Multiplicity and Ontology in Deleuze and Badiou

Becky Vartabedian

• Publisher: Palgrave Macmillan
• Year: 2018
• Print ISBN: 9783319768366
• eBook ISBN: 9783319768373
• Topic: Philosophy
• Subtopic: History & Philosophy of Mathematics

This book approaches work by Gilles Deleuze and Alain Badiou through their shared commitment to multiplicity, a novel approach to addressing one of the oldest philosophical questions: is being one or many?

Becky Vartabedian examines major statements of multiplicity by Deleuze and Badiou to assess the structure of multiplicity as ontological ground or foundation, and the procedures these accounts prescribe for understanding one in relation to multiplicity.

Written in a clear, engaging style, Vartabedian introduces readers to Deleuze and Badiou's key ontological commitments to the mathematical resources underpinning their accounts of multiplicity and one, and situates these as a conversation unfolding amid political and intellectual transformations.

© The Author(s) 2018

Becky VartabedianMultiplicity and Ontology in Deleuze and Badiouhttps://doi.org/10.1007/978-3-319-76837-3_1

Begin Abstract

1. Introduction: Lower Layers

Becky Vartabedian¹

(1)

Regis University, Denver, CO, USA

Becky Vartabedian

Email: [email protected]

End Abstract

This book investigates multiplicity in work by Gilles Deleuze (1925–1995), writing by Deleuze with his collaborator Félix Guattari (1930–1992), and in work by Alain Badiou (1937–). This investigation concerns multiplicity as an alternative to a fundamental assertion concerning the nature of being; for Deleuze writing with Guattari and for Badiou, being is neither One nor many, but multiplicity. In situating their respective work according to this shared commitment to multiplicity, a second commitment-in-common to multiplicity articulated with mathematical concepts and tools comes into view. Deleuze and Guattari deploy Bernhard Riemann ’s innovations in non-Euclidean geometry , and a particular interpretation of differential calculus . In Badiou’s case, it is his well-documented use of principles at the foundation of set theory and its revisions in the late nineteenth and twentieth century, particularly Cantor’s inconsistent and consistent multiples, the Zermelo–Fraenkel axiom system, and insights from Bourbaki .

This book is also a work of imagination. In the pages that follow, I invite the reader to consider a conversation that unfolded in the pages of pamphlets and texts. I reconstruct a series of exchanges between 1976 (with the publication of Deleuze and Guattari’s pamphlet titled, “Rhizome—Introduction”) and 1997, with Badiou’s publication of Deleuze: la clameur de l’être, a text published in English as Deleuze: The Clamor of Being (2000; hereafter Clamor). The conversation, when it turns to questions of multiplicity and ontology , reveals objections and demands concerning the structure of multiplicity, the way certain of the mathematical and conceptual tools are deployed to organize being qua being, and the procedures these chosen structures prescribe for handling any one in relation to this multiplicity. The prospects for approaching this as a conversation are aided considerably by my temporal distance from the original set of exchanges. I am, as it will become clear below, one in a lineage of thinkers that have taken up the Deleuze–Badiou knot; my contribution emphasizes the unique strategies each thinker takes when approaching the so-called “being question,” and identifies the places where these differences give way to continuous commitments, namely the demand to arrive at and maintain the multiple using some form of subtractive procedure.

In approaching Deleuze with Guattari and Badiou at the site of multiplicity, I do so with Melville in mind; I have long been fascinated with that exchange on the deck of the Pequod, in which Ahab enjoins young Starbuck to “come closer … thou requirest a little lower layer.” Ahab’s comment is something of an existential injunction and, for the purposes of this project, a useful procedural reminder. As a reader of the Deleuze–Badiou corpus and one attentive to their exchanges, it is significant to consider how, precisely, Deleuze and Badiou each bring their reader to the site of multiplicity in their ontological projects, how they identify multiplicity with a fundamental aspect of being; and how, in Melvillian parlance, they, respectively, admonish their readers to seek this lower layer underwriting that which appears, a lower layer linked to and fundamental to its operation.

Orientations

The initiating provocation for this text is found in a different conversation, begun by the father–son team of Ricardo L. and David Nirenberg . Their 2011 article, “Badiou’s Number: A Critique of Mathematics as Ontology ,” objects to the so-called “radical thesis ” Badiou proposes in Being and Event, which appears in shorthand as “mathematics = ontology ” (BE xiii). The Nirenbergs challenge the contention that mathematical ontology , in general, and especially that proposed by Badiou , produces the sorts of things it claims to; for example, they insist that Badiou’s particular—and by their lights, peculiar—use of set theory is selective in its deployment and its consequences. The Nirenbergs claim that set theory cannot be used to justify the philosophical or political conjectures Badiou draws in Being and Event, and further that the identity of ontology and mathematics Badiou proposes precludes the possibility for “pathic” elements, namely human thought, to emerge from the mathematical system (Nirenberg and Nirenberg 2011, 606–612). This approach, they insist, “will entail such a drastic loss of life and experience that the result can never amount to an ontology in any humanly meaningful sense” (Nirenberg and Nirenberg 2011, 586). The Nirenbergs operate according to the view that ontology is an inquiry into being and questions related to existence as these pertain to humans; they laud the resources of phenomenology , for example, insofar as this method derives conclusions of what it is “to be” from lived experience. By emptying ontology of these resources—a traditionally human center and the lived experience that accrues to it—the Nirenbergs see Badiou’s use of set theory to be so reckless as to endanger an entire tradition of thought.

