Lexicon, PhiloFiction

Structure and System in Badiou and Laruelle

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10 Sep , 2015  

Nothing could be more decidedly foreign to the non-standard philosophy of François Laruelle than the mathematical treatment I am about to give here. This alone seems an excellent reason to proceed. The reader should not expect a thorough formalisation of Laruelle’s theoretical machinery, nor (more likely) a failed attempt at such a formalisation. The question taken up here is simply whether there is an analogy to be drawn between the following: “suture” / “name of the void” in Badiou, “generic” / “radical concept” in Laruelle, and “initial / terminal / zero object” in category theory.

To begin with Badiou, it is well known that in Being and Event the empty set is uniquely positioned as the point of suture between “beings”, qua sets or consistent multiples, and “being” as pure inconsistent multiplicity. The suture operates via an equivocation between the notions “not a set” and “set of nothing”. In both notions, nothing is formed into a consistent multiple. But on the side of “being”, this means that no formation of the unformed has taken place, while on the side of “beings”, it means that a formation of none of the unformed has taken place. Badiou gives the equivocal name “void” to both of these cases, and calls the empty set the “name of the void” by which the void is registered within the ontological system of set theory.

The empty set has this role because it is a minimally-determined object within a system of determinations, and can thus play the part of emblem of the undetermined for that system. Badiou’s set-theoretic ontology is both stabilised by the axiomatic nomination of this object, which forms the starting point for the elaboration of the entire set-theoretic universe, and exposed at this point to the limit of its own coherence. This enables Badiou to manage the dialectic of coherence and incoherence, systematicity and a-systematicity, in a more or less disciplined way. For example, the appearance of the empty set within the power set (or “count-of-the-count”) of every set is characterised by Badiou as presenting the “errancy of the void”, or the haunting of superstructural repetition (which Badiou associates with the bureaucratic machinery of the State) by inconsistency. Or, to give another example, the “evental site” on which the “matheme of the event” supervenes is characterised as a multiple “on the edge of the void”, a set whose intersection with any of its members is the empty set.

If the meta-mathematical narrative of Being and Event is possessed of an impressive sprezzatura, a mixing together of felicity and virtuosity such that one often cannot tell where contrivance leaves off and happy coincidence begins, then much of its dialectical agility is owing to the equivocation at the heart of the system. Where structure is required, the empty set as minimal structural element is on hand to provide a sound basis. Where the system needs to be opened out to the a-systematic, the empty set as “name of the void” provides an escape hatch. The void thus functions as a kind of bellows within Badiou’s mathematical oratory, inflating and deflating as needed, supplying more “beings” here and more “being” there.

All of this Laruelle confronts with a kind of vulgar skepticism, deriding Badiou’s “ontology of the void” as an exercise in philosophical “auto-position” facilitated by a mathematical mirror. If Laruelle’s Anti-Badiou shows little sign of its author’s having comprehended the mathematical apparatus of Being and Event, its sarcasm is nevertheless effectively targeted at the entire character of the enterprise. For Laruelle, Badiou’s “ontology of the void” (“OV”) remains a hierarchical, “planifying” arrangement, secured by the privilege given to the empty set as its central operator: a strait gate through which every concept must be made to pass. By contrast, Laruelle’s non-standard philosophy, modelled (so he says) on quantum physics, holds within it no such place of privilege: there is no single point of “suture” securing a system, but rather a polyphony of theoretical “undulations” rolling in the depths of a matricial, oceanic Real.

There is nevertheless in Laruelle a characteristic operation of unstraitening or destructuration, which aims at producing a “generic” instance which can stand, amidst the other terms of a theory, as a name of the Real itself. This operation does not fix on any single foundational term, such as the empty set, but seizes on the privileged operators of whatever theoretical “material” is at hand, asset-stripping them and repurposing them as what Katerina Kolozova calls “radical concepts”. Laruelle will speak of the “non-Mandelbrotian fractal”, separated from the geometric and algorithmic affordances that give the term “fractal” its strong, specific technical sense and diverted towards a “generic” rendition in which it becomes a figure amongst others of indivisible self-similarity: a pseudonym of the Real.

Detached from context, emptied of content, the generic term does not serve as the cornerstone of a conceptual architecture but instead represents a kind of weak force of identity, according to which anything whatsoever can be disposed within immanence alongside anything else. It is thus taken up as an emblem of underdetermination. What can be a non-Mandelbrotian fractal? Anything one wishes to consider as one; that is, consider according to its generic fractality, its imbrication in the Real. If a technical term affords certain kinds of use, distinguishing one thing from another, the generic term withdraws what it names from distinction. It is a sword beaten into a plowshare.

We can now compare the two emblems, the Badiouvian “name of the void” (of which there can be only one) and the Laruellian “name of the Real” (of which there are many, in illimitable series). Each names a minimal instance of structure: the one given axiomatically, a philosopher’s stone; the other produced through a kind of inverted alchemical reduction to dross. Each represents the starting point of a procedure: the one subtractive, elaborating through purification a conceptual universe; the other subsumptive, drawing philosophical edifices down into the undertow of the Real.

