Becky Vartabedian

This chapter initiates a conversation between Deleuze and Badiou concerning multiplicity in three periods. The first (1976–1977) is political, seating multiplicity in a revised doctrine of materialism; it is initiated with Deleuze and Guattari’s “Rhizome—Introduction” (1976) and addresses Badiou’s “The Fascism of the Potato,” penned as Georges Peyrol. The terms of my analysis—multiplicity as structure and attendant procedures for producing any one—emerge in this period. The second period is conceptual, concerning Badiou’s mature ontological system in Being and Event ([1988] 2005) and its brief address by Deleuze and Guattari in “Example 12” of What is Philosophy? (1991). The third (and the one with which readers are perhaps most familiar) is martial, in which Badiou develops the antagonistic arguments in Deleuze: The Clamor of Being and claims that Deleuze is a philosopher of the One and not of multiplicity. Badiou’s criticism concerns the structure of Deleuze’s multiplicity and is the notion with which I open Chapter 10.1007/978-3-319-76837-3_3.

This Chapter attends to multiplicity as structure in Deleuze’s and Badiou’s works, structures that are not articulated according to traditional philosophical conceptions of ‘the many,’ but are rather associated with mathematical paradigms. Bernhard Riemann’s continuous manifoldness and certain of its operational accompaniments inform Deleuze’s work, which I show in his account of the virtual Idea in Difference and Repetition and to the account of smooth and striated spaces in Plateau “1440: the Smooth and the Striated” in A Thousand Plateaus. Georg Cantor’s distinction of inconsistent from consistent multiples are at the foundation of Badiou’s ontology; Zermelo’s axiom of separation solves a problem (crystallized in Russell’s Paradox) about any possible relation between inconsistent and consistent multiples. Badiou’s work in Being and Event and a short description of inconsistent multiplicity in Briefings on Existence are the texts under investigation here.

This chapter orients the reader to the ideas of multiplicity in Badiou’s and Deleuze’s work, and to the ontological paradigms these ideas support. I acknowledge a conceptual debt both programs owe to precedent uses of Mannigfaltigkeit (manifold, multiple), particularly in Kant’s discussion of the manifold in the Critique of Pure Reason. This chapter also presents my method for slow comparison—allowing the programs to be articulated on their own terms before turning to any critique—as facilitated by this moment in reflection on the debate; I also show that my own method is an alternative to the paradigm of martial opposition that characterizes much of the conversation so far. I conclude by offering a plan of the text.

This chapter recovers the work of the preceding several chapters to demonstrate my attention to structure and procedure as key components of both Badiou’s and Deleuze’s ontological multiplicity. I discuss the prospects for ‘other lineages’ in which to take up this investigation, including the discussions in Chapter 10.1007/978-3-319-76837-3_5 concerning Badiou’s relationships to Heidegger and Kant. Knox Peden (Spinoza Contra Phenomenology: French Rationalism from Cavaillès to Deleuze, Stanford University Press, Stanford, 2014) uses an entanglement with Heidegger to propose ways in which Deleuze’s alternative reinforces his deep Spinozist commitments; this recognition extends the frontier of the Badiou-Deleuze engagement beyond the boundary I identify in Kant’s notion of the manifold. I conclude by addressing the critique of multiplicity in Badiou’s review of Deleuze’s The Fold: Leibniz and the Baroque. This engagement offers an opportunity to consider the foundation of ontological inquiry not in multiplicity, but the decision or commitment to dissolution that warrants its deployment.

This chapter begins by developing an account of subtraction in Badiou’s ontological work parallel to the n-1 procedure operative in Deleuze and Guattari’s work; unlike the “special” subtraction associated with truth procedures (outlined in the Gamma Diagram), Badiou’s account also harbors a ‘general’ subtraction, largely continuous with the procedure Deleuze and Guattari describe as a mode of maintenance. I point out the success of this general subtraction’s functioning in Badiou’s well-defined ontological context and its liabilities when applied to a meta-ontological context. It is in light of this continuity over subtraction that I then return to the original engagements outlined in Chapter 10.1007/978-3-319-76837-3_2—the political, conceptual and martial connections I laid out there—from the far side of my analysis of multiplicity, multiple, and one in Badiou’s and Deleuze’s work.

