In my previous post, I argued mathematics can be considered as an highly sophisticated, fractal language in which ideas are layered on each other to build very tall mathematical buildings. Rigourous proofs are the strong mortar keeping the tower standing up. All of this was philosphical and suggestive, and stemmed from the evergreen question 'of what use is math?', and I'm still satisfied by that answer.
On the other hand, I recently come to realize that there's a more technical way in which one could argue 'math is a language'. And if my previous post might have made some people turn their nose because of the handwaving, philosophical remarks, this time we are talking actual mathematics, or at least metamathematics.
Traditionally, math is thought founded in/on sets. This means that the entities you talk about in math are assumed to be sets. In the most orthodox set theories, everything is really a set, even things we don't usually think of as sets, for instance numbers. This is called material set theory, and I think of it as axiomatizing the 'atoms' of which mathematical matter is made of. Since there's no dialectics going on between atoms and the forms they make, the first are always the same, immutable and do not see the bigger picture.
Material theories are not bad per sè, though I would argue they are far from the mathematical practice, i.e. we do not think of everything as sets. Some things are sets, sure, but some are just not. Numbers do not make sense as sets. Yeah, maybe counting numbers, right. But real numbers? Whose intuition is grounded on the concept of real numbers as Dedekind cuts? I know of no one.
This is even more apparent by the fact that (a) most people are basically oblivious of this fact and (b) nevertheless, we really don't care about 'the structure of set' on most of the objects we use. For example, when you describe a map between, say, rings, you may prove it is a well-defined map of rings, but I've never seen anyone checking it is a well-defined set as well. It'd be trivial, of course. Yet nobody even mentions it, which allows us to build a case against a material foundation as natural foundation for mathematics.
That said, facts (a) and (b) can also be read in favor of materal views. In fact a good foundation 'stays out of the way', so to speak, meaning it doesn't obstruct the study of your object of research with annoying technicalities or bookkeeping. Can this be said, say, of type theory?
Both this cases, however, have a common point: mathematical practice is usually not concerned with foundations, as long as they are solid enough to not fail us, and as long as they provide the necessary tooling to carry on working on the objects we are interested with. In other words, we could say that most of mathematics is 'foundations invariant', i.e. it doesn't really bother to switch from, say, ZFC to NBG.
What is preserved, then, in changing foundations? The answer is quite easy once we conceeded ourselves sufficient meditation. It's language.
The point is that soundness and power of tooling are properties of the language we use to describe mathematical theories. Sets have a powerful and (hopefully) sound language, which allows mathematicians to go on undisturbed much of their time. But since they never endorsed sets explicitly, we arrive to the conclusion that if we were to switch to an equivalently powerful foundation nobody will notice.
This was quite liberatory when I realized it. In fact sets impose quite a strong ontological view on the universe of discourse of mathematics, thus it is liberating to see mathematics is actually independent from them. It is awkward to think mathematics can only be made with sets, that algebra, geometry, analysis and so on are just 'emergent properties' of sets. Why would it be so?
Instead, it is now clear tht theories are independent and meaningful on their own. Given a sufficiently powerful foundation, a theory can thrive on its own.
All of this becomes more contentful in light of topos theory. A topos is a category whose internal language is sufficiently powerful to support many of the theories of everyday mathematical practice. The major drawbacks of a general topos are (1) the lack of nonconstructive principles such as LEM or AC and (2) the lack of infinite sets like the natural numbers. These could startle the reader as too big of an obstacle to ever take seriously the option of moving from sets to other toposes, yet this is nonsense as we cannot be castrated by having more choice than what we have now. If we need infinite objects, we just declare it. If we seriously need LEM/AC, we do it as well.
I'm not arguing for rebasing all of mathematics on an arbitrary topos, or for structural set theories like ETCS. I'm just noticing a simple fact: we mathematicians talk, and the objects we deal with are made, first of all, by our discourses.