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Classification of Finite Simple Groups Formalized in Lean

53 points by pfdietz 2 hours ago | 17 comments

mlmonkey 2 hours ago

Out of curiosity: is it possible to take a frontier model and just let it loose on all math textbooks out there, converting each and every proof/theorem/lemma/etc. into Lean? A frontier model should be able to do that, right? And it's a one-time task: do it once and it's there for eternity, available as a library to be used in other proofs.

Maybe my understanding of Lean is wrong; I'm thinking of it as a programming language but for Math.

jonath_laurent 2 hours ago

Axiom MTS here. What you describe is pretty close to how we formalised this proof, which is scattered across more than 500 articles and books: a swarm of agents was tasked with going through all these sources.

One challenge is assembling everything into a coherent whole, since the same informal concept can be formalised in slightly different but incompatible ways.

bananaflag 2 hours ago

Yes, but thing is, for a lot of basic stuff there are myriad ways in which it is formulated, so humans have to decide which of them goes in the library. Of course you can include two or five or ten equivalent formulations (like textbooks sometimes do) but still, not everything under the sun.

Probably this will be done in some form in the coming months.

QuesnayJr 48 minutes ago

I think people assume that it would end up not being usable by humans without a lot of hand-holding. Though maybe that's wrong, and somebody should try it.

tyromaniac 2 hours ago

Write a few proofs with it and then come back to this

sebzim4500 2 hours ago

This seems like a very good use of auto-formalization, because in the past mathematicians have incorrectly believed that they were finished with the classification only discover a missing class/sporadic group.

The original proof was a collaboration of ~100 mathematicians, if Tao is correct then we should expect to see more such collaborations in the future like we do in e.g. physics.

QuesnayJr 45 minutes ago

They didn't discover a missing sporadic group. The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.

vatsachak 2 hours ago

LLM formalization is not as useful as human formalization, but nice to know that the Classification is correct

pfdietz 2 hours ago

The formalization is 43.1M lines of Lean.

bwestergard 2 hours ago

Is that going to be mathematically useful?

sebzim4500 2 hours ago

It's nice to know that they didn't miss anything. The natural language proof is 10,000 pages and they have missed groups in the past.

curt15 57 minutes ago

More so if Lean uncovers logical errors. Note also there are already 2nd and 3rd generation proofs.

pixl97 2 hours ago

Page has some kind of redirect bug that steals the back button, may want to open it in a new tab to avoid that.

QuesnayJr 51 minutes ago

I'm surprised they only found one gap. I would have expected multiple (fixable) gaps would have been uncovered, especially since the result was spread over many papers by many different people.

bamb008 21 minutes ago

In [1] they claim:

>However, there were 34 noteworthy deviations from the published record that needed to be resolved, the most substantial of which related to the Unbalanced Group Theorem.

[1]https://axiommath.ai/research/the-enormous-theorem-formalize...

contubernio 37 minutes ago

Many many gaps had been found and patched (by humans) already over the last forty years.

v3ai 2 hours ago

based if true

Salparvezml 57 minutes ago

[flagged]