10 August 2006

Category Theory

As I work through Badiou's Logiques des mondes, I've decided that it's necessary to acquaint myself with category theory. We'll see how far I'm get. Currently I'm reading Robert Goldblatt's Topoi: The Categorial Analysis of Logic, which is billed as an introductory text accessible to those who have little or no background in formal mathematics. I have not yet gotten far enough in Logiques des mondes to know precisely how Badiou plans to put category theory to use in articulating his logic of appearances or situations, but already I can be seen just how much category theory differs from set theory. As Goldblatt writes at the very beginning of his text,
The study of topoi arises within category theory, itself a relatively new branch of mathematical inquiry. One of the primary perspectives offered by category theory is that the concept of arrow, abstracted from that of function or mapping, may be used instead of set membership relation as the basic building block for developing mathematical constructions, and expressing properties of mathematical entities. Instead of defining properties of a collection by reference to its members, i.e. internal structure, one can proceed by reference to its external relationships with other collections. The links between collections are provided by functions, and the axioms for a category derive from the properties of functions under composition.

A category may be thought of in the first instance as a universe for a particular kind of mathematical discourse. Such a universe is determined by specifying a certain kind of "object", and a certain kind of "arrow" that links different objects. (1)

The founding gesture of set theory is organized around the idea of set membership, where we think elements in terms of their membership to a particular set. From the beginning I've had problems with Badiou's proposal to found ontology on set theory, as I've always had the conviction that beings can only be understood in terms of their relations, whereas relations are irrelevant to thinking elements of a set. For instance, given a set {x, y, z}, the axiom of extension tells me that there's no difference between this set and the set {y, z, x} as they both contain the same elements.

The problem is that anyone who comes from a structuralist background knows that how elements are ordered makes a profound difference in the nature of those elements. That is, elements of a structure cannot be said to exist independently of their relations, but are what they are precisely through their relations. I cannot order phonemes of a language in any old fashion, nor can I order kinship relations in any old fashions (interestingly Goldblatt uses a quote from Levi-Strauss as an epigraph to chapter 2 of his book), nor can social relations or the Lacanian subject (as a particular way of relating to the Other) be ordered in any old fashion. Take, for instance, the following anagram: "conservation" versus "conversation". From a set theoretical point of view, both of these words are identical as they contain exactly the same elements. Yet at the level of language, the position of the elements makes a deep difference in the nature of these "sets". Something important is thus lost in set theory, which, no doubt, is precisely why Badiou had such difficulties theorizing the structure of situations in Being and Event. No doubt Badiou would respond by arguing that entities such as my anagram above pose no problem for his ontology as ontology discusses only what can be said of being qua being, not specific situations. An anagram is a situation or world appearing in language, not being qua being. Fair enough. However, can we not speak of the being of the anagram? Badiou, of course, will try to demonstrate how category theory is founded on set theory, so as to respond to precisely this sort of elementary criticism.

Interestingly, Lacan's mathemes could be thought as undertaking a sort of vulgar or very basic form of category theory, in thinking various arrows relating one object to another in a structure. What would Lacanian psychoanalysis look like when viewed through the lense of a more rigorous formulation in terms of category theory?

It seems to me that a fundamental ontological decision lies in the choice between set theory and category theory. This does not entail that the choice of one excludes the choice of the other. Both claim to be foundational, and Goldblatt spends a good deal of time showing how set theory can be founded on category theory. Rather, the ontological decision in question revolves around the question of whether membership or relation are foundational in the order of being.


Anonymous pebird said...

So, what is the parallax between set and relation?

I think the dimension of time has to be considered. Does not set theory hold time still so that a set can be established? Relations establish (incompletely in my view) a time dimension through strong/weak links.

I think events are where time is established - but I'm not completely comfortable with that. Time is a property of sets, in that the set is virtual until the event which establishes the set as real. But time requires relationships also - in that what is time without relations?

Anyway, I'm getting in a little beyond my depth. I've got the books on a reading list - thanks.

August 10, 2006 9:28 AM  
Blogger Sinthome said...

PEBird, I hadn't thought of thinking about the relationship between set and relation as a parallax. I'll have to think on this a bit more. One of the things that excites me about Badiou's ontology is the manner in which he emphasizes that there are always an excess of parts over the whole. The power set axiom authorizes me to take the set of all subsets belonging to a set, which are always great there the whole from which they come. From a socio-political idea this idea excites me in that it emphasizes the manner in which any ordered social organization is only a contigent actualization of a whole series of other possibilities (we literally see the initial set defining a social organization break down in periods of revolution, where a number of identifying features defining class identity temporarily break down). This tends to take some wind out of the sails of those who would claim that our being is thoroughly defined through internalized structural features. But I'm still thinking on it.

Generally (and I'm no expert) the dimension of time (in the phenomenological sense) is absent in maths. This is one of the major points of contention between Deleuze and Badiou, where Deleuze sees time and being as co-determining one another, whereas Badiou is strictly trying to think the present. At any rate, deductive relations are not temporal relations.

August 10, 2006 1:55 PM  
Anonymous bobo:) said...

hi Levi,

i still haven't got around to reading any badiou, but from what limited knowledge i have of set theory, there doesn't seem to be any problem deriving relations from set theory assuming the axiom of choice. a relation is simply an n-place predicate which extends over ordered sets of n-tuples.

indeed the axiom of extensionality holds that {a, b} and {b, a} are equivalent. the ordered pairs (a, b) and (b, a), however, are not equivalent. (a, b) is defined as the set {{a}, {a, b}} whereas (b, a) is {{b}, {b, a}}.

i don't know whether this idea of relations ranging over the ordering of sets can be compared to the concept of structural relations, but it does show that there is an ordering to be found in set theory. my brain isn't working well enough right now to formulate any hypotheses on the import of all this to ontology or psychoanalysis.


August 10, 2006 3:40 PM  
Blogger Sinthome said...

Hi Bobo, You're right about relations and set theory. I was reminded of your point about ordered pair this afternoon while reading and smacked my forehead. The ontological question would be whether membership is ontologically primative or whether relation is ontologically primative. That is, are there entities independent of relation or is being a being only in and through relation (in much the same way that /b/ and /p/ have no independent subsistence in the English language, but only are as relation b/p). Badiou treats membership as primative, though matters are complicated by the fact that he works from the premise that sets are always multiplicities of multiplicities without any ontologically simple and self-identical terms like atoms in the Greek sense of the word.

Once again, I don't want to make it an either/or between set theory and category theory. Siding with set theory just means that arrows or morphisms are ontologically primative. You're still able to derive set theory on this foundation. I imagine you'd get off on the category theory stuff if you can penetrate it (mathematicians often refer to it as "abstract mathematical nonsense", which isn't a dismissive term, just an affectionate way of referring to its difficulty). It resonates well with a number of other things you seem to gravitate about with regard to ordered transformations and combinatorials. Goldblatt's book can actually be found online here, if you scroll down:

August 10, 2006 5:15 PM  
Blogger Sinthome said...

My brain must be mush today too... The sentence should read "is *a* being *a* being only in terms of relation" and "primative" should, of course, be "primitive".

August 10, 2006 5:27 PM  
Anonymous r. atonement said...

"Instead of defining properties of a collection by reference to its members, i.e. internal structure, one can proceed by reference to its external relationships with other collections."

I'm struck by how this reads like the mathematical equivalent of differAnce.

August 10, 2006 8:27 PM  
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