I’ve been asked to comment on the mathematics in Badiou’s book Being and Event. Here goes.

Misunderstanding Badiou

I’m almost certainly misunderstanding Badiou. I haven’t yet had the time to investigate what he’s written in detail; it’s dense text, and I suspect most of his terminology doesn’t quite mean what it seems to. Hence the title. I’m starting to suspect that anyone who studies any philosopher probably misunderstands him in some subtle and important way. It may be truly impossible to really understand philosophy, as may be bourne out by the many disputes philosophers have about what other philosophers truly meant.

As this note will probably demonstrate, I’m not qualified to discuss his philosophy.

I am qualified to discuss his mathematics.

The text is heavily mathematical; and the mathematics he uses is well-known foundational mathematics. It appears to be fundamental to his approach, and not to be mere window-dressing. He engages the philosophical question of what mathematics means, what mathematical objects are, what mathematicians talk about when they do mathematics.

He appears to be trying use philosophy to explain Zermelo-Fraenkel set theory (hereafter abbreviated ZF). It’s as if he trusts that axiom system because it is within the jurisdiction of mathematicians (it is, but mathematics is wider thann ZF), and therefore, believing that ontology can best be approached with mathematical precision, he explicates ontology as mathematics. Or vice versa.

That said, it’s not clear to me what ontology is. Yes, you can consider that a recursion joke if you like, but I’m serious here. He defines it as the study of being qua being. This leaves me as much in the dark as before.

In philosophy, the word "ontology" has a rich history, which I’m not really familiar with. But I can see that when Badiou equates ontology with mathematics, it is a radical step. It leaves me in the dark, ecause of the three concepts here, ontology, being qua being, and mathematics, I have a clear understanding of only one, namely, mathematics.

What I do see is a presentation of ZF using obscure language involving the word "multiples" to justify it. It’s as mathematicians had convinced him that ZF was the foundation for mathematics (itself a matter under dispute by those that study foundations) and therefore a study of ontology (whatever that is) as mathamatics necessarily would be a study of ZF.

Now the virtue of ZF is that is is possible to construct sets within it that behave like just about any of the many things studied by mathematicians. But that was already true of the old, more intuitive Cantor set theory, which turned out to be inconsistent. Historically, a lot of different hacks were proposed to patch it up, to wall off the areas where the contradictions lay, without going so far as to imperil the mathematics allegedly built on it. ZF is just one of these that happened to be popular, nowadays usually in the form if its extension, ZFC (ZF with the axiom of choice).

(I find interestng that mathematicians have generally accepted the axiom of choice, even though Solovay’s axiom (all sets of real numbers are measurable) would be more convenient for many purposes. Solovay’s axiom is inconsistent with the axiom of choice.)

I’d love to see the mathematics of an alien civilisation, and what they consider to be foundational. It’s quite possible that they do it completely differently, even if they so end up with a theory of real numbers that is as useful as ours.

And I find myself wondering whether Badiou could have got as far as he did if he had used another foundational formalism. I suspect he would have, but that his book would look completely different, and would argue just as forcefully and obscurely in favour of the other formalism.

Some Other Foundational Formalisms

Category Theory

I haven’t yet seen how category theory provides a coherent foundation for mathematics, but the category theorists seem to think it does. What is clear is that ZFC is not sufficient to build all the things conceived of in category theory. In the terminology of ZFC, all the sets turn out to be too small.

So Grothendieck added another axiom: Every set is a member of some universe. A "universe", by the way, is a model for ZFC; i.e., a set whose elements collectively satisfy all the axioms of ZFC. This extension seems to be needed for cohomology theory, too.

Quine’s New Foundations

Much like set theory, but there is a set of all sets (which would cause the most horrible problems if introduced into ZF). But there is a syntactic constraint:

The variables in a formula must have subscripts attached to them (consistently — no fair having x3 and x4 in the same formula) such that whenever there’s a subformula like x member y, the subscript on x has to be one less than the subscript on y.

Given this constraint on formulas, any formula can define a set to be the set of all sets satisfying the formula. There’s no need to carve a set out of the belly of another set, as there is with ZF.

This blocks the Russell paradox, among others, because you can’t say x member of x, or not(x member of x). Those are just not syntacically valid formulas.

NF has recently been proved consistent. I’m told this has happened before. No word yet on whether there’s an error in the proof this time.

Intuitionistic Type Theory

There’s things called "types". In this formalism, everything in mathematics has a type. Even types have types. For example, integers have the type integer, true and false have the type boolean, and so forth. There are ways to build new types from old. A function type, for example, might be the type of mappings from, say, the integers to the booleans. This type would normally be written integer→boolean.

Propositions are also treated as types. The elements of type p then are, roughly speaking, the possible evidences for the proposition p. So the theory treats logical formulas uniformly with values in the domain of discourse.

An element of the function type p→q would then map the evidence for p onto the evidence for q, Thus p→q amounts to the proposition that p implies q.

I’m not going into a complete exposition here, but there are type constructors for and, or, implies, existential and universal quantifiers, and various inductive types.

This system is espoused by some computer scientists because many of its types bear a close similarity to the data types found in programming languages.

Small Theories

Instead of having a grand shebang underlying all mathematics, some prefer to have many small theories. Someone doing analysis might, for example, be pleased with a set of axioms describing the behaviour of the real numbers, rather than, say, a detailed construction of real numbers from the basics of set theory,

In this style, an axiom system isn’t a bunch of eternal truths, self-evident or not. It’s a set of defining conditions. It says, in effect, if you ever find anything that satisfies these axioms for real numbers, you can call them real numbers and use the theorems with them. These axioms never say what a real number is; instead they tell us what we can do if we ever find anything satisfying them.

Of course, in this formalism, any of the other foundations can be treated as a small theory, with its own defining axiom scheme.

— hendrik, June, 2014