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.

Argument:

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?

## 18 comments:

I'm thinking Premise 3 should be rejected. What seems true is this:

Premise 3*: If it is conceptually necessary that the truth of p depends on condition X, and it is conceptually possible that X not obtain, then it is conceptually possible that p be false.

But if it's not a conceptual necessity that p depends on X, then even if

as a matter of fact(and perhaps a matter of metaphysical necessity, for those who believe in that sort of thing) p depends on X, that shouldn't mean that the conceptual possibility of X's not obtaining lead to the conceptual possibility of p's falsehood. It might be that there are some conceptual possibilites where X not obtain and p not depend on X (and is true), and others where X does obtain and p does depend on X (and is true), but none where p depends on X and X not obtain.I'm inclined to think that the platonist/nominalist debate fits in smoothly with this here. What's up for grabs is whether or not "9 is prime" depends on any numbers existing. Unless both sides think the other is engaged in some sort of conceptual confusion (and that charge doesn't seem to be leveled often), each should think the other's claim about dependence (or non-) conceptually possible. So both sides will recognize some conceptual possibilities where 3 not exist and "9 is prime" not depend on 3 for its truth, and ones where 3 does exist and "9 is prime" is true partly thanks to 3's existence.

I'm still thinking about what the best Platonist response is, but I think the argument can be fixed to meet Jason's objection.

I think it's part of Platonism, broadly construed, that it *is* a conceptual necessity that the the truth of "9 is not prime" depends on the existence of 3. That's the point of the last paragraph of Ross's post. If that's right then even as we use Jason's Premise 3* instead of Premise 3, we can replace Premise 2 with:

Premise 2*: If Platonism is true then it is conceptually necessary that the truth of ‘9 is not prime’ depends on the existence of the number 3.

The strengthening to Premise 2* should exactly counteract the weakening to Premise 3*.

But it looks as though Jason anticipates this move in his last paragraph, with the claim that Platonists don't typically think of nominalists as conceptually confused. I certainly think that Platonism is conceptually confused, and I expect no more generosity in return. It's not as if I think that the only thing wrong with Platonism is its ontology, and I don't see what else there is to be wrong with it other than conceptual confusion. Mutatis mutandis for what the Platonist thinks of the nominalist (who isn't an error theorist).

Ok, so it didn't take me long to work out what I think. Platonists should deny Premise 1. As I understand Platonism, it's the view that it's a conceptual truth that mathematical truths depend on ontology, that there are some mathematical truths, and that (at least some) mathematical truths are conceptual truths. Together those claims amount to the rejection of Premise 1.

I guess that whether the Platonist wants to reject Premise 1 or reject my Premise 2* as Jason would suggest comes down to different flavours of Platonism. That is, the Platonist camp is likely divided on which way to go here.

I think the right way for the Platonist to respond to your argument is to challenge Premise 3 (they might also challenge premise 1. I think both of those premises are somewhat less appealing than than you've indicated).

Your premise 3 is:

If p's truth depends on condition X and it is conceptually possible that X not obtain, then it is conceptually possible that p is false.

Or, equivalently:

If p's truth depends on condition X and it is conceptually necessary that p is true, then it is conceptually necessary that X obtains.

However, there is a weaker version which can be accepted by the Platonist without running afoul of your argument:

Premise 3*:

If it is conceptually necessary that p's truth depends on condition X, and it is conceptually possible that X not obtain, then it is conceptually possible that p is false.

Or, equivalently:

If it is conceptually necessary that the truth of p depends on condition X, and it is conceptually necessary that p is true, then it is conceptually necessary that X obtains.

However, the version of the argument with 3* is invalid, because (ARG3) and P3* do not entail ARG4 (where ARGn is the nth step in your 7 step argument).

In order to get a version of the argument that works with premise 3*, you would need to replace premise 2 with:

Premise 2*: If Platonism is true then it is a conceptual truth that the truth of ‘9 is not prime’ depends on the existence of the number 3.

Thus, if we use P3* we can differentiate between those Platonists who take Platonism to be a conceptual truth and those who regard it as metaphysically necessary but not conceptually necessary. The former accept P2* (and thus, face the revised version of your argument), while the latter avoid the revised argument by rejecting P2*.

When it comes to the platonist who takes platonism about mathematical objects to be a conceptual truth (in other words, the philosopher who thinks that it is conceptually necessary that: if there is a successor of 1 and 2 is the successor of 1, then the number 2 exists), it seems obvious that such a philosopher regards your premise 1 as false (and, stronger than that, actually, they would regard the negation of premise 1 to be a conceptual truth).

Well, it would appear that the process of actually typing up my comment took too long, and I got beaten to the punch by Jason. Apologies for my (largely) redundant comment.

Daniel: I agree that, *if* a platonist thinks that the dependence claim is conceptually necessary, they should deny Premise 1. I'm less confident in the antecedent, though. The way I'm thinking of conceptual possibility, if you can make sense of an idea, it's conceptually possible. So unless the platonist is going to, van-Inwagen-style, insist he "does not understand" what the nominalist is saying when the nominalist says that "9 is prime" doesn't need 3 to be true, I'm thinking he's treating the dependence claim as conceptually contingent.

Lewis: Sorry about that! If I had known you were writing up a comment on the same lines, I would have left you to it and gone to bed instead.

