Last night, I mentioned to my roommate (who’s a mathematician) the fact that some realist philosophers of mathematics have (perhaps in a tongue-in-cheek kind of way) used the infinitude of primes to argue for the existence of mathematical entities. After all, if there are infinitely many primes, then there are primes, and primes are numbers, so there are numbers.
As might be expected, he was unimpressed, and suggested there was some sort of ambiguity in these existence claims. He wanted to push the idea that mathematically speaking, all that it means for a statement to be true (in particular, an existence statement) is that it be provable from the relevant axioms. I was eventually able to argue him out of this position, but it was surprisingly harder than expected, and made major use of the Gödel and Löwenheim-Skolem-Tarski theorems.
The picture that is appealing to the mathematician is that we’re just talking about any old structure that satisfies the axioms, just as is the case when we talk about groups, graphs, or topological spaces. However, I pointed out that the relevant axioms for the natural numbers can’t just be Peano Arithmetic, because we’ll all also accept the arithmetical sentence Con(PA), once we’ve figured out what it’s saying. This generalizes to whatever set of axioms we accept for the natural numbers. As a result, Gödel’s theorem says that we don’t have any particular recursive set of axioms in mind.
To make things worse for the person who wants to think of the natural numbers as just given by some set of axioms, there’s also the fact that we only mean to count the members of the smallest model of PA, not any model. By the LST theorem, any first-order set of axioms that could possibly characterize the natural numbers will also have uncountable models. In order to pick out the ones we really mean, we either have to accept some mathematical class of pre-existing entities (the models of PA, so we can pick out the smallest one), or accept a second-order axiomatization, or give up any distinction between standard and non-standard natural numbers.
Of course, the same argument can be run for set theory, and at this point, accepting a second-order axiomatization causes trouble. Some philosophers might argue a way around this using plural quantification, but at least on its face, it seems that second-order quantification requires a prior commitment to sets, so it can’t underlie set theory. And we really don’t want to admit non-standard models of set theory either, because they give rise to non-standard models of arithmetic, which include certain infinite-sized members as “natural numbers”.
So it seems, for number theory and set theory at least, the mathematician has to admit that something other than just axioms underlies what they’re talking about. I’ve argued that the existence of and agreement on these axioms means that mathematicians don’t often have to think about what’s going on there. And from what I can see so far, the mathematician could take a platonist, fictionalist, or some other standpoint without any trouble. And in fact, most of the reasoning is going on at a high enough level that even if there’s some problem with the axioms (say, ZFC turns out to be inconsistent) this will have almost no effect on most other areas of mathematics, just as a discovery that quantum mechanics doesn’t correctly describe the behavior of subatomic particles will have relatively little effect on biology. But the mathematician can’t insist that existence in mathematics just means provability from the axioms – she can insist on some distinction between mathematical existence and ordinary physical existence, but it will be something like the minor distinction that the platonist recognizes, or possibly something like the distinction between real and fictional existence.