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)

In the paper “Math Anxiety,” Aden Evens explores the manner by means of which concepts are implicated in the problematic Idea according to the philosophy of Gilles Deleuze. The example that Evens draws from Difference and Repetition in order to demonstrate this relation is a mathematics problem, the elements of which are the differentials of the differential calculus. What I would like to offer in the present paper is an historical account of the mathematical problematic that Deleuze deploys in his (...) philosophy, and an introduction to the role that this problematic plays in the develop- ment of his philosophy of difference. One of the points of departure that I will take from the Evens paper is the theme of “power series.”2 This 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 an historical conti- nuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus. It is in relation to the differential point of view of the infinitesimal calculus that Deleuze determines a differential logic which he deploys, in the form of a logic of different/ciation, in the development of his proj- ect of constructing a philosophy of difference. (shrink)

I would like to raise a pressing issue relating to the World Health Organization which should concern the medical profession throughout the UK. The WHO Charter advocates the provision of health to all—as a human right. It is therefore to be regretted that these Charter obligations have not been exercised with respect to Taiwan whose 23 million citizens still cannot benefit from its protection. This democratic country which is a beacon of human rights in Asia, is still excluded from the (...) WHO. It is expected that in 2005 the island will …. (shrink)

According to the reading of Spinoza that Gilles Deleuze presents in Expressionism in Philosophy: Spinoza, Spinoza's philosophy should not be represented as a moment that can be simply subsumed and sublated within the dialectical progression of the history of philosophy, as it is figured by Hegel in the Science of Logic, but rather should be considered as providing an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. 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. Deleuze presents Spinoza's metaphysics as determined according to a 'logic of expression', which, insofar as it contributes to the determination of a philosophy of difference, functions as an alternative to the Hegelian dialectical logic. Deleuze's project in Expressionism in Philosophy is therefore to redeploy Spinoza in order to mobilize his philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic. (shrink)

Corporations in the United States have been starting ethics programs for a variety of reasons both active and passive. Ethics officers are being charged with improving both company image and the level of ethical decision-making by employees. Thirty ethics officers from Fortune 500 firms were surveyed to develop a database of their duties and the companies' commitment to ethical standards. The results suggest much is being done, both in the diversity of responses and the similarities of commitment and duties.

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)

Of all twentieth century philosophers, it is Gilles Deleuze whose work agitates most forcefully for a worldview privileging becoming over being, difference over sameness; the world as a complex, open set of multiplicities. Nevertheless, Deleuze remains singular in enlisting mathematical resources to underpin and inform such a position, refusing the hackneyed opposition between ‘static’ mathematical logic versus ‘dynamic’ physical world. This is an international collection of work commissioned from foremost philosophers, mathematicians and philosophers of science, to address the wide range (...) of problematics and influences in this most important strand of Deleuze’s thinking. Contributors are Charles Alunni, Alain Badiou, Gilles Châtelet, Manuel DeLanda, Simon Duffy, Robin Durie, Aden Evens, Arkady Plotnitsky, Jean-Michel Salanskis, Daniel Smith and David Webb. (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon 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 Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic 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. This essay 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)

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)

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)

Maimon’s theory of the differential has proved to be a rather enigmatic aspect of his philosophy. By drawing upon mathematical developments that had occurred earlier in the century and that, by virtue of the arguments presented in the Essay and comments elsewhere in his writing, I suggest Maimon would have been aware of, what I propose to offer in this paper is a study of the differential and the role that it plays in the Essay on Transcendental Philosophy (1790). In (...) order to do so, this paper focuses upon Maimon’s criticism of the role played by mathematics in Kant’s philosophy, to which Maimon offers a Leibnizian solution based on the infinitesimal calculus. The main difficulties that Maimon has with Kant’s system, the second of which will be the focus of this paper, include the presumption of the existence of synthetic a priori judgments, i.e. the question quid facti, and the question of whether the fact of our use of a priori concepts in experience is justified, i.e. the question quid juris. Maimon deploys mathematics, specifically arithmetic, against Kant to show how it is possible to understand objects as having been constituted by the very relations between them, and he proposes an alternative solution to the question quid juris, which relies on the concept of the differential. However, despite these arguments, Maimon remains sceptical with respect to the question quid facti. (shrink)

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.

