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.
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.
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.
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.
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.
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 The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.
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.
>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.
>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...