Plotnitsky, Sets Badiou
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1 Introduction The mathematical grounding of Alain Badiou's philosophy is arguably the most dis- tinctive aspect of his work; it also distinguishes it from the work of other contemporary authors who have impacted recent theoretical discussions. It is true that both Jacques Lacan and Gilles Deleuze notably engage with mathematics, and they do offer some competition to Badiou in this regard. Badiou's use of mathematics is, however, more extensive and more foundational in view of the grounding roles of mathematical ontology and mathematical logic in his work. Badiou (2007, page 8) even contends that a rigorous philosophical ontology can only be mathematical, at least as ``a thesis ... about discourse'' (``mathematics ... pronounces what is expressible about being qua being''). Badiou (2006, pages 166 ^ 168) holds a similar view on rigorous philosophical logic: it could only be, or in any event should be, mathematical. (1) Two mathematical theories are especially important for Badiou: set theory, in establishing ontology, and category theory in establishing logic (or more specifically, topos theory, as grounded in category theory). Badiou argues that set and category theories are now the only ones that provide ``the apparatuses that claim to endow mathematics with its own unified space, or primordial languages'' (2006, page 105). Set theory, introduced by Georg Cantor in the late 19th century, grounds Badiou's earlier work, in particular Being and Event (2007 [1988]). This book develops a form of set-theoretical ontology, which is defined by its irreducibly multiple character ö that is, by the impossibility of finding a unity within which the multiple of this ontology could in principle be subsumed. Badiou speaks of this ontology as that of ``the multiple-without-One'' (eg, page 29; Badiou, 2006, page 35). Badiou's later Experimenting with ontologies: sets, spaces, and topoi with Badiou and Grothendieck Arkady Plotnitsky Theory and Cultural Studies Program, Purdue University, West Lafayette, IN 47907, USA; e-mail: plotnits@purdue.edu Received 29 April 2010; in revised form 16 May 2011; published online 19 January 2012 Environment and Planning D: Society and Space 2012, volume 30, pages 351 ^ 368 Abstract. The paper explores the ontology and logic of the irreducibly multiple in set theory and in topos theory by considering the differences between Badiou's logical and Grothendieck's ontological approach to topos theory. It argues that Grothendieck's ontological program for topos theory leads to a more radical concept of the multiple than does the set-theoretical ontology, which defines Badiou's view of ontology even in his later, more topos theoretically oriented work. Extending Grothendieck's way of thinking to other fields enables one to give ontological multiplicities öno longer bound by the set-theoretical ontology or ultimately by any mathematical ontology, even in mathematics öa great diversity and richness. It follows that the set-theoretical ontology is not sufficiently rich to accomplish what Badiou thinks it could accomplish even in mathematics itself, let alone elsewhere; and Badiou wants it to work elsewhere öindeed, wherever it is possible to speak of ontology. I shall also consider, in closing, some implications of the arguments for the workings of the multiple in ethics, politics, and culture. Keywords: logic, mathematics, multiplicity, ontology, philosophy, set theory, topos theory doi:10.1068/d6610 (1) I shall refer extensively to this work Briefings on Existence, because, in addition to a useful set of briefings on Badiou's thought, it provides Badiou's arguably best articulation of his use of topos theory.
description
A fine essay on Badiou's set theory.
Transcript of Plotnitsky, Sets Badiou
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