In everyday discourse and in the context of social scientific research we often attribute intentional states to groups. Contemporary approaches to group intentionality have either dismissed these attributions as metaphorical or provided an analysis of our attributions in terms of the intentional states of individuals in the group.Insection1, the author argues that these approaches are problematic. In sections 2 and 3, the author defends the view that certain groups are literally intentional agents. In section 4, the author argues that there (...) are significant reasons for social scientists and philosophers of social science to acknowledge the adequacy of macro-level explanations that involve the attribution of intentional states to groups. In section 5, the author considers and responds to some criticisms of the thesis she defends. (shrink)

The Well-Ordered Universe argues that Cavendish's natural philosophy, social and political philosophy, and medical theory share an underlying concern with order. This reveals interesting connections among Cavendish's natural philosophy and her views on gender, animals and the environment, and human health, and explains her commitment to monarchy and social hierarchy.

As regimes move from illiberal to liberal, post-transition justice methodology has been employed to engender truth and reconciliation. These normative concepts have evolved into a policy of creating truth and reconciliation commissions that trade civil and criminal amnesty with applicants in exchange for information. This bargained-for exchange can be analyzed as an imperfect information game, where the commission attempts to maximize information while the applicant seeks amnesty for the lowest possible price. Using game-theoretic analysis, the authors model the truth-amnesty game (...) and predict the optimal commission strategy. The analysis leads to the recommendation that future transitional justice commissions employ various specific lexicographic ordering strategies to minimize dead weight loss in the transaction. (shrink)

What I aim to develop in this paper is a secular foundation to the concept of reincarnation that is consistent with the different ways in which this concept is understood across a number of Buddhist traditions, drawing in particular upon the doctrinal understanding of reincarnation in the Mahāyāna or Madhyamaka tradition as presented in the work of Śāntideva and Nāgārjuna.

In the introduction to Women in Philosophy: What Needs to Change?, editors, Fiona Jenkins and Katrina Hutchison, note that women in many fields of study feel frustrated, hurt, or merely annoyed at some of their experiences in academia. However, they also note something unusual about these feelings when it comes to philosophy: the feelings have given way “to careful reflection on how to make sense of such experience, how to find an articulation of its form, structure, causes, and potential remedies” (...) . That is the beauty and power of this book. The essays in this collection are not merely stories of women philosophers feeling mistreated , nor are they an angry call to arms for a feminist revolution. The women writing in this collection are careful to look at both data and the stories of women in philosophy to try to find the cause or causes of a problem that is already well-recognized. Moreover, a central question for almost all of the women .. (shrink)

In English, two deictic space-time metaphors are in common usage: the Moving Ego metaphor conceptualizes the ego as moving forward through time and the Moving Time metaphor conceptualizes time as moving forward toward the ego . Although earlier research investigating the psychological reality of these metaphors has typically examined spatial influences on temporal reasoning , recent lines of research have extended beyond this, providing initial evidence that personality differences and emotional experiences may also influence how people reason about events in (...) time . In this article, we investigate whether these relationships have force in real life. Building on the effects of individual differences in self-reported conscientiousness and procrastination found by Duffy and Feist , we examined whether, in addition to self-reported conscientiousness and procrastination, there is a relationship between conscientious and procrastinating behaviors and temporal perspective. We found that participants who adopted the Moving Time perspective were more likely to exhibit conscientious behaviors, while those who adopted the Moving Ego perspective were more likely to procrastinate, suggesting that the earlier effects reach beyond the laboratory. (shrink)

This article examines the seventeenth-century debate between the Dutch philosopher Benedict de Spinoza and the British scientist Robert Boyle, with a view to explicating what the twentieth-century French philosopher Gilles Deleuze considers to be the difference between science and philosophy. The two main themes that are usually drawn from the correspondence of Boyle and Spinoza, and used to polarize the exchange, are the different views on scientific methodology and on the nature of matter that are attributed to each correspondent. Commentators (...) have tended to focus on one or the other of these themes in order to champion either Boyle or Spinoza in their assessment of the exchange. This paper draws upon the resources made available by Gilles Deleuze and Felix Guattari in their major work What is Philosophy?, in order to offer a more balanced account of the exchange, which in its turn contributes to our understanding of Deleuze and Guattari’s conception of the difference between science and philosophy. (shrink)

