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)

In recent years, the ideas of the mathematician Bernhard Riemann have come to the fore as one of Deleuze's principal sources of inspiration in regard to his engagements with mathematics, and the history of mathematics. Nevertheless, some relevant aspects and implications of Deleuze's philosophical reception and appropriation of Riemann's thought remain unexplored. In the first part of the paper I will begin by reconsidering the first explicit mention of Riemann in Deleuze's work, namely, in the second chapter of Bergsonism. In (...) this context, as I intend to show first, Deleuze's synthesis of some key features of the Riemannian theory of multiplicities is entirely dependent, both textually and conceptually, on his reading of another prominent figure in the history of mathematics: Hermann Weyl. This aspect has been largely underestimated, if not entirely neglected. However, as I attempt to bring out in the second part of the paper, reframing the understanding of Deleuze's philosophical engagement with Riemann's mathematics through the Riemann–Weyl conjunction can allow us to disclose some unexplored aspects of Deleuze's further elaboration of his theory of multiplicities and profound confrontation with contemporary science. This finally permits delineation of a correlation between Deleuze's plane of immanence and the contemporary physico-mathematical space of fundamental interactions. (shrink)

As a philosophical paradigm, differential heterogenesis offers us a novel descriptive vantage with which to inscribe Deleuze’s virtuality within the terrain of “differential becoming,” conjugating “pure saliences” so as to parse economies, microhistories, insurgencies, and epistemological evolutionary processes that can be conceived of independently from their representational form. Unlike Gestalt theory’s oppositional constructions, the advantage of this aperture is that it posits a dynamic context to both media and its analysis, rendering them functionally tractable and set in relation to other (...) objects, rather than as sedentary identities. Surveying the genealogy of differential heterogenesis with particular interest in the legacy of Lautman’s dialectic, I make the case for a reading of the Deleuzean virtual that departs from an event-oriented approach, galvanizing Sarti and Citti’s dynamic a priori vis-à-vis Deleuze’s philosophy of difference. Specifically, I posit differential heterogenesis as frame with which to examine our contemporaneous epistemic shift as it relates to multi-scalar computational modeling while paying particular attention to neuro-inferential modes of inductive learning and homologous cognitive architecture. Carving a bricolage between Mark Wilson’s work on the “greediness of scales” and Deleuze’s “scales of reality”, this project threads between static ecologies and active externalism vis-à-vis endocentric frames of reference and syntactical scaffolding. (shrink)