Comment by michaelfm1211
1 day ago
Reminds me of this: https://mathoverflow.net/questions/513742/are-we-stuck-with-...
I know this is an implementation bug not a meta-theory bug, but I'd almost consider the fact soundness bugs are possible as a bug in the ideology, or at least a severe drawback. Stuff like this just wouldn't happen in Metamath. In a future where AI is autogenerating formalizations, why not have the AI use a harder but airtight system like Metamath?
Took me about 90 seconds to find a Metamath implementation bug that apparently allowed proving something that shouldn't be provable: https://github.com/metamath/metamath-exe/issues/184
That's with the original C verifier only. The actual database is cross-checked by 6 independent implementations. This is the whole point of Metamath: its kernel is so tiny that you can implement a verifier in a weekend.
Metamath isn't a silver bullet in the design space of formal proof tools, but I personally think it just about nails the metatheory we want. Maybe some explicit facility around definitions would be desireable.
As it happens, the Python verifier mmverify.py has an even simpler soundness bug [0], and so far I've reviewed two independent AI-written verifiers that have replicated that bug, since they apparently really like to copy the strategy from mmverify.py. A sound verifier isn't too difficult to write in terms of architecture (I just wrote one myself for differential testing [1]), but it requires some close attention to the details.
That is to say, the Swiss-cheese approach definitely lends authority, but individual implementations are unfortunately not as foolproof as they're made out to be.
[0] https://github.com/david-a-wheeler/mmverify.py/issues/30
[1] https://github.com/LegionMammal978/mm-verifier-tests/blob/ma...
Why does the original C verifier have bugs if its supposedly so easy to implement a verifier?
2 replies →
The utility of autoformalization is not actually in confirming the correctness of human-checked results. Mathematicians have a pretty good peer review process. I'm not actually aware offhand of any mathematical results that were accepted and later found to be incorrect -- though I'm sure cases exist, it's astoundingly rare.
The mathematics community's motivation for formalizing problems like Maryna Viazovska's sphere packing results in Lean weren't because the results were in doubt -- she won a Fields Medal for it, it's an extremely examined proof -- but because formalizing those results would lead to a lot of interesting and useful mathematical objects needing to be formalized as a prerequisite, which could then be merged into Lean's Mathlib and become useful for anyone working with Lean, particularly students. Having a library of idiomatic proofs available in a formal system capable of checking your work is Really Cool! Working in a proof assistant is a great way to develop mathematical maturity, especially for people who might not have an undergraduate education available to them.
(Unfortunately in the sphere packing case, the research group working on it made the mistake of trusting one of the various "AI for Math" slop companies, who promptly rugpulled them: https://arxiv.org/html/2603.03684v3)
So the short answer is "it depends on what you want". Lean is an eminently usable system for humans and LLMs alike; Metamath is uh. Not. But yes, Metamath seems to have a more trustable kernel wrt. the independent verifiers, so if that's all you're after it would be a better pick. But... there's only so many bugs Lean's kernel can have. At some point, they'll all be found.
(I should note that there is also utility in confirming the correctness of results that are too complex to be adequately verified by humans. The recent result that NF is consistent wrt. ZFC, among others (Peter Scholze's Liquid Tensors project) are examples of this. But that's distinct from "autoformalization" as commonly applied.)
> I'm not actually aware offhand of any mathematical results that were accepted and later found to be incorrect
https://mathoverflow.net/questions/35468/widely-accepted-mat...
Yes, in particular the slides by Kevin Buzzard (linked in there or here: https://www.andrew.cmu.edu/user/avigad/meetings/fomm2020/sli... ) are funny and educational on that matter.