So my last entry mentioned a surface analogy between meta-analytic statements (statements about analytic statements) and philosophy of mathematics. What I was aiming at in that post was merely an exploration of how we might deal with these meta-analytic statements in philosophy of language and a suggestion that such explanation could throw light on philosophy of math. However, much of the interest in that post focused on what sort of specific application I had in mind so I will try to explain one possible interpretation of mathematical statements as meta-analytic statements in this post.

The motivation of this idea is kinda an attempt to combine the theory that the meaning of logical connectives is given by the deductions they support (‘and’ just means that connective which allows the derivation of A, B from ‘A and B’) and if-then-ism. While it seems there is some difficulty in just using the same technique from the logical connectives with mathematical objects themselves, because for instance we don’t in fact have words referring to every integer even if we have a procedure to produce them, it seems we might be able to make conditional statements about what would happen if we did have such words.

Note: I DON’T think this provides a good solution for philosophy of math. Rather I think it shows most of the worries about philosophy of math infect our talk about philosophy of language so an answer to these difficulties in philosophy of language might provide some help addressing them in phil of math.

Once again this is hardly a well thought out theory but here goes. Alright so one simple approach to understanding mathematical statements is if-then-ism. That is to say that the Goldbach conjecture is true (provable) in the integers is just to say that if some structure satisfies the axioms of the integers then that structure will also satisfy Goldbach’s conjecture. There are a fair number of complications I am glossing over but the appeal of the approach is obvious. It seems to automatically give an answer as to why mathematics provides useful results in counting. However, it also has several serious problems. For instance what sort of conditional are we to take the ‘if’ to be? If it is only material and the universe is finite all statements about the integers should come out trivially true. Even if it is some form of counterfactual it is unclear why we should believe there are any structures in any possible worlds that model set theory. If the existence of some structure modeling set theory is needed for the theory to work why not just take that structure and be a Platonist about it?

So it seems many of the problems in if-then-ism stem from what we mean by ‘if something satisfies the axioms’. However, it seems the same problems can be easily replicated in meta-analytic statements. For instance consider the statement (substituting some specific number for N), “If X1…XN are different words meaning the same thing as hot then very X1…very XN are different words meaning the same thing as very hot.” Should we consider this statement as trivially true if there are only N-1 different words meaning the same thing as hot? That doesn’t match with my intuitions about such statements. It seems to me we expect such statements to be true independent of the actual existence of words we can substitute in for X1…XN.

So how can this vague analogy be pressed into an interpretation of mathematical truth? Well the idea is that instead of using statements of the form ‘if blah satisfies the axioms’ which has the problems discussed above take statements of the form ‘If blah satisfies the axioms as a matter of meaning.’ Or in other words the statement of the Goldbach conjecture could be rephrased as something like, ‘If X0…XN… satisfy the axioms of arithmetic by virtue of meaning then they also satisfy the goldbach conjecture by virtue of meaning.”

Admittedly this is still mostly imprecise analogy. All I’m hoping to convey is the sense that hey statements about analytic truth seem to have alot of the same difficulties as statements about mathematical truth. The interpretation of mathematical statements in terms of meta-analytic statements is likely to suffer from serious problems. However, I’m hoping just explaining why it is wrong in a precise way would go a long way to answering questions about both problems, which I am convinced are not as far apart as most people think.