besides the Deity there now appears the understanding, which applies to absolute substance the two forms of thought and extension?’⁷¹ Hegel argues that despite his conception of a unitary absolute substance to which thought is immanent, Spinoza still implicitly appeals to a form of thought which, Redding notes, is ‘dichotomously opposed to any conceivable form of thought found within substance’.⁷² This objection implies that Spinoza overlooks carrying the negative moment of substance to its logical conclusion (as far as dialectical logic is concerned). The absolute substance of Spinozism is not yet the determination of absolute spirit. What Hegel understands by spirit both subsumes and differs from his conception of what Spinoza understands by substance. Hegel’s dialectical logic requires that substance be determined as absolute spirit.

Hegel concedes that Spinoza’s conception of substance contains thought in its unity with extension, but, he argues, ‘it contains thought itself … only in its unity with extension, that is not as separating itself from extension, hence in general not as a determinative and formative activity’.⁷³ The attributes are only determined as substance by a simple negation and therefore as a moment of a unity which takes place only in substance. Hegel’s criticism of Spinoza is that the separation of thought from extension by means of a double negation, as the condition of their ultimate reunion in substance, is lacking. Hegel argues that Spinozist thought is not commensurate with what he figures in the concept–logic as the development of the self-consciousness of the absolute idea, or spirit – this is a problem that he ascribes generally to all substance–metaphysics.

Hegel writes in The Encyclopaedia Logic that ‘Spinoza places Substance at the head of his system and defines it as the unity of thinking and extension, without demonstrating how he arrives at this distinction and how he succeeds in tracing it back to the unity of Substance’.⁷⁴ Since Spinoza’s substance is defined as the unity of thought and extension, Hegel argues that it is a necessarily abstract unity which is only formally reconstituted, by means of an external understanding or intellect. This unity implies that the world, as determined in the Spinozist system, is merely an abstract construct by an ‘external intellect’. Hegel argues that the point of view of the external intellect remains the precondition of Spinoza’s unified substance, which is then only artificially decomposed into its elements, attributes and modes.⁷⁵ If the attributes and modes subsist outside of the intellect which perceives them, they would only have reality from the point of view of this same intellect, in which case they would be only ideas or representations.⁷⁶ It is in this way that modes are figured by Hegel as lacking any determination, or reality. Hegel considers this ‘externality’ to be in contradiction with the intended immanentism of Spinoza’s philosophy.

The only coherent construal of the absolute, Hegel argues, is by means of the dialectical logic. Because neither the modes nor the attributes perform a negation of their own negative determination, which would set the precedent of a logic which would eventually lead to the determination of substance as spirit, Hegel interprets Spinoza’s system as being limited. The means by which it overcomes this limitation is via the postulation of an external intellect, which, however, is only an abstract understanding of determinateness, this determinateness only being posited in substance rather than determined as substance itself. Hegel insists that without negation as absolute, Spinoza’s substance disappears in the indifference of ‘acosmism’ (or the denial of the reality of the world of finite things).⁷⁷

According to the alternative interpretation of EID4, which posits the attributes as really distinct, the form of understanding implied in this definition, which Hegel construes as an abstract external intellect, can be conceived as an understanding which is immanent to substance. However, there remain various interpretations of Spinoza’s conception of the attribute of thought which lends support to Hegel’s argument that Spinoza’s system implicitly restricts thought to the level of external reflection. According to these interpretations, insofar as the attribute of Thought manifests all of the affections of substance as ideas, it does have an aspect which distinguishes it from the attribute of Extension. The attribute of Thought contains as many ideas as there are modes of the attribute of Extension, and as many ideas as there are individuals, insofar as individuals are composed of both an idea, in the attribute of Thought, and its object, in the attribute of Extension. This seems to indicate that the attribute of Thought has priority, or is privileged over the other attributes. There are various controversial interpretations of this aspect of Spinoza’s philosophy, many of which assert that Spinoza’s system ultimately relies on the priority of Thinking in its explication of existence, and that, in this respect, Spinoza was unable to rid his work entirely of Cartesian influences.⁷⁸ However, rather than interpreting the intellect introduced in EID4 as a reinstatement of Cartesian dualism, Hegel reads this as an insufficiency in Spinoza’s system insofar as it neglects the absolute negation required by the dialectical logic, since, for Hegel, the only veritable determination of such an intellect is through the process of the dialectical logic. Contrary to the accusation that Spinoza gives priority to the attribute of thought over the attribute of extension, Hegel argues in the History of Philosophy that, in carrying the Cartesian system to its ‘furthest logical conclusions’, Spinoza actually sets aside the dualism inherent in that system.⁷⁹ However, by reducing the attributes to external forms of reflection, of which only two are perceived, Hegel reinstalls in substance ‘a hidden duality’.⁸⁰ ‘The unity of substance is’, for Hegel, ‘both resolved and undone in the distinction of thought and extension’.⁸¹ This allows Hegel ‘to rediscover Descartes in Spinoza’.⁸² Hegel considers Spinoza to remain constrained by ‘the same problem of the relation of two distinct entities, between which it is necessary to establish the conditions of an agreement’.⁸³ ‘In this sense, Hegel can say that Spinozism is a failed effort to go beyond the limits of Cartesianism’.⁸⁴ He concludes that, insofar as absolute substance contains thought in its unity with extension, the intellect commensurate with EID4 can only be explained as an intellect abstract and external to substance.

The Spinozist thesis of ‘an infinity of attributes’

Macherey develops an alternative to this Hegelian interpretation.⁸⁵ Hegel’s error, he contends, consists in having posed the distinction of the attributes in a relation of opposition between two distinctly separate entities. According to the Spinozist system, the identity of substance in no way lets itself be reduced to the unity of two separate orders. Macherey also argues that ‘the thesis, according to which substance expresses itself in an infinity of attributes, has no real signification for Hegel’.⁸⁶ Hegel excludes this thesis by retaining only two attributes and by placing them in a relation of opposition to one another. The distinction of the attributes is thus interpreted by Hegel as a relation of opposition between thought and extension. Insofar as their nature envelopes a negation, the attributes are external negative determinations of substance. The unity of substance is then only the resolution ‘of this conflict, the reunion … of the terms which, in themselves, are separated and antagonistic’.⁸⁷ The Spinozist thesis of ‘an infinity of attributes’⁸⁸ allows this Hegelian complication of the relation between the attributes to be avoided. The reciprocal irreducibility of the infinite number of infinite attributes, the infinite number of which is the positive expression of infinite substance, is, according to Macherey, ‘perfectly coherent with their [unity] in substance’.⁸⁹ According to Macherey, the attribute belongs to the objective reality of substance, as a positive expression, and therefore as a constitutive form, of substance.

The Spinozist ‘order’, as presented by Hegel, attempts to develop a gradual process of determination that passes from the infinite to the finite according to a logic of negation. However, according to Hegel, Spinoza’s interpretation of the concept of determination tends to isolate the affections from substance, there being no ‘passage’ which would signify a process of decomposition of the infinite in the finite. It is therefore a process that Hegel considers Spinoza not to have completed. Macherey argues rather that the ‘order’ proposed by Spinoza is ‘the concrete, physically real order of substance’, which expresses itself simultaneously and identically as infinite and as finite, and ‘which is known in a contradictory way by the rational understanding and the imagination’.⁹⁰

In Letter XII, Spinoza argues that: ‘if the Modes of Substance themselves are confused with Beings of reason … , or aids of the imagination, they too can never be rightly understood. For when we do this, we separate them from Substance, and from the way they flow from eternity, without which, however, they cannot be rightly understood’.⁹¹ This is the fault committed by Hegel’s ‘external intellect’, insofar as modes, as characterized from the point of view of this intellect, lack any determination or reality. When presented from this perspective, according to Spinoza’s argument and the Spinozist system, they cannot be rightly understood since this separates them from their positive immanent expression of substance. Rather than making the contradiction in Spinoza the sole operator of determination according to a logic of negation, as Hegel does, Macherey locates the contradiction between the finite, imaginary idea of the unlimited or indefinite, and the adequate understanding of the infinite: a contradiction which does not negate the positive expression of substance in its affections. Letter XII can again be quoted in support of Macherey’s argument: ‘if you ask why we are so inclined, by a natural impulse, to divide extended substance, I reply that we conceive quantity in two ways: either abstractly, or superficially, as we have it in the imagination with the aid of the senses; or as substance, which is done by the intellect alone. So if we attend to quantity as it is in the imagination, which is what we do most often and most easily, we find it to be divisible, finite, composed of parts, and one of many. But if we attend to it as it is in the intellect, and perceive the thing as it is in itself, which is very difficult, then we find it to be infinite, indivisible and unique’.⁹²

Macherey then presents his interpretation of Spinoza contrary to that proposed by Hegel: ‘The rational point of view of the understanding is essentially affirmative: [such] that all negativity is imputed to the point of view of the imagination, which is incapable of comprehending the positive expression of substance itself’.⁹³ Macherey argues that contrary to the Hegelian dialectical logic, ‘to determine a thing positively is to perceive it in its physical, singular, reality, after the immanent necessity which engenders it in substance’.⁹⁴ According to this immanent law of causality, which is the same by which substance produces itself, it is substance which produces itself immanently and positively in its affections. Substance is therefore expressed in the modes both positively and negatively, as, respectively, infinite and finite. Macherey therefore presents two essentially different ways of comprehending things, according to whether they are related from the point of view of the imagination or from the point of view of reason. The imagination represents things inadequately, while the rational understanding comprehends things adequately. Substance and its attributes are only able to be comprehended from the point of view of reason, whereas the affections of substance are able to be comprehended positively as infinite from the point of view of reason, and negatively as finite from the point of view of the imagination. Hegel’s attempt to find negation as the sole operator of determination throughout Spinoza’s work, based on the Hegelian formula, ‘Omnis determinatio est negatio’, is thus rendered, by Macherey, as inadequate to the letter of Spinozism.

Omnis determinatio est affirmatio

Having distinguished a positive moment in Spinoza’s thought that is neglected by Hegel, Macherey then questions whether or not ‘the contrary interpretation is more satisfactory?’⁹⁵ It is in Deleuze’s Expressionism in Philosophy that Macherey finds a ‘positivism’, which, he argues, is ‘in opposition to the negativism of the Hegelian reading’.⁹⁶ Macherey quotes Deleuze as saying that ‘the philosophy of Spinoza is a philosophy of pure affirmation’.⁹⁷ From this statement Macherey erects the formula ‘Omnis determinatio est affirmatio’.⁹⁸ The gloss that he gives to this formula is as follows: Modes are determined positively in their sole capacity as immanent expressions of eternal substance, ‘they are themselves eternal, that is insofar as they cannot be destroyed except by an external cause’.⁹⁹ According to Macherey, ‘these two opposing presentations of Spinozism succeed equally in putting in evidence its non-dialectical character’.¹⁰⁰ However, he contends that, whether established according to a logic of negation, or, on the contrary, according to a logic of affirmation, determination remains devoid of content. The Hegelian formula that ‘all determination is negation’ prefigures the degradation of substance to insubstantial modes (since they are only determined abstractly by an external intellect), while the formula ascribed to Deleuze, that ‘all determination is affirmation’, affirms the identification of eternal substance with its modes (substance is eternally present, or immanent, in its affections, which are therefore the eternal expressions of substance itself). Neither interpretation, according to Macherey, offers any effective content to the modes; the concept of determination in each case therefore remains devoid of content.¹⁰¹ This leads Macherey to demand ‘what exactly does the [concept] of determination signify for Spinoza?’¹⁰² According to Macherey, Spinoza presents two contrasting ‘expositions of the [concept] of determination, “external” determination¹⁰³ and “internal” determination.¹⁰⁴’¹⁰⁵ The internal determination of a finite or singular thing involves the determination of its essence as the immanent expression of infinite substance itself. The external determination of a finite or singular thing is the determination of its existence by ‘the incessant and changing movement of external relations … which limit it’.¹⁰⁶ The essence and existence of finite or singular things ‘are therefore “determined” in completely different ways, in se and in alio’,¹⁰⁷ that is, substantially and modally.

The concept of ‘internal’ determination is, according to Macherey, essentially positive: ‘That through which things are said to be determined to produce an effect must be something positive (as is known through itself)’,¹⁰⁸ while the concept of ‘external’ determination is negative, insofar as finite modes are determined by the extent to which they limit each others existence: ‘That thing is said to be finite in its own kind that can be limited by another of the same nature’.¹⁰⁹ Macherey argues that it is only insofar as a finite or singular thing is determined externally that it ‘tends to oppose itself to all that which limits its reality, by threatening to destroy it’.¹¹⁰

The infinite, as expressed in a finite or singular thing, is defined by the same absolute affirmation that implicates the necessary existence of substance. This necessity is not limited or conditioned from the outside, but is a purely positive expression of substance. Therefore the determination of a finite or singular thing is negative only insofar as its existence is limited or conditioned by external relations with other finite or singular things. Macherey argues that the ‘negativity which marks the being of a thing expresses a limitation which can only be external to it and in no way intrinsic to the nature of this being’.¹¹¹ Finite being is therefore only ‘partially’, and not absolutely, determined by negation.¹¹²

Therefore, contrary to Hegel’s concept that all determination is negation, Macherey contends rather that finite determination is only partially negation. How then does this partial negation function in Spinoza? It is this concept of partial negation which demonstrates the implication and involvement of the infinite in the finite. Rather than establishing the internal and external determination of finite or singular things as dependent on a relation of reciprocal negation, Macherey argues that Spinoza proposes an external correlate to the internal infinite determination in his conception of the ‘most simple bodies’.¹¹³ When Spinoza speaks of the ‘most simple bodies’, he does not mean that they are really simple bodies. The Ethics, I, P15S shows that ‘the concept of substantial extension is not composed of particles isolable from one another’.¹¹⁴ Macherey argues that ‘the theory of bodies presented by Spinoza cannot … be identified with an atomism, whether this is ancient or modern’.¹¹⁵ The expression ‘the most simple bodies’ does not correspond with the idea of absolutely simple elements of material reality. Rather, the concept of ‘the most simple bodies’ designates ‘beings which are situated at the limit of bodily existence’.¹¹⁶ Following the definition given by Spinoza, that the most simple bodies are those ‘which are distinguished from one another only by motion and rest, speed and slowness’,¹¹⁷ Macherey argues that the most simple bodies are ‘pure mechanical determinations of extension’.¹¹⁸ As pure mechanical determinations, they can only be conceived as ‘rational abstractions … to which it is impossible to make correspond any existence at all’.¹¹⁹ ‘It is by hypothesis that rationality is given extremely simple bodies’.¹²⁰ Macherey argues that they are not therefore ‘less real’ than existing bodies, ‘but of a reality which cannot be directly apprehended at the level of things such as they are given immediately in experience’.¹²¹ In order to clarify this distinction, Macherey refers to Spinoza’s theory of the infinite modes of extension,¹²² which distinguishes between the infinite immediate mode (the movement and rest of the most simple bodies) and the infinite mediate mode (the facies totius universi, which always remains the same even though it varies according to an infinity of modes,¹²³ that is, ‘all the relations of motion and rest that govern the determinations of the modes as existing’.¹²⁴)¹²⁵ ‘The theory of the most simple bodies is raised to the definite order by the infinite immediate mode of extension, while the theory of composed bodies, or existing individuals, will be developed from the enchainments constituted by the infinite mediate mode of this same extension’.¹²⁶

Macherey argues that for Spinoza, ‘all that which exists in the form of finitude does not return to the idea of an elementary or unique determination, which, as such, should be isolated, but to the idea of a complex determination which carries an infinity of determinations’.¹²⁷ The existence of each body is determined by relation to another body, which is itself determined by relation to another, and so on to infinity. Spinoza can therefore say that ‘the human Body is affected in a great many ways by external bodies’,¹²⁸ a property which it shares with ‘all bodies and at the limit with all things’.¹²⁹ At the level of things such as they are given immediately in experience, there are only complex bodies or individuals, which are composed by an infinite order of causes that are interminably enchained to one another. The infinite determinations or order of causes which compose complex bodies or individuals are, according to Macherey, represented by the hypothesis of ‘the most simple bodies’. The most simple bodies, Macherey argues, are then ‘abstractions, beings of reason, which permit the construction of a discourse on reality, but which do not exist in it under a form such that they can be isolated’.¹³⁰ Nature is therefore composed of only complex bodies or individuals, because ‘all finite modes are determined by an infinite sequence of causes, which signifies that all finite determination is also infinite, both by the infinite power of its immanent cause, which is substance, and by the infinite multiplicity of its transitive causes’.¹³¹

Spinoza uses the same term to designate infinite causality, which is exercised immanently from substance, and finite causality, which is exercised in the modes by an infinite multitude of transitive causes, indicating by this that they are not two independent phenomena, but rather that the absolute infinity of substance has its correlate, not only in the absolute infinity of infinite attributes, but also in the infinite multiplicity of transitive causes of finite modes. Macherey concludes that the infinite and the finite are not ‘two independent orders’,¹³² distinguished one from the other by a dialectical logic of negation. He rather ascribes an absolute identity to the infinite and the finite which are only distinguished ‘from the abstract point of view of the imagination’.¹³³ A singular essence is united without intermediary to absolutely infinite substance which expresses itself in it positively. Insofar as this essence is expressed as an existing finite or singular thing, it is determined positively by an infinite multiplicity of transitive causes and negatively by its relation of reciprocal limitation with other finite modes. If the concept of determination, as it functions in Spinoza’s system, is taken both in its positive and negative sense, it ‘challenges the traditional opposition of positive and negative’,¹³⁴ as conceived by Hegel: the dialectical logic resolving the contradiction of the negative in the positive through the negation of the negation. Macherey argues that ‘it is not because it “goes beyond” it, or because it “resolves” it as a rational contradiction, but simply because it ignores it’.¹³⁵ ‘The mechanism of the principle of contradiction, which, [according to Hegel,] regulates … the functioning of … rational thought, … [therefore already] finds itself diverted in Spinoza’.¹³⁶ ‘For Spinoza, no thing is intrinsically determined by its contradictions, as Hegel precisely says they are’, but only extrinsically insofar as a mode exists in relation to other modes: in this sense, the dialectic, in the Hegelian sense ‘is effectively absent from Spinozism’.¹³⁷

One or Many Spinozas

Not only does Macherey’s Spinozism lack the motor of the contradiction, but it involves a ‘radical refusal of all teleology’.¹³⁸ Hegelian teleology is presupposed in the conception of the history of philosophy as having evolved and as evolving progressively,¹³⁹ according to which ‘a succession of systems … are constrained to cede their place to superior forms, which continually approach “the true philosophy”, which, in itself, would repudiate all history’.¹⁴⁰ Spinoza’s philosophy is presented by Hegel as one of these forms which is contained and abrogated in the name of the ‘true’ philosophy. Macherey argues that the absence of the logic of absolute negation in Spinozist thought ‘can no longer be interpreted only by default, as a lack to fill, as a lack which is certainly going to be filled’.¹⁴¹ Spinozist thought represents, on the contrary, ‘the positive sign of an anticipated resistance’¹⁴² to the idealism of the Hegelian dialectic.

Having concluded that there is no longer place in Spinoza for a dialectic of the Hegelian type, Hegel ou Spinoza functions as ‘an incitation to reconsider the status and the functioning of the dialectic’.¹⁴³ Macherey suggests that, if we eliminate the teleology, and, by implication, the anticipated ‘true’ philosophy from the idealism of the Hegelian dialectic, what remains is ‘a history’, which ‘opens a new sort of contradiction’,¹⁴⁴ one which does not embody the anticipation of its resolution, one whose unity is not determined by the logic of the negation of the negation. The ‘true question’ which frames Hegel ou Spinoza for Macherey then becomes ‘what is the limit which separates an idealist dialectic from a materialist dialectic?’¹⁴⁵

In his critique of Hegel ou Spinoza, Stanislas Breton points out that ‘the virtual centre of the whole work; the essential non-said which organizes it’¹⁴⁶ is the question of a Spinozist corrective to the Marxist dialectic, what Breton describes as ‘a reform (emendation) which … would consist of a Marxism “purged” of all Hegelian elements’.¹⁴⁷ Breton argues however that ‘the materialist corrective that we would inflict on the Hegelian dialectic is certainly only an impotent exorcism, a still too timid emendatio intellectus since it succeeds only in a reversal in the same genre’.¹⁴⁸ It is for this reason that Macherey denies that what he finds in Spinoza is a ‘Hegelianism without Hegel’,¹⁴⁹ suggesting instead the use of Spinoza to find a Marxism without Hegel. ‘In this sense, the text’, Breton argues, ‘opens a new space of interrogation, at the interior of Marxism itself’.¹⁵⁰ In Hegel ou Spinoza, Macherey is very careful not to find a materialist dialectic in Spinoza, but he does suggest such a dialectic as a way of repositioning, as moments of his own reading of Spinoza, the unresolved negativism of Hegel’s Spinoza and what he considers to be the equally unresolved positivism of Deleuze’s Spinoza. According to such a logic, the rejection of the Hegelian formula ‘Omnis determinatio est negatio’ also entails by implication a rejection of its opposite, the formula ascribed to Deleuze, ‘Omnis determinatio est affirmatio’. Breton acknowledges that ‘the work of Macherey marks an important stage in the contemporary reading of Spinoza’s work’, however he suggests of Macherey, just as Macherey suggests of Hegel, that ‘it is hardly sure that Spinoza inhabits the narrow circle where our debates and quarrels happen’.¹⁵¹ Breton is therefore concerned that Macherey, by returning to the question of the dialectic, risks obscuring the radical diversity established between the different interpretations of Spinoza that are presented in Hegel ou Spinoza.

By examining Hegel’s problematic reading of Spinoza, Macherey ‘restores to the history of philosophy’, and also ‘to Marxism’, ‘what is valuable in Spinoza as an alternative to Hegel’,¹⁵² and thereby demonstrates that Spinoza eludes the grasp of the dialectical progression of the history of philosophy. Rather than investigating the implications of Macherey’s reading of Spinoza to the Marxist dialectic, the present work will continue the project initiated by Macherey of investigating in detail the differences between the different interpretations of Spinoza that are presented in Hegel ou Spinoza. While I agree with Macherey’s presentation of the unresolved negativism of Hegel’s Spinoza, I consider his characterization of Deleuze’s Spinoza as an unresolved positivism to risk obscuring not only the actual difference between the respective interpretations of Spinoza by Hegel and Deleuze, but also, and more significantly, the logic that Deleuze mobilizes in Expressionism in Philosophy as an alternative to the dialectical logic. The determination of this Deleuzian logic as it functions in Expressionism in Philosophy is the problem that is addressed in chapter 2.

_______________

Notes

¹ Hegel, Hegel’s Science of Logic, p. 580.

² Ibid., p. 581.

³ Spinoza, Ethics, I, D3. The following conventional abbreviated notation will be used when referring to Spinoza’s Ethics: EI (II…V) for Ethics, Part I (roman numerals refer to the Parts of the Ethics); A for axiom; C for corollary; D for demonstration (or definition if followed by an Arabic numeral); L for lemma; Post. for postulate; P for proposition; S for scholium (Arabic numerals denote the lemma, proposition or scholium number); and, Ap for appendix. Thus EIP8S2 or Ethics, I, P8S2 refer to Ethics, Part I, Proposition 8, Scholium 2. Unless otherwise indicated, citations from the Ethics and from Spinoza’s correspondence are quoted from The Collected Works of Spinoza Volume I, ed. and trans. E. Curley (Princeton, 1985).

⁴ Spinoza, Ethics, I, D6: ‘By God I understand a being absolutely infinite, i.e., a substance consisting of an infinity of attributes, of which each one expresses an eternal and infinite essence’.

⁵ Spinoza, Ethics, I, P25C.

⁶ According to the Hegelian system of the dialectical logic, a concept is only actualized, or subsumed, as the result of its inherent contradictory conflict with its opposite which involves its own negation, that is, insofar as it ‘passes over’ into and is preserved and fulfilled by, or is sublated in relation to, its opposite.

⁷ Paul Redding, Hegel’s Hermeneutics (Ithaca, 1996), p. 146.

⁸ Hegel, Hegel’s Science of Logic, p. 577.

⁹ Hegel, The Encyclopaedia Logic (Indianapolis, 1991), §236.

¹⁰ A philosophical approach to the latter is manifest in the Philosophy of Right. See Redding, Hegel’s Hermeneutics.

¹¹ Hegel, Hegel’s Science of Logic, p. 583.

¹² Hegel, Phenomenology of Spirit (Oxford, 1977), §§17, 18, 23, 25.

¹³ Ibid., §25. This raises a controversial point of interpretation in Hegel studies: whether ‘spirit’ should be understood according to a metaphysical idealism based upon the idea of the reality of a cosmic self-actualizing mind, God; or, as based upon a non-metaphysical interpretation which determines the concept of spirit as a development of Kantian premises. For the development of the latter understanding of spirit I am indebted to Paul Redding’s Hegel’s Hermeneutics, in which he outlines an essentially hermeneutic understanding of spirit.

¹⁴ Hegel, Encyclopaedia Logic, p. 226, §151Add.

¹⁵ Redding, Hegel’s Hermeneutic, p. 147.

¹⁶ Hegel, Hegel’s Science of Logic, p. 122.

¹⁷ Redding, Hegel’s Hermeneutic, p. 148.

¹⁸ Hegel, Hegel’s Science of Logic, p. 536.

¹⁹ Ibid.

²⁰ Hegel, Hegel’s Science of Logic, p. 538. Contrary to Spinoza, Hegel proposes the inverse order, placing the Whole at the end of the process, with its determinations as moments which progressively lead to this end.

²¹ Hegel, Hegel’s Science of Logic, p. 536.

²² Ibid.

²³ Hegel, The Encyclopaedia Logic, p. 326. Hegel formulates this idea in two different ways in the two published editions of the Science of Logic. In 1812, Hegel wrote: ‘All determination is negation (determinatio est negatio) said Spinoza – a phrase which has its importance for the whole system’ (Gesammelte Werke, t. 11, p. 76). In 1831, the corresponding passage is formulated as follows: ‘Determination is the negation posed as affirmative; the sentence of Spinoza is omnis determinatio est negatio. This sentence is of incommensurable importance’ (t. 1, Lasson Edition, 1923, p. 100). Aside from Letter L (A. Wolf, The Correspondence of Spinoza. London: George Allen and Unwin, 1966), another possible source for Hegel could be F.H. Jacobi, ‘Briefe ?pinoza’, Werke. Volume IV, 1968, section 1, p. 61. See H.-C. Lucas, ‘Causa Sive Ratio’, Cahiers Spinoza. Volume 4, 1983.

²⁴ Hegel, Hegel’s Science of Logic, p. 114.

²⁵ Macherey, Hegel ou Spinoza, p. 163.

²⁶ Spinoza, Letter XII, The Collected Works of Spinoza Volume I, p. 200. ‘NS’ refers to the variations in the early Dutch translation Nagelate Schriften, 1677, and ‘LC’ refers to variations in Leibniz’s copy of Letter XII.

²⁷ Ibid.

²⁸ That is, one of which is interior to the other.

²⁹ The gloss of the first sentence that Joughin provides in his notes to Expressionism in Philosophy is very helpful in defining those aspects of the figure to which Spinoza makes reference in the example, it reads as follows: ‘The sum of the [inequalities of the] orthogonal distances between two given nested non-concentric circles’ (Deleuze, Expressionism in Philosophy, p. 419nb).

³⁰ Spinoza, Letter XII.

³¹ G.W.F. Hegel, Hegel’s Lectures on the History of Philosophy (3 vols, London, 1955), vol. 3, p. 262. The German citation is quoted from Hegel, Geschichte der Philosophie (1938), Part III, Section 2, p. 342.

³² Hegel, Hegel’s Science of Logic, p. 250.

³³ Spinoza, Ethics, I, P28.

³⁴ Hegel, Hegel’s Lectures on the History of Philosophy, p. 262.

³⁵ Ibid.

³⁶ Macherey, Hegel ou Spinoza, p. 171.

³⁷ Hegel, Hegel’s Science of Logic, p. 250.

³⁸ Macherey, Hegel ou Spinoza, p. 171.

³⁹ Hegel, Hegel’s Lectures on the History of Philosophy, p. 262.

⁴⁰ Spinoza, Letter XII.

⁴¹ Martial Gueroult, ‘Spinoza’s Letter on the Infinite’, Spinoza: A Collection of Critical Essays (Indiana, 1973), p. 190.

⁴² Hegel, Hegel’s Lectures on the History of Philosophy, pp. 262–3.

⁴³ Macherey, Hegel ou Spinoza, p. 171.

⁴⁴ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 207.

⁴⁵ Macherey, Hegel ou Spinoza, p. 172.

⁴⁶ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 210.

⁴⁷ Macherey, Hegel ou Spinoza, p. 172.

⁴⁸ Ibid., p. 170.

⁴⁹ Ibid.

⁵⁰ Ibid.

⁵¹ Ibid.

⁵² Spinoza, Letter XII.

⁵³ Macherey, Hegel ou Spinoza, p. 172.

⁵⁴ Ibid., p. 172. See the Ethics, I, P18: ‘God is the immanent, not the transitive, cause of all things’.

⁵⁵ Ibid., p. 173.

⁵⁶ Ibid., p. 173. The third kind of knowledge is introduced by Spinoza in the second part of the Ethics in relation to the imagination, or the first kind of knowledge, and to reason, or the second kind of knowledge (Ethics, II, P40S), and is developed in the fifth part of the Ethics (Ethics, V, P32C). The development of the third kind of knowledge is dealt with in detail in relation to the concept of eternity in chapter 9.

⁵⁷ Ibid., p. 164: ‘Toutes les inégalités de l’espace interposé entre deux cercles’.

⁵⁸ Ibid., p. 164.

⁵⁹ Ibid., p. 165.

⁶⁰ Ibid.

⁶¹ Spinoza, Letter XII.

⁶² Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 203.

⁶³ Macherey, Hegel ou Spinoza, p. 165.

⁶⁴ Ibid.

⁶⁵ Ibid., p. 170.

⁶⁶ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 209.

⁶⁷ Spinoza, Ethics, I, D4: ‘Per attributum intelligo id, quod intellectus de substantiâ percipit, tanquam ejusdem essentiam constituens’.

⁶⁸ Spinoza, The Collected Works of Spinoza Volume I, p. 409.

⁶⁹ Pierre Macherey, Introduction à l’Ethique de Spinoza, la première partie (Paris, 1998, p. 39.

⁷⁰ See also A.H. Wolfson, The Philosophy of Spinoza, Cambridge, Massachusetts: Harvard University Press, 1934. This interpretation has been strongly criticized by Martial Gueroult in Spinoza, Dieu (Ethique 1), (t. I, Paris, Aubier-Montaigne, 1968).

⁷¹ Hegel, Hegel’s Lectures on the History of Philosophy, p. 264.

⁷² Redding, Hegel’s Hermeneutic, p. 32.

⁷³ Hegel, Hegel’s Science of Logic, p. 537.

⁷⁴ Hegel, Encyclopaedia Logic, p. 227, §151.

⁷⁵ Macherey, Hegel ou Spinoza, p. 136.

⁷⁶ Macherey, Introduction à l’Ethique de Spinoza, la première partie, p. 42.

⁷⁷ Hegel, Encyclopaedia Logic, p. 97.

⁷⁸ Alan Donagan considers that, in spite of Spinoza’s affirmation of the identity of the mind and the object of which it is the idea, his affinities are more closely tied with the dualism of Descartes. See Alan Donagan, ‘Spinoza’s Dualism’, in Richard Kennington (ed.), The Philosophy of Baruch Spinoza (Washington, 1980). Jonathan Bennett also maintains a concept of dualism in his discussion of Spinoza’s conception of the relation between mind and body, although he does attempt to distinguish this from a distinctly Cartesian dualism. See Jonathan Bennett, A Study of Spinoza’s ‘Ethics’ (Cambridge, 1984), pp. 41–5. Curley traces the central themes of the Ethics from a critical reflection on the Cartesian system, and argues that ‘as Spinoza’s system began to take shape as a rational expression of that original intuition, it relied heavily on Cartesian assumptions to derive anti-Cartesian conclusions’(Curley, Behind the Geometrical Method, p. 140 n. 2). He considers those who think that Spinoza is as dualistic as Descartes to be ‘profoundly mistaken’. For Curley, ‘the true affinities of Spinoza’s theory of the relation between mind and body are with materialism’. See Edwin Curley, Behind the Geometrical Method: A Reading of Spinoza’s ‘Ethics’ (Princeton, 1989).

⁷⁹ Hegel, Hegel’s Lectures on the History of Philosophy, p. 252.

⁸⁰ Macherey, Hegel ou Spinoza, p. 106.

⁸¹ Ibid.

⁸² Ibid.

⁸³ Ibid.

⁸⁴ Ibid.

⁸⁵ Other alternative interpretations include, for example: Ferdinand Alquie, Le rationalism de Spinoza (Paris, 1981); Etienne Balibar, ‘Spinoza: from individuality to transindividuality’, Mededelingen vanwege het Spinozahuis 71 (Eburon Delft, 1997); Negri the Savage Anomaly.

⁸⁶ Macherey, Hegel ou Spinoza, p. 136.

⁸⁷ Ibid.

⁸⁸ Spinoza, Ethics, I, D6. See also Ethics, I, P10S.

⁸⁹ Macherey, Hegel ou Spinoza, p. 136.

⁹⁰ Ibid., p. 175.

⁹¹ Spinoza, Letter XII.

⁹² Ibid.

⁹³ Macherey, Hegel ou Spinoza, p. 175. Additional italics.

⁹⁴ Ibid., p. 175.

⁹⁵ Ibid.

⁹⁶ Ibid.

⁹⁷ Ibid.

⁹⁸ Ibid., p. 176.

⁹⁹ Ibid., p. 175.

¹⁰⁰ Ibid.

¹⁰¹ Macherey positions his own reading of Spinoza somewhere in between what he conceives to be the contradictory extremes of a Hegelian and a Deleuzian Spinozism.

¹⁰² Macherey, Hegel ou Spinoza, p. 176.

¹⁰³ See Spinoza, Ethics, I, D2, D5, D7, P11S, and P28.

¹⁰⁴ See Spinoza, Ethics, III, P5, P6, and P7.

¹⁰⁵ Macherey, Hegel ou Spinoza, p. 212.

¹⁰⁶ Ibid.

¹⁰⁷ Ibid.

¹⁰⁸ Spinoza, Ethics, I, P26D.

¹⁰⁹ Spinoza, Ethics, I, D2.

¹¹⁰ Macherey, Hegel ou Spinoza, p. 211.

¹¹¹ Macherey, Introduction à l’Ethique de Spinoza, la première partie, p. 82 n. 1.

¹¹² Spinoza, Ethics, I, P8S1: ‘Since being finite is really, in part, a negation’.

¹¹³ Spinoza, Ethics, II, P13L3A2.

¹¹⁴ Pierre Macherey, Introduction à l’Ethique de Spinoza, la seconde partie (Paris, 1997, p. 131.

¹¹⁵ Ibid., p. 132.

¹¹⁶ Ibid.

¹¹⁷ Spinoza, Ethics, II, P13L3A2.

¹¹⁸ Macherey, Introduction à l’Ethique de Spinoza, la seconde partie, p. 137 n. 1.

¹¹⁹ Ibid., p. 141.

¹²⁰ Ibid., p. 141 n. 1.

¹²¹ Ibid., p. 142.

¹²² Benedict de Spinoza, The Correspondence of Spinoza, trans. A. Wolf, (London, 1966), Letter LXIV.

¹²³ Spinoza, Ethics, II, P13L7.

¹²⁴ Gilles Deleuze, Spinoza: Practical Philosophy, trans. R. Hurley (San Francisco, 1988), p. 92.

¹²⁵ Macherey, Hegel ou Spinoza, p. 183.

¹²⁶ Macherey, Introduction à l’Ethique de Spinoza, la seconde partie, p. 142 n. 1.

¹²⁷ Macherey, Hegel ou Spinoza, p. 194.

¹²⁸ Spinoza, Ethics, III, P51D.

¹²⁹ Macherey, Hegel ou Spinoza, p. 194.

¹³⁰ Ibid., p. 193. In this sense, Gueroult was well founded to distinguish an abstract physics of the most simple bodies and a concrete physics of composed bodies, which takes as its objects effectively existing individuals (See Martial Gueroult, Spinoza, l’âme (Ethique 2), (Paris, 1974), p. 156).

¹³¹ Macherey, Hegel ou Spinoza, p. 192.

¹³² Ibid., p. 199.

¹³³ Ibid.

¹³⁴ Ibid.

¹³⁵ Ibid., p. 180.

¹³⁶ Ibid., p. 225.

¹³⁷ Ibid., p. 225. Additional italics.

¹³⁸ Ibid., p. 67.

¹³⁹ What Eugene Holland describes as an ‘a posteriori Hegelian teleologism’ (‘Spinoza and Marx’, Cultural Logic 2.1 (Fall, 1998), §21).

¹⁴⁰ Macherey, Hegel ou Spinoza, p. 258.

¹⁴¹ Ibid., p. 260.

¹⁴² Ibid.

¹⁴³ Pierre Macherey, ‘De la médiation à la constitution: description d’un parcours spéculatif’, Cahiers Spinoza, 4 (1983), p. 36.

¹⁴⁴ Macherey, Hegel ou Spinoza, p. 260.

¹⁴⁵ Ibid.

¹⁴⁶ Stanislas Breton, ‘Hegel ou Spinoza. Réflexions sur l’enjeu d’une alternative’, Cahiers Spinoza, 4 (1983), p. 64.

¹⁴⁷ Ibid.

¹⁴⁸ Ibid.

¹⁴⁹ Macherey, Hegel ou Spinoza, p. 183.

¹⁵⁰ Breton, ‘Hegel ou Spinoza. Réflexions sur l’enjeu d’une alternative’, p. 64.

¹⁵¹ Ibid., p. 87.

¹⁵² Holland, ‘Spinoza and Marx’, §7.

Chapter 2

The differential point of view of the infinitesimal calculus

The Letter on the Infinite

Having utilized the geometrical example of Spinoza’s Letter XII in chapter 1 to distinguish the respective interpretations of Spinoza by Hegel and Macherey, Deleuze’s reading of Spinoza will be distinguished from that of Hegel and Macherey insofar as it too offers an alternative reading of what Deleuze describes as the ‘very bizarre, curious’¹ geometrical example. By implicating Leibniz’s understanding of the early form of the infinitesimal calculus in his reading of the geometrical example, Deleuze argues that it is able to be characterized as an example of what had already been established of the infinitesimal calculus. He thereby traces an alternative lineage between Leibniz and Spinoza to that determined by Hegel in the development of the dialectical logic. It is therefore by means of Deleuze’s reading of Letter XII that the investigation into the logic that is mobilized in his reading of Spinoza in Expressionism in Philosophy, as an alternative to the dialectical logic, will be developed, and it is Gueroult’s reading of Letter XII which provides the point of departure for this investigation.

In ‘Spinoza’s Letter on the Infinite’, Gueroult suggests that the meaning of the geometrical example is ‘generally wrongly understood’, and that it has been ‘vitiated in its principle through errors in translation’.² Macherey continues this argument in Hegel ou Spinoza when he writes that ‘the example as reproduced by Hegel is not the same as that given in the text of Spinoza’.³ He argues that the same example is exploited by each commentator ‘in markedly different ways’.⁴ Macherey corrects Hegel’s error of ascribing to the ‘inequalities of space’ only ‘unequal distances’ with the more accurate ‘differences in distance’, thus realigning the concept of the maximum and minimum limits, which Hegel incorrectly ascribes solely to the circumferences of the greater and smaller circles rather than also to the maximum and minimum orthogonal distances, which necessitate the specificity of non-concentric circles. Macherey and Gueroult come to very similar conclusions in relation to their respective interpretations of Letter XII, and of Hegel’s misreading of it. However, in what may be seen as Macherey’s preoccupation with the question of the dialectic in Hegel ou Spinoza, he criticizes Gueroult’s fourfold distinction of the definitions of infinity in Letter XII: ‘In his commentary on the letter on infinity, Gueroult enumerates four successive cases which, in relation to the text of Spinoza, seems excessive’.⁵ Gueroult breaks the distinctions down as follows:

1. The thing Infinite by its essence or by virtue of its definition.

2. The thing without limits, not by virtue of its essence, but by virtue of its cause.

3. The thing infinite insofar as without limits.

4 The thing infinite insofar as its parts although included within a maximum and minimum known to us, cannot be expressed by any number.⁶

Macherey considers the third definition to be superfluous since for him it is merely an explanation of the second, not different to it; the difference having already been posed by the difference between an adequate understanding and an inadequate representation of the second definition. According to Macherey, the third definition of the infinite, when understood from the point of view of reason, is the same as the second definition, and when the second definition is represented inadequately from the point of view of the imagination, the result is the third definition. The significance of Macherey’s move to make the third definition implicit in the second definition is that the fourth definition, which is illustrated by the geometrical example, then becomes the example of the implied difference in the second definition, between an imaginary and a rational understanding of the infinite. For Gueroult, however, the significance of the third definition rests, on the one hand, in its distinction from the fourth definition and, on the other hand, insofar as it is significant in his criticism of the Hegelian interpretation of Letter XII. Presenting his case, Gueroult argues that the geometrical example, which illustrates the fourth definition, ‘is advanced in order to refute the false interpretation’ that ‘every infinite is such that its magnitude is so excessive … that we cannot perceive its limits, or that its variations are not contained between any boundaries’.⁷ This explanation merely repeats the third definition. To claim that the Infinite must everywhere be said to be inferred from the multitude of its parts is to explain the Infinite by the finite, which can only then ‘be avoided by the negation of the Infinite on behalf of the finite’.⁸ Gueroult contends that Hegel makes this claim in order to implicate the third definition in the dialectical logic, and that he uses the geometrical example, or the fourth definition, to support or illustrate this implication, that is, to explicate the third definition. Gueroult is critical of Hegel’s reading of Letter XII insofar as it relies on the third definition in order to secure the mediation of the dialectical relation between the finite and the infinite. Macherey’s recognition of this is the reason that he considers the third definition superfluous to an adequate understanding of Spinoza’s infinite.⁹ Gueroult argues that, contrary to Hegel’s interpretation, the third and fourth definitions present two different definitions of the infinite which do not implicate each other in a dialectical relation of negation. What Hegel presents as a dialectical relation of opposition between a Mathematical and a Philosophic infinite, Gueroult presents as a difference between a Mathematical infinite, that is, an arithmetic or algebraic infinite, as defined by the third definition,¹⁰ and a geometric infinite, as illustrated by the fourth definition.¹¹ Hegel’s reliance on the third definition renders the geometric infinite, as characterized by Spinoza in the geometrical example, superfluous to his interpretation of Letter XII. Hegel implicates the Mathematical infinite solely as an algebraic infinite. Spinoza’s depreciation of arithmetic, which, along with number, is reduced to a finite being of the imagination,¹² is set against what Gueroult describes as Spinoza’s ‘correlative exaltation of geometrical thought’.¹³ What Hegel recognizes as Spinoza’s rejection of the Mathematical infinite is rather a rejection of the algebraic infinite in the name of a geometric infinite. Spinoza’s geometric infinite is therefore different to Hegel’s mathematical algebraic infinite. Gueroult suggests that ‘this … establishes a vigorous contrast between the mathematical philosophy of Spinoza and those of the philosophers of his time’,¹⁴ a contrast which could well be extended to include the interpretations of both Macherey and Hegel. Spinoza here sets geometry against algebra, a properly geometric infinite against an algebraic infinite. Gueroult suggests that this ‘opens an abyss between arithmetic and geometry’,¹⁵ which he then maps onto the epistemological difference, which is exploited by Macherey, between an adequate understanding of the geometric infinite as actually infinite from the point of view of reason, and an inadequate representation of this geometric infinite as an algebraic infinite from the point of view of the imagination.

The geometric infinite as ‘an infinite sum of differentials’

Deleuze’s interpretation of Letter XII in Expressionism in Philosophy takes up this distinction between algebra and geometry in order to redeploy the definitions of the infinite from the point of view of the geometrical example. Deleuze elicits the support of Leibniz to argue that Spinoza’s exposition of the geometric infinite in Letter XII goes ‘further on this point than many mathematicians’.¹⁶ According to Deleuze, the properly geometric infinite characterized in the fourth definition, as distinct from the algebraic infinite of the third definition, defines ‘a strictly extensive infinity’, which is a properly ‘modal and quantitative’ infinite.¹⁷ A thorough analysis of Deleuze’s interpretation of Letter XII is required to illustrate how he comes to this conclusion.

Deleuze distinguishes his interpretation of Letter XII from that of Hegel by arguing that ‘two concentric circles of which one is interior to the other lack the greatest and smallest distance of one circle to the other’.¹⁸ In order to explicate what happens across this maximum and minimum orthogonal distance, Deleuze offers a translation of the sentence with which Spinoza introduces the example. Spinoza writes: ‘omnes inæqualitates spatii duobus circulis AB, & CD, interpositi’.¹⁹ Deleuze translates this as ‘the sum of the inequalities of distance included between two circles, AB and CD’.²⁰ Deleuze considers this example to define ‘a very particular geometric infinite’.²¹ Hegel considers only the algebraic infinity of unequal distances, that is, if all the lines or segments that go from one circle to the other, the orthogonal distances, are placed end to end, the sum would be infinite.²² This sum ‘would be equally infinite if the two circles were concentric and if all the segments … were equal, it is [therefore] clear that [for Hegel] the infinite sum of the unequal segments … is not related to their inequality and is not delimited by the maximum and the minimum of their variations’.²³ Deleuze, on the contrary, considers it to be a question, not of the sum of the segments, or orthogonal distances, but rather of the sum of their inequalities, that is, the sum of the differences between the orthogonal distances.

Macherey also distinguishes his reading of Letter XII from that of Hegel in this way, however, he does not conceive of the infinite variation of differences, understood adequately from the point of view of reason, to be a geometric infinite. Macherey reserves the use of the term ‘sum’ for references associated with the inadequate representation of this infinite from the point of view of the imagination. An infinite sum which is only inadequately supposed to be infinite, or unlimited, by the multitude of its parts, is an algebraic infinite. Deleuze considers the adequate understanding of the infinite from the point of view of reason to be an infinite sum which is distinctly geometrical. He argues that the geometric infinite ‘is not strictly speaking “unlimited”: for it relates to … the … maximum (AB) and … minimum (CD) distance between two non-concentric circles, and these distances attach to a perfectly limited and determinate space’.²⁴ When the orthogonal distances vary continuously from AB to CD, the sum of the successive differences is not simply a matter of ‘an infinite sum of finite quantities’, as Hegel contends, nor is it ‘the infinite variation of differences’ as Macherey contends, but, Gueroult argues, and Deleuze backs up this argument, it should rather be understood as ‘an infinite sum of differentials’.²⁵

The concept of the differential was introduced by developments in the infinitesimal calculus during the later part of the seventeenth century. Deleuze positions Spinoza’s geometric infinite within the early stages of this development, which Carl Boyer, in The history of the calculus and its conceptual development, describes as being ‘bound up with concepts of geometry … and with explanations of … the infinitely small’.²⁶ Boyer describes the infinitesimal calculus as dealing with ‘the infinite sequences … obtained by continuing … to diminish ad infinitum the intervals between the values of the independent variable. … By means of [these] successive subdivisions … the smallest possible intervals or differentials’ are obtained.²⁷ The differential can therefore be understood to be the infinitesimal difference between consecutive values of a continuously diminishing quantity. Boyer refers to this early form of the infinitesimal calculus as the infinitesimal calculus from ‘the differential point of view’.²⁸

Gueroult provides a succinct definition of the differential in relation to the geometrical example of Letter XII when he suggests that the determinate distance ‘AB minus CD’ ‘includes an actual infinity of infinitely small distances, and consequently is indivisible into discontinuous parts’.²⁹ Each of these infinitely small distances corresponds to a differential. From the differential point of view of the infinitesimal calculus, Boyer argues that, ‘the derivative would in this case be defined as the quotient of two such differentials, and the integral would then be the sum of a number (perhaps finite, perhaps infinite) of such differentials’.³⁰ The geometric infinite of Spinoza’s Letter XII, as ‘an infinite sum of differentials’, can therefore be understood to be an example of an integral from the differential point of view.

Gueroult identifies the integral in the geometrical example as a definite integral when he argues that ‘the distance CD is a minimal quantity obtained by continuously diminishing the infinitely small parts of AB, the sum of these diminishments being a definite integral, that is, a finite quantity … resulting from an infinite summation of differentials’.³¹ According to Gueroult, the infinite sum of the differentials is a definite integral, and a definite integral is a finite quantity. In this way, the geometrical example illustrates the implication and envelopment of the infinite in the finite.

What Deleuze then does with the concept of the infinitesimal calculus from the differential point of view distinguishes his interpretation of Letter XII from that of Gueroult. Whereas Gueroult only suggests the implication of the infinitesimal calculus in his interpretation of Spinoza’s Letter XII,³² Deleuze explores this implication in his interpretation of Spinoza’s work as a whole. It is here that we begin to approach not only the originality of Deleuze’s interpretation of Spinoza, but also one of the points where Deleuze departs from Spinozism by tracing an alternative lineage in the history of philosophy in relation to Leibniz.

The differential point of view

The infinitesimal calculus consists of two branches which are inverse operations: differential calculus, which is concerned with calculating derivatives, or differential relations; and integral calculus, which is concerned with integration, or the calculation of the infinite sum of the differentials. The derivative, from the differential point of view of the infinitesimal calculus, is the quotient of two differentials, that is, a differential relation, of the type dy/dx. The differential, dy, is an infinitely small quantity, or what Deleuze describes as ‘a vanishing quantity’;³³ a quantity smaller than any given or givable quantity. Therefore, as a vanishing quantity, dy, in relation to y, is, strictly speaking, equal to zero. In the same way dx, in relation to x, is, strictly speaking, equal to zero, that is, dx is the vanishing quantity of x. Given that y is a quantity of the abscissa, and that x is a quantity of the ordinate, dy = 0 in relation to the abscissa, and dx = 0 in relation to the ordinate. The differential relation can therefore be written as dy/dx = 0/0. However, although dy is nothing in relation to y, and dx is nothing in relation to x, dy over dx does not cancel out, that is, dy/dx is not equal to zero.³⁴ When the differentials are represented as being equal to zero, the relation can no longer be said to exist since the relation between two zeros is zero, that is 0/0 = 0; there is no relation between two things which do not exist. However, the differentials do actually exist. They exist as vanishing quantities insofar as they continue to vanish as quantities rather than having already vanished as quantities. Therefore, despite the fact that, strictly speaking, they equal zero, they are still not yet, or not quite equal to, zero. The relation between these two differentials, dy/dx, therefore does not equal zero, dy/dx ≠ 0, despite the fact that dy/dx = 0/0.³⁵ Instead, the differential relation itself, dy/dx, subsists as a relation. ‘What subsists when dy and dx cancel out under the form of vanishing quantities is the relation dy/dx itself’.³⁶ Despite the fact that its terms vanish, the relation itself is nonetheless real. It is here that Deleuze considers seventeenth-century logic to have made ‘a fundamental leap’, by determining ‘a logic of relations’.³⁷ He argues that ‘under this form of infinitesimal calculus is discovered a domain where the relations no longer depend on their terms’.³⁸ The concept of the infinitely small as vanishing quantities allows the determination of relations independently of their terms. ‘The differential relation presents itself as the subsistence of the relation when the terms vanish’.³⁹ According to Deleuze, ‘the terms between which the relation establishes itself are neither determined, nor determinable. Only the relation between its terms is determined’.⁴⁰ This is the logic of relations that Deleuze locates in the infinitesimal calculus of the seventeenth century, which he then mobilizes in his reading of Spinoza’s Letter XII, and in his reading of Spinoza’s work as a whole, particularly in relation to the physics of bodies in the second part of the Ethics.⁴¹

The differential relation, which Deleuze characterizes as a ‘pure relation’⁴² because it is independent of its terms, and which subsists insofar as dy/dx ≠ 0, has a perfectly expressible finite quantity designated by a third term, z, such that dy/dx equals z.⁴³ Deleuze argues that ‘when you have a [differential] relation derived from a circle, this relation doesn’t involve the circle at all but refers [rather] to what is called a tangent’.⁴⁴ A tangent is a line that touches a circle or curve at one point. The gradient of a tangent indicates the rate of change of the curve at that point, that is, the rate at which the curve changes on the y-axis relative to the xaxis. The differential relation therefore serves in the determination of this third term, z, the value of which is the gradient of the tangent to the circle or curve. In the case of the geometrical example, it is the tangents to the circumferences of the two circles at A and C which define the maximum and minimum limits of the orthogonal distances AB and CD.⁴⁵

When referring to the geometrical study of curves in his early mathematical manuscripts, Leibniz writes that ‘the differential calculus could be employed with diagrams in an even more wonderfully simple manner than it was with numbers’.⁴⁶ Leibniz presents one such diagram in a paper entitled ‘Justification of the Infinitesimal Calculus by That of Ordinary Algebra’,⁴⁷ when he offers an example of what had already been established of the infinitesimal calculus in relation to particular problems before the greater generality of its methods were developed. An outline of the example that Leibniz gives is as follows:

since the two right triangles, ZFE and ZHJ, that meet at their apex, point Z, are similar, it follows that the ratio y/x is equal to (Y – y)/X. As the straight line EJ approaches point F, maintaining the same angle at the variable point Z, the lengths of the straight lines FZ and FE, or y and x, steadily diminish, yet the ratio of y to x remains constant. When the straight line EJ passes through F, the points E and Z coincide with F, and the straight lines, y and x, vanish. Yet y and x will not be absolutely nothing since they preserve the ratio of ZH to HJ, represented by the proportion (Y – y)/X, which in this case reduces to Y/X, and obviously does not equal zero. The relation y/x continues to exist even though the terms have vanished since the relation is determinable as equal to Y/X. In this algebraic calculus, the vanished lines x and y are not taken for zeros since they still have an algebraic relation to each other. ‘And so’, Leibniz argues, ‘they are treated as infinitesimals, exactly as one of the elements which … differential calculus recognizes in the ordinates of curves for momentary increments and decrements’.⁴⁸ That is, the vanished lines x and y are determinable in relation to each other only insofar as they can be replaced by the infinitesimals dy and dx, by making the supposition that the ratio y/x is equal to the ratio of the infinitesimals, dy/dx. In the first published account of the calculus,⁴⁹ Leibniz defines the ratio of infinitesimals as the quotient of first-order differentials, or the differential relation. He says that ‘the differential dx of the abscissa x is an arbitrary quantity, and that the differential dy of the ordinate y is defined as the quantity which is to dx as the ratio of the ordinate to the subtangent’.⁵⁰ Leibniz considers differentials to be the fundamental concepts of the infinitesimal calculus, the differential relation being defined in terms of these differentials.

Fig. 2.1

Deleuze considers Spinoza’s Letter XII to provide an example of what had already been established of the infinitesimal calculus in relation to a particular problem at the level of geometry; the geometric infinite. Spinoza’s example is schematic in just the right proportion to be representable using this Leibnizian formulation of the early form of the infinitesimal calculus. The example of two non-concentric circles (Fig. 2.2) can be transposed onto Leibniz’s example (Fig. 2.1) by letting Z be the point on the maximum orthogonal distance AB (Fig. 2.3), such that AZ equals AB minus CD.⁵¹ The points A and Z represent the points of tangency of the minimum and maximum orthogonal distances respectively on AB. Since the values of the differences of distance of Spinoza’s example vary between A and Z, then, according to Leibniz’s justification of the infinitesimal calculus by algebra, each of these differences of distance can be represented by an algebraic ratio of infinitesimals, that is, by a differential relation, dy/dx. The infinite sum of these differential relations being the infinite sum that Spinoza refers to in Letter XII. The geometric infinite, as ‘an infinite sum of differentials’, can therefore be understood to be an example of an integral from the differential point of view of the infinitesimal calculus. Insofar as this infinite sum is understood to function as such an integral, it provides an example of what Deleuze describes as the ‘integral calculus performed unknowingly’.⁵²

Fig. 2.2

Fig. 2.3

The differential point of view of the infinitesimal calculus approaches integration as a process of summation by considering the problem of finding the area under the graph of a function. This problem is dealt with by dividing up the area under the curve into a large number of rectangles. The area under the curve is the sum of the infinitely small and infinitely numerous rectangles. The difference between this sum and the actual area is considered small enough to be neglected. The integral is therefore the finite magnitude of the area.

A new theory of relations

Leibniz recognized integration to be a process not only of summation, but also of the inverse transformation of differentiation, so the integral is not only the sum of differentials, but also the inverse of the differential relation. In the early nineteenth century, the process of integration as a summation was overlooked by most mathematicians in favour of determining integration, instead, as the inverse transformation of differentiation. Hegel continued this tendency by defining the integral solely as the inverse of the differential relation. According to Hegel ‘the integral calculus has been simplified and more correctly determined merely by the fact that it is no longer taken to be a method of summation in which it appeared essentially connected with the form of a series’.⁵³ The main reason for this was that by extending sums to an infinite number of terms, problems began to emerge if the series didn’t converge. The value or sum of an infinite series is only determinable if the series converges. Divergent series have no sum. It was considered that reckoning with divergent series would therefore lead to false results. However, the problem of integration as a process of summation from the differential point of view of the infinitesimal calculus did continue to be explored. It was Augustin Cauchy (1789 – 1857) who first introduced specific tests for the convergence of series, so that divergent series could henceforth be excluded from being used to try to solve problems of integration because of their propensity to lead to false results.⁵⁴ This method was later reformulated by Cauchy as the determination of the definite integral, which is distinguished from the indefinite integral only insofar as the definite integral is used to determine the limit of the sum of the area under the curve between two points on the x-axis, whereas the indefinite integral, as the primitive function whose derivative is the given function, determines the whole curve. Both integrals, definite and indefinite, are determinable by the inverse transformation of the differential relation; the definite integral being determined within a limited interval. Whereas Gueroult equates the infinite sum of Spinoza’s example with this definite integral, Deleuze rather appeals to the ‘barbaric or prescientific interpretations of the differential calculus’,⁵⁵ as presented by the differential point of view of the infinitesimal calculus, according to which integration is a method of summation in the form of a series.

The object of the process of integration in general is to determine from the coefficients of the given function of the differential relation the original function from which they were derived. Put simply, given a relation between two differentials, dy/dx, the problem of integration is how to find a relation between the quantities themselves, y and x. This problem corresponds to the geometrical method of finding the function of a curve characterized by a given property of its tangent. The differential relation is thought of as another function which describes, at each point on an original function, the gradient of the line tangent to the curve at that point. The value of this ‘gradient’ indicates a specific quality of the original function; its rate of change at that point. The differential relation therefore indicates the specific qualitative nature of the original function at the different points of the curve.

The inverse process of integration is differentiation, which, in geometrical terms, determines the differential relation as the function of the line tangent to a given curve. Put simply, to determine the tangent of a curve at a specified point, a second point that satisfies the function of the curve is selected, and the gradient of the line that runs through both of these points is calculated. As the second point approaches the point of tangency, the gradient of the line between the two points approaches the gradient of the tangent. The gradient of the tangent is, therefore, the limit of the gradient of the line between the two points.

It was Newton who first came up with this concept of a limit. He conceptualized the tangent geometrically, as the limit of a sequence of lines between two points on a curve, which he called a secant. As the distance between the points approached zero, the secants became progressively smaller, however they always retain ‘a real length’. The secant therefore approached the tangent without reaching it. When this distance ‘got arbitrarily small (but remained a real number)’,⁵⁶ it was considered insignificant for practical purposes, and was ignored. What is different in Leibniz’s method is that he ‘hypothesized infinitely small numbers – infinitesimals – to designate the size of infinitely small intervals’.⁵⁷ For Newton, on the contrary, these intervals remained only small, and therefore real. When performing calculations, however, both approaches yielded the same results. But they differed ontologically, because Leibniz had hypothesized a new kind of number, a number Newton did not need since ‘his secants always had a real length, while Leibniz’s had an infinitesimal length’.⁵⁸

For the next two hundred years, various attempts were made to find a rigorous arithmetic foundation for the calculus. One that didn’t rely on either the mathematical intuition of geometry, with its tangents and secants, which was perceived as imprecise because its conception of limits was not properly understood; or on the vagaries of the infinitesimal, which made many mathematicians wary, so much so that they refused the hypothesis outright, despite the fact that Leibniz ‘could do calculus using arithmetic without geometry – by using infinitesimal numbers’.⁵⁹ It was not until the late nineteenth century, that an adequate solution to this problem of rigour was posed. It was Karl Weierstrass (1815 – 1897) who ‘developed a pure nongeometric arithmetization for Newtonian calculus’,⁶⁰ which provided the rigour that had been lacking. ‘Weierstrass’s theory was an updated version of Cauchy’s earlier account’,⁶¹ which had also had problems conceptualizing limits. Cauchy actually begs the question of the concept of limit in his proof.⁶² In order to overcome this problem of conceptualizing limits, Weierstrass ‘sought to eliminate all geometry from the study of … derivatives and integrals in calculus’.⁶³ In order to characterize calculus purely in terms of arithmetic, it was necessary for the idea of a function, as a curve in the Cartesian plane defined in terms of the motion of a point, to be completely replaced with the idea of a function that is, rather, a set of ordered pairs of real numbers. The geometric idea of ‘approaching a limit’ had to be replaced by an arithmetized concept of limit that relied on static logical constraints on numbers alone. This approach is commonly referred to as the epsilon–delta method. Deleuze argues that ‘It is Weierstrass who bypasses all the interpretations of the differential calculus from Leibniz to Lagrange, by saying that it has nothing to do with a process … Weierstrass gives an interpretation of the differential and infinitesimal calculus which he himself calls static, where there is no longer fluctuation towards a limit, nor any idea of threshold’.⁶⁴ The calculus was thereby reformulated without either geometric secants and tangents or infinitesimals; only the real numbers were used.

Because there is no reference to infinitesimals in this Weierstrassian definition of the calculus, the designation ‘the infinitesimal calculus’ was considered to be ‘inappropriate’.⁶⁵ Weierstrass’s work not only effectively removed any remnants of geometry from what was now referred to as the differential calculus, but it eliminated the use of Leibnizian inspired infinitesimal arithmetic in doing the calculus for over half a century. It wasn’t until the late 1960s, with the development of the controversial axioms of non-standard analysis by Abraham Robinson (1918 – 1974), that the infinitesimal was given a rigorous foundation, and a formal theory of the infinitesimal calculus was constructed,⁶⁶ thus allowing Leibniz’s ideas to be ‘fully vindicated’,⁶⁷ as Newton’s had been thanks to Weierstrass. It is specifically in relation to these developments that Deleuze’s appeal to the ‘barbaric or pre-scientific interpretations of the differential calculus’,⁶⁸ should be understood.

Although it is only later in the development of the infinitesimal calculus that the tangent comes to be considered as a limit, and that the differential relation is used to calculate ‘limits’, Deleuze contends that the maximum and minimum illustrated in Spinoza’s geometrical example are suggestive of such limits. He introduces the concepts of the differential relation and limits into his interpretation not only of Letter XII, but also into his interpretation of the physics of bodies presented in the second part of the Ethics. So, according to Deleuze, the value of z, which was determined by Leibniz in relation to the differential relation (dy/dx) as the gradient of the tangent, functions as a limit. When the relation establishes itself between infinitely small terms, it does not cancel itself out with its terms, but rather tends towards a limit. In other words, when the terms of the differential relation vanish, the relation subsists because it tends towards a limit, z. Since the differential relation approaches closer to its limit as the differentials decrease in size, or approach zero, the limit of the relation is represented by the relation between the infinitely small. Of course, despite the geometrical nature of the idea of a variable and a limit, the former ‘decrease in size’ or ‘approach zero’, and the differential relation ‘approaches’ or ‘tends towards’ the latter, they are not essentially dynamic, but involve purely static considerations, that is, they are rather ‘to be taken automatically as a kind of shorthand for the corresponding developments of the epsilon–delta approach’.⁶⁹ It is in this sense that the differential relation between the infinitely small refers to something finite. Or, as Deleuze suggests, it is in the finite itself that there is the ‘mutual immanence’⁷⁰ of the relation and the infinitely small.

Given that the method of integration provides a way of working back from the differential relation, the problem of integration is, therefore, how to reverse this process of differentiation. This can be solved by determining the inverse of the given differential relation according to the inverse transformation of differentiation. Or, a solution can be determined from the differential point of view of the infinitesimal calculus by considering integration as a process of summation in the form of a series, according to which, given the specific qualitative nature of a tangent at a point, the problem becomes that of finding, not just one other point determinative of the differential relation, but a sequence of points, all of which together satisfy, or generate, a curve and therefore a function in the neighbourhood of the given point of tangency, which therefore functions as the limit of the function.

Deleuze considers this to be the base of the infinitesimal calculus as understood or interpreted in the seventeenth century. The formula for the problem of the infinite that Deleuze extracts from the geometrical example of Letter XII, by means of this seventeenth-century understanding of the infinitesimal calculus, is that ‘something finite consists of an infinity under a certain relation’.⁷¹ Deleuze considers this formula to mark ‘an equilibrium point, for seventeenth-century thought, between the infinite and the finite, by means of a new theory of relations’.⁷²

The differential calculus from the point of view of the dialectical logic

Hegel’s discussion of Letter XII in the Science of Logic also involves a rearticulation of the implication and envelopment of the infinite in the finite in relation to the history of the differential calculus. Hegel also considers the early stages of the development of the differential calculus to be illustrated by Spinoza’s example, however, the account that he gives of this relation is markedly different to that of Deleuze. Where Deleuze argues that ‘the inequalities of distance’ should be understood to be differentials from the differential point of view of the infinitesimal calculus, Hegel argues, on the contrary, that, from the point of view of the differential calculus, Spinoza’s example does not involve the concept of the differential. Hegel argues rather that it is an adequate concept of the differential that is specifically lacking from Spinoza’s example. Hegel brings his discussion of Letter XII and of the differential calculus together in the following paragraph: ‘Spinoza’s space enclosed between two circles, with its inequalities, … can be made larger or smaller. But this does not involve the absurdity of a larger or smaller infinite; for this quantum of the whole does not concern the relation of its moments, the nature of the fact, that is, the qualitative determination of magnitude; what is actually present in the infinite series is equally a finite quantum, but it is also still deficient. Imagination on the contrary stops short at quantum as such and does not reflect on the qualitative relation which constitutes the ground of the existing incommensurability’. With this passage, Hegel denies that Spinoza’s example is able to be interpreted in terms of the concept of the differential from the point of view of the differential calculus. He excludes ‘the relation of its moments … that is, the qualitative determination of magnitude’, which are each alternative expressions of ‘the differential relation’, from his interpretation of the ‘quantum of the whole’, which he regards as ‘the infinite sum of unequal segments’. The ‘qualitative relation’, upon which the imagination does not reflect, is the differential itself. These references introduce the qualitative and quantitative distinction into the discussion of the relation between the concept of the differential and the problem of the infinite. The implications of the qualitative and quantitative distinction to this relation are significant in distinguishing the readings of Spinoza presented by Hegel and Deleuze.

What Hegel considers to be represented in Spinoza’s example is, on the one hand, the bad infinite, that is, an infinite series, and on the other, an actual infinite represented as a finite quantity. However, according to Hegel, the example is deficient insofar as it does not function to represent the mediation between the different infinities that it represents. Although Hegel agrees that, according to the example, ‘the infinite series is equally a finite quantum’, he argues that this relation is simply posited in the example, without being developed as such. The example is therefore deficient insofar as this posited relation is not mediated by the differential relation, that is, insofar as the example does not postulate the differential relation as mediator between the different infinities that it represents. So, according to Hegel, the differential relation is conspicuous in the example because of its absence. That which constitutes the ground of the incommensurability which Hegel finds in Spinoza’s example, is therefore the differential relation.

Hegel argues that ‘the incommensurability which lies in Spinoza’s example embraces in general the functions of curved lines and more precisely, leads to the infinite which mathematics has introduced with such functions, in general, with the functions of variable magnitudes. This infinite is the genuine mathematical qualitative infinite which Spinoza also had in mind’.⁷³ The infinity that Hegel is referring to here is the actual infinite of Spinoza’s example which Hegel considers to be a ‘qualitative infinite’, and which he referred to previously as ‘the quantum of the whole’. What Hegel contends is that it is only later developments in mathematics, specifically in the differential calculus, subsequent to Spinoza’s work, which permit ascribing to Spinoza’s example elements of the differential calculus. In terms of the differential calculus, the differential relation, as the determination of the limit of the function of a curve, leads to or allows the subsequent determination of the integral of the function, insofar as differentiation and integration are inverse operations; the integral being the mathematical expression of the actual or qualitative infinite as a finite quantum. Hegel argues that Spinoza ‘also had in mind’ this qualitative infinite which is ‘the quantum of the whole’. The actual infinite of Spinoza’s example, the actual infinite represented as a finite quantity, can therefore be understood to correlate with the integral from the point of view of the differential calculus. However, according to Hegel, the example in no way expresses the transition from the bad infinite to the actual infinite, that is, from infinite series to the integral, since Hegel specifically excludes the applicability of the differential to Spinoza’s example. As far as he is concerned, it is only later developments in mathematics which provide the means to express this transition.

