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> If it is known that A is provably true then one can study the consequences of A being true But one can already study the consequences of P=NP right now. You

> If it is known that A is provably true then one can study the consequences of A being true But one can already study the consequences of P=NP right now. You don't need to know that it's provably true in order to do that. Knowing an actual proof would be useful, but an oracle revealing merely that it's true (or even provable) without telling you the proof does not let you do anything you couldn't do before.

czgov 1mo ago
Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. In the present discussion it’s not about letting me do something I can’t do now but about whether or not the endeavor is worthwhile. A lot of people spent a lot of time and effort to prove or disprove the Jacobian Conjecture. AI solved it easily. It is increasingly becoming the case that humans are not as good at mathematics as computers. You are free to ignore computer generated proofs but I don’t think this position will win out in the long run.
- skinner_ 1mo ago
  > Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.
  - czgov 1mo ago
    If a counterexample to a conjecture is found then all work toward proving consequences of the conjecture will cease. No one is trying to discover consequences of the Jacobian Conjecture now. At some point an AI will prove a result that is so long and complicated that no human will understand it. This should not preclude people from using that result. In general, whenever the body of knowledge is increased it is a good thing. Even if it isn’t increased by humans.

czgov 1mo ago

Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. In the present discussion it’s not about letting me do something I can’t do now but about whether or not the endeavor is worthwhile. A lot of people spent a lot of time and effort to prove or disprove the Jacobian Conjecture. AI solved it easily. It is increasingly becoming the case that humans are not as good at mathematics as computers. You are free to ignore computer generated proofs but I don’t think this position will win out in the long run.

skinner_ 1mo ago
> Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.
- czgov 1mo ago
  If a counterexample to a conjecture is found then all work toward proving consequences of the conjecture will cease. No one is trying to discover consequences of the Jacobian Conjecture now. At some point an AI will prove a result that is so long and complicated that no human will understand it. This should not preclude people from using that result. In general, whenever the body of knowledge is increased it is a good thing. Even if it isn’t increased by humans.