Thursday, 19 June 2008

Badiou and the mathematical group-subject.

A fascinating footnote in Hallward's Badiou: A Subject to Truth relating the advantages of category theory for his project:

Among other things, category theory allows Badiou to reformulate his theory of the subject, conceived as a group in the mathematical sense (Court traité d'ontologie transitorie, p.165-77). In categorial terms "a group is a category that has one single object, and whose arrows are all isomorphisms" (p.172). This sole object provides the name of the group, its "instantiating letter". Call it "G". Its arrows all go from G and end in G, and these provide its "operative substance." they emphasise the fact that what makes a group a group, what makes G, G, is "the set of different ways in which the object-letter G is identical to itself." Any such "group-subject is infinite" that is, the number of ways it has of being identical to itself is inexhaustible" (Court traité d'ontologie transitorie, p.176).

Badiou: A Subject to Truth (p.418, n.32)

What we can see here are definite vestigial traces, albeit in highly mathematised forms, of Badiou's Sartrean legacy at work. If, as I understand it, a subject is here defined as a group through its manner of self-identity, expressed via the "sole object", then there are parallels with Sartre's non-mathematical group's identity being forged via the relations with the third party, seen by each member as "the common individual". Obviously, I think this may well be somewhat of a simplification, but it would be an intriguing argument to trace elements of Badiou's underlying mathematical ontology back towards a CDR-era Sartrean schema. Adding to these complications however is the fact that here we are clearly dealing with topos/category theory, rather than the earlier and slightly more transparent set theory (used as the basis of Badiou's discussion of being-qua-being in Being and Event) and hence why this begins to stray into the territories explored by Badiou in Logics of Worlds. Referring again to Hallward on the distinctions between these methodologies:

Category theory and set theory offer opposing approaches to "all the decisive questions of the thought of being (acts of thought, forms of immanence, identity and difference, logical framework, admissible rationality etc...)" (Topos, ou logiques de'lonto-logique: Une Introduction pour philosophes, tome 1 p.5) They are, in short, different ways of conditioning philosophy. If the first volume of l'Etre et l'envenement was written under the sole mathematical condition of set theory, the second volume will be written under the double condition of both theories... Where set theory directly articulates being-as-being, category theory is "the science of appearing, the science of what signifies that every truth of being is irredemably a local truth" (Court traité d'ontologie transitorie, p.199)... in categorial logic "relation precedes being" (Court traité d'ontologie transitorie, p.168)... The theory offers "a highly formalized language especially suited for stating the abstract property of structures." ... the theory is designed to allow for the most general possible description of logical relations or operations between such structures and entities... Methodologically, the way these structures are represented is essentially geometric: isolated manageable parts of a category are expressed in diagrams... These diagrams are made up of objects, on the one hand, and of arrows or "morphisms" on the other. Arrows are "oriented correlations between objects; an arrow 'goes' from object a (its source) to object b (its target)." ... The single most important principle of category theory is that all individuating power or action belongs to these dynamic arrows alone...

Badiou: A Subject to Truth (pp. 303-306)

All this certainly goes someway to explaining how a category theory "group" can be constructed. The type of category known as a group is specifically designated by a structure consisting of only one object, but an infinite number of isomorphic arrows (each beginning and ending with the object). Diagrammatically perhaps (!):

What remains to be seen however is the manner in which Badiou applies this: is he getting at a kind of internal structure to the group-subject? Certainly it seems his deployment of category theory (specifically the world-building device of the sub-species of category theory known as topos) is designed to enable him to think through the local logical implications of decisions set in motion by truth process sequences. Further this brings us dangerously near towards the vexed question of relationality, (for that is what the logic of appearance in Badiou's system ultimately enables him to do, to think relations rather than merely bare being-qua-being as in the ontological discourse of set theory). This idea of relationality in many ways can be seen as operating at the core of dialectical reason and transversality also, and hence will have to be a central theme of this work.