What Hegel considers Spinoza’s example to be lacking, along with those early progenitors of the infinitesimal calculus, is an adequate expression of the differential in the process of integration. He argues that ‘the infinitely small which presents itself in the differential calculus as dx and dy, does not have merely the negative, empty meaning of a non-finite, non-given magnitude, as when one speaks of “an infinite multitude”, “to infinity”, and the like’.⁷⁴ The infinitely small that Hegel refers to is that of Spinoza’s most simple bodies represented as constituting a bad infinite in expressions such as ‘and so on to infinity’. On the contrary, the infinitely small, from the point of view of the differential calculus has, according to Hegel, ‘the specific meaning of the qualitative nature of what is quantitative, of a moment of a ratio as such’.⁷⁵ When Hegel correlates the infinitely small, as a moment of a ratio, and the differential, as a component of a differential relation, he is no longer referring to Spinoza’s example, but to subsequent developments in the differential calculus.

Hegel defines the qualitative character of what is quantitative in the differential by arguing that, in the differential, ‘a quantitative difference, the definition of which is that it not only can, but shall be smaller than any given difference, is no longer a quantitative difference, this is self-evident, as self-evident as anything can be in mathematics’.⁷⁶ The quantitative difference of a differential thus constitutes its qualitative character. Hegel argues that ‘the demonstrated qualitative character as such of the form of magnitude here under discussion in what is called the infinitesimal, is found most directly in the category of limit of the ratio’.⁷⁷ The limit of the ratio is determined by Hegel as the differential relation. However, for Hegel, ‘the idea of limit … impl[ies] the true category of the qualitatively determined relation of variable magnitudes’.⁷⁸ The differential relation functions therefore as the qualitative determination of quantity. Hegel can therefore argue that ‘the so-called infinitesimals express the vanishing of the sides of the ratio as quanta, and that what remains is their quantitative relation solely as qualitatively determined; far from this resulting in the loss of the qualitative relation the fact is that it is just this relation which results from the conversion of finite into infinite magnitudes’.⁷⁹ The finite magnitudes that Hegel is referring to are x and y; and the infinite magnitudes into which these finite quanta are converted are the differentials dx and dy, that is, the resulting qualitative relation of each of the finite quanta.

According to Hegel, ‘x and y as such are still supposed to signify quanta; now this significance is altogether and completely lost in the so-called infinitesimal differences’.⁸⁰ What Hegel contends is that the infinitesimal calculus failed to adequately consider the x of dx and the y of dy as quanta, instead it simply posited the infinite value of dx and dy themselves, without accounting for the conversion or transition from x to dx or from y to dy. Hegel argues that ‘it is this concept which has been the target for all the attacks made on the fundamental determination of the mathematics of this infinite, that is, of the differential and integral calculus’,⁸¹ and he contends that it is the inability of mathematics ‘to justify the object as Concept⁸² which is mainly responsible for these attacks’.⁸³ He argues that ‘the originators of the definitions did not establish the thought as Concept and found it necessary in the application to resort again to expedients which conflict with their better cause’.⁸⁴ According to Hegel, the object, which is determined by the limit of the ratio, is justified as the Concept by means of the dialectical logic. The product of the primary negation of the finite quantum, x, is the infinite qualitative relation, or differential, dx. The absolute negation of dx is effected in the differential relation, dy/dx, according to which the finite quanta are realized in the finite determinateness of the limit of the ratio.

What Hegel dismisses as ‘the so-called infinitesimal’ is for him the relation of primary negation of the finite quanta, x and y, in dx and dy. He argues that ‘the infinitesimal signifies, strictly, the negation of quantum as quantum, that is, of a so-called finite expression, of the completed determinateness possessed by quantum as such’.⁸⁵ The absolute negation of the infinitesimal, in the differential relation, he describes as ‘the vanishing of the sides of the ratio’. He argues that ‘the specific nature of the notion of the so-called infinitesimal is the qualitative nature of determinations of quantity which are related to each other primarily as quanta’.⁸⁶ What remains of the finite quantum, x and y, in the differential relation, he argues, is their quantitative relation solely as qualitatively determined, that is, the limit of the ratio solely as determined by dx and dy as dy/dx.

The operation of differentiation ‘does not confine itself to the finite determinateness of its object’, that is, determining the differential relation as the limit of the ratio. ‘On the contrary’, Hegel argues, ‘it converts it into an identity with its opposite, for example converting a curved line into a straight line, the circle into a polygon, etc’.⁸⁷ Differentiation determines the limit of the ratio as the tangent to the circle or curve in what Hegel refers to as ‘the moment of quantitative transition’. As with differentiation, the operation of integration converts the differential relation into an identity with its opposite, however it effects this conversion in the opposite direction, from the limit of the ratio as the tangent, that is, from dy/dx, to the curve itself, as a function of x and y. Integration expresses ‘the transition of straight lines which are infinitely small, into curved lines, and their relation in their infinity as a relation of curves’.⁸⁸ A straight line is defined as ‘the shortest distance between two points, its difference from the curved line is [therefore] based on the determination … of a quantum’.⁸⁹ Hegel argues that ‘this determination vanishes in the line when it is taken as an intensive magnitude, as an infinite moment’.⁹⁰ The finite quantum, x, is determined as having vanished, or as having been negated, in the differential dx, when dx is taken as an intensive magnitude or infinite moment. Insofar as the differential is taken as an intensive magnitude, ‘the straight line and arc no longer retain any quantitative relation nor consequently … any qualitative difference from each other either; on the contrary, the former passes into the latter’,⁹¹ in what Hegel refers to as ‘the moment of qualitative transition’.⁹² Integration is therefore the determination, from the differential, dx, as an intensive magnitude, of the finite quantum on the curve of the original function.

‘Consequently’, Hegel argues, ‘the operations which [the mathematics of the infinite] allows itself to perform in the differential and integral calculus are in complete contradiction with the nature of merely finite determinations and their relations and would therefore have to be justified solely by the Concept’.⁹³ The relation between the finite determinations, x and y, and their relations, for example y/x, is in complete contradiction with the operations dealing with the differentials, dy and dx, and their relation, dy/dx. Insofar as both differentiation, as the transformation of x and y to dy/dx, and integration, as the inverse transformation of dy/dx to x and y, deal with the relation between finite determinations, and their relations, and differentials, and their relation, they are contradictory operations, which are, furthermore, in contradiction with each other. The reference to the mechanism of the principle of contradiction in relation to the inverse processes of differentiation and integration belies Hegel’s consideration of the relation between differentiation, as the moment of quantitative transition, and integration, as the moment of qualitative transition, as being justified solely by the Concept, that is, the transformation from one to the other is determined solely according to the dialectical logic. So, according to the dialectical logic, the moment of quantitative transition, which determines the limit of the ratio, is negated in the moment of qualitative transition, which determines the differential as an intensive magnitude, the absolute negation of which determines the finite determination of the differential relation, that is, its quantitative determinateness, as actually infinite; in other words, the differential relation realizes itself as actually infinite. It is ‘in this concept of the infinite’, Hegel argues, that ‘the quantum is genuinely completed into a qualitative reality; it is posited as actually infinite; it is sublated not merely as this or that quantum but as quantum generally’.⁹⁴ In Chapter 3 of the Science of Logic, Hegel defines the differential relation, which he refers to as ‘the quantitative relation or quantitative ratio’, from the point of view of the dialectical logic, when he declares that ‘the infinity of quantum has been determined to the stage where it is the negative beyond of quantum, which beyond, however, is contained within the quantum itself. This beyond is the qualitative moment as such. The infinite quantum as the unity of both moments, of the quantitative and qualitative determinateness, is in the first instance a ratio’.⁹⁵

According to Hegel, just as Spinoza’s substance–metaphysics failed to make the absolute negation associated with the concept–logic, Spinoza’s example, and the infinitesimal calculus, both lack an adequate expression of the differential in the process of integration as an intensive magnitude, that is, of the differential in the differential relation as an intensive magnitude, or as the qualitative moment of the quantitative relation, and therefore as an infinite quantum. Hegel argues that ‘the infinite which is associated with infinite series, the indeterminate expression of the negative of quantum in general, has nothing in common with the affirmative determination belonging to the infinite of this calculus’.⁹⁶ Hegel considers the operation of the differential calculus according to the dialectical logic to provide the logical basis for such an adequate expression of the differential.⁹⁷

The concept of the differential as either ‘dx’ or ‘not-A’?

Deleuze recognizes that Hegel grasped what is at stake in dx when he writes that ‘Hegel seems to recognize the presence of a genuine infinite in the differential calculus, the infinity of “relation”‘.⁹⁸ Indeed, it is difficult to grasp a discordance between each of their declarations on the differential, each considering the differential as that moment where the terms of the relation, as vanishing quantities, are only determinable according to their relation. In Difference and Repetition, Deleuze writes:

In relation to x, dx is completely undetermined, as dy is to y, but they are perfectly determinable in relation to one another. For this reason, a principle of determinability corresponds to the undetermined as such. The universal is not a nothing since there are, in Bordas’s expression, “relations of the universal” … The relation dy/dx is not like a fraction which is established between particular quanta in intuition, but neither is it a general relation between variable algebraic magnitudes or quantities. Each term exists absolutely only in its relation to the other.⁹⁹

And in the Science of Logic Hegel writes:

Dx, dy, are no longer quanta, nor are they supposed to signify quanta; it is solely in their relation to each other that they have any meaning, a meaning merely as moments. They are no longer something (something taken as a quantum), not finite differences; but neither are they nothing; not empty nullities. Apart from their relation they are pure nullities, but they are intended to be taken only as moments of the relation, as determinations of the differential co-efficient dx/dy.¹⁰⁰

However, Deleuze affirms the existence of a discordance with the suggestion that Hegel only ‘seems’ to recognize the presence of a genuine infinite in the differential calculus. Deleuze argues that ‘for Hegel, infinite representation cannot be reduced to a mathematical structure: there is a non-mathematical or supramathematical architectonic element in continuity and in the differential calculus’.¹⁰¹ As far as Deleuze is concerned, the Hegelian concept of the genuine infinite, rather than being determined by the differential calculus, is determined, on the contrary, by the implication of the differential calculus in the dialectical logic.

Indeed, far from explaining the nature of the differential calculus Hegel’s presentation presupposes it in the form of the dialectical logic. In order to differentiate his thought from that of Hegel, Deleuze writes that ‘just as we oppose difference in itself to negativity, so we oppose dx to not-A, the symbol of difference [Differenzphilosophie] to that of contradiction’.¹⁰² Deleuze refers to the Hegelian concept of the differential as ‘not-A’, rather than ‘dx’, to indicate the implication of the differential as a moment of the dialectical logic, which results from the negation of a finite quantum, ‘A’. The implications of this discordance are illustrated by their respective readings of Spinoza, in particular their divergent interpretations of Spinoza’s example, and the significant position that each affords Spinoza in the development of their respective philosophies.

Although Hegel ascribes a principle of determinability to the vanishing quantities in the Science of Logic, he does not consider the progressive diminution of Spinoza’s ‘inequalities of space’ to refer to the concept of the differential. The only adequate understanding of the diminution that he imputes to Spinoza is that of homogeneous subtraction, which does not incorporate the significant moment of creation where two vanishing quantities combine to form a heterogeneous term which is generated by the determinability of their relation. Hegel thus inscribes Spinoza in the tradition which adheres to the realism of the infinitely small. It is this tradition which motivates Hegel’s reticence in regard to the infinitesimal. The differential, from the point of view of this tradition, is represented as an unlimited series, or bad infinite, and is thereby condemned to the unsatisfactory status of justified approximation or negligible error.

Deleuze argues that ‘to invoke here the infinitely small, and the infinitely small magnitude of error (if there is “error”), is completely lacking in sense and prejudges infinite representation’.¹⁰³ The assumption of the realism of the infinitely small begs the question of the tendency of the imagination to represent the infinite in the finite as the bad infinite. To this extent it merely ‘repeats the point of view of the imagination from which it seeks to distinguish itself’.¹⁰⁴ The establishment of the actual infinite, according to the dialectical logic, is therefore dependent upon this irremediable imaginary conflict.

What is at stake in the debate on the legitimacy of the infinitesimal is ‘the integration of the infinitesimal into the register of quantity’,¹⁰⁵ that is, of the infinite in the finite, which comes down to the alternative between infinite and finite representations. This is precisely what is at issue in what Deleuze describes as ‘the “metaphysics” of the calculus’.¹⁰⁶ Throughout the eighteenth century, there was disagreement as to the particular kind of ‘metaphysics’ by which ‘to rescue the procedures of the calculus’ from the vagaries of the infinitesimal. In speaking of the history of the differential calculus, Giorello argues that ‘it was indeed a matter of rival metaphysical frameworks that provided the basis for widely differing programs’.¹⁰⁷ The Hegelian programme is therefore determined by the following question: ‘is the fate of calculus tied to infinitesimals, or must it not be given a rigorous status from the point of view of finite representations?’¹⁰⁸

Deleuze acknowledges that ‘the interpretation of the differential calculus has indeed taken the form of asking whether infinitesimals are real or fictive’.¹⁰⁹ However, for Deleuze, the question is rhetorical, for it is of little importance whether the infinitely small are real, and if they aren’t this doesn’t signify the contemptible fictive character of their position. As far as Deleuze is concerned, it is no longer a question of reluctantly tolerating their ‘inexactitude’. They must rather be ‘separated from [their] infinitesimal matrix’,¹¹⁰ that is, from their static representation as numbers, which eludes even the axioms of non-standard analysis, and it is by means of their implication in differential relations according to the logic of the differential from the differential point of view of the infinitesimal calculus that this is effected. The undetermined differentials, or infinitesimals, dy or dx, are only determinable insofar as each is involved in a differential relation with another, that is, in reciprocal relation to one another, dy/dx. What counts is that it is within the differential relation itself that the differential possesses rigour and coherence; that the undetermined are determinable, by a process of reciprocal determination.

The Deleuzian solution to the debate on the legitimacy of the infinitesimal distinguishes itself from the Hegelian solution insofar as it is not resolved according to the dialectical logic. Rather than being involved in what Deleuze describes as the ‘circulation of opposing representations which would make their coincidence in the identity of a concept’,¹¹¹ Deleuze argues that the alternative between infinite and finite representations, and therefore the metaphysics of the calculus, are ‘strictly immanent to the techniques of the calculus itself’.¹¹² It is in Spinoza’s example that Deleuze finds evidence of the strictly immanent character of the calculus. Deleuze sets up Spinoza’s example as a rival metaphysical framework for the resolution of the problem of the infinitesimal, a rival to that provided by Hegel in the dialectical logic. Deleuze’s thesis is that the differential, dx or dy, which he considers to feature in the discourse of the pioneers as illustrated in Spinoza’s example, cannot be classified within the dialectical logic, which rather asserts the opposition of the finite and the infinite. Deleuze finds in Spinoza’s example not simply the primitive expression of the differential calculus according to which the differential as a moment of the dialectical logic is conspicuous because of its absence, but rather the logic of the differential from the differential point of view of the infinitesimal calculus. ¹¹³

By implicating Leibniz’s understanding of the early form of the infinitesimal calculus in his interpretation of the geometrical example of Spinoza’s Letter XII, Deleuze not only demonstrates how Spinoza eludes the grasp of the dialectical progression of the history of philosophy, but also nominates Leibniz as one of the figures with whom he engages in his project of renewing the history of philosophy by constructing an alternative lineage in the history of philosophy. Deleuze reads Spinoza’s geometrical example as providing an example of what had already been established by the infinitesimal calculus in response to the problem of the infinite. He considers seventeenth-century thought, and this includes Spinoza, to have developed a new theory of relations by means of the infinitesimal calculus, one which is determined according to the logic of the differential from the differential point of view of the infinitesimal calculus. It is this logic of relations that provides a starting point for the investigation into the logic that Deleuze deploys in Expressionism in Philosophy and which can be traced through Difference and Repetition as a part of his project of constructing a philosophy of difference; a logic which I propose functions as an alternative to the dialectical logic. The next stage of the investigation involves the examination of the subsequent developments that this logic undergoes in relation to some of the breakthroughs in this branch of the infinitesimal calculus, before exploring the way in which Deleuze implicates this logic in his reading of Spinoza’s work as a whole, specifically in relation to the physics of bodies in the second part of the Ethics.

_______________

Notes

¹ Deleuze, ‘sur Spinoza’, 20 January 1981, trans. S. Duffy.

² Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 206.

³ Macherey, Hegel ou Spinoza, p. 168.

⁴ Ibid.

⁵ Ibid., p. 169.

⁶ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 183.

⁷ Ibid., p. 202.

⁸ Ibid., p. 192.

⁹ Hegel’s use of the third definition betrays his reliance on the imaginary concept of the bad infinite to secure the movement of the dialectic.

¹⁰ Pierre Dedron and Jean Itard offer the following definition of the Mathematical infinite in Mathematics and Mathematicians, trans. J.V. Field (2 vols, London, 1974): ‘the idea of an infinite number is merely an abstract idea which expresses an imagined limit which no finite number can ever reach’. Whether the infinite approaches the infinitely small, insofar as formed, for example, by the series of fractions ‘1/2, 1/4, 1/8, 1/16, …, each term being half the previous one’, whose sum, if continued to infinity would be equal to 1, the number 1 being ‘the limit of the sum of the series;’ or the infinitely large, insofar as formed, for example, by ‘the series 2, 4, 6, 8, 16 …, or any other increasing series … the more terms of the series … the greater is their sum, and this sum can exceed any number however large’ (vol. 2, p. 297). An infinite quantity is merely the imagined limit of a finite one, that is, the finite quantity tends towards the infinite quantity, whether infinitely large or infinitely small, without ever reaching it.

¹¹ Dedron and Itard put forward the argument that geometry is able to ‘attach clear, simple and precise meanings to expressions which make use of the term or the idea of infinity’ (Ibid., p. 298). Spinoza’s geometric infinite can therefore be understood to be the adequate expression, from the point of view of reason, of the actual infinite.

¹² Spinoza, Letter XII: ‘Measure, Time, and Number are nothing but Modes … of imagining’.

¹³ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 206.

¹⁴ Ibid., p. 201.

¹⁵ Ibid.

¹⁶ Deleuze, Expressionism in Philosophy, p. 203. Leibniz, who was acquainted with the letter to Meyer, criticizes various details, but on the subject of the infinity that may be greater or less, he declares: ‘This, of which most mathematicians … are ignorant, is remarkably observed, and very carefully inculcated by our author’ (G.W. Leibniz, Die Philosophischen Schriften von G.W. Leibniz (Berlin, 1875–90), p. 137 n. 21). For Deleuze, the text is that much more interesting with the benediction of Leibniz, particularly because he was the first to publish the discovery of the infinitesimal calculus, which Deleuze utilizes in his explication of Spinoza’s geometric infinite. See Deleuze, Expressionism in Philosophy, p. 380 n. 8.

¹⁷ Ibid., p. 202.

¹⁸ Deleuze, ‘sur Spinoza’, 20 January 1981.

¹⁹ Spinoza, Spinoza Opera, Carl Gebhardt (ed.), (Heidelberg, 1925), Letter XII, p. 59.

²⁰ This is my translation from the French: ‘la somme des inégalités de distances comprises entre deux cercles’ (Gilles Deleuze, Spinoza et le problème de l’Expression (Paris, 1968), p. 184), which, I contend, is mistranslated by Martin Joughin as ‘the sum of the unequal distances between two … circles’, in Deleuze, Expressionism in Philosophy, p. 202.

²¹ Deleuze, ‘sur Spinoza’, 20 January 1981.

²² The two other significant French collections of the works of Spinoza translate the passage following the Hegelian interpretation: ‘la somme des distances inégales’. See Œuvres de Spinoza, trans. C. Appuhn (t. III, Paris, 1966), pp. 154–5, and Œuvres completes. trans. R. Misrahi et al. (Paris, 1954), p. 1100.

²³ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 207.

²⁴ Deleuze, Expressionism in Philosophy, p. 202.

²⁵ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 207.

²⁶ Carl Boyer, The history of the calculus and its conceptual development (New York, 1959), p. 11.

²⁷ Ibid., p. 12.

²⁸ Ibid.

²⁹ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 209.

³⁰ Boyer, The history of the calculus and its conceptual development, p. 12.

³¹ Gueroult, ‘Spinoza’s Letter on the Infinite’, p. 209.

³² In his personal correspondence Martial Gueroult writes: ‘I had wished to avoid the formulas of the differential calculus because it had not been invented when Spinoza wrote his 12th Letter. But his thought can only be rendered by the formulas of this calculus’ (21 December 1970), (Spinoza, The Collected Works of Spinoza Volume I, p. 159).

³³ Deleuze, ‘sur Spinoza’, 17 February 1981, trans. T.S. Murphy.

³⁴ Ibid.

³⁵ Note: dy/dx = 0/0 but not 0, that is, dy/dx ≠ 0.

³⁶ Deleuze, ‘sur Spinoza’, 17 February 1981.

³⁷ Deleuze, ‘sur Spinoza’, 10 March 1981, trans. S. Duffy.

³⁸ Ibid.

³⁹ Ibid.

⁴⁰ Ibid.

⁴¹ Spinoza, Ethics, II, P13.

⁴² Deleuze, ‘sur Spinoza’, 17 February 1981.

⁴³ Note: dy/dx = z.

⁴⁴ Deleuze, ‘sur Spinoza’, 17 February 1981.

⁴⁵ See Fig. 1 on page 52, which is reprinted from the section of chapter 1 entitled ‘Letter XII and “the problem of the infinite.”’

⁴⁶ See G.W. Leibniz, The Early Mathematical Manuscripts, trans. J.M. Child (Chicago, 1920), p. 53.

⁴⁷ G.W. Leibniz, ‘Letter to Varignon, with a Note on the “Justification of the Infinitesimal Calculus by that of Ordinary Algebra”’, 1701, Philosophical papers and letters, ed. and trans. Leroy E. Loemker (Dordrecht: Reidel, 1969). Figure 2. is a reproduction of the diagram on p. 545. Deleuze reproduces and discusses this diagram in The Fold: Leibniz and the Baroque, trans. T. Conley (Minneapolis, 1993), p. 18.

⁴⁸ Leibniz, ‘Letter to Varignon’, p. 545. For a similar example see also G.W. Leibniz, ‘Historia et origo calculi differentialis’, in Mathematische Schriften (t. I–VII, Hildesheim; New York, 1962), t. IV, p. 104.

⁴⁹ G.W. Leibniz, ‘Nova Methodus pro Maximis et Minimis’, 1684, Mathematische Schriften, t. V, pp. 220–26.

⁵⁰ See Boyer, The history of the calculus and its conceptual development, p. 210.

⁵¹ Note: AZ = AB – CD.

⁵² Deleuze, Difference and Repetition, p. 174. Although this reference is to Archimedean geometry, and the ‘method of exhaustion’, my claim is that, just as Archimedean geometry functioned as a kind of precursor to the kinds of problems dealt with by the calculus in its early development, Deleuze considers Spinoza’s geometrical example to function as a precursor to the kinds of problems dealt with by the calculus from the seventeenth century onwards.

⁵³ Hegel, Hegel’s Science of Logic, p. 294.

⁵⁴ See Boyer, The history of the calculus and its conceptual development, p. 287.

⁵⁵ Deleuze, Difference and Repetition, p. 171.

⁵⁶ George Lakoff and Rafael Núñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (New York, 2000), p. 224.

⁵⁷ Ibid., p. 224.

⁵⁸ Ibid.

⁵⁹ Ibid.

⁶⁰ Ibid., p. 230.

⁶¹ Ibid., p. 309.

⁶² For a thorough analysis of this problem with limits in Cauchy, see Boyer, The history of the calculus and its conceptual development, p. 281.

⁶³ Lakoff and Núñez, Where Mathematics Comes From, p. 309.

⁶⁴ Deleuze, ‘sur Leibniz’, 22 February 1972, trans. C.J. Stivale.

⁶⁵ Boyer, The history of the calculus and its conceptual development, p. 287.

⁶⁶ See John Bell, A Primer of Infinitesimal Analysis (New York, 1998).

⁶⁷ Abraham Robinson, Non-Standard Analysis (Princeton, 1996), p. 2.

⁶⁸ Deleuze, Difference and Repetition, p. 171.

⁶⁹ Robinson, Non-Standard Analysis, p. 277.

⁷⁰ Deleuze, ‘sur Spinoza’, 17 February 1981.

⁷¹ Ibid.

⁷² Ibid.

⁷³ Ibid.

⁷⁴ Ibid., p. 268.

⁷⁵ Ibid.

⁷⁶ Ibid.

⁷⁷ Ibid., p. 266.

⁷⁸ Ibid.

⁷⁹ Ibid., p. 269.

⁸⁰ Ibid., p. 253.

⁸¹ Ibid.

⁸² The reasoning of Jean-Michel Salanskis is followed in translating Begriff as Concept: ‘The term for the German Begriff in the English translation is Notion, whereas I prefer to follow the French usage in translating it as Concept’. See Jean-Michel Salanskis, ‘Idea and Destination’, in P. Patton (ed.), Deleuze, A Critical Reader (Oxford and Cambridge, 1996), p. 79.

⁸³ Hegel, Hegel’s Science of Logic, p. 254.

⁸⁴ Ibid., p. 255.

⁸⁵ Ibid., p. 299.

⁸⁶ Ibid., p. 275.

⁸⁷ Ibid., p. 254.

⁸⁸ Ibid., p. 271.

⁸⁹ Ibid.

⁹⁰ Ibid.

⁹¹ Ibid.

⁹² Ibid., p. 304.

⁹³ Ibid., p. 254.

⁹⁴ Ibid., p. 253.

⁹⁵ Ibid., p. 314.

⁹⁶ Ibid., p. 301.

⁹⁷ For a balanced assessment of the mathematics developed by Hegel in the Science of Logic, see Terry Pinkard, ‘Hegel’s Philosophy of Mathematics’, Philosophy and Phenomenological Research (vol. XLI, Buffalo, 1981); and Hegel’s Dialectic: The Explanation of Possibility (Philadelphia, 1988).

⁹⁸ Deleuze, Difference and Repetition, p. 310, n. 9.

⁹⁹ Ibid., p. 172.

¹⁰⁰ Hegel, Hegel’s Science of Logic, p. 253. Juliette Simont juxtaposes the same two citations in order to demonstrate the difficulty in differentiating the respective philosophical logics of Hegel and Deleuze. See Simont, p. 281.

¹⁰¹ Deleuze, Difference and Repetition, p. 310 n. 9.

¹⁰² Ibid., p. 170.

¹⁰³ Ibid., p. 177.

¹⁰⁴ Macherey, Hegel ou Spinoza, p. 172.

¹⁰⁵ Salanskis, ‘Idea and Destination’, p. 71.

¹⁰⁶ Deleuze, Difference and Repetition, p. 176.

¹⁰⁷ Guilio Giorello, ‘The “fine structure” of mathematical revolutions: metaphysics, legitimacy and rigour. The case of the calculus from Newton to Berkeley and Maclaurin’, Revolutions in mathematics (Oxford, 1992), p. 160.

¹⁰⁸ Deleuze, Difference and Repetition, p. 177.

¹⁰⁹ Ibid.

¹¹⁰ Ibid., p. 171.

¹¹¹ Ibid., p. 178.

¹¹² Ibid., p. 176.

¹¹³ According to Deleuze, the ‘finitist interpretations’ of the calculus given in modern set-theoretical mathematics – which Salanskis considers to be congruent with what Penelope Maddy calls ‘Cantorian finitism’: ‘namely the idea that infinite entities are so to speak seen and considered to be finite within set theory’ (Salanskis, ‘Idea and Destination’, p. 66) – betray the nature of the differential no less than Hegel and Weierstrass, since they each ‘fail to capture the extra-propositional or sub-representative source … from which calculus draws its power’(Deleuze, Difference and Repetition, p. 264). See Penelope Maddy, ‘Believing the axioms’, The Journal of Symbolic Logic, 53.2 (1988): 481–511.

Chapter 3

The metaphysics of the calculus: Extensive quantity

The logic of the differential

Having located the logic of the differential from the differential point of view of the infinitesimal calculus in the geometrical example of Spinoza’s Letter XII, with the assistance of Leibniz, the subsequent developments that this logic undergoes will now be examined in relation to the work of some of the key figures in the history of this branch of the infinitesimal calculus. These figures are implicated in an alternative lineage in the history of mathematics by means of which the differential point of view of the infinitesimal calculus is aligned with the differential calculus of contemporary mathematics, a lineage which effectively bypasses the methods of the differential calculus which Hegel uses in the Science of Logic to support the development of the dialectical logic. The logic of the differential from the differential point of view of the infinitesimal calculus is then implicated in Deleuze’s project of constructing a philosophy of difference by means of Spinoza’s discussion of the physics of bodies in the second part of the Ethics. According to Deleuze, the most simple bodies of Spinoza’s Ethics correspond directly to the infinitely small differentials of the geometrical example in Letter XII. The logic of the differential from the differential point of view of the infinitesimal calculus is thereby implicated in Deleuze’s interpretation of Spinoza’s theory of relations in the Ethics, and therefore in the development of Deleuze’s project of constructing a philosophy of difference. The manner by means of which the figures in the history of the differential point of view of the infinitesimal calculus are implicated in an alternative lineage in the history of mathematics will now be examined.

Ironically, one of the mathematicians who develops the differential point of view of the infinitesimal calculus is Karl Weierstrass, who considers the differential relation to be logically prior to the function in the process of determination associated with the infinitesimal calculus, that is, rather than determining the differential relation from a given function, the kinds of mathematical problems that Weierstrass dealt with involved investigating how to generate a function from a given differential relation. Weierstrass develops a theory of integration as the approximation of functions from differential relations according to a process of summation in the form of series. Despite Weierstrass having eliminated both geometry and the infinitesimal from the calculus, Deleuze recovers this theory in order to restore the Leibnizian perspective of the differential as the genetic force of the differential relation to the differential point of view of the infinitesimal calculus, by means of the infinitesimal axioms of non-standard analysis. Hegel, on the contrary, locates the genetic force of the calculus in the dialectical logic.

According to Deleuze’s reading of the infinitesimal calculus from the differential point of view, a function does not precede the differential relation, but is rather determined by the differential relation. The differential relation is used to determine the overall shape of the curve of a function primarily by determining the number and distribution of its distinctive points, which are points of articulation where the nature of the curve changes or the function alters its behaviour. For example, when the differential relation is zero, the gradient of the tangent at that point is horizontal, indicating that the curve peaks or dips, determining therefore a maximum or minimum at that point. These distinctive points are known as stationary or turning points. The differential relation characterizes or qualifies not only the distinctive points which it determines, but also the nature of the regular points in the immediate neighbourhood of these points, that is, the shape of the branches of the curve between each distinctive point. Where the differential relation gives the value of the gradient at the distinctive point, the value of the derivative of the differential relation, that is, the second derivative, indicates the rate at which the gradient is changing at that point, which allows a more accurate approximation of the nature of the function in the neighbourhood of that point. The value of the third derivative indicates the rate at which the second derivative is changing at that point. In fact, the more successive derivatives that can be evaluated at the distinctive point, the more accurate will be the approximation of the function in the immediate neighbourhood of that point.

This method of approximation using successive derivatives is formalized in the calculus by a Taylor series or power series expansion. A power series expansion can be written as a polynomial, the coefficients of each of its terms being the successive derivatives evaluated at the distinctive point. The sum of such a series represents the expanded function provided that any remainder approaches zero as the number of terms becomes infinite; the polynomial then becomes an infinite series which converges with the function in the neighbourhood of the distinctive point.1 This criterion of convergence repeats Cauchy’s earlier exclusion of divergent series from the calculus. A power series operates at each distinctive point by successively determining the specific qualitative nature of the function at that point. The power series determines not only the specific qualitative nature of the function at the distinctive point in question, but also the specific qualitative nature of all of the regular points in the neighbourhood of that distinctive point, such that the specific qualitative nature of a function in the neighbourhood of a distinctive point insists in that one point. By examining the relation between the differently distributed distinctive points determined by the differential relation, the regular points which are continuous between the distinctive points, that is, the branches of the curve, can be determined. In general, the power series converges with a function by generating a continuous branch of a curve in the neighbourhood of a distinctive point. To the extent that all of the regular points are continuous across all of the different branches generated by the power series of the other distinctive points, the entire complex curve or the whole analytic function is generated.¹

So, according to Deleuze’s reading of the infinitesimal calculus, the differential relation is generated by differentials and the power series are generated in a process involving the repeated differentiation of the differential relation. It is due to these processes that a function is generated to begin with. The mathematical elements of this interpretation are most clearly developed by Weierstrassian analysis, according to ‘the theorem on the approximation of analytic functions’. An analytic function, being secondary to the differential relation, is differentiable, and therefore continuous, at each point of its domain. According to Weierstrass, for any continuous function on a given interval, or domain, there exists a power series expansion which uniformly converges to this function on the given domain. Given that a power series approximates a function in such a restricted domain, the task is then to determine other power series expansions that approximate the same function in other domains. An analytic function is differentiable at each point of its domain, and is essentially defined for Weierstrass from the neighbourhood of a distinctive point by a power series expansion which is convergent with a ‘circle of convergence’ around that point. A power series expansion that is convergent in such a circle represents a function that is analytic at each point in the circle. By taking a point interior to the first circle as a new centre, and by determining the values of the coefficients of this new series using the function generated by the first series, a new series and a new centre of convergence is obtained, whose circle of convergence overlaps the first. The new series is continuous with the first if the values of the function coincide in the common part of the two circles. This method of ‘analytic continuity’ allows the gradual construction of a whole domain over which the generated function is continuous. At the points of the new circle of convergence which are exterior to, or extend outside of the first, the function represented by the second series is then the analytic continuation of the function defined by the first series; what Weierstrass defines as the analytic continuation of a power series expansion outside its circle of convergence. The domain of the function is extended by the successive adjunction of more and more circles of convergence. Each series expansion which determines a circle of convergence is called an element of the function.² In this way, given an element of an analytic function, by analytic continuation one can obtain the entire analytic function over an extended domain. The analytic continuation of power series expansions can be continued in this way in all directions up to the points in the immediate neighbourhood exterior to the circles of convergence where the series obtained diverge.

Power series expansions diverge at specific ‘singular points’ or ‘singularities’ that may arise in the process of analytic continuity. A singular point or singularity of an analytic function is any point which is not a regular or ordinary point of the function. They are points which exhibit distinctive properties and thereby have a dominating and exceptional role in the determination of the characteristics of the function. The distinctive points of a function, which include the turning points, where dy/dx = 0, and points of inflection, where d²y/dx² = 0, are ‘removable singular points’, since the power series at these points converge with the function. A removable singular point is uniformly determined by the function and therefore redefinable as a distinctive point of the function, such that the function is analytic or continuous at that point. The specific singularities of an analytic function where the series obtained diverge are called ‘poles’. Singularities of this kind are those points where the function no longer satisfies the conditions of regularity which assure its local continuity, such that the rule of analytic continuity breaks down. They are therefore points of discontinuity. A singularity is called a pole of a function when the values of the differential relation, that is, the gradients of the tangents to the points of the function, approach infinity as the function approaches the pole. The function is said to be asymptotic to the pole, it is therefore no longer differentiable at that point, but rather remains undefined, or vanishes. A pole is therefore the limit point of a function, and is referred to as an accumulation point or point of condensation. A pole can also be referred to as a jump discontinuity in relation to a finite discontinuous interval both within the same function, for example periodic functions, and between neighbouring analytic functions. Deleuze writes that ‘a singularity is the point of departure for a series which extends over all the ordinary points of the system, as far as the region of another singularity which itself gives rise to another series which may either converge or diverge from the first’.³ The singularities whose series converge are removable singular points, and those whose series diverge are poles.

The singularities, or poles, that arise in the process of analytic continuity necessarily lie on the boundaries of the circles of convergence of power series. In the neighbourhood of a pole, a circle of convergence extends as far as the pole in order to avoid including it, and the poles of any neighbouring functions, within its domain. The effective domain of an analytic function determined by the process of the analytic continuation of power series expansions is therefore limited to that between its poles. With this method the domain is not circumscribed in advance, but results rather from the succession of local operations.

The continuity between the differential point of view and the differential calculus of contemporary mathematics

Power series can be used in this way to solve differential relations by determining the analytic function into which they can be expanded. The theory of Weierstrass is developed alongside of the integral conception of Cauchy, which further developed the inverse relation between the differential and the integral calculus as the fundamental theorem of the calculus. The fundamental theorem maintains that differentiation and integration are inverse operations, such that integrals are computed by finding anti-derivatives, which are otherwise known as primitive functions. There are a number of rules, or algorithms, according to which this reversal is effected.

Hegel enshrined the early form of the inverse relation between integration and differentiation in the Science of Logic, insofar as the differential calculus is implicated in the dialectical logic.⁴ To this extent, Hegel’s presentation of the differential calculus foreshadows the subsequent developments made by Cauchy and Riemann. However, Hegel excludes the concept of integration in the form of a series from his conception of the differential calculus. He argues rather that ‘the form of a series has nothing to do with the peculiarities principle of integration’,⁵ since, as far as he is concerned, ‘what is involved is not a sum but a relation’.⁶ Instead, Hegel argues that ‘the differential is completely given by the first term’ of the series. Although he acknowledges that ‘further terms, the differentials of higher order, are required’, he argues that ‘their determination involves not the continuation of a series as a sum, but the repetition of one and the same relation which alone is desired and which is thus already completely given in the first term’.⁷ Hegel singles out the first term of the series, the first order differential or differential relation, as the quantitative moment of transition, and recognizes that the original function can be determined directly from it, the derived function, according to the process of integration as the inverse transformation of differentiation. Hegel argues therefore that ‘the need for the form of a series, its summation and all that is connected with it, must then be wholly separated from the said interest of the relation’.⁸

Contrary to Hegel’s argument, Deleuze presents Weierstrass’ theorem as an effective method for determining the characteristics of a function from the differential point of view of the infinitesimal calculus. The mathematician Albert Lautman (1908 – 1944) refers to this process as integration from ‘the local point of view’, or simply as ‘local integration’.⁹ This form of integration does not involve the determination of the primitive function, which is generated by exercising the inverse operation of integration. The development of a local point of view, rather, requires the analysis of the characteristics of a function at its singular points. The passage from the analytic function defined in the neighbourhood of a singular point, to the analytic function defined in each ordinary point is made according to the ideas of Weierstrass by analytic continuity. This method was eventually deduced from the Cauchy point of view, such that the Weierstrassian approach was no longer emphasized. The unification of both of these points of view, however, was achieved at the beginning of the twentieth century when the rigour of Cauchy’s ideas, which were then fused with those of Georg Riemann (1826 – 1866), the other major contributor to the development of the theory of functions, was improved. Deleuze is therefore able to cite the contribution of Weierstrass’ theorem of approximation in the development of the differential point of view of the infinitesimal calculus as an alternative point of view of the differential calculus to that proposed by Hegel. Deleuze thereby establishes a historical continuity between Leibniz’s differential point of view of the infinitesimal calculus and the differential calculus of contemporary mathematics, thanks to the axioms of non-standard analysis which allow the inclusion of the infinitesimal in its arithmetization; a continuity which effectively bypasses the methods of the differential calculus which Hegel uses in the Science of Logic to support the development of the dialectical logic.¹⁰

The development of a differential philosophy

In Difference and Repetition, the differential point of view of the infinitesimal calculus is developed by Deleuze into a ‘differential philosophy’. The mechanism by means of which Deleuze effects this development will now become ‘the object of a rigorous exposition’.¹¹ The logic of the differential, which characterizes this differential philosophy, incorporates three different aspects: the undetermined, the determinable and determination. To each of these aspects corresponds a principle: ‘a principle of determinability corresponds to the undetermined as such (dx, dy); a principle of reciprocal determination corresponds to the really determinable (dy/dx); a principle of complete determination corresponds to the effectively determined (values of dy/dx)’.¹²

It is in the work of Höené Wronski (1778 – 1853) that Deleuze finds the established expression of the first principle of the differential philosophy. Wronski was ‘an eager devotee of the differential method of Leibniz and of the transcendental philosophy of Kant’.¹³ Wronski argued that the differential calculus constituted ‘a primitive algorithm governing the generation of quantities, rather than the laws of quantities already formed’.¹⁴ According to Wronski, the differential is determined as ‘an ideal difference’,¹⁵ ‘without which’, Deleuze argues, the undetermined ‘could not carry out the determination expected of it’.¹⁶ The work of Wronski represents an extreme example of the differential point of view of the infinitesimal calculus which recurs throughout the nineteenth century. Wronski’s objections against the work of Lagrange and Carnot, both of whom figure significantly in Hegel’s analysis, are from the differential point of view of the infinitesimal calculus rather than from the point of view of the dialectical logic. In regarding the infinitesimal calculus as a means of explaining the ‘generation of quantities’, the concept of the infinitely small is retained, ‘not as an extensive quantity but as an intensive magnitude’.¹⁷ Whereas Hegel excludes the fixed infinitely small because it alone fails to establish the concept of the differential calculus logically, that is, from the point of view of the dialectical logic, according to Wronski, however, the differential as an ‘ideal difference’ is interpreted ‘as having an a priori metaphysical reality associated with the generation of magnitude’.¹⁸ As an ideal difference, the differential is expressed as a pure element of quantitability; insofar as it prepares for the determination of quantity. It is in the differential relation that the differential is realized as an element of quantitability. The principle of reciprocal determination corresponds to the determinability of the differential relation as ‘a reciprocal synthesis’; according to which each term of the relation, that is, each differential, each pure element of quantitability, ‘exists absolutely only in its relation to the other’.¹⁹

The question then becomes: ‘in what form is the differential relation determinable?’²⁰ According to Deleuze, it is determinable primarily in qualitative form, as a pure element of qualitability, insofar as it is the reciprocal relation between differentials, and then secondarily, insofar as it is the function of a tangent whose values give the gradient of the line tangent to a curve, or the specific qualitative nature of this curve, at a point. In this capacity it expresses a pure element of potentiality. So, ‘a pure element of quantitability’²¹ is expressed by the undetermined, ‘a pure element of qualitability’²² is incorporated by the differential relation insofar as it is a relation between differentials, and ‘a pure element of potentiality’²³ is expressed by the values of this relation insofar as it is a function of a tangent.

As the function of a tangent, the differential relation ‘expresses a function which differs in kind from the so-called primitive function’.²⁴ Whereas the primitive function, when differentiated, expresses the whole curve directly,²⁵ the differential relation, when differentiated, expresses rather the further qualification of the nature of the function at, or in the immediate neighbourhood of, a specific point. The primitive function is the integral of the function determined by the inverse transformation of differentiation, according to the differential calculus from the point of view of the dialectical logic. From the differential point of view of the infinitesimal calculus, the differential relation, as the function of the tangent, is a pure element of potentiality, which determines the existence and distribution of the distinctive points of a function, thus preparing for its further qualification. Unlike the primitive function, the differential relation remains tied to the specific qualitative nature of the function at those distinctive points, and, as the function of the tangent, it ‘is therefore differentiable in turn’.²⁶ When the reciprocal determination generates the distinctive points as power series expansions by the repeated differentiation of the differential relation, what is increasingly specified is the qualitative nature of the function in the immediate neighbourhood of that point. Deleuze argues that this convergence of a power series with an analytic function, in its immediate neighbourhood, satisfies ‘the minimal conditions of an integral’.²⁷

Deleuze refers to the difference between the differential relation and the primitive function as a ‘qualitative difference or “change of function” within the differential’.²⁸ This difference can be understood to characterize the difference between not only the differential point of view of the infinitesimal calculus, or the differential calculus from the local point of view, and the differential calculus as implicated in the dialectical logic, but also that between the logic of Deleuze’s differential philosophy and the logic of Hegel’s dialectical philosophy.

Power as capacity and the logic of ‘differentiation’

The first act of the differential calculus is the operation of differentiation, which consists in a ‘depotentialisation’²⁹ of a given function, that is, the derived function is of a lower degree than the given function.³⁰ Whereas the derived function is directly repotentialized by the inverse transformation of differentiation, that is, by integration, the differential from the differential point of view of the infinitesimal calculus, on the contrary, is increasingly potentialized in the form of a power series. As a power series expands, by means of the repeated differentiation of the differential relation, the exponent, or power, of each successive term increases. As the degree of the power series increases, the series increasingly approximates or converges with an analytic function in its immediate neighbourhood. The degree of a power series expresses the highest exponent, or power, of that series. Therefore, as a power series successively expands, it undergoes a corresponding increase in degree because of an increase in power. Since an expanding function also increasingly converges with an analytic function, its increasing potential, or capacity, to converge can be expressed as an increasing degree, or power of convergence; where the power of a series can be understood to become an expression of its capacity for convergence. So, the mathematical exponent, or power, becomes an expression of capacity: power as capacity. In Difference and Repetition, Deleuze characterizes this concept of power (puissance) as ‘the form of reciprocal determination according to which variable magnitudes are taken to be functions of one another’.³¹ That is, the concept of power is characterized as the capacity of a differential to be reciprocally determined in a differential relation and to be repeatedly differentiated in the form of a power series, the power or capacity of which therefore increases with the expansion of the series. ‘In this sense’, Deleuze argues, ‘the differential is indeed pure power, just as the differential relation is a pure element of potentiality’.³²

The theory of the analytic continuity of power series expansions, according to which a whole analytic function is effectively determined, corresponds to the third principle of complete determination. Each power series, which is an element of the locally determined analytic function, can be understood to be an element of potentiality in the complete determination of the whole analytic function.

Deleuze argues that ‘complete determination must not be confused with reciprocal determination’.³³ Whereas reciprocal determination concerns the differential relations and their degrees of expansion, or degrees of power, complete determination concerns rather ‘the composition of a form’.³⁴ This occurs by means of the determination of the continuity between the branches of the curves previously defined by the convergence of power series at the singular points which they characterize. According to the three principles characteristic of the logic of the differential, the differential, in the differential relation, can be understood to have the power (as capacity) to effectively determine the whole analytic function. The logic of these three principles characterize what is for Deleuze the process of ‘differentiation’.³⁵

From differentiation to differenciation: the logic of meromorphic functions

The differential relation expresses the qualitative relation between, not only curves and straight lines, but also between linear dimensions and their functions, and plane or surface dimensions and their functions. The domain of the successive adjunction of circles of convergence, as determined by analytic continuity, actually has the structure of a surface. This surface is constituted by the points of the domain and the direction attached to each point in the domain, that is, the tangents to the curve at each point and the direction the curve goes at that point. Such a surface can be described as a field of directions or a field of vectors. A vector is a quantity having both magnitude and direction. It is the surface of such a vector field that provides the structure for the local genesis of functions. It is within this context that the example of a jump discontinuity in relation to a finite discontinuous interval between neighbouring analytic or local functions is developed by Deleuze, in order to characterize the generation of another function which extends beyond the points of discontinuity which determine the limits of these local functions. Such a function would characterize the relation between the different domains of different local functions. The genesis of such a function from the local point of view is initially determined by taking any two points on the surface of a vector field, such that each point is a pole of a local function determined independently by the point wise operations of Weierstrassian analysis. The so determined local functions, which have no common distinctive points or poles in the domain, are discontinuous with each other; each pole being a point of discontinuity, or limit point, for its respective local function. Rather than simply being considered as the unchanging limits of local functions generated by analytic continuity, the limit points of each local function can be considered in relation to each other, within the context of the generation of a new function which encompasses the limit points of each local function and the discontinuity that extends between them. Such a function can initially be understood to be a potential function, which is determined as a line of discontinuity between the poles of the two local functions on the surface of the vector field. The potential function admits these two points as the poles of its domain. However, the domain of the potential function is on a scalar field, which is distinct from the vector field insofar as it is composed of points (scalars) which are non-directional; scalar points are the points onto which a vector field is mapped. The potential function can be defined by the succession of points (scalars) which stretch between the two poles. The scalar field of the potential function is distinct from the vector field of the local functions insofar as, mathematically speaking, it is ‘cut’ from the surface of the vector field. Deleuze argues that ‘the limit must be conceived not as the limit of a [local] function but as a genuine cut [coupure], a border between the changeable and the unchangeable within the function itself. … the limit no longer presupposes the ideas of a continuous variable and infinite approximation. On the contrary, the concept of limit grounds a new, static and purely ideal definition’,³⁶ that of the potential function. To cut the surface from one of these poles to the next is to generate such a potential function. The poles of the potential function determine the limits of the discontinuous domain, or scalar field, which is cut from the surface of the vector field. The ‘cut’ of the surface in this theory renders the structure of the potential function ‘apt to a creation’.³⁷ The precise moment of production, or genesis, resides in the act by which the cut renders the variables of certain functional expressions able to ‘jump’ from pole to pole across the cut. When the variable jumps across this cut, the domain of the potential function is no longer uniformly discontinuous. With each ‘jump’, the poles which determine the domain of discontinuity, represented by the potential function sustained across the cut, seem to have been removed. The more the cut doesn’t separate the potential function on the scalar field from the surface of the vector field, the more the poles seem to have been removed, and the more the potential function seems to be continuous with the local functions across the whole surface of the vector field. It is only insofar as this interpretation is conferred on the structure of the potential function that a new function can be understood to have been generated on the surface. A potential function is only generated when there is potential for the creation of a new function between the poles of two local functions. The potential function is therefore always apt to the creation of a new function. This new function, which encompasses the limit points of each local function and the discontinuity that extends between them, is continuous across this structure of the potential function; it completes the structure of the potential function, in what can be referred to as a ‘composite function’. The connection between the structural completion of the potential function and the generation of the corresponding composite function is the act by which the variable jumps from pole to pole. When the variable jumps across the cut, the value of the composite function sustains a determined increase. Although the increase seems to be sustained by the potential function, it is this increase which actually registers the generation or complete determination of the composite function.

The complete determination of a composite function by the structural completion of the potential function is not determined by Weierstrass’ theory of analytic continuity. A function is able to be determined as continuous by analytic continuity across singular points which are removable, but not across singular points which are non-removable. The poles that determine the parameters of the domain of the potential function are non-removable, thus analytic continuity between the two functions, across the cut, is not able to be established. Weierstrass, however, recognized a means of solving this problem by extending his analysis to meromorphic functions. A function is said to be meromorphic in a domain if it is analytic in the domain determined by the poles of analytic functions. A meromorphic function is determined by the quotient of two arbitrary analytic functions, which have been determined independently on the same surface by the point wise operations of Weierstrassian analysis. Such a function is defined by the differential relation:

where X and Y are the polynomials, or power series of the two local functions. The meromorphic function, as the function of a differential relation, is just the kind of function which can be understood to have been generated by the structural completion of the potential function. The meromorphic function is therefore the differential relation of the composite function. The expansion of the power series determined by the repeated differentiation of the meromorphic function generates a function which converges with the composite function. The graph of a composite function, however, consists of curves with infinite branches. The representation of such curves posed a problem for Weierstrass which he was unable to resolve.

The qualitative theory of differential equations

Henri Poincaré (1854 – 1912) took up this problem of the representation of composite functions, by extending the Weierstrass theory of meromorphic functions to what was called ‘the qualitative theory of differential equations’.³⁸ In place of studying the properties of complex functions in the neighbourhood of their singularities, Poincaré was primarily occupied with determining the properties of complex functions in the whole plane, that is, the properties of the entire curve. This qualitative method involved the initial investigation of the geometrical form of the curves of functions with infinite branches, only then was the numerical determination of the values of the function able to be made. While such divergent series do not converge, in the Weierstrassian sense, to a function, they may indeed furnish a useful approximation to a function if they can be said to represent the function asymptotically. When such a series is asymptotic to the function, it can represent an analytic or composite function even though the series is divergent.

When this geometrical interpretation was applied to composite functions, Poincaré found the values of the composite function around the singularity produced by the function to be undetermined and irregular. The singularity of a composite function is the point at which both the numerator and denominator of the quotient of the meromorphic function determinative of the composite function vanish. The peculiarity of the meromorphic function is that the numerator and denominator do not vanish at the same point on the surface of the domain. The points at which the two local functions of the quotient vanish are at their respective poles. The determination of a composite function therefore requires the determination of a new singularity in relation to the poles of the local functions of which it is composed. Poincaré called this new kind of singularity an essential singularity. Observing that the values of a composite function very close to an essential singularity fluctuate through a range of different possibilities without stabilizing, Poincaré distinguished four types of essential singularity, which he classified according to the behaviour of the function and the appearance of the solution curves in the neighbourhood of these points. The first type of singularity is the saddle point or dip (col), through which only two solution curves pass, acting as asymptotes for neighbouring curves. A saddle point is neither a maximum nor minimum, since it either increases or decreases depending on the direction taken away from it. The second kind of singularity is the node (noeud), which is a point through which an infinite number of curves pass. The third type of singularity is the point of focus (foyer), around which the solution curves turn and towards which they approach in the same way as logarithmic spirals. And the fourth, called a centre, is a point around which the curves are closed, enveloping one another and the centre.

Fig. 3.1³⁹

The type of essential singularity is determined by the form of the curves constitutive of the meromorphic function. Whereas the potential function remains discontinuous with the other functions on the surface from which it is cut, thereby representing a discontinuous group of functions, the composite function, on the contrary, overcomes this discontinuity insofar as it is continuous in the domain which extends across the whole surface of the discontinuous group of functions. The existence of such a continuous function, however, does not express any less the properties of the domain of discontinuity which serves to define it. The discontinuous group of local functions and the continuous composite function attached to this group exist alongside of each other, the transformation from one to the other being determined by the process of the generation of the meromorphic function. The potential function is actualized in the composite function when the variable jumps from one pole to the other. Its trajectory, in the form of a solution curve, is determined by the type of essential singularity created by the composite relation. The essential singularity determines the behaviour of the composite function, or the appearance of the solution curve, in its immediate neighbourhood by acting as an attractor for the trajectory of the variable across its domain. It is the value of this function which sustains a determined increase with each jump of the variable. Insofar as the trajectory of each variable is attracted to the same final state represented by each of the different essential singularities, these essential singularities can be understood to represent what Manuel DeLanda describes as the ‘inherent or intrinsic long-term tendencies of a system, the states which the system will spontaneously tend to adopt in the long run as long as it is not constrained by other forces’.⁴⁰

Deleuze distinguishes this differential point of view of the infinitesimal calculus from the Weierstrassian theory of approximation when he writes that: ‘No doubt the specification of the singular points (for example, dips, nodes, focal points, centres) is undertaken by means of the form of integral curves, which refer back to the solutions for the differential equations. There is nevertheless a complete determination with regard to the existence and distribution of these points which depends upon a completely different instance – namely, the field of vectors defined by the equation itself. The complementarity of these two aspects does not obscure their difference in kind – on the contrary’.⁴¹ The equation to which Deleuze refers is the meromorphic function, which is a differential equation or function of a differential relation determined according to the Weierstrassian approach, from which the essential singularity and therefore the composite function are determined according to Poincaré’s qualitative approach. This form of integration is again characterized from the local point of view, and ‘is by no means the inverse of differentiation but, rather, forms’ what Deleuze describes as ‘an original process of differenciation’.⁴² Differenciation is the complete determination of the composite function from the reciprocal synthesis of local functions or the structural completion of the potential function. It is the process whereby a potential function is actualized as a composite function.

The logic of local and global integration

According to the logic of differentiation, the principles of determinability, reciprocal determination and complete determination correspond respectively to: the determination of the undetermined, or ‘ideal difference’, in a differential relation; the repeated differentiation of the differential relation in the generation of a power series expansion, what Deleuze refers to as ‘the variety of differential relations’, or ‘order of relations’; and the determination by analytic continuity of the limit points or poles of the whole local function, what Deleuze refers to as ‘the distribution of correlative singularities’.⁴³ According to the logic of differenciation: a principle of determinability applies to the specific qualitative nature of the poles, or infinite branches, of local functions which, from the point of view of differentiation, remain undetermined, but which are reciprocally determinable in relation to one another, such a relation being represented by a potential function; a reciprocal determination generates the meromorphic function and its power series expansion, which, for Deleuze, concerns the determination of ‘the qualities … which actualize the varieties’, that is, the specific qualitative nature of the function which actualizes the power series of the meromorphic function; and a complete determination generates the composite function, or the ‘distinct parts actualizing the singular points’,⁴⁴ that is, the essential singularities determined by the meromorphic function. In relation to differentiation, Deleuze argues that ‘complete determination carries out the differentiation of singularities’, that is, of the poles of local functions, ‘but bears only upon their existence and their distribution. The nature of these singular points’, Deleuze continues, ‘is specified only by the form of the neighbouring integral curves – in other words, by virtue of the … differenciated … spaces’.⁴⁵ These integral curves are the curves of a composite function determined in a differenciated or vector space, and the specific qualitative nature of the poles of the local functions are determined by the form of these integral curves according to the process of differenciation.

Deleuze states that ‘actualization or differenciation is always a genuine creation’, and that to be actualized is ‘to create divergent lines’.⁴⁶ The expanded power series of a meromorphic function is actualized in the composite function insofar as the meromorphic function converges with, or creates, the divergent lines of the composite function. Deleuze characterizes the process of actualization as ‘the local differenciation of parts’;⁴⁷ the infinite parts or branches of local functions are actualized in the infinite parts or branches of a composite function. Actualization is therefore ‘the incarnation of the order of relations and distribution of singularities peculiar to … differenciated … parts’.⁴⁸ Differenciation, therefore, creates an essential singularity, whose divergent lines actualize the specific qualitative nature of the poles of the group of discontinuous local functions, represented by a potential function, in the form of a composite function. These complex functions can be understood to be what Poincaré called ‘Fuschian functions’, which, as Georges Valiron points out, ‘are more often called automorphic functions’.⁴⁹ The discontinuous group of local functions can therefore also be understood to be Fuschian groups. Poincaré’s pioneering work in this area eventually lead to the definitive founding of the geometric theory of analytic functions, the study of which ‘has not yet been completely carried out’,⁵⁰ but continues to be developed with the assistance of computers.⁵¹

The process designated by differentiation incorporates the three components of the logic of the differential governed by the principles of determinability, reciprocal determination and complete determination. The process designated by differenciation is the extension of this logic to the discontinuities of the local functions generated by differentiation, which are then actualized by differenciation in the form of a continuous composite function. This whole process is characterized by the determination of the existence and the distribution of singular points by means of differential relations. Singular points are primarily removable, which allow the determination of local functions that are analytically continuous over an extended domain. Poles determine the limits of these local functions, which together constitute a discontinuous group of local functions whose domain is represented by a potential function. Essential singularities are generated by the actualization of such a discontinuous group of local functions in the form of a composite function.⁵² This process is not arrested with the generation of a composite function, but rather continues, generating those functions which actualize the relations between different composite functions, and those functions which actualize the relations between these functions, and so on. The conception of differenciation is extended in this way when Deleuze states that ‘there is a differenciation of differenciation which integrates and welds together the differenciated’,⁵³ each differenciation is simultaneously ‘a local integration’, which then connects with others, according to the same logic, in what is characterized as a ‘global integration’.⁵⁴

Extensive parts: infinite collections of the infinitely small

In the section of chapter 1 entitled ‘Omnis determinatio est affirmatio’, Macherey was presented as arguing that each individual is composed of an infinity of what Spinoza calls ‘the most simple bodies’, that is, that all complex bodies are composed of an infinity of the most simple bodies. By means of the geometrical example, Deleuze arrives at a definition of ‘the most simple bodies’ different to that of Macherey. In addition to ascribing to them the precise enough status of not being atoms, meaning finite bodies, Deleuze contends that neither are they simply indefinites, or imaginary abstractions. According to Deleuze, the most simple bodies of Spinoza’s Ethics are the infinitely small differentials of Letter XII. Unlike an atom, the infinitely small have neither figure nor quantity, but are smaller than all given quantity, or are vanishing quantities. However, that which does have figure and quantity is a collection, itself infinite, of the infinitely small, that is, an infinite collection of the infinitely small. According to the terminology of the differential point of view of the infinitesimal calculus that Deleuze ascribes to Letter XII, this infinite collection of the infinitely small is ‘an infinite sum of differentials’, in the form of a power series expansion. According to the logic of differentiation, the principle of determinability deems that the infinitely small differentials remain undetermined. It is only in the differential relation, according to the principle of reciprocal determination, that the infinitely small differentials are determined by a reciprocal synthesis, which, when expanded in a power series, constitutes an infinite collection of the infinitely small. As such, Deleuze argues that the most simple bodies exist collectively and not distributively; they exist therefore by infinite collections. Insofar as an infinite collection of the most simple bodies corresponds to each individual, an individual is not a simple body, but rather a complex body composed by an infinite collection of the infinitely small.⁵⁵

According to the terminology of Spinoza the most simple bodies are those ‘which are distinguished from one another only by motion and rest, speed and slowness’.⁵⁶ From this Deleuze argues that the most simple bodies therefore only ‘react on one another, they have no interiority, they have only strictly external or extrinsic relations with one another’.⁵⁷ Deleuze therefore defines the infinitely small as ‘extensive parts’ whose interactions with one another are determined by purely extensive relations. An individual is therefore defined by a relation of movement and rest because it is under this relation that an infinite collection of extensive parts correspond or belong to it. This relation of movement and rest that Spinoza invokes is primarily between infinitely small extensive parts, which have no distributive existence, but which enter necessarily into infinite collections. The interactions between extensive parts are determined by the logic of differentiation according to which an infinite sum of infinitely small extensive parts is a power series expansion, which converges with a local function, the limits, or poles, of which are determined by the extent to which the local function can be extended by analytic continuity, according to the principle of complete determination. The complete local function corresponds to an existing complex body or individual, and the poles of the function to the limits of existence of such a complex body or individual.

The logic of pure and composite relations in Spinoza’s Letter XII

The interpretation of the geometric infinite as an extensive infinite, composed by an infinite collection of infinitely small extensive parts, is considered by Deleuze to be ‘at one with the understanding of an actual infinite’.⁵⁸ He argues that the infinity of the infinitely small, which constitutes the geometric infinite, is an ‘actual infinite’.⁵⁹ Whereas Macherey and Gueroult maintain that the infinitely small are imaginary abstractions, Deleuze rather considers the actual infinite of the geometric example to be a strictly extensive infinite, one which can be understood adequately from the point of view of reason. This Deleuzian interpretation adds a whole new dimension to the problem of the infinite. Deleuze argues that this extensive infinite is ‘necessarily conceived as greater or less’.⁶⁰ Contrary to Hegel, who ignores this distinction by relegating the greater or lesser infinities to the same bad infinite, and to Macherey, who considers greater or lesser infinities to be inadequate representations of the actual infinite from the point of view of the imagination, Deleuze argues that this extensive infinite is necessarily constituted by infinite collections (greater infinities) of the infinitely small (lesser infinities).

Rather than distinguishing an abstract physics, or metaphysics, of the most simple bodies, from a concrete physics of complex bodies, according to the distinction between the internal and external determination of singular things, as do both Gueroult and Macherey,⁶¹ Deleuze considers both the most simple bodies, in the form of extensive parts, and complex bodies or individuals, to be involved in extrinsic relations, each being expressed respectively from the local point of view by the processes of differentiation and differenciation.

The sum which Deleuze extracts from Letter XII, is not simply an addition of finite quantities, as Macherey contends, but rather a power series whose expansion increasingly determines the specific qualitative nature of a composite function, that is, a complex body or individual. Even though the applicability of an infinite sum works in relation to both Macherey’s interpretation of Letter XII and that of Deleuze, an infinity composed by a sum of differences is different to one composed by a sum of differentials. For Macherey, the infinite sum of differences is indefinite or unlimited, a bad infinite, an infinite of a series by the multiplicity of its parts, which can only be represented inadequately from the point of view of the imagination. Whereas for Deleuze, the infinite sum of differentials, as a power series expansion, which converges with either a local function (according to the logic of differentiation), or a composite function (according to the logic of differenciation), is understood adequately from the point of view of reason as constitutive of an actual infinite. The composite function or individual is not infinite from the number or multitude of its parts, but rather being infinite, it ‘divides into a multitude of parts exceeding any number’.⁶²

Gueroult argues that the geometrical example of Letter XII can be understood, in relation to modern theories of the differential calculus, to be an illustration of a process of integration. The limits upon which this process depends are given in the example as the maximum and minimum orthogonal distances. Given these specific limits, integration, as the inverse operation of differentiation, is the determination of a definite integral, which is the determination of the area under the curve as an infinite sum of differentials, which subsists within the given maximum and minimum limits. By giving a value for this integral, integration determines the infinite sum of the differentials as a finite quantity. This method of integration is dependent upon already determined limits, which Gueroult considers to have been given in the example. As the area under the graph of the curve, Gueroult considers the determination of the value of the definite integral as a finite quantity to be an inadequate representation of the actual infinite from the point of view of the imagination. This is the extent of Gueroult’s interpretation of Letter XII in relation to the differential calculus.

Deleuze’s exploration of the differential calculus in relation to Letter XII is rather from the differential point of view of the infinitesimal calculus. Instead of assuming the limits to have been given, as two arbitrary points on the x-axis, Deleuze’s focus is on just how these limits are determined mathematically. For this reason he deals with the early developments of the infinitesimal calculus from the differential point of view, and with the significance of the differential relation to this development. To distinguish his conception of the ‘infinite sum of differentials’ from that of Gueroult, Deleuze further characterizes it as ‘the kind of integration of differential relations’.⁶³ This kind of integration is primarily characterized by the process of differentiation which determines the specific limits according to which a local function is completely determined. These limits are determined by the manipulation of differential relations according to the principles of differentiation. It is therefore the significance of the differential relation to the development of this process that Deleuze extracts from the geometrical example of Letter XII.

Rather than these limits being determined by a process of reciprocal negation between other infinite collections of the most simple bodies, as Macherey contends, Deleuze suggests that the infinite collections of extensive parts ‘don’t exist independently of the differential relations which they effect’.⁶⁴ That is, according to Deleuze, the manner by means of which extensive parts exist in infinite collections is determined by the differential relations which they effect. He calls these ‘pure relations’, which are determined according to the logic of differentiation. And, just as a complete local function, being an infinite collection of extensive parts, is primarily determined by the pure relations of which it is composed, so too are the relations between these kinds of functions determined by the differential relations which they effect with each other, according to the logic of differenciation. That is, the pure relations of the complete local functions are then actualized in what Deleuze calls a ‘composite relation’.⁶⁵ The composition of these composite relations between completely determined local functions is determined according to the same logic that determines the pure relations between extensive parts. Just as extensive parts cannot exist independently of a relation, since the concept of the infinitely small cannot be defined independently of a differential relation, so too does the existence of a complete local function depend upon the composite relations which it effects with other local functions, that is, the principle of determinability also operates within differenciation. The specific qualitative nature of the limits of a complete local function, and therefore the finitude of a local function, remains undetermined by the process of differentiation. Insofar as this function is engaged in a composite relation determinative of a meromorphic function, the specific qualitative nature, and therefore the finitude, of each local function is actualized in the form of the curves of the composite function.

To each composite relation corresponds an individual as a complex body composed by the pure relations of the most simple bodies. The singularities generated by the pure relations of differentiation are characterized by Deleuze as ‘pre-individual singularities’.⁶⁶ The individualizing singularities determined during the process of differenciation are essential singularities, which take the form of the curves of composite functions. The nature of the limit of one local function, or preindividual singularity, is actualized by means of the essential singularity generated by the composite relation, in the form of the divergent lines of the infinite branches of the composite function, or individual.

The specific qualitative nature, or finitude, of the individual is also only determined in relation to other individuals. It is only in the composite relations between individuals that there is a limit that marks the finitude of each individual; and that there is ‘an infinite of a certain order’, that is, an actual infinite, that is involved by the relation.⁶⁷ The infinite that is involved in this relation is determined by the expansion of the composite relation in a power series. This series is actualized in a composite function, or individual, as an actual infinite. The limit that marks the finitude of each individual is determined by the existence and distribution of the essential singularity in the form of a composite function, which is determined by the expansion of the composite relation.

Rather than being distinguished by relations of reciprocal negation, individuals are therefore distinguishable according to the logic of differenciation. Deleuze uses Letter XXXII where Spinoza discusses the composition of blood as an example. ‘Chyle is an infinite collection of very simple bodies. Lymph is another infinite collection of very simple bodies. What distinguishes these two infinite collections? The differential relation! You have here a dy/dx which is: the infinitely small parts of chyle on the infinitely small parts of lymph, and this differential relation tends towards a limit: the blood. Chyle and lymph compose blood’.⁶⁸ The finitude of an individual is therefore characterized by the differential relation in which it is implicated, or, in other words, the nature of the limit of a local integration is determined by the global integration in which it is implicated, according to the logic of differenciation. So, according to Deleuze, there is an infinity of differential relations which compose the blood, the bones, and the flesh of an individual. These refer to systems of differential relations, or composite relations, which are integrated as composite bodies or individuals of increasing complexity, according to the logic of differenciation.

The logic of different/ciation and the philosophy of difference

The logic of the differential, as determined according to both differentiation and differenciation designates a process of production, or genesis, which has, for Deleuze, the value of introducing a general theory of relations which unites the structural considerations of the differential calculus to the concept of ‘the generation of quantities’.⁶⁹ ‘In order to designate the integrity or the integrality of the object’, whether considered as a composite function from the differential point of view of the infinitesimal calculus, or as an individual from the point of view of the physics of bodies, Deleuze argues that, ‘we require the complex concept of different/ciation. The t and the c here are the distinctive feature or the phonological relation of difference in person’.⁷⁰ Deleuze argues that differenciation is ‘the second part of difference’,⁷¹ the first being expressed by the logic of the differential in differentiation. Where the logic of differentiation characterizes a differential philosophy, the complex concept of the logic of different/ciation characterizes Deleuze’s ‘philosophy of difference’.

