Vagueness and the foundations of mathematics

There are many set-theoretic constructions of the natural numbers. For instance, one might let 0 be the empty set ∅, 1 be {0}, 2 be {1,2}, and so on. Or one might let 0 be ∅, 1 be {∅}, 2 be {{∅}}, and so on. (The same point goes for the rationals, the reals, the complex numbers, and so on.) Famously, Benacerraf used this to argue that none of these constructions could be the natural numbers, since there is no reason to prefer one over another.

My graduate student John Giannini suggested to me that one might make a move of insisting that there really is a correct set of numbers, but we don't know what it is, a move analogous to epistemicism about vagueness. (Epistemicists say that there is a fact of the matter about exactly how much hair I need to lose to count as being bald, but we aren't in a position to know that fact.)

It then occurred to me that one might more strongly take the Benacerraf problem literally to be a case of vagueness. The suggestion is this. Provable intra-arithmetical claims like that 2+2=4 or that there are infinitely many primes are definitely true. Claims dependent on one particular construction of the naturals, however, are only vaguely true. Thus, it is vaguely true that 1={0}. Depending, though, on what sorts of naturalness constraints our usage might put on constructions, it could be that some conditional claims are definitely true, such as that if 3={0,1,2}, then 4={0,1,2,3}.

There are some choices about how to develop this further on the side of foundations of mathematics. For instance, one might wonder if some (all?) unprovable arithmetical claims might be vague. (If all, one might recover the Hilbert program, as regards the definite truths.) Likewise, extending this to set theory, one might wonder whether "set" and "member of" might not be vague in such a way that the Axiom of Choice, the Continuum Hypothesis and the like are all vague.

Vagueness, I think, comes from to our linguistic practices undeterdetermine the meanings of terms. Likewise, our arithmetical practices arguably undetermine the foundations.

The above account neatly fits with our intuition that intra-mathematical claims are much more "solid" than meta-mathematical claims. For the meta-mathematical claims are all vague.

The next step would be to consider what happens when plug the above into various accounts of vagueness. Epistemicism is one option: our arithmetical terminology does have reference to one particular choice of foundation, but we aren't in a position to see what it is. I find promising a theistic variant on epistemicism. Supervaluationism seems particularly neat here. There will be one precification which precisifies things consistently with one foundational story, and another with another. can also consider other options.

There might even be some elements of epistemicism and some of supervaluationism. For there might be facts beyond our ken that say that some foundational stories are false—the epistemicism part of the story—but these facts may be insufficient to determine one foundational story to be right.

That said, I think I still prefer a more ordinary structuralism, though this story has the advantage that it takes the logical form of mathematical claims at face value rather than as disguised conditionals.