alt Hacker NewsThe proofs may not yet be understood, but to claim they should not yet be accepted as solved is to claim there is a bug in Lean, which is a big claim.
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Isn't it also possible that the written proofs themselves are incorrect, say, for example, by mis-specification of constraints? Let's say I publish what I claim is a proof that 2+2=4, but in the Lean specification, I proved 1+1=2.
the impression i get is that people don’t per se assert that exploitable bugs exist in Lean, but that the volume of (code) proofs published represent a new magnitude of challenge to the integrity of Lean.
the obverse would be to assert that Lean (or any other theorem prover) is infallible under all circumstances, present and future - an even weightier claim in my opinion.
Hmm, am no expert at Lean, but on one hand, since it's a programming language, guessing it necessarily means like all other programming languages, it has bugs.
But on the other hand, agree with you that the chances that one its bugs causes a solved proofs to be wrong is probably very, very low. Yeah, each version of Lean obviously must be run against hundreds if not thousands tests of existing proofs to ensure it's working correctly...
... Although, one thing that is a little worrying is that some mathematicians have said the Lean code generated in these OpenAI proofs is supposedly really complex and undecipherable. It might be pushing Lean to corner cases that are not well tested.
There's been many bugs found in Lean already, and there are probably still a few. So that's not the main issue.
Plenty of these AI proofs have been accepted. On the most important result, on the Quasi-Riemann Hypothesis, there is no serious belief it's wrong. Mathematicians have already dissected and reformulated the claims.
The proof with the 11/12 constant, the simplest one, has already prompted what looks like chagrin from some. It's an unexpected combination of known results/techniques, a combination that apparently fell into a blind spot so it wasn't picked up before (even though it could have been, even 20 years ago).
I think these results are showing there was a lot of low hanging fruit that human mathematicians didn't have the breadth or mental bandwidth to find.