Take for example the effect of the two procedures on the philosophical category of “Life” (as organic process, as intrinsic value, as that which resists the power of death and so on). The subtractive Badiouvian approach refuses to accept “Life” as a name of the Real, assigning it instead to procedures which partition it and develop the pieces according to their own separate logics: thus we find “life” as poetic figure of organic integrity; “life” as the ultimately manipulable material of the sciences which are on their way to knowing how to remake it in their own image; “life” as a field of inert subjectivity fecundated by the amorous encounter; or “life” as the renewed enthusiasm of the political convert, incorporated into the body of a political truth. Finally, “life” for Badiou is the object of a maxim: to live according to an Idea, which alone can bring it to the fruition of which it is capable. What authorises this refusal of authority to “Life” as a philosophical first concept, its handing-over to procedures which take it as a mere skeleton to be invested with glittering logical raiment? Nothing other than the name of the void: the only acceptable final name, precisely because it is the name of nothing.

What is the Laruellian approach? To turn from the strong philosophical concept of “Life” to the weak names of “the lived” (le vécu) and “the enjoyed” (le joui), to that which is lived by the living and enjoyed in their enjoyment (jouissance). Rather than a metaphysics of vital energy, or of desire capillarising pathways and lines of leakage, Laruelle proposes through these generic terms an infraphysics, a principle of underdetermination which gives not laws but occasions, instances of the Real. “Life”, the life of the philosophers, is then neither refused nor mathematically purified, but subjected to its own insufficiency to the lived as it is lived. This is not the philosophical ruse according to which Life is posited as always in excess of “its” concept (the excess thus being philosophically specified and controlled, as philosophy’s own), but the non-philosophical disposition according to which Life, however you conceptualise it, is thought according to the lived, of which it is never more than a model occasioned by (some) living.

For Badiou, the minimal term — the empty set — is an absolute minimum: this is what gives it its structuring power. The generic term in Laruelle is only ever relatively minor: just “minoritised” enough, in the circumstances, which may indeed be varied and admit of multiple generic operators. This is then the difference between determinate indetermination and underdetermined underdetermination.

We come now to our formalisation, which will no doubt seem all the more perverse given the preceding discussion. We suppose that a category (in the mathematical sense) is given in which the morphisms between objects are understood to carry some determination between their domain and codomain. In other words, the existence of a morphism f: A -> B implies, in the category at hand, that “A determines B” in some sense. In Lawvere and Schanuel’s usage, for example, such a morphism in the category of abstract sets represents a “general element” of B: that part of the set B which bears the image of the set A under the mapping f. The collection of all morphisms from other objects into some object A is thus understood to carry all of the possible determinations of A within the category: everything that is determinate about A, from the point of view of the category in which it appears, is determined extrinsically through these morphisms. (This is, very hand-wavily, one way of taking the general import of the Yoneda Lemma).

What does it mean, in such a category, for an object to be “minimally determined”? The answer must be given by way of the morphisms connecting that object to other objects. An “initial object”, if one exists in the category, is an object such that, for every object in the category, there is a single uniquely determined morphism from the initial object to that object. It is provable that if more than one such object exists, all such objects are “isomorphic”, or functionally indistinguishable from each other in terms of the roles they can play within the category.

It so happens that in the category “Set”, the initial object is the empty set. For every set, there exists one and one only mapping from the empty set to that set: the empty mapping which takes no elements of the empty set to no elements of the target set. There is no mapping from any set other than the empty set into the empty set: it is thus both minimally determined and unilaterally determining.

Other categories may have multiple (isomorphic) initial objects, or none. An example worth paying attention to is that of the category of pointed sets. A pointed set is a pair (A, a) of a non-empty set and a single element of that set, which “anchors” it. The morphisms between pointed sets are the mappings between them that take the “anchor” of the source set to that of the target set. In this category, every singleton pointed set ({a}, a), containing only its anchor element, is initial. Unlike the category of sets, however, in the category of pointed sets there is also a unique mapping from every pointed set back into the initial object; which is also, therefore, a “terminal” object. An object which is both initial and terminal is known as a “zero” object, and represents a kind of maximally stable point within a category, being both universally determined and universally determining.

It is clear that initial, terminal and zero objects represent, within a category, privileged structural positions. The Real is of course not a mathematical object; but might not such a minimally-structured object be considered as an emblem of indetermination, and hence a point of conformance to the Real, within the mathematical system regulated by a category? We have seen that this analogy holds with regard to Badiou’s set-theoretic ontology, inasmuch as the empty set Badiou takes as his privileged point of suture with the Real is also the initial object in the category of sets. But does it communicate in any way with Laruelle’s selection of generic terms, local or relative minoritisations which serve to re-orient the systems to which they belong “according to” the Real?

The question can be put more generally: why, among all the languages and “modelisations” considered by Laruelle, do some terms present themselves for genericisation more readily than others? What, given Laruelle’s desire to operate a “democracy of thought”, picks out just these terms as structurally privileged, and therefore apt cases for the “dualysing” treatment? Here the category theoretic analogy suggests a possible answer: a “minimal instance of structure” is always minimal relative to some particular system of structuration. If Laruelle disdains the notion of a universal minimum, a single foundational point of suture, he nevertheless practices a selection of local minima guided by the system at hand. If Laruelle is not himself a systematic thinker, a builder or maintainer of systems as Badiou unarguably is, he is nevertheless inexorably tied to systematic evaluation in his non-philosophical practice: a “heretic”, yes, but loyal to the last.

 

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