Logical difficulties in modern mathematics (2012)

In my experience, people won't come out and say it, but this seems to be what everyone is thinking. :)

The problem with this is that it is wrong.

Classical ZFC in particular is a very strong and specific set of assumptions* with a very tenuous link to any practical application. If you actually want to develop a useful bit of mathematics it makes sense to consider the "foundations" as a moving piece. It's a part of the design space for modeling your problem domain, not some god-given notion of truth.

You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things.

---

*) I mean that second-order ZFC has basically no non-trivial models, so there is no real way of extending ZFC to talk about domain specific aspects of your problems.

Everything in mathematics is divorced from reality. Unbounded integers are divorced from reality. (Once you move beyond naive realism, bounded integers are divorced from reality, but that's a deeper philosophical debate.) The only question is whether these models are more or less effective for their various theoretical and applied purposes.

> There are better approaches to measure theory which live in different "foundations". For example, you can build measure and probability theory based on the locale of valuations on a locale instead of a sigma-algebra on a topological space.

Better by what definition? According to the practical needs of students, pure and applied mathematicians, etc? I've studied some topos theory and know a little bit about locales from the Topology Via Logic book, but it's hard for me to see that as anything more than a fun curiosity when considering the practical needs of mathematics as a whole. In my mind that kind of thing is much closer to navel-gazing than something like measure theory.

> It is really not the most straightforward approach.

The onus is on critics to do better. Dieudonne/Bourbaki made a valiant and elegant attempt even if they intentionally snubbed the needs of probability theory. And "better" will obviously be judged by the broader community.

> You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things.

I don't see how these two are consistent. Almost everything most mathematicians do can be done both in ZFC and your favourite non-kooky axiom system. Certain Powers That Be seem to have decided that ZFC is the foundation of mathematics, so they say that what they're doing follows from ZFC even if they have a very hazy idea of what it is, but why does it matter? Most mathematics probably won't be formalized in their lifetime anyway, so whether it ends up being formalized on top of ZFC or something else doesn't affect them.

Mathematics is an abstraction, but it is still useful for talking about concrete problems. Your mathematical assumptions can be either close or far away from your problem domain. Sometimes we introduce idealized objects, such as unbounded integers, in order to abstract further and simplify our reasoning.

These ideal objects can then either be "compiled away" in specific instances, or really do ignore corner cases which might invalidate your results.

For an example of the former, you can assume that there is an algebraically closed field containing a specific field, give an argument in terms of this closure and then translate this argument to one which does not construct the closure explicitly. The translation is mechanical and does not represent additional assumptions you made.

The second kind of ideal object is something like the real numbers applied to physics. We can think of a real number as an arbitrarily good approximate result. In practice we can only ever work with finite approximations. At the scales we are operating on the difference is usually not relevant, but there might, for example, be unstable equilibria in your solutions which are not physically realizable.

> Better by what definition?

Informally, better because it is "simpler". There are fewer corner cases to consider, theorems are more inclusive, constructions are more direct.

Formally, the theory has more models and is therefore more widely applicable. Theorems have fewer assumptions (but talk about a different and incompatible type of objects).

> The onus is on critics to do better. Dieudonne/Bourbaki made a valiant and elegant attempt even if they intentionally snubbed the needs of probability theory. And "better" will obviously be judged by the broader community.

Oh, sure, but that's not what I want to argue about.

I can tell you with certainty that classical measure theory is complicated by the interplay of excluded middle and the axiom of choice. This is a technical result. You can see this yourself in textbooks every time the author presents an intuitive "proof idea" which then has to be refined because of problems with the definitions. In alternative models, or in a metatheory with alternative assumptions, the simple proof idea usually works out fine.

> "Almost everything most programmers do can be done both in x86 assembly and your favorite non-kooky programming language. Certain Powers That Be seem to have decided that x86 is the foundation of computer science. [...] Why does it matter?"

The problem is that it is difficult to translate results in a theory built in ZFC to other "architectures". In mathematics, the architectures in question are not different axiom systems, they are different branches of mathematics.

Let me give you an example. There is a large body of work on differential geometry with many useful constructions. Classical differential geometry works directly in a model where manifolds are certain subspaces of (countable products of) R^n. These constructions have been successfully imported into many different areas of mathematics. In most cases people just had to tweak the definitions slightly and adapt the proofs by keeping the basic strategy and changing all the details.

What is happening here is that the underlying ideas of differential geometry are not specific to this particular model.

When faced with such a concrete model, our first instinct should be to abstract from it and ask which assumptions are required. This is difficult in ZFC, because in the end you have to encode everything into sets. It's not possible to reason about "abstract datatypes" directly, without literally building a notion of logic (generalized algebraic theory) and models of it within ZFC. Even then, the existence of choice means that you usually have to exclude unwanted models of your theory.

Coming back to differential geometry: You can generalize a lot of it by working in a differentially cohesive (infinity-)topos. This is terribly indirect (in my opinion) and looses a lot of the intuitions. A topos is literally a model of a certain logic. Alternatively you can work directly in this logic (the "internal language" of the topos), where the differential geometric structure is available in the form of additional logical connectives. You are now talking in a language where it makes sense to talk about two points being "infinitesimally close" and where you can separate the topological from the infinitesimal structure.

At the same time you reap the benefits that there are many more models of differential cohesion than there are models of "R^n modeled in classical set theory". You can easily identify new applications, which might in turn suggest looking into different aspects of your theory. It's a virtuous cycle. :)

This approach is deeply unnatural when working in set theory or arithmetic. You have to encode everything into sets or numbers and then these sets or numbers become the thing you are studying.