I'm pretty much in agreement with everything Daniel said. I had thought about the objection that Jason and Lewis raise against premise 3 but didn't discuss it because I still think 3 is true as it stands, and that even if it isn't the argument could be repaired in the manner Daniel suggests. Jason suggests the Platonist will reject that it is a conceptual necessity that the truth depends on the existence of the number - maybe we need to hear from some Platonists, but I'd be surprised if this was the route they took - and I'd be interested to know what reason they think there is to hold that the dependence claim is true if its truth is not guaranteed by the concepts of the numbers. But as I said, I still think the weaker 3 is true as it stands: if it's merely true that p depends on the conceptually contingent conditions X but not conceptually necessary then you might *think* that p is conceptually necessary, but I think you'd be mistaken: what's conceptually necessary is just that if X obtain then p.

(There may be some amount of us talking past each other here. Jason says he means by conceptual possibility that you can make sense of the idea; I mean by conceptual necessity that the truth is guaranteed by the concepts involved. Those may connect, but they're not obviously the same thing.)

I also agree with Daniel that the Platonist and nominalist should think of the other as conceptually confused. (I also think they tend to, even if it's not always explicit.)

I agree that the premises might be less appealing than I admit. I guess I kind of think there are looser standards in a blog post. Daniel suggests the Platonist should reject Premise 1. I suppose that would be what the neo-Fregean would do. But it seems to me like a massive cost: P1 looks independently appealing to me, for any domain of entities. Okay, the Platonist believes stuff that entails its negation - but without independent reason to deny it, I think that's a cost for Platonism.

Hey guys,

I'm inclined to think that we can view this kind of argument as a choice point. Some Platonists (neo-Fregeans?) might think that Premise 1 is false, particularly if they think that it's analytic that numbers exist (in some sense). Others (Proper Quineans?) might reject all the chat about conceptual truth. Others might do other things like reject (3) or some suitably finessed version of it. Pick your poison.

Incidentally, there is something bugging me about (5). If it's a conceptual truth, it's a truth. And it has a singular term in it. Put those things together, and it's natural to think there is something the singular term refers to. But then it surely refers to the number 9. So there is a bit of a weird aspect to the argument if this is meant to be an argument that the nominalist is offering against the Platonist.

Yeah, if "conceptual necessity" means "truth is guaranteed by the concepts involved", then I think the Platonist (traditional, non-neo-Fregean) should deny Premise 4. (It's a conceptual truth, in this sense, that *if* there are numbers, then..." but not the unconditional thing.) I take it this is one way to take Rich's point...

Yeah, I'm sure everyone here knows this, but there is quite a bit of discussion relating to this stuff in the Field vs. Hale-Wright exchange about conceptual contingency and the like.

Yeah, it was through re-reading the Field v Hale/Wright stuff that I started thinking about this (after getting frustrated about Field being totally misrepresented in some of the metaphysics of maths literature - not by H&W obviously!) I think Field's view is pretty much what you should hold *if* you accept the metaontology that tells you that the existence of numbers is a demand of the truth of mathematical claims. I don't accept that metaontology, but I think the only reason to be a Platonist is that you do. So really, I think Platonism is kind of unstable: the metaontology that drives you to it should really drive you to be Field. (Of course, I don't take myself to have argued for this here!)

I can see where you're coming from. But I do think that if your playing Field's game --- and I don't think that the neo-Fregean is playing that game --- then (5) is off limits for both the platonist and the nominalist. The very best we get is "if there are numbers, then 9 is not prime", or somesuch. Only the neo-Fregan thinks we have the unconditional thing. But the neo-Fregean will reject (3).

Yeah, but it's one thing to conditionalize the truth of the mathematical claim on the existence of the numbers it's *about* - it looks to me a lot odder to conditionalize the truth on some *other* number. (This is why I'm going for the example of primeness, rather than just any old mathematical property of any old number.) I agree different theorists might, post theorising, have theoretical reason to abandon or accept certain claims - but if those claims are intuitive, that reveals a cost to the theory.

It doesn't seem to odd to me to have someone in agreement with Field's methodology to say 'Well, whether or not there are truths about 9 depends on whether or not 9 exists. So it's contingent whether or not there are such truths in whatever sense of contingency that applies to 9's existence'. It seems to me *a lot* odder to hold open the possibility that 9 does exist, and hence there are truths about 9, but that it is not prime is not one of those truths. As far as I can think myself into the Fieldian view, it's all or nothing: either you've got the number and all the orthodox mathematical truths about it, or you don't have the number and hence nothing is true of it. That mathematicians might be right to say that 9 is the successor of 8 but wrong to say that 9 is not prime (in the relevant sense of 'might') seems really odd to me. Just my intuition, of course, but there it is.

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 matterSo 3 necessarily exists, but it is not apriori that 3 necessarily exists. But then, presumably, '3' might pick out something other than 3, or in any case might not refer to 3 (since it might not refer to anything). But it is hard to imagine how that might have happened. You have 3 necessarily existing, so in whatever world from the point of view of which '3' refers to nothing at all, 3 exists. How might it have happened that '3' failed to pick out 3?

I've included this post in the latest Philosophers' Carnival. I hope that's ok!

Mike: I wasn't making any claim about aprioricity - as far as I'm concerned, it may well be apriori that 3 necessarily exists - all I would have to admit is that not all apriori truths are conceptually necessary - but I think there's good reason to hold that anyway (if you believe in apriori truths).

But it any case, I'm not sure that the problem is. *Everyone* thinks '3' might not have referred to 3, since it's contingent how we use our words - a fortiori it's contingent what names denote.

Tuomas: of course that's okay - thanks for including it!

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RdRsaid...I'm bothered by the ontology of 4. If from (3) the number 3 doesn't exist, then would the number 9 exist?

If the number 9 doesn't exist, then the term in the sentence '9 is not prime' has no reference. That could be interpreted as a "nonsense", or interpreted as 'a nonexistent number is not prime'. The latter can be interpreted as true.

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