Gilles Deleuze’s engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems – for example, the problem of individuation – and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze’s philosophy, (...) as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. -/- In the wake of Alain Badiou’s recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze’s work, Simon Duffy offers a robust defence of the structure of Deleuze’s philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze’s seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze’s philosophy, rebuffing the recent challenges against it. -/- Simon B. Duffy is a Senior Lecturer in Philosophy at Yale-NUS College, Singapore, and Honorary Research Associate in the Department of Philosophy at the University of Sydney, Australia. He is the author of The Logic of Expression: Quality, Quantity, and Intensity in Spinoza, Hegel and Deleuze (2006). (shrink)

What I plan to do in this paper is to provide a survey of the ways in which Spinoza’s philosophy has been deployed in relation to early modern thought, in the history of ideas and in a number of different domains of contemporary philosophy, and to offer an account of how some of this research has developed. The past decade of research in Spinoza studies has been characterized by a number of tendencies; however, it is possible to identify four main (...) domains that characterize these different lines of research: studies of Spinoza’s individual works, of its problematic concepts, from the point of view of the history of ideas, and comparative studies of Spinoza’s ideas. (shrink)

In Difference and Repetition, Deleuze explores the manner by means of which concepts are implicated in the problematic Idea by using a mathematics problem as an example, the elements of which are the differentials of the differential calculus. What I would like to offer in the present paper is a historical account of the mathematical problematic that Deleuze deploys in his philosophy, and an introduction to the role played by this problematic in the development of his philosophy of difference. One (...) of the points of departure that I will take from the history of mathematics is the theme of ‘power series’ (Deleuze 1994, 114), which will involve a detailed elaboration of the mechanism by means of which power series operate in the differential calculus deployed by Deleuze in Difference and Repetition. Deleuze actually constructs an alternative history of mathematics that establishes a historical continuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus. It is in relation to this differential point of view that Deleuze determines a differential logic which he deploys, in the form of a logic of different/ciation, in the development of his project of constructing a philosophy of difference. (shrink)

In the chapter of Difference and Repetition entitled ‘Ideas and the synthesis of difference,’ Deleuze mobilizes mathematics to develop a ‘calculus of problems’ that is based on the mathematical philosophy of Albert Lautman. Deleuze explicates this process by referring to the operation of certain conceptual couples in the field of contemporary mathematics: most notably the continuous and the discontinuous, the infinite and the finite, and the global and the local. The two mathematical theories that Deleuze draws upon for this purpose (...) are the differential calculus and the theory of dynamical systems, and Galois’ theory of polynomial equations. For the purposes of this paper I will only treat the first of these, which is based on the idea that the singularities of vector fields determine the local trajectories of solution curves, or their ‘topological behaviour’. These singularities can be described in terms of the given mathematical problematic, that is for example, how to solve two divergent series in the same field, and in terms of the solutions, as the trajectories of the solution curves to the problem. What actually counts as a solution to a problem is determined by the specific characteristics of the problem itself, typically by the singularities of this problem and the way in which they are distributed in a system. Deleuze understands the differential calculus essentially as a ‘calculus of problems’, and the theory of dynamical systems as the qualitative and topological theory of problems, which, when connected together, are determinative of the complex logic of different/ciation. (DR 209). Deleuze develops the concept of a problematic idea from the differential calculus, and following Lautman considers the concept of genesis in mathematics to ‘play the role of model ... with respect to all other domains of incarnation’. While Lautman explicated the philosophical logic of the actualization of ideas within the framework of mathematics, Deleuze (along with Guattari) follows Lautman’s suggestion and explicates the operation of this logic within the framework of a multiplicity of domains, including for example philosophy, science and art in What is Philosophy?, and the variety of domains which characterise the plateaus in A Thousand Plateaus. While for Lautman, a mathematical problem is resolved by the development of a new mathematical theory, for Deleuze, it is the construction of a concept that offers a solution to a philosophical problem; even if this newly constructed concept is characteristic of, or modelled on the new mathematical theory. (shrink)

To make sense of what Gilles Deleuze understands by a mathematical concept requires unpacking what he considers to be the conceptualizable character of a mathematical theory. For Deleuze, the mathematical problems to which theories are solutions retain their relevance to the theories not only as the conditions that govern their development, but also insofar as they can contribute to determining the conceptualizable character of those theories. Deleuze presents two examples of mathematical problems that operate in this way, which he considers (...) to be characteristic of a more general theory of mathematical problems. By providing an account of the historical development of this more general theory, which he traces drawing upon the work of Weierstrass, Poincaré, Riemann, and Weyl, and of its significance to the work of Deleuze, an account of what a mathematical concept is for Deleuze will be developed. (shrink)

