The Mathematical Model of Smooth and Striated Spaces and the Connectivity Problem

Abstract

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: (1) the heterogeneity and homogeneity of spaces, and the local/global duality; (2) the connectivity of spaces and the intrinsic/extrinsic duality; and (3) 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.