In constructing this theory of relations characteristic of a philosophy of difference, Deleuze draws significantly from the work of Albert Lautman, who refers to this whole process as ‘the metaphysics of logic’.⁷² It is in Difference and Repetition that Deleuze formulates a ‘metaphysics of logic’ that corresponds to the differential point of view of the infinitesimal calculus. However, he argues that ‘we should speak of a dialectics of the calculus rather than a metaphysics’,⁷³ since ‘each engendered domain, in which dialectical Ideas of this or that order are incarnated, possesses its own calculus. … It is not mathematics which is applied to other domains but the dialectic which establishes … the direct differential calculus corresponding or appropriate to the domain under consideration’.⁷⁴ It is not the early form of the inverse transformation of differential calculus which is applied to the dialectical logic to support its development, but rather the dialectical logic which determines the direct differential calculus which corresponds or is appropriate to its own development. Similarly, Deleuze finds in the differential point of view of the infinitesimal calculus a form of the differential calculus appropriate to the development of a philosophy of difference in Expressionism in Philosophy and Difference and Repetition. However, Deleuze develops the differential point of view of the infinitesimal calculus as an alternative point of view of the differential calculus to that proposed by Hegel. The differential point of view of the infinitesimal calculus represents not a moment that can be simply sublated and subsumed within the dialectical progression of history, but rather an opening, providing an alternative trajectory for the construction of an alternative history of mathematics. Deleuze thereby establishes a historical continuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus which effectively bypasses the methods of the differential calculus which Hegel uses in the Science of Logic to support the development of the dialectical logic. ‘That is why the differential calculus belongs to mathematics, even at the very moment when it finds its sense in the revelation of a dialectic which points beyond mathematics’.⁷⁵ While Hegel is interested in using advances in mathematics to secure the development of the dialectical logic, Deleuze is interested in using mathematics to secure not only the development of a philosophy of difference, but in the process, to undermine the mathematical support of the Hegelian project, by historically bypassing it and providing an alternative trajectory, not only in the history of mathematics, but simultaneously in the history of philosophy.

The alternative lineage in the history of mathematics is implicated in Deleuze’s alternative lineage in the history of philosophy by means of a convergence between the logic of the differential point of view of the infinitesimal calculus and the logic of the theory of relations in Deleuze’s philosophy of difference. The manner by means of which a mathematical function – the relation between infinitely small differentials – is implicated in the mathematical logic which determines it, converges with, or serves as a function, or mathematical representation of, the manner by means of which a philosophical concept – the relation between the most simple bodies – is implicated in the philosophical logic which determines it. There are ‘correspondences without resemblance’ between them, insofar as both are determined according to the logic of different/ciation. The philosophical implications of this convergence are developed by Deleuze in Expressionism in Philosophy in relation to his reading of Spinoza’s theory of relations in the Ethics, which will be examined in chapter 4.

The main problematic that has determined the direction of the investigation so far, and which continues to do so throughout the following chapters, is the question of ‘the generation of quantities’,⁷⁶ which takes the form of asking whether or not an alternative to the solution posed by the dialectical logic is able to be determined. The alternative, proposed by the logic of the differential from the differential point of view of the infinitesimal calculus, is deployed by Deleuze, in the form of the logic of different/ciation, as a component in the development of his project of constructing a philosophy of difference.

Before determining in detail the relation between an extensive quantity and the infinite collection of infinitely small extensive parts of which it is composed, which will be addressed in detail in chapter 5, it is necessary to determine the intensive processes by means of which these extensive parts are determined, which involves, for Deleuze, an investigation of the problem of intensive quantity, or intensity, in Spinoza, the development of which is addressed in chapter 4.

_______________

Notes

¹ Given a function, f(x), having derivatives of all orders, the Taylor series of the function is given by

where f (k)(a) is the ‘kth’ derivative of ‘f’ at ‘a’. A function is equal to its Taylor series if and only if its error term Rn can be made arbitrarily small, where

The Taylor series of a function can be represented in the form of a power series, which is given by

where each ‘a’ is a distinct constant. It can be shown that any such series either converges at x = 0, or for all real ‘x’, or for all ‘x’ with –R < x < R for some positive real ‘R’. The interval (–R, R) is called the circle of convergence, or neighbourhood of the distinctive point. This series should be thought of as a function in ‘x’ for all ‘x’ in the circle of convergence. Where defined, this function has derivatives of all orders. See also H.J. Reinhardt, Analysis of approximation methods for differential and integral equations (New York, 1985).

² See Morris Kline, Mathematical thought from ancient to modern times (London and New York, 1972), pp. 643–4.

³ Deleuze, Difference and Repetition, p. 278.

⁴ See the section of chapter 2 entitled ‘The differential calculus from the point of view of the dialectical logic’, p. 58.

⁵ Hegel, Hegel’s Science of Logic, p. 295.

⁶ Ibid., p. 265.

⁷ Ibid.

⁸ Ibid.

⁹ Albert Lautman, Essai sur les notions de structure et d’existence en mathématiques (Paris, 1938), p. 38.

¹⁰ In ‘Hegel’s Philosophy of Mathematics’, Pinkard suggests that ‘Hegel would, so it would seem, accept the interpretation which moderns like Weierstrass gave to the infinite’ (p. 463). Perhaps he would. However, the subsequent development of this infinite from the differential point of view of the infinitesimal calculus runs counter to the manner by means of which the Hegelian infinite is implicated in the differential calculus from the point of view of the dialectical logic. Weierstrass’ theorem, on the contrary, actually anticipates the return of the infinitesimal in the differential calculus, or non-standard analysis, of contemporary mathematics. In Hegel’s Dialectic, Pinkard writes that ‘Hegel would not be pleased with the rise of non-standard analysis, in which the notion of the infinitesimal has made its reappearance. He would no doubt side with those philosophers and mathematicians who view this with only the greatest suspicion’ (p. 52).

¹¹ Deleuze, Difference and Repetition, p. 171.

¹² Ibid.

¹³ Boyer, The history of the calculus and its conceptual development, p. 261.

¹⁴ Ibid., p. 262.

¹⁵ Deleuze, Difference and Repetition, p. 175.

¹⁶ Ibid.

¹⁷ Boyer, The history of the calculus and its conceptual development, p. 262.

¹⁸ Ibid.

¹⁹ Deleuze, Difference and Repetition, p. 172.

²⁰ Ibid.

²¹ Ibid., p. 171.

²² Ibid., p. 173.

²³ Ibid., p. 175.

²⁴ Ibid., p. 172.

²⁵ Note: the primitive function ∫f(x)dx, expresses the whole curve f(x).

²⁶ Deleuze, Difference and Repetition, p. 172.

²⁷ Ibid., p. 174.

²⁸ Ibid., p. 172.

²⁹ Ibid., p. 174.

³⁰ The exponent, or power, of the variable of a function decreases by one degree when the function is differentiated. That is, if f(x)=xn, then f’(x)=nxn–1.

³¹ Deleuze, Difference and Repetition, p. 174.

³² Ibid., p. 175.

³³ Ibid.

³⁴ Ibid.

³⁵ Ibid., p. 209.

³⁶ Ibid., p. 172.

³⁷ Lautman, Essai sur les notions de structure et d’existence en mathématiques, p. 8.

³⁸ Kline, Mathematical thought from ancient to modern times, p. 732.

³⁹ June Barrow-Green, Poincaré and the Three Body Problem (Rhode Island, 1997), p. 32, figure 3.2.i.

⁴⁰ Manuel DeLanda, Intensive science and virtual philosophy (New York, 2002), p. 15.

⁴¹ Deleuze, Difference and Repetition, p. 177.

⁴² Ibid., p. 209.

⁴³ Ibid., p. 221.

⁴⁴ Ibid., p. 212.

⁴⁵ Ibid., p. 210.

⁴⁶ Ibid., p. 212.

⁴⁷ Ibid., p. 211.

⁴⁸ Ibid., p. 212.

⁴⁹ Georges Valiron, ‘The Origin and the Evolution of the Notion of an Analytic Function of One Variable’, in François Le Lionnais (ed.), Great Currents of Mathematical Thought (2 vols, New York, 1971), vol. 1, p. 171.

⁵⁰ Ibid., p. 173.

⁵¹ Benoit Mandelbrot (b. 1924) considers Poincaré, with his concept of essential singularities, to be ‘the first student of fractal (“strange”) attractors’, that is, of the kinds of attractors operative in fractals which occur in mathematics, and cites certain theories of Poincaré as having ‘led [him] to new lines of research’, specifically ‘the theory of automorphic functions’, which made Poincaré and Felix Klein (1849 – 1925) famous. Mandelbrot qualifies these statements when he says of Poincaré that ‘nothing I know of his work makes him even a distant precursor of the fractal geometry of the visible facets of Nature’ (Mandelbrot, The Fractal Geometry of Nature (San Francisco, 1982), p. 414).

⁵² See Albert Lautman, Essai sur l’unité des sciences mathématiques dans leur développement actuel (Paris, 1938), p. 138.

⁵³ Deleuze, Difference and Repetition, p. 217.

⁵⁴ Ibid., p. 211.

⁵⁵ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁵⁶ Spinoza, Ethics, II, P13L3A2.

⁵⁷ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁵⁸ Ibid.

⁵⁹ Ibid.

⁶⁰ Deleuze, Expressionism in Philosophy, p. 202.

⁶¹ The most simple bodies being the abstract correlates of the infinite determination of substance.

⁶² Deleuze, Expressionism in Philosophy, p. 203.

⁶³ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁶⁴ Ibid.

⁶⁵ Deleuze, ‘sur Spinoza’, 24 January 1978. Translated by Timothy S. Murphy.

⁶⁶ Deleuze, Difference and Repetition, p. 176.

⁶⁷ Deleuze, ‘sur Spinoza’, 17 February 1981.

⁶⁸ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁶⁹ Deleuze, Difference and Repetition, p. 175.

⁷⁰ Ibid., p. 209.

⁷¹ Ibid.

⁷² Lautman, Essai sur les notions de structure et d’existence en mathématiques, p. 3.

⁷³ Deleuze, Difference and Repetition, p. 178.

⁷⁴ Ibid., p. 181.

⁷⁵ Ibid., p. 179.

⁷⁶ See the section of this chapter entitled ‘The development of a differential philosophy’, p. 76.

Chapter 4

From Scotist univocity to Spinozist immanence: Intensive quantity

Scotus’ formal distinction as real

The problem of intensive quantity, or intensity, is developed by Deleuze in Expressionism in Philosophy by means of an engagement between Spinoza and Duns Scotus. According to Deleuze, the problem of intensive quantity, or intensity, has a long Scholastic tradition, which is particularly developed in the work of Scotus. By reading Spinoza alongside of the Scholastics, in particular Scotus, Deleuze provides a Spinozist context for the elaboration of this problematic, which functions as an alternative to that presented by Hegel’s dialectical philosophy. Scotus thereby becomes another figure in Deleuze’s alternative lineage in the history of philosophy. It is by means of determining the relation between the early stages of the development of the infinitesimal calculus and the Scholastic examples of intensive quantity, which Deleuze considers to be directly related to the problem of intensive quantity for Spinoza, that the different concepts of intensive quantity developed by Hegel and Deleuze are distinguished. The Scotist concept of intensive quantity is first distinguished from the Hegelian concept of intensive quantum. However, before explicating the Deleuzian concept of intensive quantity, the passage from the infinite collection of infinitely small extensive parts to the intensive processes which produce them will be determined. These intensive processes are determined by what Deleuze describes as ‘intensive parts’, the relations between which are constitutive of intensive quantities. The mechanism by means of which the relations between intensive parts are determinative of intensive quantities will then be examined in chapter 5.

Deleuze purports to find in Spinoza’s work, if not specific references to Scotus, at least references to specific problems raised and developed by philosophers of the Middle Ages, in particular those problems which circulated amongst the Scholastics and were elaborated in the work of Scotus. The metaphysics of Scotus establishes a univocal concept of being which is neither a genus nor a category and which ‘is predicated in the same sense of everything that is’,¹ whether finite or infinite, singular or universal, created or uncreated. Univocal being is therefore understood to be neutral, or indifferent, to such a change in ‘modality’, that is, when the concept is predicated of infinite being and of finite being, etc., the distinction between them, from the point of view of their being, is neutralized. Being, according to Scotus, expresses primarily a quidditative notion, according to which being is understood to be a common nature or essence. In the article ‘Duns Scotus’ Teaching on the Distinction Between Essence and Existence’, Andrew O’Brien argues that ‘being … becomes universal when it is abstracted by the human intellect, which confers on it universal predicability. This absolute essence becomes particularized when it is united with matter, and it becomes a singular individual when it receives its ultimate determination, the “thisness” (haecceity) which, though not of itself a form, is, for Scotus, a “formality.”’² So, in relation to the distinction between the universal, the particular, and the singular, Deleuze notes that, for Scotus, ‘univocity does not lead to any confusion of essences’,³ that is, insofar as the essence referred to is the common nature of being.⁴

Scotus’ concept of the univocity of being leads to his conception of the univocity of divine attributes which is enshrined in the formal distinction.⁵ While pondering the mysteries of the Trinity and the divine attributes, Scotus came up against the problem of referring to a plurality of attributes or perfections in God when the divine nature was supposedly devoid of any real distinction. He first postulated the formal distinction to elucidate these theological problems. Apart from the concept of haecceity as the principle of individuation and the concept of the univocity of being, no thesis is more distinctly Scotist than the formal distinction. However, although he was the first to postulate the formal distinction, the concept of such a distinction did not originate with him. The Scholastics generally recognized the need for some intermediary distinction between a real and a purely rational distinction. In The philosophical theology of John Duns Scotus, Allan Wolter argues that ‘Medieval philosophers generally admitted a threefold distinction’:⁶ the real distinction, which entails a distinction between individuals in actual existence; the rational distinction, which is purely conceptual with no real foundation in objective reality; and an intermediate distinction, which, though defined rationally, has some kind of basis in reality. ‘Historically the formal distinction had its origin in the attempt to clarify the precise nature of that foundation in reality. … The need for some intermediary distinction, so the scholastics argued, stems from the fact that individuals are characterized by certain objective properties which, though inseparable in reality, are separable in concept, since neither notion includes the other’.⁷ Scotus refers to the distinction that obtains between such objective properties, or propria, which he calls divine attributes, such as justice, goodness, wisdom and so on, as ‘formal’, but remarks that it may also be referred to as a ‘virtual distinction’,⁸ since the distinction determines not two things for example, but ‘one thing which is virtually or preeminently, as it were, two realities’.⁹ The peculiar properties of the divine attributes must be ‘really identical with, but formally distinct from, the divine nature they have in common, … that is, although really one are virtually many’.¹⁰ Therefore, not only is the univocity of being extended in the univocity of its attributes, but divine essence possesses a plurality of formally distinct univocal attributes without losing anything of its unity; the plurality of the former being perfectly in accordance with the simplicity of the latter. Scotus considers the formal distinction to be ‘logically prior to actual thought’,¹¹ and a necessary condition for the conception of one divine attribute without the other. However, although he recognized that the divine essence could be conceived without these attributes, he nevertheless defined it in relation to them, ‘by intrinsic perfections, understanding and will’.¹²

The main criticism that Deleuze brings to bear upon the concept of univocity in Scotus, both in relation to being and to the divine attributes, or the formal distinction, is that it ‘seems compromised by a concern to avoid pantheism’,¹³ which, of course, is ‘in accordance with the requirements of Christianity’.¹⁴ In order to do this, Deleuze argues, Scotus’ concept of univocity ‘neutralized being … in an abstract concept’.¹⁵ According to Deleuze, Scotus ‘only thought univocal being’,¹⁶ that is, the formally distinct divine attributes ‘are referred by understanding to an ontologically single substance’.¹⁷ Scotus was thus still dealing with propria as the supposed attributes of divine essence.

In the Introduction à l’Ethique de Spinoza, La première partie, Macherey argues that ‘the attribute, as Spinoza conceives it, is not a property or a “propria” attached or attributed to a substance, but a constitutive form of substance’.¹⁸ Spinoza therefore removes from the attribute the general character of ‘a quality of things’,¹⁹ which, according to Macherey, remained the characteristic with which Descartes defined the attribute. In addition to distinguishing Spinoza’s conception of the relation between substance and its attributes from that of Descartes, Deleuze also argues that ‘Spinoza marks a considerable progress’ in relation to Scotus. What is of primary interest for Deleuze is the way Spinoza ‘uses and transforms’²⁰ the Scotist concepts of univocity and formal distinction. In Difference and Repetition, Deleuze argues that ‘instead of understanding univocal being as neutral or indifferent, [Spinoza] makes it an object of pure affirmation’.²¹ Therefore, rather than being an abstract concept, with Spinoza ‘univocal being becomes identical with unique, universal and infinite substance’.²² To distinguish this infinite substance from its attributes, Deleuze argues that ‘Spinoza restores formal distinction’, however, just as he redefines univocity, Spinoza gives formal distinction a range it didn’t have in Scotus. Indeed, Spinoza doesn’t use the term ‘formal distinction’ but rather ‘real distinction’, which he borrows from Descartes. According to Deleuze, it is the distinctly Spinozist reworking of the ‘formal distinction that provides an absolutely coherent concept of the unity of substance and the plurality of attributes, and gives real distinction a new logic’.²³ By using the term ‘real distinction’ rather than ‘formal distinction’, Spinoza effectively neutralizes the theological content of Scotus’ ‘formal distinction’, that is, its concern to avoid pantheism. It is for this reason that Deleuze suggests that the term ‘real distinction’ ‘allowed great audacity without stirring up old controversies’.²⁴

What then is at stake in Spinoza’s use of the Cartesian term? According to Descartes’ concept of ‘real distinction’, ‘two substances are distinguished from one another, whether [or not] they have the same attributes … This is known from the fact that each can be conceived, and consequently, can exist, without the aid of the other’.²⁵ For Spinoza, however, there is only one substance, and real distinction is rather between the attributes, in relation to this one substance. The attributes, each of which satisfies the definition of substance, are really distinct from one another. Therefore, contrary to the Cartesian concept, an attribute cannot exist and cannot be conceived without the aid of another, insofar as they are really distinct.²⁶ Deleuze considers Spinoza’s use of the term ‘real distinction’ to be the extent of his Cartesianism, arguing that Spinoza’s ‘whole theory of distinctions is profoundly Anticartesian’.²⁷ However, Deleuze considers Spinoza to be no more Scotist than Cartesian, indeed he argues that ‘to picture Spinoza as Scotist rather than Cartesian is to risk certain distortions’.²⁸ Deleuze means only to suggest that ‘Scotist theories were certainly known to Spinoza, and played a part, along with other themes, in forming his pantheism’.²⁹

Modal distinction and intensive quantity in Duns Scotus

The identification by Spinoza of an attribute as a constitutive form of infinite substance, rather than as an objective property or propria, is interpreted by Deleuze from the point of view of quality: the quality of the attribute is no longer attributable to things, nor referred by the understanding to divine essence, but is rather constitutive of substance. The attributes thus become the component qualities of substance itself. With Spinoza, formal distinction is no longer a distinction between certain objective properties which, though supposedly inseparable in reality, are separable in concept; but a distinction between a plurality of attributes in relation to the single substance they constitute. Formal distinction therefore becomes, in effect, a real distinction between the attributes. However it is not necessarily a numerical distinction, since it is established between qualities which allow the persistence of the unity of substance. Deleuze concludes therefore that ‘there is one substance per attribute from the viewpoint of quality, but one single substance for all attributes from the viewpoint of quantity’.³⁰ The attributes are therefore ‘qualitatively, but not quantitatively, distinct’.³¹ The distinction between the different attributes in relation to the substance they constitute is real qualitatively while not being numerical, or quantitative. All attributes, as qualifications of substance, ‘form only one substance from the point of view of quantity. So that attributes themselves have at once identity of being and distinction of formality, that is, the attributes ‘are “formally”, … and not “ontologically” distinct’.³² ‘All formally distinct attributes … [constitute] an ontologically single substance’.³³ ‘Ontologically one, formally diverse, such is their status’.³⁴ Deleuze argues therefore that in Spinoza’s ‘conception of a non-numerical real distinction, it is not hard to discern Scotus’ formal distinction’.³⁵ Formal and yet not ontological, qualitative and yet not quantitative, ‘real and yet not numerical, such is the status of formal distinction’³⁶ in Spinoza. Furthermore, according to Deleuze, Spinoza’s understanding of the constitutive nature of the attributes is such that ‘all formal essences [that is, the essence of each attribute taken together,] form the essence of an absolutely single substance’.³⁷ This is the new logic of the real distinction that Deleuze attributes to Spinoza. This new logic allows Spinoza ‘to reconcile the ontological unity of substance with the qualitative plurality of its attributes’.³⁸ As Deleuze argues, ‘the distinction of attributes is [therefore] nothing but the qualitative composition of an ontologically single substance; substance is distinguished into an infinity of attributes, which are as it were its actual forms or component qualities’.³⁹

Deleuze argues that this new logic of the real distinction, which is not numerical, is also ‘capable of expressing differences within Being, so bringing about the restructuring of other distinctions’.⁴⁰ The other type of distinction that Deleuze considers to have been restructured by Spinoza is the modal distinction. Scotus’ concept of modal distinction is elaborated by O’Brien in ‘Duns Scotus’ Teaching on the Distinction Between Essence and Existence’. In terms of his univocal concept of being, Scotus explains what he means by formal distinction in relation to its difference from a real and a purely rational distinction. Scotus ‘then seems to interpose between this formal and purely rational distinction another, a modal distinction, the significance of which’, O’Brien argues, ‘is somewhat ambiguous in his system’.⁴¹ According to O’Brien, the precise question at issue is the nature of the distinction between an individual’s essence and its existence. Many commentators on Scotus and historians of philosophy simply disregard this modal distinction when they explain the Scotist doctrine on distinctions. Other commentators, like Gilson, are careful to point out that the modal distinction is a distinction made by the Scotist school rather than by Scotus himself’.⁴² Those commentators who interpret Scotus as holding a formal distinction between an individual’s essence and existence assume that this is so because Scotus ‘clearly maintains such a distinction between the common nature and its “haecceity” (formality) or principle of individuation’.⁴³ ‘However’, O’Brien argues, ‘this would be to confuse the principle of individuation’, or the being of an actually existing individual, with its actual existence.⁴⁴ O’Brien argues that, according to Scotus, ‘an essence exists just as much as it is, and its existence is exactly defined by the mode of being which belongs to that essence’.⁴⁵ The being of an actually existing individual actually exists as an essence which is distinct from its being, insofar as its being is univocal. The existence of this essence is defined by the mode of being, or ‘individuating modality’,⁴⁶ which belongs to that essence. An ‘infinite individuating modality’⁴⁷ is the mode of being of the essence of an actually existing individual which actually exists within a distinct attribute or qualitative form, and a finite individuating modality is the mode of being of that actually existing individual.

According to the principle of individuation, the individualizing modalities of being are formally distinct from the attributes, or qualitative forms, and do not modify the essence of the attributes insofar as this is being, that is, the distinction between being and its individuating modalities is formal. However, ‘from the point of view of its own neutrality, univocal being … does not implicate distinct attributes or qualitative forms’,⁴⁸ within which actually existing individual essences are distinguishable. Therefore, when univocal being is ‘considered in its individuating modalities (infinite and finite), rather than in its nature as Being, it ceases to be univocal’.⁴⁹

The concept of modal distinction deals with the question of the very distinction between the infinite and the finite individuating modalities, which is the distinction between ‘an actually existing individual essence and its actual existence’.⁵⁰ The actual existence of an individual, as a finite individuating modality, cannot be distinguished from its actually existing essence by the formal distinction for the simple reason that the essence of the actually existing individual is an essence within a distinct infinite individuating modality, and not an absolute essence, that is, in its capacity as an individuating modality it is not univocal, and therefore cannot be a ‘formality’ (haecceity).⁵¹ If, according to Scotus’ metaphysics, the infinite actual existence of an individual can be understood only as an individuating modality of being, and not as a formality or haecceity, then the essence and existence of an individual, its infinite and finite individuating modalities, can be understood to be distinct by a modal distinction.

Modus intrinsecus: the quantitative treatment of variation in quality, or intensity

When Scotus speaks of form, he is referring to the philosophy of Aristotle, according to which an actually existing individual, or what Aristotle would have referred to as the matter component or effect of a form, acquires the character that makes it the kind of thing that it is only in relation to its formal reason or essence.⁵² Within this framework, the form of any actually existing individual remains invariable, even though there are variations in the effects of form, that is, the different ways in which that form exists. Therefore when Scotus asserts that a form has ‘intrinsic modes’, or modi intrinseci, it is strictly incomprehensible within the framework of the philosophy of Aristotle. The problem of whether ‘a quality’ can vary without changing its form, and whether these variations belong to the qualitative form itself, or only to its existence resurfaces in the Middle Ages, and is particularly developed in Scotism. Instead of distinguishing only two terms, form and matter or existence, Scotus distinguishes three. According to Scotus, if a form effects itself in extrinsic modes, it is necessary to distinguish form from both intrinsic and extrinsic modes of being. O’Brien argues that Scotus considers the actual existence of the essence of an individual, or its infinite individuating modality, to be ‘an intrinsic mode of being’,⁵³ and the actual existence of an individual, or its finite individuating modality, to be an extrinsic mode of being. The distinction between intrinsic and extrinsic modes of being is therefore modal. Scotus argues that a form also has a kind of latitude, or, more precisely, that there is ‘a latitude of form’⁵⁴ (‘latitudo formæ’⁵⁵). A latitude of form has distinctions of gradus, or degree, which he describes as ‘the intrinsic degrees of form’.⁵⁶ Scotus’ theory of intrinsic modes or degrees addresses the problem of the increase and decrease (intensio and remissio), or the variation, of the quality of a form. In The physics of Duns Scotus, Richard Cross argues that according to Scotus, a distinct qualitative form or attribute, insofar as it is an infinite individuating modality, ‘is infinitely divisible’ into ‘homogeneous parts’,⁵⁷ which ‘count as discrete items only after the process of division’.⁵⁸ Scotus therefore holds that any quality contains an infinity of homogeneous parts, and, furthermore, these parts, being homogeneous, are all of the same sort, such that changes in the degree or variations in the intensities of quality can be explained by the addition or subtraction of these parts.⁵⁹ The point being that Scotus postulates a quantitative account of the different degrees or variations in the intensities of quality.⁶⁰ As Wolter argues, ‘Scotus was one of the first Scholastics to show how variations in quality, or intensity, might be treated quantitatively’.⁶¹

In The Science of Mechanics in the Middle Ages, Clagett argues that ‘it was accepted by the successors of Scotus … that the increase or decrease of qualitative intensity takes place by the addition or subtraction of degrees of intensity’.⁶² The Scotists developed a clear distinction between the measure of the intensity of a quality and its quantitative extension. This distinction was expressed in terms of one between virtual or intensive quantity and corporeal or extensive quantity.⁶³ There are therefore degrees or intensities of a quality, referred to in general as ‘intensive quantities’, which are just as different from quality as from extensive quantity. Just as Scotus’ distinction between intrinsic and extrinsic modes of being is modal, the Scotists can be understood to have distinguished intensive quantities from extensive quantities by a modal distinction.

The Hegelian distinction between intensive and extensive quantum

Before determining the significance of this distinction to Deleuze’s reading of Spinoza, the Hegelian distinction between intensive and extensive quantum will be explicated in order to differentiate the concept of intensive quantity, or intensity, that Deleuze develops, from that which operates according to Hegel’s dialectical logic. In the Science of Logic, Hegel formulates a critique of what he refers to as the concept of intensive magnitude conceived as a type of exceptional quantity, mediating between the qualitative and quantitative concept of being. He employs the Latin terms quantitas and quantum to distinguish between two senses of the word magnitude: quantitas, which is pure quantity or infinite quantum, ‘is to be distinguished from itself as a determinate quantity, from quantum’.⁶⁴ Hegel describes the moments of the dialectical logic which articulate this difference as follows: ‘Quantum differentiates itself … into extensive quantum … and intensive quantum or degree, … [before it] passes over … into quantitative infinity’.⁶⁵

Hegel’s concept of intensive quantum has already been introduced in the section of chapter 2 entitled ‘The differential calculus from the point of view of the dialectical logic’, where it was argued that the differential is determined as an intensive magnitude, or quantum, in the moment of qualitative transition of the differential calculus. Extensive magnitude, or quantum, correlates with the moment of quantitative transition, which determines the limit of the ratio or differential relation. The relation between extensive and intensive quantum correlates with that between differentiation, as the moment of quantitative transition, and integration, as the moment of qualitative transition, which is determined according to the dialectical logic.

Hegel argues that ‘when considering intensive and extensive quantum we found that it is the same quantum which is present, once in the form of intensity and again in the form of extension’.⁶⁶ In the form of extension, quantum is considered to be limited by its amount, that is, by the multiplicity of units of which it is composed, as such it is extensive. However, this limit is not considered as a simple determinateness, but rather as an aggregate, for example, one hundred is not conceived of as a unit, but rather as one plus one etc., each unit able to be taken as the hundredth. If, on the other hand, quantum is considered as a simple unit, then it is intensive, and the quantitative limit is a degree. As the ‘simple determination of the limit, quantum is intensive magnitude; and the limit or determinateness which is identical with the quantum is now also thus posited as unitary – degree’.⁶⁷ For example, the temperature of one hundred degrees is not the same as one plus one etc, but is rather a simple unit. Hegel argues that ‘the determinateness of degree must … be expressed by a number, the completely determined form of quantum, but the number is not an amount but unitary, only a degree’.⁶⁸ When Hegel speaks of ‘ten or twenty degrees, the quantum that has that number of degrees is the tenth or twentieth degree, not the amount and sum of them – as such, it would be an extensive quantum – but it is only one degree, the tenth or twentieth’.⁶⁹ While extensive quantum consists in nothing other than its multiplicity, which is obtained by the addition or sum of its parts, intensive quantum or degree resolves this multiplicity in unity. ‘Extensive magnitude passes over into intensive magnitude because its many spontaneously collapse into oneness’.⁷⁰ This happens as though the uniqueness of the degree actually precedes what is always a retrospective division of the 20th degree into 20 degrees. The uniqueness of the 20th degree, which is always given in totality before its always extensive parts, is constituted by excluding from intensive quantum the extensive multiplicity of the 20 degrees, that is, the unity of intensive quantum is negated by its extensive multiplicity according to the dialectical logic.

Rather than determining the extensive and the intensive as two essentially distinct types of quantum, according to which only the former would be quantitative, the latter instead constituting a sort of qualitative enclave in quantity, Hegel considers the determination of quantum to remain ‘identically quantitative’ whatever the ‘position’ that it adopts in the relation between unity and multiplicity.⁷¹ The two types of quantum only differ by the manner of accentuation of one term rather than the other. This ‘manner of accentuation’, Hegel maintains, is determined by the dialectical logic. THE 20th degree, which is an intensive quantum apprehended in its unity, is divisible into the extensive multiplicity of 20 degrees. Just as there is in 100 centimetres not only the intensive unity of one metre in THE hundredth centimetre, but also the extensive sum of 100 centimetres, which, apprehended in their multiplicity, together constitute a metre. The intensive unity of one metre in THE 100th centimetre being divisible into the extensive multiplicity of 100 centimetres, the one being determined in relation to the other by a process of exclusion or negation according to the dialectical logic.

Hegel resolutely maintains that the degree is an element of the quantitative, by this he refuses any special qualitative status to the intensive. He argues against the consideration of intensive quantum as some kind of a species of quantity that accounts for ‘the qualitative uniqueness of being’. Hegel’s argument therefore repeats the Scotist distinction between the principle of individuation, or haecceity, which is univocal with being, and the infinite individuating modality, which is not. However, rather than considering the distinction to differentiate a formal distinction from a modal distinction, Hegel considers the distinction to be resolvable only according to the dialectical logic.

In the ‘Addition’ to §103 of The Encyclopaedia Logic, Hegel clarifies the distinction that he makes between what he refers to as the intensive quantum, or degree, and extensive quantum, with the following example: ‘Any intensive magnitude is also extensive, and conversely. So, a certain degree of temperature, for instance, is an intensive magnitude, to which, as such, there corresponds a wholly simple sensation; and if we then go to the thermometer we find that a certain expansion of the column of mercury corresponds to this degree of temperature, and this extensive magnitude changes together with the temperature taken as an intensive magnitude’.⁷²

By presenting the relation between intensive and extensive quantity in this manner, Hegel considers the degree of intensive quantity to vary in direct proportion to the amounts of extensive quantity. That is, as the temperature increases in degree, or intensive quantity, the variation of the amount of heat, or extensive quantity, as measured by the thermometer, is in direct proportion with the change in intensity of heat, or degree. It is this direct proportionality, according to which intensive and extensive quantum are identically quantitative, that allows Hegel to pose this relation as one moment in the process of the dialectical logic, where the ‘many spontaneously collapse into oneness’.⁷³ The intensity of heat and the amount of heat are for Hegel identically quantitative, differing only in their manner of accentuation; as either a unity, in THE degree, or a multiplicity, in the number of degrees. The unity of THE degree, or the intensity of heat (as a wholly simple sensation) is only determinable in the multiplicity, as the number of degrees, or amount of heat.

However, for the Scholastics, variations in intensity are not necessarily measurable extensively, that is, intensive quantity does not necessarily vary in direct proportion to extensive quantity. Although the quantitative treatment of degrees of intensity does enable degrees to be ordered linearly, it does not necessarily entail that they are additive, that is, that the amount by which one intensity is greater than, or differs from, another can be directly specified extensively, as Hegel posits in the example of the Encyclopaedia Logic.

Scotus’ variable degrees of intensity

In Medieval Logic and the Rise of Mathematical Physics, Curtis Wilson, in support of the Scholastics, argues that ‘in a scale of intensity … the numbers lack this quantitative significance; they indicate simply positions in a linearly ordered series’.⁷⁴ An example of such a scale is given by Wilson in the scale of hardness in minerals. He writes that ‘a mineral to which is assigned a higher degree of hardness is able to produce a scratch on a mineral to which is assigned a lower degree, but it is operationally meaningless to speak of a degree of hardness of eight differing from a degree of hardness of seven by the same amount as a degree of hardness five differs from a degree of hardness four’.⁷⁵

The problem then becomes a question of how, if not numerically or extensively, differences in degree are able to be distinguished from one another? ‘What distinguishes them?’ Deleuze asks in the Spinoza Seminars, ‘How is one degree distinguished from another?’⁷⁶ By reading Spinoza alongside of the Scholastics, in particular Scotus, Deleuze provides a Spinozist context for the elaboration of this problematic, which functions as an alternative to that presented by Hegel according to the dialectical logic.

In The physics of Duns Scotus, Richard Cross argues that ‘the crucial move to an account according to which (some sort of) numerical value can be assigned to qualitative intensities can be reasonably traced to Scotus’.⁷⁷ The examples which Scotus considered include the ‘intensity of illumination, of thermal content, [and] of density’.⁷⁸ Boyer argues that these variations in intensity were measured by the Scholastics as ‘time rates of change’,⁷⁹ and therefore mark an early stage in the development of the infinitesimal calculus. In the Spinoza Seminars, Deleuze refers to this very connection when he states that ‘the theory of intensive quantities is like the conception of differential calculus of which I have spoken. It is determining in all of the Middle Ages’.⁸⁰ This connection between the infinitesimal calculus and intensity is pivotal in distinguishing the different concepts of intensive quantity developed by Hegel and Deleuze. According to Deleuze, the concept of ‘intensity’ in Spinoza refers to a long Scholastic tradition, which is particularly developed in the work of Scotus. Indeed Deleuze considers the early stages of the development of the infinitesimal calculus in the Scholastic examples of intensive quantity to be ‘directly related to the problem of modal essence for Spinoza’.⁸¹ It is here that Deleuze appeals in particular to the concept of the variation in the intensity of illumination, or whiteness, as developed by Duns Scotus, in relation to the example of a white wall.⁸²

The example is presented in Expressionism in Philosophy as follows: ‘Whiteness … has various intensities; these are not added to whiteness as one thing to another thing, like shape added to the wall on which it is drawn; its degrees of intensity are intrinsic determinations, intrinsic modes, of a whiteness that remains univocally the same under whichever modality it is considered’.⁸³ In the example of the white wall, the whiteness of the wall represents a ‘distinct qualitative form or attribute’. Since, according to Scotus, a ‘distinct qualitative form or attribute’, insofar as it is an infinite individuating modality, is infinitely divisible,⁸⁴ it follows that the whiteness is divisible into an infinity of intrinsic modes, which are the degrees or intensities of white. While the whiteness of the wall has an infinity of degrees of intensity of white, the determination of a figure on the white wall is the determination of a body or thing which is white. Such a white body is an extrinsic mode or extensive quantity.

According to Deleuze, the variable degrees of intensity of white are distinct from their corresponding white bodies by a modal distinction: there are degrees which do not confound themselves with the figures, that is, a degree of white is not a white figure or body. Therefore, even though two different degrees are distinguishable, they are not distinguished from one another as different white figures on the wall, as occurs in Hegel’s example, which posits that the difference between different degrees is directly determinable extensively, and therefore numerically. Instead, according to Deleuze, Scotus argues that ‘although figures are distinguished from one another externally, taking account of their common parts, degrees involve a different type of distinction’.⁸⁵ In relation to the white wall example, Deleuze argues that ‘one degree of whiteness is distinguished from another degree of whiteness in a totally different way to that by which a figure on the white wall is distinguished from another figure on the white wall’.⁸⁶ In the example there is ‘an intrinsic distinction’⁸⁷ between degrees which is different to the extrinsic distinction between figures. ‘There is a distinction of degrees which does not confound itself with the distinction of figures’.⁸⁸ An intrinsic distinction is a distinction within the infinite individuating modality itself, according to which actually existing individual essences are distinguishable from one another as different degrees of intensity or intensive quantities. This distinction is different and prior to the extrinsic distinction between actually existing individuals, or extensive quantities.

The immanent existence of singular modal essence

Deleuze finds in the work of Spinoza the development of the same problematic, according to which the distinction between modal essences is different to, and does not confound itself with, the distinction between existing finite modes.⁸⁹ In the Short Treatise, Spinoza writes that: ‘there is no thing in Nature of which there is not … an idea proceeding from it according to its essence and its existence together. … Since the essence, without existence, is conceived as belonging to the meanings of things, the Idea of the essence cannot be considered as something singular. That can only happen when the existence is there together with the essence, and that because then there is an object which did not exist before. E.g., when the whole wall is white, then there is no this or that in … etc.’.⁹⁰

Spinoza often finishes passages in this way, with ellipses, thereby referring the reader to the argument just demonstrated in order to complete the sentence. In this case the last sentence would read: when the whole wall is white, then there is no this or that in [the wall], that is, no thing, of which there is not an idea proceeding from it according to its essence and its existence together.

If a figure is drawn on the white wall, for example the figure of a man, and then next to it another, then two white figures exist on the white wall. This or that figure exists only insofar as each is there together with its respective essence. In this instance, following the text of the Short Treatise, Deleuze contends that Spinoza would argue that each of these essences can be considered to be singular.

In relation to Spinoza’s example Deleuze asks: ‘While nothing existed on the wall, did something exist which would be distinct from the white wall?’⁹¹ Spinoza’s initial response would seem to be that nothing would exist on the white wall before the figures have been traced, since an idea proceeds from both essence and existence together, therefore ‘no idea can exist unless the thing also exists’.⁹² However, as far as Deleuze is concerned, Spinoza has already introduced the concept of singular essence in this passage. Therefore, if the essence belonging to each of the figures respectively is singular, Deleuze argues that ‘it is necessary to distinguish something on the wall without the figures necessarily [having been] traced’,⁹³ that is to say, there is an essence of each figure on the wall, that is, a singular modal essence, which does not confound itself with the existence of each figure, that is, with its corresponding finite modal existence.

Deleuze considers Spinoza’s distinction between a singular modal essence and its finite modal existence to be a modification of the Scotist modal distinction between infinite and finite individuating modalities. In order to demonstrate this, Deleuze argues that, just as Scotus considers the essence of an actually existing individual to exist, so too does Spinoza consider singular essences to actually exist. ‘A modal essence has an existence distinct from that of the corresponding mode’.⁹⁴ To support this claim Deleuze cites Ethics, II, P8C, in which Spinoza writes that ‘singular things are said to exist, not only insofar as they are comprehended in God’s attributes but insofar also as they are said to have duration’. Deleuze understands this to mean that singular things or modes have two simultaneous, and yet distinct, modes of existence, both insofar as they are understood to be in the attribute, as singular modal essences, and insofar as they endure, as finite modal existences. In the seminars on Spinoza, Deleuze distinguishes these two modes of existence as ‘immanent and durational’ existence respectively.⁹⁵

Deleuze considers the Corollary to Ethics, I, P24 to prepare for the introduction of the question of the immanent existence of singular modal essence and therefore for the ‘modal distinction’ between the immanent and durational existence of a mode. Spinoza writes that: ‘God is not only the cause of things beginning to exist, but also of their persevering in existing, or (to use a Scholastic term) God is the cause of the being of things. For – whether the things [NS: produced] exist or not – so long as we attend to their essence, we shall find that it involves neither existence nor duration. So their essence can be the cause neither of their existence nor of their duration, but only God, to whose nature alone it pertains to exist, [can be the cause] (by P14C1)’.⁹⁶ Deleuze considers this passage to present Spinoza’s critique of the Scholastic concept of univocal being. Unlike the univocal being of the Scholastics, which, when considered in its individuating modalities (infinite and finite), rather than in its nature as Being, ceases to be univocal; for Spinoza, substance is not only the ‘being of things’, in the Scholastic sense of the term, but it is also ‘the cause of the being of things’. This extension of the Scholastic concept of univocal being is clearly expressed in the Ethics, I, D1 where substance is said to be cause of all things in the very sense (eo sensu) that it is said to be cause of itself.⁹⁷ Spinozist substance produces things through the very forms that constitute its own essence, that is, through the attributes, such that things in general, or modes, can be understood to be modifications of substance by means of the attributes. As Spinoza says: ‘Particular things are nothing but affections of God’s attributes, or modes by which God’s attributes are expressed in a certain and determinate way’.⁹⁸ Therefore, whether considered in its nature as being, that is, as substance, or in its individuating modalities, that is, as the immanent existence of a modal essence and the durational existence of its corresponding finite mode, Deleuze argues that Spinozist substance remains univocal, and that it is ‘in immanence that univocity finds its distinctly Spinozist formulation’.⁹⁹ According to Deleuze, ‘the concept of univocal Being is perfectly determinate, as what is predicated in one and the same sense of substance in itself, and of modes that are in something else’.¹⁰⁰ It is in the attributes that the univocal character of being is absolutely common to substance and to modes, such that modes implicate the same attributes that constitute the essence of substance and these same attributes contain all of the essences of modes. Modes have an essence and existence of their own, but do not exist, and have no being, apart from the attributes in which they are produced. The immanence of univocal being is therefore affirmed by Spinoza in the attributes. Deleuze argues that ‘it is the idea of immanent cause that takes over, in Spinoza, from univocity’,¹⁰¹ and this immanent cause is expressed in the immanent existence of singular modal essence.

Substance, for Spinoza, is therefore not only the being of things, but the immanent cause of the being of things. It is the immanent cause of ‘the things beginning to exist’, that is, of the immanent existence of singular modal essence, and of ‘their persevering in existence’, that is, of the durational existence of the corresponding finite mode. Spinoza’s critique of the Scholastic concept of univocal being in Ethics, I, P24C can therefore be understood to proceed in the following way. So long as ‘their essence’¹⁰² is considered solely in its nature as Being (as in the univocal being of things in the Scholastic sense of the term), it will not involve the mode of being of the things, in their infinite and finite individuating modalities, that is, it will involve neither their existence nor their duration. So ‘their essence’ (or the being of things in the Scholastic sense) is the cause of neither ‘their existence’, which in Spinozist terms would be the immanent existence of singular modal essence, nor of ‘their duration’, the durational existence of the corresponding finite mode. Deleuze actually retranslates the final sentence of Ethics, I, P24C as follows: ‘So their essence can be the cause neither of its own existence nor of its own duration’,¹⁰³ in order to emphasize that the Scholastic sense of ‘their essence’ is not the cause of the existence and duration of what would be, according to Spinoza, its own individuating modalities. The actual corollary to the argument being that only Spinozist substance, in the very sense that it is cause of itself (causa sui), can be understood to be the immanent cause of all things, even of the singular essences of things.

Although Spinoza affirms that modal essences are distinguishable, insofar as they are singular, the question of exactly ‘how they are distinguishable from one another is not directly addressed in the Ethics’.¹⁰⁴ While admitting that Spinoza ‘does not explicitly develop such a theory’, Deleuze contends that, ‘Spinoza is looking forward towards the idea of a distinction or singularity belonging to modal essences as such’.¹⁰⁵ If modal essences are singular, then the immanent existence of one singular modal essence should be distinguishable from that of another. This is where Deleuze returns to the example of the white wall, introduced by Spinoza in the Short Treatise, and its relation to the Scotist (modal) distinction between the variable degrees of intensity of white and their corresponding white bodies. Just as a variable degree of intensity of white is, according to Scotus, an intrinsic mode, Deleuze maintains that ‘this seems also to be the case for Spinoza: a modal essence is an intrinsic mode or intensive quantity’.¹⁰⁶ And, just as existing things, or white bodies, cannot be distinguished except insofar as their essences are distinct, as variable intensities of white, similarly, Deleuze maintains that for Spinoza, any extrinsic distinction between finite modes seems to presuppose a prior intrinsic distinction between singular modal essences. ‘So’, he continues, ‘a modal essence should be singular in itself, even if the corresponding mode does not exist’.¹⁰⁷

However, what distinguishes Spinoza’s singular modal essences from Scotus’ variable degrees of intensity is the immanent existence of a singular modal essence. Deleuze argues that ‘the status of modal essence [therefore] relates to a strictly Spinozist problem’, which he describes as that ‘of passing from infinite to finite’¹⁰⁸ while maintaining the immanent nature of the relation between substance and modes. This problematic is characterized by Deleuze as the passage from the distinction between the immanent existence of singular modal essences, to that between the immanent existence of a singular modal essence and its corresponding finite mode. It is the Spinozist modification of the modal distinction which characterizes the dynamics of this problematic, and it is to the letter of Spinoza’s texts that Deleuze turns in order to chart its unfolding.

Modal distinction and intensive quantity in Spinoza

In the Short Treatise, Spinoza speaks of substance as ‘God’s infinite power’,¹⁰⁹ and in the Ethics, as ‘God’s supreme power, or infinite nature’, from which ‘infinitely many things in infinitely many modes, that is, all things, have necessarily flowed, or always follow’.¹¹⁰ The reference to ‘all things’ can be understood to imply all finite things. What then constitutes the passage from a concept of the infinite power of substance to a concept of finite things? Following Spinoza, Deleuze argues that substance is ‘the absolutely infinite power’¹¹¹ insofar as it is causa sui, and it is ‘the absolutely infinite power of existing’¹¹² insofar as it is the immanent cause of all finite things. So, the attributes, insofar as they are the infinite component qualities of substance, formally correspond to ‘the absolute power of God’,¹¹³ and the immanent existence of singular modal essences, as collectively contained within an attribute, and insofar as they are expressive of the immanent cause of all things, correspond to the infinite power of existing of substance. What exactly is the nature of this relation between the immanent existence of singular modal essences and the power of existing of substance? According to Deleuze, as infinite qualities, attributes are indivisible: ‘Each attribute is indivisible qua quality. But’, he continues, ‘each attribute–quality has an infinite quantity that … constitutes a … purely modal matter’.¹¹⁴ Any concept of the finite will therefore be neither substantial nor qualitative, but rather modal and quantitative. What distinguishes Deleuze’s argument from that of Hegel in this respect is that Deleuze goes on to argue that each attribute–quality ‘has intensive modal quantity, … which actually divides into an infinity of intrinsic modes’.¹¹⁵ Intensive quantity is therefore divisible intensively rather than extensively, as maintained by Hegel in the example in the Encyclopaedia Logic.¹¹⁶ And, unlike the distinct individuating modalities of the Scholastics, which are not univocal with being, what Deleuze has identified as Spinozist intrinsic modes necessarily participate in God’s power as parts thereof, within the attribute that contains them. ‘These intrinsic modes, contained together as a whole in an attribute are’ what Deleuze calls the ‘pars intensiva’, or ‘intensive parts of the attribute itself’.¹¹⁷

According to Deleuze, modal essences are composed by these intensive parts. Therefore he argues that, for Spinoza, a singular modal essence ‘has an expressive power’¹¹⁸ insofar as it is ‘part of God’s power, as an intensive part, or a degree of power’.¹¹⁹ A singular modal essence is always ‘a certain degree, a certain quantity, of a quality’, and it is precisely in this way that it is ‘a part … of God’s power’.¹²⁰ Thus a singular modal essence is immanently expressive of God’s power ‘according to the degree of power that constitutes its essence’.¹²¹

Deleuze distinguishes his argument from that of Hegel by characterizing singular modal essences as ‘neither qualitative nor extrinsic, but [as rather] quantitative and intrinsic, intensive’.¹²² ‘Modal essences are not distinct in any extrinsic way, being contained in their attribute, but they have nonetheless a type of distinction or singularity proper to them, within the attribute that contains them’.¹²³ He describes this distinction as ‘an internal difference, a difference of intensity’.¹²⁴ There is therefore, according to Deleuze, ‘a distinction of modal essences, both from the attribute that contains them, and one from another’.¹²⁵ Singular modal essences are ‘distinguished from their attribute as intensities of its quality, and from one another as different degrees of intensity’.¹²⁶ So that each finite existing mode ‘express[es] the absolute, according to the intensive quantity that constitutes its essence, according, that is, to the degree of its power’.¹²⁷

An attribute remains a univocal quality, formally distinct from substance, containing all the modal essences that affect it as different degrees of power, or intensity. As contained together in an attribute, as intensities of its quality, ‘modal essences are not distinct from it’.¹²⁸ They are rather ‘inseparable, and are characterized by their total agreement’,¹²⁹ and together ‘form a total system, an actual infinite whole’.¹³⁰ In this way, within an attribute, ‘all essences are involved in the production of each’.¹³¹ According to Spinoza, modal essences are nevertheless singular, and they are ‘distinguished from one another intrinsically’, such that ‘each essence is produced as an irreducible degree, necessarily apprehended as a singular unity’.¹³² However, unlike Hegel, Deleuze considers this intensive infinity to be divisible into an infinity of intrinsic or intensive parts, rather than into extrinsic or extensive parts.

The intensive quantity of an attribute is infinite, not from the number of intensive parts, but rather because ‘it divides into a multitude of parts exceeding any number’.¹³³ In fact Deleuze argues that ‘number does not well explain the nature of such quantity’¹³⁴ at all. Following Spinoza’s Letter XII, which states that ‘Measure, Time, and Number are nothing but Modes of thinking, or rather, of imagining’, Deleuze argues that ‘number is, in fact, only a way of imagining quantity’, it ‘never adequately expresses the nature’ of intensive quantity. Instead, ‘number separates modal essences from one another, and from the principle of their production’, that is, from their immanent expression of substance, ‘and thereby grasps them abstractly’,¹³⁵ or inadequately.

As with the example of a scale of intensity given by Wilson in the scale of hardness in minerals, although the quantitative treatment of different degrees of intensity, or hardness, enables these degrees to be ordered linearly, it does not necessarily entail that they are numerically additive, that is, that the amount by which one intensity differs from another is able to be directly specified numerically or extensively, as Hegel posits in the example of the Encyclopaedia Logic. In a scale of intensity number lacks this quantitative significance, it rather indicates simply the position of any particular degree in a linearly ordered series. Deleuze considers the immanent existence of singular modal essences, as different degrees of power, to be implicated in such a scale of intensity, and, therefore, that the relation between them should be considered to be ‘quantitative, rather than numerical’.¹³⁶ Different degrees of power are therefore distinguished from one another quantitatively and intensively, rather than numerically or extensively.

The intensive parts into which this infinite intensive quantity actually divides, and which it ‘intrinsically includes’, are therefore ‘not separable from one another’ numerically, being neither additive nor distributive.¹³⁷ They are, rather, ‘differentiated’ from one another ‘quantitatively’ as different degrees of power or intensity, or as different intensive quantities. For Deleuze, it is ‘only in and through such quantitative differentiation’ that substance, or ‘God’s power’, actually ‘expresses or explicates itself modally’.¹³⁸

How then, are intensive parts ‘differentiated’ from one another quantitatively, such that they are determined as different degrees of power or intensity, or as different intensive quantities? What are the characteristics of such a ‘quantitative differentiation’? Before responding to these questions, it is necessary to distinguish these intensive parts from the extensive parts that Deleuze introduced by means of the logic of differentiation in the section of chapter 3 entitled ‘Extensive parts: infinite collections of the infinitely small’. By determining the mechanism which distinguishes intensive parts from extensive parts, the other relation introduced in chapter 3 that is yet to be determined in detail, that is, the relation between an extensive quantity and the infinite collection of infinitely small extensive parts of which it is composed, can also be elaborated. The investigation of the mechanism by means of which both of these relations are determined, that is, that between intensive parts and the intensive quantities which they compose, and that between extensive parts and the extensive quantities which they compose, will be elaborated in chapter 5.

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Notes

¹ Deleuze, Expressionism in Philosophy, p. 63.

² Andrew O’Brien, ‘Duns Scotus’ Teaching on the Distinction Between Essence and Existence’, The New Scholasticism, 38 (1964), p. 69. Haecceity is from the Latin haecceitas, meaning ‘thisness’.

³ Deleuze, Expressionism in Philosophy, p. 63.

⁴ Deleuze, Difference and Repetition, p. 39.

⁵ John Duns Scotus, Ordinatio I d.2 q.4, d.8 q.4, in Opera Omnia (t. I–IXX, Vatican City, 1950–93), t. II, 1954, which is the same text but a different edition to that referred to by Deleuze as Opus Oxoniense, I.ii.4, viii.4. Where possible references to both the Vatican edition of the Ordinatio and to the Opus Oxoniense are also given. See Etienne Gilson, Jean Duns Scot (Paris, 1952), chapter 3.

⁶ Allan Wolter, The philosophical theology of John Duns Scotus (Ithaca, 1990), p. 28.

⁷ Ibid.

⁸ Duns Scotus, Ordinatio I d.2 p.2 q.1–4 n.402 (Vatican edn, t. II, paras 355–6).

⁹ Ibid., §356. See A. Wolter, The philosophical theology of John Duns Scotus, p. 33.

¹⁰ Wolter, The philosophical theology of John Duns Scotus, p. 11.

¹¹ Ibid., p. 48.

¹² Deleuze, Expressionism in Philosophy, p. 67.

¹³ Ibid.

¹⁴ Deleuze, Difference and Repetition, p. 39.

¹⁵ Ibid.

¹⁶ Ibid.

¹⁷ Deleuze, Expressionism in Philosophy, p. 67. Additional italics.

¹⁸ Macherey, Introduction à l’Ethique de Spinoza, la première partie, p. 143.

¹⁹ Ibid.

²⁰ Deleuze, Expressionism in Philosophy, p. 67.

²¹ Deleuze, Difference and Repetition, p. 40.

²² Ibid.

²³ Deleuze, Expressionism in Philosophy, p. 67.

²⁴ Ibid.

²⁵ Spinoza, Descartes’ ‘Principles of Philosophy’, Appendix Part 2, Chapter 5, in Spinoza, The Collected Works of Spinoza Volume I, p. 323.

²⁶ See Deleuze, Expressionism in Philosophy, pp. 27–39.

²⁷ Deleuze, Expressionism in Philosophy, p. 66.

²⁸ Ibid., p. 67.

²⁹ Ibid.

³⁰ Ibid., p. 37.

³¹ Ibid.

³² Ibid.

³³ Ibid., p. 65.

³⁴ Ibid., p. 67.

³⁵ Ibid., p. 65.

³⁶ Ibid., p. 64.

³⁷ Ibid., p. 65.

³⁸ Ibid., p. 182.

³⁹ Ibid.

⁴⁰ Ibid., p. 39.

⁴¹ O’Brien, p. 66.

⁴² Ibid., p. 66. See E. Gilson, Jean Duns Scot, p. 244 n. 1.

⁴³ Ibid., p. 66.

⁴⁴ Ibid.

⁴⁵ Ibid., p. 71.

⁴⁶ Deleuze, Expressionism in Philosophy, p. 359 n. 19. See also Gilson, Jean Duns Scot, pp. 89, 629.

⁴⁷ Ibid.

⁴⁸ Deleuze, Difference and Repetition, p. 39.

⁴⁹ Deleuze, Expressionism in Philosophy, p. 359 n. 19.

⁵⁰ O’Brien, p. 66.

⁵¹ Ibid., p. 75.

⁵² In the notes to his translation of Aristotle’s Nicomachean Ethics, Terence Irwin offers an explanation of the Aristotelian notion of form as the cause, or reason for being, of things: ‘Here form is closely associated with essence and definition; something achieves its form to the extent that it acquires the character that makes it the kind of thing that it is’ (Aristotle, Nicomachean Ethics, trans. T. Irwin (Indianapolis, 1985), p. 375, n1174a19). See also Aristotle’s Physics, trans. R. Waterfield, intro. D. Bostock (Oxford; New York, 1996), ii, 3.

⁵³ O’Brien, p. 76.

⁵⁴ Boyer, The history of the calculus and its conceptual development, p. 73.

⁵⁵ Pierre Duhem, Le systeme du monde: histoire des doctrines cosmologiques de Platon à Copernic, (7 vols, Paris, Hermann, 1954), vol. 7, p. 483.

⁵⁶ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁵⁷ Richard Cross, The physics of Duns Scotus: the scientific context of a theological vision (Oxford, 1998), p. 170.

⁵⁸ Ibid., p. 147. See Duns Scotus, Ordinatio II d.3 q.1, paras 5–6.

⁵⁹ Ibid., p. 171.

⁶⁰ See also Jean Duns Scotus, Quaestiones super libros Metaphysicorum Aristotelis, VIII q.2–3, VI. 43–245, in Opera Philosophica (t. I–IV, St. Bonaventure, 1997), t. IV, paras 426–88.

⁶¹ Wolter, The philosophical theology of John Duns Scotus, p. 37.

⁶² Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, 1961), p. 206.

⁶³ See Ibid., p. 212.

⁶⁴ Hegel, Hegel’s Science of Logic, p. 185.

⁶⁵ Ibid., p. 202.

⁶⁶ Hegel, Hegel’s Science of Logic, p. 338.

⁶⁷ Ibid., p. 218.

⁶⁸ Ibid.

⁶⁹ Ibid.

⁷⁰ Ibid., p. 220.

⁷¹ For further analysis of the manner by means of which intensive and extensive quantum are determined as identically quantitative in Hegel see Simont, p. 278.

⁷² Hegel, Encyclopaedia Logic, pp. 164–5.

⁷³ Hegel, Hegel’s Science of Logic, p. 220. Hegel’s ‘analysis of “intensive quantum”, or “degree”, … prepares [for the subsumption] of quality and quantity as moments of “measure”’ (Ramond, qualité et quantité dans la philosophie de Spinoza, p. 201). Although there is a stage further than this, ‘in which measure takes the form of a quantitative relationship between two qualities’ (Paul Johnson, The critique of thought: a re-examination of Hegel’s Science of Logic (Aldershot, 1988), p. 71), what is determined at this stage is the ‘quality of distinct things’ (p. 78), that is, how already determined extensive quantities are distinguished by their primary and secondary qualities. According to this dialectic of measure, intensity or intensive quantity has already been subordinated to these qualities. In Difference and Repetition, Deleuze argues that ‘intensity itself is subordinated to the qualities which fill extensity (primary physical qualities or qualitas, and secondary perceptible qualities or quale). … we know intensity only as already developed within an extensity, and as covered over by qualities. Whence our tendency to consider intensive quantity as a badly grounded empirical concept, an impure mixture of a sensible quality and extensity, or even of a physical quality and an extensive quantity’ (p. 223). The problem of the subordination of intensity to the qualities that fill extensity is further elaborated in the section of chapter 10 entitled ‘The concept of intensity from the point of view of the expression itself’, p. 240.

⁷⁴ Curtis Wilson, William Heytesbury: Medieval Logic and the Rise of Mathematical Physics (Madison, 1956), p. 145.

⁷⁵ Ibid., p. 145.

⁷⁶ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁷⁷ Cross, The physics of Duns Scotus: the scientific context of a theological vision, p. 192.

⁷⁸ Boyer, The history of the calculus and its conceptual development, p. 73.

⁷⁹ Ibid.

⁸⁰ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁸¹ Ibid., 20 January 1981.

⁸² In response to the question of whether or not Spinoza read Scotus, Deleuze replies that ‘this is of no interest, because I am not sure at all that it is Scotus who invented this example! It is an example which can be found throughout the Middle Ages’ (Deleuze, ‘sur Spinoza’, 10 March 1981). As far as Deleuze is concerned, Scotus is the figure most representative of the Scholastic tradition, and therefore of the Scholastic concepts to which he is referring. For an elaboration of the relation between Scotus and the Scholastics concerning the concept of intensive quantity, see Duhem, Le systeme du monde, vol. 7, pp. 462–534.

⁸³ Deleuze, Expressionism in Philosophy, p. 196. ‘Whiteness has variable intensities’ (Duns Scotus, Ordinatio I d.3; Vatican edn, t. III, para. 58). ‘Intensity adds nothing extensive to the essence of the considered whiteness’ (Ibid.; para. 55). ‘A certain degree of intensity is the exclusive mode under which is seen this whiteness’ (Ibid.). ‘Whiteness’ is distinct from ‘a particular being which is white’ (Ibid.; para. 161). See also Opus Oxoniense I.3.i, ii.4 n. 17.

⁸⁴ Cross, The physics of Duns Scotus: the scientific context of a theological vision, p. 170. See the section of this chapter entitled ‘Modus intrinsecus: the quantitative treatment of variation in quality, or intensity’, p. 103.

⁸⁵ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁸⁶ Ibid.

⁸⁷ Ibid.

⁸⁸ Ibid.

⁸⁹ In making this comparison between Spinoza and Duns Scotus, Deleuze states that it ‘bears only on the theme of intensive quantities or degrees’ (Deleuze, Expressionism in Philosophy, p. 379 n. 14).

⁹⁰ Spinoza, Short Treatise, Part 2, Chapter 20, note c, lines 7–9/13, 1660–62. Additional italics.

⁹¹ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁹² Spinoza, Short Treatise, Part 2, Chapter 20, n. c, line 11/13.

⁹³ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁹⁴ Deleuze, Expressionism in Philosophy, p. 192.

⁹⁵ Deleuze, ‘sur Spinoza’, 10 March 1981.

⁹⁶ Spinoza, Ethics, I, P24C. Additional italics.

⁹⁷ Deleuze, Expressionism in Philosophy, p. 67.

⁹⁸ Spinoza, Ethics, I, P25C.

⁹⁹ Deleuze, Expressionism in Philosophy, p. 67.

¹⁰⁰ Ibid.

¹⁰¹ Ibid.

¹⁰² Spinoza, Ethics, I, P24C. Deleuze considers Spinoza’s use of ‘their essence’ to refer to the Scholastic concept of the being (univocal) of things, it is therefore singular. It is here that Deleuze locates the connection between Spinoza and Scotus. This paragraph repeats the argument of the first few pages of the chapter: the Scotus distinction between the being (univocal) of things and their mode of being (not univocal). Immanence in Spinoza overcomes this distinction in univocity, or so Deleuze contends, and he purports to find that argument in the Ethics, I, P24C.

¹⁰³ Deleuze, Expressionism in Philosophy, p. 378 n. 8.

¹⁰⁴ Deleuze, ‘sur Spinoza’, 10 March 1981.

¹⁰⁵ Deleuze, Expressionism in Philosophy, p. 197.

¹⁰⁶ Ibid., p. 196.

¹⁰⁷ Ibid.

¹⁰⁸ Ibid., p. 198.

¹⁰⁹ Spinoza, Short Treatise, Part 1, Chapter 2.

¹¹⁰ Spinoza, Ethics, I, P17S1.

¹¹¹ Deleuze, Expressionism in Philosophy, p. 198.

¹¹² Spinoza, Ethics, I, P11S.

¹¹³ Deleuze, Expressionism in Philosophy, p. 183.

¹¹⁴ Ibid., p. 191.

¹¹⁵ Ibid., p. 198.

¹¹⁶ See the section of this chapter entitled ‘The Hegelian distinction between intensive and extensive quantum’, p. 104. See also the Science of Logic where Hegel writes that ‘the ascent and descent in the scale of degrees is a continuous progress, a flux, which is an uninterrupted, indivisible alteration’ (Hegel, Hegel’s Science of Logic, p. 219).

¹¹⁷ Deleuze, Expressionism in Philosophy, p. 198. Deleuze argues that ‘the notion of ‘intensity’ appears in the Ethics (EIVP9–12)’ (p. 417), however, ‘Spinoza himself never speaks directly of “parts” as “intensive” or “extensive” (and it is not clear just what the Latin adjectives would be). He only qualifies different orders of composition contextually, by using the genitive of the particular “whole” in question: pars extensionis, pars substantiae extensae, pars naturae, pars Dei … pars mentis, pars imaginationis, pars totius universi and so on (and extrinsecus qualifies only denominatio and ratio)’ (p. 417 n. 4b). So, although the expression ‘intensive parts’ to designate modal essence, is not used by Spinoza, Deleuze defends his use of the expression in his notes to the English translation of Spinoza et le problème de l’Expression, when he writes that ‘It is quite true that one doesn’t, strictly speaking, find intensity in Spinoza. But potentia and vis cannot be understood in terms of extension. And potentia, being essentially variable, showing increase and diminution, having degrees in relation to finite modes, is an intensity. If Spinoza doesn’t use this word, current up to the time of Descartes, I imagine this is because he doesn’t want to appear to be returning to a precartesian physics. Leibniz is less concerned by such worries. And does one not find in Spinoza the expression “pars potentia divinae”?’ (p. 418).

¹¹⁸ Deleuze, Expressionism in Philosophy, p. 198.

¹¹⁹ Ibid., p. 199.

¹²⁰ Ibid., p. 183.

¹²¹ Ibid., p. 199. Spinoza also doesn’t speak directly of modal essences as ‘degrees of power’, however, support for Deleuze’s use of the expression can be found in the Ethics when Spinoza speaks of ‘degrees’. In Ethics, II, P13S, Spinoza writes that ‘the things we have shown so far are completely general and do not pertain more to man than to other Individuals, all of which, though in different degrees, are nevertheless animate omina … diversis gradibus animata … sunt’. Deleuze can therefore be understood to employ the expression ‘degrees of power’ to define ‘modal essence’ with a certain legitimacy.

¹²² Deleuze, Expressionism in Philosophy, p. 197.

¹²³ Ibid.

¹²⁴ Ibid.

¹²⁵ Ibid.

¹²⁶ Ibid.

¹²⁷ Ibid.

¹²⁸ Ibid., p. 195.

¹²⁹ Ibid., p. 198.

¹³⁰ Ibid., p. 194.

¹³¹ Ibid., p. 198.

¹³² Ibid.

¹³³ Ibid., p. 203.

¹³⁴ Ibid., p. 183.

¹³⁵ Ibid., p. 203.

¹³⁶ Ibid.

¹³⁷ Ibid.

¹³⁸ Ibid., p. 183.

Chapter 5

The distinction between intensive and extensive parts

The relation between the successive terms of a power series expansion

The Deleuzian concept of intensive quantity is explicated in this chapter as determined according to the logic of differentiation. It is according to this logic that intensive parts are differentiated from one another such that they are each determined as different degrees of power or intensity, each of which thereby constitutes a different intensive quantity. The mechanics of the logic of different/ciation is then examined in relation to the concept of individuality in order to determine the extent to which this logic is actually able to be mapped onto Spinoza’s theory of relations, the implications of which is investigated in relation to the work of other scholars working in the field of contemporary Spinoza studies in the chapters which follow. Intensive and extensive quantities are determined in relation to Spinoza’s theory of relations respectively as the singular modal essences complicated by an attribute, and the finite existing modes explicated by an attribute. Each singular modal essence that is complicated in an attribute is an intensive part of different degree of power or intensity, or a different intensive quantity. To every different intensive part, that is, to every intensive quantity or singular modal essence, there corresponds an infinite collection of extensive parts, which is explicated by an attribute as an extensive quantity or finite existing mode. The mechanics of both the complication of singular modal essence and the explication of finite modal existence, it is argued, are determined according to the same logic of different/ciation. However in order to determine the convergence of the logic of complication and explication, it is necessary to elaborate in detail how intensive parts are differentiated from one another quantitatively, and to distinguish these intensive parts from the infinite collections of extensive parts which correspond to them.

