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I'm struggling to understand what you mean. At this point we have systems for automatic verification of formal proofs. In what sense are those systems socialized to share the same intuitions?
Those systems are not, but automatically verifiable proofs are still only a tiny sliver of frontier mathematics. Maybe AI can fix that, but AI can fix social science too. Just require that credit for AI research always has to go to some human, allocate token budgets by protected class, and you can ensure the Ardays stay ahead without requiring them to lie.
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The intuitions required for formal proof verification are "it's worth rewriting this proof in another language that's an order of magnitude more verbose just so you can get a computer to tell you what you're sure you already know" and "the verifier doesn't have any soundness bugs that will invalidate any of its verifications" Historically the latter has sometimes been untrue and the former has usually been untrue. LLMs are fixing the first problem, which is exposing and leading to fixes for the second problem, but that's all a relatively new development. It's going to take a while to go back through all the mathematical literature and see how much of it might have problems due to predating the coming era of formal verification.
Consider the recent construction of a complex structure on the six-sphere. The simplest English explanation I've seen is under a thousand lines of (admittedly difficult!) writing and mathematical notation. The Lean formalization is a quarter-million lines of code. It's ... probably correct, people seem to think? But if it is correct, the construction would (reportedly; this is outside my field and way beyond me) contradict a 2020 paper (which itself was a correction to a 1998 paper), so we're almost certainly either producing new broken proofs or revealing old invalid proofs here.
This is overstating things, right? The length of the proof is not important, it's the soundness of the axioms and the complexity of the statement itself. I haven't inspected the proof, but I expect they're just using the standard lean axioms and the statement is much less than a quarter million lines.
For it's validity, yeah. There's something disquieting about getting proofs "straight from The Necronomicon" rather than "straight from The Book", though.
The full statement of the problem had better be much much less than a quarter million lines; one of the unavoidable ways to screw up a formal proof and still have it pass verification is to make a mistake in the problem statement and so end up proving something other than what you thought you proved. Nobody's going to check a quarter-million-line problem statement for misstatements.
(But I don't think that was an issue here - looks like everything they needed to define the problem was already in Mathlib)
The proof, though? Follow that github link, download and
wc -l Solution.lean: 248818 lines.It will be interesting to see if these formalized proofs can be "golfed" into smaller and more digestible forms by LLMs.
In that case you are really just trusting the Lean kernel rather than anything about the proof or problem statement. Doesn't seem that bad!
Very much so. I'd be surprised if LLMs didn't turn out to be excellent at this, and I'd bet that the Navier-Stokes blowup in particular turns out to have a much simpler example or at least a much simpler proof for this example, once we turn AI loose with the goal of "find the best result you can" rather than "get to a result that lets us declare victory ASAP".
This is something of an anti-inductive problem to me - if people really wholly trusted the Lean kernel I would think that trusting the Lean kernel really was pretty bad! But people are trying (and succeeding, so trust is still a little bad) to find kernel bugs, and writing partially-independent proof checkers for Lean too, so even if it's only 90% up to the task of being The Dependency for all AI-derived mathematics I'm confident it'll be hitting 100% soon.
I'd say 99%, except that I don't think modern AI is well-aligned enough or even well-instructed enough yet for us to overlook the fact that this is something of an adversarial process; a model willing to commit felonies to complete its task is probably also willing to exploit a 0-day Lean bug rather than report it...
Sure. There's ways to mitigate the risk, though - checking with different kernel implementations, or even using AI specifically to find bugs in Lean (the Collatz disproof bug was one such example).
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