Deleuze’s philosophy is permeated with the problem of the continuum. The idea that the coexistence of durations is implied in the concept of duration itself allows Deleuze to offer a fresh perspective on multiplicity, which is distinct from Bergson’s approach, and which proposes new perspectives on the continuum. While Deleuze critiques Leibniz’s view on this concept by highlighting the non-uniform nature of the continuum, the infinitesimal still plays a significant role in his analysis. However, in his late reading of Leibniz, (...) Deleuze emphasises that folds, rather than infinitesimals, should be considered as the smallest components of the continuum’s labyrinth. This implies that there is a union of indiscernible cuts in the continuity, cuts that do not create gaps or breaks in the overall coherence, but rather a labyrinth. I will show how, in exploring this problem, Deleuze and Guattari draw inspiration from Kafka, in order to relate continuity to contiguity. This relation reveals an internal difference that defines the distinction between what is continuous and what is contiguous. This, in our view, marks a considerable shift between Deleuze’s early reading of Bergson and his late reading of Leibniz, and it allows Deleuze to further develop his idea of the continuum. (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)

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that quantity, in (...) the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. (shrink)