According to the logic of differentiation, the infinitely small differentials of Letter XII, which correspond to the most simple bodies of Spinoza’s Ethics, and which Deleuze characterizes as ‘extensive parts’, are only determined by the differential relations which they effect with each other. When considering extensive infinity, Deleuze argues that ‘it is of course divisible into the extrinsic parts that compose existing things’, such as white bodies or finite existing modes. ‘But’, he continues, ‘these extrinsic parts always come in infinite collections’.¹ While the principles of determinability and reciprocal determination correspond to the determination of extensive parts, it is the principle of complete determination which corresponds to the effective determination of infinite collections of extensive parts. So, according to Deleuze, extensive parts exist collectively, that is to say they have no distributive existence but rather enter necessarily into infinite collections, as infinite collections of infinitely small extensive parts. What are these ‘infinite collections’ into which the extensive parts necessarily enter? According to the logic of differentiation, an infinite collection of infinitely small extensive parts is expressed by a power series expansion.²

A power series expansion is an infinite series, which, insofar as it converges with a local function according to the principle of complete determination, is constitutive of an actual infinite. Now, because it actually ‘divides into a multitude of parts exceeding any number’,³ a power series is not a sum in numerical terms, its parts being neither additive nor distributive. The parts into which this actual infinite is divisible are rather the successive terms of the power series, each of which is determined by the repeated differentiation of a differential relation. Each part of the series therefore consists of this differential relation, differentiated to various successive degrees, and a corresponding variable, whose exponent is the same, or has been raised to the same degree or power.⁴ So each part of the power series, each successive term, implicates those extensive parts determined by the included differential relation. What then is the nature of this implication? What is the relation between each successive part of the power series and the extensive parts which they implicate? According to Deleuze, each of these parts, both successive and extensive, are distinguished as different kinds of parts. There are ‘extensive parts’ which are determined in and by the differential relation, which is itself only a component of each ‘successive part’ of the series. If the successive parts of a power series implicate extensive parts but are not themselves extensive parts, what kind of parts are they? Insofar as a power series expansion is actually infinite, it corresponds to the actually infinite quantity of intensive parts into which the intensive quantity of an attribute is divisible. So, the successive terms of a power series expansion correspond to the intensive parts of an attribute. Each successive term of a power series can therefore be understood to be an intensive part. A power series therefore involves an infinity of intensive parts as its successive terms. Just as extensive parts cannot exist independently of a differential relation, intensive parts cannot exist independently of an actually infinite power series expansion in which they are complicated as its successive terms. Each intensive part of such a series implicates a differential relation which has been repeatedly differentiated in each successive term of the series. It is this differential relation which is determinative of the extensive parts generated by the series. Therefore, to each intensive part of the series there corresponds those extensive parts generated by the differential relation which it implicates.

It would therefore seem that intensive and extensive parts correspond to one another within each successive term of a power series. However, according to Deleuze, ‘extensive parts and … intensive parts … in no way correspond term for term’.⁵ On the contrary, Deleuze maintains that to every intensive part there corresponds ‘an infinity of extensive parts’.⁶ The extensive parts determined by the differential relation implicated in each intensive part are infinitely small extensive parts, thus constituting what is for Deleuze a ‘lesser’ infinity of extensive parts. Insofar as a power series expansion converges with a local function, it is actually infinite, and it divides into an infinite quantity of successive terms or intensive parts. The quantity of infinitely small extensive parts corresponding to such an infinite quantity of intensive parts constitutes a ‘greater infinity’ of extensive parts. To each intensive part of a series there corresponds a lesser infinity, and to each power series expansion there corresponds an infinite collection of lesser infinities which together constitute a ‘greater’ infinity. So when Deleuze writes that ‘extensive infinity is thus an infinity necessarily conceived as greater or less’,⁷ he is arguing that extensive infinity, or an infinity of extensive parts, is always constituted by infinite collections (greater infinities) of the infinitely small (lesser infinities). The actual infinite of a power series expansion, although determinative of greater and lesser infinities of extensive parts, is therefore rather an intensive infinite, composed of, or divisible into, an actually infinite quantity, or collection, of intensive parts.

As a power series expands, the exponent, or power, of each successive term increases in degree, and the series increasingly approximates or converges with its function, which in this case would be an attribute. Insofar as each of the terms of the series corresponds to one of the actually infinite intensive parts into which the intensive quantity of an attribute is divisible, there are an actually infinite quantity of terms, the degrees of which range from zero to infinity. Each intensive part, as a successive term of a power series expansion, therefore has a degree, or a degree of power, and the power series expansion can be expressed as the infinite collection of these different degrees of power. So, the intensive parts of a power series expansion are determined collectively as the different degrees of power or intensity, or the different intensive quantities, that constitute the intensive quantity of an attribute. These intensive parts can therefore be understood to be differentiated from one another quantitatively as the successive terms of a power series expansion. Quantitative differentiation is therefore the distinction between the different degrees of power, or the different intensive quantities, of an attribute in the form of a power series expansion. The concept of a power series expansion, whose terms are quantitatively, rather than numerically, differentiated, can be used as a Deleuzian model for the scale of intensity. As a linearly ordered series, a power series expansion is composed by successive terms, each of which has an increasing degree of power. The actually infinite quantity, or collection, of terms of the series are quantitatively differentiated as different intensive parts, which correspond to the different intensive quantities of an attribute as different degrees of power.

Deleuze maintains that within such a power series expansion, ‘all of the intensive parts of the series are comprised in the production of each’.⁸ Not only insofar as they are together constitutive of an actual infinite, but also insofar as they are determined collectively as the infinite quantity of successive parts of such a series. In Expressionism in Philosophy, he argues that ‘modal essences are … parts of an infinite series’,⁹ and that ‘an attribute contains, that is, complicates, the essences of all its modes as the infinite series of degrees corresponding to its intensive quantity’.¹⁰ Insofar as the infinite series of degrees, as a power series expansion, corresponds directly with the intensive quantity of an attribute, the infinite collection of intensive parts, as different degrees of power, actually correspond to the modal essences that the attribute complicates. Each of these modal essences is an intensive part which has a different degree of power, and thereby constitutes a different intensive quantity. As the actually infinite quantity of intensive parts of a power series expansion, modal essences ‘do not form a hierarchical system in which the less powerful depend on the more powerful, but an actually infinite collection, a system of mutual implications, in which each essence conforms with all of the others, and in which all essences are involved in the production of each’.¹¹ This system of mutual implication, or complication, as Deleuze predominantly refers to it,¹² which ‘applies not merely to the lowest degree of essence, but to the highest also, since the series is actually infinite’,¹³ is characteristic of quantitative differentiation. According to this system, modal essences are complicated within an attribute as the quantitatively differentiated intensive parts, or degrees of power, of an actually infinite power series expansion. if modal essences are quantitatively differentiated from one another within an attribute as the different intensive parts, or degrees of power, of a power series expansion, how are they then distinguished as singular in relation to their corresponding finite existing modes? What actually constitutes the modal distinction between a singular modal essence and its corresponding finite existing modes?

The passage from absolutely infinite substance to ‘the individuation of the finite’, in the form of finite modal existence, proceeds, for Deleuze, ‘from an infinite quality to a corresponding quantity, which divides into irreducible intrinsic or intensive parts’.¹⁴ To this infinite quantity of intensive parts there corresponds an extensive infinity which consists of the infinite collections of infinitely small extensive parts. The divisibility of intensive quantity is characteristic of the system of complication of modal essences within an attribute, and is determined by means of quantitative differentiation. It is only when modes are determined extrinsically, that is, ‘outside the attribute’,¹⁵ as finite existing modes, that they ‘cease to exist [solely] in the complicated form that they have while their essences are contained … in their attribute’.¹⁶ It is only then that the passage from infinite to finite is actually effected. So to each modal essence that is complicated within an attribute, there corresponds a finite existing mode outside the attribute. However, according to Deleuze, ‘not only Spinoza’s physics, but Spinozism as a whole, becomes unintelligible if one doesn’t distinguish what belongs to essences, what belongs to existences, and the correspondence between them, which is in no way term for term’.¹⁷ According to the system of complication, every modal essence is an intensive part, or degree of power or intensity, to which ‘there correspond an infinity of extensive parts’.¹⁸ It is this infinity of extensive parts that is modified into an extensive quantity, or finite existing mode. The modification of infinite collections of extensive parts into finite existing modes is determined extrinsically, outside the attribute. These finite existing modes, as extensive quantities, are together determinative of extensive quantity. Addressing the question of the extrinsic nature of finite existing modes, Deleuze argues that ‘extensive quantity belongs to an attribute no less than intensive quantity, but … it is a strictly modal form of exteriority. It presents existing modes as external to the attribute, and as external to one another. It is nonetheless contained, along with all existing modes, in the attribute it modifies’.¹⁹ Deleuze describes the extrinsic determination of finite modal existence, or extensive quantity, as the ‘explication’ of an attribute.

The double movement of complication and explication

The modal expression of substance, or ‘modal expression as a whole’, is, according to Deleuze, ‘constituted by this double movement of complication and explication’.²⁰ Modal essences are expressive of substance insofar as they are determined by the complication of an attribute, and finite modal existence is expressive of substance insofar as it constitutes the explication of an attribute. ‘An existing mode is thus no less expressive than its essence, but is so in another manner. An attribute’, Deleuze argues, ‘no longer expresses itself only in the modal essences that it complicates or contains according to their degrees of power; it also expresses itself in existing modes that explicate it in’ what he describes as ‘a certain and determinate manner, that is, according to the relations corresponding to their essences’.²¹ The ‘relations’ which correspond to modal essences are those differential relations belonging to each intensive part of the power series expansion constitutive of an attribute. Within each intensive part, there is a differential relation which explicates that intensive part as a finite existing mode. Each intensive part is therefore explicated as a finite existing mode by the differential relation specific to it. According to Deleuze, ‘each existing mode explicates the attribute in the relation that characterizes it, in a way extrinsically distinct from other ways in other relations’.²² Each of the quantitatively differentiated intensive parts of an attribute is explicated as a finite existing mode, in such a way that finite existing modes remain extrinsically distinct from one another. The way in which finite existing modes are explicated is such that they are not only external to the attribute, but also external to one another. Each intensive part is explicated ‘in a certain and determinate way’²³ as an extrinsically distinct finite existing mode by the differential relation specific to it.

The specific characteristics of the ‘certain and determinate way’ in which finite existing modes are explicated can be determined from the differential point of view of the infinitesimal calculus. According to the logic of differenciation, the differential relation specific to an intensive part is a meromorphic function, or composite relation. Both the denominator and numerator of a composite relation are discontinuous local functions, each determined by means of the logic of differentiation. According to the principles of this logic, infinite collections of infinitely small extensive parts are generated by power series expansions which converge with a branch of a local function. The poles or limits of each local function are the infinite branches of each curve, which are determined by the process of analytic continuity. It is these infinite branches which characterize the discontinuous nature of local functions. The complete determination of a local function, which is the determination of the specific qualitative nature of the infinite branches of its poles, remains undetermined by the process of differentiation.

The specific qualitative nature of the curve of any function is the gradient of the line tangent to the curve at a point, which is determined by the values of the differential relation of the function at that point.²⁴ The specific qualitative nature of the curve of a function is increasingly specified as the differential relation is repeatedly differentiated in the form of a power series expansion. However, in the case of the poles of local functions, the value of the differential relation at these points remains undetermined, the poles rather designating the points of discontinuity, or the limits, of a local function. Deleuze considers each point of discontinuity of a local function to be ‘a pure element of potentiality’.²⁵ It is only insofar as a local function is engaged in a composite relation, according to the logic of differenciation, that the specific qualitative nature of the infinite branches of its poles are determinable. When determined in reciprocal relation to one another in the form of a composite relation, Deleuze considers the poles of local functions to be ‘pre-individual singularities’,²⁶ which are together constitutive of ‘an individualizing’ or ‘essential singularity’. The reciprocal determination of the poles of local functions creates an essential singularity. The essential singularity is the point around which the infinite branches of the poles of local functions are determined in the form of the curves of the infinite branches of the composite function. The curves of a composite function are generated by the power series expansion of a composite relation. The determination of a composite function requires the creation of an essential singularity in relation to the poles of the local functions of which the composite relation is composed. The type of essential singularity (a saddle point, node, focal point, or centre) is determined by the form of the curves of the local functions constitutive of the composite relation. The specific qualitative nature of the infinite branches of local functions are therefore only determined in the infinite branches of a composite function.

Just as a differential relation is the function of a line tangent to a curve, whose values determine the specific qualitative nature or gradient of that line, a composite relation is represented by a potential function, whose domain is determined by the poles of the local functions of which it is constituted, which are points of discontinuity or pure elements of potentiality; otherwise known as ‘accumulation points’. The values of the composite relation determine the specific qualitative nature of the line of discontinuity which stretches across the domain of the potential function.

Deleuze considers the potential function to be ‘a genuine cut [coupure], a border between the changeable and the unchangeable within the [local] function itself’.²⁷ The composite relation is therefore distinguished from the local functions of which it is composed by the potential function. The discontinuous local functions remain unchangeable in relation to the infinite branches of their respective poles, or their limits. Whereas, when determined in the form of a composite relation, these limits, as pure elements of potentiality, have the potentiality, or capacity, to generate change. In the form of a potential function, the line of discontinuity between these points is representative of a threshold beyond which a composite function can be generated. The potential function is actualized in the composite function when the variable jumps across the threshold, from one pole to the other. As a composite relation is expanded in a power series, there is an increase in the degree of power (exponent) of each term, and therefore of the series itself. This increase in degree of power is represented by an accumulation in the potential of the variable to jump across the threshold of the potential function, following a trajectory which converges with the curves of a composite function. The type of trajectory followed by the variable is determined by the type of essential singularity created by the composite relation. As the power series expands, the divergent lines of the potential trajectory of the variable, across the threshold, increasingly converge with the curves of the composite function, and the specific qualitative nature of each of the local functions is thereby increasingly determined.

The differenciation of the differenciated

The degree of power of the power series expansion of a composite relation varies between a minimum and a maximum. The minimum degree of power corresponds with the simplest term of the series, or that determined by the composite relation itself, which is represented by the variable with the least potential to jump the threshold of the potential function along a trajectory which converges with the curves of the composite function. The maximum degree of power is determined by the convergence of the power series expansion with the composite function, or the actualization of the potential function in the infinite branches of the composite function. The variation in degree of power corresponds with the variation of the specific qualitative nature of the infinite branches of the local functions, which are represented by the divergent lines of the different potential trajectories of the variables across the threshold. The potential trajectories range from the straight line of the potential function to the curves of the infinite branches of the composite function. The determination of the specific qualitative nature of the infinite branches of the poles of the local functions corresponds with the actualization of the potential function in the curves of the infinite branches of the composite function.

Now, just as in the case of the specific qualitative nature of the infinite branches of the poles of local functions, the specific qualitative nature of the infinite branches of the composite function is only actualized insofar as the composite function is itself implicated in a composite relation, that is, the specific qualitative nature of the infinite branches of a composite function, whose form is determined by the type of essential singularity proper to it, is only determinable in reciprocal relation to that of another composite function, whose form is again determined by the type of essential singularity proper to it. The logic of differenciation is in this way compounded, such that ‘there is a differenciation of differenciation which integrates and welds together the differenciated’.²⁸ A differenciated composite function is therefore simultaneously ‘a local integration’ when it connects with another differenciated composite function, or local integration, in what is characterized as a ‘global integration’.²⁹ The power series of each of these differenciated composite functions, or local integrations, together form a meromorphic function, or a composite relation, the expansion of which generates, or is actualized in, a global integration. The specific qualitative nature of the infinite branches of this global integration is again however only determined by means of the logic of differenciation, insofar as it becomes a new local integration in reciprocal relation to another local integration, which together constitute a new global integration, and so on. The specific qualitative nature of any local integration is therefore only determined within the context of the global integrations in which it is implicated. Each new global integration thereby incorporates an increasingly composite, multipli-differenciated assemblage of local integrations.

The essential singularities created by the composite relation of each global integration are created within the quantitatively differentiated structure of an intensive part, of which each composite relation is the component differential relation. Essential singularities are quantitatively differentiated from one another insofar as they are components of the different intensive parts of an attribute. The singular modal essence that corresponds to an intensive part of an attribute is therefore expressed as the essential singularity created by the composite relation of the global integration belonging to that intensive part. Each quantitatively differentiated intensive part involves a global integration, the composite relation of which creates an essential singularity that is complicated within an attribute as a singular modal essence.

While the complication of an attribute is determined by means of quantitative differentiation, the explication of an attribute is determined from the differential point of view of the infinitesimal calculus according to the logic of different/ciation. It is by means of the pure relations of the logic of differentiation that discontinuous local functions are generated as infinite collections of infinitely small extensive parts, the poles or limits of which are characterized as ‘preindividual singularities’.³⁰ It is by means of the principles of the logic of differenciation that these pre-individual singularities together create individualizing or essential singularities.

The composite relation of each global integration brings together, or explicates, the infinite collections of extensive parts belonging to its discontinuous local functions, or local integrations, to generate a finite existing mode. The composite relation of each global integration simultaneously creates an essential singularity which is complicated within an attribute as an intensive part or singular modal essence; and generates an infinity of extensive parts which is explicated by an attribute as a finite existing mode. Global integrations are thereby complicated as quantitatively differentiated singular modal essences, and explicated as extrinsically different/ciated finite existing modes.

The individuation of the finite

The individuation of the finite, which is characterized, for Deleuze, by Spinoza’s theory of relations, consists of three different components, or levels. It involves ‘a singular essence, which is a degree of power or intensity; a particular existence, always composed of an infinity of extensive parts; and an individual form that is the characteristic or expressive relation which corresponds eternally to the mode’s essence, but through which also an infinity of parts are temporarily related to that essence’.³¹ The ‘individual form’ that Deleuze refers to in this passage is characterized by a global integration that not only corresponds to a mode’s ‘singular essence’, insofar as its composite relation creates the essential singularity expressive of the quantitatively differentiated singular modal essence which is complicated in an attribute as ‘a degree of power or intensity’, but is simultaneously that through which a ‘particular existence’ is generated by bringing together, or explicating, the infinity of extensive parts belonging to the local integrations of which it is constituted, as a finite existing mode or individual. The individuation of each finite existing mode is therefore expressed in an individual form as a global integration which is determinative of both the creation of a singular essence and the generation of a particular existence. And, it is by means of the individual form of the global integration that the infinity of extensive parts of a particular existence are related to a singular essence. ‘The individual’, Deleuze argues, ‘is a singular essence’, insofar as ‘each singular essence expresses itself in the characteristic relations of the differential relation type, and under these differential relations, the infinite collections of the infinitely small belong to the individual’.³² The characteristic relations of the differential relation type are the global integrations of the individual form, which are determined by the logic of differenciation. Each quantitatively differentiated singular modal essence expresses itself in a global integration, and it is under this global integration that an infinity of extensive parts belong to the individual.

According to Deleuze, a singular modal essence expresses itself in a global integration only insofar as ‘the essence is a degree of power’.³³ Each singular modal essence is quantitatively differentiated as a degree of power or intensity, belonging to a power series expansion which is constitutive of an attribute. He argues that ‘this degree expresses itself in a relation; and the relation subsumes an infinity of parts’.³⁴ That is to say that this degree of power expresses itself in a global integration, and it is this global integration which subsumes an infinity of extensive parts which constitute the particular existence of the corresponding finite existing mode.

Although a singular modal essence expresses itself, in this way, in a global integration, Deleuze emphasizes that ‘it is not the essence that determines an infinity of extensive parts to enter into that relation’.³⁵ Instead, he maintains that ‘there is a reality of the essence which expresses itself in the relation independently of the actually given parts’.³⁶ This reality is the specific qualitative nature of the infinite branches of a global integration, whose form is determined by the type of essential singularity created by its composite relation. The specific qualitative nature determines how the infinite collections of extensive parts, belonging to the local integrations of which it is constituted, are brought together, or explicated, by the global integration as a finite existing mode. According to Deleuze, this explains Spinoza’s formulation in Letter XXXII, according to which the infinite collections of extensive parts belonging to each of the reciprocally determined local integrations, ‘being under the domination of one and the same nature’,³⁷ that is, the specific qualitative nature of the infinite branches of the global integration, are ‘forced, as [this] nature … demands, to adapt themselves to one another, so as to harmonize with one another in a certain way’.³⁸ It is the specific qualitative nature of the infinite branches of the global integration, the form of which is determined by the type of essential singularity created by its composite relation, which forces the infinity of extensive parts to enter into that relation. However, Deleuze goes on to argue that ‘a mere nature does not establish its dominance by itself, or itself force the parts to adapt themselves to one another so as to conform with the relation in which it expresses itself’.³⁹ The specific qualitative nature of the infinite branches of one global integration is rather only determinable in reciprocal relation to that of another global integration, that is, when the global integration is itself further differenciated.

Each of these global integrations is therefore reciprocally determined as a local integration, which together form a new, more composite, global integration. Each global integration only being conceivable as the reciprocal relation between two local integrations, and each local integration only being conceivable in reciprocal relation with another to form a global integration. So the specific qualitative nature of the infinite branches of a global integration does not itself force the infinite collections of extensive parts, belonging to the local integrations of which it is constituted, to adapt themselves to one another according to its relation, but rather, it is only insofar as the specific qualitative nature of the infinite branches of the global integration is determined that the infinite collections of extensive parts belonging to each of its local integrations are forced to adapt themselves to one another. And the specific qualitative nature of the infinite branches of the global integration is only determined insofar as it is further differenciated as a local integration, in reciprocal relation to another local integration.

The infinite collections of extensive parts belonging to the local integrations of one global integration are only brought together as the infinity of extensive parts of that global integration insofar as the specific qualitative nature of the infinite branches of this global integration is determined, and this is only determinable insofar as this global integration is in reciprocal relation with another to form a more composite global integration. It is only then that the ‘infinite collections of extensive parts’ are determined as an ‘infinity of extensive parts’, belonging to the global integration, in the individuated form of a particular existence. It is in this way that ‘extensive parts determine one another from outside and ad infinitum; they have none but an extrinsic determination’.⁴⁰

So, according to Deleuze, ‘a mode comes into existence, not by virtue of its essence, but by virtue of purely mechanical laws which determine an infinity of some extensive parts or other to enter into a precise given relation, in which its essence expresses itself’.⁴¹ The particular existence of a finite existing mode comes about, not by virtue of its singular modal essence, but by virtue of the purely mechanical laws of the logic of different/ciation. The infinite collections of extensive parts belonging to reciprocally determined local integrations are determined, or forced, by means of the differenciation of the differenciated, to enter into a precise global integration, in which a singular modal essence expresses itself. The form of the infinite branches of this global integration is determined by the type of essential singularity created by its composite relation. The specific qualitative nature of the infinite branches of this global integration is only determined insofar as this global integration is determined as a local integration in reciprocal relation to another local integration to form a new, more composite, multipli-differenciated, global integration. However, it is only by determining the specific qualitative nature of the infinite branches of a global integration that the infinite collections of extensive parts are determined to enter into the precise global integration of an individual form; as the infinity of extensive parts whose particular existence is that of a finite existing mode. It is therefore only by determining the specific qualitative nature of the infinite branches of a global integration that the finitude of an existing mode is determined.

From the differenciations of the differentiated to the actually infinitely composite differenciations of the differenciated

The specific qualitative nature of the infinite branches of any global integration, the form of which is determined by the type of essential singularity created by its composite relation, or the quotient of the two local integrations that it incorporates or differenciates, is determinable solely in reciprocal relation to that of another global integration, in the form of the infinite branches of a more composite global integration. The determination of the specific qualitative nature of the infinite branches of a global integration corresponds with the actualization of the potential function, which is generated across the threshold that divides or limits this global integration and the one with which it is further differenciated from being actualized in a more composite global integration. The form of the divergent lines of the different potential trajectories of the variables across the threshold, which range from the straight line of the potential function to the curves of the infinite branches of the actualized more composite global integration, are determined by the type of essential singularity created by the composite relation between the global integrations. The variation of the divergent lines of the different potential trajectories across this domain corresponds with the variation in the degree of power of the power series expansion of the more composite global integration. The minimum degree of power of the power series expansion of the more composite global integration, that is, at intensity = 0, corresponds with the maximum degree of power of the power series expansion of each of the global integrations of which it is composed, that is, with their limit or maximum intensity. The maximum degree of power of a global integration is only determined in reciprocal relation with another global integration insofar as they are together incorporated in the power series expansion of a more composite global integration.

The declaration by Deleuze that: ‘in Spinoza, there is no existing mode that is not actually infinitely composite, whatever be its essence or degree of power’,⁴² can be understood insofar as a global integration, or finite existing mode, incorporates the composite, multipli-differenciated assemblage of local integrations of which it is composed. And the assertion that: ‘in greater or lesser infinities [extensive parts] compose, in different relations, the existence of modes whose essences are of greater or lesser degree’,⁴³ can be understood insofar as the greater infinities of extensive parts, belonging to global integrations, incorporate the lesser infinite collections of extensive parts, belonging to local integrations, to compose the existence of modes. So, existing modes are affirmed as having ‘an infinity (a very great number) of parts’.⁴⁴ Now, the singular modal essences of these existing modes can be understood to be ‘of greater or lesser degree’ only insofar as ‘their essences or degrees of power always correspond to a limit (a maximum or minimum)’.⁴⁵ The singular modal essence of an existing mode is of a greater degree when it corresponds to the maximum degree of power or intensity, of a global integration, that is, with the limit of its power series expansion; and of lesser degree when it corresponds to the minimum degree of power, or intensity = 0, of the power series expansion of the more composite global integration in which it is implicated. The limit to which the singular modal essence of an existing mode corresponds is the maximum degree of power or intensity, of the power series expansion of its global integration, which is reciprocally determined, or further differenciated, in relation to another; or, the minimum degree of power, or intensity = 0, of the power series expansion of the so determined more composite global integration.

The limit of a local integration is only determined by its implication in a global integration, the limit of which is also only determined insofar as it is implicated, in reciprocal relation with another global integration, in a more composite global integration. It is therefore only in the more composite global integrations between global integrations that there is a limit that marks the finitude of each global integration, or of each finite existing mode or individual. The limit, or finitude, of a finite existing mode (individual or human being⁴⁶) is therefore characterized by the minimum degree of power, or intensity = 0, of the power series expansion of the more composite global integration in which it is implicated. The limit to which the singular modal essence of a finite existing mode corresponds is the maximum degree of power of the power series expansion of its global integration, or the minimum degree of power of the more composite global integration in which it is further differenciated.

It has been argued that insofar as singular modal essences are contained together in an attribute, as intensities of its quality, they are ‘inseparable, and are characterized by their total agreement’,⁴⁷ together forming ‘a total system, an actual infinite whole’.⁴⁸ The way in which all singular modal essences, within an attribute, ‘are involved in the production of each’⁴⁹ is simultaneously expressed, not only by their quantitative differentiation, as the actually infinite quantity of intensive parts of an attribute, or as the different degrees of power of its power series expansion, but also by the way in which they are differenciated in relation to one another as global integrations, each global integration incorporating an increasingly composite, multipli-differenciated assemblage of local integrations, which range from the differenciations of the differentiated, that is, of Spinoza’s most simple bodies, to the actually infinitely composite differenciations of the differenciated.

It is not until chapter 8 that the actual mechanics of global integrations, which incorporate an increasingly composite multipli-differenciated assemblage of local integrations, and which are themselves incorporated by an increasingly composite multipli-differenciated assemblage of global integrations, are explicated in detail. Before doing so, it is necessary to determine the extent to which this logic is actually able to be mapped onto Spinoza’s theory of relations. In order to do so the next two chapters examine some of the more controversial aspects of this theory which have led to varying interpretations among scholars in the field of Spinoza studies. What has been determined so far, however, is that both the complication of singular modal essence and the explication of finite modal existence are determined according to the logic of different/ciation, and that the individuated form of a particular existence, that is, the finitude of a finite existing mode, is only determined insofar as it is further different/ciated in reciprocal relation to another. The determination of the finitude of a finite existing mode is therefore secondary to the relations of further different/ciation in which it is implicated.

_________________

Notes

¹ Deleuze, Expressionism in Philosophy, p. 203.

² See the section of chapter 3 entitled ‘Extensive parts: infinite collections of the infinitely small’, p. 86.

³ Deleuze, Expressionism in Philosophy, p. 203.

⁴ Recall that each term of a Taylor Series is comprised of a differential relation multiplied by a variable: See the section of chapter 3 entitled ‘The logic of the differential’, p. 69.

⁵ Deleuze, Expressionism in Philosophy, p. 207.

⁶ Ibid.

⁷ Ibid., p. 202.

⁸ Ibid., p. 198.

⁹ Ibid.

¹⁰ Ibid.

¹¹ Ibid., p. 184.

¹² ‘The system of complication of essences’ (Ibid., p. 198).

¹³ Ibid., p. 198.

¹⁴ Ibid., p. 199.

¹⁵ Ibid., p. 213.

¹⁶ Ibid., p. 215.

¹⁷ Ibid., p. 207.

¹⁸ Ibid.

¹⁹ Ibid., p. 214.

²⁰ Ibid., p. 215.

²¹ Ibid.

²² Ibid.

²³ Spinoza, Ethics, II, P10DII.

²⁴ Recall that a differential relation is the function of a tangent to a curve.

²⁵ Deleuze, Difference and Repetition, p. 175.

²⁶ Ibid., p. 176.

²⁷ Ibid., p. 172.

²⁸ Ibid., p. 217.

²⁹ Ibid., p. 211. See the section of chapter 3 entitled ‘The logic of local and global integration’, p. 84.

³⁰ Ibid., p. 176.

³¹ Deleuze, Expressionism in Philosophy, p. 209.

³² Deleuze, ‘sur Spinoza’, 10 March 1981.

³³ Deleuze, Expressionism in Philosophy, p. 209.

³⁴ Ibid.

³⁵ Ibid.

³⁶ Deleuze, ‘sur Spinoza’, 10 March 1981.

³⁷ Deleuze, Expressionism in Philosophy, p. 209.

³⁸ Spinoza, Letter XXXII.

³⁹ Deleuze, Expressionism in Philosophy, p. 209.

⁴⁰ Ibid.

⁴¹ Ibid.

⁴² Ibid., p. 207.

⁴³ Ibid.

⁴⁴ Ibid., p. 204.

⁴⁵ Ibid.

⁴⁶ Throughout the present work, the main example given of a finite existing mode or individual will be an individual human being.

⁴⁷ Ibid., p. 198.

⁴⁸ Ibid., p. 194.

⁴⁹ Ibid., p. 198.

Chapter 6

Spinoza’s theory of relations

Are modal essences fixed, or do they admit a certain degree of variation?

There is in Spinoza’s Ethics what could be described as a double point of view of singular modal essence: sometimes it seems to be fixed to a precisely determined degree; sometimes it seems to admit a certain degree of variation. The problem of resolving this apparent contradiction has been responsible for a variety of interpretations among scholars working in the field of Spinoza studies. In a passage from Spinoza, L’âme (Ethique 2), Martial Gueroult speaks of inferior and superior ‘limits’ of modal essence: ‘All individual humans’, writes Gueroult, ‘have … their own essence which differs from the essence of others by a different relation of movement and rest between the parts of their bodies, but these singular differences remain, in one as in the others, within the limits of the relation which defines the essence of all human bodies’.¹ In quantité et qualité dans la philosophie de Spinoza, Charles Ramond suggests that ‘the text of Spinoza argues to the contrary’.² The argument that Ramond constructs in response to this passage is that: ‘All variation of the precise relation of movement and rest which characterizes an individual should be interpreted as a change of essence, and therefore as the destruction of the individual, the “characteristic relation” of which has changed’.³

Contrary to what Ramond seems to read into this passage, when Gueroult speaks of inferior and superior ‘limits’ of essence, he speaks generally of human essence and not of the specific singular essence of a finite existing mode or individual human being. Gueroult argues that logically all human essences exist within inferior and superior limits, which is an altogether different argument to that which Ramond extracts from the passage. Even though Ramond’s argument in reference to the fixity of modal essence seems plausible, his reading of Gueroult does not. In fact there is nothing in this passage to indicate that Gueroult would disagree with Ramond’s argument for the fixity of individual essences. Rather than defending Gueroult against Ramond, Ramond’s interpretation can function as a point of departure for the introduction of the problem of resolving the apparent contradiction posed by the double point of view of singular modal essence. Any resolution of this problem requires an understanding of Spinoza’s theory of relations, which deals with the relation between the singular modal essence of an individual human being and that human being’s finite modal existence. The example of Ramond’s reading of Gueroult serves to illustrate the problematic nature of this aspect of Spinoza’s work, and will be used to further distinguish the interpretations of Spinoza’s theory of relations offered by both Macherey and Deleuze. The similarities and differences between their respective interpretations will be examined, in particular how this contributes to the way that they each resolve the apparent contradiction presented by the problematic of the double point of view of singular modal essence.

In Introduction à l’Ethique de Spinoza: la troisième partie, when Macherey speaks of ‘the conatus’ of a finite mode as constituting its ‘actual essence’, he writes that ‘the power (puissance) and the energy of the conatus of each thing is invested by being deployed within the thresholds of intensity which are distributed between a minimum and a maximum, the first corresponding to a pole of extreme passivity, the second to a pole of extreme activity’.⁴ When he writes ‘between a minimum and a maximum’, he seems to be in accord with Ramond’s (mis)interpretation of Gueroult, that a singular modal essence varies between two limits. If not, Macherey at least seems to agree with an interpretation of singular modal essence which admits a certain margin of variation. Even Deleuze seems to be open to criticism from Ramond for the same reason when he writes that, with Spinoza, ‘the relation that characterizes an existing mode as a whole is endowed with a kind of elasticity;’⁵ and that singular modal ‘essences or degrees of power always correspond to a limit (a maximum or minimum)’.⁶ However, it is yet to be determined exactly what each of these interpreters understands by a maximum and a minimum, and exactly how they reconcile the singular modal essence and the finite modal existence of an individual with the concept of variation.

Certa ratio motus et quietis

Although Ramond mistakenly disputes Gueroult’s interpretation of the movement and rest of the singular essence of each mode, in doing so he emphasizes an important point. In order to support his argument he retranslates a passage in the Ethics. Ramond notes that the foundation of Spinoza’s physics appears in proposition 13 of the Ethics, II, and that it is here that Spinoza determines the essence of a body by a <certa ratio motus et quietis>.⁷ Ramond translates this sentence as: a ‘precise relation of movement and rest’, and justifies this translation by citing precedent from the definition of the proposition. ‘When a number of bodies’, writes Spinoza, ‘whether of the same or of different size, are so constrained by other bodies <a reliquis … coercentur> that they lie upon one another, or if they so move, whether with the same degree or different degrees of speed, that they communicate their motions to each other in a certain [or precise] fixed manner [or relation] <ut motus suos invicem certa quadam ratione communicent>’.⁸ Ramond argues that the ‘translation of the term certa in the expression certa ratione is imperative, because Spinoza never leaves it out. And this is why the translation of Appuhn is at fault in relation to the sense of the text, because “a certain relation” may translate quandam ratione, but certainly not certa quandam ratione. It happens to be a “precise” relation of movement and rest, that is to say, distinct from all other relations’.⁹ The point that Ramond is making is more significant than the disputed translation, and, contrary to the criticism of Ramond, who accuses Deleuze of not adhering to this point,¹⁰ it is this point which can be used as a common point of reference for the interpretations of both Macherey and Deleuze.

Continuing his discussion of Spinoza’s physics, Ramond argues that the ‘precise relation’ of what he refers to as ‘the essence of a singular thing’ is indifferent to certain types of variation. Nutrition, growth, and all forms of movement and displacement in space are described as being able to be produced without altering the individual essence. ‘By this, then’, writes Spinoza, ‘we see how a composite Individual can be affected in many ways, and still preserve its nature’.¹¹ On the condition that the precisely determined relation of movement and rest remains the same as before, the changes resulting from nutrition, growth and movement are no longer, ‘absurdly, metamorphosis’.¹² The variety of movements accomplished by singular things detracts nothing from the precision of the relation which expresses their essence. It is from here that the permanence of individual essence becomes conceivable, and, consequently, Ramond argues that ‘all variation of this relation will involve a change of essence’.¹³ Ramond concludes his interpretation by emphatically stating his understanding of ‘the determined, fixed, if not to say locked, aspect of the essence of singular things, without fluctuations between a “maximum” and “minimum”’.¹⁴

Ramond’s reading of Spinoza can be used to introduce two further aspects of Spinoza’s theory of relations. The first is the conatus of a singular thing, and the next is its power to act. Ramond argues that it is in the Ethics, III, that the essence of a singular thing is no longer determined as a ‘precise relation of movement and rest’, but rather as an ‘effort’. ‘The striving [effort] <conatus> by which each thing strives to persevere in its being’, writes Spinoza, ‘is nothing but the actual essence of the thing’.¹⁵ Ramond argues that the ‘determination of the essence as “effort” is at base equivalent to its determination as a “precise relation of movement and rest”’.¹⁶ Therefore, for Ramond, the relation between the ‘actual essence’ and the ‘conatus’ of a finite mode is a relation of equivalence. Ramond argues that Spinoza characterizes the two principal determinations of the essence of singular things as a ‘precise relation of movement and rest’ and as the ‘effort to persevere in being’.¹⁷

Ramond next introduces the relation between conatus and the power to act by saying that ‘there is not a “precise relation of movement and rest” without a certain power (puissance); and that, at the same time, all “effort” supposes a power’.¹⁸ In both cases, the definition of essence refers in effect to the expression of a same power which Ramond interprets as ‘a quantum of the power to act (puissance d’agir)’.¹⁹ Therefore, for Ramond, the essence of a singular thing is determined by a ‘precise relation’ of movement and rest, or by a quantum of the power to act, and all variation is forbidden, since all augmentation and diminution of this power to act would create another individual. Despite this conclusion that all augmentation and diminution of the power to act of the same individual is inconceivable, Ramond believes that it leaves man ‘absolutely incomprehensible’, and, moreover, that it is ‘contrary to good sense’.²⁰ This is the form that the determining problematic of this chapter takes in Ramond’s reading of Spinoza. In order to resolve the apparent contradiction Ramond suggests redefining one of the components of the contradiction. He maintains the concept of a singular thing’s fixed power to act, while denying that there is room for the concept of a margin of variation ‘of’ its power to act. He proposes to do this in the following way: ‘if we want to be able to maintain a certain capacity of variation of the power to act of an individual who remains identical in its essence despite these variations, it will be necessary to separate in some way essence and power’.²¹ Ramond considers this to be possible only by changing the way in which an individual’s power to act is understood to vary. Ramond proposes to do this by making a distinction between changes ‘of’ the power to act and changes ‘in’ the power to act. By interpreting the Ethics according to the latter formulation, Ramond retains the concept of an unchanging power to act. ‘There are therefore positive variations in the power to act of the body, and negative variations’,²² while there are no variations ‘of’ the power to act itself. Ramond justifies this formulation by arguing that the ‘variations “in” the power to act’ occur within the theory of the passions and are therefore imaginary, that is, the power to act itself remains unchanged by these variations within the theory of the passions. ‘This moreover’, writes Ramond, ‘does not surprise us, since such a distinction is at the base of the spinozist theory of the passions’.²³ In concluding his analysis, Ramond argues that the ‘relation between the active and passive affections cannot however be reduced to a simple comparison of powers’,²⁴ because, as will be determined, it is the affections which cause the variations in the theory of the passions. The definition of the affections and their significance in defining a singular thing’s power to act, as foreshadowed by Ramond, will be pivotal in differentiating the interpretations of Deleuze and Macherey.

Individuation as a determinate level of integration

The interpretation that Deleuze offers of a singular modal essence, its conatus and its power to act further complicates that of Ramond. In Expressionism in Philosophy, Deleuze invites us not to ‘confuse’ the essence of a mode and the ‘relation in which it expresses itself’. ‘A modal essence expresses itself eternally in a relation but we should not confuse the essence and the relation in which it expresses itself’.²⁵ Contrary to this Deleuzian interpretation, Ramond considers the two to be identical, he argues that ‘the “precise relation of movement and rest”, which characterizes a singular thing does not “express”, but “is” exactly its essence’.²⁶ For Deleuze, however, when Spinoza defines modes as modifications of the attributes of substance, he refers only to singular modal essences, not to the existence of finite modes, which is determined rather solely by the effects of existing modes on each other. Singular modal essences, as modifications of an attribute’s quality, are together complicated within an attribute as ‘intensive parts’. Each mode is composed, therefore, of both a singular modal essence, within an attribute, and, corresponding to this, a finite existing mode, which is composed by an infinity of extensive parts. As already determined in the section of chapter 5 entitled ‘The relations between the successive terms of a power series expansion’, Deleuze understands intensive parts to be ‘parts of power (puissance)’, that is, ‘intrinsic or intensive parts, true degrees’, which are distinguished from one another as different ‘intensities’ or ‘degrees of power’.²⁷

Corresponding to any determined degree of power, or singular modal essence, is a finite existing mode, each of which involves its attribute precisely in the form belonging to that attribute, that is, as an idea or body when represented respectively from the point of view of thought or extension. Existing modes are composed by an infinity of extensive parts, which are external to one another, and which determine one another from outside, that is, they come to exist by virtue of an external cause, which is another existing mode, whose own cause is another existing mode, and so on ad infinitum.²⁸ As Spinoza writes in the Ethics, I, P28: ‘Every singular thing, or any thing which is finite and has a determinate existence, can neither exist nor be determined to produce an effect unless it is determined to exist and produce an effect by another cause, which is also finite and has a determinate existence; and again, this cause also can neither exist nor be determined to produce an effect unless it is determined to exist and produce an effect by another, which is also finite and has a determinate existence, and so on, to infinity’. According to the logic of different/ciation, a mode comes to exist, or is actualized, in the form of a global integration by virtue of an external cause, or the differenciation of the local integrations of which it is composed. Each local integration being in turn the global integration of the local integrations of which it is composed, and so on, thus incorporating, or individuating, the differentiated preindividual singularities which are characteristic of the most simple bodies, or infinitely small extensive parts. The finitude of an existing mode is also only determined in relation to that of another existing mode, insofar as they are further differenciated in relation to one another, that is, insofar as each is implicated as a local integration in the production of a more composite global integration. According to the logic of different/ciation, the component extensive parts of an existing mode are external to the mode’s essence,²⁹ however these extensive parts exist directly in relation to the mode’s intensive part, that is, to its singular essence or degree of power. The primary elements of Spinoza’s theory of relations, according to Deleuze, are therefore a mode’s singular essence, which is a degree of power or an intensive part, and, corresponding to this, a mode’s particular, or finite, existence, which is determined by the individual form of the extensive parts of which it is composed.

As noted in the discussion of Ramond, it is in the Lemma to Proposition 13 of Ethics, II, that Spinoza discusses the physics of bodies. Deleuze argues that with Spinoza, the relations between the extensive parts which constitute the existence of a mode or composite body and which correspond to that mode’s essence, are determined by ‘purely mechanical laws’.³⁰ Such a mode comes to exist when an infinity of extensive parts enter into a ‘given’ relation which corresponds to a given modal essence, when its parts ‘actually belong to it … in a certain relation of movement and rest’.³¹ Deleuze refers to this relation as ‘a characteristic relation’.³² A mode continues to exist as long as the ratio or proportion of relations and relative movement of the extensive parts is maintained. By arguing in this way, Deleuze suggests that the individual form of an existing body will not change if its component parts are at each moment renewed according to the ‘characteristic’ relation of movement and rest. The whole remains the same insofar as it is composed by the same ratio or proportion of extensive parts that are the expression of that particular modal essence. An existing mode or composite body is thus open to continual alteration of motion and rest between its extensive parts, but it will continue to exist as long as the same ratio or proportion of extensive parts subsists in the whole.³³ ‘Each mode endures’, writes Deleuze, ‘as long as its parts remain in the relation that characterizes it’.³⁴ Deleuze is therefore in accord with the emphasis that Ramond places on the specificity or the precision of this relation.³⁵ However, for Deleuze, the relation is characterized rather by a global integration, which is determined according to the logic of different/ciation. The global integration in which the mode’s essence is expressed constitutes a whole present in its parts. It does this insofar as these extensive parts are determined to enter into the characteristic relation, or are actualized in the global integration, by means of the operation of ‘an extrinsic determination’,³⁶ which is, for Deleuze, the operation of the purely mechanical laws of the logic of different/ciation.

The infinity of extensive parts which correspond to a singular modal essence, insofar as actualized in a global integration, are only temporarily related to that singular modal essence. If these parts are determined to enter into a different relation corresponding to a different singular modal essence, and therefore constitute the existence of a different mode or composite body, they will form part of a different whole. Stated differently, if the extensive parts belonging to one of the reciprocally determined local integrations is reciprocally determined in relation to a different local integration, then the global integration characteristic of the mode would cease to exist. ‘A mode ceases to exist as soon as its parts are determined to enter into another relation, corresponding to another essence’.³⁷ Therefore, a finite existing mode only exists insofar as ‘an infinity of parts are temporarily related to that essence’.³⁸ It is in this way that it is determined ‘as having duration, as having a relation with a certain extrinsically distinct time and place’.³⁹ As Spinoza writes in the Ethics, II, P30D: ‘Our body’s duration … (by IP28) … is determined to exist and produce an effect from such [NS: other] causes as are also determined by others to exist and produce an effect in a certain and determinate manner, and these again by others, and so to infinity’.

For Deleuze, the finitude, or finite existence, of such a composite body, is determined both by the characteristic, precise, relation of its composition, that is, in the form of a global integration, and by the more composite relations in which this global integration is implicated, that is, on it remaining a part of a larger more composite whole. The finite existence of a mode or composite body is therefore not solely bound to the relations between the simple bodies of which it is composed, as Ramond holds. The duration of a mode’s finite existence, rests not only with the preservation of the proportion or ratio of its extensive parts rather than with the particular extensive parts incorporated by it at any moment, but for any composite body, its finite existence is determined both by the spatial and dynamic characteristics of the extensive parts belonging to it, ‘and’ by it being further differenciated as a component part, or local integration, in more composite global integrations. The finite existence of a composite body is thus determined by its incorporation in an increasingly composite multipli-differenciated assemblage of global integrations, with only its characteristic global integration enduring while the extensive parts of which it is constituted come and go. This is the concept of the individual as a determinate level of integration, or ‘individuation’,⁴⁰ which, as a whole, incorporates other individuals or bodies, and is itself incorporated as a part into more composite, although not necessarily more complex, individuals or bodies.⁴¹

A mode’s ‘capacity to be affected’

The spatial and dynamic characteristics determinative of the finite existence of a mode, which function according to the mechanical laws of the logic of different/ciation, correspond to a certain ‘capacity’ of the mode’s extensive parts ‘to be affected’ by other extensive parts. Deleuze notes that for Spinoza, the motion and rest of the extensive parts belonging to the differenciated local integrations, or composite bodies, of which the human Body is composed, are further differenciated insofar as some ‘are fluid, some soft, and others, finally, are hard’.⁴² Those bodies ‘whose parts lie upon one another over a large surface’ and whose position is therefore difficult to change, Spinoza calls ‘hard’; those bodies ‘whose parts lie upon one another over a small surface’ and whose position can be changed with less difficulty, Spinoza calls ‘soft’; and those whose parts are in motion, Spinoza calls ‘fluid’ bodies’.⁴³ Spinoza goes on to describe what happens when these different parts of the human Body interact with an external body. He says that ‘when a fluid part of the human Body is determined by an external body so that it frequently thrusts against a soft part of the body, it changes its surface and, as it were, impresses on the soft part certain traces of the external body striking against the fluid part’.⁴⁴ Such a determination by an external body is what Spinoza understands to be an ‘affection’ of the human Body, that is, the way in which the human Body is affected by another body. Deleuze argues that if an individual is able to undergo, or withstand, such affections without them changing the proportion or ratio of its extensive parts, that is, without the composite multiplidifferenciated assemblage of local integrations of which it is composed being destroyed or decomposed by them, then this constitutes what Deleuze calls its ‘capacity to be affected’. Insofar as a composite body is an individual human being, whose structure is constituted by the composition of a composite multiplidifferenciated assemblage of local integrations, this assemblage of local integrations corresponds to that individual’s capacity to be affected. Furthermore, Deleuze argues that an affection in Spinozist terminology may be passive or active, depending on whether the affection is determined from without by an external body, or whether it is determined by the mode’s own degree of power. Therefore, the degree of power of a mode always corresponds to a certain capacity to be affected, which is always exercised, either in affections produced by external things or in affections determined by its own essence. Those affections produced by external things are called passive affections, and those determined by the mode’s own essence are called active affections. Deleuze argues that to the extent that an individual’s affections can be explained by passive affections, it is said to suffer, or undergo things, and its degree of power is expressed by a power of suffering. To the extent that an individual’s affections can be explained by active affections, it is said to act, and its degree of power is expressed by a power of acting.

Continuing the complicity that has been established between Deleuze and Ramond in relation to the concept of a fixed modal essence, which, as has been argued, Ramond would deny, Deleuze writes that the ‘capacity to be affected remains constant, whatever the proportion of active and passive affections’.⁴⁵ However, Deleuze argues that within this fixed capacity to be affected, the proportion of active and passive affections is open to variation. ‘For a given essence, for a given capacity to be affected, the power of suffering and that of acting should be open to variation in inverse proportion one to the other. Both together, in their varying proportions, constitute the capacity to be affected’.⁴⁶ The production of active affections will have as a result a corresponding reduction of passive affections, and, reciprocally, the continuation of passive affections will inhibit proportionally the power to act. Thus for Deleuze, unlike Ramond, the power to act of an essence is open to variation. Ramond confounds what is for Deleuze a mode’s variable power to act with what Deleuze determines here as its fixed capacity to be affected.

When Deleuze first discusses the conatus of a mode he seems to agree with Ramond, that it remains fixed for any given mode. He writes that the conatus of an existing mode is ‘the effort to preserve the relation of movement and rest that defines it, that is, to maintain constantly renewed parts in the relation that defines its existence’.⁴⁷ However, rather than stating the strict equality between conatusand modal essence, Deleuze argues that a mode comes to exist, or is actualized, when its extensive parts are extrinsically determined to enter into the relation, or global integration, that characterizes the modal essence, and he emphasizes that ‘then, and only then, is its essence itself determined as a conatus: … A conatus is indeed a mode’s essence (or degree of power) once the mode has begun to exist’. From this Deleuze argues that conatus is the ‘affirmation of essence in a mode’s existence’.⁴⁸ Both of these statements seem to indicate that there is a relation different to that of strict equality between conatus and essence. But before attempting to determine what Deleuze understands by this concept of affirmation, Macherey’s interpretation of the relation between a modal essence, its conatus, and its power to act will be examined.

The affective life of a mode as a state of uninterrupted affective flux

Macherey bases his argument on the foundation that the conatus of a finite mode constitutes its ‘actual essence’. Macherey’s discussion of the ‘thresholds of intensity distributed between a minimum and a maximum’ within which the power (puissance) and energy of the conatus is deployed would seem to indicate that both essence and conatus are open to variation between the minimum and maximum. This would leave Macherey open to the criticism which Ramond levels at Gueroult, that all augmentation and diminution of a singular thing would create another individual, that is to say would be, according to Ramond, ‘absurdly, metamorphosis’.⁴⁹ As has been determined in regards to Deleuze, there is room for an interpretation of a mode’s power to act which allows for its variation while maintaining the concept of a fixed modal essence. What is surprising is that Macherey would not only agree with Ramond’s criticism that any concept of variation of modal essence is absurd, but he would also agree with Ramond that neither the conatus nor the power to act of a finite mode are variable. How then does Macherey incorporate a concept of variation into his interpretation of Spinoza? Macherey already accounts for the point of view held by Ramond when he says that ‘we are here very close to the exposition of a problem of logic, which concerns essences insofar as they are fixed once and for all by definitions which they evidently cannot contravene’.⁵⁰ This problem of logic is resolved by what Macherey considers to be ‘an economic perspective on the system of the affective life (la vie affective) as a whole’.⁵¹

According to Macherey, the affective life of a mode is characterized by the power (puissance) of its conatus, which is deployed between the poles of extreme passivity and activity. It can be characterized in part by the theory of the passions of which Ramond speaks, there is therefore a certain convergence of their different points of view in regards to Spinoza’s theory of relations. However, their readings are distinguished by their different points of view of the role that the affections play in the determination of modal existence as such.

Macherey’s analysis of the affective life begins with the question ‘What is an affect?’ It is in Definition 3 of the Ethics, III, that Spinoza defines what he understands by affect, which Macherey explains in the following way: ‘it is the idea of an affection of the body which corresponds to an augmentation or diminution of its power to act: this variation is in relation to the fact that the body is affected, sometimes by itself, sometimes by an external body’.⁵² That is to say, the body is affected sometimes by itself, resulting in the idea of an augmentation of its power to act, and sometimes by an external body, resulting in the idea of a diminution of its power to act. In the General Definition of the Affects, Spinoza notes that: ‘when I say a greater or lesser force of existing than before, I do not understand that the Mind compares its Body’s present constitution with a past constitution, but that the idea which constitutes the form of the affect affirms of the body something which really involves more or less of reality than before’.⁵³ Macherey argues that this idea expresses only ‘a momentary state of our body … in rupture with [its] preceding state’,⁵⁴ in the sense of an augmentation or diminution, which marks a development or a restriction of its power to act. As Spinoza says, all affectivity is based on the foundation of ‘joy’ and ‘sadness’ which, according to Macherey, expresses the ‘transformations’(mutationes) associated with the fact that the soul⁵⁵ is, without end, exposed to ‘passing sometimes to a greater sometimes to a lesser perfection’. These transformations, or ‘transitions’ (transitio), are experienced as ‘passions’, at the heart of which the ‘soul’, Macherey writes, ‘is completely subject to the mechanisms of the imagination’.⁵⁶ The fact of passing to a greater perfection leaves the soul ‘euphoric’,⁵⁷ however, Macherey emphasizes that even though this ‘joy’ experienced by the soul is perfectly real, the base upon which it rests remains imaginary.

The idea of a mode’s perfection of which Spinoza speaks is a measure of that proportion of the mode’s power to act which is expressed actively by this mode at any precise moment. In accord with Ramond’s concept of a mode’s fixed essence and power to act, Macherey suggests that whatever the margin of variation within which the power (puissance) of the soul is expressed, it ‘remains in all cases, and by definition, the same’,⁵⁸ and, the soul ‘maintains this conformity to its nature without which it would simply cease to be’.⁵⁹ Therefore, to pass to a greater perfection is to express ‘actively’ a greater proportion of what is a mode’s fixed power to act.

Passing to a lesser perfection is neither to be deprived of a greater perfection, in the sense of a lack, nor is it ‘a pause in the pursuit of the movement which effects the fundamental impetus of the conatus’.⁶⁰ Sadness is the inverse of joy, but it is not the absence of joy. As Macherey writes, sadness is ‘a contraction of the power to think of the soul, which deprives it momentarily, but which would be unthinkable without the persistence of this power, of which it continues to give a paradoxical expression’.⁶¹ Sadness therefore is an integral part of the movement of the conatus. What Macherey wants to make understandable by reasoning in this way is that, whatever the orientation of the variation which affects the soul, whether joy or sadness, the soul continues in all cases to be animated by the pressure of the conatus, and the ‘transitions’ that the soul does not cease to experience, in no way alter the constancy of the conatus. It is, writes Macherey, ‘constitutionally inalterable, and persists imperturbably across these series of transitory states’.⁶² In postulating that the soul can be confronted at any moment by the alternative between two contrary orientations, Macherey maintains that there is no room to think that the soul ‘would have more control of itself when it is occupied by those sentiments which elate it than when it is occupied by those sentiments which depress it’.⁶³ The character of the ‘transitions’ therefore leave the affections ‘fundamentally ambiguous’.⁶⁴ As none of these states of the soul have in themselves a guarantee of stability, the affective life remains in a ‘state of permanent instability’.⁶⁵ This state, Macherey suggests, prevails in what he refers to as the ‘uninterrupted affective flux (le flux affectif ininterrompu), which varies continuously between the two extreme poles of a maximum and a minimum’.⁶⁶

Macherey concludes that the states of activity and passivity of the soul are not absolute states, and as such radically exclusive one of the other. They are rather measured against one another at the interior of a gradual series of states described as the ‘uninterrupted affective flux’, which tends to realize all intermediate states between the two extremes. The variation of the active expression of a mode’s power to act is therefore inversely related to the proportion of a modes whole power to act that is occupied by the ‘uninterrupted affective flux’.

The distinction that Ramond makes between the affections and the power to act of a singular thing when he writes that ‘the relation between the active and passive affections cannot however be reduced to a simple comparison of powers (puissances)’,⁶⁷ is expressed by Macherey as the difference between the ‘uninterrupted affective flux’ and the power to act of a singular thing. However, there is more to Macherey’s distinction than Ramond’s. It is here that Macherey introduces the concept of negativity into his interpretation of the constitution of a singular thing. Since we can never escape the ‘uninterrupted affective flux’, Macherey argues that ‘there is in our nature … something incomplete and unfinished which is the mark of our impotence, that is, of the limited and finite character of our power (puissance), which is always exposed to being measured against other powers (puissances) which exceed and oppose us negatively’.⁶⁸ Macherey considers Spinoza to have developed the concept of power ‘to understand the way in which the representation of unaccomplished virtualities, which would traverse this power (puissance) as a sort of internal negativity, are definitively excluded from one’s field’.⁶⁹ The ‘unaccomplished virtualities’, which are the imaginary ideas that correspond to the ‘uninterrupted affective flux’, function as ‘a sort of internal negativity’ only insofar as they inhibit or limit the full expression of the power to act of a singular thing. However, as has already been determined, the affects of the ‘uninterrupted affective flux’ flow from relations with external bodies. The negativity of a singular thing is therefore only determined from the outside. As Macherey argues: ‘reality can only be understood negatively in regards to its extrinsic relations, not to that which concerns its intrinsic constitution’.⁷⁰ To confuse this negativity as something fundamental to singular things, ‘as if it actually constituted their cause or origin’,⁷¹ would be a mistake. Therefore the negativity of a singular thing does not fundamentally define it. Macherey writes that ‘it is in reference to this point of view that Hegel could say … that Spinoza “did not give justice to the negative”’.⁷² Spinoza’s negativity is something fundamentally extrinsic to the nature of things. Negation only occurs, or intervenes, between things, as the affections of modes which are reciprocally limited by one another, but it does not actually belong to the things themselves. It cannot in any way be determined from within a singular thing. From this point of view, the reference that Macherey makes to an ‘internal negativity’ is immediately relativized. Negativity is only manifested in singular things in the form of its passive affections, but these actually come from the outside. This negativity of the ‘uninterrupted affective flux’ draws the affirmation of a singular thing’s power to act towards a minimum. Its power to act is only more actively expressed, on the contrary, when the ‘uninterrupted affective flux’ is drawn to a minimum, bringing its power to act back towards its maximum.

The physical view of modal existence

While Macherey analyses affectivity from the point of view of the imagination, in relation to the attribute of thought or the soul, Deleuze’s explication remains from the point of view of the attribute of extension, in relation to the body’s power to act.⁷³ Macherey considers the effect of individual affects, whether joy or sadness, to remain fundamentally ambiguous, absorbed within the ‘uninterrupted affective flux’, which functions as a hindrance or limit to the active expression of a mode’s fixed power to act.⁷⁴ Deleuze’s interpretation of Spinoza’s theory of the affections, however, further complicates that offered by Macherey by introducing another level of variation. Even though Deleuze considers the capacity to be affected of a ‘given’ mode to be fixed, he maintains that it does not remain fixed at all times and from all points of view. ‘Spinoza suggests’, Deleuze maintains, ‘that the relation that characterizes an existing mode as a whole is endowed with a kind of elasticity’.⁷⁵ Contrary to the interpretation of Ramond, Deleuze maintains that the individuation of a mode changes on leaving behind childhood, or on entering old age, and also after the permanent effects of illness.⁷⁶ Such changes as growth, aging and illness may be understood as though a mode’s capacity to be affected, which corresponds to both the composite multipli-differenciated assemblage of local integrations of which it is composed and to the global integrations in which it is further differenciated, enjoys ‘a margin, a limit within which’ these global integrations ‘take form and are deformed’.⁷⁷ What Deleuze understands by this is that, even though quantitatively there has been a change in the way that the intensive part, or singular modal essence of an individual, is actualized in a global integration; qualitatively, insofar as the intensive part, or singular modal essence, belonging to this individual, is a modification of an attribute’s quality it remains unchanged. So although the individual’s intensive part or singular essence remains unchanged, there is a change in the individual’s capacity to be affected, which corresponds to a change in the extent to which it is further differenciated as a component local integration, in larger, more composite global integrations.

Deleuze argues that the power of suffering and the power of acting of a finite existing mode can only be considered as two distinct principles, which are inversely proportional to one another within a ‘given’, or fixed, capacity to be affected, insofar as the affections are considered ‘abstractly, without concretely considering the essence of the affected mode’.⁷⁸ Macherey’s theory of the affective life as a state of ‘uninterrupted affective flux’ remains abstract in this way. Although Deleuze maintains that this is a primary thesis of Spinoza, he suggests that ‘this thesis, if physically true, is not metaphysically true’.⁷⁹ Deleuze maintains that, for Spinoza, the power of suffering expresses nothing positive or real. On the contrary, ‘in every passive affection there is something imaginary which inhibits it from being real’,⁸⁰ that is, which inhibits it, the passive affection, from being real for the finite existing mode itself. As has been demonstrated, Macherey is in agreement on this point with Deleuze. When Spinoza writes in the General Definition of the Affects, Ethics, III, that ‘the idea which constitutes the form of the affect affirms of the body something which really involves more or less of reality than before’, Macherey argues that although the affection that the soul experiences is perfectly real, the base upon which affectivity rests, that is, the uninterrupted affective flux, is imaginary. Despite this point of convergence in their interpretations of Spinoza’s physics, Deleuze further develops Spinoza’s theory of the affections differently to that presented so far of Macherey.

Deleuze’s argument hinges on what he describes as ‘a coincidence in the development’ of the respective philosophies of Leibniz and of Spinoza in relation to the theory of the affections, or the concept of action and passion. The coincidence that Deleuze finds in their respective terminology, which he considers to be ‘more remarkable’ than an influence,⁸¹ is the most crucial aspect of the argument. According to Deleuze, the Leibnizian pre-Newtonian concept of ‘force’ is ‘coincident’ with the Spinozist concept of the ‘power to act’. Deleuze presents the findings of Leibniz as follows: ‘only active force is strictly real, positive and affirmative. Passive force … expresses nothing but … the mere limitation of active force. There would be no such force without the active force that it limits’.⁸² This is then transposed onto Spinoza as: ‘our force of suffering is simply … the limitation of our force of acting itself. Our force of suffering asserts nothing, because it expresses nothing at all: it “involves” only … the limitation of our power of action’.⁸³

The way that ‘the affections’, in a state of ‘uninterrupted affective flux’, hinder or ‘limit’ the expression of a mode’s power to act is very different to the way that passive affections ‘limit’ the expression of active affections. For Macherey, the term ‘limit’ functions to explain the impact of ‘the affections’ on the full expression of a finite existing mode’s power to act, the uninterrupted affective flux thereby limiting the active expression of the mode’s power to act to only a proportion of its range, which varies between a maximum and a minimum, that is, to less than maximum. Whereas for Deleuze, the term ‘limit’ defines a margin or threshold beyond which a mode’s capacity to be affected ceases to be animated by active affections and therefore ceases to be expressed altogether, that is to say, beyond which a finite existing mode ceases to exist.

Ramond criticizes Deleuze for using the term ‘limit’ in the singular (‘a limit’, writes Deleuze, ‘a margin’) ‘since’, according to Ramond , ‘what is actually designated are two limits (“a maximum and a minimum”), or two margins, “between which”, Deleuze should have written, and not … “in which” the bodies “take form and are deformed”’.⁸⁴ This criticism highlights the difference between the two concepts of ‘limit’ that are used by Macherey and Deleuze respectively. Macherey considers there to be two limits, a maximum and a minimum, the active expression of a mode’s power to act being hindered by the ‘uninterrupted affective flux’ to varying within, or between, these fixed limits. Whereas Deleuze uses the term ‘limit’ in the singular to define a point beyond which a finite existing mode ceases to exist.

How, then, can Deleuze speak of a limit in the singular when he also speaks of a maximum and a minimum?⁸⁵ Deleuze argues that it is only in the ‘physical view’ that the capacity to be affected remains fixed for a given essence, and can therefore be represented by a fixed range of variation between a maximum and a minimum. Macherey’s point of view of modal existence fits this description. However Deleuze introduces another point of view which he calls the ‘ethical view’, in which a mode’s capacity to be affected ‘is fixed only within general limits’.⁸⁶ Before determining the difference between these ‘general limits’ and ‘the limit’ of which he speaks in the singular, it is necessary to explicate Deleuze’s ‘ethical view’.

The ethical view of modal existence

The distinction between the physical and ethical views made by Deleuze involves further distinguishing passive affections from active affections. In the ethical view, passive affections function solely as a limit to the existence of a finite mode. They no longer express modal essence, as all passive affections continue to do with Macherey, but actually limit its expression. What changes from the physical view is the definition of a mode’s capacity to be affected, which was previously understood to be the combination of a mode’s power to suffer and its power to act. According to the ethical view, the power to suffer is no longer considered to be expressive of a mode’s capacity to be affected. A mode’s capacity to be affected is rather expressed solely by its power to act. Deleuze describes the effect of the passive affections on a mode’s capacity to be affected in the following manner: ‘While exercised by passive affections, it is reduced to a minimum; we then remain imperfect and impotent, cut off, in a way, from our essence or our degree of power, cut off from what we can do’.⁸⁷ The passive affections, which, according to the physical view, contribute to a mode’s power of suffering, now only reduce or limit a mode’s power to act to its ‘lowest degree’.⁸⁸ Insofar as active affections express a mode’s power to act, they are the only affections which exercise this new ethical concept of a mode’s capacity to be affected. ‘The power of action is, on its own’, Deleuze argues, ‘the same as the capacity to be affected as a whole’, and this newly defined power to act, ‘by itself, expresses essence’.⁸⁹ Therefore, according to the ethical view, an existing mode’s essence is expressed by its power to act, and its power to act is the same as its capacity to be affected.

To determine how a mode’s conatus functions according to the ethical view it is necessary to return to the previous reference made to Deleuze’s understanding of the relation between a mode’s conatus and its essence. It has been demonstrated that Macherey holds there to be a strict equality between these two aspects of a singular thing. He would therefore not disagree with Deleuze’s statement that the ‘conatus is … always identical to the power of acting itself’,⁹⁰ since Macherey also considers the relation of equality to hold between a singular thing’s power to act and its modal essence. However, their respective interpretations of conatus differ in relation to the ethical view. When Deleuze states that the conatus is the ‘affirmation of essence in a mode’s existence’,⁹¹ what he understands by this is that conatus is the affirmation of the expression of a singular modal essence in the corresponding finite existing mode only insofar as the finite mode actually exists, that is, only ‘once it has begun to exist’.⁹² Now it has been determined that a finite mode only exists for Deleuze insofar as it has a capacity to be affected, which is determined by the extent to which the composite multipli-differenciated assemblage of local integrations, of which it is composed, is further differenciated in more composite global integrations. A mode’s conatus is, therefore, the affirmation of its capacity to be affected, that is, of the extent to which it is further differenciated as a component local integration in more composite global integrations. Insofar as a mode’s conatus is the ‘effort to maintain the body’s ability to be affected in a great number of ways’,⁹³ which, for Deleuze, is the effort to maintain its capacity to be affected, it is the effort to maintain the extent to which it is further differenciated in a great number of ways. The greater the extent to which the global integration characteristic of a mode is further differenciated, the more composite are the global integrations in which it is implicated; the greater is its capacity to be affected, and, consequently, the greater is its power to act. The conatus of a mode thus affirms the expression of its capacity to be affected as its power to act. Deleuze is therefore able to conclude that ‘the variations of conatus as it is determined by this or that [active] affection are the dynamic variations of our power of action’.⁹⁴ The conatus therefore affirms the expression of the dynamic variations of a mode’s power to act.

The three different types of expressive change

Deleuze maintains that finite existing modes are open to three different types of expressive change. To begin with there are mechanical changes determined by the affections that a finite existing mode experiences. These changes correspond to those envisaged by Ramond in his argument about a singular thing’s precisely determined relation of movement and rest, which remains the same despite the variety of displacements undergone by its extensive parts. All three interpretations converge on this point, that Spinoza’s concept of singular modal essence as fixed. There are also dynamic changes in the active expression of a singular thing’s power to act. For Macherey, dynamic changes are incorporated by a singular thing according to the varying degree to which the ‘uninterrupted affective flux’ inhibits or limits the active expression of its power to act. All of a singular thing’s power to act is expressed between the range of a maximum and a minimum, however it is expressed within this range both actively and passively. The active expression of a singular thing’s power to act is therefore open to variation in inverse proportion to its passive expression, the latter taking the form of the limiting uninterrupted affective flux.

