In October 1984, Bruno Huisman stated with regards to Jean Cavaillès, ‘Let us be honest, or at least realistic: today, one can be a professor of philosophy without ever having read a single line of Cavaillès. Often invoked, sometimes quoted, the oeuvre of Cavaillès is little attended for itself’ (Huisman 1984). As for Albert Lautman, it would seem that the situation is even more extreme. In 1994, the publisher Hermann, under the impetus of Bruno Huisman and George Canguilhem, collected almost (...) the totality of the Jean Cavaillès papers in one volume (Oeuvres complètes de philosophie des sciences (Cavaillès 1994)). But, the Essai sur l’unité des mathématiques et divers écrits (Lautman 1977), published by the Union générale d’Éditions in 1977, had all but disappeared by the early 1980s and yet was never republished! This will remain one of the great indignities of French publishing, for as Jean Petitot rightly affirms: ‘Regarded as too speculative, in spite of his exceptional mathematical scholarship and his close connection with Hilbertian axiomatic structuralism, his mathematical philosophy has, until now, been devoid of any particular attention …. We would like to state clearly from the start, Albert Lautman represents, in our view, without exaggeration, one of the most inspired philosophers of this century’ (Petitot 1987, 79-80). (shrink)
God Owes Us Nothing reflects on the centuries-long debate in Christianity: how do we reconcile the existence of evil in the world with the goodness of an omnipotent God, and how does God's omnipotence relate to people's responsibility for their own salvation or damnation. Leszek Kolakowski approaches this paradox as both an exercise in theology and in revisionist Christian history based on philosophical analysis. Kolakowski's unorthodox interpretation of the history of modern Christianity provokes renewed discussion about the historical, intellectual, and (...) cultural omnipotence of neo-Augustinianism. "Several books a year wrestle with that hoary conundrum, but few so dazzlingly as the Polish philosopher's latest."--Carlin Romano, Washington Post Book World "Kolakowski's fascinating book and its debatable thesis raise intriguing historical and theological questions well worth pursuing."--Stephen J. Duffy, Theological Studies "Kolakowski's elegant meditation is a masterpiece of cultural and religious criticism."--Henry Carrigan, Cleveland Plain Dealer. (shrink)
_God Owes Us Nothing_ reflects on the centuries-long debate in Christianity: how do we reconcile the existence of evil in the world with the goodness of an omnipotent God, and how does God's omnipotence relate to people's responsibility for their own salvation or damnation. Leszek Kolakowski approaches this paradox as both an exercise in theology and in revisionist Christian history based on philosophical analysis. Kolakowski's unorthodox interpretation of the history of modern Christianity provokes renewed discussion about the historical, intellectual, and (...) cultural omnipotence of neo-Augustinianism. "Several books a year wrestle with that hoary conundrum, but few so dazzlingly as the Polish philosopher's latest."—Carlin Romano, _Washington Post Book World_ "Kolakowski's fascinating book and its debatable thesis raise intriguing historical and theological questions well worth pursuing."—Stephen J. Duffy, _Theological Studies_ "Kolakowski's elegant meditation is a masterpiece of cultural and religious criticism."—Henry Carrigan, _Cleveland Plain Dealer_. (shrink)
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)
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 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)
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)
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)
Across cultures, people employ space to construct representations of time. English exhibits two deictic space–time metaphors: the “moving ego” metaphor conceptualizes the ego as moving forward through time and the “moving time” metaphor conceptualizes time as moving forward towards the ego. Earlier research investigating the psychological reality of these metaphors has shown that engaging in certain types of spatial-motion thinking may influence how people reason about events in time. More recently, research has shown that people’s interactions with cultural artifacts may (...) also influence their representations of time. Extending research on space–time mappings in new directions, three experiments investigated the role of cultural artifacts, namely calendars and clocks, in the interpretation of metaphorical expressions about time. Taken together, the results provide initial evidence that, in their interpretation of ambiguous metaphorical expressions about time, people automatically access and use spatial representations of absolute time, whereby moving forward in space corresponds with moving later in time. Moreover, asking participants to use a reverse space–time mapping causes interference, which is reflected through their temporal reasoning. (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)
