I was re-reading an old post where I expressed doubt that there are any necessary existents at all, not even numbers, and I saw Richard Chapell’s comment,
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.
I guess I was being dense that month 🙂 because I didn’t see what the objection was supposed to be. But now I do. If numbers are the kinds of things that are not necessary then they must depend on something to exist. But what could numbers depend on? Since there is no plausable candidate we should conclude that if numbers exist they would do so necessarily. Here is what I should have said.
Whether a given possible world has non-physical elements is surely a contingent fact about that possible world. The worlds which do have non-physical elements will most likely have numbers and those that don’t won’t. If this were the case then the existence of numbers is contingent on which possible world best describes the actual world (or if one doesn’t like this way of characterizing the actual world as a possible world we can say it depends on what is actually the case). In some worlds, the existence of numbers may be contingent on whether they were created by some non-physical being. In short there are a couple of different ways in which the existence of numbers could turn out to be contingent.
Richard Brown 
“Whether a given possible world has non-physical elements is surely a contingent fact about that possible world.”
That’s an odd premise. Abstracta are not meant to be so perfectly extricable and superfluous. If we need numbers to make sense of (e.g.) physics and simple counting truths like “the number of planets is 8”, then presumably there couldn’t be a world like ours but without the numbers. Conversely, if we don’t have or need numbers after all, then it’s dubious whether speaking of numbers as possibly existing abstracta even makes sense. They’re either necessary or impossible, given their theoretical role; no in-between status makes sense.
I don’t see why it’s odd.
If we need numbers to make sense of (e.g.) physics and simple counting truths like “the number of planets is 8″, then presumably there couldn’t be a world like ours but without the numbers.
What’s the argument for this? There are those, like Field, who think that all that needs to be true is that there are numbers in some possible world, it doesn’t have to be this one. What’s wrong with that view?
Conversely, if we don’t have or need numbers after all, then it’s dubious whether speaking of numbers as possibly existing abstracta even makes sense.
Again, I don’t see the argument for this. If it turns out that the actual world is a completely physical world then there will actually be no numbers but the story we tell about numbers is enough for us to do science and count. It seems easy to deny that we need numbers here and also deny that it is hard to imagine them existing in some possible world where there are non-physical objects.
I would think that numbers ‘transcend’ all possible worlds. Seems weird to say that they are located “in” those possible worlds.
After all, the whole “possible world” semantics wouldn’t be possible if abstracta are located within the worlds, no?
I’m not sure I follow, Tanasije. All of the things that exist do so actually. Possible worlds are (complete) descriptions of ‘ways things might have been/will be’ Any complete description of the way things are/might have been/will be will have to include a pronouncement about what exists.
Richard,
I’m thinking of e.g. quantifying over possible worlds, or saying “in half of the possible worlds” or “in two possible worlds” , etc…
If the very possibility to use the possible world semantics depends on those, I’m not sure how can we imagine them as “inside” possible worlds.
Oops, “those” = set theory, numbers, and other abstracts.
Yeah but we say that stuff outside any possible world, in the actual world. A possible world is just a complete description of how the actual world might have been/will be (they may be abstract objects themselves if one goes infor that sort of thing…Kripke does, others don’t). To say that the description is complete is to say that any statement p is either true or false (and not both) ‘according to that description’ or ‘in that possible world’. So there is at least one complete description of the way things are which includes in it a commitment to the existence of numbers and other abstract objects. If the actual world turns out to be a world in which numbers do not exist it will because of that possible world (that could have been actual but is not) that we will get true counter-factual statements like ‘If there were numbers, 2+2=4″, we could in fact give a detailed account of the rules which would govern these things as for instance by postulating that ‘if there were numbers, they would be goverened by things like Peano’s axioms’. In that way we could reason ‘as if’ numbers existed in the actual world. Feild’s claim is that this is all that we need in order to do science and talk about abstract objects.