Deleuze’s interpretation of the dynamic changes in a mode’s power to act diverges somewhat from that of Macherey. Deleuze introduces the concept of a mode’s capacity to be affected, which, according to the physical view, incorporates its power of acting and power of suffering as the active and passive expression of its degree of power. A mode’s capacity to be affected incorporates the concept of variation in a mode’s power to act as the active expression of its degree of power. However, this arrangement is further complicated in the ethical view, when Deleuze directly correlates a mode’s capacity to be affected with its power to act. For Deleuze, a mode’s capacity to be affected, which is affirmed by its conatus as the expression of its power to act, is open to a third type of change, what he calls ‘metaphysical’ or ontological changes. These changes are those which Deleuze refers to when he writes that mode’s ‘are endowed with a kind of elasticity’.⁹⁵ In the physical view, as far as Deleuze is concerned, the maximum and minimum of the range of variation determined by a mode’s degree of power function only as ‘general limits’. In the ethical view, Deleuze emphasizes instead that ‘while a mode exists, its very essence is open to variation, according to affections that belong to it at a given moment’.⁹⁶ As has been noted, Deleuze argues that a singular modal essence is only fixed qualitatively, insofar as it corresponds to the qualitative modification of an attribute in the form of a complicated intensive quantity or intensive part. It nevertheless remains open to variation under certain conditions quantitatively, insofar as the essential singularity of its ‘intensive part’, or of its singular modal essence, is further differenciated in the form of a component local integration within a more composite global integration. The maximum and minimum are determined as ‘general limits’ because Deleuze uses the term ‘limit’ in the singular to indicate that a finite mode is not so much limited between a maximum and minimum, than it is by the passive affections that it experiences in its interactions with other more composite bodies, which, at any given moment, have the potential to limit its further differenciation, and, therefore, the further deployment of its power to act, and by consequence, its actual existence. Passive affections, for Deleuze, therefore not only function as a limit of the expression of a mode’s active affections, but also of the existence of the finite mode itself. This limit determines the margin of variation of the expression of the given finite mode’s power to act, which varies from a minimum, below which it would cease to exist (intensity = 0), to a maximum, which would be the extent to which its power to act is further differenciated at any given moment in more composite global integrations.

Macherey and Deleuze therefore come to different conclusions concerning the resolution of the apparent contradiction between the double point of view of the singular essences of finite existing modes. According to Macherey, this contradiction is dissipated as soon as the fact that the two theses in question are not situated on the same plane is taken into account. He considers the concept of the margin of variation of a mode’s power to act to be a problem of logic, which is resolved by taking an economic perspective on the system of the affective life of a mode as a whole. Macherey argues that when Spinoza characterizes the elementary forms of affectivity in the unfolding of a mode’s affective life, they happen to be imaginary ideas of its power to act. It is these imaginary ideas which determine the variation of a mode’s power to act between two alternate poles of extreme activity and extreme passivity, a maximum and a minimum. A mode’s affective life is expressed by these ideas which are determined as a state of ‘uninterrupted affective flux’. Macherey argues that a mode’s affective life limits the active expression of its power to act, the associated ideas of the augmentation and diminution of which are experienced only at the level of the imagination, while its essence, conatus and power to act remain fixed once and for all.

Deleuze considers the problematic of the apparent contradiction to be determined by two different perspectives within the Ethics: a physical view and an ethical view. Deleuze agrees with Macherey in regards to the perspective of the physical view. But the differences in their respective interpretations are determined by the different role given to the passive affections. For Macherey, they remain an integral part of the mode’s existence, being expressed by the conatus of a mode although hindering its ability to express fully, or more perfectly, its fixed power to act. Deleuze’s ethical view serves to resolve the apparent contradiction differently. One of the fundamental aspects of the distinction between the physical view and the ethical view is, for Deleuze, a change in perspective of the relation between passive affections and active affections. In the ethical view, only active affections function integrally as constitutive of finite modal existence. The active affections are solely determinative of the mode’s capacity to be affected, which is affirmed by its conatus as the expression of its power to act. Passive affections function solely to limit the expression of the mode’s power to act, and therefore of its existence. By affirming the variability of a mode’s capacity to be affected, as expressed in its power to act, while maintaining the concept of its fixed singular modal essence, Deleuze’s ethical view resolves the apparent contradiction in Spinoza’s Ethics; in fact, according to the ethical view there is no apparent contradiction.

_______________

Notes

¹ Gueroult, Spinoza, l’âme, p. 351.

² Ramond, qualité et quantité dans la philosophie de Spinoza, p. 224.

³ Ibid., p. 194.

⁴ Pierre Macherey, Introduction à l’Ethique de Spinoza, la troisième partie (Paris, 1995), p. 24.

⁵ Deleuze, Expressionism in Philosophy, p. 222.

⁶ Ibid., p. 204.

⁷ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 189.

⁸ Spinoza, Ethics, II, P13L3A2D1. Translated from Ramond, qualité et quantité dans la philosophie de Spinoza, p. 189.

⁹ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 189.

¹⁰ Ibid., p. 226.

¹¹ Spinoza, Ethics, II, P13L7S, For Spinoza, the nature of a mode is determined by its essence, just as for Deleuze, the specific qualitative nature of a global integration is determined by its essential singularity.

¹² Ramond, qualité et quantité dans la philosophie de Spinoza, p. 224.

¹³ Ibid., p. 226.

¹⁴ Ibid., p. 228.

¹⁵ Spinoza, Ethics, III P7.

¹⁶ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 191.

¹⁷ Ibid., p. 189.

¹⁸ Ibid., p. 191.

¹⁹ Ibid., p. 194.

²⁰ Ibid.

²¹ Ibid.

²² Ibid., p. 193. Additional italics.

²³ Ibid., p. 193.

²⁴ Ibid., p. 292.

²⁵ Deleuze, Expressionism in Philosophy, p. 191.

²⁶ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 202.

²⁷ Deleuze, Expressionism in Philosophy, p. 173.

²⁸ See Ibid., p. 210: ‘For extensive parts determine one another from outside and ad infinitum’.

²⁹ ‘Modes come into existence, and cease to exist, by virtue of laws external to their essences’ (Ibid., p. 210).

³⁰ Ibid., p. 209.

³¹ Ibid., p. 208.

³² Ibid., p. 209.

³³ Spinoza, Ethics, II, P13L4–7.

³⁴ Deleuze, Expressionism in Philosophy, p. 213.

³⁵ Contrary to the argument that Ramond constructs against Deleuze in Ramond, qualité et quantité dans la philosophie de Spinoza, p. 226.

³⁶ Deleuze, Expressionism in Philosophy, p. 209.

³⁷ Ibid., p. 210.

³⁸ Ibid., p. 209.

³⁹ Ibid., p. 213. See the Ethics, II, P8C.

⁴⁰ I am indebted to Etienne Balibar and the concept of transindividuality that he introduces to Spinoza studies in his paper Spinoza: from individuality to transindividuality, which provided the initial impetus to determine the process of individuation according to the logic of different/ciation. In the section of his paper entitled ‘Transindividuality as a concept of integration’, Balibar introduces ‘the concept of the individual as a determinate level of integration’ (p. 16). He argues that the concept of the individual in Spinoza ‘is dependent on the consideration of individuation as a process’, and that this process of individuation is always already transindividual (p. 16). It is this idea of the always already transindividual nature of the process of individuation as determined according to the logic of different/ciation that is being developed in the present work.

⁴¹ Matheron discusses how the human being is more complex though less composite than a political society: ‘There are … individuals that are very integrated yet not very complex. For example, a stone: nearly all that happens in it concerns it’s structure, but nearly nothing happens to it; it’s essence is therefore very poor [pauvre]. Inversely, there are individuals that are very complex yet not very integrated. For example, a political society: lots of things happen to it, it’s possibility of internal variation assure it of a great chance of survival; but the majority of these variations (private life, loves and personal hatreds, etc.: all things indifferent to the eyes of the law) do not in any way concern its structure; and this is why it’s essence is very much less perfect than that of man, even though it has a higher degree of composition. Then again, there are individuals that are both very integrated and very complex: man, in particular, whose essence, for this reason, is extremely rich’. Translated from Alexandre Matheron, Individu et communauté chez Spinoza (Paris, 1969), p. 58. In the letters to Boyle (Letters 6, 11, 13, and 16) , Spinoza expresses his doubts concerning the mechanist explanation of phenomena, which supposes the hierarchy from simple to complex, or the idea of a hierarchy of complexity based upon a hierarchy of composition. See Spinoza, The collected Works of Spinoza vol. I, pp. 173–218.

⁴² Spinoza, Ethics, II, P13PostII.

⁴³ Ibid., II, P13A3.

⁴⁴ Ibid., II, P13PostV.

⁴⁵ Deleuze, Expressionism in Philosophy, p. 222.

⁴⁶ Ibid.

⁴⁷ Ibid., p. 230.

⁴⁸ Ibid.

⁴⁹ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 224.

⁵⁰ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 74.

⁵¹ Ibid., p. 24.

⁵² Ibid., p. 355.

⁵³ Spinoza, Ethics, III, the general definition of the affects, translated from Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 357.

⁵⁴ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 355.

⁵⁵ Curley and Gueroult (See Gueroult, Spinoza, l’âme, p. 10) both note in relation to Spinoza’s usage of mens, animus and anima, that ‘he uses all three terms pretty much indifferently for the mind, conceived intellectualistically’(Spinoza, The collected Works of Spinoza Volume I, p. 645). However, commentators generally translate ‘mind’ for mens and animus, and ‘soul’ for anima. When presenting the translation and interpretation of particular passages by different commentators, I have maintained the use of either ‘mind’ or ‘soul’ where appropriate to the development of the commentators’ arguments, despite considering the concepts to be interchangeable. Therefore, during the development of a number of arguments presented in chapters 6, 7 and 9, an understanding that ‘soul’ = ‘human Mind’ will be assumed.

⁵⁶ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 121.

⁵⁷ Ibid., p. 333.

⁵⁸ Ibid., p. 125.

⁵⁹ Ibid., p. 123.

⁶⁰ Ibid., p. 124.

⁶¹ Ibid., p. 125.

⁶² Ibid.

⁶³ Ibid., p. 121.

⁶⁴ Ibid., p. 125.

⁶⁵ Ibid.

⁶⁶ Ibid., p. 121. Macherey’s formulation of the theory of the affections as a state of ‘uninterrupted affective flux’ is remarkably coincident with the Hegelian formulation of the indivisibility of intensive magnitude presented in the following phrase: ‘The ascent and descent in the scale of degrees is a continuous progress, a flux, which is an uninterrupted, indivisible alteration’ (Hegel, Hegel’s Science of Logic, p. 219).

⁶⁷ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 292.

⁶⁸ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 70.

⁶⁹ Ibid., p. 36.

⁷⁰ Ibid., p. 125n.

⁷¹ Ibid., p. 73.

⁷² Ibid., p. 73n.

⁷³ See Spinoza, Ethics, III, D3: ‘By affect I understand affections of the Body by which the Body’s power of acting is increased or diminished, aided or restrained, and at the same time, the ideas of these affections’.

⁷⁴ Macherey bases his conception of the functioning of the ‘uninterrupted affective flux’ on the Ethics, III, P17, where Spinoza introduces the theme of affective ambivalence <fluctuatio animi> (Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 163). This relation is further explicated in the sections of chapter 7 entitled ‘The relation between passivity and activity in the affective life’, p. 170, and ‘A joyful passive affection can be reversed to sadness’, p. 177.

⁷⁵ Deleuze, Expressionism in Philosophy, p. 222.

⁷⁶ Spinoza, Ethics, IV P39S: ‘what shall we say of infants? A man of advanced years believes their nature to be so different from his own that he could not be persuaded that he was ever an infant, if he did not make this conjecture concerning himself from [NS: the example of] others’. See Deleuze, Expressionism in Philosophy, p. 222.

⁷⁷ Deleuze, Expressionism in Philosophy, p. 223.

⁷⁸ Ibid.

⁷⁹ Ibid., p. 224.

⁸⁰ Ibid.

⁸¹ Ibid., p. 223.

⁸² Ibid., p. 223. See G.W. Leibniz, ‘On Nature itself …’, Philosophical Essays, trans. R. Ariew and D. Garber (Indianapolis, 1989), pp. 155–67.

⁸³ Ibid., p. 224.

⁸⁴ Ramond, qualité et quantité dans la philosophie de Spinoza, p. 226.

⁸⁵ As has already been noted in the following: ‘essences or degrees of power always correspond to a limit (a maximum or minimum)’. Deleuze, Expressionism in Philosophy, p. 204. See the section of this chapter entitled ‘Are modal essences fixed, or do they admit a certain degree of variation?’, p. 135.

⁸⁶ Deleuze, Expressionism in Philosophy, p. 225.

⁸⁷ Ibid.

⁸⁸ Ibid., p. 224.

⁸⁹ Ibid., p. 225.

⁹⁰ Ibid., p. 231.

⁹¹ Ibid., p. 230.

⁹² Ibid.

⁹³ Ibid.

⁹⁴ Ibid., p. 231.

⁹⁵ Ibid., p. 222.

⁹⁶ Ibid., p. 226.

Chapter 7

The transformation of the characteristic relations of modal existence

The distinction between joyful passive affections and sad passive affections

In the article ‘The encounter with Spinoza’, Macherey gives an account of Deleuze’s project in Expressionism in Philosophy. According to Macherey, ‘Deleuze’s expressionist reading of Spinoza … forces the text out of itself by introducing the minimal dislocations needed to get it moving’.¹ He argues that Deleuze carries out the process of introducing dislocations to the text ‘with an extraordinary precision that quite forbids any suggestion of arbitrariness’, however he does suggest that ‘Deleuze might be said to stray from the text, not by introducing completely foreign elements, but by amplifying certain themes that are not fully developed, thereby modifying the internal economy of their relations with other more prominent aspects to which historians of philosophy have devoted most attention, but which seem from this viewpoint less significant’.² One of these not ‘completely foreign elements’ that is ‘amplified’ by Deleuze is the theme of ‘joyful passions’. In his interpretation of Spinoza’s theory of relations in Expressionism in Philosophy, Deleuze assigns a specific role to joyful passions. They are characterized as a significant determinant in the dynamic changes or transformations of the characteristic relations of finite existing modes. The theme of joyful passions is pivotal in further distinguishing Deleuze’s reading of Spinoza’s theory of relations from that offered by Macherey in Introduction à l’Ethique de Spinoza, la cinquième partie.

The dynamic changes or transformations of the characteristic relations, or global integrations, determinative of a finite existing mode occur when the mode’s level of activity increases or inversely when its level of activity decreases, that is, with the changes in the extent to which it is further differenciated.³ It is while discussing the General Definition of the Affects⁴ that Macherey formulates the problem which effectively distinguishes his interpretation of this aspect of the Ethics from that of Deleuze. He raises the following question: ‘Can the soul be completely active, without at all being passive, or does it rather find itself permanently placed between the two extremes of passivity and activity, following regimes which make it lean sometimes to the side of activity sometimes to that of passivity? And then what are the thresholds which swing one of these regimes into the other?’⁵ Each interpreter approaches this problem differently. Macherey’s reading of the Ethics is progressive, moving carefully through each part of the text reconstructing the consistency of its arguments, following the developments of the relations characteristic of singular things, as the text proceeds up to the problematic transformations of the fifth and final part of the Ethics. In fact, the first of the fivevolume study to appear is devoted to the fifth part of the Ethics.⁶ Indeed Macherey argues that ‘the end coincides in a certain manner with the beginning. It is therefore no more absurd to take the Ethics by its end than by its beginning’.⁷ There is therefore a retrospective consistency to Macherey’s study which provides an interpretation that, from a Deleuzian point of view, can be considered to be characteristic of the physical view of the Ethics. Deleuze’s reading, however, follows a different trajectory, one that is characteristic of the ethical view, which already takes the transformations of the fifth part into account, offering rather an interpretation of modal existence as conceived at the end of the Ethics, or from the point of view of the Ethics as a whole.

Macherey and Deleuze are in accord in regards to the fixity of singular essence, but their interpretations differ in regards to the transformations of the characteristic relations determinative of singular things. According to Macherey, the affective life of a singular thing is constituted by its ideas or passions which are expressed as an ‘uninterrupted affective flux’.⁸ The transformations of the characteristic relations of a singular thing correspond to the varying degrees to which the uninterrupted affective flux hinders or limits the active expression of a mode’s power to act to within the range of a maximum and a minimum. All of a mode’s power to act is expressed, however, according to the uninterrupted affective flux, it is simultaneously expressed both actively and passively. The passive affections, or the passions, are the mark of a negation, and, inversely, the active affects, or actions, are the active expression, or affirmation, of a singular thing’s power to act. Macherey considers passion, with its logical mark of ‘negativity’, to be that ‘which is found most naturally in man’. The question for Macherey is therefore ‘to know whether man can ever completely escape this logic and engage in actions which are not marred by such a limitation?’⁹

Deleuze, however, considers the transformations of the relations characteristic of modal existence to implicate a mode’s capacity to be affected. A mode’s capacity to be affected is constituted by its active affections. Passive affections, on the contrary, function only to limit its capacity to be affected. This limit functions within the range of a maximum and minimum, that is, a mode’s capacity to be affected, which is affirmed by its conatus as the expression of its power to act, is open to variation within the ‘general limits’ of this range. According to Deleuze, the variation of a mode’s power to act is directly limited by the passive affections to which it is subjected, rather than proportionally limited, as Macherey proposes.

The difference between Macherey’s and Deleuze’s reading of Spinoza’s theory of relations rests with their respective interpretations of the role of passive affections. According to Macherey, they remain an integral part of the existence of a singular thing, being expressed by its conatus even though hindering its capacity to actively, or more perfectly, express its fixed power to act. According to Deleuze, on the contrary, only active affections function integrally as a part of modal existence. Passive affections function rather to limit the existence of a finite mode, that is, of the active affections constitutive of its capacity to be affected, which is affirmed by its conatus as the expression of its power to act. This however does not exhaust the differences between their respective interpretations of passive affections but rather prepares for a further distinction. Deleuze argues that ‘the opposition of actions and passions should not conceal the other opposition that constitutes the second principle of Spinozism: that of joyful passive affections and sad passive affections’.¹⁰ Spinoza first introduces the notions of joy and sadness in the Ethics, II, P11S, by making explicit reference to them as passions: ‘By joy, therefore, I shall understand in what follows that passion by which the mind passes to a greater perfection. And by sadness, that passion by which it passes to a lesser perfection’. The other reference that Deleuze cites is from the Ethics, III, P58, where Spinoza introduces joys and desires whose active character sets them apart from those joys and desires that are passions because they are determined by external encounters. Spinoza writes that ‘Apart from the joy and desire that are passions, there are other affects of joy and desire that are related to us insofar as we act’. These are the only two explicit references to passions which are joys to be found in the Ethics. In Expressionism in Philosophy, Deleuze characterizes these joyful passions as ‘joyful passive affections’, and it is from this starting point, with joyful passive affections, that Deleuze begins his account of the transformations of the characteristic relations of finite existing modes.

Macherey concedes that ‘the notion of a “joyful passion” is not in fact entirely absent from Spinoza’s text, at least at first glance’.¹¹ However, when Macherey returns to the Ethics, after this ‘first glance’, to offer an account of the transformations of the characteristic relations of singular things, the notion of a joyful passion does not retain the same significance that Deleuze assigns it in Expressionism in Philosophy. In fact, ‘joyful passions’ are implicated quite differently in Macherey’s reading of the Ethics. However, before developing Macherey’s account of what he calls ‘passionate joys’, the role played by joyful passive affections in Deleuze’s account of the transformations of the characteristic relations of finite existing modes will be explicated.

The role of joyful passive affections in Deleuze’s account of modal existence

Deleuze actually prefigures his discussion of joyful passive affections at the beginning of chapter 15 of Expressionism in Philosophy, when he argues that ‘Our passive joy is and must remain a passion: it is not “explained” by our power of action, but it “involves” a higher degree of this power’.¹² What does Deleuze understand by correlating a passion with an increase in a mode’s power to act? This would seem to contradict his concept of the role of passive affections in the determination of a finite existing mode, that is insofar as they function solely to limit its existence. This suggestion of a contradiction is reinforced by the fact that Deleuze follows his introduction of the distinction between joyful passive affections and sad passive affections by the statement ‘One increases our power, the other diminishes it’.¹³ It is this suggestion of a contradiction that allows Macherey to regard Deleuze’s joyful passive affections as occupying ‘an intermediate place between the two extremes of freedom and slavery, and thus to provide the framework for the transition from one to the other’.¹⁴ Deleuze seems to be arguing that although joyful passive affections are passions they function more or less actively and therefore can be seen to occupy an intermediate place between passions and actions, or, in effect, to mediate between them. Indeed, in the role that joyful passive affections play for Deleuze in the transformations of the characteristic relations of finite existing modes, Macherey suggests that ‘one might see the outline of a dialectic’.¹⁵

However, on returning to Expressionism in Philosophy, a different logic is beginning to unfold, one according to which joyful passive affections mediate not between active and passive affections, which would be suggestive of a Hegelianstyle dialectic, but rather solely between different active affections. In order to explicate the mechanism by means of which this logic operates, it is necessary to determine exactly what the relation is then between a joyful passive affection and the increase in power to which Deleuze relates it. Deleuze suggests that we ‘come closer to our power of action’¹⁶ insofar as we are affected by the joy of a joyful passive affection. He argues that ‘passive joy is produced by an object that agrees with us, and whose power increases our power of action, but of which we do not yet have an adequate idea’,¹⁷ however he maintains that ‘it never increases enough for us to become the adequate cause of the affections that exercise our capacity to be affected’.¹⁸ The initial affect is a passion because we are effected from the outside by an external object, however this object agrees with our nature and, consequently, is not harmful to us. We therefore do not experience the passive affection as the passion of sadness because our power to act is not diminished by the encounter. One would expect a feeling of ambivalence to be experienced since at this stage our perfection has been neither augmented nor diminished. Yet, insofar as the external body ‘agrees’ or ‘has something in common with our nature’ the potential for the combination of the power to act of the external body with our own, and therefore the increase in power that this would involve, promotes the feeling of joy which allows the overall affect to be described as a joyful passive affection. In order to determine what Deleuze understands by an object that ‘agrees’ or ‘has something in common with our nature’, it is necessary to turn to the logic of different/ciation. According to this logic, the nature of a finite existing mode or individual is characterized by the specific qualitative nature of the infinite branches of its global integration, whose form is determined by the type of essential singularity created by its composite relation. The specific qualitative nature of the infinite branches of a global integration, or the nature of the finitude of an existing mode or individual, is determinable solely in reciprocal relation to that of another global integration, in the form of the infinite branches of a more composite global integration, that is, solely insofar as it is further differenciated. The specific qualitative nature of the infinite branches of each global integration ‘agree’ solely insofar as they can be further differenciated in relation to one another, as global integrations, thus generating a new more composite global integration, the infinite branches of which incorporate, and therefore determine or actualize, the specific qualitative nature of its constitutive global integrations. The specific qualitative nature of the infinite branches of each global integration converges with the infinite branches of the more composite global integration. Therefore the concept of finite existing modes or individuals whose natures agree corresponds to the potential for the complication or convergence of their specific qualitative natures in the infinite branches of a more composite global integration. Insofar as the effect of this external body on our own is experienced as an affection which is explained by the external body, it remains an inadequate idea of the imagination, and therefore a passion. To distinguish passive joys from active joys Deleuze argues that an ‘Active joy we produce by ourselves, it flows from our power of action itself, follows from an adequate idea in us’.¹⁹

According to Deleuze’s ethical view of the constitution of a finite existing mode, joyful passive affections are passions because they limit the expression of our power to act, and yet correspond to a feeling of joy because they are somehow implicated in an increase of that power. This can only work if joyful passive affections are understood to function at the limit imposed by passive affections. The joy of a joyful passive affection can therefore be understood insofar as it affirms that limit while simultaneously announcing the potential for positive transformation, that is the surpassing of the limit, or an increase in the power to act, rather than functioning solely as a limit marking the point beyond which a finite existing mode ceases to exist, as do sad passive affections. Deleuze distinguishes his interpretation of the physical view from that of Macherey when he writes that ‘Spinoza’s project is thus realized … by the aid of joyful passions restricting passions to the smallest part of ourselves, so that our capacity to be affected is exercised by a maximum of active affection’.²⁰ It is against this point of the division of the passions that Macherey constructs his critique of joyful passive affections and presents his own account of joyful passions in ‘The encounter with Spinoza’, which will be developed in the following section.

Deleuze, however, goes on to describe how joyful passive affections are implicated in this process according to the ethical view. Joyful passive affections ‘still do not give us possession of our power of action; we have no adequate idea of objects that agree in nature with us; joyful passions are themselves born of inadequate ideas, which only indicate a body’s effect on us. We must then, by the aid of joyful passions, form the idea of what is common to some external body and our own. For this idea alone, this common notion, is adequate. This is the second stage of reason; then, and only then, do we understand and act, and we are reasonable: this not through the accumulation of joyful passions as passions, but by a genuine “leap”, which puts us in possession of an adequate idea, by the aid of such accumulation’.²¹ Only to the extent that this initially inadequate relation results in the production of active joys, and therefore in an increase in our power of acting, are joyful passive affections implicated in the transformative process. Joyful passive affections indicate a partial or inadequate idea of something common to both our own body and an external body which effects it. They indicate the potential for an increase in our power to act but are not themselves directly related to that increase in power. It is rather the active affections which follow from joyful passive affections that are directly associated with the increase in power. The suggestion of a contradiction between joyful passive affections as passions and the increase in our power associated with them is therefore unfounded.²²

In Expressionism in Philosophy, Deleuze describes the three types of relation that can occur between two different bodies. An individual or human being can encounter another existing body whose nature agrees with its own, in which case either: the individual’s conatus will strive to bring the body into composition with its own, thus the body becomes a component part of the overall relation of the human being, that is, a component relation in the composite multipli-differenciated assemblage of local integrations constitutive of a finite existing mode, individual, or human being; or, they can both strive together to combine with each other to form a third more powerful relation in which the individuals of both composing bodies are preserved, that is, they can be further differenciated in relation to one another as global integrations thereby constituting a more composite global integration. Conversely, an individual or human being can encounter a different body that does not agree with its own nature, that is, the specific qualitative nature of which diverges with its own. The extensive parts of the given finite existing mode, individual, or human being, are then determined to enter into a different relation, which is determined rather by the specific qualitative nature of the infinite branches of the global integration characteristic of the external body which is encountered. They are therefore effectively stripped from the relation characteristic of the given finite existing mode, being determined instead by that of the external body. This type of relation therefore tends to decompose or destroy the given finite existing mode, or a part of it. Deleuze argues that all of these encounters initially involve only passive affections since they involve encounters with external bodies. However, in the first two encounters, the individual experiences the initial passive affection as a joyful passive affection. The feeling is joy, since it is produced by the idea of a body with which the individual’s nature agrees, indicating the potential for the convergence of the specific qualitative nature of the infinite branches of their respective global integrations. As for the third encounter, a feeling of sadness always results from an encounter with a body whose relation does not agree with one’s own, where one’s power of acting is diminished. A sad affection is the only way of knowing that there is disagreement, or divergence, between the nature of the external body and our own. The converse of this also applies: the feeling of Joy associated with a joyful passive affection is initially the only way of knowing that our natures in some respect actually agree, or converge.

The body is continuously affected by relations with other bodies which are external to it. The ideas ordinarily formed in the mind are therefore ‘inadequate’, that is to say not properly speaking false, but ‘mutilated and confused’; this mode of thought is that of the imagination. The relations that dispose the body to experience joyful and sad passions are inadequate relations to which correspond inadequate ideas. When we experience a passive affection, we lack knowledge both of ourselves and of the body which acts or seems to act on us from the outside and which produces in us the affection. The first kind of knowledge, or imagination, is composed of those partial or inadequate ideas which involve the privation of the knowledge of their cause. An adequate idea, however, does not signify the correspondence of an idea and the object it represents, but rather the conformity, within the attribute of Thought, of an idea with the idea which is its cause, that is, between two or more ideas, each therefore being a cause of the adequate idea. Ideas within the attribute of thought are determined, as are bodies in the attribute of extension, according to the logic of different/ciation. The causality between ideas is a differencial causality, two or more ideas are reciprocally determined as global integrations in the form of a more composite global integration or adequate idea, each reciprocally determined idea itself being composed by a composite multipli-differenciated assemblage of local integrations or ideas. Each idea therefore represents, in itself, something that is common to all of the ideas which together conform in the adequate idea. This is what Spinoza describes as the second kind of knowledge, or reason.

The simplest of common notions

In the second part of the Ethics, Spinoza introduces the concept of ‘common notions’ as the ‘foundation of our reason’.²³ Deleuze distinguishes two main varieties of common notion, those which are less universal, which he describes as the most useful, and those which are most universal. He argues that ‘the most universal common notions … allow us to determine the viewpoint beyond which a very general agreement between two bodies ends; they show us how and why opposition appears when we adopt a “less universal” viewpoint on these same two bodies. We are able, by making an experiment in thought, to vary a relation up to the point where the corresponding body takes on a nature in some sense “contrary” to its own; we can thereby understand the nature of disagreements between bodies with these or those relations’.²⁴ The less universal or simple common notions represent ‘what is common to a human Body and to certain external bodies’²⁵ by which the human Body is usually effected. Their natures agree or are compatible in so far as there is something common to each of them. That which is generally considered to be common is the idea of the agreement in the proportion or ratio of motion and rest of the bodies involved, that is, in the extent to which their specific qualitative natures converge such that they are able to be further differenciated in relation to one another. A simple common notion can therefore be understood to be the idea that there is a similarity of composition between two or more bodies whose natures therefore directly agree or are compatible. Bodies whose natures agree are like parts of a whole, the whole functioning in relation to those parts, and the parts are common to the whole. The whole being the more composite global integration constituted by the reciprocal relation between the parts, or global integrations. The agreement in the attribute of extension, or commonness in the attribute of thought, is determined by the reciprocal relation between the global integrations, or parts, which are thus incorporated in the more composite global integration, or whole, as its component parts. Just as the causality between bodies is a differencial causality, from the reciprocal relation between global integrations to the more composite global integration which they together constitute, so too is the causality between ideas, or images of things. The distinction between bodies and ideas involves two different points of view of the same composite multiplidifferenciated assemblage of global integrations, one from the point of view of the attribute of extension, the other from the point of view of the attribute of thought; the laws of causality being determined, in both instances, according to the logic of different/ciation. So the ‘experiment in thought’, to which Deleuze refers, is not only to think of the relations between bodies according to the logic of different/ciation, but also to think of the relations between ideas according to this same logic. The idea of commonness, incorporated in the form of the simplest of common notions, allows the order of Nature to be understood adequately as an order of constitutive relations according to which: bodies agree with and are opposed to one another from the point of view of the attribute of extension; and ideas are either common or not from the point of view of the attribute of thought, that is, according to which the specific qualitative natures of the infinite branches of the global integrations determinative of finite existing modes either converge or diverge. The formation of these common notions is, for Deleuze, the first step in the practical constitution of reason.

Spinoza maintains that the ideas that we generally have of ourselves, and of external bodies, are only inadequate ideas or passive affections which indicate an encounter between some external body and our own. A joyful passive affection, since it is a passion, is always the result of an external cause, and is thus always indicated by an inadequate idea. However, since it is a joyful passive affection, it indicates that there is something common to an external body and our own, or that it has a nature compatible, or potentially convergent, with our own. Deleuze argues that ‘joyful passions are themselves born of inadequate ideas, which only indicate a body’s affect on us. We must then, by the aid of joyful passions, form the idea of what is common to some external body and our own. For this idea alone, this common notion, is adequate’.²⁶ According to Deleuze, the experience of a joyful passive affection can induce the formation of the corresponding common notion. The first common notions formed by an individual are those which apply to its body and to another whose nature agrees directly, or converges, with its own and therefore affects its body with joy. Deleuze argues that ‘the first common notions we form are thus the least universal, those, that is, that apply to our body and to another that agrees directly with our own and affects it with joy’.²⁷ When our Mind forms an idea of what is common to the external body and our own, it forms a common notion. The joyful affection then ceases to be passive and becomes active. By indicating that there is something common, that there is a connection between the bodies, a joyful passive affection can initiate the formation of a common notion. Deleuze maintains that it is ‘the joyful affection itself [which] induces us to form the corresponding common notion’.²⁸ A common notion is an adequate idea of the relation, which therefore incorporates the cause of the affection within the very idea of that affection, that is, insofar as the reciprocal relation between global integrations, as causes, are further differenciated in a more composite global integration. Deleuze maintains that Spinoza describes an affection which expresses its cause in this way as no longer passive, but active. The joy of a joyful passive affection no longer indicates an inadequate idea of an object that agrees with us, but the necessarily adequate idea of what is common to that object and ourselves. ‘An affect which is a passion ceases to be a passion as soon as we form a clear and distinct [adequate] idea of it’.²⁹ An adequate idea of the affection is formed when the cause of the affection is attached to what is common to the bodies involved, that is, when the potential for convergence of their specific qualitative natures is actualized, insofar as they are further differenciated, in a more composite global integration. This is the ‘leap’ that Deleuze speaks of, from inadequate to adequate ideas, from joyful passive affections to active joys, from passions to actions; the ?leap that corresponds to the actualization of the potential function between the global integrations involved, according to the logic of different/ciation.

In the Ethics, V, P8,³⁰ Spinoza argues that the more external causes that act together to provoke a particular affection, the greater is that affection. Insofar as a joyful passive affection, by virtue of indicating commonness,³¹ relates to several things or images of things at once, the greater the indication of commonness. As Spinoza writes: ‘the Mind is less acted on by an affect which determines it to consider many things together than by an equally great affect which keeps the mind engaged solely in considering one or a few objects’.³² Therefore the greater the number of objects or affects associated with the joyful passive affection, ‘the more it engages the Mind’,³³ that is, the greater the chance that the mind will form an idea of what is common, or of the commonness between them, that is, the greater the chance that the mind will form a common notion.

The accumulation of joyful passive affections

Before giving an account of the role played by joyful passive affections in the dynamic changes or transformations of the characteristic relations of an existing finite mode according to the logic of different/ciation, which is explicated in detail in chapter 8, it is necessary to distinguish Deleuze’s interpretation of Spinoza’s theory of relations from that of Macherey. According to Deleuze, the ‘natural situation’ of our existence as human beings is such that we are filled with inadequate ideas and passive affections. This is so because we ‘are continuously affected by external bodies’.³⁴ Before we can form common notions, we must learn to distinguish sad passions from joyful passions, what Deleuze describes as ‘a starting point in joyful passions’.³⁵ Sad passions are inadequate ideas that arise from the experience of random encounters with external bodies, whereas joyful passions are inadequate ideas which arise from the encounters with external bodies that have something in common with our own. The immediate idea that we have of these external bodies that have something in common with our own is partial and therefore imaginary. Insofar as this encounter is associated with the experience of joy we can form an idea of there being something common to the external body and our own. We desire to increase this initial joy by striving to determine or to form an idea of what it is that is specifically common to our Body and the external body by means of the simplest of common notions. Our chances of achieving this is improved to the extent that we relate or imagine several things at once as similarly common to our body and the external body, thereby increasing the number of affections associated with the joy of the joyful passive affection, what Deleuze describes as ‘the accumulation of joyful passive affections’.³⁶ Although joyful passive affections are inadequate ideas of the imagination, and as such, involve privation of the knowledge of their cause, they are at the same time affections that in some way, according that is to the logic of different/ciation, ‘involve’, or implicate, that cause. The imagination is composed of inadequate ideas which, through an understanding of their cause, by means of the joyful passive affections and the simplest of common notions, may be transformed into adequate ideas, thereby constituting reason. In this way, the joyful passive affection is the mechanism by which the mind moves from an inadequate idea to an adequate idea, and by which the body moves from experiencing a passion to an action. Deleuze argues therefore that ‘the active joys that flow from common notions find as it were their occasional causes in passive affections of joy’,³⁷ and, according to Deleuze, the only way of reaching an adequate idea is by means of the mechanism of joyful passive affections.

The relation between passivity and activity in the affective life

Macherey does not agree with the division of passive affections into joyful passive affections and sad passive affections. In ‘The encounter with Spinoza’, he claims ‘rather bluntly that for Spinoza all passions, without exception, are sad – even those that are or appear to be joys. Or that they are all ultimately sad, in a sort of passionate entropy’.³⁸ Macherey distinguishes what he considers Spinoza to be referring to in the Ethics, II, P11S and the Ethics, III, P58 as ‘passionate joys’ from that which Deleuze characterizes as ‘joyful passive affections’.³⁹ Contrary to what he considers to be Deleuze’s point of view, Macherey maintains that passionate joys, ‘which are in fact imaginary joys linked to encounters with external bodies, cannot be assembled into a coherent stable group, but rather tend inevitably to conflict, tending not towards composition but towards decomposition’.⁴⁰ Macherey considers passionate joys to characterize the mental state which corresponds to thefact that ‘the human Body … can be affected in a great many different ways by one and the same body’.⁴¹ He speaks therefore of a ‘sad destiny of the passions’,⁴² that is to say that although the imagination always initially asserts the presence of its object, as an image, it can be affected by causes which actually exclude the presence of this object,⁴³ thus propelling the imagination into a sort of ambivalence or confusion. Macherey maintains that ‘when this is produced, the mind no longer knows which side to turn to, but rather finds itself plunged into a state of ‘mental confusion’ fluctuatio animi), which reveals its profound disarray’.⁴⁴ Macherey considers the fluctuatio animi to be an extreme consequence of the incertitude of the affective life, and an exemplary illustration of its state of instability. In the Ethics, III, P17, Spinoza introduces the theme of the fluctuatio animi in the case of a sadness which doubles as a joy. Macherey explains this case in the following manner: Spinoza ‘starts by presenting a sad affect attached ordinarily to an object, and, if we understand well, it is in this case that the object is cause “in itself’ of the affect; then he shows how, by contamination, because the object in question appears to resemble another object which ordinarily gives us joy, this joy is artificially transferred onto the first object, which is then the cause “by accident” of this affect of which the second object would itself be the cause “in itself.” The first affect proceeds from a direct association while the second proceeds from an indirect association by transfer, according to which the object that the affect is fixed on is considered analogically, as representing another object, and not itself’.⁴⁵ Therefore, the fluctuatio animi permanently exposes the joy associated with this affect to the risk of reversing to sadness, and vice versa. This is why Macherey considers all passions, ‘including joys that are passions’, to have a ‘sad destiny’,⁴⁶ which cannot somehow be transformed into something active, which Macherey accuses Deleuze of attempting to do with joyful passive affections. Macherey argues that ‘not only is it not enough to multiply, to accumulate such joyful passions to provide a framework for the soul to accede, in a qualitative leap, to the level of activity, such an accumulation is actually impossible’.⁴⁷ Macherey maintains that a joyful passive affection, as characterized by Deleuze in Expressionism in Philosophy, is ‘a contradiction in terms, corresponding at best to a passing, unstable and literally non-viable state of our constitution’.⁴⁸ The uninterrupted fluctuation of affections, as demonstrated by the fluctuatio animi, excludes the possibility of affirming active or passive affections independently of each other.

The question, according to Macherey, is whether or not it is possible to stabilize the uninterrupted affective flux, by liberating ‘affectivity’ from its perpetual alteration. He argues that ‘in order to do this it is necessary to determine the means by which the mind … can engage itself in other forms of activity than that which consists in affirming … the actual existence of the body’.⁴⁹ One way of displacing and inverting the relation between passivity and activity in the affective life is to determine the progressive intermediate thresholds between the two. Macherey argues that the imagination, which consists of ideas which are all inadequate and therefore passive, and the understanding, which consists of thinking adequately and which restores to the mind all the active stature of its power, are opposing figures, which ‘present themselves from the outset as terms of an alternative: on one hand, the imagination subjugates the mind by preventing it from developing the plenitude of its power (puissance); on the other hand, true knowledge liberates the mind by permitting it to reappropriate the forms of its manifestation and to make of them expressions of its essential nature. The question then is to understand how the passage between these two limited states operates’.⁵⁰ Of course Macherey accedes to the fact that if the mind could be completely dispossessed of its active character it would be destroyed. The corollary of this is the question of whether or not the mind can be totally active, ‘ceasing definitively to be passive?’⁵¹ According to Macherey it is only in the fifth and final part of the Ethics, in the rational treatment of affectivity undertaken by propositions 1 to 20, that a response to this question is given.

This same question can be used to distinguish Deleuze’s reading of Spinoza from that of Macherey. Deleuze’s configuration of an existing finite mode is such that it exists solely insofar as it is active, the passive affections it experiences functioning solely to limit its existence and therefore its capacity to be affected, which is affirmed by its conatus as the expression of its power to act. In ‘The encounter with Spinoza’, Macherey argues that ‘there is in sadness nothing even potentially positive that might allow one to see it as a limiting expression of conatus’.⁵² Instead, Macherey considers sadness to be a ‘contraction of the power of thinking of the mind, which momentarily depresses it, but which would be unthinkable without the persistence of this power, of which it continues to give a paradoxical expression’.⁵³ Sadness rather results from the fluctuatio animi associated with the uninterrupted affective flux of a singular thing’s affective life, which does not affirm the active expression of its conatus. What Macherey understands is that whatever the orientation taken by the variation which affects the mind, whether active or passive, joyful or sad, the mind continues in all cases to be animated by the pressure of conatus, and these ‘transformations’ that the mind does not cease to experience in no way alter the constancy of the conatus. Macherey can be considered to be critical of Deleuze’s reading when he argues that a mind, which could cease definitively to be passive, would then achieve a ‘total intelligence of its own operations and [would] directly see these as produced in the idea of God, or infinite intellect, that which seems to exceed the power of the human mind’.⁵⁴ As far as Macherey is concerned, a mind defined as existing solely insofar as it is active could only be the mind of God, not that of a human being, since man is without end affected by encounters with other bodies which seem to act on it from the outside, therefore it will always experience passive affections. Therefore, according to Macherey, the installation between imagination and reason, or activity and passivity, of an absolute rupture which separates them abstractly one from the other, is not feasible. The only way to dominate the unfolding of the affective life is to make it pass from a regime of greater passivity to one of greater activity; to make the mind accede to a higher level of activity which expresses its true active power. The objective, according to Macherey, is therefore to progressively enact a reversal of the relation between passivity and activity. The problem for him then is how to modify this relation in such a way.

The many ways of imagining

The solution to the dilemma of the relation between passivity and activity that Macherey offers in Introduction à l’Ethique de Spinoza, la cinquième partie, rests on a distinction between the ‘many ways of imagining’, which allows a ‘comparative evaluation of the forms of the imagination’.⁵⁵ This distinction appears in the Ethics, V, between proposition 5, according to which ‘we imagine simply’, and the formulation of the scholium of proposition 6, according to which ‘we imagine more distinctly and vividly’. Macherey argues that between the two, a progress in the power of the imagination is implicitly affirmed. To imagine simply is to consider a thing as a free cause, or cause of itself, on the model of substance, which is causa sui. For Macherey, this would be to simply ‘divinize’ or to make a ‘fetish’ of it.⁵⁶ He maintains that ‘this idiotic manner of seeing things, which relies on the ignorance of their cause, is manifestly erroneous’.⁵⁷ We could however diminish our passivity by ceasing, as much as possible, to represent things to ourselves as free, and thereby diminish the extent to which we imagine things simply. We can do this by placing this ‘thing’ into a ‘more general context, where it ceases to appear as an isolated event … because it is related to causes which restore a part of its intelligibility and render it in this way more supportable’.⁵⁸ For Macherey, this corresponds to imagining more distinctly,⁵⁹ which allows us to regain a greater mastery over our affective life, that is, to develop the base of our affective life in the sense of progressively enriching its preoccupations and thereby enlarging its active expression. As Spinoza maintains in the Ethics, II, P29S, an affect which is excited by the intervention of ‘a number of things’, or causes, ‘at once’, is more powerful than an affect produced by a sole thing, or cause, and therefore such an affect renders the mind more active. By trying to consider more things at the same time, we can bring out the ‘agreements’ that there are between them, and their ‘differences and oppositions’. This does not relate to things in particular, but rather to the relations between things, which present themselves as ‘common properties’. In order for the mind to form the idea of common properties, it passes from the ideas of the affections of the properties of things in particular, to those of the common properties between things in general, that is to say, between the affecting and the affected bodies which are implicated in these affections. It is by following this path, which brings it from the consideration of the particular to that of the general, that the mind succeeds in rediscovering the ideas which, by definition, concern not only more things at once but everything. According to Spinoza, the idea of there being common properties shared by all things in general constitutes one of the most general common notions that we can have. Such a general common notion is an idea which presents itself in the same way in our mind as it is in God, or the infinite intellect. In this way, our mind actively participates in, or functions as a part of, the infinite intellect. According to Macherey, Spinoza therefore considers these common notions to really exist in the human mind, however only insofar as they are constituted by God.

The project of increasing our perfection that Macherey encounters in the Ethics, V, P1–20, ‘does not come from the demonstrative movement of rational knowledge, but is attained at the price of an extensive procedure, that is not deductive but rather inductive, which, without departing from the terrain of the imagination and experience, finishes by integrating the representation of all external causes in that of one unique cause, towards which the soul progressively orients all of its affective preoccupations’.⁶⁰ The representation of this ‘unique cause’ being the common notion of God as cause of all things. By giving an absolute point of reference to all images of things, Macherey argues that ‘the idea of God confers on these a value of necessity, which shields them from the indecision and the indetermination inherent in the hazards of the affective life’.⁶¹ The first stage of taking control of the affective life consists therefore, for Macherey, in orienting the field of the imagination towards the consideration of the properties of things in general. Grasping this general common notion prepares the soul for the activity of rational knowledge. It is in this way that the procedures of rational knowledge intervene in the unfolding of the affective life.

Amor erga Deum

The concept which is introduced in the Ethics, V, P16, and which gives rise to all of the primary developments of the fifth part, is the ‘love towards God’ (amor erga Deum), which, according to Macherey, is integrated into the movement of the affective life as the mind’s first common notion. Macherey maintains that for Spinoza, the ‘love towards God must engage the Mind most’, that is, it opens to a maximum the field of activity of the soul. The love that we have for such a being is of a particular type;⁶² it escapes the fatality of the fluctuatio animi, as it is no longer enclosed in the alternative of joy and sadness; it ceases in this way to be exposed to the perpetual reversal of the one into the other. For Macherey, ‘the love towards God is an entirely positive and active affect, which floods the soul with a joy without mix’.⁶³ Nevertheless, the human soul ‘cannot be definitively disengaged from the criteria imposed by affectivity’,⁶⁴ that is, from the hesitations tied to the phenomenon of the fluctuatio animi, since it must always form inadequate ideas which cannot be suppressed, by reason of the fact that this soul is the idea of a body which is affected ceaselessly by encounters with other bodies which seem to act on it from the outside. It maintains therefore ‘a point of view which remains that of the imagination and of desire’.⁶⁵

By initially following the inductive path, we are brought to the love towards God, which is for Macherey the first common notion which can set in motion the procedures of rational knowledge. Insofar as the soul develops love towards God, it can succeed, by means of the deductive path, in promoting the specific affects capable of overcoming the unbalanced affects engendered by the uninterrupted affective flux. This allows the soul to develop a greater regime of activity, the ideas so formed being ‘adequate’. ‘In this way’, writes Macherey, ‘all that which the soul thinks adequately, that which constitutes the specific form in which it “acts”, it thinks insofar as it is invested, at least partially, with the disposition of forming ideas the principle of which is found in God, insofar as God is himself a thing thought’.⁶⁶ As soon as the formation of the common notion of love towards God is acquired, it exercises a positive action on all our sad passions which it can convert into active affects.⁶⁷

The first common notion that we can have, according to Macherey, is then amor erga Deum, whereas for Deleuze, the first common notions that we can have are the simplest common notions, which represent what is common to our body and to certain external bodies by which we are effected. Macherey doesn’t deny that there are the simple common notions that Deleuze speaks of, however he does deny that from them we can deduce adequate ideas without first having attained the love towards God, which he therefore considers to be the first common notion capable of leading to adequate ideas. Macherey considers the love towards God to establish the basis for the regulation of the affective life and therefore to be the first step in the production of the second kind of knowledge. Whereas for Deleuze, the love towards God represents one step in the transition from the second kind of knowledge to the third kind of knowledge.⁶⁸ The idea of God as the cause of all things, that is, the general common notion of the love towards God, is for Macherey the primary point of reference for adequate ideas. All adequate ideas without exception therefore include, by means of the love towards God, the idea of God as their cause. According to Deleuze, however, adequate ideas are constituted locally by means of the simplest of common notions, or the shared knowledge that each involved idea or body is the common cause of the adequate idea. An adequate idea for Deleuze is therefore determined in direct relation to the bodies or ideas which interact with one another as causes of the adequate idea, without necessarily requiring reference to the general common notion of the love towards God.⁶⁹ Nevertheless, according to both Macherey and Deleuze, reference to God, the God of the love towards God, is required before we can have knowledge of ourselves or of other bodies from the point of view of our own specific qualitative nature or singular essence, which constitutes the third kind of knowledge.

A joyful passive affection can be reversed to sadness

The difference between the ‘passionate joys’ of Macherey and the joyful passive affections of Deleuze is brought out effectively by the discussion of the fluctuatio animi in ‘The encounter with Spinoza’. In fact Macherey argues that the ‘fluctuatio animi … completely undermines the notion of joyful passions’⁷⁰ presented by Deleuze. Macherey understands a ‘passionate joy’ to be a joy ‘by accident’, that is, a sadness which is doubled as a joy, and he maintains that Spinoza chose this case, ‘and not that where a joy is impaired by becoming tinged with bitterness’,⁷¹ in order to determine the theme of the fluctuatio animi. The joy of a passionate joy is a joy whose cause remains unknown; it is therefore associated with a passion, and, according to Macherey, must reverse to being sad. Macherey contends that Deleuze’s interpretation of a ‘joyful passion’ as a joyful passive affection presents a joy that does not reverse as expected, which leads him to ask ‘if there is in joy something stronger and more stable than in sadness, which protects it against this risk of reversal?’⁷² Macherey can be understood to be suggesting with this question that Deleuze’s response would be ‘yes’, since, for Deleuze, the sadness of a sad passive affection simply limits the existence of an existing finite mode, whereas the joy of a joyful passive affection not only affirms the limit, but simultaneously announces the potential for positive transformation, that is, to go beyond the limit imposed by the passive affections in general. In this way Deleuze does seem to interpret joy as being stronger and more stable than sadness, which could therefore protect it against the risk of reversal. Macherey responds to the question by arguing that ‘the extremely condensed way in which the content of this question is exposed in proposition 17 and in its scholium, only permits the question to be posed, but hardly gives any means to respond to it’.⁷³ In fact Spinoza only gives the example of a sadness which doubles as a joy in his explication of the fluctuatio animi.⁷⁴ Macherey therefore argues that such ‘a harmful pleasure, whether inflicted or suffered, would clearly for Spinoza be a passion imbued with fluctuatio animi, ineluctably producing a negative legacy of sadness’.⁷⁵ By referring to a joyful passion as ‘a harmful pleasure’, or simply a sadness which doubles as a joy, Macherey reduces the Deleuzian concept of a joyful passive affection to that of a passionate joy.

Deleuze, on the contrary, considers the ethical view to provide a means of responding to the question raised by Macherey. Deleuze doesn’t deny that ‘passionate joys’, as described by Macherey, are experienced by finite existing modes, nor that such a joy can be doubled or reversed to sadness and therefore be lost to the fluctuatio animi. And Deleuze in no way guarantees that every joyful passive affection will always produce an active joy. According to Deleuze, a joyful passive affection ‘may always … be interrupted by destruction, or even simply by the sadness of the loved object itself’,⁷⁶ that is, insofar as a joyful passive affection is a passion, its cause can be confused with another external cause or image of an object or body, which effaces the joy and renders the joyful passive affection sad. There is therefore nothing inherently stable or coherent in a joyful passive affection which stops it from falling pray to the fluctuatio animi. Instead, what Deleuze seems to be arguing is that, despite the difficulty in distinguishing a passionate joy from a joyful passive affection, the joy of a joyful passive affection can be isolated before it becomes prey to the fluctuatio animi, and in this way contribute to the formation of a common notion. Macherey does not at all agree with Deleuze on this point. Macherey argues, on the contrary, that nothing can turn a passionate joy into an action since being a passion it necessarily tends towards a fluctuatio animi, that is, for him, ‘all passions without exception … [tend] towards a fluctuatio animi’⁷⁷

Deleuze’s concept of a joyful passive affection is the concept of a joy which can be reversed to sadness or, conversely, which can contribute to the formation of common notions. The uncertainty of a joyful passive affection is carried over into the common notions which it can form when Deleuze maintains that, the isolation of a joyful passive affection does not bypass the need for common notions ‘to be formed, and formed either more or less easily, and so being more or less common to different minds’.⁷⁸ Macherey is in agreement with Deleuze on this point when he writes that: ‘that which distinguishes the souls of different men, is the place occupied by those common notions in relation to other ideas, inadequate ideas’.⁷⁹ However, the theme of joyful passions remains one of the points around which their respective interpretations of Spinoza’s theory of relations diverge.

What remains to be explicated in detail is exactly how adequate ideas are constituted locally by means of the simplest of common notions according to the logic of different/ciation. What is the mechanism by means of which joyful passive affections are isolated and accumulated, thereby contributing to the formation of the simplest of common notions, the manner by means of which this mechanism operates according to the logic of different/ciation is explicated in chapter 8.

______________

Notes

¹ Macherey, ‘The encounter with Spinoza’, p. 148.

² Ibid., p. 149

³ See the section of chapter 6 entitled ‘The ethical view of modal existence’, p. 152, where Deleuze argues that ‘The greater the extent to which the global integration characteristic of a mode is further differenciated, the more composite are the global integrations in which it is involved; the greater is its capacity to be affected, and, consequently, the greater is its power to act’.

⁴ Spinoza, Ethics, III, the General Definition of the Affects.

⁵ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 20.

⁶ Macherey’s study of the Ethics is in five volumes entitled Introduction à l’Ethique de Spinoza, published between 1994 and 1998. The study on the fifth part of the Ethics was the first to be published, followed by the study on the third part, and then that on the second, the fourth and the first.

⁷ Pierre Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie (Paris, 1994), p. 19.

⁸ Ibid., p. 121.

⁹ Ibid., p. 71.

¹⁰ Deleuze, Expressionism in Philosophy, p. 246.

¹¹ Macherey, ‘The encounter with Spinoza’, p. 153.

¹² Deleuze, Expressionism in Philosophy, p. 240.

¹³ Ibid., p. 246: ‘the opposition of actions and passions should not conceal the other opposition that constitutes the second principle of Spinozism: that of joyful passive affections and sad passive affections. One increases our power, the other diminishes it. We come closer to our power of action insofar as we are affected by joy’. Additional italics.

¹⁴ Macherey, ‘The encounter with Spinoza’, p. 152.

¹⁵ Ibid., p. 153.

¹⁶ Deleuze, Expressionism in Philosophy, p. 246.

¹⁷ Ibid., p. 274.

¹⁸ Ibid., p. 241.

¹⁹ Ibid., p. 274.

²⁰ Ibid., p. 285.

²¹ Ibid., p. 283.

²² Instead of suggesting that Deleuze repeatedly exposes his argument to the charge of contradiction whenever it seems as though joyful passive affections are equated with an increase in power, Deleuze’s use of the concept ‘joyful passive affections’ should rather be understood to be the articulation of the process of transformation, or increase in power, that actually takes place in the generation of active joy by means of the accumulation of joyful passive affections.

²³ Spinoza, Ethics, II, P40S.

²⁴ Deleuze, Expressionism in Philosophy, p. 276. Additional italics.

²⁵ Spinoza, Ethics, II, P39.

²⁶ Deleuze, Expressionism in Philosophy, p. 283.

²⁷ Ibid., p. 282.

²⁸ Ibid., p. 283.

²⁹ Spinoza, Ethics, V, P3.

³⁰ Ibid., V, P8: ‘The more an affect arises from a number of causes concurring together, the greater it is’.

³¹ That is, that there is something common to an external body and our own.

³² Ibid., V, P9D.

³³ Ibid., V, P11.

³⁴ Ibid., II, P47S.

³⁵ Deleuze, Expressionism in Philosophy, p. 307.

³⁶ Ibid., p. 283.

³⁷ Ibid., p. 307.

³⁸ Macherey, ‘The encounter with Spinoza’, p. 153.

³⁹ Although Deleuze distinguishes his ethical view from a physical view, Macherey engages with Deleuze’s interpretation solely in the form of a physical view of the Ethics. See the section of this chapter entitled ‘The distinction between joyful passive affections and sad passive affections’, p. 158.

⁴⁰ Macherey, ‘The encounter with Spinoza’, p. 154.

⁴¹ Spinoza, Ethics, III, P17S.

⁴² Macherey, ‘The encounter with Spinoza’, p. 154.

⁴³ Spinoza, Ethics, V, P7.

⁴⁴ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 166.

⁴⁵ Ibid., pp. 162–3.

⁴⁶ Macherey, ‘The encounter with Spinoza’, p. 154.

⁴⁷ Ibid.

⁴⁸ Ibid.

⁴⁹ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 361.

⁵⁰ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 67.

⁵¹ Ibid.

⁵² Macherey, ‘The encounter with Spinoza’, pp. 155–6.

⁵³ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 125.

⁵⁴ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 67.

⁵⁵ Ibid., p. 68.

⁵⁶ ‘la diviniser, ou encore, pourrions-nous dire, la fétichiser’ (Ibid., p. 69).

⁵⁷ Ibid., p. 69.

⁵⁸ Ibid., p. 71.

⁵⁹ Macherey argues that ‘It is not necessary to imagine less, but to imagine better’ (Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 30).

⁶⁰ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 92.

⁶¹ Ibid., p. 94.

⁶² From the definition of the nature of love which is given in the Ethics, III, love is ‘joy accompanied by the idea of an external cause’, that is, it is a secondary affect, which is the result of an objective fixation of desire. But the object of the love towards God is the idea of God as external cause, which is the property common to all the things of the first general common notion.

⁶³ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 97.

⁶⁴ Ibid., p. 108.

⁶⁵ Ibid.

⁶⁶ Ibid., p. 168.

⁶⁷ Spinoza, Ethics, V, P18D: ‘The idea of God which is in us is adequate and perfect (by IIP46, P47). So insofar as we consider God, we act (by IIIP3). Consequently (by IIIP59), there can be no Sadness accompanied by the idea of God’. Ethics, V, P18S: ‘… insofar as we understand the causes of Sadness, it ceases (by P3) to be a passion, i.e. (by IIIP59), to that extent it ceases to be Sadness. And so, insofar as we understand God to be the cause of Sadness, we rejoice’.

⁶⁸ A discussion of the different interpretations of the third kind of knowledge given by Macherey and Deleuze is presented in the next section of this chapter entitled ‘A joyful passive affection can be reversed to sadness’.

⁶⁹ The mechanism by means of which adequate ideas are constituted locally in relation to the simplest of common notions is explicated according to the logic of different/ciation in chapter 8.

⁷⁰ Macherey, ‘The encounter with Spinoza’, p. 156.

⁷¹ Macherey, Introduction à l’Ethique de Spinoza, la troisième partie, p. 166.

⁷² Ibid.

⁷³ Ibid.

⁷⁴ Spinoza, Ethics, III, P17: ‘If we imagine that a thing which usually affects us with an affect of Sadness is like another which usually affects us with an equally great affect of Joy, we shall hate it and at the same time love it’. Ethics, III, P17S: ‘This constitution of the Mind which arises from two contrary affects is called vacillation of mind [fluctuatio animi], which is therefore related to the affect as doubt is to the imagination (see the Ethics, II, P44S); nor do vacillation of mind and doubt differ from one another except in degree. But it should be noted that in the preceding Proposition I have deduced these vacillations of mind from causes which are the cause through themselves of one affect and the accidental cause of the other. I have done this because in this way they could more easily be deduced from what has gone before, not because I deny that vacillations of mind for the most part arise from an object which is the efficient cause of each affect’.

⁷⁵ Macherey, ‘The encounter with Spinoza’, p. 156.

⁷⁶ Deleuze, Expressionism in Philosophy, p. 244.

⁷⁷ Macherey, ‘The encounter with Spinoza’, p. 155.

⁷⁸ Deleuze, Expressionism in Philosophy, p. 307.

⁷⁹ Macherey, Introduction à l’Ethique de Spinoza, la seconde partie, p. 291.

Chapter 8

The mechanics of joyful passive affections

Does the intensive part of a finite existing mode remain unchanged?

The mechanism by means of which joyful passive affections are implicated in the dynamic changes or transformations of the characteristic relations of modal existence, is determined according to the logic of different/ciation. According to the ethical view developed by Deleuze in Expressionism in Philosophy,¹ these transformations implicate a finite existing mode’s, or human being’s, capacity to be affected, which is constituted by its active affections, while the passive affections it is subject to function only to limit this capacity. The limitative effect of passive affections operates within the ‘general limits’ of the range of singular modal, or human, essence, that is, between a maximum and minimum. The minimum would be the least point determinative of the global integration of a finite existing mode, or individual human being, below which it would cease to exist. The maximum would be the actually infinitely composite assemblage of multipli-differenciated global integrations in which a finite existing mode, or individual human being, can be further differenciated; which would be the body of the whole universe, or Nature (Substance) as a whole. The ‘limit’ denotes the extent to which the global integration of a finite existing mode, or individual human being, is further differenciated at any given moment as a component local integration in the more composite global integrations constitutive of the actually infinitely composite assemblage of multipli-differenciated global integrations, or Nature (Substance) as a whole. The ‘limit’ of a finite existing mode’s, or individual human being’s, capacity to be affected is therefore open to variation within the ‘general limits’ of this range. A mode’s capacity to be affected, which is affirmed by its conatus as the expression of its power to act, is therefore determined both by the composite multipli-differenciated assemblage of local integrations of which it is composed, and by the more composite global integrations in which it is further differenciated.²

The extensive parts determinative of a finite existing mode are determined to enter into a characteristic relation through the operation of the purely mechanical laws of the logic of different/ciation. It has been determined that the characteristic relation of a finite existing mode, as the proportion or ratio of motion and rest between the extensive parts of which it is composed, remains fixed, while the extensive parts that belong to it at any given moment are able to come and go. According to Deleuze, the characteristic relation of a finite existing mode is a global integration, which is determined by the composite, multipli-differenciated assemblage of local integrations of which it is composed. And that which determines the extensive parts to come and go in relation to this global integration is the degree to which it is further differenciated in larger more composite global integrations. What Deleuze understands by this is that, even though quantitatively there are dynamic changes in the extent to which the global integration of a finite existing mode is further differenciated; qualitatively, insofar as the intensive part, or singular essence, of a finite existing mode is a modification of an attribute’s quality, it remains unchanged.

Each quantitatively differentiated intensive part of an attribute involves a global integration which creates an essential singularity that is complicated within that attribute as a singular modal essence. As intensities of its quality, singular modal essences are quantitatively differentiated as the actually infinite quantity of intensive parts of that attribute, or as the different degrees of power of its power series expansion.³ Now, just as the extensive parts constitutive of a finite existing mode are explicative of an attribute according to the logic of different/ciation, so too are the intensive parts determinative of singular modal essences complicated within an attribute according to the logic of different/ciation, that is to say that the mechanism of quantitative differentiation is also determined according to the logic of different/ciation. The global integration of an intensive part, the composite relation of which is creative of its essential singularity, is determined as a composite multipli-differenciated assemblage of local integrations, each of which also incorporates the composite relation creative of the essential singularity of an intensive part. Each global integration incorporates the intensive parts of the increasingly composite, multipli-differenciated assemblage of local integrations of which it is composed. Intensive parts are therefore differenciated in relation to one another by means of their global integrations. An attribute is therefore composed of an actually infinitely composite multipli-differenciated assemblage of intensive parts. These complicated intensive parts include those involving the differenciations of the differentiated, and those involving the differenciations of the differenciated, up to the actually infinitely composite multipli-differenciated assemblage of global integrations.⁴ The former include those intensive parts whose global integrations create essential singularities from composite relations between the local integrations determinative of pre-individual singularities, which would correspond to the first term of the power series expansion of an attribute, or to its minimum degree of power, or intensity = 0. The latter include the actually infinite quantity of intensive parts constitutive of an attribute, which would correspond to the maximum degree of power or intensity, of the power series expansion of an attribute. The differenciations of the differentiated are incorporated, insofar as they are repeatedly differenciated, in each successive term of the series, up to and including the actually infinite differenciations of the differenciated. Each successive intensive part, or degree of power, of the power series expansion of an attribute effectively incorporates all of those intensive parts of which it is composed, each intensive part remaining unchanged regardless of the global integration in which it is implicated in the composite multipli-differenciated assemblage of global integrations.

The reciprocal relation between two local integrations constitutive of a global integration creates an essential singularity which determines the intensive part or singular essence of a mode. The expansion of this reciprocal relation in the power series of a global integration generates a finite existing mode by determining the relations between its extensive parts. The extensive parts of a global integration are the redeployed extensive parts of the local integrations of which the global integration is composed. The global integration of a finite existing mode is therefore determinative of both its intensive part, insofar as the local integrations of which it is composed are creative of the essential singularity belonging to its intensive part, and also of the relations between the extensive parts belonging to it, insofar as it is generative of the finite existing mode itself. It is therefore the global integration of a finite existing mode which determines how an attribute is both complicated and explicated according to the logic of different/ciation.

This characterizes the mechanism by means of which intensive parts or singular modal essences are complicated within an attribute in the form of a power series expansion according to the logic of different/ciation. However, the question remains as to how joyful passive affections are implicated in the mechanism by means of which extensive parts are explicated in the dynamic changes or transformations of the characteristic relations, or the global integrations, of finite existing modes.

The mechanics of further differenciation

Although a finite existing mode’s intensive part or singular essence remains unchanged, there are dynamic changes or transformations of the characteristic relations determinative of the extensive parts belonging to it, which correspond to changes in the extent to which its global integration is further differenciated as a component local integration in larger more composite global integrations. Just as the power of each successive intensive part, or singular modal essence, of an attribute increases in degree, each dynamic change or transformation of the global integration of a finite existing mode, insofar as it is explicative of the attribute, incorporates these increases in degree of power, that is, insofar as the global integration determinative of a given finite existing mode is further differenciated, it is implicated in an increase in degree of power. By being further differenciated, the global integration determinative of a given finite existing mode creates the essential singularity which is incorporated by the next successive intensive part of the power series expansion of the attribute, whose exponent, or power, is therefore of greater degree. So although the intensive part of the given finite existing mode remains unchanged, by being further differenciated it is incorporated as a component part of a more composite global integration, whose intensive part is of greater degree of power. It is with the intensive part of greater degree of power that the intensive part of the given finite existing mode is now implicated. This increase in the degree of power of the intensive part with which the given global integration is now implicated corresponds to an increase in the given finite existing mode’s power to act. A finite existing mode’s power to act ranges from a minimum to a maximum. The minimum degree of power of a finite existing mode would be the minimum degree of power, or intensity = 0, of the power series expansion of the global integration of its intensive part, which corresponds to the reciprocal relation between the local integrations of which it is composed. The maximum degree of power of a finite existing mode would be the actual degree of power or intensity, of the intensive part in which it is incorporated, or with which it is implicated, which is determined by the extent to which its global integration is further differenciated as a component local integration in more composite global integrations. Each subsequent further differenciation of such a more composite global integration results in a further increase in the degree of power of the intensive part in which the intensive part of the given finite existing mode is implicated, and corresponding to this is a further increase in its power to act. So the further differenciation of a global integration corresponds to an increase in the degree of power of the intensive part in which it is implicated and therefore of the power to act of the finite existing mode initially determined by it.⁵

The implication of joyful passive affections in the mechanics of further differenciation

In order to explicate how joyful passive affections are implicated in the mechanism of the dynamic changes or transformations of the characteristic relations, or the global integrations, of finite existing modes, which can result in an increase in their power to act, the mechanism by means of which a global integration is further differentiated requires further elaboration. The question that needs to be asked then is what determines the further differenciation of a global integration, and by consequence the increase in the power to act of its finite existing mode? How are these dynamic changes or transformations actually effected?

In Expressionism in Philosophy, Deleuze argues that a finite existing mode’s power to act enjoys ‘a limit within which’ the global integrations in which it is further differenciated, ‘take form and are deformed’.⁶ The limit within which these global integrations take form and are deformed is not determined by the passive affections encountered by the finite existing mode, but rather by the extent to which its global integration can be further differenciated. From the point of view of the creative and generative capacity of the global integrations of a finite existing mode, that which corresponds to the limit of a finite existing mode’s power to act is the specific qualitative nature of the infinite branches of its global integration. The specific qualitative nature of the infinite branches of any global integration is determinable solely in reciprocal relation to that of another global integration, in the form of the infinite branches of a more composite global integration. The mechanism by means of which the infinite branches of the more composite global integration take form and are deformed determines the specific qualitative nature of the given global integrations, and corresponds to the mechanism by means of which the given mode’s power to act is limited.

The specific qualitative nature of the infinite branches of any global integration is only determinable insofar as that global integration is implicated in a more composite global integration, the specific qualitative nature of which is also only determinable insofar as it is further implicated, as a local integration, in a more composite global integration, that is, insofar as it is in turn further differenciated. It is therefore only in the global integrations between multipli-differenciated assemblages of local integrations that there is a specific qualitative nature that marks the limit of each of the implicated local integrations involved. The specific qualitative nature of a local integration is therefore characterized by the global integrations in which it is further differenciated, whether in the global integration in which it is immediately differenciated with another local integration, or in the more composite global integration in which this global integration is further differenciated in reciprocal relation with another global integration, and so on. The specific qualitative nature of any global integration is therefore only determinable in relation to the more composite global integrations in which it is further differenciated; from the immediate further differenciation of the global integration whose composite relation is creative of the essential singularity determinative of its intensive part, to the extent to which this global integration is further differenciated in the actually infinitely composite assemblage of multipli-differenciated global integrations.

