Is math "falsifiable"?

Ooooh, another chance to babble on about philosophy of science!

Kevin Bryan writes:

Arrow’s (Im)possibility Theorem is, and I think this is universally acknowledged, one of the great social science theorems of all time. I particularly love it because of its value when arguing with Popperians and other anti-theory types: the theorem is “untestable” in that it quite literally does not make any predictions, yet surely all would consider it a valuable scientific insight.

But Arrow's Theorem is a math result. It takes some definitions of mathematical objects as given, and it shows a relationship between those mathematical objects. Of course you can't "falsify" it with empirical observation, any more than you can "falsify" Jensen's Inequality.

I really hope there aren't "Popperians and other anti-theory types" running around out there complaining that math results are useless because they're non-falsifiable. There are at least two reasons that would be silly.

Reason 1: Pure math results tell us about what we can and can't do with math itself. For example, suppose we knew that it was impossible to factor an integer in polynomial time. That would have important implications for cryptography. Math itself is a technology, so math results can give us useful technological advancements.

Reason 2: Pure math results are necessary for math to be useful for engineering. One big reason - the main reason, I'd argue - that we make mathematical theories is because the math seems to correspond to real observable phenomena. As long as that correspondence holds, then we can predict things about the world just by doing math. To "use" a theory means to assume that the correspondence holds - to simply do the math and assume that it's going to give you useful results, without having to go re-test the theory each and every time. If you don't let yourself make that assumption, then all mathematical theories are useless for engineering purposes.

Modern engineers can do a hell of a lot of cool stuff just by doing math using theories from physics and chemistry, without re-testing those theories every time they want to build an airplane or synthesize a polymer. And computer scientists can do a hell of a lot of cool stuff just by telling their computers to do pure math. So if there are "Popperians" going around saying pure math isn't useful, they should think again.

Anyway, the rest of Kevin's post is quite interesting, and the philosophy-of-science literature it links to is even more interesting - here are a couple more papers in that literature: (paper 1, paper 2, paper 3). And here's the paper that started the discussion. Neat stuff. As blog readers must have already guessed, I've actually considered just quitting finance and working on this stuff instead, and maybe someday I will. When I'm old, perhaps...