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is (...) to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization. (shrink)

This book examines a widespread, and often misunderstood, doctrine within the medieval Aristotelian tradition, namely the inclusion of Aristotle's Rhetoric and Poetics within the scope of the Organon. It studies this doctrine, as presented by the Islamic philosophers Al- Fārābī, Avicenna, and Averroes, from a purely philosophical perspective, and argues that the logical construal of the arts of rhetoric and poetics is both interesting and illuminating. The book begins by examining some prevalent misconceptions regarding the logical interpretation of the Rhetoric (...) and Poetics. Chapter two considers the Greek background of the doctrine, first through an examination of the Aristotelian divisions of the sciences, and then through an examination of the beginnings of the logical classification of the Rhetoric and Poetics among the Greek commentators from the school of Alexandria. The remainder of the work is devoted to a detailed consideration of the Arabic philosophers' development of the doctrine, both their understanding of its general epistemological and logical underpinnings, and their elaboration of the specific logical structures upon which poetical and rhetorical discourse is based. Consideration is also given to the relationship between contemporary philosophical views of rhetoric and poetics, and the views of these medieval authors. (shrink)

Floridi and Sanders, seminal work, “On the morality of artificial agents” has catalyzed attention around the moral status of computer systems that perform tasks for humans, effectively acting as “artificial agents.” Floridi and Sanders argue that the class of entities considered moral agents can be expanded to include computers if we adopt the appropriate level of abstraction. In this paper we argue that the move to distinguish levels of abstraction is far from decisive on this issue. We also argue that (...) adopting certain levels of abstraction out of context can be dangerous when the level of abstraction obscures the humans who constitute computer systems. We arrive at this critique of Floridi and Sanders by examining the debate over the moral status of computer systems using the notion of interpretive flexibility. We frame the debate as a struggle over the meaning and significance of computer systems that behave independently, and not as a debate about the ‘true’ status of autonomous systems. Our analysis leads to the conclusion that while levels of abstraction are useful for particular purposes, when it comes to agency and responsibility, computer systems should be conceptualized and identified in ways that keep them tethered to the humans who create and deploy them. (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)

The concept of agency as applied to technological artifacts has become an object of heated debate in the context of AI research because some AI researchers ascribe to programs the type of agency traditionally associated with humans. Confusion about agency is at the root of misconceptions about the possibilities for future AI. We introduce the concept of a triadic agency that includes the causal agency of artifacts and the intentional agency of humans to better describe what happens in AI as (...) it functions in real-world contexts. We use the VW emission fraud case to explain triadic agency since in this case a technological artifact, namely software, was an essential part of the wrongdoing and the software might be said to have agency in the wrongdoing. We then extend the case to include futuristic AI, imagining AI that becomes more and more autonomous. (shrink)

In Hegel ou Spinoza,1 Pierre Macherey 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. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be 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. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising 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. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

Theodor W. Adorno and Jürgen Habermas both champion the goal of a rational society. However, they differ significantly about what this society should look like and how best to achieve it. Exploring the premises shared by both critical theorists, along with their profound disagreements about social conditions today, this book defends Adorno against Habermas' influential criticisms of his account of Western society and prospects for achieving reasonable conditions of human life. The book begins with an overview of these critical theories (...) of Western society. Both Adorno and Habermas follow Georg Lukács when they argue that domination consists in the reifying extension of a calculating, rationalizing form of thought to all areas of human life. Their views about reification are discussed in the second chapter. In chapter three the author explores their conflicting accounts of the historical emergence and development of the type of rationality now prevalent in the West. Since Adorno and Habermas claim to have a critical purchase on reified social life, the critical leverage of their theories is assessed in chapter four. The final chapter deals with their opposing views about what a rational society would look like, as well as their claims about the prospects for establishing such a society. Adorno, Habermas and the Search for a Rational Society will be essential reading for students and researchers of critical theory, political theory and the work of Adorno and Habermas. (shrink)