The determination of a finite existing mode’s power to act

The question that remains however is what determines this extent? What determines the infinite branches of more composite global integrations to mark the limit of the given global integration, and thereby determine the power to act of its finite existing mode? What is the mechanism by means of which the infinite branches of these more composite global integrations take form and are deformed?

Insofar as the specific qualitative nature of the infinite branches of any global integration corresponds to a threshold, or limit, it is characterized by a potential function. In the section of chapter 3 entitled ‘The logic of the differential’, it was determined that the domain of a potential function is determined by the poles of the local integrations of which a global integration is constituted. The poles are the points of discontinuity or pure elements of potentiality of each local integration, otherwise known as ‘accumulation points’.

The infinite branches of a global integration are increasingly determined as the composite relation between the two local integrations of which it is composed is repeatedly differentiated in the form of a power series expansion. As this power series is expanded, there is an increase in the degree of power (exponent) of each successive term of the series, and therefore of the series itself. This increase in degree of power can be characterized by an accumulation in the potential of the variable to jump across the threshold of the potential function, following a trajectory which converges with the infinite branches of the global integration. The kind of trajectory followed by the variable is determined by the type of essential singularity created by the reciprocal relation between the local integrations constitutive of the global integration. As the power series expands, the divergent lines of the potential trajectory of the variable across the threshold increasingly converge with the infinite branches of the global integration.

The degree of power of the power series expansion of a global integration can therefore also be understood to vary between a minimum and a maximum. The minimum degree of power corresponds to the simplest term of the series, or to that determined by the composite relation between the local integrations determinative of the global integration, which is characterized by the variable with the least potential to jump the threshold of the potential function. The maximum degree of power of this power series expansion is determined by its convergence with the global integration, or the actualization of the potential function in the infinite branches of the global integration. The potential function is actualized when the variable jumps across its threshold, from the pole of one of the local integrations to that of the other, along a trajectory which converges with the infinite branches of the global integration. The terms of the power series expansion of a global integration can be referred to as being bound by a maximum and a minimum degree of power so long as this does not detract from an understanding of the transformation of this same global integration in relation to the degrees of power of the successive terms of the power series expansion of the attribute. The minimum degree of power of the power series expansion of a global integration registers the creation of the essential singularity determinative of its intensive part, or singular modal essence. The maximum degree of power of the power series expansion of a global integration registers the generation of the extensive parts determinative of the finite existing mode corresponding to its intensive part. The specific qualitative nature of the infinite branches of a global integration, or the finitude of the existing mode it generates, is however only determinable in relation to the subsequent further differenciations in which it is implicated with the global integrations of other modes, which determines both the complications and explication of an attribute according to the logic of different/ciation.

The successive terms of the power series expansion of a global integration are characterized by the variety of divergent lines of the different potential trajectories of the variable across the threshold of the potential function, each successive term corresponding to the divergent lines of a different potential trajectory; which range from the straight line directly between the poles of its constitutive local integrations, to the infinite branches of the actualized global integration. The minimum degree of power of the power series expansion of a global integration, which is characterized by the straight line between the poles of its local integrations, incorporates the maximum degree of power of the power series expansion of each of these local integrations. The maximum degree of power of the power series expansion of the global integration is incorporated, in reciprocal relation with that of another, in the minimum degree of power of the power series expansion of the more composite global integration which they together determine. The specific qualitative nature of the divergent lines of the infinite branches of the global integration is determinable solely insofar as the global integration is further differenciated with another global integration in a more composite global integration. The determination of the specific qualitative nature of the infinite branches of a global integration therefore always involves both the local integrations of which the global integration is composed, and the more composite global integrations in which it is further differenciated. There is therefore a maximum and a minimum in relation to the power series expansion of each global integration which is distinct from the general limits of the power series expansion of the attribute in which each global integration is determined.

It has been determined that intensive parts are complicated within an attribute in the form of a power series expansion according to the logic of different/ciation. Intensive parts are created by composite relations between global integrations; each global integration being determined by the intensive part created by the composite relation between the local integrations of which it is composed. The extensive parts explicative of an attribute and determinative of extensive modal existence are generated by the expansion of the composite relation, creative of the essential singularity of each intensive part, in the form of a power series; the power series expansion of a global integration. The transformation from the global integration determinative of one intensive part of an attribute to a more composite global integration determinative of another intensive part is characterized by the further differenciation of the given global integration with the global integration determined by yet another intensive part. Each of these transformations, or further differenciations, from local integrations to global integrations to more composite global integrations etc., which are determinative of the different terms of the power series expansion of an attribute, is characterized by the actualization of a potential function in the infinite branches of a global integration. The degree of power of the intensive part of a given global integration remains at a minimum, at intensity = 0, unless its global integration is further differenciated in composite relation to the global integration determined by another intensive part. When further differenciated in a more composite global integration, the degree of power of the intensive part determinative of the given global integration sustains a determined increase insofar as it is now implicated in the intensive part of greater degree of power which is determinative of the more composite global integration, and which corresponds to another successive term in the power series expansion of the attribute. Each further differenciation of any global integration therefore corresponds to its implication in another successive term of the power series expansion of an attribute.

Just as the degree of power of the intensive part of each of the global integrations constitutive of a more composite global integration sustains a determined increase, the finite existing mode explicated by each of these global integrations sustains a corresponding increase in its power to act. The increase in degree of power sustained by a mode’s global integration is registered as an increase in its power to act insofar as it is further differenciated, in composite relation to the global integration of another finite existing mode, in a more composite global integration, whose intensive part is of greater degree. Each further differenciation is characterized by the jump of a variable across the threshold of the potential function generated between the essential singularities determinative of each of the global integrations involved. The potential function is thereby actualized in a more composite global integration, whose intensive part is of greater degree. It is to this increase in degree of power that the increase in the power to act of the finite existing mode generated by each of the implicated global integrations corresponds.

The correspondence between the mathematical and the ontological concepts of accumulation

A potential function is characterized by the potential trajectory of the variable of one local integration in relation to another local integration. That which determines two local integrations to form a composite relation, and therefore to generate a potential function, is the potential for the convergence of the specific qualitative nature of their respective infinite branches in the infinite branches of a global integration. It is in this sense that local integrations are said to agree in nature, or to have something in common. The expansion of the power series of the composite relation between the local integrations characterizes the accumulation of potential for the variable to jump the threshold of the potential function and thereby be actualized in the infinite branches of a global integration. The accumulation of the potential for the variable to jump across the threshold of a potential function corresponds to what Deleuze refers to as the accumulation of joyful passive affections.⁷ The ‘jump’ of the variable, which actualizes the potential function in the infinite branches of the global integration, corresponds to the ‘leap’ that Deleuze refers to when an adequate idea of the joyful passive affection is formed.⁸ Such an adequate idea is formed when the cause of the joyful passive affection is understood to be that which is common to the finite existing modes involved, that is, when the agreement or potential for the convergence of the specific qualitative nature of the infinite branches of their respective global integrations is actualized in the infinite branches of a more composite global integration. Therefore the accumulation of the potential for the variable to jump across the threshold of a potential function, which corresponds to the accumulation of joyful passive affections, prepares for the actualization of the potential function in the infinite branches of a more composite global integration, which characterizes the ‘leap’ from inadequate to adequate ideas, from joyful passive affections to active joys, from passions to actions.

A finite existing mode, or an individual human being, which encounters another whose nature agrees with its own, experiences the joy of a joyful passive affection. The feeling of Joy associated with a joyful passive affection is initially the only indication that their natures agree or that there is the potential for convergence. They can both strive to combine with each other to form a third more powerful relation, or more composite global integration, in which the individuals of both composing finite existing modes are preserved. This corresponds with the expansion of the power series of the composite relation between their global integrations, such that each global integration is further differenciated in relation to the other, thereby generating a more composite global integration, in which the intensive parts of both global integrations are preserved. This is the mechanism by means of which joyful passive affections are implicated in the transformation of the characteristic relations, or global integrations, of finite existing modes, that is, in the further differenciation of global integrations, thereby generating more composite global integrations. According to Deleuze, the only way of reaching or of coming into possession of an adequate idea is by means of the mechanism of joyful passive affections.

The determination of an existing mode as finite

An existing mode is determined as finite only insofar as the global integration determinative of its extensive parts is further differenciated in composite relation with the global integration of another mode. It is with each further differenciation that the specific qualitative nature of the infinite branches of the given global integration, or its finite modal existence, increasingly takes form. This occurs solely in relation to the more composite global integrations of which it is a part. And it is the extent to which a global integration is further differenciated that determines the degree of power of the intensive part with which it is implicated, and therefore the power to act of its finite existing mode.

The potential function determined by the agreement in nature between two global integrations characterizes the threshold, or limit, which functions as a border between that which remains unchangeable, or the intensive part of each global integration, and that which is changeable, or the extent to which each global integration is further differenciated.⁹ A joyful passive affection either contributes to the further differenciation of the global integration of a finite existing mode, insofar as it indicates the potential for its convergence with another global integration; or conversely, it is reversed to sadness, insofar as this potential for convergence is not actualized. The reversal to sadness of the joy of a joyful passive affection can be understood to occur when the other local integration with which a given local integration is in composite relation is determined to enter into composite relation with a different local integration. The initial composite relation is thereby effectively deformed. The unactualized potential for convergence, and the decrease in degree of power associated with it, is experienced by the finite existing mode of the given local integration as sadness. The potential for convergence is established instead between the local integrations of the latter composite relation. There is nothing inherently stable or coherent in the joy experienced by virtue of the initial relation which prevents it from being deformed and thus converted to the latter relation. Any composite relation that a local integration enters into with another is determined rather by the potential for the actualization of the global integration generated by this composite relation, and in its further differenciation in relation to other global integrations, thereby forming more composite global integrations. It is these more composite global integrations into which the generated global integrations enter that determine the specific qualitative nature of the implicated local integrations, and therefore the finitude of each of the existing modes corresponding to these local integrations. The power to act of each finite existing mode is determined by the greater degree of power of the intensive part of the more composite global integration in which they are further differenciated. The intensive parts of each of the local integrations of a composite relation, and the intensive part of greater degree of power created by this relation, continue to exist alongside of each other, the transformation from the former to the latter being determined by the mechanism of joyful passive affections according to the logic of different/ciation.

The joy of a joyful passive affection affirms the extent to which a global integration has already been further differenciated, while simultaneously announcing the potential for positive transformation, that is, the potential for convergence, or further differenciation, with other global integrations in more composite global integrations. By affirming the extent to which a global integration is further differenciated, the joy of a joyful passive affection affirms both the specific qualitative nature of the multipli-differenciated assemblage of local integrations of which the global integration is composed, which marks the finitude of the existing mode, or individual human being; and the degree of power of the intensive part in which this global integration is implicated, or further differenciated, which then corresponds to the finite existing mode’s power to act.

The mechanics of joyful passive affections

Any outline of a dialectic that one might see in Expressionism in Philosophy functions according to a logic different to ‘a Hegelian-style dialectic’.¹⁰ It is a logic according to which joyful passive affections mediate not between active and passive affections, but rather solely between different active affections. An active affection, or active joy, is produced with the actualization of a potential function in a global integration. The expansion of the composite relation between two such global integrations is characterized by the accumulation of the potential of the variable to ‘jump’ the threshold of the potential function or by the accumulation of joyful passive affections in preparation for the ‘leap’ determinative of an increase in a finite existing mode’s power to act. Joyful passive affections therefore mediate between the different active affections of finite existing modes which are further differenciated in relation to one another, thereby generating the active joy associated with the actualization of a more composite global integration. So the increase in the power to act of a finite existing mode, which Deleuze ascribes to the mechanics of joyful passive affections, is determined by the further differenciation of its global integration with the global integration of another finite existing mode.

On the other hand, the decrease in the power to act of a finite existing mode is determined by the decomposition of one of the more composite global integrations in which its global integration is further differenciated as a component part. This occurs when one of these more composite global integrations encounters another whose specific qualitative nature is divergent from its own, and where there is no potential for convergence. In such an encounter, the extensive parts of one of these global integrations are determined to enter into a different relation, one which is determined rather by the specific qualitative nature of the global integration of greater degree of power. The extensive parts are effectively stripped from the global integration of lesser degree of power and are captured by, or incorporated in, the global integration of greater degree of power.¹¹ When the more composite global integration of the given finite existing mode is the global integration of lesser degree of power, this relation is decomposed. The intensive part in which the global integration of the given finite existing mode is now implicated is of lesser degree of power, its power to act therefore corresponding to this lesser degree.

However, rather than this decrease being understood to have been determined directly by a passive affection, it is actually determined instead by the extent to which some of the other component parts, or global integrations, formerly constitutive of the more composite global integration in which the global integration of the finite existing mode is further differenciated, are now implicated in the different more composite global integration of greater degree of power. Any decomposition therefore involves the recomposition of these component parts in a different relation. Only a partial understanding, one from the point of view of the imagination of the finite existing mode itself, would represent this transformation solely in terms of a decomposition determined by a passive affection. The adequate alternative would be to recognize that a different relation has been generated, one which is determined by the specific qualitative nature of the more composite global integration of greater degree of power, in which the component parts formerly constitutive of the global integration of lesser degree of power are now implicated.

What happens when this process of decomposition/recomposition occurs at the level of the finite existing mode itself? What would happen to the intensive part of a finite existing mode whose extensive parts were all captured by different relations of greater degree of power, that is, a mode which effectively ceases to exist extensively as a result of being subject to such relations of decomposition? Would its degree of power be diminished to the extent that it would equal zero, that is, intensity = 0? This problem is investigated in chapter 9 in relation to the distinction between the duration of a finite existing mode and its eternity.

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Notes

¹ See the section of chapter 6 entitled ‘The ethical view of modal existence’, p. 152.

² See the section of chapter 6 entitled ‘The three different types of expressive change’, p. 154.

³ Recall that, as a power series expands, the exponent, or power, of each successive term increases in degree, and the series increasingly approximates or converges with its function, which in this instance would be an attribute. See the section of chapter 3 entitled ‘Power as capacity and the logic of “differentiation”’, p. 77.

⁴ The actually infinitely composite multipli-differenciated assemblage of global integrations is characteristic of what Spinoza refers to in Letter LXIV as the ‘infinite mediate mode’, which, from the point of view of the attribute of extension, constitutes the body of the whole universe (facies totius universi), and, ‘as concerns thought, [constitutes] the ideal relations governing the determinations of ideas as ideas of existing modes’ (Deleuze, Spinoza: Practical Philosophy, p. 92). All of the complicated intensive parts of an attribute are characteristic of the ‘infinite immediate mode’. The composite relations of these intensive parts, each of which is constituted by the reciprocal relation between two local integrations and is therefore determined as a composite multipli-differenciated assemblage of local integrations, are constitutive of the essential singularities which determine the relations between the extensive parts, that is, the movement and rest of the most simple bodies, of the global integrations of finite existing modes, or individual human beings. Deleuze argues that ‘there is an order ... of the expression of attributes. Each attribute is first expressed in ... an immediate infinite mode ... Then the modified attribute expresses itself, in a mediate infinite mode. Finally the attribute is expressed “in a certain and determinate way”, or rather in an infinity of ways which amount to finite existing modes’ (Deleuze, Expressionism in Philosophy, p. 105). The discussion of Macherey’s understanding of this distinction between immediate and mediate infinite modes is in the section of chapter 1 entitled ‘Omnis determinatio est affirmatio’, p. 35. See Spinoza, The Correspondence of Spinoza, Letter LXIV.

⁵ The process of further differenciation is non-linear insofar as a given global integration can simultaneously be in reciprocal relation with multiple different global integrations.

⁶ Deleuze, Expressionism in Philosophy, p. 223. See the section of chapter 6 entitled ‘The physical view of modal existence’, p. 149.

⁷ Deleuze, Expressionism in Philosophy, p. 283.

⁸ Deleuze writes of ‘the accumulation of joyful passions’, and of ‘a genuine “leap”, which puts us in possession of an adequate idea, by the aid of such accumulation’ (Ibid., p. 283).

⁹ See the section of chapter 6 entitled ‘The ethical view of modal existence’, p. 152.

¹⁰ Contrary to Macherey’s initial suggestion that ‘there is something [in Deleuze’s logic] reminiscent of a Hegelian-style dialectic’ (Macherey, ‘The encounter with Spinoza’, p. 147). See the section of chapter 7 entitled ‘The role of joyful passive affections in Deleuze’s account of modal existence’, p. 160.

¹¹ ‘Parts that enter into some relation must formerly have existed in other relations. These initial relations have to combine if the parts subsumed in them are to enter into the new relation. The latter is thus in this sense composite. Conversely, it decomposes when it loses its parts, which must then enter into other relations’ (Deleuze, Expressionism in Philosophy, p. 236 n. 6). In other words, Deleuze argues that the extensive parts that enter into a relation, or that are actualized in a global integration, must have existed in other relations.

Chapter 9

The distinction between the duration of a finite existing mode and its eternity

Amor intellectualis Dei

The concept of amor intellectualis Dei, or the intellectual love of God, which follows from adequate ideas of the third kind of knowledge, and the concept of the eternity of the soul are introduced by Spinoza in the fifth and final part of the Ethics. Throughout this part Spinoza is occupied with charting the transition from ‘the love towards God’ to ‘the intellectual love of God’, and with the discussion, in relation to the intellectual love of God, of the eternity of the soul as distinct from life in the present, which is commonly mistaken for a Spinozist argument in favour of the immortality of the soul. The interpretation of these two notions by Macherey and Deleuze diverge according to their different readings of the love towards God. According to Macherey, the common notion which is the love towards God is the primary point of reference for all adequate ideas, whether they are of the second or third kind of knowledge. Therefore all adequate ideas without exception include, by means of the love towards God, an idea of God as their cause. Deleuze maintains, rather, that by means of the simplest common notions the adequate ideas of the second kind of knowledge are constituted locally, without it being necessary to make reference to the general common notion of the love towards God. Macherey therefore reads the love towards God as the point of departure for the second kind of knowledge, whereas Deleuze reads it rather as the limit of the second kind of knowledge. Nevertheless the two interpreters agree that it is necessary to make reference to the God of the love towards God before being able to have a knowledge that is constituted by adequate ideas of the third kind.

According to Macherey, ideas of one’s existence and of the relations in which one finds oneself remain imaginary, that is more or less ‘distinct and vivid’,¹ unless abstracted or universalized by means of the love towards God. By following the path of the ‘common notions’ defined by the love towards God, the ideas which form in the soul become ‘adequate’. Such adequate ideas belong to the demonstrative knowledge of the second kind, described by Macherey as ‘an abstract knowledge, which explains the nature of things from a general point of view by submitting it to rules or to laws which are modes of thought and not physical realities’.² But the love towards God does not cut off ‘definitively all relation with the conditions of concrete life’,³ that is, it cannot be definitively disengaged from the criteria imposed by affectivity, since it maintains a point of view which remains that of the imagination, which, in itself, cannot be suppressed by reason because of the fact that this soul is the idea of a body which is limited necessarily by its nature as a finite mode. By giving all the images of things an absolute point of reference, the love towards God ‘confers on them a value of necessity, which extracts them from the indecision and the indetermination inherent to the hazards of the affective life’,⁴ permitting a regime of affective stability to be established in this life. Spinoza writes that the love towards God ‘is the most constant of all the affects, and insofar as it is related to the Body, cannot be destroyed, unless it is destroyed with the Body itself’.⁵ This point of view, of the irreversibility of the affective movement associated with the love towards God, indicates that, once the soul is engaged in this movement, it is rendered increasingly more active and therefore stable. The idea of God which is embodied in the love towards God constitutes therefore, according to Macherey, ‘the common element at the interior of which all our ideas, all our representations, and all our experiences develop’.⁶

Macherey maintains that by virtue of its basis in common notions, the love towards God leads us to ‘associate in the imagination ... other individuals, and increasingly more individuals, engaged with [us] in this common, shared affective impulse’;⁷ as Spinoza writes, ‘the more men we imagine to be joined to God by the same bond of Love, the more it is encouraged’.⁸ Macherey describes this as a process of ‘depersonalization’⁹ whereby we are delivered from our exclusive attachment to ourselves, according to which we imagine simply - where to imagine simply is to consider a thing as a free cause, or cause of itself, on the model of substance, which is causa sui - that is to say, according to which we make a fetish of ourselves. Because of the limitations that this procedure implies, it is ‘necessarily a cause of sadness’.¹⁰ With the love towards God, we discover that the affects we understand as our own do not belong only to us in particular, but that the causes from which they derive are common in God and can therefore be experienced ‘collectively, within the context of the union of a multiplicity of individuals’. It is in this way, Macherey maintains, that the practice of love towards God acquires ‘a political and social dimension’.¹¹

From the Ethics, V, P21, the love towards God becomes the point of departure for a new process. One which is animated by the dynamic tendency of the love towards God, insofar as it fulfils a function of transition which prepares the soul to elevate itself to knowledge of the third kind, during the course of which the love towards God ‘is related to the soul alone’. Macherey interprets this sentence as describing the state of the soul as ‘detached from any consideration of the body’.¹² This concept is central to his interpretation of the relation between the second kind of knowledge and the third kind of knowledge, and the role that the third kind of knowledge plays in the affective life of a human being.

The duration of the soul without relation to the body

Two of the notions elaborated by Spinoza in the second half of part five of the Ethics are that of eternity, which gives a content to the idea of a life or of a duration of the soul distinct from the ‘present life’¹³; and that of the intellectual love of God, which is substituted for the love towards God.¹⁴,¹⁵ The distinction between the duration of the soul and its eternity corresponds to the difference between the second and third kinds of knowledge, and the transition from one to the other. But the presentation of this relation by Spinoza in Ethics, V, P23, is, according to Macherey, ‘manifestly ambiguous’,¹⁶ because Spinoza, by affirming that ‘the human Mind cannot be absolutely destroyed with the Body, but something of it remains which is eternal’,¹⁷ introduces the idea that the soul is itself only partially destroyed with the body, leaving something of it that subsists, which, precisely, is eternal. Macherey points out that ‘between that which is destroyed and that which subsists, a temporal relation of succession can be represented, permitting the establishment of a distinction between that which the soul is before this destruction and that which it is after’.¹⁸ By interpreting the theme of the eternity of the soul as it seems to be developed here by Spinoza, Macherey admits that it is difficult to escape an interpretation which reinstates this theme ‘on the terrain of traditional conceptions of the immortality of the soul, which imply the survival of the soul after the life of the body’.¹⁹ However, such an interpretation does not take into account the fact that, at least in relation to a body’s actual existence, there is also something of the body which cannot be absolutely destroyed, the idea of which is precisely that which remains of the soul. According to Spinoza, when a body is destroyed, all of its parts, and their corresponding ideas, continue to exist in different relations, constituting other bodies, and therefore other souls. ‘I understand the body to die when its parts are so disposed that they acquire a different relation of motion and rest’.²⁰ Macherey therefore argues that ‘that something of the soul’ which remains detached from the consideration of the body, ‘even while the body of which it is the idea is exposed to the risk of being destroyed by external causes, must not be interpreted in terms of survival and immortality, in relation to the affirmation of a previous or future existence’.²¹

The second kind of knowledge, which the love towards God makes available to the soul, is a mode of thinking that thinks of things in ‘duration, according to the principle of a necessity external to their own nature, which supposes that the soul affirms ... the actual existence of the body’.²² This is simply a knowledge that the body actually exists, without contributing anything to a knowledge of the body itself. Whereas the third kind of knowledge is a ‘knowledge of singular things’,²³ which it comprehends directly, both from the point of view of their existence in duration, and from that of their essence. It distinguishes itself from knowledge by common notions insofar as it engages the soul from a different point of view: the love towards God engages the soul from the point of view of duration, whereas the third kind of knowledge engages the soul from the point of view of eternity. Therefore, according to the logic of reasoning followed by Spinoza, the third kind of knowledge concerns the soul insofar as it is the ‘idea of the essence of the body and not of its existence’.²⁴ The activity of this knowledge does not coincide with the affirmation of the actual existence of the body but with the affirmation of the essence of the body, that which constitutes ‘the duration of the soul without relation to the body’.²⁵ In the Ethics, V, P23S, Spinoza characterizes this kind of knowledge as ‘a certain mode of thinking which pertains to the essence of the Mind, and which is necessarily eternal’. This mode of thinking is not the soul itself, nor its essence, but it ‘belongs to the essence of the soul’, that is, it belongs to its essence insofar as being the active tendency of the soul to form adequate ideas of the third kind of knowledge. It is in this way that the soul consists of something eternal. According to Macherey, ‘if there is something eternal in the human soul, it is therefore in this sense that it is inhabited, possessed, by a certain practice of eternity, which occupies it at least in part’.²⁶

The ‘idea of the idea’

The mode of thinking of the soul ‘without relation to the existence of the body’²⁷ should be considered as a ‘form of idea’, that is, insofar as the soul, as idea of the object that is the body, itself ‘constitutes the object of another idea’.²⁸ The form of idea that Macherey is referring to here is that of the ‘idea of the idea’, which is introduced in the Ethics, II, P21. According to the reasoning followed by Spinoza, ‘a thing, a body for example, the idea of this thing, which is the soul to which the body is united ... and finally the idea of this idea ... are one and the same thing envisaged from different points of view’.²⁹ According to Macherey, the paradox is that the ‘idea’ and the ‘idea of the idea’ are the same thing considered, not from the point of view of different attributes, as a body in the attribute of extension and the idea of this body in the attribute of thought, but rather from the point of view of ‘one and the same attribute; Thought’.³⁰ The knowledge that the soul has of itself is deployed dynamically within the attribute of thought between two extremes: one of these corresponding to inadequate ideas of the imagination, the inadequate ideas of inadequate ideas which are mixed in with the ideas that are their objects, that is, with the inadequate ideas that are the ideas which correspond partially or inadequately to the existence of the body; the other corresponding to the adequate ideas of the third kind of knowledge, adequate ideas of adequate ideas, being that to which Spinoza refers with the concept of the ‘idea of the idea’. The third kind of knowledge is developed by passing from the status of the idea as idea of a body, to that of the idea of the idea, that is, ‘to an idea which reflects its form of idea by detaching itself from the object of which it is the immediate idea’.³¹ The idea of the idea is therefore not a knowledge directly mixed with the others, but it develops on what Macherey describes as ‘a completely different plane’.³² An adequate idea of the third kind of knowledge, and not an inadequate idea of the imagination, ‘would be in this way the place [of the soul, or human Mind,] where the idea of the idea is formed, according to a principle of order and connection distinct from that which controls the formation of ideas properly speaking’,³³ that is, according to the order and connection of essences which is an eternal order, in place of the order and connection of existences which is the order of duration. However, these two distinct orders are situated in the ‘same system of causal determinations’,³⁴ therefore, the idea of the idea is presented as ‘a sort of simultaneous reading’ of the idea.³⁵ This conception, according to which the activities of knowledge, or ideas, develop simultaneously on many planes in the attribute of thought, assists in understanding what Spinoza refers to in the Ethics, V, P23, as that of the soul or human Mind which is eternal.

The practice of the knowledge of the third kind, and the intellectual love of God which follows from it,³⁶ leads the soul to its actual nature or essence. It corresponds to ‘the most perfect part of the soul’, such that no inadequate ideas of the imagination, ‘which are directly associated with the actual existence of the body, intervene in this process’.³⁷ The intellectual love of God is not dependant upon external relations of determination, but is rather the eternal element with which the soul effects the transition towards the conditions which permit it to develop a full regime of activity. According to Macherey, this is precisely what is expressed by the formula: ‘the duration of the soul without relation to the body’, at the end of the scholium to proposition 20. It is a duration which ‘must be thought without relation to time and incorporates in this way the paradoxical figure of an eternal duration, or of a duration which is deployed in a context which is not that of change but that of permanence and of eternity’.³⁸ The intellectual love of God is eternal because its function maintains no structural relation to the duration of the body. However, Macherey indicates that this ‘evidently does not signify that its eternity commences with the death of the body, nor that it would be subordinated to this’.³⁹ But rather, what Spinoza understands by this, Macherey argues, is that ‘the soul cannot liberate itself nor achieve eternity in the absence of the body, body and soul being necessarily maintained at the same time, of a same movement, by the process of emancipation of their power to act ... [that is] the eternity of the soul must equally concern the body’.⁴⁰

During the practice of the knowledge of the third kind, the soul conceives itself directly, in its singularity, as an integral part of the divine intellect, it is in this way that it shares its eternity. The actual, and not future, practice of the third kind of knowledge modifies the affective life by creating the conditions for the regulation of the affects, liberating the soul progressively ‘from all of the hindrances preventing the full development of its immanent power (puissance)’.⁴¹ The intellectual love of God establishes the precise conditions which allow the soul, and also the body, to accede to another form of life which is not only rooted in the singular reality of things, but in the reality of all things, that is, it is invested in ‘the universal impetus which carries in its movement the entirety of all reality’.⁴² Therefore, Macherey can conclude that ‘the experience of the intellectual love of God does not have in any way the value of a solitary mystical exaltation’, but, on the contrary, it is ‘a practice that tends to be collective’ according to which one experiences oneself as ‘a man among others’, who are ‘directly implicated in the unfolding of this experience’.⁴³ This constitutes in a certain way its ‘political’ dimension, which can therefore be understood to be no longer solely concerned with existences but rather also with essences, such as they are considered from the point of view of eternity.

Macherey describes the activity of knowledge, which ‘leads it to develop simultaneously on many planes’, as ‘a constitutive reflexivity’.⁴⁴ Given that both corporeal and mental determinations are one and the same thing, considered from the point of view of different attributes, he raises the question of ‘whether or not an identical reflexivity is to be found in the corporeal order’.⁴⁵ Macherey argues that if ‘the events of the body also simultaneously unfold on many strata, the superimposition of which increases the depth of its organization: [then] there would also be a body of the body, which is eternal, as there is a soul of the soul, which is eternal’.⁴⁶ Despite the fact that ‘Spinoza does not explicitly develop this point’,⁴⁷ Macherey suggests that ‘all hypotheses on this subject are permitted’.⁴⁸ In Introduction à l’Ethique de Spinoza: la cinquième partie, Macherey does not provide such a hypothesis, since the aim of his reading is to follow as directly as possible the argument of the text of the Ethics. Instead, he maintains that, as of the Ethics, V, P29, the question of the role attributed to the body in the unfolding of this process is no longer addressed, and it is only with proposition 39 that Spinoza ‘reestablishes a strict correlation between the nature of the body and that of the soul, conforming to the definition of the soul as “idea of body”’.⁴⁹ Spinoza writes that someone ‘who has a Body capable of a great many things has a Mind whose greatest part is eternal’.⁵⁰ According to Macherey, the necessity of reaffirming this strict correlation is due to the fact that, in the Ethics, it is ‘ceaselessly called into question, or quite simply forgotten’.⁵¹ He writes that ‘the presence of the body is indicated in the text of the Ethics by a stippled effect, on a sort of line of accompaniment, the execution of the principal melody remaining reserved for the soul’.⁵² Macherey’s reading of the different parts of the text of the Ethics provides, as close as possible, a re-presentation of Spinoza’s arguments. It proceeds by finding the aporias in the text which illustrate its limits. Macherey introduces the paradoxical figure of an eternal duration, which he considers to be repeated in the idea of the idea, such that they can be determined together as the eternal idea of the idea in duration, to represent the limit of the third kind of knowledge in the fifth part of the Ethics.

The argument from powers

The paradoxical figure of an eternal duration does not emerge in Deleuze’s reading of Spinoza. Instead Deleuze eliminates the need to postulate such a figure by providing a reading of the Ethics that allows the characterization of a particular hypothesis of a body of the body which is eternal. This divergence of their respective readings of the Ethics is determined according to their different interpretations of the Ethics, II, P7C: ‘God’s power of thinking is equal to his actual power of acting’. Deleuze understands Spinoza to have defined substance as being composed of two equal powers: the power of existing or acting, and the power of thinking. The power of thinking has as its sole condition the attribute of Thought. The attribute of Thought is the condition for assigning to substance an absolutely infinite power of thinking. The difference between their respective readings emerges with the question of whether or not it is necessary to identify ‘the power of thinking’ with the ‘attribute of Thought’? Macherey affirms their identity, whereas Deleuze affirms their difference.

According to Deleuze, the attribute of Thought has an aspect different to the attribute of Extension insofar as it manifests all of the affections of substance as ideas. The attribute of Thought contains as many ideas as there are modes of the attribute of Extension, and as many ideas as there are individuals,⁵³ insofar as an individual is composed of both an idea, in the attribute of Thought, and its object, in the attribute of Extension. The idea of the idea according to Deleuze is the idea not only of the idea but also of the object to which it corresponds, insofar as the two constitute an ‘individual’. Therefore, according to Deleuze’s reading, the attribute of Thought seems to have priority or to be privileged over the other attributes. In response to this aspect of Spinoza’s argument Macherey introduces the paradoxical figure of the eternal duration of the idea of the idea, while at the same time affirming that both the idea and the idea of the idea, insofar as they are ideas, are the same thing considered in one and the same attribute of Thought. Deleuze does not disagree with this affirmation, however his argument from powers eliminates the need for such a figure. Deleuze argues rather that the power of thinking, in which all ideas participate, is not greater than the corresponding power of existing, or acting, in which all things participate. He quotes Letter XL of the correspondence of Spinoza, in which Spinoza maintains that ‘the power of Thought to think about or to comprehend things, is not greater than the power of Nature to exist and to act’. Deleuze argues that what Spinoza understands by this is that all attributes are equal in relation to the power of existing, yet this power of existing is only one side of the absolute power of substance; the other side being a power of thinking, which is equal to it. It is in relation to the power of thinking that the attribute of Thought can be understood to have certain privileges. The attribute of Thought itself conditions a power equal to that conditioned by all the other attributes. Deleuze suggests that ‘it is the equality of powers that confers special capacities in a domain which is no longer that of the equality of attributes’.⁵⁴ The attribute of Thought is related to the power of thinking in the same way in which all of the attributes (including Thought) are related to the power of existing. It is important to note that the power of existing is not the same as the attribute of Extension. All attributes have a power of existing, including the attribute of Thought. The extensive parts which belong to the attribute of Thought are ideas, that is, those ideas which are ideas of objects in a different attribute. The distinction between ideas in the power of thinking and simple ideas, as, for example, the idea of a body, is explained by Spinoza as follows: ‘The idea that constitutes the formal being of the human mind is the idea of a body, which is composed of a great many highly composite individuals. But of each individual composing the body, there is necessarily an idea in God’.⁵⁵ The idea of each of these individuals, which is an idea in God, or in the power of thinking, is an idea both of the Body and of the idea of the Body, which is an idea of the human Mind, insofar as they both exist. Therefore ideas exist no less than do bodies, and Thought, no less than Extension, is a form of existence: ‘when singular things are said to exist, not only insofar as they are comprehended in God’s attributes, but insofar also as they are said to have duration, their ideas also involve the existence through which they are said to have duration’.⁵⁶

According to this argument from powers, the problem of the attribute of Thought having priority or being privileged over the other attributes is resolved. It renders any difference between the attributes ‘ontologically insignificant’.⁵⁷ It allows Deleuze to make the assertion of an identity of being between objects and ideas, such that objects and ideas can be affirmed as ontologically identical. Therefore it can be argued that an individual, and the idea of that individual, are one and the same thing expressed in two ways, in two powers. Deleuze argues that every idea that belongs to the attribute of Thought is the object of an idea that participates in the power of thinking, this is what he understands to be the idea of the idea. According to Deleuze, the idea and the idea of the idea, which are referred to the same attribute, are referred also to the power of existing and the power of thinking respectively, since the attribute of Thought is, on the one hand, a form of existence, and on the other, the condition of the power of thinking. The idea of the idea in the power of thinking is the idea of both the mind and the body it corresponds to, insofar as these are composed of extensive parts in the power of existing.

The idea of the essence of the body

Contrary to Macherey, Deleuze maintains that ‘Every common notion leads us to the idea of God’,⁵⁸ and since common notions already belong to the second kind of knowledge, the idea of God, incorporated in the love towards God, constitutes the limit of this kind of knowledge. ‘Every common notion’, he argues, ‘expresses God as the source of the relations combined together in the bodies to which the notion applies’,⁵⁹ that is, the commonness of a common notion, once the common notion has been formed, can be understood to express, or lead to, the idea of God. It is in relation to this that he concludes that: ‘We … come to the idea of God … through the second kind of knowledge’.⁶⁰ Therefore, the ‘second kind of knowledge does indeed give us adequate ideas; but these are only ideas of properties common to our body and external bodies’,⁶¹ they in no way amount to an adequate idea of ourselves, of the essence of our body, nor of the essence of any other thing. Deleuze argues that in arriving at the idea of God, of the love towards God, we are determined to leave behind the second kind of knowledge and enter into what he calls ‘a new state’, according to which ‘the idea of God will itself then change in content’.⁶² He explains that ‘the idea of God, which is joined to, or “accompanies”, all common notions, leads us to a reappraisal of attributes and modes’.⁶³ The attributes, when understood according to the third kind of knowledge, are no longer grasped simply as the common properties of existing modes, but as univocal forms common both to substance, whose essence they each equally express, and to modes, insofar as each attribute contains all of the essences of its modes. The third kind of knowledge is eternal insofar as the systematic relation between the essence of substance and the modal essences, which is made univocal through the attributes, is understood as eternally given, rather than being understood in an occasional sense in duration, that is, in relation to the extensive parts of the finite existing modes, as are the ideas of the first and second kinds of knowledge. Deleuze maintains that ‘the “transition” is only an appearance; in reality we are simply finding ourselves as we are immediately and eternally in God’.⁶⁴ As Spinoza writes: ‘The mind has had eternally the same perfections which, in our fiction, now come to it’.⁶⁵ Deleuze describes the third kind of knowledge of an individual as ‘a faculty, or a power’, or as an active power of thinking or of understanding things sub specie aeternitatis. He argues that when we have an idea of the third kind of knowledge, ‘our soul is thus affected by the idea of itself, by the idea of God, and by the idea of other things sub specie aeternitatis’.⁶⁶ The thought of an individual within the power of thinking is in this way immanent to substance. In relation to the third kind of knowledge, Deleuze maintains that ‘insofar as it expresses the body’s actual existence in duration, the soul has the power to conceive other bodies in duration; insofar as it expresses the body’s essence, it has the power to conceive other bodies sub specie aeternitatis’.⁶⁷

According to Deleuze’s concept of individuation, as determined by his interpretation of Spinoza’s theory of relations, a singular modal essence, or intensive part, expresses its degree of power as a power of existing or acting in the extensive parts of its finite existing mode, which consist of both ideas in the attribute of Thought, and the objects of these ideas in the attribute of Extension. A singular modal essence simultaneously expresses this degree as its power of thinking, insofar as its essence is univocal with the essence of Substance by means of the attributes; an idea of this univocity would be an idea sub specie aeternitatis. The singular modal essence of an individual is the essence of its soul from the point of view of the attribute of Thought, and the essence of its body from the point of view of the attribute of Extension. The essence of the soul in the attribute of Thought is expressed both in the power of thinking, as an idea in God or an idea according to the third kind of knowledge; and in the power of existing, as simple ideas. The essence of the body in the attribute of Extension is only expressed in the power of existing as the component parts of an individual’s body. In the power of existing, the essence of the soul and the essence of the body are expressed by extensive parts in different attributes. However, from the point of view of the power of thinking, they are each expressions of the same intensive part. An intensive part in the power of thinking is an idea in God of its corresponding extensive parts in the power of existing, whether the extensive parts are an expression of the essence of the soul as ideas in the attribute of Thought, or an expression of the essence of the body as objects of those ideas in the attribute of Extension. An idea in the power of thinking is therefore an idea of the essence of the soul, insofar as the essence of the soul is expressed by an idea in the power of existing, that is, an idea of the essence of the soul is an idea in the power of thinking of this idea in the power of existing; it is therefore the idea of the idea. An idea in the power of thinking is also an idea of the essence of the body, insofar as the essence of the body is expressed by the object, in the attribute of Extension, of an idea, in the attribute of Thought, both of which are in the power of existing; both of which together constitute the object of the idea of the essence of the body in the power of thinking. An idea in the power of thinking is therefore both the ‘idea of the idea’ and the ‘idea of the essence of the body’. An idea in the power of thinking is necessarily an idea in God, and an idea that we have in the power of thinking, as individual human beings, is an idea according to the third kind of knowledge. This is what Spinoza understands by the ‘idea of the idea’. Such an idea is a specific instance of an idea in the power of thinking that we have according to the third kind of knowledge. It is an idea of the essence of our body insofar as it is univocal with substance. To the extent that we have ideas according to the third kind of knowledge, they are shared with God in the power of thinking, and as such are eternal.

‘We become completely expressive’

Given the differences between the respective interpretations of Spinoza by Macherey and Deleuze, it is not surprising to find differences in the way that each resolves the ‘extremely elliptical character of the reasoning of the fifth part of the Ethics’,⁶⁸ in particular, the relation between the death of a human being and the eternal character of the third kind of knowledge. The comments that Macherey makes towards the end of his article, ‘The encounter with Spinoza’, can be used to characterize this difference. According to Macherey’s reading, ‘there is nothing to suggest that the soul can ever cease being the idea of a body, and so continue to exist without the body, … the experience of eternity available to us while alive is for Spinoza an absolute actuality and thus completely independent of any beginning or end of our existence’.⁶⁹ He writes that ‘Deleuze’s discussion … of the problem of the soul’s immortality in terms of the state in which the soul finds itself “after death” (“We become completely expressive”⁷⁰) is hardly convincing’.⁷¹ By articulating this relation as ‘the problem of the soul’s immortality’, Macherey misleadingly suggests that Deleuze resolves the problem in these terms. Deleuze, however, considers Spinoza not to have used the concept of immortality in the Ethics specifically in order to avoid ‘the most tiresome of confusions’, insofar as ‘theories of immortality always involve a confusion of duration with eternity’.⁷² In order to avoid this confusion Deleuze argues that ‘Spinoza … asserts a difference of nature between duration and eternity’.⁷³ Contrary to Macherey’s articulation of the paradoxical figure of an eternal duration, Deleuze maintains ‘that while the body exists, duration and eternity themselves “coexist” in the soul as two elements different in nature’.⁷⁴ What Deleuze understands by this is that ‘while the soul is the idea of the existing body, there coexist in it extensive parts that belong to it in duration, and an intensive part that constitutes it in eternity’.⁷⁵ However, in agreement with Macherey, Deleuze affirms that ‘we should not imagine that the soul endures beyond the body: it endures while the body itself endures, and it is eternal insofar as it expresses the body’s essence’.⁷⁶ Deleuze also agrees that the soul’s eternity can indeed be the object of a direct experience. According to him, ‘we experience that we are eternal’⁷⁷ to the extent that we form the idea that expresses our body’s essence, that is, form the idea of ourselves as it is ‘in God’⁷⁸ according to the power of thinking.

To talk of the eternal element of the soul of an individual after its death is, according to Macherey, to introduce a concept of immortality into the Ethics, which, both Macherey and Deleuze agree is absent. Therefore, the argument of the Ethics, V, according to the reading of Macherey, does not go beyond a discussion of the eternal element of the soul of an individual while it endures. For Macherey, that ‘something of the soul which is eternal’ can only be characterized by making reference to the body which is ‘exposed to the risk of being destroyed’⁷⁹ but which is not yet dead. Deleuze, however, considers there to be traces of a further argument in the Ethics that does address the death of an individual and the implications that this has for the eternal element, the intensive part, of that individual’s soul. On the one hand, Deleuze seems to be in agreement with Macherey when he affirms that ‘the eternity of essence does not come afterwards; it is strictly contemporaneous, coexistent with existence in duration’;⁸⁰ however, on the other hand, he maintains that ‘the soul eternally remains what it already is in its essence during the body’s existence: an intensive part, a degree of power or power of understanding [or thinking], an idea that expresses the body’s essence sub specie aeternitatis’.⁸¹ From the point of view of that which remains, Deleuze argues that, with death, the soul of an individual only ceases to be expressed by extensive parts, and that which ‘remains’ is its intensive part.

In Ethics, IV, P39S, Spinoza outlines what happens when an individual is destroyed or decomposed.⁸² According to Deleuze, what Spinoza understands by this is that, with death, an individual loses all the extensive parts that belonged to it in ‘a certain relation’,⁸³ that is, the extensive parts generated by the global integration determinative of the individual. What has been ‘done away with’ is not the composite relation which creates the intensive part determinative of the individual’s singular modal essence, but rather ‘the extensive parts generated by the precise relation’, or global integration, ‘which have now assumed another relation’.⁸⁴ He argues that ‘to some extent we lose common notions and active affections of the second kind, for these have no independent role except as they relate to existence’.⁸⁵ Only those ‘affections of the third kind’ that we succeeded in experiencing during our existence, will ‘be exercised (after death)’.⁸⁶ Deleuze suggests that ‘death seems to put us in a situation where we can only be affected by affections of the third kind, which are themselves explained by our essence’.⁸⁷ What Deleuze understands by an ‘affection of the third kind’ is the implication of the intensive part, or the essence of our soul, in intensive parts of greater degree of power, that is, its further differenciation in more composite global integrations. The more affections of the third kind that we experience during our existence, ‘the less we lose in losing existence, in dying or even in suffering’,⁸⁸ that is, the greater is ‘the part that remains and the greater the part that is not touched by bad affects’.⁸⁹ Therefore, according to Deleuze, the essence of the soul of an individual ‘loses none of its perfection’⁹⁰ when it loses the elements of extension of which its finite existence was composed.⁹¹ What remains is the intensive part, which is the essence of the individual’s soul in the power of thinking, as implicated in those intensive parts of greater degree of power, in which it has been further differenciated. Deleuze concludes by suggesting that in this capacity, ‘our essence adequately expresses God’s essence, and the affections of our essence⁹² adequately express our essence. We become completely expressive’.⁹³ As already indicated, Macherey is critical of this conclusion. However, given the differences between their interpretations of Spinoza, the question remains as to what exactly Deleuze understands by the phrase ‘We become completely expressive’? This question can be addressed by means of two examples: the first taken from Deleuze’s Spinoza: Practical Philosophy; and the second from the correspondence of Spinoza, Letter XVII.

Adequate relations and the experience of active joy

In Spinoza: Practical Philosophy, Deleuze suggests that ‘whenever an idea is adequate, it precisely captures at least two bodies, mine and another, insofar as they compound their relations (“common notion”)’.⁹⁴ The constitutive nature of an adequate idea, in the form of an adequate relation or more composite global integration, establishes the parameters for the following example of a determined relation between two individual human beings. In a marriage, ‘the characteristic relation of a couple, … though it is an instituted, contractual social relation, nevertheless constitutes’, according to Deleuze, ‘an individuality of a certain type’.⁹⁵ A couple finds themselves in a relation, explains Deleuze, when the experience of a joyful passive affection by each potential partner indicates that they have something in common, or that their natures agree, and that there is the potential of a composite increase in each of their powers of acting. The experience of the joy of a joyful passive affection can therefore initiate the development of a relation which could in turn lead to the development of an adequate idea of the relation, and the experience of active joy. If each partner has the idea of her or himself as a cause of the relation, that is, as the common cause of the relation, then this idea constitutes an adequate idea of the properties or affections of the relation, the relation would thereby become an adequate relation.⁹⁶

Insofar as each partner has the idea of their relation as generated by the further differenciation of their respective global integrations in a more composite global integration, they share the idea of their relation as an adequately determined relation in the power of thinking. Such an adequate relation would correspond to the incorporation of the institutionalized contractual arrangement by their relation in the form of a social relation, or social body. Deleuze suggests that a couple involved in an adequate relation constitute an individuality of a certain type, which is characterized by the more composite global integration into which the global integrations determinative of each partner involved is further differenciated.

Insofar as each partner makes a fetish, in her or his own way, of an institutionalized contractual social relation as the determinant of the relation in which they are involved, their relation would rather remain an inadequately determined or imaginary relation, insofar as each partner has a different inadequate or imaginary idea of an institutionalized contractual social relation as the cause of their relation. Rather than the relation adequately incorporating the institutionalized contractual social relation, each potential partner has instead been incorporated or determined by it, insofar as they have each been effected from the outside by what becomes from this point of view a larger more powerful external body, which is consequently only partially understood by them, that is, of which they each have only a partial, or inadequate, idea. The initial joy of the joyful passive affection experienced by each potential partner is thereby exposed to the risk of reversal to sadness.

Aside from his disagreement with the role or function that Deleuze ascribes to joyful passive affections, Macherey would agree with the distinction that Deleuze makes between an inadequately determined or imaginarily institutionalized relation and an adequately determined institutional or social relation. However, he would argue that the latter relation would only be that which is imagined ‘more distinctly and vividly’,⁹⁷ both relations nevertheless remaining imaginary. The only way to initiate the development of an adequate idea of a relation, according to Macherey, would be to extrapolate from the more distinct and vivid imaginary idea of the relation to that which is universally common to each partner in the relation, that is, to the idea of God, thereby developing amor erga Deum. It is only by means of having an idea of their commonness in God that each partner can develop an adequate idea of ‘the properties of’ the relation.⁹⁸

Post Mortem

In Letter XVII to Pieter Balling, Spinoza discusses the relation between Balling and his son. What is of interest in this letter is the way Spinoza describes the relation and how this description can be interpreted given Deleuze’s argument about the relation between the eternal character of the third kind of knowledge and death. Spinoza writes: ‘a father so loves his son that he and his beloved son are, as it were, one and the same. According to what I have demonstrated on another occasion, there must be in thought an idea of the son’s essence, its affections, and its consequences. Because of this, and because the father, by the union he has with his son, is a part of the said son, the father’s soul must necessarily participate in the son’s ideal essence, its affections, and consequences’.⁹⁹ The relation that Spinoza describes here is one that functions at the level of the third kind of knowledge since it involves the knowledge of essences. The relation between father and son can be understood to be another example of an adequately determined, or individuated, relation between two individual human beings, and is therefore continuous with the previous example. According to Macherey, when an individual experiences itself as an individual among others who are directly implicated in the unfolding of this experience, in this case the experience of the relation with one’s partner or son, an idea of such a relation according to the third kind of knowledge is an idea not only from the point of view of one’s existence, but also from the point of view of the relation of one’s essence to reality as a whole, sub specie aeternitatis, that is, from the point of view of eternity. It is an actual, and not future, experience of eternity, which is therefore only an experience of freedom and emancipation which ends with duration.

When Spinoza writes in Letter XVII that ‘there must be in thought an idea of the son’s essence, its affections, and its consequences’, he is referring to what is, according to Deleuze’s interpretation of Spinoza, an idea in God, or in the power of thinking. Insofar as the father’s soul ‘must necessarily participate in the son’s ideal essence, its affections, and consequences’, the father can be said to have an idea in the power of thinking, that is, of the third kind of knowledge, not only of the essence of the son, which follows necessarily according to the third kind of knowledge from the idea of his own essence, but also of his son’s existence in duration which follows necessarily according to the third kind of knowledge from the idea of his own existence in duration. Spinoza also refers to the father and son as ‘one and the same’, insofar as ‘the father, by the union he has with his son, is a part of the said son’, which can also be understood conversely as the union that the son has with his father, insofar as he is a part of the father. What does Spinoza mean when he writes that ‘father and son are one and the same’? The father and son are one and the same insofar as there is an idea in the power of thinking of their relation, according that is to the third kind of knowledge. The idea that the father has of their relation is therefore an idea that is shared in God. This idea in God, or in the power of thinking, is the idea of the union of the two, father and son. Therefore, father and son are one and the same insofar as the father has the idea in the power of thinking of their individuality, which is an idea of the relation composed by the union of the two of them. Together, father and son constitute ‘an individuality of a certain type’.¹⁰⁰ The discussion of the hypothesis of the ‘body of the body which is eternal’ can be reintroduced here to assist in the explanation of this relation. According to the hypothesis, such an individuality would be a type of body composed by the particular adequate relation between father and son, that is, it would be constituted by the further differenciation of their global integrations in a more composite global integration, which would be understood adequately by the father, according to the third kind of knowledge in the power of thinking. According to Deleuze’s reading of Spinoza, the idea of the intensive part of this relation is eternal. The body of the body is therefore the more composite global integration generated by the union of father and son, which has as its essence the intensive part created by the composite relation between the global integrations of the existing bodies of both father and son. The ‘ideal essence’ of the son of which Spinoza speaks can be understood to be the intensive part of the son insofar as it constitutes the intensive part of the more composite global integration together in composite relation with that of the father. The intensive part of the more composite global integration is eternal in the power of thinking independently of the extensive parts which it redeploys with the expansion of its power series in the power of existing. The intensive parts of both father and son are implicated in the intensive part of the more composite global integration which is of greater degree of power. The power of acting of both father and son sustain a determined increase by virtue of this relation.

In Letter XVII, Spinoza is addressing an event which occurred just before the death of Balling’s son. Spinoza writes that ‘the father’s soul participates ideally in the things which follow from the son’s essence’, that is, the father has an idea in the power of thinking of the more composite global integration composed by both he and his son. Given the hypothesis of the body of the body, which expresses the relation between father and son as an individuality in the form of a more composite global integration, the question that remains is what actually transpires with the death of one of the partners, in this instance specifically with the death of Balling’s son? According to the argument put forward by Deleuze, the son loses all of the extensive parts constitutive of his existence, that is, which express his power of existing, or power to act. All that remains is his intensive part which, along with the intensive part of his father, has created the intensive part of the more composite global integration. This intensive part, insofar as it is eternal in God, remains unaffected by death; it is still constituted by the intensive part of the father together in composite relation with the intensive part of the son, despite the extensive parts of the more composite global integration, which was characteristic of the father– son relation, having been redeployed in the relations of different global integrations. Insofar as the intensive part of the son continues to be implicated in the intensive part of the more composite global integration, the son can be understood to function solely in this capacity, that is, the son can be understood to have become, as Deleuze writes, ‘completely expressive’. Both the intensive part of the father, together in composite relation with the intensive part of the son, continue to express the intensive part of the more composite global integration, despite the fact that the extensive parts of this more composite global integration, those specifically of the son, cease to exist in the same form. That which remains of the soul of the son is its intensive part, which has become completely expressive, together with that of the father, in the intensive part, or ‘ideal essence’, which was determinative of their adequate relation, or the more composite global integration in which they were implicated.¹⁰¹ The eternity of the intensive part created by the composite relation between father and son continues to be the object of a direct experience for the father.¹⁰²

Both Macherey and Deleuze agree that the eternity of the third kind of knowledge is immanent to the duration of an individual. However, to speak of the eternal element of the soul of an individual after its death is, for Macherey, to introduce a concept of immortality to the Ethics. Deleuze, on the contrary, suggests that, with death, the soul of an individual only ceases to be realized by extensive parts, that which remains being its intensive part. According to Deleuze, it is only insofar as the soul of an individual is immanent to duration in the power of thinking that it can ‘remain’. Macherey suggests that the eternity of the soul occupies the soul at least in part, but not as a separate part, rather it belongs simultaneously to the soul insofar as it endures. This paradoxical relation is introduced by Macherey in the figure of an eternal duration, which functions as a limit to the third kind of knowledge. In this way, Macherey emphasizes the actual, and not future, practice of the third kind of knowledge. According to Deleuze however, the intensive part of the soul remains only insofar as it has been further differenciated in a more composite global integration, thereby being implicated in the creation of an intensive part of greater degree of power, during its lifetime. The intensive part of the soul, once created, continues to function in its capacity as an intensive part, that is, as a degree of power of the power series expansion of an attribute, expressing, as it does a singular idea, in the power of thinking, of an actual or potential individual, or social body, in the power of existing. According to whether one adopts with the Ethics the point of view of an uninterrupted affective flux or that of the joyful passive affections, two different readings of Spinoza arise. Macherey’s reading presents ‘the essentially subjective character of the affects’,¹⁰³ from the point of view of the imagination of the soul, while Deleuze explains affectivity from the point of view which remains that of the body and its power to act. Macherey considers the effect of the individual affects of the affective life of a finite mode, whether joy or sadness, to remain fundamentally ambiguous, absorbed by the uninterrupted affective flux, which hinders the expression of a mode’s fixed power to act to within the range of a maximum and minimum. All of a mode’s power to act is expressed, however, according to the uninterrupted affective flux, it is simultaneously expressed both actively and passively. Deleuze’s reading complicates these aspects of Macherey’s reading. The concept of a mode’s capacity to be affected implies the concept of a variable power to act as the active expression of a mode’s power of existence. Deleuze assimilates directly a mode’s capacity to be affected with its power to act as the expression of its active affections. The passive affections function to limit these to within the range of a mode’s fixed power of existence, that is, between a maximum and minimum. Therefore, according to Deleuze, the three aspects of a mode, its capacity to be affected which is expressed by its conatus as its power to act, are all together proportionally open to variation within this range, and their expression is limited by the passive affections to which they are subject.

The difference between their interpretations of passive affects does not stop here. According to Deleuze passive affections are divided into on the one hand, joyful passive affections, and on the other, sad passive affections. Insofar as they indicate a ‘chance composition of our being with that of something else that agrees with it in nature’, the joyful passive affections ‘prefigure or prepare the way for the transition to greater activity’.¹⁰⁴ To this extent, without actually being reasonable, ‘they agree with reason’.¹⁰⁵ However, Deleuze in no way guarantees that all of the joyful passive affections that are experienced will always form adequate ideas. He suggests rather that the joy of a joyful passive affection can be isolated before becoming the prey of the fluctuatio animi, and can in this way contribute to the formation of a common notion.

According to Macherey, the first common notion that we can have is the love towards God, while according to Deleuze the first common notions that we can have are the simplest common notions, which are therefore constituted locally, without it being necessary to make reference to the general common notion of the love towards God. Macherey however denies that we can deduce adequate ideas without having in the first place acquired the love towards God. He considers that the love towards God establishes the base of an internal regulation of the affective life which therefore constitutes the first step in the production of the second kind of knowledge. Whereas Deleuze considers the love towards God rather as a limit of the second kind of knowledge which prepares for the transition to the third kind of knowledge. It is therefore only necessary to make reference to the God of the love towards God before we can have a knowledge which is constituted by the adequate ideas of the third kind.

Insofar as the eternity of the third kind of knowledge is immanent to the duration of the individual, Macherey explains it with the paradoxical figure of an eternal duration, that is, of the eternal idea of the idea which endures. He affirms that the idea, and the idea of the idea, insofar as they are ideas, are one and the same thing considered under one and the same attribute of Thought. Deleuze does not oppose this affirmation, he does, however, introduce the argument from powers which eliminates the need for such a paradoxical figure. According to Deleuze, the two kinds of ideas are not only related to the same attribute, but also to the power of existing and to the power of thinking respectively. The attribute of Thought is related to the power of thinking in the same way that all the attributes (including thought) are related to the power of existing. The paradoxical figure functions as a limit of the third kind of knowledge insofar as this kind of knowledge is an actual, and therefore not future, experience or knowledge. Deleuze, on the contrary, maintains that, with death, the soul of an individual only ceases to be realized by extensive parts, that which remains being its intensive part. It is only insofar as the intensive part of an individual is immanent to duration in the power of thinking that its soul can be understood to remain. The intensive part of an individual remains only insofar as it has been further differenciated in a more composite global integration, thereby being implicated in the creation of an intensive part of greater degree of power, during its lifetime. Insofar as the intensive part of an individual remains implicated in the intensive parts of the more composite global integrations that were generated during its lifetime, it can be understood to have become ‘completely expressive’.

The mechanism by means of which joyful passive affections operate in Deleuze’s reading of the Ethics has been determined according to the logic of different/ciation, and it is this logic which is determinative of Deleuze’s reading of the system of the Ethics as a whole. The project of the present work has been to develop an understanding of Deleuze’s Spinoza by providing a Deleuzian reading of Expressionism in Philosophy, which not only charts the mechanism of operation of the logic of different/ciation in Expressionism in Philosophy, but which also positions Deleuze’s Spinoza and Expressionism in Philosophy, and this logic, within the context of the development of Deleuze’s wider philosophical project. Such a Deleuzian reading of Expressionism in Philosophy is therefore important for the determination of the manner by means of which Spinoza becomes for Deleuze a significant figure in his project of tracing an alternative lineage in the history of philosophy, and serves well to situate Expressionism in Philosophy within the context of Deleuze’s project of constructing a philosophy of difference. The context of this dual Deleuzian project, specifically in relation to the Deleuzian reading of Expressionism in Philosophy offered in the present work, is explicated in chapter 10 in relation to Deleuze’s other major text from the same period, Difference and Repetition.

_______________

Notes

¹ See Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 68; a translation of Spinoza, Ethics, V, P6S.

² Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 175.

³ Ibid., p. 97.

⁴ Ibid., p. 94.

⁵ Spinoza, Ethics, V, P20S.

⁶ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 98.

⁷ Ibid., p. 103.

⁸ Spinoza, Ethics, V, P20.

⁹ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 103.

¹⁰ Ibid.

¹¹ Ibid., p. 104.

¹² Ibid., p. 113.

¹³ Spinoza, Ethics, V, P21-31.

¹⁴ Ibid., V, P32-37.

¹⁵ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 117.

¹⁶ Ibid., p. 127.

¹⁷ Spinoza, Ethics, V, P23.

¹⁸ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 127.

¹⁹ Ibid., p. 128.

²⁰ Spinoza, Ethics, V, P39S.

²¹ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 133. Additional italics.

²² Ibid., p. 123.

²³ Spinoza, Ethics, V, P36S.

²⁴ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 131.

²⁵ Spinoza, Ethics, V, P20S.

²⁶ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 131.

²⁷ Spinoza, Ethics, V, P40.

²⁸ Macherey, Introduction à l’Ethique de Spinoza, la seconde partie, p. 199 n. 1.

²⁹ Ibid., p. 201.

³⁰ Ibid., p. 200.

³¹ Ibid., p. 199 n. 1.

³² Ibid., p. 196 n.

³³ Ibid., p. 202 n. 1.

³⁴ Ibid., p. 201.

³⁵ Ibid., p. 196 n.

³⁶ According to Spinoza, ‘From the third kind of knowledge there necessarily arises an intellectual Love of God. For from this kind of knowledge there arises joy, accompanied by the idea of God as cause’ (Ethics, V, P32C).

³⁷ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 189.

³⁸ Ibid., p. 158.

³⁹ Ibid., p. 187.

⁴⁰ Ibid., p. 182.

⁴¹ Ibid., p. 203.

⁴² Ibid., p. 175.

⁴³ Ibid., p. 172.

⁴⁴ Macherey, Introduction à l’Ethique de Spinoza, la seconde partie, p. 200 n. 1.

⁴⁵ Ibid.

⁴⁶ Ibid., p. 202 n. 1.

⁴⁷ Ibid.

⁴⁸ Ibid., p. 200 n. 1.

⁴⁹ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 182 n. 2.

⁵⁰ Spinoza, Ethics, V, P39.

⁵¹ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 182 n. 2.

⁵² Ibid.

⁵³ According to Spinoza, in each attribute, a mode together with the idea that corresponds to it in the attribute of Thought forms an ‘individual’; that is, an idea in Thought, and its object in a different attribute, form one and the same individual.

⁵⁴ Deleuze, Expressionism in Philosophy, p. 121.

⁵⁵ Spinoza, Ethics, II, P15D.

⁵⁶ Ibid., II, P8C.

⁵⁷ Michael Hardt, An Apprenticeship in Philosophy: Gilles Deleuze (London, 1993), p. 85.

⁵⁸ Deleuze, Expressionism in Philosophy, p. 299.

⁵⁹ Ibid., p. 298.

⁶⁰ Ibid., p. 299.

⁶¹ Ibid., p. 306.

⁶² Ibid., p. 299.

⁶³ Ibid., p. 300.

⁶⁴ Ibid., p. 308.

⁶⁵ Spinoza, Ethics, V, P33S.

⁶⁶ Deleuze, Expressionism in Philosophy, p. 316.

⁶⁷ Ibid., p. 313.

⁶⁸ Macherey, ‘The encounter with Spinoza’, p. 155.

⁶⁹ Ibid., p. 154.

⁷⁰ Deleuze, Expressionism in Philosophy, p. 315.

⁷¹ Macherey, ‘The encounter with Spinoza’, p. 154.

⁷² Deleuze, Expressionism in Philosophy, p. 314.

⁷³ Ibid., p. 313.

⁷⁴ Ibid., p. 315.

⁷⁵ Ibid.

⁷⁶ Ibid.

⁷⁷ Ibid.

⁷⁸ Spinoza, Ethics, V, P30: ‘Insofar as our Mind knows itself and the Body under a species of eternity, it necessarily has knowledge of God, and knows that it is in God and is conceived through God’.

⁷⁹ Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 133.

⁸⁰ Deleuze, Spinoza: Practical Philosophy, p. 40.

⁸¹ Deleuze, Expressionism in Philosophy, p. 314.

⁸² Spinoza, Ethics, IV, P39S: ‘I understand the body to die when its parts are so disposed that they acquire a different relation of motion and rest’.

⁸³ Ibid., II, P13D1. See Deleuze, Expressionism in Philosophy, p. 315. Or, in ‘a precise relation of motion and rest’, translated from Ramond, qualité et quantité dans la philosophie de Spinoza, p. 189.

⁸⁴ Deleuze, Spinoza: Practical Philosophy, 1981, p. 32.

⁸⁵ Deleuze, Expressionism in Philosophy, p. 319.

⁸⁶ Ibid., p. 317.

⁸⁷ Ibid., p. 315.

⁸⁸ Deleuze, Spinoza: Practical Philosophy, p. 43.

⁸⁹ Spinoza, Ethics, V, P40C.

⁹⁰ Deleuze, Expressionism in Philosophy, p. 315.

⁹¹ Spinoza, Ethics, V, P40C: ‘The eternal part of the mind is the understanding, through which alone we are said to act. But what we have shown to perish is the imagination, through which alone we are said to suffer action’.

⁹² That is, those affections in which our essence is further differenciated.

⁹³ Deleuze, Expressionism in Philosophy, p. 315.

⁹⁴ Deleuze, Spinoza: Practical Philosophy, p. 36.

⁹⁵ Ibid., p. 34.

⁹⁶ Amelie Rorty explores the implications of such a relation in ‘Spinoza on the Pathos of Idolatrous Love and the Hilarity of True Love’, in R.C. Solomon and K.M. Higgins (eds), The Philosophy of (Erotic) Love (Lawrence, 1991).

⁹⁷ Spinoza, Ethics, V, P6S. Translated from Macherey, Introduction à l’Ethique de Spinoza, la cinquième partie, p. 68.

⁹⁸ See the section of chapter 7 entitled ‘The many ways of imagining’, p. 171.

⁹⁹ Spinoza, Letter XVII, The Collected Works of Spinoza Volume I, p. 354. Genevieve Lloyd writes extensively on this letter of Spinoza’s in Spinoza and the Ethics (London and New York, 1996).

¹⁰⁰ Deleuze, Spinoza: Practical Philosophy, p. 34.

¹⁰¹ In Expressionism in Philosophy, Deleuze writes that ‘The obscure formulation reflects the difficulties of a finite understanding rising to the comprehension of absolutely infinite substance’ (p. 37).

¹⁰² This theme is further explicated in the section of chapter 10 entitled ‘The concept of Substance as immanent cause’, p. 223.

¹⁰³ Macherey, Introduction à l’Ethique de Spinoza; la troisième partie, p. 355.

¹⁰⁴ Macherey, ‘The encounter with Spinoza’, p. 153.

¹⁰⁵ Deleuze, Expressionism in Philosophy, p. 274.

Chapter 10

The logic of expression and the construction of a philosophy of difference

The wider context of Deleuze’s philosophical project

In the collection of Dialogues bearing the same title, in which both Deleuze and Claire Parnet engage with one another as interlocutors, Deleuze is cited as saying that ‘It is on Spinoza’s philosophy that I worked most seriously according to the norms of the history of philosophy’.¹ The norms to which Deleuze refers would be those determined by the structural–genetic method of Gueroult. The present work has attempted to determine the manner by means of which Deleuze renews the history of philosophy in relation to his reading of Spinoza by charting the structural–genetic method deployed throughout Expressionism in Philosophy; the contention being that this structural–genetic method is determined according to the logic of different/ciation. An understanding of the mechanism by means of which this logic operates has been the problem that has guided the development of the previous nine chapters. Far from considering the determination of the mechanism of operation of this logic in relation to Spinoza’s Ethics to resolve all of the problems inherent in the text of the Ethics itself,² what Deleuze’s reading of the Ethics in Expressionism in Philosophy according to the logic of different/ciation does is to simultaneously demonstrate the operation of this logic and to set up this logic as that which is then redeployed within the wider context of his own philosophical project, the explication of which is concurrently undertaken by Deleuze in Difference and Repetition. A Deleuzian reading of Expressionism in Philosophy must therefore offer an explication of the operation of this logic not only in relation to Deleuze’s reading of the text of the Ethics, but also in relation to the wider context of Deleuze’s philosophical project, which includes that of renewing the history of philosophy by tracing an alternative lineage, in which Spinoza becomes a significant figure, and the manner by means of which this project of tracing an alternative lineage in the history of philosophy culminates in the construction of a philosophy of difference. The context of this dual Deleuzian project is explicated in this chapter in relation to Difference and Repetition.

