I have been reading Jason Stanely’s paper on names and rigid designation from the Oxford Companion to the Philosophy of Language in the course of doing some research for my frigidity v. rigidity axe-grinding. It is an interesting and informative, though technical, introduction to issues about rigidity and I will come back to its relation to frigidity in a later post… but one thing caught my attention early on. He says,

consider Kripke’s class of strongly rigid designators (Kripke, 1980, p. 48). This class contains the rigid designators of necessary existents. That is, this class contains all and only those designators d of an object x which exists in all possible worlds, which designate the same thing in all possible worlds (viz. x). For example, the descriptive phrase “the result of adding two and three” is a strongly rigid designator, since its actual denotation, namely the number five, exists in all possible worlds, and the phrase denotes that number with respect to all possible worlds.

Is it really the case that ‘the number five exists in all possible worlds’? Isn’t there a possible world where fictionalism about math is true? In that world 2+2=4 is not true because ‘2’ stands for an existing object, viz. The Number Two, it is true because ‘in the story we tell about mathematics’ ‘2’ stands for The Number Two in just the same way that ‘Santa wears a red suit’ is true, not because ‘Santa’ picks out some guy who wears a red suit but because ‘in the story about Santa’ ‘Santa’ picks out a guywho wears a red suit. Maybe fictionalism about math isn’t actual, but surely it’s possible, isn’t it?

We can give the same kind of argument for any proposed ‘strongly rigid’ designator. Take God for instance. It take it that Atheism is a legitimate possibility for the actual world. That is, it migt actually turn out to be the case that there is no God. Of course it might also turn out to be the case that there isn’t one. Each of these seems to me to be a metaphysical possibility, not merely an epistemic possibility. If so then there is a possible world where God does not exist (it may or may not be the actual world). Isn’t this some reason to prefer, when faced with the possiblility of a proof of necessary existence, to take my view (fix it) rather than Williamson’s (accept it)? That is, isn’t there an issue here about whether there are any ‘strongly rigid’ designators?

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I’d expect the ontological status of numbers to be non-contingent — after all, what in the world could they be contingent

on? Abstract objects seem to be the kind of things that exist necessarily, if they exist at all.Suppose we know that mathematical realism rather than fictionalism is actually true. Whatever considerations base this knowledge presumably do not vary from world to world. Rather, such philosophical truths are a priori and necessary. (Their negations are only “possible” in a very loose epistemic sense — kind of like the possibility that water might not be H2O.)

I’d be interested in hearing more about your thoughts on the contingency of the existence of God. It seems that traditionally God is defined as a necessarily existent being. Therefore, although it is epistemically possible that God doesn’t exist, if in fact God exists he would exist necessarily.

I think the real issue here is what you mean by “God”.

Hi Richard,

Thanks for the comment!

I am not sure if I get the point of your first question, so this may not be an answer, but I assume that the ontological status of numbers would be contingent on the existence of numbers.

OK, let’ suppose that number actually exist. What is the argument that ‘whatever considerations base this knowledge presumably don’t vary from world to world’? So, I can grant that the math truths are a priori (this is an epistemological notion) and even nexessarily true (i.e. true in all possible worlds) because in the possible worlds where numbers don’t actually exist math fictionalism is true, and so the math truths are true. My claim is that the necessity of the truth of ‘2+2=4’ does not necessitate the necessity of the existence of numbers if we really take fictionalism seriously.

Hi Andy,

Thanks for the interest and the comment!

When I talk about God around here I mean the usual mono-theistic God of the Jews, Christens and Muslims. I mean the one and only all-powerful, all-knowing, supremely loving being.

You are right that it is traditional to stipulate that if God exists then he necessarily exists. But this is problematic. I think that you need an actual argument to show that God, should he exist, necessarily exists. Here is a little argument that is supposed to show that he can’t be a necessary existent.

1. It is epistemically possible that there is no God. That is to say, for all that we know right now God does not exist. Of course, for all that we know right now, He does exist. It is an open question at this point in time whether there is actually such a being as I earlier described.

2. But, if it is epistemically possible that God does not exist then it might turn out that we discover that God does not in fact exist. The Aethiest might turn out to be right about the actual world and the theist might turn out to be wrong. That is, the actual world may turn out to be a world where God does not exist.

3. But if 2 is right then it is metaphysically posible that God does not exist, for how else could we come to know that He does not exist if not by finding out that He doesn’t?

