Since the proof already exists in human-written form, I'm wondering, can't OpenAI's IOM gold winning algorithm not translate the blueprint to lean?
In particular, note that a key lemma of crystalline cohomology rests on a mistake. Experts think that it is fixable by virtue that results have depended on it for a long time and no issue was found, but it is not fixed.
I admit that my original comment was inaccurate, as it seems to suggest that the gap still exists.
Research level mathematics like this is as hard as it gets, and this proof is famously difficult: uses many branches of advanced mathematics, required thousands of pages of proofs, years of work.
It's somewhat like taking rough pseudo code (the informal proof, a mixture of maths and English) and translating that into a bullet-proof production app (the formal proof, in Lean), where you're going to have to specify every step precisely traced back to axioms, handle all the edge causes, fix incorrect assumptions, and fill in the missing parts that were assumed to be straightforward but might not be.
A major part is you also have to formalise all the proofs your informal proof relied on so everything is traced back to the initial axioms e.g. you can't just cite Pythagorus theorem, you have to formalise that too.
So it's an order of magnitude more difficult to write a formal proof compared to an informal one, and even when you have the informal proof it can teams many years of effort.
But I thought it was a widely celebrated result.
That's part of the motivation to formalise it. When a proof gets really long and complex, relies on lots of other complex proofs, and there's barely any single expert who has enough knowledge to understand all the branches of maths it covers, there's more chance there's a mistake.
There's a few examples here of errors/gaps/flaws found while formalizing proofs that were patched up:
https://mathoverflow.net/questions/291158/proofs-shown-to-be...
My understanding is it's common to find a few problems that then need to be patched or worked around. It's a little like wanting to know if a huge codebase is bug-free or not: you might find some bugs if you formalized the code, but you can probably fix the bugs during the process because it's generally correct. There can be cases where it's not fixable though.
And then, when I raised concerns in Zulip about Lean's metaprogramming facilities being used to trick the pipeline into accepting false proofs, he said the opposite. He even emphasized that the formalization efforts are not for checking proof correctness, but for cataloguing truths we believe in.
This kind of equivocation turned me away from that community, to be honest. That was an extremely frustrating experience.
I imagine one bias is because formal verification is such a huge effort, you're only going to do it for really interesting and impactful proofs, which means the proofs that get formal verified will already have been reviewed and critiqued a lot, and will be less likely to have glaring critical errors.
It takes thorough review by advanced mathematicians to verify correctness.
This is not unlike a code review.
Most people vastly underestimate how complex and esoteric modern research mathematics are.
The systems we deal with in software are massive compared with your typical mathematical framework though. But FLT is probably on similar scope.
That's actually where LLMs are already quite good at.
There is a pseudocode app that depends on a bunch of pseudocode libraries. They want to translate that pseudocode app into a real runnable app. They can do that, and it's a good amount of work, but reasonable. The problem is to get the app to run they also need to translate the hundreds or thousands of pseudocode libraries into actual libraries. Everything from OS APIs, networking libs, rendering libs, language standard libs all need ro be converted from specs and pseudocode to real code to actually run the app. And that's a ton of work.
Plus, Kevin Buzzard is a world expert with some ideas for how to better organize the proof. In general, formalization leads to new understanding about mathematics.
Something people outside of mathematics don't tend to appreciate is that mathematicians are usually thinking deeply about what we already know, and that work reveals new structures and connections that clarify existing knowledge. The new understanding reveals new gaps in understanding, which are filled in, and the process continues. It's not just about collecting verifiably true things.
Even if somehow the OpenAI algorithm could apply here, we'd get less value out of this whole formalization exercise than to have researchers methodically go through our best understanding of our best proof of FLT again.
It's, as far as I know, quite hard to teach an LLM things it doesn't know.
It's a bit frustrating because, before Copilot, new users would come with proofs and you could spend some time helping them write better proofs and they'd learn things and gain skills, but now it's not clear that this is time well spent on my part. Copilot is not going to learn from my feedback.
A "two-page document" (perhaps 200-300 lines of code) would typically check in a matter of seconds to, at most, a few minutes. If it took hours, it would mean the file contains a pathological case that a developer would be expected to identify and fix. It is absolutely not a normal or "not unusual" occurrence.
1. Load a complex data structure, e.g. from JSON. (I do not consider typing out the JSON to be part of the human's job.)
2. Process the data, using the magic of functional programming. (Performance? What's that? Performance is not a priority.)
3. Add some helpful lemmas.
4. Prove the data processing stage was correct… but proving this in general is haaaard, so just make automation that's capable of proving it for most cases… eventually.
5. Great! It takes 3 minutes to run on my tiny examples. Now run it on the real data.
And that's where hours come from!
> Lean's metaprogramming capabilities enable users to extend the language with domain-specific notations and new proof automation techniques.