Much has been made of Deleuze’s Neo-Leibnizianism,3 however not very much detailed work has been done on the specific nature of Deleuze’s critique of Leibniz that positions his work within the broader framework of Deleuze’s own philo- sophical project. The present chapter undertakes to redress this oversight by providing an account of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold. Deleuze provides a systematic account of the structure of Leibniz’s metaphys- ics in terms of its mathematical underpinnings. (...) However, in doing so, Deleuze draws upon not only the mathematics developed by Leibniz – including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus – but also the developments in mathematics made by a number of Leibniz’s contemporaries – including Newton’s method of fluxions – and a number of subsequent developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work – including the theory of functions and singularities, the theory of continuity and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the theory of continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this chapter is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

The aim of this chapter is to test the degree to which Deleuze’s philosophy can be reconciled with the subject naturalist approach to pragmatism put forward by Macarthur and Price.

The theme of the conflict between the different interpretations of Spinoza’s philosophy in French scholarship, introduced by Christopher Norris in this volume and expanded on by Alain Badiou, is also central to the argument presented in this chapter. Indeed, this chapter will be preoccupied with distinguishing the interpretations of Spinoza by two of the figures introduced by Badiou. The interpretation of Spinoza offered by Gilles Deleuze in Expressionism in Philosophy provides an account of the dynamic changes or transformations of the (...) characteristic relations of a Spinozist finite existing mode, or human being. This account has been criticized more or less explicitly by a number of commentators, including Charles Ramond. Rather than providing a defence of Deleuze on this specific point, which I have done elsewhere, what I propose to do in this chapter is provide an account of the role played by “joyful passive a affections” in these dynamic changes or transformations by distinguishing Deleuze’s account of this role from that offered by one of his more explicit critics on this issue, Pierre Macherey. An appreciation of the role played by “joyful passive affections” in this context is crucial to understanding how Deleuze’s interpretation of Spinoza is implicated in his broader philosophical project of constructing a philosophy of difference. The outcome is a position that, like Badiou in the previous chapter, rules out “intellect in potentiality” but maintains a role for the joyful passive affects in the development of adequate ideas. (shrink)

The collection Virtual Mathematics: the logic of difference brings together a range of new philosophical engagements with mathematics, using the work of French philosopher Gilles Deleuze as its focus. Deleuze’s engagements with mathematics rely upon the construction of alternative lineages in the history of mathematics in order to reconfigure particular philosophical problems and to develop new concepts. These alternative conceptual histories also challenge some of the self-imposed limits of the discipline of mathematics, and suggest the possibility of forging new connections (...) between philosophy and more recent developments in mathematics. This component of Deleuze’s work has, to date, been rather neglected by commentators working in the field of Deleuze studies. One of the aims of this collection is to address this critical deficit by providing a philosophical presentation of Deleuze’s relation to mathematics; one that is adequate to his project of constructing a philosophy of difference and to the exploration of some of its applications. This project developed as a result of encounters with an increasing number of researchers who have been working on the mathematical aspects of Deleuze’s work and the key role that these play in his philosophy. By bringing this work together for the first time, this collection makes an important contribution to the emerging body of work that endeavours to explore the broad range of Deleuze’s philosophy. (shrink)

A renaissance in Spinoza studies took place in France at the end of the 1960s, which gave new impetus to the study of Spinoza’s work and continues to have a marked effect on the direction of research in the field today. The effect of this renewed interest and direction did not remain isolated to France but quickly spread across the continent. Although certain of the figures involved in this event have become rather well known in some academic circles, and their (...) work widely read, the details of these developments and the specific texts that contributed to and sustained this new direction in research have remained largely unknown in the English-speaking world. The aim of this essay, therefore, is to provide a survey of this event, to review the background to these developments, to introduce the main protagonists – along with some of the lesser known but equally important figures – and to single out and asses the key texts, their specific focus and their contribution to the new direction in research. (shrink)

Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes on to characterize this (...) reality as an ‘ideal reality’ that ‘governs’ the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by a what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems. (shrink)

