The University of QueenslandCentre for the History of European Discourses

Abstracts

Arkady Plotnitsky. Manifolds: On the Concept of Space in Riemann and Deleuze

Bernhard Riemann is arguably the most significant mathematical influence upon and presence in Gilles Deleuze's work. This paper will explore the key reasons for this coming together of Deleuze's philosophy and Riemann's mathematics, and the significance of this conjunction for both philosophy and mathematics. My aim is not only to see how Riemann's ideas shape Deleuze's thought or why they were so significant for Deleuze, but also what Deleuze's philosophy tells us about Riemann's work and, through Riemann, mathematics in general.

My point of departure is a view of Riemann's mathematics as conceptual mathematics, as opposed to the set-theoretical mathematics that dominated the field in the wake of the work of Georg Cantor later in the nineteenth century. More recently, roughly since 1960, the so-called category theory may be seen as more dominant, and, as I shall explain, this approach is closer to that of Riemann, via such mathematical fields as topology and algebraic geometry, both developed in the wake of and influenced by Riemann's work. On the other hand, Deleuze sees the primary nature of philosophy in its invention of new concepts, a view which is underlined by a different understanding or concept of philosophical concept. This view is especially developed in his late book (with Felix Guattari) What is Philosophy?, but it is traceable to much earlier texts and defines most of his own philosophical work. Conjoining these two understandings of mathematics and philosophy, this article will explore the significance of Riemann's mathematical concepts and mathematics in general, to the conceptual architecture of Deleuze's key concepts and conversely, or reciprocally, what Deleuze's understanding of philosophy and his concepts themselves tell us about Riemann's mathematical concepts, and about the nature of mathematical conceptuality in general.

J-M Salanskis. Mathematics, Metaphysics, Philosophy

The intimate connexion between mathematics and philosophy is well known and well illustrated throughout the history of philosophy, from Plato to Husserl. There are nevertheless two reasons for no longer seeing, or, even worse, for deciding to neglect this connection nowadays. The first reason is that being an analytical philosopher, one believes more in the partnership of logic and philosophy than in the old Platonic pact; and the other one is that being a follower of Heidegger and phenomenology, one believes that mathematics hinders us from understanding and seeing what stands beyond the objective form. The aim of giving an impulse of new life to the relation between mathematics and philosophy today therefore means just as much as to be liberated from the alternative between a Heideggerian and an analytical mood. In this paper, I will describe in what way we can try to give such an impulse. I will argue that Deleuze thought that he could do so by proposing a general doctrine of what is as such a metaphysics in the classical sense, which would be directly expressed in mathematical rather than logical terms; and that we can also promote the couple of mathematics and philosophy in a more Kantian way, in reference to a renewed conception of the transcendental.

Daniel W. Smith. Axiomatics and Problematics as Two Modes of Formalisation: Deleuze's Epistemology of Mathematics

Throughout his work, Gilles Deleuze has made a distinction between two modes of formalization, in mathematics and elsewhere, which he terms, respectively, "axiomatics" and "problematics." This paper will examine the nature of this distinction, and the role it plays in Deleuze's philosophy. The axiomatic (or "theorematic") method of formalization is a familiar one, already having a long history in mathematics, philosophy, and logic, from Euclid 's geometry to Spinoza's philosophy to the formalized systems of modern symbolic logic. Problematics

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has an equally strong trajectory in the history of mathematics, but one that is slightly less visible, though it has recently become the object of analysis by a number of contemporary epistemologists (Canguilhem, Bouligand, Vuillemin, Lautman). The fundamental difference between these two modes of formalization can be seen in their differing methods of deduction: in axiomatics, a deduction moves from axioms to the theorems that are derived from it, whereas in problematics a deduction moves from the problem to the ideal accidents and events that condition the problem and form the cases that resolve it. More generally, Deleuze characterizes axiomatics as belonging to a "major" or royal form of science, which constantly attempts to effect a reduction or repression (or more accurately, an arithmetic conversion) of the problematic pole of mathematics, itself wedded to a "minor" or nomadic conception of science. The goal of the paper will be to analyze the epistemological and ontological mportance of the problematic/axiomatic and major/minor distinctions in Deleuze's work.

Simon Duffy. The mathematics of Deleuze's differential logic and metaphysics

This paper offers an historical account of one the mathematical problematics that Deleuze deploys in Difference and Repetition (1994), and an introduction to the role that this problematic plays in the development of Deleuze's philosophy of difference. The episode in the history of mathematics from which this mathematical problematic is extracted is the history of the calculus and its various (alternative) lines (or lineages) of development, which were only put on a rigorous algebraic foundation towards the end of the nineteenth century. Arguments constructed on the basis of developments in Set Theory in the 1960s, specifically the controversial Abraham Robinson axioms that determine the distinction between Standard and Non-Standard analysis and which allow the pre-foundational proofs of the calculus to be verified, allow for the reintroduction of the relationship between mathematical and metaphysical developments of the calculus that were marginalised, to say the least, as a result of the determination of its rigorous algebraic foundation. It will be argued that it is by means of the development of such an argument in Difference and Repetition that Deleuze determines a differential logic which is deployed, in the form of a logic of different/ciation, in the development of the logical schema of a theory of relations characteristic of a philosophy of difference. This logical schema provides one of the keys to understanding the relationship between Deleuze's philosophy of difference and the mathematical problematics with which it engages.

David Webb.

Abstract forthcoming

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