Simon B. Duffy

In the wake of the numerous translations of Badiou’s works that have appeared in recent years, including the translation of the second volume of his major work, Logic of Worlds: Being and Event II, there has been a marked increase in interest in the philo- sophical underpinnings of his oeuvre. The papers brought together in this volume provide a range of incisive and critical engagements with Badiou’s philosophical heritage and the philosophical prob- lems his work engages, both directly and indirectly. Our aim is to offer a balanced assessment of the ways in which Badiou responds to various philosophical thinkers, arguments and traditions, and of the arguments that he employs to do so.

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.

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.

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.

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.

Spinoza, according to common opinion, could only have written lamentable platitudes on sexual love, narrowly inspired by the prejudices of his time and without serious philosophical foundation: that for which, in the past, he has been congratulated,1 he is now reproached; or, at best, excused. He would even have, some believe to be able to add, increased the pervading puritanism: sexuality, as such, would give rise in him to a deep repulsion and women would horrify him. The second of these two assertions, if one sticks to the manifest content of the texts, actually rests on nothing; if one calls upon their latent content, it would require, to be established with a minimum of rigor, a study of which we will not dispute the theoretical possibility, but which, in fact, has not yet been undertaken. The first, on the other hand, has all the appearance of the obvious: that men love women for their beauty and do not support their attachment to someone else,2 that they desire them more the more admirers they have,3 that the jealousy of the male is exacerbated by the representation of the pudenda and of the excrementa of his rival,4 that sensual attachment is unstable and conflictual,5 that it often turns to obsession,6 that Adam loved Eve because of their similarity of nature,7 that he who remains insensitive to the generosity of a courtesan does not offend by ingratitude,8 that only free men and free women can marry one another and only if they want children,9 well, it seems, that isn’t anything sensational; now these eight passages, if the two definitions of the libido are added to them,10 are the only, if I’m not mistaken, that Spinoza expressly devoted to the question! He would therefore, apparently, only have drawn up a report of deficiency.

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.