Rather than defend each instance of problem suggested by Williams, what I propose to do is to respond by making two clarificatory points: first, I rule out two ways of understanding mathematical problems that might be clouding the water; and then, second, I further characterise how Deleuze thinks some mathematical problems, two in particular, are not just examples of mathematical problems, but provide mathematical models for what a mathematical problem in general can be understood to be. This is important because (...) it is how mathematical problems can be modelled in mathematics that provides Deleuze with a model for how problems can be modelled in other discourses. This will then get to the bottom of what is at stake in my approach to Deleuze and to Deleuze studies in general. (shrink)

In this paper, two different ways of thinking about individuality in Spinoza are presented to draw out what is at stake in trying to make sense of what could be described as a double point of view of the degree of the power to act of a singular thing in Spinoza’s Ethics: 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, notably the different interpretations of Spinoza’s theory of relations offered by Pierre Macherey and Gilles Deleuze. Deleuze’s interpretation of the dynamic changes in an individual’s power to act diverges somewhat from that of Macherey. For Macherey, dynamic changes are incorporated by an individual according to the varying degree, or proportion, to which the active expression of its fixed power to act are inhibited or limited. Whereas for Deleuze, an individual’s power to act is open to “metaphysical” or ontological changes. An individual for Deleuze is limited 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 integration, and, therefore, the further development of its power to act, and by consequence, its actual existence. This limit determines the margin of variation, or proportion, of the expression of the given individual’s power to act, which varies from a minimum, below which it would cease to exist (intensity = 0), to a maximum, which would only be limited by the extent to which its power to act is further integrated at any given moment in more composite relations, expressing the affective life of the individual. The paper will examine the implications of this difference to their respective interpretations of Spinoza, with a view to characterising the role that Spinoza plays in Deleuze’s broader project of constructing a philosophy of difference. (shrink)

Gilles Deleuze maintains that an individual’s power to act is open to “metaphysical” or ontological changes. An individual for Deleuze is limited 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 integration, and, therefore, the further development of its power to act, and by consequence, its actual existence. This limit determines the margin of variation of the expression of the given individual’s power (...) to act, which varies from a minimum, below which it would cease to exist (intensity = 0), to a maximum, which would only be limited by the extent to which its power to act is further integrated at any given moment in more composite relations, expressing the affective life of the individual. The paper will examine the implications of this interpretations of Spinoza, with a view to characterising the role that Spinoza plays in Deleuze’s broader project of constructing a philosophy of difference. (shrink)

Michael Hunter, The Boyle Papers: Understanding the Manuscripts of Robert Boyle. With contributions by Edward B. Davis, Harriet Knight, Charles Littleton and Lawrence M. Principe. Aldershot, England; Burlington, VT: Ashgate, 2007. Pp. xiii + 674. US$139.95/£70.00 HB. -/- The publication by Michael Hunter of this revised edition of the catalogue of the Boyle Papers contributes admirably to the renaissance in Boyle studies which has taken place over the past decade and a half. Robert Boyle (1627–91), arguably the most influential British (...) scientist of the late seventeenth century, was a pioneering experimenter, profound thinker, and figure-head of the new science in its early years of development. This volume brings together the materials necessary for understanding the Boyle archive, one of the most important archives from this period, which has been at the Royal Society since 1769. (shrink)

Michiel Wielema: The March of the Libertines. Spinozists and the Dutch Reformed Church (1660–1750). ReLiC: Studies in Dutch Religious History. Hilversum: Uitgeverij Verloren, 2004; pp. 221. The Dutch Republic of the seventeenth century is famous for having cultivated an extraordinary climate of toleration and religious pluralism — the Union of Utrecht supported religious freedom, or “freedom of conscience”, and expressly forbade reli- gious inquisition. However, despite membership in the state sponsored Calvinist Dutch Reformed Church not being compulsory, the freedom to (...) gather and worship, or “to air anti-Christian or atheistic opinions” was little tolerated “within” the organized structure of the church, which functioned more as “an exclusive organisation for those willing to submit freely to certain confessional canons and to the disciplinary author- ity of the church’s governing bodies” (10): the consistories, classes, and synods. Those not prepared to submit to Reformed doctrine were free to leave the church without fear of any legal or political repercussions. However, for those not prepared to leave for reasons of personal belief, matters turned out to be quite different. Because the Reformed Church enjoyed full State protection, matters of doctrinal conflict could well evolve into political affairs. And, contrary to the Union of Utrecht, religious inquisition was in some cases actually applied with political approval for “heretics” within the Reformed Church. The main focus of The March of the Libertines is an investigation of this obvious tension. (shrink)