Why does local validity work as well as it does in math? <...> In other words, why aren’t most interesting math questions like P=NP, or how to win a game of chess?
Why do you think that it works well? Are you sure most possible mathematical questions aren’t exactly like P=NP, or worse? The set of “interesting” questions isn’t representative of all questions, this set starts with “2+2=?” and grows slowly, new questions become “interesting” only after old ones are answered. There is also some intuition about which questions might be answerable and which might be too hard, that further guides the construction of this set of “interesting” questions.
I think the key difference between math and chess is that chess is a two player game with a simple goal. In math there is no competitive pressure to be right about statements fast. If you have an intuition that says P=NP, then nobody cares, you get no reward from being right, unless that intuition also leads to a proof (sometimes it does). But if you have an intuition that f3 is the best chess opening move, you win games and then people care. I’m suggesting that if there was a way to “win” math by finding true statements regardless of proof, then you’d see how powerless local validity is.
Why do you think that it works well? Are you sure most possible mathematical questions aren’t exactly like P=NP, or worse? The set of “interesting” questions isn’t representative of all questions, this set starts with “2+2=?” and grows slowly, new questions become “interesting” only after old ones are answered. There is also some intuition about which questions might be answerable and which might be too hard, that further guides the construction of this set of “interesting” questions.
I think the key difference between math and chess is that chess is a two player game with a simple goal. In math there is no competitive pressure to be right about statements fast. If you have an intuition that says P=NP, then nobody cares, you get no reward from being right, unless that intuition also leads to a proof (sometimes it does). But if you have an intuition that f3 is the best chess opening move, you win games and then people care. I’m suggesting that if there was a way to “win” math by finding true statements regardless of proof, then you’d see how powerless local validity is.
This is all a bit off topic though.