4. So, since it is metaphysically possible that God does not exist, His existence is not necessary.

Of course, I can see someone potentially balking at my characterization of epistemic possiblity. They may implicitly be assuming that by ‘epistemically possible’ we mean ‘merely ignorant of the right answer’ as opposed to ‘both could really turn out to be true’…to illustrate the former, consider yourself before you learned how to do square roots, say when you were 10 or 11. It would then be epistemically possible for you then that the square root of 9 be other than 3, but it isn’t metaphysically possible. That is because you are merely ignorant of the right answer and so you can think that the square root of 9 is other than 3 without any absurdity manifesting in your imagined scenerio. To illustrate the latter kind of epistemic possibility think about Golbach’s conjecture (i.e. the conjecture that every even number greater than 2 is the sum of two prime numbers). This has never been formally proved, though it seems reasonable enough and easy to satisfy oneself that it holds true for very many even numbers greater than 2, so it could turn out to be true. But it might also turn out that there is some number N which is even and greater than two and which is not the sum of two prime numbers. Either of these is at this point a real possibility. So, I claim that our position with respect to the existence of God is more like the latter than the former.

Or maybe, if the conjecture is true then it is true in all possible worlds, and if it is false then it is false in all possible worlds, jsut as in the case of the positive square root of 9… But I’ve been wondering rather vaguely about similar questions, e.g. since it is possible that we are being deceived by a demon, making us think that 2 + 2 equals 4 when in fact it doesn’t (as Descartes and Wittgenstein emphasized), so if possible worlds are the right way to think about possibility, there ought to be a possible world where 2 + 2 does not equal 4. There seems to be a primitive sense of ‘logical’ in which that is a logical possibility; and the alternative seems to be to build an awful lot of metaphysical assumptions into our possible worlds semantics.

Hi Enigman.

Yeah I agree that the conjecture, if true, is necasarily true just like the square root of nine example. The difference between the two examples is that in the one case we have a proof procedure that allows us to show what the right answer is and so someone who does not know the answer is merely ignorant of the existing proof procedure. In the Goldbach case we do not have any such procedure, and it is not for lack of trying. So whatever the math truths turn out to be they are necessary…even Descartes agrees with that. His point is not that it is not the math truths are not necessary. His point is that we may not actually know what the math truths are. So whatever 2+2 actually equals it will necessarily equal…what isn’t possible is a world where 2+2=4 and a different world where 2+2=(say) 5.

Oh…ooopppsss…I accidently hit submit without finishing the thought…

So, Descartes is wondering whether it is possible to doubt that the proof procedue has been completed adequately (his worry is that we may be being tricked into thinking a proof is correct when it is actually incorrect because we are being made not to notice some flaw in the proof)…an so his worry

islike the square root of nine example and not like the Golbach example…Oh, sorry, by ‘ontological status’ I simply meant ‘whether they *really* exist’.

Here’s another way to get at my point: precisely what is it about a possible world in virtue of which either fictionalism or mathematical realism is made true? It doesn’t seem to me that this question – whether numbers really exist – depends on any contingent feature of the world at all.

Here’s an argument that ‘whatever considerations base this knowledge [of mathematical realism] presumably don’t vary from world to world’: the metaphysics of mathematics is not an empirical field. We don’t discover that realism is true by investigating the contingent world we find ourselves in. So, whatever it is that makes realism true, it isn’t the contingent world.

Hi Richard,

You say:

I’m not sure why you point to this example…

If we use ‘2’ to refer to existing The Number Two, and in that possible world people use ‘2’ to refer to an imaginary The Number Two, our ‘2’ and their ‘2’ refer to different things.

Same as if in a possible world there was real detective on Baker Street 221b, and he was called ‘Sherlock Holmes’. That name used in that possible world has different meaning from the one that we use.

So, even in some possible world, in that possible-world language, the statement “2+2=4” is true “fictionally”, it picks-out proposition different than our “2+2=4”. So when we speak our language we can’t say that in that possible world 2+2=4 is still true, but for different reasons.

Probably I’m missing something.

…for some reason this comment went into the spam section, which I just happened to check, I hope other comments are not suffering a similar fate…:(

Anyways, thanks Richard for the clarification.

I am not sure about your argument. Consider: we came to find out that water is H20, indeed that it is necessarily so, by investigating the contingent world so I don’t know why I should buy the claim that the metaphysics of mathematcis won’t turn out to be similar. So also it might turn out that we come to find out that the mathematical truths are contingent if we ever had irrefutable empirical evidence that 2 objects added to 2 other objects resulted in (say) 5 objects. Now, I don’t think that this could happen (though some die hard Quinians do, like David Rosenthal) my point is only that investigating the contingent world around us seems like a convievable way of confirming or disconfirming whether something is necessary or not…

Also, it may

seemto a knower in a world where math realism is true that he knows it is true (if he does) via some non-contingent feature of numbers and yet that knower is wrong, couldn’t it? It seems possible in other areas, like for instance consciousness. If a higher-order thoeyr is right then when people consciously experience pain they are conscious of a mental state that they pick out via a contingent feature of the mental state (i.e. the way it feels to have it) and yet it seems to the person that they are picking it out via a non-contingent property (i.e. the way it feels!). So claiming without argument that pain is necessarily conscious is begging the question against higher-order theories of consciousness.I think something like that is going on here. If P is true then P is the case in some possible world. Just as a higher-order theory of conscious may turn out to be the right metaphysical theory of consciousness so math fictionalism may turn out to be the correct account of the metaphysics of math. So MF is true (where MF is ‘math fictionalism), so MF must be true at some possible world but MF says that numbers don’t really exist. So there is a possible world where numbers don’t exist. So unless there is some reason to think that MF is false then numbers existence is contingent. What do you think?

