### Alain Badiou, part II

[I recently stumbled across this series of video lectures on *Being and Event* and was quite impressed by the presenter's clarity and passion. Unfortunately they are somewhat marred by the presence of one asshole who is constantly interjecting with pretentious comments; but I chalked this up to the ubiquity of "that kid" in any conference class. I was understandably upset when the asshole turned out to be the lecturer for Parts II and III of the book. At any rate, I recommend the lecture on the first part as a great introduction to Badiou's program, which after all is very ambitious and takes a good half-hour to summarize coherently.]

In the last post we looked at how Badiou thinks math gets Being right- that mathematics is ontology. Another major theme of *Being and Event* is of course the event. Unsurprisingly, Badiou also thinks that math has a lot to tell us about what the event must be as well.

**Badiou’s event**

What doe Badiou mean by the event? He is not talking about everyday happenings (the sum of which I think is what he terms “history”); rather Badiou means revolutionary events in any of the conditions of art, love, politics, or science. Examples include the development of atonal music under Schoenberg, the development of set theory under Cantor and Frege, and, Badiou’s favorite example, the French Revolution. (Badiou never gives an example of revolutionary love, or indeed has much to say about this condition in *Being and Event*; but I recommend the delicious and short interview *In Praise of Love* for anyone who wants a refreshing perspective on the possibility of real human contact.)

To Badiou, the event is ontologically interesting because it contains itself. Consider the French Revolution. The Revolution contains more than just the *Declaration of Rights of Man and of Citizen*, the assassination of Jean-Paul Marat, and so on; it contains more than just all the happenings in France between 1789 and 1799; it contains, in an important way, *itself*. Even as it was ongoing, participants would refer the French Revolution by name as conceived whole. In this way, the French Revolution is different from an arbitrary collection like the set that contains Alaska, the Moon, and Plato. The participants were aware that they were creating a particular Revolution; compare this to the way an art movement like Impressionism is only fully formed when it acquires its name and artists identify themselves as belonging (or not) to the movement. Movements that fail to be revolutionary often dissolve into merely a collection of happenings.

Badiou denotes the event *X* as follows: *X* = {*e _{X}*,

*X*}. Here

*e*is the evental site (e.g. France 1789-1799), and

_{X}*X*is the proper name of the event. Badiou wants to say that the event is a paradoxical object because of how it belongs to itself.

**Self-containment and Russell’s paradox**

Early set theorists (Cantor, Frege) had a naive conception of set whereby a set is merely the collection of entities picked out by a concept, and any concept defines some (possibly empty) set. This is after all how most working mathematician still conceive of sets when they use, e.g., the set of prime numbers. But such a permissive definition in fact leads to many famous paradoxes, chief among them Russell’s paradox.

Russell’s paradox precedes as follows. Consider the concept ‘is not an element of itself.’ Surely this is a valid property, one which most ‘normal’ sets (like the set of all prime numbers) enjoy. However, consider the potential set *S* corresponding to this property; that is, let *S* be the set of all sets that don’t contain themselves. The paradox arises from asking whether *S* contains itself. If it contains itself, it ought to not; and if it doesn’t, it should. Another way of putting it is as follows: the Barber of Seville cuts the beards of all men who do not cut their own beard. Does the Barber of Seville cut his own beard? (Solution: the Barber of Seville is a woman.)

Russell’s paradox is the basis of all analytic philosophy. This is not because it is especially profound; in fact, it is an inane and obvious problem (so obvious that apparently Zermelo knew of it before Russell communicated it to Frege in 1903.) Indeed, Russell’s paradox leads to analytic thinking precisely because it so inanely demonstrates that the use of naive language in any field (even math!- where we would expect things to be clearest) is suspect and likely riddled with incoherence and vagueness. Thus we must carefully build up an ideal language free of paradoxes, full of clarity.

It is sometimes said that Russell’s paradox shows self-containment to be an inherently contradictory concept. But this is not quite right. Russell’s solution to his paradox was to create a system of types for sets so that sets can only contain sets of lower type; this is like saying rank 1 barbers can cut the beards of rank 0 barbers only, and rank 2 barbers can cut the beards of rank 1 or 0 barbers, and so on. So Russell certainly wants to outlaw self-containment. However, Zermelo put forward a simpler and more lasting solution, which is to allow properties not to define sets but only to carve out sets from other sets. For instance, if *A* is some already existing set, we can let *B* be the set of all things in *A* which have property *P*. Now letting *P* = ‘does not contain itself’ is no problem.

But note that this separation axiom does not explicitly disallow self-containment. It is only another axiom, another *choice* on the part of the set theorist, called the Axiom of Foundation, that says no set can contain itself. Whereas separation is constantly used in ‘everyday’ math, the axiom of foundation is almost never used outside of logic and its truth is not at all self-evident.

**What does this mean for Badiou?**

Badiou wants to say the event is not a part of Being because it is self-containing and self-containment is disallowed by set theory, the language of ontology. It would be better to say that self-containment is outlawed by *certain* set theories like ZFC (the most commonly used one.) However other set theories like Quine’s New Foundations allow for a universal ‘set of all sets’ that in particular contains itself.

Badiou recognizes all this, I think. (Certainly New Foundations is mentioned in the introduction.) He would say that the Axiom of Foundation is the right choice to make; it is a true axiom that no structure contains itself. Perhaps. But often it would be convenient in math to think of a structure as containing itself (see for instance this recent question on a forum dedicated to research math questions.) And *self-similarity* is a very important concept in math; indeed, the correct definition of an infinite set is one which is similar (in number) to a part of itself.

**What’s next**

Right now I’m reading Badiou’s *Number and Numbers* which is more straightforwardly a book on the philosophy of mathematics. In it, Badiou proposes that the surreal numbers of John Conway finally describe all that is meant by “number.” I may talk about the surreal numbers in a future post, because at any rate they are a fun concept and are more related to combinatorics (my research interest) than a lot of what I’ve talked about lately.

I also hope to discuss the philosophical writings of Gian-Carlo Rota at some point, but first I have to read them!