Here’s a quick argument against Platonism about mathematical ontology.
Premise 1: For everything that exists, it is conceptually possible that it not exist.
Premise 2: If Platonism is true then the truth of ‘9 is not prime’ depends on the existence of the number 3.
Premise 3: If the truth of p depends on condition X and it is conceptually possible that X not obtain, then it is conceptually possible that p is false.
Premise 4: No conceptual truth is such that it’s conceptually possible that it’s false.
1. Platonism is true. (Assumption.)
2. The truth of ‘9 is not prime’ depends on the existence of the number 3. (From (1) and Premise 2.)
3. It is conceptually possible that the number 3 not exist. (From (1) and Premise 1.)
4. It is conceptually possible that ‘9 is not prime’ is false. (From (3) and Premise 3.)
5. ‘9 is not prime’ is a conceptual truth. (Assumption.)
6. Contradiction. (From (4), (5) and Premise 4.)
7. Platonism is not true. (From (1) and (6).)
This argument is valid and rests only on premises 1-4 and the assumption that it’s a conceptual truth that 9 is not prime. I won’t consider challenging that assumption here: it seems to me absurd to deny that it’s a conceptual truth about 9 that it is divisible by a factor other than itself or 1. So the Platonist must deny one of the four premises. Premise 4 is analytic, so it’s premises 1-3 that are of interest.
Premise 3 seems to me to be overwhelmingly plausible. How could it be conceptually necessary that something be true whilst being conceptually possible that the conditions required for its truth not obtain? If it’s conceptually necessary that a thing, A, is a certain way, F, then the truth of ‘A is F’ is guaranteed by our very concept of what A (and F) is; so if there is any further condition on the truth of ‘A is F’ it simply must be the case that the fact that this condition obtains is also guaranteed by our very concept of what A (and F) is. If it’s conceptually possible that this condition not obtain then either it’s not a condition on A’s being F after all or we should hold off on a judgment as to whether A is in fact F until we know whether the condition is met, in which case ‘A is F’ is not conceptually necessary.
So I think the real action is on premises 1 and 2. Premise 1 is the claim that there’s nothing such that our very concept of that thing guarantees its existence. It would be denied by proponents of the ontological argument for the existence of God – but that doesn’t bother me, since that argument is hopeless. And in any case, defenders of it will likely hold that God is the only being that constitutes such a counterexample, and so the argument will still go through if we build in the assumption that the number 3 is not God. (Even Trinitarians, in declaring that God is 3, probably don’t literally mean that God and the number 3 are numerically identical!) The possible exception of God aside, Premise 1 seems plausible to me. Even if the existence or otherwise of mathematical ontology is a metaphysically non-contingent matter, it still seems to me that it is a conceptually contingent matter: whatever the truth of the matter is between nominalism and Platonism, nothing about our concepts of mathematical ontology rules out that it be otherwise.
Premise 2 is, I think, the best premise to give up. But the thought in favour of it is very simple: when a number is not prime we explain why by appealing to an existential – there is some number other than it or 1 that is its factor. But if existence is in general a conceptually contingent matter then we can make sense of all the numbers existing except those factors. As far as our concepts go, we can make sense of all the numbers existing except 3, in which case there simply is no factor of 9 other than 1 or 9 itself. Were that the case, 9 would be prime. Now, of course, the Platonist can rightly insist that this is metaphysically impossible – but the argument here is that the conceptual possibility is worrying enough.
I think the best thing for the Platonist to do is to resist the thought that the non-primeness of 9 is hostage to fortune to the existence of the number 3: that is, to deny the dependence claim. This would be to claim that the existence of the factors is not a necessary condition on it being true that the number in question is so divisible. So the conceptual possibility of the non-existence of 3 doesn’t threaten the conceptual necessity of 9’s being divisible by 3. But once one does this, one starts to pull apart the truth of mathematical claims from the apparent ontological demands of those truths, which undercuts the motivations for Platonism in the first place. If ‘9 is divisible by 3’ doesn’t depend for its truth on the existence of 3, why think it depends on the existence of 9 either? If we start going down this route, why not just follow it to its natural non-Platonistic end, where the truth of mathematical claims doesn’t depend on the existence of mathematical ontology?