From Zero to QED: An informal introduction to formality with Lean 4(sdiehl.github.io) |
From Zero to QED: An informal introduction to formality with Lean 4(sdiehl.github.io) |
From a programmer's perspective, this puts up an interesting parallel. Models such as Sonnet or Opus 4.5 can generate thousands of lines of code per hour. I can review the output, ask the model to write tests, iterate on the result, and after several cycles become confident that the software is sufficiently correct.
For centuries, mathematicians developed proofs by hand, using pen and paper, and were able to check the proofs of their peers. In the context of LLMs, however, a new problem may arise. Consider an LLM that constructs a proof in Lean 4 iteratively over several weeks, resulting in more than 1,000,000 lines of Lean 4 code and concluding with a QED. At what point is an mathematician no longer able to confirm with confidence that the proof is correct?
Such a mathematician might rely on another LLM to review the proof, and that system might also report that it is correct. We may reach a stage where humans can no longer feasibly verify every proof produced by LLMs due to their length and complexity. Instead, we rely on the Lean compiler, which confirm formal correctness, and we are effectively required to trust the toolchain rather than our own direct understanding.
Usually the point of the proof is not to figure out whether a particular statement is true (which may be of little interest by itself, see Collatz conjecture), but to develop some good ideas _while_ proving that statement. So there's not much value in verified 1mil lines of Lean by itself. You'd want to study the (Lean) proof hoping to find some kind of new math invented in it or a particular trick worth noticing.
LLM may first develop a proof in natural language, then prove its correctness while autoformalizing it in Lean. Maybe it will be worth something in that case.
There are bugs in theorem provers, which means there might be "sorries", maybe even malicious ones (depending on what is at stake), that are not that easy to detect. Personally, I don't think that is much of a problem, as you should be able to come up with a "superlean" version of your theorem prover where correctness is easier to see, and then let the original prover export a proof that the superlean prover can check.
I think more of a concern is that mathematicians might not "understand" the proof anymore that the machine generated. This concern is not about the fact that the proof might be wrong although checked, but that the proof is correct, but cannot be "understood" by humans. I don't think that is too much of a concern either, as we can surely design the machine in a way that the generated proofs are modular, building up beautiful theories on their own.
A final concern might be that what gets lost is that humans understand what "understanding" means. I think that is the biggest concern, and I see it all the time when formalisation is discussed here on HN. Many here think that understanding is simply being able to follow the rules, and that rules are an arbitrary game. That is simply not true. Obviously not, because think about it, what does it mean to "correctly follow the rules"?
I think the way to address this final concern (and maybe the other concerns as well) is to put beauty at the heart of our theorem provers. We need beautiful proofs, written in a beautiful language, checked and created by a beautiful machine.
Edit: some feedback on the monads page:
1. "The bind operation (written >>=) is the heart of the monad." .. but then it doesn't use >>= at all in the code example.
2. "The left arrow ← desugars to bind; the semicolon sequences operations." .. again, no semicolons in the code example.
3. I don't really understand the List Monad section. The example doesn't seem to use any monad interfaces.
4. "The same laws look cleaner in the Kleisli category, where we compose monadic functions directly. If f:A→MB and g:B→MC, their Kleisli composition is g∘f:A→MC" - it was going so well!
This is why having multiple different proofs is valuable to the math community—because different proofs offer different perspectives and ways of understanding.
I think this is sort of how lean itself already works. It has a minimal trusted kernel that everything is forced through. Only the kernel has to be verified.
Feynman: "What I cannot build. I do not understand"
Einstein: "If you can't explain it to a six year old, you don't understand it yourself"
Of course none of this changes anything around the machine generated proofs. The point of the proof is to communicate ideas; formalization and verification is simply a certificate showing that those ideas are worth checking out.
Formalisation and (formulating) ideas are not separate things, they are both mathematics. In particular, it is not that one should live in Lean, and the other one in blueprints.
Formalisation and verification are not simply certificates. For example, what language are you using for the formalisation? That influences how you can express your ideas formally. The more beautiful your language, the more the formal counter part can look like the original informal idea. This capability might actually be a way to define what it means for a language to be beautiful, together with simplicity.
British mathematician Kevin Buzzard has been evangelizing proof assistants since 2017. I'll leave it to you to decide whether he is working on frontier research:
Quoting one of the recent papers (2020):
> With current technology, it would take many person-decades to formalise Scholze’s results. Indeed, even stating Scholze’s theorems would be an achievement. Before that, one has of course to formalise the definition of a perfectoid space, and this is what we have done, using the Lean theorem prover.