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mlmonkey 4 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 3 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 3 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.
thaumasiotes 1 hours ago [-]
> Of course you can include two or five or ten equivalent formulations (like textbooks sometimes do) but still, not everything under the sun.
Note that mathlib currently includes duplicates of many theorems; I believe deduplication is an active goal (if not one seeing much labor right now).
Write a few proofs with it and then come back to this
QuesnayJr 3 hours 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.
sebzim4500 4 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 3 hours 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 The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.
senderista 19 minutes ago [-]
Filling in well-known gaps like this is the kind of thing the AI companies should be doing, not trying to scoop research in progress.
esafak 6 minutes ago [-]
Looking at the visualizations made me wonder if the embeddings themselves can be used to suggest, if not derive, mathematical results. I realize it depends on the quality of the embeddings.
pfdietz 4 hours ago [-]
The formalization is 43.1M lines of Lean.
bwestergard 3 hours ago [-]
Is that going to be mathematically useful?
sebzim4500 3 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.
Panoramix 52 minutes ago [-]
How do we know the formalization itself doesn't have any errors? AFAIK it's very far from trivial to formalize something.
sebzim4500 43 minutes ago [-]
In general checking that the definitions are correct isn't that hard and assuming at least one of the lean kernels you verify with is correct (which tbf is an assumption, the implementations are based on each other so could have a common bug) that's all you need to verify.
In this case the statement is unusually complicated but still much much easier to verify than the proof itself.
pfdietz 2 hours ago [-]
The proof was of such complexity and length there were lingering doubts. This dispels them.
2 hours ago [-]
curt15 3 hours ago [-]
More so if Lean uncovers logical errors. Note also there is already an ongoing 2nd generation proof.
cubefox 51 minutes ago [-]
I suppose this is the longest Lean proof by a wide margin.
vatsachak 3 hours ago [-]
LLM formalization is not as useful as human formalization, but nice to know that the Classification is correct
QuesnayJr 3 hours 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 2 hours 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.
Maybe my understanding of Lean is wrong; I'm thinking of it as a programming language but for Math.
One challenge is assembling everything into a coherent whole, since the same informal concept can be formalised in slightly different but incompatible ways.
Probably this will be done in some form in the coming months.
Note that mathlib currently includes duplicates of many theorems; I believe deduplication is an active goal (if not one seeing much labor right now).
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.
In this case the statement is unusually complicated but still much much easier to verify than the proof itself.
>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...