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Comment by n4r9 | Hacker News Reader

Why would an AI confirmation or rejection be more convincing than the proof itself?

Presumably an AI would formalise the proof in a system such as Lean, then you only need to trust the kernel of that proof system.

Rejecting a proof would be more complicated, because while for confirming a proof you only need to check that the main statement in the formalisation matches that of the conjecture, showing that a proof has been rejected requires knowledge of the proof itself (in general).

> requires knowledge of the proof itself (in general)

Why? If a proof is wrong it has to be locally invalid, i.e. draw some inference which is invalid according to rules of logic. Of course the antecedent could have been defined pages earlier, but in and of itself the error must be local, right?

Human-written proofs are not written in Lean to be checked easily and there'll be potentially many formalizations for written prose and only some of them will be what the author intended. You need to pick the right formalization before you can say that this proof has local errors.

If a human reviewer rejects a proof, what do they consider apart from local errors? Formal proofs have formal local errors (which can be checked mechanically), while informal proofs have informal local errors (which require humans and some degree of hermeneutics to check). Am I missing something?

It might be using some nonlocal context e.g. something similar was proved earlier in the proof and the reader is assumed to remember and/or generalize this or there's some assumptions that are stated only in the beginning. There's also probably a bit of skipped proofs of the kind "I'll be able to do this if pressed, but so will the reviewer".

Rejection: an incredibly complex proof can fall for (comparatively) simple reasons. If the AI scans the entire proof and says yep, there's the flaw, page 126 theorem X contradicts <well established known theorem> then a human can verify this without having to understand the whole proof.

This could lead to the proof being rejected entirely, or fixed and strengthened.

Confirmation: if the AI understands it well enough that we're even considering asking it to confirm the proof, then you can do all kinds of things. You can ask it to simplify the entire proof to make it easier for humans to verify. You can ask it questions about parts of the proof you don't understand. You can ask it if there's any interesting corollaries or applications in other fields. Maybe you can even ask it to rewrite the whole thing in LEAN (although, like the author, I know nothing about LEAN and have no idea if this would be useful).

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