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.
Metamath has done quite a bit or primary research into this. Theorems often have a slightly non-standard, but more general form.
One can ask things like, "what is the equivalent of functional programming?" Or, "is there global state?" It's a perspective on reverse mathematics.
Note that mathlib currently includes duplicates of many theorems; I believe deduplication is an active goal (if not one seeing much labor right now).
LLMs can do smarter searches, and will likely find proofs for most of them, but are not guaranteed to find a proof.
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.
What you mean it say is that Lean is not a HoTT, and Mathlib has tons of classical reasoning (excluded middle, axiom of choice).
I don't think it's too late at all for HoTT. In fact, I think the AI companies will find trying to crank out more new math which is out of distribution in a funky way a highly useful exercise.
An error in the formalization would be a bug in lean. Which, as we know, certainly seems plausible (I've mused on if these giant proofs are incentivizing LLMs to find lean bugs to exploit), but at least it's a smaller surface area.
In this case the statement is unusually complicated but still much much easier to verify than the proof itself.
Humans have to establish trust in this.
After that, the rest of the language is written. Yes it is possible the type inference engine is broken but in that case the kernel will reject it
Also don't discount the fact that people would learn how to do this kind of project, currently no one really knows how to decide what definitions work best, what should be reducible, what should be part of a public api and what should be local to a module, etc.
The article refers to its use of the "first-generation proof". It would seem this proof is too large to be truly known by one person, and the group of people who "collectively knew" it grew old and started to die. Since its original formulation, mathematicians began to compile, consolidate, and revise the proof into a 2nd generation digested form (estimated to be 5000 pages instead of tens of thousands). But that effort was still so monumental in its own right that a 3rd gen proof project began to digest the 2nd gen proof before the 2nd gen proof was even finished. I believe both the 2nd and 3rd gen projects are still ongoing.
It will be interesting to see how small the proof gets once we finish learning how to obtain these results "efficiently" with the benefit of hindsight.
>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...