7 comments:

David said...

I may be entirely wrong about this, it's 7 years since I did any set theory after all, but your diagram of an isomorphism don't look right to me. Or rather, it's kind of missing the point. From what I recall...

The isomorphism is a kind of function, a mapping between two sets, but one with two special features:

- One-to-one (a bijection): each element in the initial set is "paired off" with one in the resulting set. E.g. If you take the set of integers and apply the morphism to "add 1", you get the same set of integers with each element in the initial set having a single element in the resultant set associated with it (and visa versa for the inverse of the function, -1). In contrast, the morphism to square root something is not a bijection: the square root of 4 could be +2 or -2.

- Structure-preserving (a homomorphism). The key relations and properties of the initials set are preserved.

Hope that helps. Keen to see how Badiou uses this!

Alex said...

David- thanks for the comment. The way I understand it is that this is Category theory, not set theory... so it is the morphisms which actually give any identity to a given object (and hence two ostensibly different objects with identical morphisms operating upon them are basically the same object). The interesting thing is that category theory places relationality as primary, whereas set theory takes (in Badiou's understanding at least) being as primary. In his formulation in Logics of Worlds relationality remains a secondary feature to being (ie: he subordinates category theory to set theory)-- which is kind of an abuse of category theory from what I can tell. However I'm almost certain my hastily constructed diagram is bunk! I need to read up on this stuff as its kind of crucial to my argument I think.

Alex said...

Or maybe I'm just confused. Any help here would *really* be appreciated...

David said...

Isomorphisms aren't distinct in category theory and set theory though - it's a general definition of a mapping that's generic to all abstract algebra.

As I understand it, the distinction between set theory and category theory is that set theory is about axiomatising pure being, which runs into the CH hypothesis problems, whereas category theory is a bit more specific in looking at how a priori defined groups are structured. Crucially, I think in category theory you would always have a set defined by some collective definition that brings the group together (the set of all obese cats, maybe), but in set theory the set is (or can be) defined simply by it being a collection of elements.

Category theory could therefore fall foul to Russell's paradox, but the ZFC axioms avoid that under set theory.

Alex said...

Ah I see reading elsewhere that categories are already defined as sets. Interesting. Another question: How close does category theory come to a Saussure-esque chain of structural signification? It seems fairly similar but this may be to a non-mathematically trained eye.

David said...

Thinking about this again for a moment (I'm avoiding doing work, as per), I remember my initial impressions of what I anticipate Logique du mondes to be about, from the notes in Subject to Truth, were that where Being and Event was a ontological work, Logique... is more epistemological and practical. Being and Event uses set theory as THE grounding for pure (inconsistent) being (equating ontology and set theory), to talk about the structure of the world-as-it-is, where Logique... uses category theory as the grounding for consistent entities (i.e. categories = institutions, norms, the state, etc.), to talk about the structure of the world-as-it-appears and is understood, and therefore offers a more practical ethics (rather than just waiting around to be touched by the hand of THE EVENT).

Perhaps?

Alex said...

Ok, yes I think you're along the right lines here re: BE vs LW, though I would add that surely set theory/ontology for Badiou doesn't tell us much about the structure (in the more usual sense of the word) of the internal components of a set? The "structure" (count as one) and "metastructure" (state) operate merely to secure the appearance of unicity or that the things being counted are a set at all. They don't actually as far as I can tell detail the internal relations between things. Set theory, as Badiou presents it anyway, merely counts collections of items. To orientate the particular interrelations within them you need CT, which will be able to tell you a bit more than merely "in or out" "belongs or does not belong" etc.

Your last point is interesting- re: CT grounding a more practical ethics. I think it is a bit of a misreading to suggest that we merely have to wait around for an event to occur: there are already truth sequences in process, we are quite free to accede to them surely? But whether CT enables a more active description of what a truth process might be in practice (rather than the somewhat abstract idea of connection/disconnection)...? I think I need to read LW to answer that though...