Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article link text that says:
This effort leads him, in Being and Event, to combine rigorous mathematical formulae with his readings of poets such as Mallarmé and Hölderlin and religious thinkers such as Pascal.
Badiou's got some mathematical training; reading back and forth between the relevant sections of Goldblatt's Topoi and Badiou's account of $\Omega$-sets in Logics of Worlds, for example, you can see that the one tracks the other closely. It's not just blind quotation, followed by hand-wavy inference-drawing either: you could actually learn about $\Omega$-sets from Badiou's presentation of them alone, and not be too horribly surprised or confused when you came to read the technical presentation in Goldblatt (this was, in fact, the order in which I did it).
On the axiom of choice and "infinite liberty":
The AoC says that given a set $\{A,B,C,\ldots\}$ none of whose members are the empty set, there exists a set $\{x\in A,y\in B,z\in C,\ldots\}$ which takes one element from each of the first set's members. The point here is that the AoC "freely" chooses an element from each set rather than (for example) identifying a "least" element and choosing that: even when there's no rule that can tell you which element should be chosen, the AoC says that a set exists representing some choice.
The AoC only has any work to do in situations where no rule can be found (for example, no-one knows of a rule that will well-order the reals, but the AoC entails that a real can be chosen, then another from the remaining reals, then another etc. - so "axiomatically" a well-ordering of the reals exists, provided one accepts AoC) - hence it represents, in this sense, the possibility of a predicatively undetermined choice. That's the "infinite liberty" he's on about. It is nowhere asserted that AoC "proves" that such a liberty exists, but rather that introducing AoC into ZF makes such a liberty thinkable within the confines of its axiomatic system (this is in line with Badiou's general program of treating mathematics as "ontology", as a means for systematically demarcating what is thinkable of "being as such").
In terms of "interest to mathematicians": Badiou's early text The Concept of Model is a good philosophical introduction to model theory, and his Number and Numbers is an interesting and accessible guide to the philosophy of number, covering Frege, Peano, Cantor, Dedekind and Conway (surreal numbers).
There's an interesting review of Badiou's "Number and Numbers" at the Notre Dame Philosophical Review by John Kadvany.
I don't know if the paragraph that talks about the relation between set theory and his philosophy is actually representative of what the man truly thinks but I'll give it a try assuming that it is.
There are moments of lucidity in those paragraphs about set theory being an axiomatic system for talking about collections but the rest is founded on shaky assumptions and a very loose interpretation of the lessons that set theorists learned some time ago about unrestricted comprehension and the confounding of types. In general it is not a good idea to use axiomatic systems and their properties to argue for or against the existence of certain things since axiomatic systems are by design meant to be interpreted and existence even in the mathematically precise sense is still not precise enough. I mention this because at some point Russell's paradox is used to argue against the existence of god, which kinda means whoever wrote this article doesn't really know much about axiomatic and logical systems. So all the parts that reference any kind of mathematical theory should really be thrown out and there isn't much left if you do that.
No, there is absolutely nothing at all. It's empty metaphor. I read through the wikipedia page, and the thing about the axiom of choice stood out as nonsense, so I looked it up. Badiou associates the axiom of choice with "anarchic representation" and "a principle of infinite liberty." That clinched it for me.
I don't know if this was the answer you were looking for, but it looks to me like standard philosophical tripe.
About blog discussions: it is true that mathoverflow far from ideal for discussion but it does provide many opportunities for discussions and blogs often cannot compete. I tried to transport a discussion about planar graphs from MO to my blog with little success. There are only few blogs with genuine mathematical discussions. Discussions in MO is like sex in cars; it is a terrible platform, but often it provides the only available opportunity.
– Gil Kalai Dec 09 '09 at 12:15