What is a multiplicity?

A fundamental question! Philosophies of the multiple or manifold today abound, as philosophy recovers from its linguistic detour. So much has been said on the idea of the pure multiple, and yet, it seems, what really needs to be said has not been. Far be it from me to enunciate philosophy’s subject; but I think I can put forth some suggestions, if only in clarification.

Deleuze is the first true, systematic thinker of multiplicity in the philosophical / ontological sphere. For Deleuze, everything is multiplicity. Most importantly, however, “multiplicity” usually refers to a virtual Idea or problem, the unextended, creative plenum of ideality folded into the material world as we know it in a banal sense. The notion that “everything is multiplicity,” and especially in the latter sense, is what allows Deleuze to move beyond the structuralism with which he begins. (For Deleuze’s seminal ontological works, Difference & Repetition and The Logic of Sense are undoubtedly attempts to provide ontological orientation to the basic operations of structural anthropology, ethnology, and linguistics; what emerges from this attempt is a theory that supersedes structuralism tout court: “post-structuralism” might be less accurate than “genetic structuralism,” however.) Though the two works just mentioned contain much more rigorous presentations of the idea of multiplicity, it is in Deleuze & Guattari’s What is Philosophy? that we obtain the most basic insights into the nature of multiplicity. One could turn to practically any page in this text and find a slightly different “definition” of multiplicity on each one. Yet there is a constancy of thought as well, as each supplementation or erasure can be assimilated with the claims put forward in Difference & Repetition concerning multiplicities as Ideas, in the discussion that begins with an analysis of Kant (“Ideas and the Synthesis of Difference”). I will quote an excerpt that is all the more revealing in that it comes at the end of Deleuze & Guattari’s brief inquiry into Badiou’s set-theoretical ontology in Being & Event:

By starting from a neutralized base, the set, which indicates any multiplicity whatever, Badiou draws up a line that is single, although it may be very complex, on which functions and concepts will be spaced out, the latter above the former: philosophy thus seems to float in an empty transcendence, as the unconditioned concept that finds the totality of its generic conditions in the functions (science, poetry, politics, and love). Is this not the return, in the guise of the multiple, to an old conception of the higher philosophy? It seems to us that the theory of multiplicities does not support the hypothesis of any multiplicity whatever (even mathematics has had enough of set-theoreticism). There must be at least two multiplicities, two types, from the outset. This is not because dualism is better than unity but because the multiplicity is precisely what happens between the two. Hence, the two types will certainly not be one above the other but rather one beside the other, against the other, face to face, or back to back. Functions and concepts, actual states of affairs and virtual events, are two types of multiplicities that are not distributed on an errant line but related to two vectors that intersect, one according to which states of affairs actualize events and the other according to which events absord (or rather, adsorb) states of affairs.”

-Deleuze & Guattari, What is Philosophy?, 152-153. Emphasis added.

Even the One, they claim at some earlier point in this text, is a multiplicity, having multiple elements which grant it consistency (in this case, being and non-being, themselves multiplicities, and so on ad infinitum). Everything is multiplicity, but are we by that sweeping claim to understand that everything consists of smaller elements, can be broken down into component parts? Is this a new mereology? Not at all. And the reason is deceptively simple. Let us take an analytic approach and follow this thread nevertheless. Take any empirical thing or state of affairs. At the very least, it is constructed of physical components: chemical elements, protons and neutrons, etc. Now strip this physical item of all its properties, down to the single, smallest possible component particle (it does not matter what it is – quark, lepton, something smaller still). It is still a multiplicity by virtue of its relation to other quarks, leptons, whatevers. Let us take another step: strip this particle of all relation to other things. It remains a multiplicity by virtue of its genetic history and its relation to the potentials insisting in it, which are just as real as this physical particle, though non-actual, unextended. Furthermore, as any physicist will tell you, even this (purely academic) minute particle is in a state of flux, and can thereby be said to constitute a field of individuation in itself, which field can only be understood as a multiplicity in becoming (different/ciating into ever more multiplicities).

What is a multiplicity? A field of exterior relations – what I prefer to call proximities.

Deleuze & Guattari point up a competing theory of multiplicities: Badiou’s. A brief detour: Badiou takes Deleuze & Guattari’s criticism seriously, and his more recent writings on Deleuze reflect this more than does his work stamped with the proper name of his critic. The intensive, the entre-deux is appreciated as the “beyond” of the One and the Multiple: “‘[B]eyond’ clearly does not mean a synthesis or a third transcendent term. ‘Beyond’ means in the middle, wherein Being is what activates the essential falsehood of the true and virtualizes the truth of the false through the rhizomatic-network shifter between virtualization and actualization.” (Badiou, “Deleuze’s Vitalist Ontology,” Briefings on Existence, 65.) In short, as Deleuze & Guattari claim, there must be at least two types of multiplicity because there is a third.

Badiou’s idea of multiplicity, the set (which, moreover, refuses definition because to define necessitates a counting procedure, where the parts or elements of a set, pure multiples or sets themselves, can only be thought as what-will-have-been-counted), always comprises the name of the void as minimum of Being, the “point of Being” of any situation. The void or multiple of nothing subtracts itself from any presentation, unpresents itself, and hangs over as remainder, “phantom of inconsistency.” In this way, all sets or multiples are themselves multiples of multiples: even the multiple of nothing is a multiple of multiples insofar as the void “insists” but resists presentation.

What is a multiplicity? Insofar as belonging, inclusion, and subtraction are relations, a field of relations.

Badiou, unlike Deleuze & Guattari, does not suggest that all relations are exterior. Indeed, for Badiou, there are intrinsic relations that, by virtue of this thesis, expose the entire ontological theory of sets as multiples of multiples to the criticism that a dialectic of negation is operative, even if synthesis cannot ever be complete (if only because there is no set of all sets). Subtraction is not negation, to be sure. But the logic of sets is indeed a logic of contraries. After all, the unpresented grounds (abyssally, it is true) the presented and reacts in relation to the presentation of the presented, precisely by a dialectical procedure of unpresentation or subtraction. Subtraction or unpresentation itself escapes the charge of dialecticism, but its relation with presentation is unquestionably dialectical, and this because of its intrinsic predisposition to negate the presented, because it includes within itself its other as itself. The same claims hold a fortiori for the presented, for every presented implies an unpresented and a process of unpresentation.