The Nirenbergs’ claims occasion replies from A.J. Bartlett , Justin Clemens , and Badiou himself. In a subsequent volume of Critical Inquiry, the discussion unfolds with accusations that one camp has fundamentally misunderstood the other. Bartlett and Clemens insist that the Nirenbergs have not read Badiou’s oeuvre carefully; the Nirenbergs fail to understand that ontology as mathematics, for Badiou, really functions as a “figure of philosophical fiction” (Bartlett and Clemens 2012, 368). Badiou , explaining that Bartlett and Clemens respond to the Nirenbergs with “polite irony,” calls the criticism raised by the Nirenbergs “stupid” (2012, 363–364). While significant points both in defense and critique of Badiou’s program are raised in these pages, the interlocutors seem largely to talk past one another; the insights are, unfortunately, lost in the polemical nature of the interchange.

As I was reading this exchange, however, I felt like the conversation was missing something crucial, both as an opportunity for inquiry and a chance for defense. A closer look at the “radical thesis ” Badiou proposes in Being and Event situates “mathematics is ontology ” as the consequent of a preliminary claim: “Insofar as being, qua being, is nothing other than pure multiplicity , it is legitimate to say that ontology , the science of being qua being, is nothing other than mathematics itself” (BE xiii). Badiou spends the first five meditations of Being and Event arguing for the antecedent claim that being qua being is multiplicity. This claim is not mentioned in the Nirenberg–Bartlett –Clemens–Badiou contretemps; it is neither a matter of common sense nor established fact, and it led me to ask a further question following this debate: What does Badiou mean when he claims being qua being is pure multiplicity?

Badiou’s Multiplicities

Badiou’s claim that being is pure multiplicity situates his project in the debate persisting since Parmenides as to whether being is One or many. Badiou’s opening salvo in Meditation One of Being and Event is to show this debate as having stagnated:

For if being is one, then one must posit that what is not one, the multiple, is not. But this is unacceptable for thought, because what is presented is multiple and one cannot see how there could be an access to being outside all presentation … On the other hand, if presentation is, then the multiple necessarily is. It follows that being is no longer reciprocal with one and thus it is no longer necessary to consider as one what presents itself, inasmuch as it is. This conclusion is equally unacceptable to thought because presentation is only this multiple inasmuch as what it presents can be counted as one; and so on. (BE 23)

Badiou identifies here two positions that are “unacceptable to thought.” The first is the claim that being is not multiple; this is unacceptable because we experience a kind of multiplicity and diversity of things in the world. In other words, the presence of different kinds of things suggests to us that there are different ways in which a thing can be. However, and this is the second “unacceptable” claim, each of these different sorts of things presents itself to us as unified, as one thing. These positions are so entrenched, Badiou claims, that he must enact a decision that breaks the impasse and can restart ontological questioning anew. This decision consists in the claim that “the one is not” (BE 23), which means by implication that the multiple is Badiou’s preferred solution to the question of being, at least as he has presented the available options.

Badiou observes a kind of dispositional affinity with the atomist programs associated with Democritus and Leucippus or the Epicureans and Lucretius , for whom ‘the many’ are bodies arranged in void . However, and in spite of these alignments, Badiou ’s pure multiple is not material and does not consist in the ‘bodies’ associated with these older positions. Rather, Badiou ’s multiple is articulated in three ways, each according to mathematical innovations: The first and second—inconsistent and consistent multiplicities—are derived from work by Georg Cantor . The third— generic multiplicity —comes from Paul Cohen’s more contemporary work.

Georg Cantor distinguishes two types of multiplicity as a means of avoiding a common “no set of all sets” paradox in the early development of set theory, particularly with respect to the ordinal numbers (e.g., first, second, third). To avoid the contradiction that arises from the presence of an ordinal not counted in the set of all ordinals, Cantor posits a consistent multiple that is closed and organized according to the typical rules of the set...

Table of contents

• Cover
• Front Matter
• 1. Introduction: Lower Layers
• 2. Engagements, 1976–1997: History of a Misunderstanding
• 3. Structure: Multiplicity and Multiple in Deleuze and Badiou
• 4. Procedures: One, Multiple, Subtraction
• 5. Re-engagements
• 6. Conclusion: Multiplicity, Ontology, Deleuze, Badiou
• Back Matter