One reason why existence matters

Let's consider two statements: "Michael has 13 blue hats" and "Tom Sawyer has 13 blue hats."

The first one might be true, or it might be false. It would depend on whether I have 13 blue hats or not. There's a fact of the matter--it's either the case or it is not. It's objective, so to speak. And if you felt that the sentence was true and I believed it to be false, that disagreement would matter. After all, we can't both be right. And then we could begin to offer arguments and counter-arguments to attempt to establish the truth and falsity of that statement.

The second sentence is...messier, to say the least. For starters, it's not at all clear that the sentence is either true or false if Mark Twain didn't mention anything about it in his book. That is, if we don't know how many hats Tom Sawyer has from the author, then the disagreement about how many hats he has doesn't seem to amount to very much. We could disagree, but there's no reason for thinking that we'll get anywhere. The discourse about Tom Sawyer isn't objective.

What's the difference between these two sentences? It seems to be that the difference is just whether the subject of the sentence exists or not. Michael exists; Tom Sawyer doesn't exist.

This is true in math as well. Suppose that numbers do exist (and they are abstract objects). Now, it may seem that this is completely inconsequential. After all, we're talking about abstract objects here, and it's very unlikely that I'll trip over a number of a set anytime soon (the point: they can't cause stuff to happen). So does it matter whether they exist or not? It seems, from the example above, that if that which is being referred to in a sentence exists, then that sentence is capable of truth and falsity; otherwise, not. Then mathematical statements can easily said to be true as long as that which they discuss exists. It seems that mathematical statements talk about numbers and sets, so it would seem that numbers and sets need to exist in order for mathematical statements to be capable of truth and falsity.

Now, what other options are there? We could reanalyze the rest of language and see if there is some other factor that makes statements objective other than existence of what's discussed in the sentence. Further, we could accept the claim that existence is necessary for objectivity, but reject that what is being refered to are abstract obejcts such as numbers, and instead insist that what's being refered to are material objects or more acceptable abstract objects (and this is the nominalistic program).

The point is that whether numbers exist or not seems to be equivalent to asking whether mathematical statements are capable of truth and falsity, or if they are not. It seems that what gives sentences their objectivity is their "aboutness." That is, the sentences that are about the world seem to have right and wrong answers (and see Frege on this issue, and Goldfarb's insistence that the attempt to prserve the objectivity of discourse is what motivates Frege's work). So when we ask whether numbers exist as an important question, according to this line of thought what we're really asking are several questions: (1) Is math objective, capable of truth and falsity? (2) What gives it this objectivity? Is being "about" something real what makes it objective? (3) Are the "about" things numbers, or are the objectivity-granting things something besides numbers?