Hi Tanasije,

As I understand it, fictionalism is the claim that ‘2’ purports to refer to The Number Two, it just turns out that there isn’t one. So 2+2=4 is literally false but it is true in the stroy that we tell about numbers. In the story there is a number called ‘2’ and in the story when added to itself it is equal to another number, 4 and so on. So our sentence which, let us suppose for a second does refer to The Number Two, is not literally true in this world, but it will be true in the sense that they tell a story which is supposed to about The Number Two and in it ‘2+2=4’ is true. So it does seem to meet the criterea: If “2+2=4” is true then “2” refers to The Number Two. “2” does do this,

in the story…but why should that little qualification at the end matter? We have to have an ‘in the story’ operator prefixed to the assertion and then we can say that the same sentence I say here is true there…Compare Newtonian mechanics. Nothing it says is literally true, though its theoretical terms purport to refer (i.e. ‘mass’, in Newtonian Mechanics, is supposed to pick out a property). When we use this theory to plot a course to Mars it seems as though we adopt a fictionalist stance towards it. So if someone asks us if the inverse square law is true here in the actual world we might say that it was true in Newtonian theory but not ‘really’ true…but if it is true…now of course, your worry is that in the possible world when math fictionalism is true we can make the same ‘true in the story’ ‘REALLY true in REAL LIFE’ distinction. But then maybe that is some reason to think that if math fictionalism is true then deflationism about truth has to be true at that world too (in order to kleep ‘2+2=4’ as a necessary truth…if one did not want to save that then they could reject this claim about deflationism)….hmmm interesting…I gotta go now, but I would be interested to hear what you think…

Also, after just reading some footnotes in the Stanley article I have been talking about, I see that there is a camp of people who take ‘necessarily true’ to mean ‘non-falsity in all possible worlds’ (as opposed to ‘true in all possible worlds’) so someone could say that ‘2+2=4’ is necessarily true if it not-false in all possible worlds which means that it is either true or that it lacks a truth value (i.e. in worlds where 2 does not exist)…so I think there are options for people who want to claim that 2+2=4 is necessarily true even though the existence of numbers is a contingent matter.

“So unless there is some reason to think that MF is false then numbers existence is contingent. What do you think?”

No, I think that if MF is true then it is necessarily true. Indeed, I’m agnostic on the question whether numbers really exist, but I’m sure that the answer (whatever it turns out to be) is non-contingent. Note that epistemic skepticism is no grounds whatsoever for inferring a positive claim of metaphysical possibility.

“

Consider: we came to find out that water is H20, indeed that it is necessarily so, by investigating the contingent world”Not really. Of course,

that water is H2Ois an empirical discovery. But it’s a priori that if water is H2O then it is necessarily H2O. The modal aspect — the recognition of non-contingency — is a priori. Indeed, it’s even a priori that learning about water’s chemical composition requires empirical investigation. Philosophical analysis can tell us what empirical work needs to be done. In the mathematical case, there’s none.Hi Richard, I don’t know why your comments keep going in the spam section…sorry about that!

Anyways I see that you think that about math fictionalism, but I don’t see why someone has to think that or should think that. It seems to me that math fictionalism is a metaphysical possibility, not merely an epistemological possibility (whatever that is) even if it happens to be false here in the actual world. I don’t see any reason to be so sure that the answer to ‘do numbers exist?’ must be non-contingent.

You suggest that one reason to think this is that whatever we would actually show that numbers actually exist can’t be something that is part of the contingent world, but this itself stands in need of an argument or else it seems you are just saying ‘it is necessary because it is necessary’. Why can’t it be the case that we discover that numbers actually exist and that we do so by some property that numbers have here in this world and that we take this to be acess to something necessary but yet we are wrong?

Finally, I guess I just don’t share your convictions about the a priori! (x=y) –> [](x=y) is not a priori! In fact its truth depends on what version of modal semantics one adopts…so for instance if one takes a Lewisian kind of modal semantics that endorses real counter-part theory it won’t turn out that identies are necessary and so that won’t be a theorem…which is nice since there are people who think that there are contingent identies still to this day! (like, Jack Smart)…also I don’t see why it’s a priori that we need empirical investigation to learn about water’s composition. Perhaps there is a possible being that can ‘just see’ the composition, or ‘apprehends its form’…

[…] Filed under: Philosophy of mathematics — Richard Brown @ 6:00 pm I was re-reading an old post where I expressed doubt that there are any necessary existents at all, not even numbers, and I […]