In this paper I present an interpretation of Deleuze's concept of the virtual. I argue that this concept is best understood in relation to the problematic of individuation or differentiation, which Deleuze inherits from Duns Scotus. After analysing Scotus' critique of Aristotelian or hylomorphic approaches to the problem of individuation, I turn to Deleuze's account of differentiation and his interpretation of the calculus in chapter 4 of Difference and Repetition. The paper seeks thereby to explicate Deleuze's dialectics or theory of (...) Ideas, his approach to the principle of sufficient reason, and the concept of the virtual itself. (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)

The mathematical model of smooth and striated spaces in Deleuze and Guattari's A Thousand Plateaus essentially rests on Riemann's manifold theory and Lautman's account of Riemannian spaces. In this paper I provide a detailed analysis of Lautman's account, arguing, however, that to discern its intricacies, and particularly what I shall call the connectivity problem, it is crucial to consider the fundamental work of Élie Cartan. This allows to address three key problems of Deleuze and Guattari's model: the heterogeneity and homogeneity (...) of spaces, and the local/global duality; the connectivity of spaces and the intrinsic/extrinsic duality; and the translatability of spaces and their concrete mixes. As a result, Lautman's and Cartan's philosophical-mathematical questioning proves decisive to Deleuze and Guattari's appropriation of Riemann's theory, and thus the mathematical model of smooth and striated spaces. (shrink)