The author uses census data to assess the consequences of two alternative theoretical formulations of care work for understanding the intersections of gender, race, and economic inequalities in paid care. The nurturance conceptualization focuses on care as relationship while the reproductive labor framework includes both relational and nonrelational jobs that maintain and reproduce the labor force. An empirical application of both models to the labor market shows that placing increasing theoretical emphasis on nurturant care privileges the experiences of white women (...) and excludes large numbers of very-low-wage workers from consideration. (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)
To understand H.L.A. Hart's general theory of law, it is helpful to distinguish between substantive and methodological legal positivism. Substantive legal positivism is the view that there is no necessary connection between morality and the content of law. Methodological legal positivism is the view that legal theory can and should offer a normatively neutral description of a particular social phenomenon, namely law. Methodological positivism holds, we might say, not that there is no necessary connection between morality and law, but rather (...) that there is no connection, necessary or otherwise, between morality and legal theory. The respective claims of substantive and methodological positivism are, at least on the surface, logically independent. Hobbes and Bentham employed normative methodologies to defend versions of substantive positivism, and in modern times Michael Moore has developed what can be regarded as a variant of methodological positivism to defend a theory of natural law. (shrink)
Situating narrative: philosophical and theological context -- Ethical being: the storied self as moral agent -- Reconciled being: narrative and pardon -- Pedagogies of pardon in praxis -- Towards a narrative pedagogy of reconciliation -- Ricoeur's legacy: A Praxis of Peace.
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.
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)
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)
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)
This essay deals with property rights in body parts that can be exchanged in a market. The inquiry arises in the following context. With some exceptions, the laws of many countries permit only the donation, not the sale, of body parts. Yet for some years there has existed a shortage of body parts for transplantation and other medical uses. It might then appear that if more sales were legally permitted, the supply of body parts would increase, because people would have (...) more incentive to sell than they currently have to donate. To allow sales is to recognize property rights in body parts. To allow sales, however, makes body parts into “commodities”—that is, things that can be bought and sold in a market. And some view it as morally objectionable to treat body parts as commodities. (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)
Language is an imperfect and coarse means of communicating information about a complex and nuanced world. We report on an experiment designed to capture this feature of communication. The messages available to the sender imperfectly describe the state of the world; however, the sender can improve communication, at a cost, by increasing the complexity or elaborateness of the message. Here the sender learns the state of the world, then sends a message to the receiver. The receiver observes the message and (...) provides a best guess about the state. The incentives of the players are aligned in the sense that both sender and receiver are paid an amount which is increasing in the accuracy of the receiver’s guess. We find that the size of the language endogenously emerges as a function of the costs of communication. Specifically, we find that higher communication costs are associated with a smaller language. Although the equilibrium predictions do not perform well, this divergence occurs in a manner which is consistent with the experimental communication literature: overcommunication. We find that the sender’s payoffs, relative to equilibrium payoffs, are decreasing in the cost of communication. We also find that the receiver’s payoffs, relative to equilibrium payoffs, are increasing in the cost of communication. Finally, we find imperfections in coordination on the basis of the experimental labels. (shrink)
[Stephen Makin] Aristotle draws two sets of distinctions in Metaphysics 9.2, first between non-rational and rational capacities, and second between one way and two way capacities. He then argues for three claims: [A] if a capacity is rational, then it is a two way capacity [B] if a capacity is non-rational, then it is a one way capacity [C] a two way capacity is not indifferently related to the opposed outcomes to which it can give rise I provide explanations (...) of Aristotle's terminology, and of how [A]-[C] should be understood. I then offer a set of arguments which are intended to show that the Aristotelian claims are plausible. \\\ [Nicholas Denyer] In De Caelo 1: 11-12 Aristotle argued that whatever is and always will be true is necessarily true. His argument works, once we grant him the highly plausible principle that if something is true, then it can be false if and only if it can come to be false. For example, assume it true that the sun is and always will be hot. No proposition of this form can ever come to be false. Hence this proposition cannot be false. Hence it is necessarily true, and so too is anything that follows from it. In particular, it is necessarily true that the sun is hot. Moreover, if the sun not only is and always will be hot, but also always has been, then it follows by similar reasoning that the sun not only cannot now fail to be hot, but also never could have failed. Anything everlastingly true is therefore, in the strictest sense of the term, necessarily true. (shrink)
