February 23, 2016
From: Reddit AMA
What do you think is the most interesting open problem in mathematics?
One that I’m definitely very interested in is really a metaproblem: how much math is doable?
Gödel’s Theorem tells us that there are mathematical questions that are undecidable from existing axioms of math… but those questions often seem very artificial, and most working mathematicians merrily proceed without worrying about undecidability. On the other hand, in my studies of the general computational universe of possible programs, I’ve found that undecidability is quite rampant. So the question is: how does mathematics avoid that?
My guess is that it’s a reflection of the human aspect of doing mathematics: that human mathematicians specifically choose to extend the field in directions where its methods work, and they don’t fall into the morass of undecidability. But it’d be nice to really understand this. There are practical questions, like where the boundary of undecidability is, say, for Diophantine equations. (We know complicated examples of equations involving integers where it’s undecidable if they have solutions… but can equations as simple as x3 + y3 + z3 == 20 have this issue too?) [ See https://www.wolframscience.com/nksonline/section-12.9]
Another question is how inevitable the “tower” of current pure mathematics is. If one starts from arithmetic and basic geometry, is there a definite set of concepts that one gets by generalizing those? If so, then one should be able to do “predictive metamathematics”, and work out what the limit of all this generalization will be…