Use, Meaning and Number Theory

I like to joke that all philosophical questions in their clearest and most beautiful form philosophy of math. But I think this is actually true, in the case of questions about how much our use of a word has to "determine" the meaning of that word.

Consider the relationship between our use of the language of number theory, and the meaning of claims in this language.

I think the following two claims are about as uncontroversial as anything gets in philosophy:

a) The collection of sentences we are disposed to assert about the number theory (for any reasonable sense of the word disposition) is recursively enumerable.

b) The collection of truths of number theory is not. In particular there's a fact of the matter about all claims of the form "Every number has X recursively checkable property" e.g. whether fictionalism or platonism is the correct view about how to understand talk of the numbers, mathematicians are surely wondering about something when they ask "Are there infinitely many twin primes" (maybe something about what would have to be of any objects that had the structures which we take the numbers to have).

But what emerges from these two claims is a nice, and perhaps suprizing, picture of the relationship between use and meaning.

I use the words "all the numbers" in a way that (is r.e. and hence by Godel's theorem) only allows me to derive certain statements about the numbers. We can picture my reasoning about the numbers as what could be gotten by deriving things from a certain limited collection of axioms.

BUT in listing these limited collection of statements, I count as talking about some collection of objects/structure that objects could have. And, there are necessary truths about what those objects are like/what anything that has that structure must be like, which are not among the claims my use allows me to derive.

[If you're daunted by the mathematical example, here's another one inspired (oddly) by Wittgenstien on phil math. You use the words "bauhaus style" and "ornate" in a certain way, mostly to describe particular objects. This gives your words a meaning (though perhaps there is some vaguness). They would apply to some objects but not to others. Hence the question "Can any thing be both bauhaus style and ornate?" is either true false or perhaps indeterminate, if e.g. objects could be ornate in a way that makes it vague whether they are in the bauhaus style or not. But your use (e.g. your ability to say, when presented w/ one particular thing whether it is bauhaus style/ornate) does include anything which allows you to arrive at one answer to the question or another.]

So, there's a nice clear sense in which it strongly appears that: even if use determines meaning, facts about the meaning of our sentences can go beyond what our use of the words contained in them allows us to derive.

And, any philosophy that wishes to deny this claim will have to do quite alot to make itself more plausible than a) and b) above!