lunes, 5 de julio de 2010

Badiou's Mathematical Platonism

The Meta-Ontological Exception:
Notes on Badiou's Mathematical Platonism


“The concept of model is strictly dependent, in all its successive stages, on the (mathematical) theory of sets. From this point of view, it is already inexact to say that the concept connects formal thought to its outside. In truth, the marks ‘outside the system’ can only deploy a domain of interpretation for those of the system within a mathematical envelopment, which preordains the former to the latter. […] Semantics here is an intramathematical relation between certain refined experimental apparatuses (formal systems) and certain ‘cruder’ mathematical products, which is to say, products accepted, taken to be demonstrated, without having been submitted to all the exigencies of inscription ruled by the verifying constraints of the apparatus." (AB, The Concept of Model)

Zachary-Luke Fraser advances a nice rejoinder to Ray Brassier’s outstanding analysis from Nihil Unbound. There Brassier asks what precise role metaontology comes to play vis the distinction between ontological and non-ontological situations. On the one hand, metaontology is clearly not ontology itself, since the latter only speaks of sets while the former speaks of presentations in general. As such metaontology cannot be said to be ‘founded on the void’ in the same way as ontology, since it operates with resources strictly external to the latter. On the other hand, metaontology suspends Leibniz’s thesis, which asserts the identity of being and the One (or being and unity), declaring the latter to be a mere operation (the count-as-one which structures every presentation). So it seems that metaontology stands somewhere inbetween the two ‘fields’ of presentation, enacting the transitivity of the very concept of presentation across the two domains, affirming the identity of ontology and mathematics. Luke Fraser thus seeks to dissolve the pertinence of the polarity between discourse and world in Badiou's mathematical Platonism by arguing that non-ontological presentations, thought in their being, must be already mathematized, i.e. must be thought of as models for set theory.
This way, it is not that set-theory qua singular discourse is ‘connected’ to its outside in non-ontological presentations via metaontology. Rather, all presentations are thinkable ontologically only as mathematically treated as a domain for the testing of the set-theoretical axioms: thus stipulating that insofar as ontology thinks of the form of presentations as sets, it thinks them in their being. This is part of the ‘mathematical Platonism’ that dissolves the transcendent bond between a formal language and its outside, and thereby dissolves the tension between a true materialism and what would appear to be a discursive brand of idealism anchored in set theoretical mathematics. So ontology and non-ontological situations are at once indiscernible as immanent (every presentation is immanently presupposed as having being mathematically intelligible in its being) and as transcendent (ontology is just one situation among others in itself;no situation contains all others; the concept of general presentation formally described by ontology is bridged through the meta-ontological decision):
“The point to which this brings us is this: To the extent that a mathematical ontology of concrete situations is possible, it must be possible to treat these as ‘models’ of set theory. Accordingly, these situations must be apprehended as being already mathematical in some sense, however crudely or vaguely understood. To the extent that ontology avoids the ‘empiricist’ mandate of being an ‘imitative craft’ (a characterisation against which Badiou rails in The Concept of Model), the correspondence between the ontological situation and its outside can be classified as neither a relation of transcendence nor immanence, but must be thought as a point of indiscernibility between the two. This is the source of all the obscurity attributable to the ‘Platonist’ position of metaontology, which forces us to ask, as Brassier does”,
“Where is Badiou speaking from in these decisive opening editations of Being and Event? Clearly, it is neither from the identity of thinking and being as effectuated in ontological discourse, nor from within a situation governed by knowledge and hence subject to the law of the One. […M]etaontological discourse seems to enjoy a condition of transcendent exception vis-àvis the immanence of ontological and non-ontological situations”(NU: Chapter IV)”
The relation between ontology/non-ontology is not strictly transcendent because non-ontological situations can only be thought in their being as already mathematized, i.e. for ordinary situations to be treated as models requires their mathematization into domains for set theory. It is not strictly immanent either because the ontological situation does not present all others, but only their general form as ‘sutured to their being’, inside the characteristic mode of thought that is ontology, i.e. there is no presentation encompassing all others, but only presentations of presentations, and void/nothingness.
However, the equation of mathematics with ontology, and thus the affirmation that set theory alone renders the form of presentation in all non-ontological situations, can only be performed by metaontology as declaring the equation of being qua presentation and inconsistent multiplicity qua the inertia of the domain of set theory. This in turn requires the primitive inconsistency of presentation to be foundational for the axiomatic, which Badiou perceives in the existential inscription of the pure name of the void as the primitive and radically non-phenomenological sign from which the entire stock of operations are woven in continuity with the axioms of set theory and on the basis of the primitive relation of ‘belonging’ alone. Metaontology is thus prerequisite to establish the indiscernibility of set theory as a unique situation and the wealth of possible non-ontological situations insofar as they are thought in their being.
Why must we assume that inconsistent multiplicity underlies the consistency of the pure multiple, which amounts to thinking being as fundamentally ‘without unity’ or as being-nothing? It is because the primitive subtraction of being from the count-as-one is thought in accordance with the Parmenidean statement that whatever is not One must be by necessity multiple. Since multiplicity resists unity, and given that being is essentially multiple, any discourse on being qua being will begin from the assumption of the non-being of the One, or the lack of any foundational figure of oneness.
Badiou assigns the multiplicity of presentation to its properly discursive (ontological) domain through the unique consistency of ZF set theory, which proceeds in the assumption of existence of a set with an empty extension. Once the operation of belonging to x can be said to be tantamount to ‘being presented to x’ via the speculative move, set theoretical strictures appear fittingly to depart from the sole assumption of the lack of a phenomenological given. Badiou explains the presupposition of lack required to render multiple being discursively through the sterility of the presentational domain set theory inscribes with the mark of the void. This identifies presentation with inconsistent multiplicity, and as such tethers being to set theory as ontology, enacting the thinking of that which primitively proceeds from the absence of unity, i.e. from accepting Parmenides’ embrace of the form of presentation as essentially multiple. The ‘suture to being’, again, thus remains strictly meta-ontological.
Of course, the additional assumption, also spotted by Brassier, is that non-ontological presentation is distinguished insofar as it presents the One, which necessarily makes the ontological situation qua theory of the multiple as the unique situation in which presentation is thought of in ‘its being’. The fictive being of the one is expressed as set theory operates consistently on the basis of the primordial lack of the void itself. It is thus that all unity is given in ontology solely against the impenetrable background of the void’s empty inertia. Each set is constructed on the basis of the primitive lack indexed by the void’s proper name, and is determined purely extensionally in terms of what it presents (which is always nothing but a function of replacement operating over the proliferation of subsets woven from the void alone -this is guaranteed via the axioms of void, powerset and replacement). Oneness remains thus the result of the operation of belonging, which presents sets whose being is nevertheless indexed to the void in the last instance.
“This splitting of unity into operation and effect is integral to the thinking of presentation and the metaontological delivery of the formal thinking of presentation to mathematics, and specifically, to set theory. It is set theory itself that formalizes this split, and provides us with a figure of multiplicity adequate to the thinking of presentation, and, more dramatically, to the univocal determination of the existent as presentation (and so of presentation as the presentation of presentations). To speak of a presentation is to speak of presentation affected by an operation of the count-as-one, and not of a presentation solidified according to the intrinsic unity characteristic of entity. The unity of a presentation is always extrinsic.” [ZLF Pg. 68]
It is this split between the one as an operation and as effect which becomes occluded in non-ontological presentations, where being attains fictive unity in closing this gap. It thus assumes itself identical to what it presents, or to its own singleton: x = {x}, thus violating the constraint set to it in well founded set theory by the axiom of foundation and extensionality (the couple of which require a set’s extensional determination or identity, and its incapacity to belong to itself). This indistinction between the one as operation and the one as result is thus the violation of what Zachary Luke Fraser designates as the two principles of multiplicity for Badiou:
a) Material component - Every set is extensionally determined.[1]
b) Formal unity – Every set is different from itself by a pure differentiation ( grounded in the axiom of foundation).
Consistency is thus the formal unity in which a presentation is given, its being-counted-as-one (yet different from itself) in the situation (one as result). On the other hand, presentation itself remains necessarily inconsistent as the retroactive presupposition of the multiple gathered is but merely counted-as-one, and thus presupposes its prior existence, not exhausted by what it unifies or presents in its formal unity (one as operation). In ordinary situations where this gap is closed, we do not think according to the being of what is presented, which necessarily differs from itself formally but on the basis of the fictional consistency of unity in which it appears.
The evaluation of such situations in the ‘moment of the One’, as we know, becomes properly the subject of Badiou’s Greater Logic in Logics of World and the phenomenology of objects.