The concept of reproductive labor is central to an analysis of gender inequality, including understanding the devaluation of cleaning, cooking, child care, and other “women's work” in the paid labor force. This article presents historical census data that detail transformations of paid reproductive labor during the twentieth century. Changes in the organization of cooking and cleaning tasks in the paid labor market have led to shifts in the demographics of workers engaged in these tasks. As the context for cleaning and (...) cooking work shifted from the dominance of private household servants to include more institutional forms, the gender balance of this reproductive labor workforce has been transformed, while racial-ethnic hierarchies have remained entrenched. This article highlights the challenges to understanding occupational segregation and the devaluation of reproductive labor in a way that analyzes gender and race-ethnicity in an intersectional way and integrates cultural and structural explanations of occupational degradation. (shrink)

Two studies tested the relationship between three facets of personality—conscientiousness, agreeableness, and openness to experience—as well as moral identity, on individuals’ ethical ideology. Study 1 showed that moral personality and the centralityof moral identity to the self were associated with a more principled (versus expedient) ethical ideology in a sample of female speech therapists. Study 2 replicated these findings in a sample of male and female college students, and showed that ideology mediated therelationship between personality, moral identity, and two organizationally (...) relevant outcomes: organizational citizenship behavior and the propensity to morally disengage. Implications for business ethics are discussed. (shrink)

Contemporary analytic epistemology exhibits an individualistic bias. The standard analyses of knowledge found in current epistemological discussions assume that the only epistemic agents worthy of philosophical consideration are individual cognizers. The idea that collectives could be genuine knowers has received little, if any, serious consideration. This individualistic bias seems to be motivated by the view that epistemology is about things that go on inside the head. In this paper I challenge this type of epistemic individualism by arguing that certain groups (...) are intentional agents and that cognition occurs at the level of groups. In section I, I extend a plausible and well-defended account of individual intentionality to organizations. In section II, I appeal to research on distributed cognition in order to challenge the view that cognition occurs exclusively in the individual mind-brain. Having established that groups can be believers and cognizers, I turn to the issue of the justification of group belief. In section III, consider whether a reliabilist account of justification can be extended to groups. (shrink)

Plato’s philosophy is important to Badiou for a number of reasons, chief among which is that Badiou considered Plato to have recognised that mathematics provides the only sound or adequate basis for ontology. The mathematical basis of ontology is central to Badiou’s philosophy, and his engagement with Plato is instrumental in determining how he positions his philosophy in relation to those approaches to the philosophy of mathematics that endorse an orthodox Platonic realism, i.e. the independent existence of a realm of (...) mathematical objects. The Platonism that Badiou makes claim to bears little resemblance to this orthodoxy. Like Plato, Badiou insists on the primacy of the eternal and immu- table abstraction of the mathematico-ontological Idea; however, Badiou’s reconstructed Platonism champions the mathematics of post-Cantorian set theory, which itself af rms the irreducible multiplicity of being. Badiou in this way recon gures the Platonic notion of the relation between the one and the multiple in terms of the multiple-without-one as represented in the axiom of the void or empty set. Rather than engage with the Plato that is gured in the ontological realism of the orthodox Platonic approach to the philosophy of mathematics, Badiou is intent on characterising the Plato that responds to the demands of a post-Cantorian set theory, and he considers Plato’s philosophy to provide a response to such a challenge. In effect, Badiou reorients mathematical Platonism from an epistemological to an ontological problematic, a move that relies on the plausibility of rejecting the empiricist ontology underlying orthodox mathematical Platonism. To draw a connec- tion between these two approaches to Platonism and to determine what sets them radically apart, this paper focuses on the use that they each make of model theory to further their respective arguments. (shrink)

The role of mathematics in the development of Gilles Deleuze's (1925-95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze's interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770-1831) presents in the Science of Logic . Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in his (...) treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz's account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908-44), which plays a significant role in Deleuze's project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense. (shrink)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. 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 paper 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)