This chapter attends to the prescribed procedures for relating one to multiple and multiplicity in Badiou’s and Deleuze’s work. I begin with the work of constructing consistent multiples in Badiou’s work, a procedure requiring attention to the void set and its mark Ø, their background in Zermelo and Bourbaki, and axioms of Zermelo-Fraenkel set theory. I attend to this procedure’s relation to Badiou’s earlier account of scission, a technique for ‘cutting’ a one from the multiple. I address one-production in Deleuze’s account by discussing the differential relation (dy/dx) and its mobilization toward the pre-individual singularity, which Deleuze develops in his account of the virtual idea. My analysis focuses on those procedures by which the relation of one to multiple is maintained, namely subtraction, apparent in the analysis of the war machine in A Thousand Plateaus and in their assessment of the cogito in “Example 1”. I show how Deleuze’s earlier concern with overturned Platonism expresses this subtractive impulse; in short, one sees that where multiplicity appears in Deleuze’s oeuvre, subtraction seems to follow.

Jane Bennett’s vital materialism develops positive ontological commitments to lively matter and resistant vitality, articulated using notions of actant and assemblage, thing-power and the out-side. I show that these ontological commitments reveal a limit for traditional modes of human knowing, favoring an emergent epistemology that attends to the ways actants and assemblages express themselves. I then argue for an account of acting that positions humans as guests of vibrant matter. Compacts of guest-friendship in Plato’s Crito and Kant’s To Perpetual Peace indicate that to be a guest is to be embedded in an asymmetrical system. The compact that binds the guest in a world of vibrant matter is the prospect of friendship with nonhuman others, a prospect I discuss following the work of Nick Bingham. I conclude by addressing Axelle Karera’s recent critique of Anthropocenean discourses, explaining the role guest-friendship can play in addressing certain of the weaknesses Karera identifies.

This Collision discusses the cyanotypes produced by photographer Meghann Riepenhoff, in particular those in the Littoral Drift project; these cyanotypes are direct positive prints produced from the action of sea water washing over photo-sensitive paper placed at shorelines in the United States. I discuss these images and the cyanotype process Riepenhoff uses to produce these. I then discuss philosopher Fred Evans's concept of "voice" and its contemporary extension to nonhuman others. Arguing that an intermediate aesthetic intervention would assist Evans's affordance of voice to nonhuman others, I position Riepenhoff as a facilitator of the sounding of wind, water, salt, sand, and chemicals expressed in the cyanotype process. I conclude by linking the Littoral Drift series and Riepenhoff's prospective position of facilitator to recent aesthetic and ontological meditations on the Anthropocene, that epoch of geologic time in which human activity is the primary shaping force on the earth; I situate Riepenhoff's work in conversation with insights from Joanna Zylinska's account of nonhuman photography and Jane Bennett's vital materialism.

In this article, I discuss Alain Badiou’s 2008 address titled “The Three Negations.” Though the text was originally presented in a symposium concerning the relationship of law to Badiou’s theory of the event, I discuss the way this brief address offers an introduction to the broad sweep of Badiou’s metaphysics, outlining his accounts of being, appearing, and transformation. To do so, Badiou calls on the resources of three paradigms of negation: from classical Aristotelian logic, from Brouwer’s intuitionist logic, and in paraconsistent logics developed by DaCosta. I explain Badiou’s use of negation in the three primary areas of his metaphysics, as well as to diagnose the degrees of transformation that may have occurred in a situation. My analysis of Badiou’s use of negation in this text is aided by examples from his broader ontological oeuvre. I also explain the underlying requirement in Badiou’s work that formal considerations - mathematical or logical - get their sense by being tethered to readily-identifiable political, aesthetic, scientific, or interpersonal concerns. I conclude by addressing the foundation Badiou’s work establishes for pursuing a new metaphysics, and by discussing certain of the liabilities that remain in the wake of his account.

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 mathematical 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, including the mathematical resources underpinning their accounts of multiplicity and one, and situates these as a conversation unfolding amid political and intellectual transformations.

This paper develops a notion of allowances, a designation for speech-acts indicative of the collaborative behaviour I engage in with others. Such behaviour marks a facet of my individual identity that I could not create on my own. I ground this notion of allowance in the view of the self as intentional body-consciousness developed by Maurice Merleau-Ponty and John Russon. I explain the relation of this self to others on their phenomenological paradigm, and then explain the functions of speech and gesture Merleau-Ponty offers in the _Phenomenology of Perception_. In order to present my concept of allowances, I turn to J.J. Gibson's concept of affordances. Gibson's concept provides an analogue for the notion of allowance. I then explain two examples of allowances in action: gossip and truth-telling. I conclude by drawing attention to the ways these allowances are markers of the collaborative constitution of identity.