- Annotations on Meditation 26 (The Concept of Quantity) in Alain Badiou’s Being and Event
In Alain Badiou’s theoretical framework set-theory as ontology comes to explicate the notion of quantity through some technical concepts worth elucidating in close detail, even if, as Badiou admits that the formal exposition of the ontological operations can exceed philosophical (and therefore meta-ontological) interest. In particular, it is easy to overlook Badiou’s explication of the concept of a ‘function’, since it is delegated to a short (but doubtlessly crucial) Annex at the end of the book. There, a function is described straightforwardly as a particular kind of multiple, in unproblematic continuity with the strictures of set-theory and the pure multiple. In what follows we’ll try to elucidate the surrounding notions, since the prose in the book lends itself to easy confusion.
A function f of a given set α to a set β, which can be written f(α) = β, establishes a one-to-one correspondence between the two sets, where it is understood that:
- For every element of α there corresponds via f an element of β.
- For every two different elements of α there corresponds two different elements of β
- For every element of β there corresponds via f, an element of α.
At this point, the set-theoretical grounding becomes quite necessary to follow Badiou’s argumentation, since the concept of ‘function’ outlined above is defined, after all, as simply a particular kind of multiple. What kind of a multiple is at stake here? Here we must move to Appendix 2 of the book, which provides the sought for clarification.
Badiou begins by describing multiples which constitute relations between other multiples. These are structured as series of ordered pairs, and are written as follows:
Let’s assume the existence of a relation R between two given multiples α and β: R(α, β). Badiou describes relation as getting behind two ideas: that of the pairing of the two elements, and that of their sequence or order. This second condition guarantees that even if R (α, β) obtains in a given situation, it is possible that R(β, α) does not. The first condition entails that all relations can be expressed as consisting of two element multiples, written in the form <α, β>, so that to say that there exists a relation R between two existing elements α and β finally amounts to no more than saying: <α, β> ε R. Given that for any two existing elements α and β there exists necessarily the set which has α and β as its sole elements {α, β}[2], although se will see right away that this set is not identical to <α, β> . The only problematic aspect pending is finally that of order, and thus of the stipulated asymmetry between R (α, β) and R(β, α):
Interestingly enough, the ‘ordered pair’ solicited by Badiou is not simply the pairing of α and β, but actually the pairing of the singleton of α, and the pairing of α and β. So we get:
<α, β> ↔ { {α}, {α, β} }
This set must exist, given that the existence of α and β guarantees the existence of their respective singletons, as well as their pair. Therefore the union of either of the first terms with their conjunction must also exist. In other words, for any given two multiples α and β there exist two different possible ordered pairings, which are not identical:
<α, β> ≠ < β, α> .↔. { {α}, {α, β} } ≠ { {β}, {β, α} }
Notice, however, that both ordered pairings are completely indifferent with respect to order in the terms of the set {β, α} / {α, β}; which are transparently identical sets. The impossibility of substitution and thus the asymmetry of the two orderings laid above occurs in the difference occasioned by the choice between {α} or {β}. This must mean that an ordered pair always consists, for any two elements supposed existent, of the two-element set consisting of the singleton of one of the two elements and the two-element set consisting of the two already given elements. Additionally, it is implied that:
<α, β> = <г, у> .->. (α = г) & (β = y)
Finally, to say a relation R obtains between two given sets α and β entails:
<α, β> ε R or <β, α> ε R
Having established that a relation is a multiple composed of ordered pairs, Badiou proceeds to explain how a function may be described a particular kind of relation. The trick here consists in grasping adequately the abovementioned idea of ‘correspondence’. Let us assume a function f that makes a multiple β correspond to α: f(α) = β. Having established functions are relations, and relations are sets of ordered pairs, it plainly follows that functions are sets of ordered pairs. If we then allow Rf to stand for the function of α to β, we can write as follows:
<α, β> ε Rf
But the peculiarity of the function resides on the uniqueness of β, so that no other element can correspond to it by it. This means that for any two multiples β and y that correspond to α via a function R, it must be the case that β and y are identical. Formally we write:
[f(α) = β .&. f(α) = y] -> β = y
Or, alternatively:

(<α, β> ε R f .&. <α, y > ε Rf) -> β = y

If we want to unpack this formula, we write:

({{α}, {α, β}} ε Rf .&. {{α}, {α,y}} ε Rf) -> β = y
With this Badiou completes his reduction of the concept of relation to pure set-theoretical constructed multiplicities. The next step is to ground the comparison between sets in the series of ordinals (natural multiples[3]). With respect to a multiple’s ‘size’ or ‘magnitude’, there always exists an ordinal which is equal to it (which is not to say only natural multiples exist; we know this isn’t true given the existence of historical multiples). Badiou claims that thus ‘nature includes all thinkable orders of size” [BE: Pg. 270]. Here things turn a confusing, since Badiou doesn’t really provide an example until later. We can, however, give a very simple case to illustrate how exactly this happens.
First, recall that the series of ordinals are woven from the void alone, as the structured sequence or passage from the void into its singleton, and thus consecutively in serial manner. If we repeat the basic example laid above where Rf stands for the function of α to β. We got:
[R(α, β)] ↔ [f(α) = β] ↔ [< α, β> ε Rf]

Or, more explicitly:

{ {α}, {α, β} } ε R f

However, we can easily see that the multiple thus produced has the same power as the ordinal which composes the Von Neumann ordinal Two, and which is guaranteed given the sequence of ordinals:
Π: {{Ø}, {Ø, {Ø}}
Notice, however, that although this ordinal certainly has the same power as the given set, there’s an infinity of ordinals with the same power as the laid set: we can easily imagine the ordinal: {{{Ø}}, {{Ø}, {{Ø}}} and successive variants, all with the same power. The requirement is merely that there will be at least one ordinal with the same power. Identity as such is guaranteed through the comparison of a set's extensions, where the axiom of foundation guarantees the void lingers within each form of presentation (forbidding non-wellfounded sets from proliferation indefinately; self-belonging becomes forbidden). I will continue with these notes later.

[1] See the annotations below to explain the procedure of the determination of the identity of a set on the basis of the extensional determination of each set; which delivers us to the concept of quantity.
[2] See Being and Event, Meditation 12.
[3] Meditations 11-12.

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