> Some philosophical ideas seem increasingly unsustainable in the face of mounting scientific knowledge. Vitalism, for instance.
OK, but the philosophy of mathematics is a more slippery beast.
> But if you do include it, don't you have to commit arbitrarily to either AC or ¬AC?
No. You can have a theory that's consistent with either.
> Isn't this 'just' a question of knowing that the correspondence is valid?
Correspondence to what? Are those objects platonic ideals? Are they abstractions of physical reality? You can answer this question in many ways. But whichever way, this question of soundness (formal provability corresponds to semantic truth) rests on us knowing what is true.
Some philosophies of mathematics say that the question is unimportant: it doesn't matter if we get the "truth" part right, all that matters is that mathematics is a useful tool for whatever we choose to apply it to (this doesn't even mean applied mathematics; one valid "use" is intellectual amusement).
> No. You can have a theory that's consistent with either.
I think I see your point. If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all.
> Are those objects platonic ideals? Are they abstractions of physical reality?
My intuition is to favour the former, as the latter seems a much stronger claim.
It seems reasonable to say that any mathematical system exists as a platonic ideal, but it's plain that not all mathematical systems abstract physical systems, and we shouldn't be quick to say that we know that any mathematical system does so. That's an empirical question that - true to Popper - can never be positively proved.
> If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all.
That's not what I meant. I meant that choice can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false.
> My intuition is to favour the former, as the latter seems a much stronger claim.
Well, Platonism is a popular philosophy of mathematics among mathematicians, but many strongly reject it.
> can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false
So it's similar to, but not the same thing as, modifying the LEM to permit Unknowable as a third category (beyond the proposition is true and its negation is true). Instead we maintain that it has a truth value one way or the other, but that it happens to be both unknowable and of no consequence.
> modifying the LEM to permit Unknowable as a third category
LEM doesn't need to be modified, and it's not a third category. In classical logic (with LEM), unlike in constructive/intuitionistic logic (no LEM), truth and provability are simply not the same -- in fact, it is LEM that creates that difference.
> Instead we maintain that it has a truth value one way or the other, but that it happens to be both unknowable and of no consequence.
Precisely! That is the difference between classical and constructive logic. In constructive logic truth and provability are the same, and something is neither true nor false until it has been proven or refuted. But before you conclude that this kind of logic, without LEM, is obviously better (philosophically), you should know that any logic has weirdnesses. What happens without LEM, for example, is that it is not true that every subset of a finite set is necessarily finite. ¯\_(ツ)_/¯
OK, but the philosophy of mathematics is a more slippery beast.
> But if you do include it, don't you have to commit arbitrarily to either AC or ¬AC?
No. You can have a theory that's consistent with either.
> Isn't this 'just' a question of knowing that the correspondence is valid?
Correspondence to what? Are those objects platonic ideals? Are they abstractions of physical reality? You can answer this question in many ways. But whichever way, this question of soundness (formal provability corresponds to semantic truth) rests on us knowing what is true.
Some philosophies of mathematics say that the question is unimportant: it doesn't matter if we get the "truth" part right, all that matters is that mathematics is a useful tool for whatever we choose to apply it to (this doesn't even mean applied mathematics; one valid "use" is intellectual amusement).