It is in Difference and Repetition that Deleuze specifically positions his reading of Spinoza within the wider context of his own philosophical project. The alternative lineage in the history of philosophy that he traces between Spinoza and Scotus, which is explicated in chapter 4, is extended in Difference and Repetition to the work of Nietzsche, specifically in relation to the Nietzschean concept of the eternal return. The dynamics of this relation are explicated in the section of this chapter entitled ‘Univocity, immanence and the eternal return’. By reading the Nietzschean concept of the eternal return in relation to Spinoza’s Ethics, Deleuze considers the concept of the eternal return to not only provide a correction to a certain form of Spinozism influential within the field of Spinoza studies, one which Deleuze is specifically critical of, but he also considers Spinoza’s Ethics to provide the ideal context for the adequate explication of the concept of the eternal return. It could therefore be argued that the engagement between Spinoza and Nietzsche that is elaborated by Deleuze in Difference and Repetition, and which is explicated in the section of this chapter entitled ‘The Nietzschean correction to Spinozism’, is as much a Spinozist correction to Nietzsche as a Nietzschean correction to Spinozism. How can Spinoza be a correction to Nietzsche when Spinoza was writing at least a century before him? In order to adequately explicate the Nietzschean concept of the eternal return Deleuze considers it necessary to return to what he describes as the Spinozist concept of Substance as immanent cause which antedates it. Such a return to Spinoza defies the linear progression in the history of philosophy advocated by the Hegelian dialectical logic, and betrays the Deleuzian concept of temporal succession as being rather different, or at least non-linear. Deleuze proposes to ‘give up the narrowly historical point of view of before and after in order to consider the time rather than the history of philosophy’.³ The alternative lineage in the history of philosophy only makes sense in relation to the demands of this Deleuzian concept of temporality, which is explicated according to the logic of different/ciation in the section of this chapter entitled ‘The duration of a finite existing mode’.

In What is Philosophy?, Deleuze argues that ‘philosophy is the discipline that involves creating concepts’.⁴ Concepts are created by the encounters between the concepts belonging to philosophies of the past which generate problems of which the new concepts are the solution. The concepts of philosophies of the past, to which Deleuze refers, are determined in relation to the history of philosophy and include, for example, the Scotist concept of univocity, the Spinozist concept of Substance as immanent cause, and the Nietzschean concept of the eternal return. It is by means of the encounters between these concepts (of the past) that, on the one hand, new concepts are created, and that, on the other hand, the problem, onto which each of these concepts (of the past) is grafted, is generated. The determination of these concepts (of the past), and of the manner by means of which the encounters between them create new concepts and generate problems is explicated according to the logic of different/ciation in the section of this chapter entitled ‘In short, dx is the Idea’. Deleuze’s understanding of the structure of the Ethics, which is explicated in the section of this chapter entitled ‘The demonstrative and the polemical series’, serves as an example of the relation between problems and concepts, and of the manner by means of which the relations between problems and concepts are determined according to the logic of different/ciation. The logic of different/ciation is actually explicated in the text of Expressionism in Philosophy as a logic of expression. The determination of the logic of expression according to the logic of different/ciation is explicated in the section of this chapter entitled ‘The logic of expression’. Deleuze’s reading of the Ethics in Expressionism in Philosophy follows this logic in problematizing the concept created by the Ethics, which he considers to be the concept of amour intellectualis Dei, in relation to the Scotist concept of the univocity of being. The concept created by this relation is the Spinozist concept of Substance as immanent cause. It is by means of the subsequent further differenciation of this concept, in Difference and Repetition, in relation to the Nietzschean concept of the eternal return, that the Deleuzian concept of intensity is created. This concept is distinguished from the concept of intensity determined by the Hegelian dialectical logic in the next two sections of this chapter entitled ‘The concept of intensity from the point of view of the expression itself’, and ‘The concept of intensity according to the logic of expression’. It is by means of the development of this problematic relation according to the logic of expression, rather than from the point of view of the expression itself, that the logic of expression is determined as an alternative logic to the Hegelian dialectical logic. The logic of expression is thereby determined as the logic according to which the philosophy of difference is constructed, and according to which the philosophy of difference functions as an alternative to the dialectical philosophy determined by the Hegelian dialectical logic. However, before embarking on this project of determining the logic of Deleuze’s philosophy of difference, a final word on the logic of the decomposition of a finite existing mode is required.

The logic of the decomposition of a finite exiting mode

Finite existing modes are defined by their degree of power, or their capacity to be affected, which includes both those active affections into which they are capable of entering, by being further differenciated in relation to another finite existing mode, and those passive affections which, in exceeding their capacity, they are able to withstand. A finite existing mode is determined to exist and act insofar as an infinity of already existing extensive parts are determined from without to enter into a relation that corresponds to its singular modal essence, or degree of power. So determined, a finite existing mode interacts with other modes, either combining with them, or being decomposed by them, it thereby passes to a greater or lesser degree of power, which it experiences as either joy or sadness.

When a finite existing mode passes to a lesser degree of power, it can be conceived as being determined by negation only when it is considered in itself abstractly, apart from its cause; that is, apart from the singular modal essence or degree of power to which it corresponds. Although a finite existing mode, when considered abstractly, from the point of view of the imagination, can be understood to be determined in its existence by being negated by another more powerful mode, or by something which denies its existence at some moment, when considered adequately from the point of view of reason it is always as perfect as it can be in terms of the affections that pertain to its essence at any particular moment. For Spinoza, negation is never determinative of the nature of a thing: ‘To say that the nature of the thing required this limitation … is to say nothing. For the nature of the thing cannot require anything unless it exists’.⁵ This aspect of Spinoza’s philosophy again throws into question the accuracy of Hegel’s interpretation of it according to the dialectical logic.⁶

The decomposition of a finite existing mode can be understood adequately when conceived according to the logic of different/ciation. The infinity of extensive parts which correspond to a singular modal essence, insofar as actualized in the global integration of a finite existing mode, are only temporarily related to that singular modal essence. If these parts are determined to enter into another relation corresponding to another singular modal essence, that is: if the extensive parts determined by one of the more composite global integrations in which a finite existing mode is further differenciated are determined in relation to a different global integration, then the finite existing mode is partially decomposed, which it experiences as sadness; or, if the extensive parts belonging to one of the reciprocally determined local integrations constitutive of the global integration of a finite existing mode is reciprocally determined in relation to a different local integration, then the finite existing mode’s capacity to be affected can be overwhelmed, such that it ceases to exist: ‘A [finite existing] mode ceases to exist as soon as its parts are determined to enter into another relation, corresponding to another [singular modal] essence. [Finite existing] Modes come into existence, and cease to exist, by virtue of laws external to their essences’.⁷ It is by means of laws which function according to the logic of different/ciation that both the creation of singular modal essences and the generation of finite existing modes is determined. A singular modal essence, or intensive part, can only be created, not destroyed. Once created, it continues to function in its capacity as an intensive part, that is, as a degree of power of the power series expansion of an attribute. Whereas the infinity of extensive parts of which the corresponding finite existing mode is constituted are able to come and go, that is, they are renewable, however only potentially so, thereby leading to its eventual decomposition. It is therefore also by means of laws which function according to the logic of different/ciation that decomposition can be understood to be the result of a different more powerful creative and generative process, rather than that of negation. The obvious extreme example of which would be the creative and generative processes of Nature in its entirety; Deleuze describes Nature as ‘an infinitely transformable and deformable structure, universal rhythm, Facies totius Naturae, infinite mode’.⁸ In ‘Spinoza and the three “Ethics”’, Deleuze further elaborates his position in relation to the composition and decomposition of finite existing modes or bodies when he writes that ‘there is in each body an infinity of relations that are composed and decomposed, in such a way that the body in turn enters into a more vast body under a new composite relation or, on the contrary, makes smaller bodies come out from under their composite relations’.⁹ The smaller bodies are determined solely in relation to the bodies with which they are implicated in different relations of further differenciation. They are therefore determined in relation to these different creative and generative relations rather than by a negation associated with the decomposition. In Difference and Repetition, Deleuze argues that ‘we should nevertheless speak of “smaller” and “greater”, according to whether the nature of a given part presupposes a given change of nature or is presupposed by it. Thus, the acceleration or deceleration of a movement defines within it intensive parts that must be called greater or smaller, even while these undergo a change of nature and following the order of these changes (ordered differences)’.¹⁰ An intensive part or quantity is therefore greater insofar as it presupposes the change of nature (acceleration or creation) from the smaller intensive quantities of which it is composed, and smaller insofar as it is presupposed by the change of nature (deceleration or decomposition) from a greater intensive quantity out from under which it comes. Although presupposed by the decomposition of the greater intensive quantity, the smaller intensive quantities are determined solely by the different creative and generative relations in which they are further differenciated.

Univocity, immanence, and the eternal return

Deleuze reclaims the logic of the differential relation insofar as it effectively characterizes the conditions according to which the differential or composite relation constitutive of each singular modal essence has a consistency even though it ceases to be effected by extensive parts. In the seminars on Spinoza, he argues that there is ‘a consistency of the relation … independent of the terms which effect it’, such that ‘that which disappears with death is the effectuation of the relation, it is not the relation itself’.¹¹ Deleuze further characterizes this distinction between the effectuation of the differential relation and the differential relation itself with the claim that it is ‘the relation and the essence [it creates that] are said [by Spinoza] to be eternal, or to have a species of eternity’.¹² The singular modal essence and the composite relations in which it expresses itself, that is, in which it is further differenciated, can therefore also be characterized as ‘eternal’.¹³ However, the eternity that Spinoza speaks of is ‘a species of eternity’(sub specie aeternitatis), according to which ‘the relation and the essence’, which are for Deleuze the differential or composite relation and the essential singularity or singular modal essence it creates, are each ‘eternal by virtue of its cause, rather than by virtue of itself’.¹⁴ The ‘cause’ referred to by Deleuze is the ‘immanent cause’, which has already been explicated in chapter 4 in relation to the concept of the univocity of being, where Deleuze argues that ‘it is the idea of immanent cause that takes over, in Spinoza, from univocity’.¹⁵ The concept of immanent cause characterizes the immanent nature of the expressive relation between substance and modes such that Spinozist substance, in the sense that it is cause of itself (causa sui), is the immanent cause of all things, that is, of the immanent existence of singular modal essence and of the durational existence of the corresponding finite existing mode. Therefore, as far as Deleuze is concerned, to be eternal is for Spinoza to have a species of eternity, which is to be eternal by virtue of the immanent causality that characterizes the expressive relation between substance and modes, a relation which functions according to the logic of different/ciation.

It is in Difference and Repetition, in relation to the concept of the univocity of being, that the immanent nature of the expressive relation between substance and modes in the Ethics is further explicated. In the first chapter, entitled ‘Difference in Itself’, Deleuze argues that there are ‘three principal moments in the history of the philosophical elaboration of the univocity of being’,¹⁶ which he locates respectively in the work of Scotus, Spinoza and Nietzsche. As already determined in chapter 4, univocal being is understood by Scotus to be neutral, ‘indifferent to the distinction between the finite and the infinite’.¹⁷ The two types of distinction which characterize that indifferent, neutral being, and which relate it to difference are: formal distinction, established between qualitative forms which ‘allow the persistence of the unity of being’; and modal distinction, established between the qualitative forms and ‘the intensive variations of which these are capable’.¹⁸ According to Deleuze, ‘these variations, like degrees of whiteness, are individuating modalities of which the finite and the infinite constitute precisely singular intensities’.¹⁹ The problem with the Scotist concept of univocal being is that it ‘does not implicate distinct … qualitative forms which are themselves univocal’, it rather relates the formally distinct qualitative forms ‘to intensive factors or individuating degrees which vary the mode of these … forms without modifying their essence insofar as this is being’.²⁰ Nevertheless, Deleuze considers the univocal position to have continued to rest upon these ‘two fundamental theses’²¹ from Scotus to Spinoza. However, with Spinoza, ‘univocal being ceases to be neutralized’, instead it is made ‘an object of pure affirmation’, that is, it ‘becomes expressive’.²²

According to Deleuze, Spinoza demonstrates that the attributes, while formally distinct, remain ontologically one, introducing no division into substance, which is expressed through them. ‘In other words’, Deleuze argues, ‘the real distinction between the attributes is a formal, not a numerical distinction’.²³ And the modes are complicated within the attributes as differences in intensity, or degree of power. ‘In other words, the numerical distinction between “beings” is a modal, not a real distinction’.²⁴ Deleuze can therefore conclude that ‘substance is equally [expressed through] all the attributes in accordance with their essence, and equally expressed by all the modes in accordance with their degree of power’.²⁵ It is in the immanent nature of the expressive relation between substance and modes, which is characterized by the concept of substance as immanent cause, that univocity finds what is for Deleuze ‘its distinctly Spinozist formulation’.²⁶

The Nietzschean correction to Spinozism

The concept of substance as immanent cause distinguishes Deleuze’s interpretation of Spinoza from what he refers to as a certain ‘Spinozism’ which he relates to a problem with the concept of the univocity of being; a problem which he considers his reading of Spinoza in Expressionism in Philosophy to offer a solution. The problem with this form of Spinozism, Deleuze argues, is that, for it, ‘there still remains a difference between substance and the modes’.²⁷ According to Deleuze, ‘Spinoza’s substance appears independent of the modes, while the modes are dependent on substance, but as though on something other than themselves’.²⁸ It is important to note that Deleuze considers substance only to ‘appear’ to be independent of the modes. As far as Deleuze is concerned this is an interpretative problematic attributable to a certain Spinozism, as distinct from the text of the Ethics. The Spinozism of which Deleuze speaks includes that determined by a Hegelian reading of the Ethics, which, together with Cartesian readings of Spinoza, continue to propagate this problematic as one determined by the text of the Ethics rather than as one determined by the problematic nature of the interpretative strategies which are deployed in relation to the text.²⁹ From the point of view of Deleuze’s interpretation of Spinoza, this problematic is not only resolvable in the concept of substance as immanent cause, but it is immanently resolved in this concept, that is, it ceases to have been a problem. However, in order to demonstrate the resolution of the problem from the point of view of the Spinozism which determines it, Deleuze introduces a third moment in the history of the philosophical elaboration of the univocity of being, which he locates in the Nietzschean concept of the eternal return. This moment, the development of which is historically subsequent to the production of Spinoza’s text, Deleuze considers to be immanent to the text of the Ethics itself. Deleuze characterizes this moment when he writes that: ‘All that Spinozism needed to do [but didn’t] for the univocal to become an object of pure affirmation was to make substance turn around the modes’.³⁰ Deleuze declares that ‘Substance must itself be said of the modes and only of the modes’.³¹ The crucial argument is that Substance, or being, ‘not be first, that it exist as a principle but as a second principle, as a principle become’, such that ‘being is said of becoming’.³² Deleuze contends that ‘Nietzsche meant nothing more than this by eternal return’.³³ Deleuze characterizes the being which returns in the concept of the eternal return as ‘the being of becoming’: ‘Returning is being, but only the being of becoming’.³⁴ The Deleuzian argument is that, just as the being which returns, and which is therefore univocal, is characterized as the being of becoming, so too can substance be said to return insofar as it is understood to be characterized as the substance of the modes. A mode, of which substance is said, can be characterized as ‘becoming’, of which being is said, insofar as it is understood to be generated by the expansion of the power series of its global integration in composite relation to the power series expansion of another. The substance that returns can be characterized as that part of substance belonging to those modes which have either ‘become completely expressive’, or which continue to be determined by active affections, that is, to be involved in relations of further differenciation.³⁵ Substance, can therefore be characterized as the substance of the modes according to the logic of different/ciation by the actually infinitely composite assemblage of multipli-differenciated global integrations. It is therefore only insofar as substance is understood to be determined by the concept of the univocity of being, in its distinctly Spinozist formulation as immanent cause, that it can be characterized as the substance that returns, or as the substance of the modes.

The concept of substance as immanent cause

The concept of substance as immanent cause is characteristic of the expressive relation between substance and modes insofar as it determines both the creation of essential singularities by the differential or composite relations between the global integrations of finite existing modes; and, the generation of the corresponding more composite global integrations, in which these finite existing modes are further differenciated, by means of the expansion of the power series of the composite relations. The concept of substance as immanent cause is therefore characteristic of both the differenciations of the differentiated, and the differenciations of the differenciated, that is, the further differenciated, up to and including the actually infinitely composite multipli-differenciated assemblage of global integrations.

It is by means of the concept of ‘repetition’³⁶ that Deleuze characterizes the return of the being of becoming, which returns eternally according to the concept of eternal return, and the return of the substance of the modes, which returns eternally, insofar as sub specie aeternitatis, according to the concept of substance as immanent cause. He argues that insofar as ‘repetition in the eternal return’ consists in conceiving being ‘on the basis of’ becoming,³⁷ repetition in relation to the concept of substance as immanent cause consists in conceiving substance ‘on the basis of’ the modes.

It is in repetition that the concept of difference in Difference and Repetition has its correlate. The differential or composite relations of the logic of different/ciation, according to which the global integrations of different modes are reciprocally determined in relation to one another, are characteristic of the concept of difference. Difference only exists in these differential or composite relations, and it returns in each of the further differenciations in which the global integrations of these relations are implicated. However, this can only occur insofar as difference has repetition as its correlate. Repetition operates between differences, that is, from one differential or composite relation to the more composite relation in which its global integration is further differenciated. This occurs insofar as the former differential or composite relation is repeatedly differentiated in the expansion of its power series,³⁸ or the generation of its global integration, which is then able to be further differenciated in relation to the power series expansion of other global integrations in more composite relations. The operations of difference and repetition are therefore characteristic of the processes of creation and generation that function according to the logic of different/ciation.

Deleuze further characterizes the operations of difference and repetition as ‘at once both production of repetition on the basis of difference and selection of difference on the basis of repetition’.³⁹ According to the logic of different/ciation, the ‘production of repetition on the basis of difference’ is characteristic of the expansion of the power series of a composite relation (production of repetition) on the basis of the essential singularity created by this composite relation (difference). This is the process of production, or generation, of the corresponding global integration. The ‘selection of difference on the basis of repetition’ is characteristic of the selection of potential global integrations with which to be further differenciated (selection of difference), on the basis of the above generation of global integrations (repetition). This is the process of selection, or creation, of the corresponding essential singularity. It is the ‘selection of difference’ (creation) of the latter operation which completely determines the former operation, and it is the ‘production of repetition’ (generation) of the former operation which provides the basis of the latter, each operation being repeated in the other. It is between the ‘basis of difference’, or the composite relation, of the former operation and the ‘selection of difference’, or the more composite composite relation, of the latter operation that repetition operates and continues to operate as the processes continue to repeat one another. The latter operation can also be characterized in terms of Spinoza’s theory of relations, according to which those affections (joyful passive affections) are selected (selection of difference) that can be accumulated (on the basis of repetition) and potentially lead to active joy.

Deleuze argues that it is by means of the concept of repetition in the eternal return that ‘univocal being is not only thought and even affirmed, but effectively realized’.⁴⁰ ‘[I]n other words’, in order to resolve the problematic of Spinozism, it is necessary ‘to realize univocity in the form of repetition in the eternal return’.⁴¹ Deleuze considers his reading of the Ethics in Expressionism in Philosophy to demonstrate the effective realization of the univocity of being in the form of repetition in substance as immanent cause according to the logic of different/ciation. It is the global integrations of finite existing modes that effectively realize substance as the actually infinitely composite assemblage of multipli-differenciated global integrations.

The example of Letter XVII functions as an example of this form of repetition insofar as the active joy actualized in the relation between father and son continues to be the object of a direct experience for the father.⁴² Deleuze claims that ‘relations and their laws of composition’, which function according to the logic of different/ciation, ‘can be the object of a true knowledge’, that is, of the knowledge of the third kind, ‘although, under natural conditions’, he continues, ‘we need to go by way of an experience of the parts that realize these relations’.⁴³ Such an experience would be that of the active joy associated with the intensive part created by the composite or composite relation between the father and son in the example of Letter XVII. The active joy can continue to be the object of a direct experience for the father insofar as he is able to create the intensive parts of more composite global integrations by becoming further differenciated in composite relation with the global integrations of other finite existing modes or individuals, both while his son exists and once his son ceases to exist. The father would have to select from the affections experienced as a result of these relations with external bodies those joyful passive affections (selection of difference) that can be accumulated (on the basis of repetition) in preparation for the leap to active joy. The active joy that the father experienced with his son can be used as a reference in the process of selection against which the experience of each subsequent interaction can be evaluated, that is, in order to determine the kind of affection experienced, whether joyful or sad, and, if joyful, whether the joy is of the kind that can be accumulated. What is repeated is not the relation with the son, but the logic of that relation. The experience of active joy is repeated in each relation of further differenciation insofar as it continues to function as a reference for the process of selection, or evaluation, of these relations. Just as each intensive part, as complicated in an attribute, conforms with all of the others insofar as all are implicated in the production of each, each experience of active joy would then return in all of the others, and each would, in its turn, make those return that are implicated in its production, insofar as all are implicated in the production of each according to the logic of different/ciation.

The duration of a finite existing mode

In the section of chapter 8 entitled ‘The determination of an existing mode as finite’, it was argued that modal existence is determined as finite only insofar as it is further differenciated in relation to another according to the logic of different/ciation. The finitude of an existing mode is determined as the specific qualitative nature of its global integration, which is only determinable insofar as it is further differenciated in relation to the global integration of another existing mode. The specific qualitative nature of a global integration is increasingly specified as the composite or composite relation in which it is further differenciated is expanded in a power series. The finitude of an existing mode is therefore increasingly determined with the expansion of the power series in which it is further differenciated. It is also only insofar as the global integration of a finite existing mode is further differenciated, and the so determined composite relation expanded in a power series, that a finite existing mode is determined ‘as having duration, as having a relation with a certain extrinsically distinct time and place’.⁴⁴ Deleuze considers ‘the introduction of the temporal factor’ to be ‘essential’.⁴⁵ It is in relation to the rate of expansion of the power series in which an existing mode’s global integration is further differenciated that it can be said to endure. So long as the power series continues to expand, the finite mode continues to exist, or to have duration, that is, it continues to endure as finite. The duration of its finitude is measured by the rate of expansion of the power series in which it is further differenciated. So the duration of a finite existing mode corresponds to the rate of expansion of the power series in which it is further differenciated.

This duration is further characterized when it is noted that the power series expansions of each of the further differenciations in which the global integration of a finite existing mode is implicated has its own duration. Each different further differenciation in which the global integration of a finite existing mode is implicated can therefore be understood to have a characteristic duration, which is determined by the rate of expansion of the corresponding power series, or the rate of actualization of the corresponding more composite global integration. These further differenciations in which the global integration of a finite existing mode is implicated are determined concurrently. Therefore, in the variety, or multiplicity, of relations in which the global integration of a finite existing mode is further differenciated there coexist a multiplicity of durations. An individual or finite existing mode is therefore composed of a multiplicity of durations each of which is associated with one of the more composite global integrations in which it is further differenciated.

When considering Deleuze’s concept of time in Deleuze, une philosophie de l’événement, François Zourabichvili attests that ‘to the traditional image of time’ – both as ‘a homogeneous, unidimensional continuity’, that is, one constituting a 4th Dimension to 3-Dimensional Space; or as ‘a line’ representative of a chronological ‘flow of time’ that is ‘already divided into instants of a given … duration’, which are sequentially juxtaposed as multiple ‘nested sets’ of relative scales of time of different duration: seconds, minutes, hours, days, etc.⁴⁶ – ‘is substituted the idea of a time which progresses by intensities, by an augmentation of the number of its dimensions’,⁴⁷ such that time itself has ‘more dimensions than space’.⁴⁸ Instead of the chronological succession of one given instant by another, temporal succession should rather be understood to be ‘the actualization of a new dimension’⁴⁹ of time, in the more composite global integration of a relation of further differenciation. Successive durations or temporal dimensions are accumulated in the variety, or multiplicity, of more composite global integrations in which the global integration of a finite existing mode is further differenciated. Succession as accumulation according to the logic of different/ciation is determined by the differential or composite relations between different, or ‘heterogeneous’, temporal dimensions, which include both the differenciations of the differentiated and the differenciations of the differenciated; and it is determinative of the actually infinitely composite assemblage of multipli-differenciated global integrations ‘where they coexist’.⁵⁰ It can therefore be concluded that ‘time is heterogeneous before being successive’,⁵¹ and that succession is determined according to the logic of different/ciation by the accumulation of heterogeneous temporal dimensions, rather than as the course of (chronological) time, which, Zourabichvili notes, ‘is only … the manner by which time is represented’.⁵² The logic of different/ciation therefore determines an unfolding of time that is different to a chronological succession. Deleuze writes that ‘Time itself unfolds….instead of things unfolding within it’.⁵³

The rate of expansion of the power series generating a global integration is potentially of unlimited duration (time of generation), limited only by the relations of further differenciation in which it is implicated, while the creation of an essential singularity by a composite relation between global integrations is of zero duration (time of creation).⁵⁴ It is with the creation of an essential singularity that there is the correspondence of the minimum time that can occur in the instant (time of creation: zero duration) with the maximum time that can be thought in accordance with the temporal dimension incorporated by each of the global integrations of which it is constituted. The completely determined power series expansions of each of these global integrations determines the duration (maximum time) of the local integrations of which each global integration is constituted. This temporal dimension is heterogeneous to that determined by the expansion of the power series of the essential singularity, which determines rather the duration (time of generation) of each of the global integrations of which it is constituted.

Deleuze considers the power series expansions of global integrations to be ‘no less temporal than spatial. They constitute a time of … differenciation no less than they outline spaces of actualization’.⁵⁵ These spaces of actualization are determined by the power series expansions of global integrations insofar as they ‘begin to incarnate differential relations between elements of the reciprocally and completely determined structure’,⁵⁶ that is, insofar as global integrations differenciate the differentiated, which are, in turn, further differenciated. The power series expansion of a global integration ‘thus comprises its own power of determining space and time’.⁵⁷ It is therefore in the power series expansions of global integrations that ‘we discover pure spatio–temporal dynamisms’.⁵⁸ A global integration ‘is not a qualified space–time but a spatio–temporal dynamism, which does not empirically combine two preexisting space–times but presides on the contrary at their genesis’,⁵⁹ that is, at the actualization of the space–times of its constitutive local integrations in the spatio–temporal dynamism of the global integration that incorporates them. ‘It is the birth of a space–time, “beginning of the world” or “the birth of time itself” in a spatializing dynamism’.⁶⁰ In The Logic of Sense, Deleuze argues that ‘there is always a space which condenses and precipitates singularities, just as there is always a time which progressively completes the event through fragments of future and past events’.⁶¹

In other words, there is always a space associated with the creation of essential singularities, just as there is always a time associated with the power series expansion of the composite relation between two local integrations (fragments of past events) which create these essential singularities (fragments of future events). Any global integration which is local in relation to a further differenciation constitutes a fragment of a past event. Any essential singularity created but not yet actualized in a global integration constitutes a fragment of a potential future event. The present is constituted by the process of expansion of the power series of such a global integration, in a spatializing dynamism.⁶²

Zourabichvili also notes that ‘Time … only becomes sensible … at the encounter between the different fluxes of duration that incorporate it’,⁶³ that is, it is in the accumulation of joyful passive affections in the generation of the power series expansion of the composite relation between the global integrations of two finite existing modes, which have been further differenciated in relation to one another, that time becomes sensible, that is, it is in the accumulation of joyful passive affections that time is experienced by each of the finite existing modes implicated in a composite relation.

The Deleuzian project of renewing the history of philosophy by tracing an alternative lineage, in which Scotus, Spinoza and Nietzsche become significant figures, only makes sense in relation to the demands of this complex concept of temporality. The manner by means of which Deleuze is able to ‘give up the narrowly historical point of view of before and after in order to consider the time rather than the history of philosophy’⁶⁴ is investigated in the following section where the same logic determinative of this concept of temporality is mobilized in relation to Deleuze’s understanding of ‘Ideas’ and their ‘concepts’.

‘In short, dx is the Idea’

In Difference and Repetition, Deleuze argues that ‘the metaphysics of differential calculus finds its true signification when it escapes the antinomy of the finite and the infinite in representation in order to appear in the Idea’.⁶⁵ This occurs insofar as there is a correspondence between the mathematics of the logic of different/ciation and Spinoza’s theory of relations, as characterized by Deleuze’s argument from powers,⁶⁶ such that the differential, dx, corresponds to the Idea. ‘In short’, Deleuze states, ‘dx is the Idea’.⁶⁷ He argues that ‘Ideas appear in the form of a system of ideal connections – in other words, a system of differential relations between reciprocally determined genetic elements’.⁶⁸ Just as dx is only determinable in a differential relation, so too are Ideas only reciprocally determinable in ideal connections which are characterized by the differential relations of the logic of different/ciation. When a differential relation is expanded in a power series, it constitutes an infinite collection of extensive parts, which are the genetic elements to which Deleuze refers. According to the argument from powers, these extensive parts are constitutive not only of bodies, from the point of view of the attribute of Extension, but also of the Ideas of these bodies, from the point of view of the attribute of Thought. Extensive parts incorporate the Idea just as much as the body of which it is the Idea. Deleuze argues that ‘differentials certainly do not correspond to any engendered quantity, but rather constitute an unconditioned rule for the production of knowledge of quantity, and for the constitution of series or the generation of discontinuities which constitute its material’.⁶⁹ The unconditioned rule constituted by differentials is the logic of different/ciation; the knowledge produced consists of Ideas in the attribute of Thought, which is knowledge of the first kind; and, the series, or discontinuities generated, are power series expansions, which, according to the ‘unconditioned rule’, constitute the extensity of bodies and the Ideas of these bodies, which together constitute their power of existence.

The Idea and the differential relation in which it is reciprocally determined correspond respectively to the first two aspects of the logic of differentiation: the pure element of quantitability and the pure element of qualitability.⁷⁰ The third aspect of the logic of differentiation, complete determination, corresponds to ‘the pure element of potentiality in the Idea’⁷¹ which, Deleuze argues, is conditioned by ‘an evolution of the function of a variable series constituted by the powers of i (undetermined quantity) and the coefficients of these powers (new functions of x) …, in such a way that the evolution function of that variable be comparable to that of the others’.⁷² The ‘powers of i’ are the different degrees of power of the successive terms of a power series expansion, and the ‘coefficients of these powers’ consist of the repeatedly differentiated differential relation which constitutes each new term of the series. The complete determination, or pure element of potentiality in the Idea, is conditioned by an evolution function, or power series expansion, only insofar as it is ‘comparable to that of the others’, that is, only insofar as it is further differenciated according to the logic of differenciation as a local integration in reciprocal or composite relation to the local integrations or the pure elements of potentiality of other Ideas, or to at least that of one other Idea.

Deleuze considers an Idea to possess ‘a comprehension all the more vast as its extension is great’.⁷³ As the differential relation ‘made up of relations between differential elements’,⁷⁴ or Ideas, is repeatedly differentiated in a power series expansion according to the logic of differentiation, the degree of comprehension of an Idea increases, where the degree of comprehension of an Idea corresponds to the degree of power of the coefficient of the power series expansion in which it is implicated. It is this repetition, Deleuze argues, that ‘testifies … to the power of Ideas to give rise to Ideas of Ideas’.⁷⁵ While the expansion of the differential relation of an Idea in a power series according to the logic of differentiation determines the accumulation of understanding of the Idea, the further differenciation of this power series expansion as a local integration, which is only immanently determinable in composite relation to a local integration determined in relation to another Idea, creates ‘a space’ corresponding to the composite relation ‘to be actualized’.⁷⁶ What is created is an essential singularity, whose space is traced by the expansion of the composite relation in a power series which is to be actualized in a global integration according to the logic of differenciation. The composite relation between the local integrations determined by the differential relations made up of differential elements or Ideas, corresponds to a common notion, which is knowledge of the second kind. It is the Idea of such a common notion that is an Idea of the Idea, or an Idea in the power of thinking, which would be knowledge of the third kind.⁷⁷ Common notions are characterized by Deleuze in ‘Spinoza and the Three “Ethics”’ as concepts. He writes that ‘common notions are concepts of objects’,⁷⁸ where ‘The structure or object is formed by at least two bodies’,⁷⁹ that is, the two local integrations which constitute the composite relation. Common notions are therefore concepts of the structure of a composite relation. An Idea of the Idea, or an Idea in the power of thinking, is therefore characteristic of an Idea of a concept. Deleuze argues that ‘the reciprocal dependence of the degrees of the relation, and ultimately the reciprocal dependence of the relations themselves’ define ‘the universal synthesis of the Idea (Idea of the Idea, etc.)’.⁸⁰ According to the logic of differentiation, the different degrees of the power series expansion of a differential relation are determined by the repeated differentiation of the differential relation, each of which is therefore dependent upon the reciprocal determination of the differential relation itself, which is made up of differential elements or Ideas. And, the complete determination of such a local integration is ultimately dependent upon its reciprocal relation or further differenciation with another local integration according to the logic of differenciation. In other words, the universal synthesis of the Idea, or the Idea of the Idea, which is an Idea in the power of thinking, or an Idea of a concept, is an Idea of both the reciprocal determination of Ideas in differential relations according to the logic of differentiation, and of the composite relation between the power series expansions of these differential relations, as local integrations, according to the logic of differenciation. An Idea in the power of thinking of a concept is therefore an Idea of the relational structure by means of which an essential singularity is created according to the logic of different/ciation, which ‘immediately incarnates the [composite] relations, the [essential] singularities and the [power series expansions] immanent in the Idea’,⁸¹ that is, in the Ideas as genetic elements.

Ideas and their power series expansions are referred to by Deleuze in Difference and Repetition as ‘problems, or as schemas of problematization’,⁸² and concepts are considered to ‘determine the cases of solution for problems’.⁸³ Deleuze argues that ‘all concepts are connected to problems without which they would have no meaning’,⁸⁴ and that these problems, or schemas of problematization, ‘can themselves only be isolated or understood as their solution emerges’.⁸⁵ A solution emerges as an Idea, or its problem, is further differenciated in reciprocal relation to another Idea, or its problem. The composite relation formed between two Ideas, or their problems, according to the logic of differentiation, which creates the essential singularity corresponding to a concept, is expandable according to the logic of differenciation in a power series in reciprocal relation to another newly created concept. Such a power series expansion recasts the newly created concepts in a new schema of problematization. ‘A concept’, he continues, ‘requires not only a problem through which it recasts or replaces earlier concepts but a junction of problems where it combines with other coexisting concepts’.⁸⁶ The composite relations between such schemas of problematization constitute the junction of problems where the recast concepts combine with other coexisting, recast concepts to again create new concepts. The structure of the Ethics characterized by Deleuze in ‘Spinoza and the Three “Ethics”’ functions as an example of the composite relation between coexisting concepts which have been recast as schemas of problematization and the new concepts that they thereby together create.

The demonstrative and the polemical series

Deleuze considers most of the Ethics to be ‘written in common notions’.⁸⁷ He argues that common notions are ‘expounded for themselves’,⁸⁸ and that the ‘consequences’ are ‘ceaselessly’ developed throughout the Ethics. He therefore declares that ‘the Ethics is the discourse of the concept’, describing it as a ‘discursive and deductive system’.⁸⁹ The logic of such a system corresponds to the logic of different/ciation, according to which each concept is further differenciated in relation to another thereby generating a ‘demonstrative chain’,⁹⁰ or series of concepts. The demonstrative series determines the creation of concepts, and corresponds to the development of the second kind of knowledge.

Accompanying this demonstrative series is another series, one constituted by the ‘scholia’ of the Ethics. According to Deleuze, it is a ‘discontinuous … broken’ series, which ‘irregularly’ interrupts the demonstrative series. Its elements are distinct from those of the demonstrative series insofar as the scholia predominantly refer only to other scholia. Each of these elements is further differenciated in relation to another thereby generating another kind of series which is different to the demonstrative series, and which Deleuze describes as being ‘ostensive and polemical’.⁹¹ Rather than operating by means of the creation of concepts, the series of scholia operates through the generation of affects (joyful passive affections), the Ideas of which are the inadequate Ideas of the imagination associated with the partial comprehension of the corresponding concepts. The polemical series of the scholia determines the generation of Ideas, as schemas of problematization, and corresponds to the first kind of knowledge.

The structure of the Ethics presented so far in this example gives ‘the impression that [the common notions or concepts] owe nothing to’⁹² the affects or ideas. Of course this is not so. The affects and the inadequate ideas of them are, in fact, ‘precursors of the notions’.⁹³ The selection and accumulation of joyful passive affections is the very process that determines the actualization of common notions, or concepts, according to the logic of different/ciation. Deleuze argues that ‘the Ethics cannot dispense with this passional form of expression that operates through [affects], for it alone is capable of bringing about the indispensable selection without which we would remain condemned to the first kind [of knowledge]’.⁹⁴ Once the common notions are attained, the affects and the inadequate ideas of them ‘will not disappear entirely, nor even the inevitable sadnesses. They will subsist, they will double the notions, but will lose their exclusive or tyrannical character to the profit of notions and actions’.⁹⁵ Therefore, despite being actualized in their corresponding common notions or concepts, the affects and the inadequate ideas of them subsist in the form of the polemical series of the affect, which never ceases to accompany the demonstrative series of the concept.

Deleuze considers the first four parts of the Ethics to be comprised of these two series which, insofar as they ‘intersect and reintersect with’⁹⁶ one another according to the logic of different/ciation, actually constitute one series in the form of a composite assemblage of multipli-different/ciated global integrations, each of which consists of a created concept and its generated schema of problematization. In ‘Spinoza and the Three “Ethics”’, Deleuze refers to the demonstrative and polemical series as ‘books’. He considers the series of the affect to form one book of the Ethics, and ‘the more visible Ethics’, the series of the Concept, to form another, such that ‘the two books, the two Ethics coexist’.⁹⁷ It is in relation to these two series, or books, that Deleuze suggests that ‘perhaps there is a third’,⁹⁸ which would be constitutive of the fifth remaining part of the Ethics. He argues that this fifth part ‘must be seen as being coextensive with all the others’,⁹⁹ and it is so insofar as the logic according to which it operates is the logic of different/ciation.

Deleuze considers the concept created in the fifth part of the Ethics, that is, the concept of the intellectual love of God (amor intellectualis Dei),¹⁰⁰ to be ‘the focus, the focal point that was already at work before it appeared’.¹⁰¹ Although there is ‘the impression of reaching it at the end’, he maintains that ‘it was there from the start’, or in other words, ‘it was there all the time, for all time’.¹⁰² This concept determines from the start the relations of further differenciation characteristic of the assemblage generated by the concepts and affects of the demonstrative and polemical series. The fifth part of the Ethics bears witness to the type of thought determined by the creation of this concept.¹⁰³ This type of thought, which is constituted by Ideas (of concepts) in the power of thinking, is knowledge of the third kind. Deleuze therefore maintains that there is a third series that constitutes the fifth part of the Ethics, the series of Ideas in the power of thinking, which ‘is irreducible to the one developed by the common notions in the course of the first four books’.¹⁰⁴ It is according to the argument from powers¹⁰⁵ that the series of Ideas in the power of thinking ‘coexist[s] with the others and is taken up in the others despite their differences in kind’,¹⁰⁶ that is, despite the demonstrative and polemical series being in the power of existing. Deleuze contends that the coexistence of the demonstrative and polemical series developed throughout the first four parts of the Ethics, and their coextension in the series of Ideas in the power of thinking generated in the fifth part of the Ethics, which is determined by means of the argument from powers according to the logic of different/ciation, is characteristic of the structure of the text of the Ethics itself. It is in relation to this structure of the text that Deleuze’s reading of the Ethics in Expressionism in Philosophy is generated; the structure of which is examined in the following section.

The logic of expression

The concept with which Deleuze orients his reading of the Ethics in Expressionism in Philosophy is that of expression. The concept of expression is explicated throughout the text by means of its further differenciation in relation to the concept created in the fifth part of the Ethics, that is, with the concept of an idea in the power of thinking or of the Idea of a concept. Insofar as the process of further differenciation involves the repeated differentiation of the composite relation between two concepts in the form of a power series expansion, each repeated differentiation corresponds to the further elaboration of the implication of these two concepts in the corresponding ideas that are explicated throughout the text. Expressionism in Philosophy therefore bears witness to the type of thought determined by the further differenciation of these two concepts, the explication of which is the defining problematic of the text. The logic of this Deleuzian reading strategy, which is determined according to the logic of different/ciation, is actually explicated in the text in relation to the concept of expression as a logic of expression. Expressionism in Philosophy takes the form of an explication of the logic of expression in relation to the Ethics, and an explication of the Ethics from the point of view of this logic.

Despite the fact that the idea of expression only appears in the Ethics in the form of the verb exprimere, not in the nominal substantive form expressio,¹⁰⁷ and that it is neither the object of a definition nor of a demonstration,¹⁰⁸ Deleuze considers it to play an important role in the explication of the logical structure of Spinoza’s theory of relations. As with the logic of different/ciation, Deleuze considers the structure of the logic of expression to be ‘essentially triadic’.¹⁰⁹ He distinguishes between ‘what expresses itself, the expression itself and what is expressed’.¹¹⁰ At the level of Substance, the attributes, in which Substance, as causa sui, expresses itself, are characterized as expressions, and the essence of Substance is characterized as that which is expressed.¹¹¹ The logic of expression also operates at the level of the modes: ‘each attribute expresses itself, the dependent modes are expressions, and a modification is expressed’.¹¹² The dependent modes of an attribute, in which an attribute expresses itself, for example, the ideas of the attribute of thought and the bodies of the attribute of extension, are characterized as expressions, and it is a modification of substance that is expressed. Insofar as a modification of substance corresponds to a singular modal essence,¹¹³ this level of expression can be recast from the point of view of a singular modal essence as follows: each attribute expresses itself, the explicated finite existing modes are expressions, and the complicated singular modal essences are what is expressed. Each attribute expresses itself in the complicated singular modal essences of which it is composed, and these complicated singular modal essences are what is expressed in expressions which correspond to explicated finite existing modes.

Deleuze further characterizes the elements of the logic of expression when he argues that ‘“what is expressed” has no existence outside its expression, yet bears no resemblance to it, but relates essentially to what expresses itself as distinct from the expression itself’.¹¹⁴ At the level of substance, the essence of substance (what is expressed) has no existence outside its expression in the attributes, however it relates essentially to substance (what expresses itself) rather than to its expression in the attributes (the expression itself). At the level of the modes, a singular modal essence (what is expressed) has no finite existence outside its expression in a finite existing mode, to which it bears no resemblance, it rather relates essentially to the attribute in which it is complicated (what expresses itself) as distinct from the finite existing mode in which it is explicated (the expression itself).

There is however another point of view from which Deleuze considers the logic of expression to operate in relation to the modes; that from which Substance is said of the modes and only of the modes, that is, from the point of view of the concept of Substance as immanent cause.¹¹⁵ The further differenciation of the concept of expression in relation to the concept created in the fifth part of the Ethics develops the conditions for the determination of the problem of immanence. It is by means of the subsequent further differenciation of the concept of expression in relation to the Scotist concept of univocity that the problem of immanence is developed, or generated, in the text.¹¹⁶ Deleuze argues that ‘in Spinoza the whole theory of expression supports univocity; and its whole import is to free univocal Being from a state of indifference or neutrality, to make it the object of a pure affirmation, which is actually realized in an expressive … immanence’.¹¹⁷ The further differenciation of the concept of expression in relation to the concept of univocity generates an expressive immanence, which is characterized by the ‘immanence of expression in what expresses itself, and of what is expressed in its expression’.¹¹⁸ This ‘expressive immanence’ operates not only both at the level of expression of Substance and at the level of expression of the modes, but also between the different levels of expression. Deleuze argues that the logic of expression is ‘always, and everywhere, a causal explication’, such that between the ‘different levels of expression’ there is ‘only the necessary concatenation of the various effects of an immanent cause’.¹¹⁹ The immanent cause to which Deleuze refers is the same immanent cause that is operative in the concept of substance as immanent cause.¹²⁰ From the point of view of the expressive immanence determined by the concept of Substance as immanent cause, that which expresses itself, rather than being Substance, or one of its attributes, is understood to be each and every complicated singular modal essence. In Expressionism in Philosophy, Deleuze writes that ‘A modal essence expresses itself in a characteristic relation; this relation expresses a capacity to be affected; this capacity is exercised by changing affections, just as the relation is affected by parts which are renewed’.¹²¹ As an intensive part, a singular modal essence expresses itself in a characteristic differential relation.¹²² What is expressed by this relation is a capacity to be affected, a degree of power or intensity. The characteristic expression of the differential relation in which a singular modal essence expresses itself is the power series in which the relation is expanded, the terms of which correspond to the extensive parts of a finite existing mode. It is in relation to the power series expansion of the global integration of a finite existing mode that the degree of power expressed by a singular modal essence is ‘exercised by changing affections’¹²³ (joy or sadness) which correspond to the effects of extensive parts which are renewed. A mode’s degree of power is therefore implicated as what is expressed both in the complicated singular modal essence which expresses itself, and in the explicated finite existing mode as an expression. Such an implicated degree of power (what is expressed) has no finite existence outside the power series expansion which explicates it as a finite existing mode, that is, outside the more composite global integration in which it is further differenciated, however, it relates essentially to the intensive part (what expresses itself) of this global integration as distinct from the power series expansion of the global integration itself (the expression), to which it ‘bears no resemblance’. In Deleuze, une philosophie de l’événement, Zourabichvili writes that, from the point of view of a degree of power, ‘Complication is an implication in itself, explication an implication in other things. The set forms a logic of expression’.¹²⁴ In other words, there is an immanence of expression of what is expressed (a mode’s degree of power) in both what expresses itself (the complicated singular modal essence) and in its expression (the explicated finite existing mode), such that, what expresses itself (the complicated singular modal essence) implicates what is expressed (the mode’s degree of power) in itself, while the expression (the explicated finite existing mode) implicates what is expressed (the mode’s degree of power) in other things, that is, in the global integrations of the explicated finite existing mode. It is this set that forms a logic of expression.

Insofar as the complication of singular modal essences as the intensive quantities or parts of an attribute and the explication of these intensive parts in the extensive quantities or parts of finite existing modes is understood to operate according to the logic of expression, there is a correspondence between the elements of the logic of expression and the components of Spinoza’s theory of relations which are determined according to the logic of different/ciation: ‘what expresses itself’ corresponds to a complicated singular modal essence; ‘the expression’ corresponds to its explicated finite existing mode; and ‘what is expressed’ corresponds to the implicated degree of power deployed by the singular modal essence and exercised by its finite existing mode.¹²⁵ There is however more than just a correspondence here. Just as the components of Spinoza’s theory of relations are determined according to the logic of different/ciation, so too can the elements of the logic of expression be understood to be determined according to the logic of different/ciation. And, insofar as the elements of the logic of expression are understood to be determined according to the logic of different/ciation, so too can the logic of the logic of expression be understood to operate according to the logic of different/ciation. Consequently, it can be concluded that, for Deleuze, there is in Spinoza a logic of expression that operates according to the logic of different/ciation. Such a logic determines a complication of intensive quantity corresponding to singular modal essences (what expresses itself); an explication of extensive quantity corresponding to the mechanism through which finite modes come into existence (the expression itself); and an implication of degrees of power corresponding to the dynamism through which a singular modal essence asserts itself in existence, determining the variations of its power to act (what is expressed).¹²⁶ The explication of this logic is the defining problematic of Expressionism in Philosophy. The mechanism by means of which this logic operates is further elaborated in relation to the concept of intensity in the following section.

The concept of intensity from point of view of the expression itself

Rather than taking ‘what is expressed’ as ‘involved, implicit, wound up in its expression’, the strategy with which Deleuze deploys the logic of expression in Expressionism in Philosophy ‘unfolds, explicates, unwinds expression so as to restore what is expressed’.¹²⁷ Deleuze restores the concept of ‘what is expressed’ to Spinoza’s theory of relations understood according to the logic of expression. Rather than taking ‘what is expressed’, that is, a mode’s degree of power or intensity, as solely implicated in its expression, in a finite existing mode, Deleuze considers a mode’s degree of power to be concurrently implicated in both an explicated finite existing mode (the expression) and a complicated singular modal essence (what expresses itself) as the dynamism through which the latter expresses itself in existence in the former, determining the variations of its power to act (what is expressed). It is according to the logic of expression, that the variations of a finite existing mode’s degree of power, or power to act, are determined by the dynamic nature of the relations in which it is further differenciated; including both the variety of these relations and the extent to which the more composite global integrations determined by these relations are themselves further differenciated in relation to other more composite global integrations. This dynamism, which determines the variations of a mode’s power to act and therefore of ‘what is expressed’, is actualized in the concept of intensity.

The concept of intensity is introduced in the section of chapter 4 entitled ‘The Hegelian distinction between intensive and extensive quantum’, where it is argued that according to the Hegelian dialectical logic, intensity (what is expressed) ‘is subordinated to the qualities which fill extensity’.¹²⁸ Deleuze maintains that, from the point of view of the Hegelian dialectical logic, intensity is only ‘known … as already developed within an extensity, and as covered over by qualities’.¹²⁹ A quality covers over or subordinates intensity insofar as it fills extensity, or ‘occupies this extensity, either in the form of a qualitas’, a primary physical quality, ‘or in the form of a quale’, a secondary perceptible quality.¹³⁰ It is precisely as a result of the subordination of intensity to primary and secondary qualities that the concept of intensive quantity can be misunderstood to be what Deleuze describes as ‘a badly grounded empirical concept, an impure mixture of a sensible quality and extensity, or even of a physical quality and an extensive quantity’.¹³¹ This problematic point of view lends support to the partial understanding of intensity (what is expressed) as solely ‘involved, implicit, wound up in its expression’, that is, as solely implicated in an explicated finite existing mode or extensive quantity. Rather than considering intensity to be ‘an impure mixture’ of a sensible or physical quality and an extensive quantity, Hegel maintains that intensity is able to be quantified as an intensive quantity precisely in the form of the primary and secondary qualities of an extensive quantity. This is so because he considers intensive and extensive quantity to be identically quantitative, differing only in their manner of accentuation.¹³² The Hegelian dialectical logic therefore resolves the problem of considering intensity as a badly grounded empirical concept, however, in doing so, it affirms that intensive quantity (what is expressed) is only able to be considered from the point of view of the extensive quantity (the expression) in which it is implicated, in the form of primary and secondary qualities.

The subordination of the concept of intensive quantity incorporated by the Hegelian dialectical logic can however be understood from the point of view of the logic of expression insofar as it is related to what Deleuze describes as the ‘tendency on the part of differences of intensity to cancel themselves out in qualified extended systems’.¹³³ The ‘differences of intensity’ to which Deleuze refers correspond to the differences in degree between the different intensive quantities implicated in a composite relation. A composite relation is constituted by the further differenciation of the power series expansions of the differential relations belonging to at least two different intensive parts or intensive quantities. The difference between the degree of power of each of the given intensive quantities constitutive of a composite relation and the new intensive quantity of greater degree of power created by this composite relation corresponds to a difference of intensity. Each composite relation therefore incorporates at least two different differences of intensity, corresponding to each of the given intensive quantities constitutive of the composite relation. Differences of intensity are therefore determined according to the same logic that determines the development of composite relations, that is, the logic of expression.

In the section of chapter 8 entitled ‘The determination of a finite existing mode’s power to act’, it is argued that the finitude of a global integration is determinable solely insofar as it is further differenciated in relation to another global integration, which always involves both the power series expansion of the composite relation between the intensive quantities of each of the implicated global integrations and the intensive quantity created by this composite relation. The differences between each of the given intensive quantities constitutive of the composite relation and the new intensive quantity created by the composite relation constitute ‘differences of intensity’. And, just as the finitude of a global integration is only determined by the relations of further differenciation in which it is implicated, the degree of power of the intensive quantity to which a global integration corresponds is only implicated in the explicated global integration by the relations of further differenciation in which this global integration is implicated.

When Deleuze writes that ‘intensity is explicated only in being cancelled in this differenciated system that it creates’,¹³⁴ the intensity to which he refers is a ‘difference of intensity’. The differences of intensity between the degree of power of each of the given intensive quantities and the degree of power of the new intensive quantity they together create are cancelled in the process of the explication of each given intensive quantity in its corresponding global integration. Each difference of intensity is explicated in a global integration only insofar as it is cancelled as a difference, leaving the degree of power of each of the given intensive quantities as the degree of power that is expressed (what is expressed) by the explicated global integration (the expression) corresponding to it. However, it is only by taking ‘what is expressed’ as solely implicated in its expression, that is, as solely considered from the point of view of the expression itself, that the degree of power of each given intensive quantity can be understood to be solely implicated in the explicated global integration which corresponds to it. Differences of intensity only ‘cancel themselves out’ as differences from the point of view of the expression itself.

This tendency on the part of differences of intensity, when understood solely from the point of view of the expression itself, to cancel themselves out in the generation of global integrations, or in the explication of what Deleuze refers to as ‘qualified extended systems’, correlates with the Hegelian argument that intensity is only able to be considered from the point of view of the extensive quantity in which it is implicated.

The concept of intensity according to the logic of expression

However, when the expression of a global integration is understood as explicated according to the logic of expression, rather than solely from the point of view of the expression itself or according to the Hegelian dialectical logic, ‘what is expressed’ is restored insofar as it is determined by the differences of intensity concurrently implicated in the composite relation in which a global integration (the expression) is further differenciated. The degree of power of the intensive quantity to which a global integration corresponds is only implicated in the explicated global integration insofar as the global integration is determined as finite by means of its further differenciation in relation to another global integration. This only occurs insofar as its intensive quantity is concurrently implicated in the composite relation, in which it is further differenciated, as a difference of intensity with the new intensive quantity created by the composite relation. The degree of power of an intensive quantity (what is expressed) is therefore only implicated in the explicated finite global integration (the expression) which corresponds to it insofar as it is concurrently implicated in a composite relation as a difference of intensity. It is as differences of intensity that different intensive quantities are complicated in an attribute as intensive parts of different degree. Intensive quantities are therefore determined as different degrees of power only insofar as they are implicated in composite relations as differences of intensity. Intensive quantities can therefore be understood either according to Spinoza’s theory of relations as the complicated intensive parts of an attribute, or according to the logic of expression as complicated in the differences of intensity of composite relations. The degree of power of an intensive quantity is therefore implicated in an explicated global integration only insofar as it is concurrently implicated as a complicated intensive quantity in one of the differences of intensity of a composite relation. The degree of power implicated in an explicated global integration is determined by the difference of intensity of the composite relation in which its intensive quantity is complicated, and in which its global integration is further differenciated.

Although differences of intensity are cancelled in being explicated in global integrations when considered from the point of view of the global integration itself, according to the logic of expression these differences remain as differences insofar as they are concurrently implicated in the composite relations determinative of the finitude of such global integrations. The difference of intensity in which an intensive quantity is complicated determines the degree of power expressed by the explicated finite global integration. ‘What is expressed’ according to the logic of expression is therefore the difference of intensity in which the intensive quantity corresponding to an explicated finite global integration is complicated.

Rather than the degree of power of an extensive quantity being the same as that of its intensive quantity as a result of intensive and extensive quantity being identically quantitative, Deleuze considers the degree of power of an intensive quantity to be implicated in an extensive quantity, or explicated finite global integration, only insofar as it is concurrently implicated in the difference of intensity complicated by the composite relation in which the global integration is further differenciated. Deleuze therefore maintains that there is a difference of quantity between intensive and extensive quantity.¹³⁵ Extensive quantities, or explicated global integrations, can be considered independently of the composite relations in which they are determined as finite only insofar as their intensive and extensive quantities are taken to be identically quantitative, that is, only insofar as the difference of quantity is cancelled. Intensive quantities, on the contrary, cannot be considered independently of the composite relations in which they are complicated as differences of intensity. Differences of intensity therefore affirm the difference of quantity.

Although differences of intensity seem to be cancelled with the explication of an intensive quantity, what is actually cancelled is the difference of quantity. It is only as a result of the cancellation of the difference of quantity that differences of intensity are also cancelled, and they are only cancelled when considered from the point of view of the expression itself. According to Deleuze, ‘difference in the form of intensity remains implicated in itself, while it is cancelled by being explicated in extensity’.¹³⁶ Difference in the form of difference of intensity remains insofar as it is complicated in composite relations according to the logic of expression. However, when considered from the point of view of the expression itself, the difference that is cancelled includes both the difference of quantity and the differences of intensity.

Deleuze distinguishes between the difference of quantity and the differences of intensity when he argues that ‘Intensity is the uncancellable in difference of quantity, but this difference of quantity is cancelled by extension, extension being precisely the process by which intensive difference is turned inside out and distributed in such a way as to be dispelled, compensated, equalized and suppressed in the extensity which it creates’.¹³⁷ According to the logic of expression, difference of intensity is the uncancellable in difference of quantity, insofar as the difference of the difference of quantity is determined by the differences of intensity. In other words, it is the differences of intensity between different intensive quantities that distinguishes intensive quantity from extensive quantity. However, when considered from the point of view of the expression itself, this difference of quantity is cancelled, that is, intensive and extensive quantity are considered to be identically quantitative. The cancellation of the difference of quantity also cancels the difference determinative of the difference of quantity, which is difference of intensity. Although according to the logic of expression difference of intensity creates the system in which it is explicated by explicating itself,¹³⁸ when considered from the point of view of the expression itself, difference of intensity is cancelled as a difference, and is instead distributed in the system in the form of the primary and secondary qualities of the corresponding extensive quantities or finite global integrations.

According to the Hegelian dialectical logic, there is no difference of quantity, therefore difference of intensity has always already been cancelled. What is expressed by a difference of intensity according to the logic of expression is subordinated by the dialectical logic to the primary and secondary qualities expressed by the extensive quantity which is identically quantitative with the intensive quantity to which it corresponds. The cancellation of the difference of a difference of intensity, when considered from the point of view of the expression itself, can also be characterized, on the one hand, as the reduction of difference to identity, since the given intensive quantity is considered from this point of view to be identically quantitative with its extensive quantity (cancellation of difference of quantity), such an extensive quantity is considered by the Hegelian dialectical logic to be ‘identical’ or ‘equal to itself’;¹³⁹ and, on the other hand, as the reduction of the Other¹⁴⁰ to the Same, since the greater intensive quantity of the difference of intensity, that is, the Other, is reduced to, or represented by the Same primary and secondary qualities exhibited by each of the extensive quantities of the differential relation, each of which is identically quantitative with the given intensive quantity of the difference of intensity in which it is implicated (cancellation of difference of intensity).

Whereas the reduction of difference to identity cancels, or excludes, the difference of a difference of intensity, the reduction of the Other to the Same determines difference instead as a relative difference between the reciprocally implicated extensive quantities of a differential relation. Each of the extensive quantities of a differential relation, each of which is identical to itself, is determined as relatively different solely insofar as they each exhibit the Same primary and secondary qualities.¹⁴¹ Insofar as the cancellation of the difference of quantity correlates with the consideration that intensive and extensive quantity are identically quantitative, Hegel’s dialectical logic can be understood according to the logic of expression to be a logic determined solely from the point of view of the expression itself.

Difference and intensity

The concept of intensity, understood according to the logic of expression, refers to ‘an implicated order of constitutive differences’,¹⁴² where ‘constitutive differences’ are the differences of intensity belonging to a composite relation. According to Deleuze, these ‘differences are not composed of differences of the same order but imply series of heterogeneous terms’.¹⁴³ The constitutive differences composed of different intensive quantities, complicated in composite relations as differences of intensity, are complicated as the intensive parts of an attribute, where each intensive part is a heterogeneous term of the power series expansion of the attribute. The composite relation of each constitutive difference is ‘constructed on at least two series, one superior and one inferior, with each series referring in turn to other implicated series’.¹⁴⁴ In speaking of superior and inferior series, Deleuze is not outlining ‘an “Ethics” of intensive quantities’¹⁴⁵ but rather an ontology that is determined according to the logic of expression.¹⁴⁶ The inferior series to which he refers is the power series expansion between each of the given intensive quantities that are reciprocally related to one another as the denominator and numerator of a composite relation. The superior series is the power series expansion determined between the newly created intensive quantity that the given intensive quantities together create, and another intensive quantity with which it is further differenciated. Each of the given intensive quantities of the inferior series is referred to as smaller in relation to the intensive quantity of the superior series which is referred to as greater. The smaller intensive quantities of the inferior series are only differentiated from one another in relation to the greater intensive quantity of the superior series in the form of the different differences of intensity of their composite relation, that is, in the form of a constitutive difference. Differences of intensity incorporate the creative changes of nature, that is, the variations of a finite existing mode’s capacity to be affected, as expressed in its power to act and by the extent to which it is further differenciated,¹⁴⁷ that distinguish different intensive quantities as either greater or smaller in relation to one another.¹⁴⁸ Between the two intensive quantities of a difference of intensity there is therefore only ‘heterogeneity or difference of nature’.¹⁴⁹

There are therefore three kinds of difference: differences of nature or intensity between heterogeneous intensive quantities; differences of degree between the degrees of power explicated in different extensive quantities; and differences in kind between the primary and secondary qualities of different extensive quantities. Deleuze argues that ‘Difference is a matter of degree only within the extensity in which it is explicated; it is a matter of kind only with regard to the quality which covers it within that extensity’.¹⁵⁰ Differences of intensity are implicated as degrees solely in the extensive quantities in which they are explicated; and, it is only from the point of view of the expression itself that differences in kind tend ‘to cancel out those differences in the extended order in which they are explicated’.¹⁵¹ ‘Between the two’, Deleuze continues, ‘are all the degrees of difference – beneath the two lies the entire nature of difference – in other words, the intensive’.¹⁵² The intensive, or the implicated order of constitutive differences, therefore constitutes qualitative differences, or differences in kind in qualities, and quantitative differences, or differences of degree in extensity, however, it does so ‘at the risk of appearing to extinguish itself in both’.¹⁵³ According to Deleuze, the risk or tendency of a difference of intensity to be extinguished or cancelled renders it ‘imperceptible’ when considered from the point of view of the expression itself. However, when considered according to the logic of expression, ‘intensity is simultaneously the imperceptible and that which can only be sensed’.¹⁵⁴ So, although imperceptible when considered from the point of view of the expression itself, according to the logic of expression difference of intensity is restored as ‘that which can only be sensed’. Difference of intensity is sensed by a finite existing mode, initially as the joy of each of the joyful passive affections accumulated or repeated with the expansion of the power series of the composite relation in which it is further differenciated, and then as the joy of the active joy associated with the actualization of the superior series of a greater intensive quantity in a more composite global integration when further differenciated in relation to the intensive quantity of another finite existing mode. Deleuze argues that ‘it is … the differenciation of their difference’, that is, the further differenciation of the differences of intensity created by the composite relation between these two finite existing modes, ‘which affirms them in intensity’.¹⁵⁵ When further differenciated in relation to another, the accumulation of joyful passive affections by a finite existing mode is affirmed in intensity insofar as the joy of a joyful passive affection is actualized in an active joy, resulting in an increase in the power to act of each of the implicated finite existing modes. Deleuze is therefore able to argue that ‘Difference in intensity’, which is characterized by the differences of intensity belonging to composite relations, or constitutive differences, ‘constitutes the being “of” the sensible’,¹⁵⁶ where the being of the sensible corresponds to the greater intensive quantity associated with the active joy and with the increase in the power to act of the implicated finite existing modes.

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Notes

¹ Deleuze, Dialogues, p. 22.

² Indeed Deleuze is cited in Dialogues as making the following comment in relation to his reading of Spinoza: ‘Spinoza, we have not yet started to understand, and me not more than the others’ (Ibid.).

³ Deleuze and Guattari, What is Philosophy?, p. 58.

⁴ Ibid., p. 5.

⁵ Spinoza, Short Treatise, chapter 2, part 5, note 2.

⁶ See in particular the section of chapter 1 entitled ‘Et determinatio negatio est’, p. 18.

⁷ Deleuze, Expressionism in Philosophy, p. 210.

⁸ Deleuze, Essays Critical and Clinical, trans. D.W. Smith and M.A. Greco (Minneapolis, 1997), p. 142. The Spinozist concept of Nature is interchangeable with that of both Substance and God. See the section of chapter 1 entitled ‘Spinoza’s metaphysical philosophy’, p. 11.

⁹ Ibid., p. 142.

¹⁰ Deleuze, Difference and Repetition, p. 237.

¹¹ Deleuze, ‘sur Spinoza’, 10 March 1981.

¹² Ibid.

¹³ Ibid.

¹⁴ ‘The relation and the essence are said to be eternal, or to have a species of eternity, that is, insofar as each is eternal by virtue of its cause and not by virtue of itself’ (Ibid.).

¹⁵ Deleuze, Expressionism in Philosophy, p. 67. See the section of chapter 4 entitled ‘The immanent existence of singular modal essence’, p. 109.

¹⁶ Deleuze, Difference and Repetition, p. 39.

¹⁷ Ibid.

¹⁸ Ibid.

¹⁹ Ibid.

²⁰ Ibid. See the first and second sections of chapter 4, ‘Scotus’ formal distinction as real’, and ‘Modal distinction and intensive quantity in Duns Scotus’.

²¹ Deleuze, Difference and Repetition, p. 304.

²² Ibid., p. 40.

²³ Ibid., p. 304.

²⁴ Ibid.

²⁵ Ibid., p. 40.

²⁶ Deleuze, Expressionism in Philosophy, p. 67.

²⁷ Deleuze, Difference and Repetition, p. 41.

²⁸ Ibid., p. 41. Additional italics.

²⁹ Hegel writes that ‘in Spinozism … it is precisely the mode as such which is untrue; substance alone is true and to it everything must be brought back’ (Hegel, Hegel’s Science of Logic, p. 328.). Macherey’s reading of Spinoza can also be included in the Spinozism to which Deleuze refers. See the section of chapter 1 entitled ‘Omnis determinatio est affirmatio’, p. 35, where it is argued that Macherey positions his own reading of Spinoza somewhere in between what he considers to be the contradictory extremes of a Hegelian and a Deleuzian Spinozism.

³⁰ Deleuze, Difference and Repetition, p. 304.

³¹ Ibid., p. 41.

³² Ibid.

³³ Ibid.

³⁴ Ibid.

³⁵ See the section of chapter 9 entitled ‘We become completely expressive’, p. 204.

³⁶ Deleuze, Difference and Repetition, p. 41.

³⁷ Ibid.

³⁸ See the section of chapter 3 entitled ‘Power as capacity and the logic of “differentiation”’, p. 77, where it is determined that ‘a power series expands by means of the repeated differentiation of the differential relation’.

³⁹ Deleuze, Difference and Repetition, p. 41.

⁴⁰ Ibid., p. 40.

⁴¹ Ibid., p. 304.

⁴² See the section of chapter 9 entitled ‘Post Mortem’, p. 209.

⁴³ Deleuze, Spinoza: Practical Philosophy, p. 33.

⁴⁴ Deleuze, Expressionism in Philosophy, p. 213. See Spinoza, Ethics, II, P8C.

⁴⁵ Deleuze, Difference and Repetition p. 214.

⁴⁶ See M. Delanda, Intensive science and virtual philosophy, p. 84.

⁴⁷ François Zourabichvili, Deleuze, une philosophie de l’événement (Paris, 1994), p. 79.

⁴⁸ Ibid., p. 75. Zourabichvili argues that ‘Time … is neither one nor multiple, it is a multiplicity’ (p. 83).

⁴⁹ Ibid., p. 80.

⁵⁰ Ibid.

⁵¹ Ibid., p. 71.

⁵² Ibid.

⁵³ Deleuze, Difference and Repetition, p. 88.

⁵⁴ In The Logic of Sense, Deleuze further characterizes the time of generation and the time of creation in the respective figures of Chronos and Aion. He writes that: ‘In accordance to Chronos, only the present exists in time. Past, present and future are not three dimensions of time; only the present fills time, whereas past and future are two dimensions relative to the present in time. In other words, whatever is future or past in relation to a certain present (a certain extension or duration) belongs to a more vast present which has a greater extension or duration. There is always a more vast present which absorbs the past and the future. Thus, the relativity of past and future with respect to the present entails a relativity of the presents themselves in relation to each other….Chronos is an encasement, a coiling up of relative presents…’. (Deleuze, The Logic of Sense, trans. M. Lester with C.J. Stivale, C.V. Boundas (ed.), (New York, 1990), p. 162). ‘Chronos … makes of the past and future its two oriented dimensions, so that one goes always from the past to the future – but only to the degree that presents follow one another inside partial worlds or partial systems. Aion is the past–future, which in an infinite subdivision of the abstract moment endlessly decomposes itself in both directions at once and forever sidesteps the present. For no present can be fixed in a Universe which is taken to be the system of all systems, or the abnormal set. To the oriented line of the present, which “regularizes” in an individual system each singular point which it takes in, the line of Aion is opposed. This line leaps from one pre– individual singularity to another and recovers them all, each one of them within the others’. (Ibid., p. 77)

⁵⁵ Deleuze, Difference and Repetition, p. 217.

⁵⁶ Ibid.

⁵⁷ Ibid., p. 218.

⁵⁸ Ibid., p. 214.

⁵⁹ Zourabichvili, Deleuze, une philosophie de l’événement, p. 119.

⁶⁰ Ibid.

⁶¹ Deleuze, The Logic of Sense, p. 121.

⁶² The creation of an essential singularity according to the logic of different/ciation is characteristic of an event. The concept of an event is further explicated by Deleuze in The Logic of Sense.

⁶³ Zourabichvili, Deleuze, une philosophie de l’événement, p. 90.

⁶⁴ Deleuze and Guattari, What is Philosophy?, p. 58.