In recent years there has been a great deal of discussion about the prospects of developing a ‘naturalized epistemology’, though different authors tend to interpret this label in quite different ways. One goal of this paper is to sketch three projects that might lay claim to the ‘naturalized epistemology’ label, and to argue that they are not all equally attractive. Indeed, I'll maintain that the first of the three—the one I'll attribute to Quine—is simply incoherent. There is no way we (...) could get what we want from an epistemological theory by pursuing the project Quine proposes. The second project on my list is a naturalized version of reliabilism. This project is not fatally flawed in the way that Quine's is. However, it's my contention that the sort of theory this project would yield is much less interesting than might at first be thought. (shrink)
The idea for the Innovative Conservatoire was first proposed at the Reflective Conservatoire Conference in 2006. An international collaboration which stimulates knowledge exchange, innovation and reflective practice in conservatoires, ICON has opened up an area of work that is often carried out behind closed doors. Working via creative methods in an experimental, artistically orientated and safe environment, it has enabled both scrutiny and development of established pedagogies. It has become a beacon of innovative thinking and practice in specialist professional development (...) for conservatoire teachers and leaders. This essay will draw on the author’s experiences as an ICON participant and Creative Director and interviews with a number of senior conservatoire personnel. It will reflect on future directions for ICON and for professional and leadership development in conservatoires. (shrink)
Perhaps the most salient feature of Rawls's theory of justice which at once attracts supporters and repels critics is its apparent egalitarian conclusion as to how economic goods are to be distributed. Indeed, many of Rawls's sympathizers may find this result intuitively appealing, and regard it as Rawls's enduring contribution to the topic of economic justice, despite technical deficiencies in Rawls's contractarian, decision-theoretic argument for it which occupy the bulk of the critical literature. Rawls himself, having proposed a “coherence” theory (...) of justification in metaethics, must regard the claim that his distributive criterion “is a strongly egalitarian conception” as independently a part of the overarching moral argument. The alleged egalitarian impact of Rawls's theory is crucial again in normative ethics where Rawls is thought to have developed a major counter-theory to utilitarianism, one of the most popular criticisms of which has been its alleged inadequacy in handling questions of distributive justice. Utilitarians can argue, however, as Brandt recently has, that the diminishing marginal utility of money, along with ignorance of income-welfare curves, would require a utility-maximizing distribution to be substantially egalitarian. The challenge is therefore for Rawls to show that his theory yields an ethically preferable degree of equality. (shrink)
Man and machine are rife with fundamental differences. Formal research in artificial intelligence and robotics has for half a century aimed to cross this divide, whether from the perspective of understanding man by building models, or building machines which could be as intelligent and versatile as humans. Inevitably, our sources of inspiration come from what exists around us, but to what extent should a machine's conception be sourced from such biological references as ourselves? Machines designed to be capable of explicit (...) social interaction with people necessitates employing the human frame of reference to a certain extent. However, there is also a fear that once this man-machine boundary is crossed that machines will cause the extinction of mankind. The following paper briefly discusses a number of fundamental distinctions between humans and machines in the field of social robotics, and situating these issues with a view to understanding how to address them. (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 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)
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)
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)
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)
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)
Between the later views of Wittgenstein and those of connectionism 1 on the subject of the mastery of language there is an impressively large number of similarities. The task of establishing this claim is carried out in the second section of this paper.
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)