We all have an intuitive grasp of what it is for entities to exist. My parents exist whereas Santa doesn't, and all that. But what of abstract objects? When philosophers argue about whether numbers truly exist, what is in dispute here? Even ontological debates about material entities seem dubious: does there exist an individual entity which is a table, or are there merely particles arranged table-wise? What's the difference? These don't seem to be debates about how the world is. Everyone agrees that there is table-ish stuff in the world. They merely dispute how to count or describe it.
Of particular concern are too-easy arguments like the following:
(P) There are nine planets.
(C1) So, nine is the number of planets.
(C2) So, there is a number that is the number of planets
(C3) So, there exist numbers.
They start with some undisputed fact, and show that it trivially entails a (seemingly substantive) ontological conclusion. But surely that's cheating! Trivial entailments can't produce substantive new results. They merely serve to highlight what is already contained in the premise. But counting planets shouldn't commit us to the existence of numbers in any deep sense, should it? At least, if the above argument is sound, then it's a marvel that so many smart philosophers could make such a simple blunder. Ontology would be easy!
At this stage, many philosophers appeal to a distinction between kinds of existence claims -- I'll follow Cian Dorr in calling these "superficial" and "fundamental". The idea, then, is that the above argument is valid only if 'existence' is used in the same sense throughout. The Platonist conflates the two, invalidly jumping from premises about superficial existence to a conclusion about fundamental existence. We can all agree that numbers "exist" in the superficial sense that follows analytically from sentences like "there are nine planets". But that says nothing about fundamental existence, which is what philosophers (at least, ontologists) are interested in.
So what do these two senses of 'exist' really amount to? I think the "superficial" sense is tolerably clear. It concerns those claims we can arrive at through conceptual analysis, analytic entailments from commonsense truths, and so forth. In this sense, the existence of abstract objects is an entirely trivial matter. It's not to claim anything substantive about how the world is. Rather, claims like "there exist numbers" are analytic: true simply in virtue of meaning, without needing any input from the world. They're more semantic than metaphysical in nature, telling us only about language and not reality. (Of course, they might be combined with a worldly component to form synthetic claims, e.g. "there are nine planets".) Scientists can look into the empirical component of such claims, but there's nothing of interest here for philosophers.
What of "fundamental" existence? This seems harder to characterize. It is meant to involve a substantive claim about how reality is. As such, claims of fundamental existence are never merely analytic. The trivial argument given above has no place in "serious ontology". Instead, perhaps, we may use inference to the best explanation, or Quinean indispensibility arguments, to conclude that we should posit some class or other of abstract objects. I follow that much. My worry is this: what, exactly, are we "positing" here? ("The existence of numbers." "Um, okay, and that means...?")
To restate the problem: how would a world with numbers be any different from a world without them? I take it the answer must be that my question is ill-formed. There are not two such possibilities to compare. Whether numbers exist or not, they have this status necessarily, and there simply is no sense to be made of the alternative proposal. But then it's still hard to see what the ontologists are disputing. (Perhaps "which of us is speaking nonsense"?)
Here's something I found helpful: In last week's reading group, Brendan pointed out that when we're speaking in the superficial sense (i.e. all the time in everyday life), we have a limited concern for the ways in which what we say might not be literally true. We're only interested in a restricted class of "relevant alternatives". His example: when I say "The board is white", we would consider it relevant if this turned out to be false because the board is really black or blue, or perhaps if I was merely hallucinating the board in the first place. But we are not concerned about the possibility that the claim is strictly false because colours don't objectively exist, say, or because there exists no fundamental entity that is a "board", but merely atoms arranged board-wise!
What I take away from this is that, in communication, we seek to narrow down the list of (epistemically) possible worlds which are candidates for actuality. When I say "the board is white", this serves to knock out those possible worlds which lack white board-ish presences. Think of it this way: we are given the various possible worlds, and we have to sort them into an "in" pile and an "out" pile. We all know how to do this, or what kind of instructions "the board is white" is meant to convey here, even if we don't know exactly how to define or describe the contents of the possible worlds that have been given to us. In particular, we don't know whether those white-boardish presences should be described fundamentally as individual objects that are white. Perhaps they shouldn't -- perhaps such macroscopic commonsense terms do not make it into "the final analysis". But we can still identify which worlds they are meant to pick out. We can tell which worlds contain stuff that fills the whiteboard role.
Another example: When I say that Santa Claus does not exist, I mean to reject those possible worlds which contain a certain qualitative character. This is the character we all associate with Santa: a red and white humanish presence usually located at the North Pole, which flies around the world delivering presents each Christmas Eve. I could draw a picture, if that'd help. ;-) Anyway, if you picture a possible world in your mind's eye, you can tell whether it contains the sorts of qualities I'm talking about here. (Or if you read a sufficiently thorough description in some idealized language.) It doesn't matter whether "Santa" is really just atoms arranged Santa-wise. So long as there's something(s) playing "the Santa role" in a world, then that counts as Santa "existing", for my purposes.
I welcome suggestions for how to express this notion more clearly, but hopefully you get the rough idea. (I feel like it's related to this post, but I can't say exactly how.) Substantive claims of superficial existence (say of concrete entities) serve to distinguish between various possible worlds. Fundamental existence claims are different. Ontologists don't discriminate between possibilities, telling us that world w1 is actual rather than w2. Rather, they fill out the possible worlds' contents, specifying what (exactly) we find in the given worlds w1 and w2.
If an ontologist says that Santa couldn't possibly exist because there are no composite objects but only arrangements of atoms, they haven't really said anything which narrows the space of possibilities. It's like Putnam's lesson from Twin Earth: the world we qualitatively imagine is still possible, it's just that we were misdescribing it. That watery presence shouldn't be called "water", and that Santa-Clausish presence shouldn't be called an individual.
Note that my analogy might be a little misleading, in that the ontologist isn't making a merely semantic claim about the meaning of our word "individual", or "exists". Rather, he's making a (purportedly) substantive claim about the contents of possible worlds. ("That Santa-ish presence is really not an individual entity! And there really are numbers -- and I'm talking about reality, not about our words!")
But we can grant that while recognizing my point that he's not really narrowing the possibilities in my sense. Since he's making claims about the necessary contents of possible worlds, if he's right then the alternative view doesn't invoke any genuine possibilities at all. It has the same kind of status as the "possibility" that there are finitely many prime numbers. This doesn't describe any coherent scenario -- it's just that not everyone realizes that.
Compositional nihilists don't really believe in a more restricted space of possible worlds than the rest of us, I take it. They simply dispute what those worlds contain. Pointing to a molecule of hydrogen gas in some possible world, they will deny that it is a third thing in addition to the two Hydrogen atoms that compose it. (Let's pretend that our so-called "atoms" really are indivisible.) They don't deny that this world (*points to a spot in the North-East of modal space*), that this world we're discussing is a possible one. They simply think we mistake its contents when we say it contains molecules as well as atoms.
I'm not convinced the difference between these views really is a substantive one though, since I can't see what the difference is. Sure, one philosopher says the two atoms compose a molecule, and the other denies this. But again, what is the content of this disagreement? What difference does it make whether we say there are three things here or just two? It seems to come down to the arbitrary matter of how you choose to count! So I'm skeptical that existence in this "fundamental" sense really amounts to much. Perhaps we should be pluralists about it, allowing that adopting different ontological frameworks might be useful for different philosophical purposes, but there's not really any deep fact of the matter. The important (even if "superficial"!) existence questions concern the differences between possibilites, and the empirical question of which one is actual.
(Though cf. Chalmers' more moderate view, which allows that there might be some determinate ontological truths, as well as some indetermine matters.)