⁶⁵ Deleuze, Difference and Repetition, p. 280.

⁶⁶ See the section of chapter 9 entitled ‘The argument from powers’, p. 200.

⁶⁷ Deleuze, Difference and Repetition, p. 171.

⁶⁸ Ibid., pp. 173–4.

⁶⁹ Ibid., p. 175.

⁷⁰ See the section of chapter 3 entitled ‘The development of a differential philosophy’, p. 75.

⁷¹ Deleuze, Difference and Repetition, p. 221.

⁷² Ibid., p. 175.

⁷³ Ibid., p. 173.

⁷⁴ Deleuze, Difference and Repetition, p. 244.

⁷⁵ Ibid., p. 172.

⁷⁶ Ibid., p. 216.

⁷⁷ See the section of chapter 9 entitled ‘The idea of the essence of the body’, p. 202.

⁷⁸ Deleuze, Essays Critical and Clinical, p. 141.

⁷⁹ Ibid., p. 142.

⁸⁰ Deleuze, Difference and Repetition, p. 173.

⁸¹ Ibid., p. 218.

⁸² Deleuze and Guattari, What is Philosophy?, p. 18 n. 19.

⁸³ Deleuze, Difference and Repetition, p. 244.

⁸⁴ Deleuze and Guattari, What is Philosophy?, p. 16.

⁸⁵ Ibid.

⁸⁶ Ibid., p. 18.

⁸⁷ Deleuze, Essays Critical and Clinical, p. 145.

⁸⁸ Ibid., p. 149.

⁸⁹ Ibid., p. 145.

⁹⁰ Ibid.

⁹¹ Ibid.

⁹² Ibid., p. 144.

⁹³ Ibid.

⁹⁴ Ibid., p. 145.

⁹⁵ Ibid., p. 144.

⁹⁶ Ibid., p. 146.

⁹⁷ Ibid.

⁹⁸ Ibid., p. 148.

⁹⁹ Ibid.

¹⁰⁰ See the section of chapter 9 entitled ‘The idea of the essence of the body’, p. 202.

¹⁰¹ Deleuze, Essays Critical and Clinical, p. 148.

¹⁰² Ibid.

¹⁰³ See Deleuze and Guattari, What is Philosophy?, where Deleuze and Guattari characterizes this ‘type of thought’ determined by the creation of concepts as an ‘image of thought’, which is not a concept ‘that is or can be thought, but rather the image of thought, the image that gives itself of what it means to think, to make use of thought, to find ones bearings in thought’ (Ibid., p. 37). It is according to the logic of different/ciation that the created concepts characteristic of an image of thought enter into resonance with one another on what Deleuze and Guattari call ‘a plane of immanence’ (Ibid., p. 35).

¹⁰⁴ Deleuze, Essays Critical and Clinical, p. 150.

¹⁰⁵ See the section of chapter 9 entitled ‘The argument from powers’, p. 200.

¹⁰⁶ Deleuze, Essays Critical and Clinical, p. 151.

¹⁰⁷ See André Robinet, ‘Expression ou expressivité selon Ethica 77’, Revue de Synthèse, 89–91 (janvier–septembre, 1978), p. 261.

¹⁰⁸ See Deleuze, Expressionism in Philosophy, p. 19.

¹⁰⁹ For an explication of the triadic structure of the logic of different/ciation see the section of chapter 3 entitled ‘The development of a differential philosophy’, p. 75.

¹¹⁰ Deleuze, Expressionism in Philosophy, p. 333.

¹¹¹ ‘Substance expresses itself in attributes, each attribute was an expression, the essence of substance was expressed’ (Ibid., p. 110).

¹¹² Ibid.

¹¹³ See the section of chapter 6 entitled ‘Individuation as a determinate level of integration’, p. 138, where it is argued that when Spinoza defines modes as modifications of the attributes of substance, he refers only to singular modal essences, not to their finite existing modes.

¹¹⁴ Deleuze, Expressionism in Philosophy, p. 333.

¹¹⁵ See the sections of this chapter entitled ‘The Nietzschean correction to Spinozism’, p. 222, and ‘Univocity, immanence, and the eternal return’, p. 220.

¹¹⁶ See the last two sections of chapter 4 entitled ‘The immanent existence of singular modal essence’, p. 110, and ‘Modal distinction and intensive quantity in Spinoza’, p. 116.

¹¹⁷ Deleuze, Expressionism in Philosophy, p. 333.

¹¹⁸ Ibid., p. 180.

¹¹⁹ Ibid., p. 234.

¹²⁰ See the section of this chapter entitled ‘The concept of substance as immanent cause’, p. 226.

¹²¹ Deleuze, Expressionism in Philosophy, p. 234.

¹²² The characteristic differential relation of a singular modal essence creates the essential singularity belonging to its intensive part. See the section of chapter 5 entitled ‘The differenciation of the differenciated’, p. 126.

¹²³ Deleuze, Expressionism in Philosophy, p. 234.

¹²⁴ Zourabichvili, Deleuze, une philosophie de l’événement, p. 87.

¹²⁵ For an elaboration of the mechanism by means of which complication and explication operate according to the logic of different/ciation see the section of chapter 5 entitled ‘The double movement of complication and explication’, p. 124.

¹²⁶ See Deleuze, Expressionism in Philosophy, p. 233.

¹²⁷ Ibid.

¹²⁸ Deleuze, Difference and Repetition, p. 223. Deleuze notes that ‘the extensity whose genesis we are attempting to establish is extensive magnitude’ (Ibid., p. 230.), or extensive quantity. The distinction is therefore between intensity, or intensive quantity, and extensity, or extensive quantity. See footnote 52 of the section of chapter 4 entitled ‘The Hegelian distinction between intensive and extensive quantum’, p. 103.

¹²⁹ Deleuze, Expressionism in Philosophy, p. 223.

¹³⁰ Ibid., p. 232.

¹³¹ Ibid., p. 223.

¹³² See the section of chapter 4 entitled ‘The Hegelian distinction between intensive and extensive quantum’, p. 103, where it is argued that ‘Hegel considers the degree of intensive quantity to vary in direct proportion to the amounts of extensive quantity. … The intensity of heat and the amount of heat are for Hegel identically quantitative, differing only in their manner of accentuation; as either a unity, in THE degree, or a multiplicity, in the number of degrees. The unity of THE degree, or the intensity of heat (as a wholly simple sensation) is only determinable in the multiplicity, as the number of degrees, or amount of heat’.

¹³³ Deleuze, Difference and Repetition, p. 224. Additional italics. Deleuze remarks that ‘Intensity is suspect only because it seems to rush headlong into suicide’ (Ibid.).

¹³⁴ Deleuze, Difference and Repetition, p. 256.

¹³⁵ See the section of chapter 4 entitled ‘Modal distinction and intensive quantity in Spinoza’, p. 113, where it is argued that different intensive quantities ‘are distinguished from one another quantitatively and intensively, rather than numerically and extensively’.

¹³⁶ Deleuze, Difference and Repetition, p. 228.

¹³⁷ Ibid., p. 234.

¹³⁸ Deleuze argues that ‘Difference of intensity is cancelled or tends to be cancelled in this system, but it creates this system by explicating itself’ (Ibid., p. 228).

¹³⁹ Ibid., p. 301.

¹⁴⁰ See Deleuze, Difference and Repetition, p. 281, where Deleuze characterizes ‘the Other’ as ‘a structure which is implemented only by variable terms in different perceptual worlds – me for you in yours, you for me in mine’.

¹⁴¹ These remarks are further elaborated upon by Deleuze in the final chapter and the conclusion of Difference and Repetition, see in particular pp. 260–61 and 281–2.

¹⁴² Ibid., p. 228.

¹⁴³ Ibid., p. 237.

¹⁴⁴ Ibid., p. 234.

¹⁴⁵ To restore a concept of intensity according to the logic of expression he argues that ‘we must advance further, at the risk of falling into an “Ethics” of intensive quantities’ (Ibid., p. 234).

¹⁴⁶ The ‘differences of intensity’ characteristic of Deleuze’s concept of intensity as determined according to the logic of expression can be understood to offer a solution to Heidegger’s ontological problematic, according to which traditional metaphysics conceals the ontological difference between the beingness of beings and the truth of being, that is, between our relation to beings and our understanding of being. For further discussion of the relation between Deleuze and Heidegger see Daniel W. Smith, ‘The Doctrine of Univocity: Deleuze’s Ontology of Immanence’, in Mary Bryden (ed.), Deleuze and Religion, (London, 2001).

¹⁴⁷ See the section of chapter 6 entitled ‘The three different types of expressive change’, p. 154.

¹⁴⁸ See the section of this chapter entitled ‘The logic of the decomposition of a finite existing mode’, p. 219, where it is argued that ‘an intensive part or quantity is … greater insofar as it presupposes the change of nature (acceleration or creation) from the smaller intensive quantities of which it is composed, and smaller insofar as it is presupposed by the change of nature (deceleration or decomposition) from a greater intensive quantity out from under which it comes’.

¹⁴⁹ Zourabichvili, Deleuze, une philosophie de l’événement, p. 88.

¹⁵⁰ Deleuze, Difference and Repetition, p. 239.

¹⁵¹ Ibid., p. 228.

¹⁵² Ibid., p. 239.

¹⁵³ Ibid.

¹⁵⁴ Ibid., p. 230.

¹⁵⁵ Ibid., p. 236.

¹⁵⁶ Ibid.

Conclusion

Expressionism in Philosophy

The alternative lineage in the history of philosophy

The question of whether or not one can find a dialectic operating in the Ethics is one of the defining problematics that Macherey brings to bear on Hegel’s reading of Spinoza. He argues that Hegel transposes the Ethics by using the notions of opposition and contradiction which are evidently not those of Spinoza, ‘implicitly making the dialectic, in the Hegelian sense, intervene’ in the Spinozist system.¹ The simple negation that Hegel locates in the Ethics serves to position the philosophy of Spinoza as one moment in the linear progression of the history of philosophy that is determined according to the Hegelian dialectical logic. Macherey considers such a logic to be ‘manifestly absent’² from Spinoza’s work, he suggests rather that ‘It is Spinoza who constitutes the real alternative to the Hegelian philosophy’.³ What Macherey proposes is ‘to rethink the dialectic, starting with Spinoza’,⁴ such a project would require responding to the question of whether or not a concept of ‘historical contradiction free from dialectical negativity’⁵ is able to be determined in relation to Spinoza. Despite not finding such a materialist dialectic operating in the Ethics, Macherey does nevertheless suggest a materialist dialectic as a means of repositioning, as moments of his own reading of Spinoza, what he considers to be the unresolved negativism of Hegel’s Spinoza and the equally unresolved positivism of Deleuze’s Spinoza. However, the characterization of Deleuze’s Spinoza as an unresolved positivism risks obscuring not only the actual difference between the respective interpretations of Spinoza by Hegel and Deleuze, but also, and more significantly, the logic that Deleuze deploys in Expressionism in Philosophy as an alternative to the Hegelian dialectical logic. Indeed, a more appropriate question in relation to Deleuze’s reading of Spinoza would be what sort of dialectic is able to be found operating in the Ethics? Rather than determining that the traces of a logic reminiscent of a Hegelian style dialectic, which attempts to resolve contradiction according to a logic of negation, are nowhere to be found in the Ethics, as Macherey effectively does in Hegel ou Spinoza, Deleuze purports to find instead an alternative logic actually operating in the Ethics. Whether the structure of this logic is referred to as dialectical or not, it is quite different to the Hegelian style dialectical logic. Indeed, Deleuze considers the Ethics to contain the outline of a dialectic whose logic is that of affirmation rather than negation. In Difference and Repetition, Deleuze claims that ‘the long history of the distortion of the dialectic … culminates with Hegel and consists in substituting the labour of the negative for the play of difference and the differential. … The false genesis of affirmation, which takes the form of the negation of the negation and is produced by the negative, is substituted for the complementarity of the positive and the affirmative, of differential positing and the affirmation of difference’.⁶ One of the projects that Deleuze undertakes in his reading of Spinoza in Expressionism in Philosophy is to offer a correction to this distortion by developing a logic that renews the relationality between these substituted characteristics.

Macherey is highly critical of this aspect of Deleuze’s work, questioning whether or not Deleuze’s reading is ‘consistent with the original sense of the work he purports to analyze, or does it rather misrepresent Spinoza’s philosophy’?⁷ In order to respond to this kind of questioning it is necessary to be clear about the conception of the history of philosophy that is being brought to bear on both the text of the Ethics, and on Deleuze’s reading of the Ethics in Expressionism in Philosophy. If a study in the history of philosophy solely strives ‘after a faithful, correct reading, attempting merely a risk-free identical reproduction or charting of what is written in the Ethics as though it belonged to a realm of past thoughts’,⁸ and as though Spinoza’s thought ‘could be captured once and for all, grasped definitively in the ideological context in which he lived and died’,⁹ then the presupposition that there is an original sense of a work accessible only to the erudite historian of philosophy would be acceptable as unproblematic, and any problematization of this presupposition would thereby be determinable as a misrepresentation of the ‘original sense of the work’.

According to this conception of the history of philosophy, one way to understand the ‘importance’ or ‘influence’ of the different figures in the history of philosophy on contemporary thought would be to determine ‘the citations, the references, and the borrowings (acknowledged and unacknowledged) that bind contemporary thought to the texts of’ these figures, which would thus put each of them ‘in the position of a predecessor or forebear whose thought “anticipated” the concerns’ of contemporary thought.¹⁰ ‘Another way’, specifically in relation to Spinoza, ‘would be to situate the contemporary “reception” of Spinoza in the history of Spinoza studies, as the most recent in a series of “readings” of Spinoza from the atheistic Spinoza of the seventeenth century to the pantheist Spinoza of the eighteenth and early nineteenth centuries to the monist of the twentieth century’.¹¹ When Ramond argues that: ‘According to Deleuze, Spinoza locates, by using the notion of “intensity”, “a long Scholastic tradition”, of which only “Scotism”, without more precision, is evoked’; and that ‘When Deleuze … declares that, in Spinoza, “modal essences are … intensive parts”, he utters an assertion strictly incomprehensible within the framework of Spinozism’,¹² he is critical of Deleuze from the point of view of each of these different ways of representing this particular conception of the history of philosophy. The presuppositions determinative of each of the points of view from which he is critical of Deleuze include: that Deleuze doesn’t establish enough of a connection between Scotus and Spinoza, that is, that there are not enough citations or references, neither quoted by Deleuze nor in the text of the Ethics itself, to justify the connection or to determine the connection as historically relevant; and, that the value of an interpretation of Spinoza is determinable solely in relation to the parameters or criteria of Spinoza interpretation already established by the tradition of Spinoza studies.

There is however a different way of understanding the relation between the different figures in the history of philosophy and contemporary thought, the elaboration of which is one of the other projects undertaken by Deleuze in his reading of Spinoza in Expressionism in Philosophy. This project is that of renewing the history of philosophy by tracing an alternative lineage that challenges the Hegelian concept of the dialectical progression in the history of philosophy determined by the dialectical logic.

In Difference and Repetition, Deleuze gives a general outline of this project when he writes that ‘The task of modern philosophy is to overcome the alternatives temporal / nontemporal, historical / eternal and particular / universal. Following Nietzsche we discover, as more profound than time and eternity, the untimely: philosophy is neither a philosophy of history, nor a philosophy of the eternal, but untimely, always and only untimely – that is to say, “acting counter to our time and thereby acting on our time and, let us hope, for the benefit of a time to come.”’¹³ It is in this context that Deleuze raises ‘the question of the utilization of the history of philosophy’.¹⁴ Deleuze considers each of the figures of the alternative lineage in the history of philosophy that he traces to ‘bring to philosophy new means of expression’.¹⁵ It is in Expressionism in Philosophy, in relation to Spinoza, that the logic of this new means of expression is explicated as a logic of expression. Rather than Expressionism in Philosophy providing a representation of Spinoza’s metaphysics, Deleuze instead wants ‘to put [Spinoza’s] metaphysics in motion, in action … to make it act, [or to] make it carry out immediate acts’.¹⁶ Expressionism in Philosophy therefore does not offer an alternative representation of the movement of the Hegelian dialectical logic but rather an alternative logic that is ‘capable of affecting the mind outside of all representation’, a logic capable ‘of inventing vibrations, rotations, whirlings, gravitations, dances or leaps which directly touch the mind’.¹⁷ These are the affects of the logic of expression, which is not an abstract logic that merely represents the movement of these affects, but the very logic by means of which these affects are expressed. It is in Expressionism in Philosophy that Deleuze charts the metaphysics of this logic, determining the mechanism by means of which it operates in Spinoza’s philosophy.

Spinoza’s role in this project is determined in chapter 2 by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) from that which Hegel presents in the Science of Logic.¹⁸ Both Hegel and Deleuze each position the geometrical example at different stages in the early development of the differential calculus. Deleuze actually locates the differential from the differential point of view of the infinitesimal calculus in the geometrical example of Spinoza’s Letter XII with the assistance of Leibniz, whereas Hegel argues that the differential is conspicuous in Spinoza’s example because of its absence. The Deleuzian solution offered to the problem of the infinite distinguishes itself from the Hegelian solution insofar as it is not resolved according to the dialectical logic. Deleuze’s thesis is that the differential cannot be classified within the dialectical logic, which asserts the opposition of the infinite and the finite. Instead, Deleuze sets up Spinoza’s example of the relation between the infinite and the finite as a rival metaphysical framework for the resolution of the problem of the infinite, a rival to that provided by Hegel in the dialectical logic. According to Deleuze, the most simple bodies of Spinoza’s Ethics correspond directly to the infinitely small differentials of the geometrical example in Letter XII. The logic of the differential from the differential point of view of the infinitesimal calculus is thereby implicated in Deleuze’s interpretation of the discussion of the physics of bodies by Spinoza in the second part of the Ethics. Deleuze develops the differential point of view of the infinitesimal calculus as an alternative point of view of the differential calculus to that proposed by Hegel. In chapter 2, it is determined that the distinction between the differential and integral calculus that Hegel uses to support the development of the dialectical logic opposes one to the other as inverse or contradictory operations. It is then determined over the course of the next chapter that this distinction, which was later determined as ‘the fundamental theorem of the calculus’, does not necessarily have to be conceived solely as an opposition between irreducible disciplines within the differential calculus, since the operation of integration from the differential point of view of the infinitesimal calculus, according to which power series can be used to solve differential relations by determining the composite functions into which they are potentially expanded, can be recovered in the differential calculus of contemporary mathematics.

The differential point of view of the infinitesimal calculus represents not a moment that can be simply sublated and subsumed within the dialectical progression of history, but rather an opening, providing an alternative trajectory for the construction of an alternative history of mathematics. Deleuze thereby establishes a historical continuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus which effectively bypasses the methods of the differential calculus which Hegel uses in the Science of Logic to support the development of the dialectical logic. While Hegel is interested in using advances in mathematics to secure the development of the dialectical logic, Deleuze is interested in using mathematics to secure not only the development of an alternative logic, but in the process, to undermine the mathematical support of the Hegelian project, by historically bypassing it and determining an alternative trajectory, not only in the history of mathematics, but simultaneously in the history of philosophy.

The alternative lineage in the history of mathematics is implicated in Deleuze’s alternative lineage in the history of philosophy by means of a convergence between the logic of the differential from the differential point of view of the infinitesimal calculus and the logic of expression. There are what Deleuze considers to be ‘correspondences without resemblance’¹⁹ between them, insofar as both are determined according to the logic of different/ciation. The philosophical implications of this convergence are developed by Deleuze in Expressionism in Philosophy in relation to his reading of Spinoza’s theory of relations in the Ethics. By exploiting the implications of the differential point of view of the infinitesimal calculus in his interpretation of the physics of bodies in the second part of the Ethics, Deleuze is able to read the system of the Ethics as a whole as determined according to the logic of expression. The explication of this reading strategy is what constitutes the Deleuzian reading of Expressionism in Philosophy that is undertaken in the present work. This strategy of reading the Ethics as determined according to the logic of expression marks not only the originality of Deleuze’s interpretation of Spinoza, but also one of the points where Deleuze can be considered to depart from the Hegelian and Cartesian Spinoza familiar to scholars working in the field of Spinoza studies, by tracing an alternative lineage in the history of philosophy that expresses the convergence between Spinoza’s ontology and the mathematics of Leibniz.

The Deleuzian solution to the problem of the infinite

In offering an alternative solution to the problem of the infinite, or of the relation between the infinite and the finite, Deleuze draws significantly on the work of Albert Lautman. In Essai sur l’unité des sciences mathématiques dans leur développement actuel, Lautman argues that there are ‘two classic positions’ as regards the relation between the continuous and the discontinuous, or the infinite and the finite. On the one hand, ‘the continuous emanates from the discontinuous like the infinite from the finite, by a sort of progressive enrichment of the finite and the discontinuous’,²⁰ and on the other hand, ‘the priority of the continuous and of the infinite can equally be affirmed and it can be seen in the finite and the discontinuous either a limitation of the infinite, or an approximation of the infinite’.²¹ Lautman argues that the latter position is evident ‘in the mathematical discipline which is most in contact with philosophical thought … the authentic mathematical theorems of approximation’.²² This position is characteristic of what Hegel determines as the Mathematical or Bad Infinite, which is the idea of the infinite from the point of view of the finite. The relation of the infinite to the finite is resolved by Hegel according to the dialectical logic insofar as the Bad Infinite, or the latter classic position, which he argues is determined by the primary negation of the finite, or of the former classic position, is itself negated and thereby subsumed in the actual or Philosophic Infinite, such that the finite realizes itself as actually infinite.²³

Lautman argues that ‘it is possible to observe in the movement of twentieth century mathematics a third way of conceiving [the relations between] the continuous and the discontinuous, [or] the infinite and the finite … which sees in the infinite and the finite not the two extreme terms of a passage to be negotiated, but two distinct genres of being, each having its own structure’ that is sustained by the ‘relations of imitation or of expression’ between them.²⁴ This third position is characteristic of the alternative solution offered by Deleuze to the problem of the infinite, and introduces the concept of expression between the infinite and the finite that is characteristic of the logic developed by Deleuze in Expressionism in Philosophy as the logic of expression.

‘By relations of imitation’, Lautman understands the case where the ‘structure of the infinite imitates that of the finite’.²⁵ According to the logic of expression, there is a relation of imitation between: on the one hand, the intensive parts belonging to singular modal essences insofar as they are differentiated from one another in the form of a power series expansion; and, on the other hand, the power series expansions of the composite relations in which finite existing modes are further differenciated.²⁶ And, ‘by relations of expression’, Lautman understands ‘the case where the structure of a finite and discontinuous domain envelops the existence of another continuous or infinite domain, which finds itself, in this way, to express²⁷ the existence of this finite domain to which it is adapted’.²⁸ According to the logic of expression, when an existing mode or global integration is further differenciated in relation to another existing mode or global integration, the so determined composite relation, which is also known as a discontinuous group of local integrations, envelops the more composite global integration, or continuous infinite domain, in which each of the existing modes or global integrations (or the discontinuous group of local integrations) is actualized or expressed.

According to this third position, there is therefore ‘a relation of expression between the discontinuous and the continuous’, that is, between the discontinuous group of local integrations and the continuous infinite domain in which they are actualized as a global integration. Lautman argues that ‘The structure of the first envelops the existence of the second and inversely the existence of the second expresses or represents the structure of the first’.²⁹ According to the logic of expression, the actualized global integration would therefore be ‘the expression’; the continuous infinite domain would be ‘what expresses itself’; and the difference of intensity between the discontinuous group of local integrations and the actualized global integration that it envelops, or in which it is expressed, would be ‘what is expressed’ by the relation.

The existence of the continuous infinite domain, represented or expressed by the actualized global integration, does not express any less the properties of the domain of discontinuity determinative of it. In fact, the determinateness or finitude of a local integration, or existing mode, is only determined insofar as the global integration in which it is further differenciated is itself further differenciated in a more composite global integration.³⁰ The structure of a finite and discontinuous domain, or discontinuous group of local integrations, is determined by the relations of further differenciation between these local integrations, or existing modes. The finitude of each of these local integrations, and therefore of the domain itself, is determined solely insofar as the global integration attached to the domain is further differenciated in relation to another, and thereby actualized in a more composite global integration, that is, solely insofar as the global integration in which the discontinuous group of local integrations is expressed envelops the existence of another continuous or infinite domain. The global integration thereby finds itself as the expression of the finite domain to which it is adapted solely insofar as the relation of further differenciation in which it is implicated with another global integration is actualized in a more composite global integration.

The difference between the infrastructure of the discontinuous group of local integrations, which consists of the composite relation between the local integrations or existing modes determinative of it, and the structure of the continuous global integration in which the power series expansion of this composite relation is actualized, is a difference of intensity; just as the difference between: on the one hand, the structure of the relation of further differenciation between the global integration attached to, or continuous across, one domain, and the global integration attached to, or continuous across, another domain; and, on the other hand, the more composite global integration attached to this structure, and in which the global integrations are actualized, is a difference of intensity. An actualized global integration is only conceivable as a ‘global’ integration insofar as it is attached to the discontinuous group of local integrations constitutive of it. And, the local integrations of a discontinuous group are only conceivable as local insofar as the discontinuous group is actualized in a global integration. A global integration and the global integration with which it is further differenciated are therefore ‘local’ in relation to the more composite global integration in which they are actualized; which is itself ‘global’ in relation to them as ‘local’ integrations, and a ‘more composite’ global integration in relation to them as ‘global’ integrations.

The relation between the continuous and the discontinuous, or the infinite and the finite, or indeed, the global and the local, is determined according to what Lautman describes as ‘the logical schemas which preside over the organization of their edifices’.³¹ Lautman argues that ‘it is possible to recover within mathematical theories, logical Ideas incarnated in the same movement of these theories’.³² The logical Ideas to which Lautman refers include the relations of imitation and expression between the continuous and the discontinuous, the infinite and the finite, or the global and the local. He argues that these logical Ideas ‘have no other purpose than to contribute to the illumination of logical schemas within mathematics, which are only knowable through the mathematics themselves’.³³ The project of the present work has been to locate these ‘logical Ideas’ in the mathematical theory of the infinitesimal calculus from the differential point of view, in order then to demonstrate that Deleuze uses these ‘logical Ideas’, which are recast as philosophical concepts, to develop the logical schema of a theory of relations characteristic of a philosophy of difference, which, in Expressionism in Philosophy, is determined in relation to Spinoza’s theory of relations as the logic of expression.

The project of renewing the history of philosophy

In What is Philosophy?, Deleuze and Guattari define philosophy as ‘the creation of concepts’.³⁴ The manner by means of which concepts are created is determined in the present work according to the logic of expression. The concepts created enter into resonance with one another on what they define as ‘a plane of immanence’,³⁵ where a plane of immanence can be understood according to the logic of expression to consist of a composite assemblage of multipli-differenciated global integrations, the relations between which determine the creation of the concepts that are distributed over it. These created concepts must not be confused with the plane of immanence that they occupy. What Deleuze and Guattari propose instead is ‘a strict correspondence between the created concepts and the instituted plane’ which is determined by what they characterize as ‘indirect relationships’.³⁶ The concepts are what Deleuze and Guattari describe as the ‘intensive features’, or essential singularities, of a plane, which ‘require a special construction distinct from the plane’.³⁷ It is the ‘indirect’ relations of further differenciation between global integrations that create concepts as the intensive features, or essential singularities, of a plane, and these are distinct from the power series expansions characteristic of the composite relation between the global integrations, that is, the schemas of problematization, which actually institute the plane. Philosophy is therefore ‘at once concept creation and instituting of the plane’,³⁸ that is, philosophy both creates concepts and generates the plane on which they are distributed.

Deleuze and Guattari argue that ‘every great philosopher lay[s] out a new plane of immanence’,³⁹ such that ‘there are varied and distinct planes of immanence that … succeed and contest each other in history’.⁴⁰ Indeed they argue that the history of philosophy ‘exhibits so many quite distinct planes’⁴¹ that ‘we can and must presuppose a multiplicity of planes’.⁴² Every plane of immanence is related to all of the others insofar as they are all ‘complicated’⁴³ in relation to one another. ‘The plane of immanence’, which is ‘neither a concept nor the concept of concepts’,⁴⁴ is therefore ‘interleaved’,⁴⁵ and can be understood according to the logic of expression to be expressed by the actually infinitely composite assemblage of multipli-differenciated global integrations. The differences of intensity generated by the composite relations between these global integrations actually constitute an intensive multiplicity, the structure of which is determined as that in which the global integrations of each plane are implicated, and as that of which the global integrations of each plane are the actualization.

Deleuze and Guattari further qualify the image of philosophy as the creation of concepts when they argue that ‘If philosophy begins with the creation of concepts, then the plane of immanence must be regarded as prephilosophical’.⁴⁶ The distinction between the prephilosophical character of the plane of immanence and philosophy understood as the creation of concepts is characteristic, for Deleuze and Guattari, of the distinction between the history of philosophy and philosophy itself. According to Deleuze and Guattari, the difficult task of the historian of philosophy is the reproduction of a past philosophy. In Difference and Repetition, Deleuze argues that ‘In the history of philosophy, a commentary should … have a double existence and a corresponding ideal: the pure repetition of the former text and the present text in one another’.⁴⁷ However, in What is Philosophy?, Deleuze and Guattari emphasize that it is ‘not a matter of … repeating what a philosopher said but rather of producing resemblance by separating out both the plane of immanence he instituted and the new concepts he created’.⁴⁸ The practice of the history of philosophy is therefore the repetition of the generation of the plane of immanence of the philosophy studied, which always involves determining those concepts (of the past) whose further differenciation created the concepts of this philosophy, and thereby repeating the actual creation of these concepts, which are then distributed over the plane of immanence determined by the schema of problematization generated by the expansion of the composite relation between these concepts (of the past) in a power series.

The actual practice of philosophy, according to Deleuze and Guattari, consists above all in disengaging the concepts that a philosophy of the past created, along with the plane of immanence that it generated, from the supposed order of succession of achieved philosophical systems, by grafting the concepts (of the past) onto new concepts, that is, by selecting those concepts (of the past) distributed on the planes of immanence laid out by the philosophers who created them, which enter into relations of further differenciation with the signature concepts of other philosophers,⁴⁹ thereby extending each of these planes by determining their actual coexistence in relation to one another.

Rather than simply the presentation of a succession of achieved philosophical systems, Deleuze and Guattari consider the practice of the history of philosophy to be determined by the actual recreation of the concepts of a philosophy of the past and the generation of the schema of problematization or of the plane of immanence on which they are distributed. These concepts (of the past) are then redeployed by means of their further differenciation in relation to one another to create new concepts. The creation of new concepts therefore requires grafting concepts (of the past) onto new concepts, by means of their further differenciation in composite relation to one another. The practice of philosophy is therefore determined by the further differenciation of the concepts of philosophies of the past in order to create new concepts and to generate the schema of problematization or plane of immanence on which these new concepts are distributed. The practice of the history of philosophy is therefore distinct from the practice of philosophy since it doesn’t involve the creation of new concepts but rather the repetition of the creation of the concepts (of the past). However, according to Deleuze and Guattari, the history of philosophy is strictly prephilosophical insofar as it serves solely to recreate the concepts (of the past) in order that they are then further differenciated to create new concepts, and which are then grafted onto the new schemas of problematization or planes of immanence on which the new concepts are distributed. Deleuze and Guattari therefore argue that ‘philosophy cannot be reduced to its own history, because it continually wrests itself from this history in order to create new concepts that fall back into history but do not come from it’.⁵⁰

The (pre)philosophical reading of philosophers of the past, or of the history of philosophy is therefore not simply a representation of concepts, nor of the schemas of problematization generated in relation to them, but rather the repetition of the act of their creation and generation. The practice of philosophy then becomes a process of selection that involves determining those concepts (of the past) which can be further differenciated or problematized in relation to one another. In chapter 10, it is argued that the signature concept that Deleuze selects as that with which to further differenciate or problematize the Scotist concept of the univocity of being, is the Nietzschean concept of the eternal return. The new concept created by this process of selection is the Spinozist concept of substance as immanent cause. When problematized in relation to the logical schema determined in relation to the mathematics of Lautman, the Spinozist concept of substance as immanent cause creates the Deleuzian concept of intensity.⁵¹ It is this process of selection that determines the creation of new concepts and it is the operation of this process that Deleuze considers to be characteristic of the practice of philosophy, and to be determinative of, or to generate, the plane of immanence of the philosophy of difference.

The generation of the schema of problematization of a newly created concept, which is determined by expanding the composite relation between the concepts (of the past) in a power series, and which institutes the plane of immanence on which the new concept is distributed, is what Deleuze considers to be determinative of the ‘time of philosophy’.⁵² The Deleuzian definition of the time of philosophy mocks the understanding of the history of philosophy as the succession of achieved philosophical systems. The time of philosophy is rather ‘a grandiose time of coexistence that does not exclude the before and after but superimposes them in a stratigraphic order’.⁵³ The ‘time of philosophy’ is therefore not the subject of a future (after) and a past (before) (Nietzsche after Spinoza, Scotus before Spinoza), but rather constitutes a ‘stratigraphic time where “before” and “after” indicate only an order of superpositions’.⁵⁴ Spinoza can therefore come ‘after’ Nietzsche,⁵⁵ insofar as the Spinozist plane of immanence, on which the concept of substance as immanent cause is distributed, is superimposed on the Nietzschean plane of immanence, on which the concept of the eternal return is distributed. Deleuze and Guattari write that ‘very old strata can rise to the surface again, can cut a path through the formations that covered them and surface directly on the current stratum to which they impart new curvature’.⁵⁶ It is in this way that Deleuze ‘reintroduces Spinozism into the heart of the philosophical process’.⁵⁷ Philosophical concepts therefore do not enter into the linear progression of an evolutionary history, but are rather distributed over a plane of immanence that generates instead an intensive temporality of its own, one that implicates the time that it explicates, or which is actualized in the space–times that it deploys.

Rather than understanding the history of philosophy to be determined by an irreversible evolutionary progression, Deleuze renews the history of philosophy in relation to the practice of philosophy by determining a philosophy of the past instead as a plane of immanence. The Deleuzian practice of the history of philosophy therefore reactualizes, or repeats the actualization of, the plane of immanence of a philosophy of the past in order then to reintroduce it into new arrangements. This is achieved by grafting the concepts (of the past) onto new concepts, and by generating the schemas of problematization that constitute the plane of immanence on which the new concept is distributed. It is these new arrangements that constitute the ‘infinite becoming of philosophy that crosscuts its history without being confused with it’.⁵⁸ Deleuze and Guattari can therefore argue that ‘Philosophy is becoming, not history; it is the coexistence of planes not the succession of systems’.⁵⁹

The project of the present work has been to explicate the image of thought, or plane of immanence, characteristic of Deleuze’s philosophy of difference as explicated in Expressionism in Philosophy and Difference and Repetition. The plane of immanence of the philosophy of difference is therefore actually constituted by a variety of interleaved strata of the past, whose coexistence is determined by the relations of further differenciation between the concepts (of the past) distributed on each of these strata, which creates the new concepts onto which they are grafted. The mechanism by means of which new concepts are created, and new schemas of problematization generated in relation to them, thereby instituting the plane of immanence on which the new concepts are distributed, is determined in Expressionism in Philosophy according to the logic of expression and is characteristic of what Deleuze poses in Difference and Repetitionas the philosophy of difference.

The difference between ‘relative’ and ‘actual’ difference

According to the Deleuzian concept of intensity, the difference characteristic of the philosophy of difference is a difference of intensity. Rather than difference being determined as the difference between the composite terms of a differential relation, or as a relative difference, as it is according to the Hegelian dialectical logic, the difference of a difference of intensity is determined as the actual difference between the reciprocally related global integrations of a differential relation and the new essential singularity created by this relation. Insofar as such a newly created essential singularity is further differenciated in relation to another, it is actualized in a more composite global integration. Only then is the finitude, or identity, of the global integrations of which the essential singularity is composed determined. The finitude, or identity, of a global integration is therefore only determined insofar as the more composite global integration in which it is implicated is itself further differenciated in relation to another. Deleuze therefore considers the finitude, or identity, of a global integration to exist ‘as a principle but as a second principle, as a principle become’.⁶⁰ The finitude or identity of a global integration or existing mode is determined solely by the relations of further differenciation in which it is implicated. Identity is therefore not determined first, but is rather determined secondarily in relation to difference.

The Hegelian dialectical logic however cancels the difference of a difference of intensity. Difference is instead subordinated to identity, where identity is considered to be ‘the sufficient condition for difference to exist and be thought’.⁶¹ Each composite term of a differential relation is determined as identical to itself solely insofar as its intensive and extensive quantity are understood to be identically quantitative. Difference is therefore understood instead as the relative difference between the composite terms of a differential relation, each of which is always ‘already understood as identical’.⁶² Therefore, according to the Hegelian dialectical logic, with any difference there is an always already assured identity. By cancelling the difference of a difference of intensity, the Hegelian dialectical logic subordinates intensity to the qualities which fill extensity, such that intensity is only determinable in the form of the primary and secondary qualities of the extensive quantities, or terms of the differential relation, in which it is implicated. The ‘Other’, which Deleuze characterizes as the intensive quantity of greater degree of power of a difference of intensity, is thereby reduced to the ‘Same’ insofar as each of the ‘relatively different’ composite terms or extensive quantities of a differential relation, each of which is already determined as identical to itself, exhibits the ‘Same’ primary and secondary qualities. If each of the global integrations of a composite relation is determined as identical to itself, as they are according to the Hegelian dialectical logic, then each global integration is separated, or abstracted as a ‘disembodied appearance’,⁶³ from the difference of intensity in which it is implicated, or from the more composite global integration in which it is further differenciated. The greater intensive quantity of this more composite global integration is therefore subordinated to, or solely represented in, the ‘Same’ primary and secondary qualities of each of the given global integrations. According to the logic of expression, the global integrations of a differential relation are different solely in relation to the greater intensive quantity of the difference of intensity determined by the relation of further differenciation in which they are implicated; and each of these global integrations can be considered to be the ‘Same’ solely insofar as the difference of intensity of the relation of further differenciation in which they are implicated is actualized in a more composite global integration.

The process of the creation of concepts, or of the essential singularities to which they correspond, according to the logic of expression, is simultaneously pre-individual and individuating: pre-individual, insofar as it marks the genesis of the process of individuation; and, individuating, insofar as the global integrations that are selected in composite relation to one another are actualized in a more composite global integration, a process which determines the finitude or identity, and therefore the individuation, of the local integrations of which each global integration is constituted. Deleuze argues that ‘any confusion between the two …, any reduction of individuation to a limit or complication of differenciation’, that is, any reduction of individuation to either its pre-individual or its individuating components, just as any reduction of philosophy to either its prephilosophical, or its philosophical components, ‘compromises the whole of the philosophy of difference’.⁶⁴ Individuals themselves are not singular, nor are the global integrations, existing modes or bodies to which they correspond, ‘even though they are constituted “in the neighbourhood of” certain singularities’.⁶⁵ Individuation, as the determination of the finitude, or identity, of a global integration, is instead determined ‘after the fact, as the positive effect of [actual] difference rather than’ as the effect of the subordination of this difference to the Same primary and secondary qualities which are ‘presupposed for only relative differences’⁶⁶ between the always already identical. The consistency of individuation is no longer that of composite negation, characterized by the ‘sublation’ of one term, or moment, in relation to another, as determined according to the Hegelian dialectical logic, but rather composite implication, characterized instead by the process of ‘selection’, as determined according to the logic of expression.

The process of ‘selection’ determines those global integrations which are reciprocally implicated and therefore enter into composite relation with one another. According to Deleuze’s understanding of Spinoza’s theory of relations, those joyful passive affections are ‘selected’ that can be repeated, or accumulated in a power series expansion, and potentially lead to active joy. In ‘Spinoza and the three “Ethics”’, Deleuze argues that ‘This selection is extremely hard, extremely difficult. The joys and sadnesses, increases and decreases, … are often ambiguous, partial, changing, intermixed with each other. … The selection of … affects as the primary condition for the birth of the concept does not merely imply the personal effort each person must make on his or her own behalf (Reason), but a passional struggle, an inexpiable affective combat in which one risks death, in which … affects clash with affects in order that a little joy⁶⁷ might be saved that could make us [increase our power to act] and change kind’.⁶⁸ According to Deleuze, an account of the effectuations of bodies and of the relations in which they are implicated can only be given by invoking the differences of intensity that are generated by these relations and in which these bodies are implicated. The process of selection is therefore a process of evaluation of the relations in which one is implicated to determine which relations can potentially increase one’s power to act. In Expressionism in Philosophy he maintains that ‘The question is in each case: Does, say, this feeling, increase our power of action or not? Does it help us come into full possession of that power?’⁶⁹

The project of the present work has been to develop a Deleuzian reading of Expressionism in Philosophy that attempts to explicate in detail the operation of the logic of expression both in relation to Deleuze’s engagement with Spinoza and with certain concepts of the text of the Ethics, and in relation to the wider context of his project of constructing a philosophy of difference. In relation to the latter, what I have attempted to determine is the insufficiency of the Hegelian dialectical logic insofar as it ‘menaces the complexity and the multiplicity of difference’⁷⁰ by reducing it to the ‘circulation of opposing representations’ that coincide in ‘the identity of a concept’.⁷¹ Deleuze considers the Hegelian dialectical logic to have narrowly missed the actual movement of thought which he considers to be determined by a ‘difference of intensity’ according to the logic of expression rather than by an ‘identity of quantity’ according to the Hegelian dialectical logic. The logic of expression is therefore the logic according to which Deleuze constructs the philosophy of difference, and according to which the philosophy of difference functions as an alternative to the dialectical philosophy determined by the Hegelian dialectical logic.

_______________

Notes

¹ Macherey, Hegel ou Spinoza, p. 136.

² Ibid., p. 12.

³ Ibid.

⁴ Ibid.

⁵ Holland, ‘Spinoza and Marx’, §32.

⁶ Deleuze, Difference and Repetition, p. 268.

⁷ Macherey, ‘The encounter with Spinoza’, p. 142. This is one of the questions central to Macherey’s problematization of Deleuze’s reading of Spinoza in the ‘The encounter with Spinoza’.

⁸ Ibid., p. 148.

⁹ Warren Montag and Ted Stolze (eds), The New Spinoza (Minneapolis, 1997), p. x.

¹⁰ Ibid.

¹¹ Ibid.

¹² Ramond, qualité et quantité dans la philosophie de Spinoza, p. 201.

¹³ Deleuze, Difference and Repetition, p. xxii. The Nietzsche citation is quoted by Deleuze from: Friedrich Nietzsche, ‘On the uses and disadvantages of history for life’, Untimely Meditations, trans. R.J. Hollingdale (Cambridge, 1983), p. 60.

¹⁴ Deleuze, Difference and Repetition, p. xxii.

¹⁵ Ibid., p. 8.

¹⁶ Deleuze, Difference and Repetition, p. 184.

¹⁷ Ibid.

¹⁸ Hegel, Hegel’s Science of Logic, p. 251.

¹⁹ Deleuze, Difference and Repetition, p. 184.

²⁰ Lautman, Essai sur l’unité des sciences mathématiques dans leur développement actuel, p. 56.

²¹ Ibid.

²² Ibid.

²³ See the section of chapter 1 entitled ‘Infinite actu and the “bad infinite”’, p. 21.

²⁴ Lautman, Essai sur l’unité des sciences mathématiques dans leur développement actuel, p. 57.

²⁵ Ibid.

²⁶ See the section of chapter 5 entitled ‘From the differenciations of the differentiated to the actually infinitely composite differenciations of the differenciated’, p. 131.

²⁷ That is, ‘what expresses itself’.

²⁸ Lautman, Essai sur l’unité des sciences mathématiques dans leur développement actuel, p. 57.

²⁹ Albert Lautman, Nouvelles recherches sur la structure dialectique des mathématiques (Paris, 1939), p. 31.

³⁰ See the section of chapter 10.6 entitled ‘The duration of a finite existing mode’, p. 226.

³¹ Lautman, Essai sur l’unité des sciences mathématiques dans leur développement actuel, p. 58.

³² Ibid.

³³ Ibid.

³⁴ Deleuze and Guattari, What is Philosophy?, p. 41.

³⁵ Ibid., p. 35.

³⁶ Ibid., p. 58.

³⁷ Ibid., p. 39.

³⁸ Ibid., p. 41.

³⁹ Ibid., p. 51.

⁴⁰ Ibid., p. 39.

⁴¹ Ibid., p. 50.

⁴² Ibid.

⁴³ See the section of chapter 5 entitled ‘The double movement of complication and explication’, p. 124.

⁴⁴ Deleuze and Guattari, What is Philosophy?, p. 35.

⁴⁵ Ibid., p. 50.

⁴⁶ Ibid., p. 41.

⁴⁷ Deleuze, Difference and Repetition, p. xxii.

⁴⁸ Deleuze and Guattari, What is Philosophy?, p. 55.

⁴⁹ See What is Philosophy? where Deleuze and Guattari write that: ‘The life of philosophers, and what is most external to their work, conforms to the ordinary laws of succession; but their proper names coexist and shine either as luminous points that take us through the components of a concept once more or as the cardinal points of a stratum or layer that continually comes back to us’ (Deleuze and Guattari, What is Philosophy?, p. 59).

⁵⁰ Ibid., p. 96.

⁵¹ See the section of chapter 10 entitled ‘The concept of intensity according to the logic of expression’, p. 243.

⁵² This is the manner by means of which Deleuze proposes to ‘give up the narrowly historical point of view of before and after in order to consider the time rather than the history of philosophy’ (Deleuze and Guattari, What is Philosophy?, p. 58). See the section of chapter 10 entitled ‘The wider context of Deleuze’s philosophical project’, p. 215.

⁵³ Deleuze and Guattari, What is Philosophy?, p. 59.

⁵⁴ Ibid., p. 58.

⁵⁵ See the section of chapter 10 entitled ‘The Nietzschean correction to Spinozism’, p. 222.

⁵⁶ Deleuze and Guattari, What is Philosophy?, p. 58.

⁵⁷ Gilles Deleuze, ‘Immanence: A Life’, Pure immanence: essays on a life, trans. A. Boyman (New York, 2001), p. 28.

⁵⁸ Deleuze and Guattari, What is Philosophy?, p. 59.

⁵⁹ Ibid.

⁶⁰ Deleuze, Difference and Repetition, p. 40.

⁶¹ Simont, p. 251.

⁶² Deleuze, Difference and Repetition, p. 41.

⁶³ Zourabichvili, Deleuze, une philosophie de l’événement, p. 90.

⁶⁴ Deleuze, Difference and Repetition, p. 247.

⁶⁵ Zourabichvili, Deleuze, une philosophie de l’événement, p. 102.

⁶⁶ Ibid., p. 85.

⁶⁷ That is, ‘joyful passive affection’.

⁶⁸ Deleuze, Essays Critical and Clinical, p. 145.

⁶⁹ Deleuze, Expressionism in Philosophy, p. 269.

⁷⁰ Antonioli, Deleuze et l’histoire de la philosophie, p. 38.

⁷¹ Deleuze, Difference and Repetition, p. 171.

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Index

abrogated 23

abscissa 49, 52

absolute ideal 4–15, 17–18, 31

active joy 162–3, 166, 168, 176, 188, 190, 207–8, 225–6, 248, 264

actualization 1, 84–5, 126, 131, 167, 185–6, 188–90, 227, 229, 234, 248, 258, 261

affections

joyful passive 5, 7, 149, 157, 159–64, 166–9, 174–7, 179, 181–3, 185, 187–91, 208, 212–14, 225, 230, 234, 248, 264

passive 7, 139, 144, 148–9, 151–2, 155–6, 159–63, 166–8, 170–71, 175, 179, 183, 188, 190, 212, 217, 230, 248

sad passive 7, 157, 159, 160, 162, 168, 212

Aion 228

Alquie, F. 33

amor erga Deum, see God, love towards

amor intellectualis Dei, see God, intellectual love of

analytic continuity 72–4, 78–80, 84, 87, 124

Anticartesian 5, 99

Antonioli, M. 3, 4, 264, 267

approximation

of functions 69

method of 70

Aristotle 102

attractor 83

attribute 12–13, 16–17, 25–6, 30–35, 39, 96–101, 110–111, 113–4, 139, 181, 197, 199–204, 213, 221, 237–8

of extension 32, 149, 164, 165, 181, 197, 237

of thought 32, 149, 164–5, 197, 198, 200–204, 213, 231, 237

Balibar, E. 33, 142, 267

Barrow-Green, J. 82

being-in-itself 16

being–logic 14

Beiser, F. 7

Bell, J.L. 56

Bennett, J. 32

Bergson, H. 2

blood 90

body

complex 38–9, 86–89

most simple 37–9, 60, 69, 86–9, 92, 119, 133, 140, 181, 252

Boyer, C. 48, 52, 54, 56, 75–6, 102, 107, 267

Breton, S. 40–41

calculus

dialectics of the 91

differential 2, 4, 49–51, 54, 56, 58–60, 63–6, 69, 73–7, 88, 91–2, 104, 107, 230, 252–3

fundamental theorem of 73, 253

infinitesimal 4, 43, 48, 50, 52, 54, 56–7, 60, 67, 69, 74, 76, 91, 107, 252, 253

differential point of view of 4, 43, 45, 47–9, 51, 53–5, 57–9, 61, 63, 65–7, 69–70, 74–7, 83, 86, 88, 91–2, 124, 252–3

integral 49, 53–4, 61–2, 73, 253

metaphysics of the 65

capacity to be affected 143–4, 150–154, 156, 158–9, 161–2, 170, 179, 212, 217–18, 238, 247

Cartesian 4, 6, 32–3, 56, 98–9, 222, 254

Cauchy, A-L. 54, 56, 70, 73–4, 269

causa sui 12, 112–13, 171, 194, 220, 237

cause

immanent 8, 9, 26, 29, 39, 111, 112, 113, 114, 211, 216, 217, 220, 222, 223, 224, 225, 238, 260, 261

transitive 26, 39

Chronos 228

Clagett, M. 103

common notions 164–8, 172, 174, 176–7, 193–4, 196, 202–3, 206, 213, 232–5

conatus 136, 138–9, 144–7, 153–6, 159, 163, 170–171, 179, 212, 267

Concept–logic 14

convergence

circle of 71–3

Cauchy’s criterion of 70

Cross, R. 103, 107–8

Curley, E. 12, 30, 32, 146

curve 50, 53–7, 59, 61–2, 70–71, 76, 79, 81, 83, 88, 124–5

cut 79, 80, 82, 125, 194, 261

death 7, 198, 204–6, 209–11, 213, 220, 264

deconstruction 2

Dedron, P. 45

DeLanda, M. 83, 268

Deleuze, G. 1–9, 20, 35–6, 38, 41–3, 46–58, 61, 63–7, 69–71, 73–9, 83–4, 86–93, 95–103, 106–117, 119–125, 128–30, 132, 136–7, 139–45, 149–71, 174–6, 179–81, 183, 187–8, 190, 193, 200–255, 257–65, 267–74

denominator 81–2, 124, 247

Descartes, R. 2, 5, 32, 33, 98, 114, 268–9

determinability 64–5, 75–6, 84–6, 89, 120

determination

complete 75, 78, 80, 83–5, 87, 120, 124, 231–2

reciprocal 66, 75–8, 84–6, 120, 125, 232

dialectical logic 1, 4, 9, 11–18, 23, 25–6, 29, 31–2, 35, 39, 42–4, 58, 61–7, 69, 70, 73, 75–7, 91–2, 104–7, 216–8, 241–3, 245–6, 249–54, 262–5

difference

actual 41, 249, 262

ideal 75, 76, 84

of intensity 115, 242–8, 255–6, 262–4

of quantity 244–6

quantitative 60

relative 246, 262

differenciation 79, 83–6, 88–91, 124–30, 140, 182–4, 186–90, 207–8, 210, 217, 219, 223, 227–9, 231–3, 235–6, 238, 242–3, 248, 256–7, 259, 261–3

further 155, 182–4, 186–90, 207–8, 210, 217, 219, 223, 226–9, 232, 235–6, 238, 242–3, 248, 256–7, 259, 261–3

logic of 90, 127, 232

different/ciation 91

logic of 2, 4, 6–8, 90–92, 119, 128, 130, 133, 140–43, 161, 164–5, 167–8, 174, 177, 179, 180, 182, 185–6, 189, 214–19, 220, 223–9, 230–31, 233–7, 239–40, 253, 269

differential

logic of the 66–7, 69, 75, 78, 85, 90–2, 120, 184, 220, 252–3

differential relation 6, 49–50, 52–5, 57–63, 66, 69–72, 74, 76–9, 81, 83–4, 86, 89–90, 104, 120–121, 124–5, 127, 129, 220, 224, 230–232, 238–9, 246, 262–3

differentiation 54–5, 57, 59, 61–2, 71, 73–4, 77–9, 81, 83–91, 104, 117, 119–120, 122–5, 127–8, 133, 180, 224, 231–3, 236

quantitative 122

Donagan, A. 32

Duffy, S. 43, 50

Duhem, P. 102, 108

elasticity 136, 150, 154

essence

of substance 12–13, 111, 203, 237

singular modal 6, 109–13, 115, 119, 123, 127–30, 132–3, 135–6, 139–41, 150, 153–6, 180, 182, 185, 203, 206, 218, 220, 237–9, 240

essence–logic 14, 17

essential singularity 83–4, 90, 127–8, 181, 187, 223, 229, 257, 263

eternal return 8, 9, 216–17, 220, 222–5, 238, 260–261

eternity 7, 27, 34, 191, 193, 195–9, 201, 203, 205–7, 209, 211–13, 220, 251

expressio 236

expression

logic of 2, 9, 215, 217, 236–40, 242–8, 252–5, 257–8, 260, 263–5

exprimere 236

field

scalar 79–80

vector 79–80

figure 3, 8, 11–12, 19–21, 82, 86, 95, 108–10, 198, 200–201, 205, 212–14, 216

finite modal existence, see modal, existence, finite

fluctuatio animi 149, 169–70, 173, 175–6, 213

function

analytic 71–4, 77–8

composite 80–85, 88–91, 125–7

local 79–80, 84, 87–90, 120–121, 124–8

meromorphic 81–4, 89, 124, 127

potential 79–85, 125–7, 131, 167, 184–8, 190

primitive 54, 73–4, 76, 77

functions

automorphic 85

Fuschian 85

discontinuous group of 83, 85, 255–6

genetic 2, 7, 70, 215, 230, 233

geometrical example 4, 21, 24, 27–8, 43–6, 48, 50, 53, 57, 67, 69, 86, 88–9, 252

Gilson, E. 96, 100–101

Giorello, G. 65, 269

God

idea of 171–4, 193–4, 198, 202–3, 209

intellectual love of, see knowledge, third kind of

love towards 173–5, 193–6, 202, 213

Guattari, F. 8–9, 216, 230, 233, 235, 257–261, 268, 270, 273

Gueroult, M. 1–2, 6, 23–5, 28–30, 39, 43–9, 54, 87–9, 135–6, 145–6, 215, 268–9

Hardt, M. 6, 202

Hegel, G.W.F. 1, 3–4, 11–41, 43–7, 54, 58–67, 69–70, 73–7, 87, 91–2, 95, 103–9, 114–16, 148–9, 218, 222, 241, 246, 249–50, 252–4, 267, 270–73

history of philosophy

alternative lineage in the 2–8, 49, 67, 214, 216, 249, 252, 254

dialectical progression of the 1, 3, 5, 41, 67

human being, see individual

Hume, D. 2, 268

idea

adequate 28, 29, 161–2, 164, 166, 168, 174, 187–8, 197, 202, 207–9

idea of the 197–8, 200–202, 204, 213, 232

inadequate 19, 26–9, 162–3, 166, 168, 197

imagination 7, 19, 25–9, 34–5, 45, 58, 65, 147, 155, 162, 164, 168–9, 170–73, 175, 194, 197–8, 207, 234

point of view of the 19, 25–6, 28, 35, 39, 44, 46–7, 65, 87–8, 149, 191, 212, 218

immanence 9, 57, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 216, 220, 235, 238–9, 257–61, 268

expressive 238

problem of 238

immortality 7, 193, 195–6, 205, 211

indefinite 21

individual 6–7, 14, 86–91, 96, 100–102, 109–10, 125, 128–9, 131–2, 135–42, 144–5, 149–50, 161, 163, 166, 179, 181, 188–9, 200–207, 209, 211–13, 227–8, 263

individuality 5, 6, 33, 119, 142, 207–8, 210–11, 267

infinite 2, 4, 12, 19–30, 34–9, 44–50, 52–4, 57–67, 70, 75, 80–82, 84, 86–90, 92, 95–6, 98–9, 101–5, 108–14, 116–17, 119–33, 161–3, 165, 171–2, 180–81, 183–8, 219, 221, 228, 230, 252, 254–7, 261

absolutely 12, 25–6, 29, 39, 113, 123, 200, 211

actual 22–9, 45, 59, 65, 87–8, 90, 120–21

arithmetic or algebraic 45–7

bad 22–6, 28, 59–60, 65, 87–8

collection of the infinitely small, 86

extensive 87

geometric 45–8, 87

in kind 25

intensive 121

Mathematical, see infinite, bad

Philosophic, see infinite, actual

problem of the 20, 25, 29, 58, 87, 252

infinite actu, see infinite, actual

infinitely small, see magnitude, infinitely small

infinitesimal 4, 6, 43, 45–61, 63, 65–7, 69–71, 74–7, 83, 86, 88–9, 91–2, 95, 107–8, 124, 128, 252–3, 257

problem of the 66

infinities

greater and lesser 28, 121

integration 49, 53–5, 57, 59, 62–3, 65, 69, 73, 77, 83, 87–9, 104, 139, 237

global 84, 86, 90, 127–33, 137, 140–42, 145, 150, 153, 155, 157, 161–6, 179–91, 206–8, 210–13, 218, 223–4, 226–9, 232, 239, 242–4, 248, 255–6, 262–3

local 74, 86, 90, 127, 130–32, 140, 142, 150, 153, 155, 179, 182–4, 187, 189, 218, 231–2, 256

intensity 5, 103, 108, 114–15, 128, 242, 247–8, 264

Itard, J. 45

Jacobi, F.H. 18

Johnson, P.O. 106

Klein, F. 85

knowledge

first kind, or imagination 27, 164, 234

second kind, or reason 27, 164, 174, 193, 195–6, 202–3, 213, 234

third kind of 7, 27, 28, 29, 174, 175, 193, 195–200, 203–4, 209–13225, 232, 235

Lagrange, J.L. 56, 76

Lakoff, G. 55–6

Lautman, A. 74, 80, 85, 91, 254, 255, 256, 257, 260, 270

leap 50, 162, 166–7, 169, 187–8, 190, 226

Leibniz, G.W. 2, 5, 20, 43, 46, 49–57, 67, 69, 74–5, 114, 151, 252, 254, 268–71, 273

Letter XII 4, 20–21, 23, 25–9, 34, 43–50, 52, 57–8, 67, 69, 86–9, 116, 119, 252

limit 36–8, 40, 45, 54–7, 59–62, 73, 79–80, 84, 90, 104, 131–2, 136, 148–53, 155–6, 159–60, 162, 170, 175, 179, 183–4, 188, 193, 200, 202, 212–13, 263

limits 11, 20–21, 25, 33, 37, 43, 44, 50, 55–7, 79–80, 85, 87–9, 124–6, 128, 131, 135–6, 151, 152, 154, 155, 158, 159, 175, 179, 186, 200

Lloyd, G. 209

Macherey, P. 1, 3, 5, 20, 22, 24–31, 33–41, 43–7, 65, 86–9, 98, 136–7, 139, 145–62, 167–77, 181, 190, 193–202, 204–9, 211–13, 222, 249–50, 271

Maddy, P. 66–7, 271

magnitude 21–2, 44, 53, 58, 60, 62–3, 76, 79, 104–6, 148, 241

infinitely small 65

intensive 62–3, 105–6

Malabou, C. 4

Malebranche, N. 2, 269

Mandelbrot, B. 85, 271

Marxism 41

Matheron. A. 143, 271

maximum 20–22, 24, 26–7, 43–4, 46–7, 50, 52–3, 57, 70, 82, 88, 126, 131–2, 136, 138, 145, 148–9, 151–2, 154–5, 158–9, 162, 173, 179, 181–2, 185–6, 212, 228

metaphysics

of logic 91

of the calculus 66, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93

minimum 20–22, 24, 26–7, 43–4, 46–7, 50, 53, 57, 70, 82, 88, 126, 131–3, 136, 138, 145, 148–9, 151–2, 154–5, 158–9, 179, 181–2, 185–6, 212, 228

modal

essence 108, 110–15, 119, 123, 127, 129, 130, 132, 135–6, 139, 141, 144–5, 150, 152–3, 155, 203, 218, 220, 237–9

existence 123, 146, 149, 152, 157–61, 163, 165, 167, 169, 171, 173, 175, 177, 179, 183, 186, 188, 190, 226

finite 7, 110, 119, 123–4, 133, 136, 156, 188

mode

finite 19, 27, 29, 111–12, 113, 145, 153, 155, 226

finite existing 119, 123–4, 128, 132–3, 139, 150–51, 154, 162–3, 179–83, 187–91, 217–19, 223, 226–7, 230, 237, 239, 240, 248, 255

infinite immediate 38, 181

infinite mediate 38, 181

intrinsic 6, 102, 113

extrinsic 102–3, 108

Montag, W. 250

Moreau, P.F. 267, 272, 273, 275

nature

intrinsic 11

specific qualitative 55, 57, 70, 76–7, 84–5, 88–90, 125–7, 129–31, 137, 161–3, 175, 183–91, 226

negation 15–16, 18–19, 24–5, 36–7

absolute 17–19, 32, 40, 61, 63

all determination is 18–19, 36–7

determining 17

logic of 17, 34, 36, 249

primary 22–3, 25, 61, 254

negative 3, 4, 17, 30–31, 33, 36–7, 39, 60, 63, 139, 149, 176, 250, 274

Negri, A. 6, 33

Newton, I. 55–6, 65, 269

Nietzsche, F. 2, 8, 216, 221, 223, 230, 251, 260, 268, 272

non-concentric circles 20–21, 24–5, 27, 44, 47, 52

non-dialectical 16, 36

non-standard analysis 56, 66, 70, 75

numerator 81, 124, 247

Núñez, R. 55–6

O’Brien, A. 96, 100–102

ontology 5, 246, 254

ordinate 49, 52

orthogonal distance 21, 46, 52

pantheism 7, 97–9

parallelism 7

Parnet, C. 215

parts

extensive 6, 87–9, 92, 95, 105, 116–17, 119, 120–121, 123–5, 127–33, 140–145, 154, 163, 180–182, 185–6, 188, 190–191, 201–6, 210–211, 213, 218–20, 230, 239

intensive 5–6, 95, 114–17, 119–24, 127, 133, 139–40, 180–82, 186, 188–9, 207, 210, 213, 219, 225, 239, 242–3, 246, 251, 255

passions

joyful 157, 160, 162, 166–7, 169, 175, 177, 187

sad 164, 167, 174

Patton, P. 1, 3–4, 61

philosophy

critical 16

dialectical 9, 77, 95, 217, 265

metaphysical 11, 219

of difference 1, 3, 8–9, 67, 69, 90–92, 214–17, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 257, 260–65, 269

physics of bodies 4, 50, 57, 67, 69, 91, 141, 252–3

Pinkard, T. 63, 75

plane of immanence 9, 235, 257–9, 260–61

Poincaré, H. 81–3, 85

points

accumulation, or points of condensation 73

distinctive 70–72, 77, 79

limit 79–80, 84

regular 70–71

singular 72, 74, 78, 80, 83–5

non-removable 80

removable 72–73

stationary or turning 70, 72

polynomial 70

potentia 6, 114

potestas 6

power

degree of 78, 115–17, 119, 121–4, 126, 128–9, 131–3, 136, 140, 144–5, 152, 154, 180–91, 203, 206–7, 210, 212–13, 217–19, 221, 231, 239–40, 242–4, 247, 262

of existence 212, 231

of existing 12, 113–14, 200–204, 210–13, 236

of thinking 170, 200–205, 207–13, 232, 235–6

absolutely infinite 12, 200

power to act 6–7, 138–9, 144–9, 151–6, 158–63, 170, 179, 182–4, 186–91, 198, 211–12, 240–42, 247–8, 264

quantity

extensive 76, 92, 103, 106, 108, 117, 119, 123, 240–41, 243–6, 262

intensive 5, 93, 95, 99, 103, 106, 108, 113, 115–16, 119–23, 155, 219, 221, 238, 240–48, 262–3

vanishing 49

quantum 22, 59, 62, 104, 114, 241

Ramond, C. 5, 106, 135–9, 141–2, 144–8, 150–52, 154, 206, 251, 272

reason 7, 26–7, 34, 38, 41, 45, 54, 64, 89, 98, 101–2, 136, 143, 162, 164–5, 168, 171, 173, 194, 213, 267

point of view of 26, 28, 35, 44–7, 87–8, 218

reciprocal synthesis 76, 83, 86

Redding, P. 13–16, 30

Reinhardt, H.J. 71

relations

composite 83, 89–90, 124–9, 131, 161, 180, 184–90, 206, 210–11, 219–20, 223–6, 228–33, 236, 242–4, 246–8, 255–6, 258–60, 263–4

extensive 87

external or extrinsic 87, 88, 149

of movement and rest 87, 135, 137–9, 141, 144, 154

logic of 50, 67

social 207–8

theory of 54, 58, 67, 69, 91–2, 119, 128, 133, 135–41, 143, 145–7, 149, 151, 153, 155, 157, 159, 167, 177, 203, 225, 230, 236, 239, 240, 243, 253, 257, 264

Riemann, G. 73–4, 269

Robinet, A. 236

Robinson, A. 56–7, 273

Rorty, A.O. 208

Salanskis, J-M. 61, 65–6, 273

schema of problematization 9, 233, 235, 259, 260

Scotism 5, 102, 251

Scotus, D. 5–6, 8, 95–103, 107–10, 112–3, 216, 221, 230, 251, 260, 267, 269, 272, 274

self-comprehension 12

self-consciousness 14, 15, 17, 31

self-determination 14

series

demonstrative 234

divergent 54

polemical 217, 233–6

Taylor or power 70–73, 77–8, 81, 84, 86–8, 90, 119–27, 129, 131–3, 140, 180–82, 184–8, 210, 212, 219, 223–4, 226–33, 236, 239, 242, 246–8, 253, 255–7, 259, 260, 264

set theory 66

Simont, J. 1, 3–4, 64, 105, 262, 273

singularity 72–3, 81–3, 85, 90, 112, 115, 125–9, 131, 137, 155, 161, 180–82, 184–6, 198, 220, 224–5, 228–9, 232–3, 238, 262

essential 82–3, 85, 90, 125, 127, 129, 131, 180, 228, 262

pole 72–73, 79–80, 83, 126, 136, 185

Smith, D.W. 219, 247

space

3-dimensional 227

space–time 229

spatio–temporal dynamism 229

Spinoza, B. 1–9, 11–13, 15–41, 43–50, 52–4, 57–60, 63–7, 69, 86–7, 89–90, 92–3, 95, 98–100, 102–3, 106–116, 119, 123–4, 128–9, 132–3, 135–43, 145–60, 162, 164–77, 181, 190, 193–201, 203–10, 212, 214–16, 218–22, 225–6, 230, 232–3, 235–40, 243–4, 249–54, 257, 260, 264, 267–74

spirit 14–15, 17–18, 31

the Absolute as 15

Stolze, T. 250

structural–genetic

criteria 2

method 2, 215

nature 7

structuralism 2

sub specie aeternitatis 203, 206, 209, 220, 224

substance 11–19, 22, 25–37, 39, 63, 88, 97–100, 111–17, 123–4, 139, 171, 194, 200–201, 203–4, 211, 220–25, 237–8, 260–61, 267

substance–metaphysics 16, 31

subsumption 14, 16, 17, 106

subtangent 52

tangent 50

temporality 8, 216, 230, 261

thresholds 21, 136, 145, 158, 170

time 22, 24, 46, 107, 114, 138, 142, 149, 168, 172, 175, 198, 201, 216, 226–30, 235, 251, 260–61

chronological 228

transformations 7, 146–47, 157–60, 167, 171, 179, 182–3, 186

uninterrupted affective flux 145, 148–52, 154–5, 158, 170, 173, 212

univocity 5, 8, 95–99, 101, 103, 105, 107, 109, 111–13, 115, 117, 203, 216–17, 220–23, 225, 238, 260

unlimited 21–6, 28–9, 34, 47, 65, 88, 228

Valiron, G. 85, 274

variation 21–22, 25, 27–9, 47, 102–3, 106, 108, 126, 131, 135–8, 143–8, 150, 152, 154–5, 159, 171, 179, 212

view

ethical 152–8, 162, 168, 176, 179, 188

physical 149, 152–6, 158, 162, 168, 183

Verstraten, P. 3

Weierstrass, K. 55–6, 67, 69, 71–5, 80–81

Weierstrassian analysis 71, 79, 81

Wilson, C. 107, 116

Wolf, A. 18, 38

Wolfson, M.A. 30

Wolter, A.B. 96–7, 103

Wronski, H. 75–6

zero 49, 51, 55, 57, 70, 121, 191, 228

Zourabichvili, F. 227–30, 239, 247, 263