Ok, I need to refine my description of math a bit. I’d claimed that an incorrect premise gives useless conclusions; actually as you point out if we have a close-to-correct premise instead, we can have useful conclusions. The word “instead” is important there, because otherwise we can then add in a correct contradictory premise, generating new and false conclusions. In some sense this is necessary to all math, most evidently geometry: we don’t actually have any triangles in the world, but we use near-triangles all the time, pretending they’re triangles, with great utility.
Also, to look again at Kempe’s “proof”: we can see where we can construct a vertex of degree 5 where his proof does not hold up. And we can try to turn that special case back into a map. The fact that nobody’s managed to construct an actual map relying on that flaw does not give any mathematical evidence that an example can’t exist. Staying within the field of math, the Bayesian is not updated and we can discard his conclusion. But we can step outside math’s rules and say “there’s a bunch of smart mathematicians trying to find a counterexample, and Kempe shows them exactly where the counterexample would have to be, and they can’t find one.” That fact updates the Bayesian, but reaches outside the field of math. The behavior of mathematicians faced by a math problem looks like part of mathematics, but actually isn’t.
Ok, I need to refine my description of math a bit. I’d claimed that an incorrect premise gives useless conclusions; actually as you point out if we have a close-to-correct premise instead, we can have useful conclusions. The word “instead” is important there, because otherwise we can then add in a correct contradictory premise, generating new and false conclusions. In some sense this is necessary to all math, most evidently geometry: we don’t actually have any triangles in the world, but we use near-triangles all the time, pretending they’re triangles, with great utility.
Also, to look again at Kempe’s “proof”: we can see where we can construct a vertex of degree 5 where his proof does not hold up. And we can try to turn that special case back into a map. The fact that nobody’s managed to construct an actual map relying on that flaw does not give any mathematical evidence that an example can’t exist. Staying within the field of math, the Bayesian is not updated and we can discard his conclusion. But we can step outside math’s rules and say “there’s a bunch of smart mathematicians trying to find a counterexample, and Kempe shows them exactly where the counterexample would have to be, and they can’t find one.” That fact updates the Bayesian, but reaches outside the field of math. The behavior of mathematicians faced by a math problem looks like part of mathematics, but actually isn’t.