4 ms·

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?

tripplyons 1y ago
You're either underestimating the length of the proof, or overestimating the length of tasks that models can currently accomplish.
- amelius 1y ago
  The blueprint is a step-by-step outline.
  - tripplyons 1y ago
    If the goal is to formalize the proof, you would need more than an outline.
rcxdude 1y ago
This is a significantly harder problem than winning gold in IOM. A large part of it is figuring out how to represent some of the relevant ideas in Lean at all.
- Davidzheng 1y ago
  I disagree. I think it's a sequence of a huge number of modular moderately hard tasks each much easier than a hard IMO question.
  - markusde 1y ago
    IMO problems are stated in and solved by math a high schooler could understand. Getting the definitions right (one of the harder parts of mechanizing proofs IME) is a different beast altogether.
seanwilson 1y ago
"The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students", so not to undermine it but it's below university or graduate level. 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.
- amelius 1y ago
  Yes but the hard work (coming up with a human-readable proof) has already been done.
  - rcxdude 1y ago
    No, some of the harder work has been done. Translating human-readable proofs into machine-readable ones is also very hard work and an area of active research.
  - seanwilson 1y ago
    Human readable (informal) proofs are full of gaps that all have to be traced back to axioms e.g. gaps that rely on shared intuition, background knowledge and other informal proofs. 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.
    - bee_rider 1y ago
      I’m almost certain this is ignorance on my part, but it seems like this would mean the proof is… possibly wrong? I mean if there are gaps and other informal proofs in there? But I thought it was a widely celebrated result.
voxl 1y ago
Proof assistant code is high reliability, there is no room to fudge it. This is perhaps the one place where you can really see how bad LLMs are when you care about reliability.
- adastra22 1y ago
  Why? Coding assistants tend to do even better in contexts where they have tools like type checkers and linters to verify their implementations. This area seems actually uniquely well suited to LLM usage.
  - rocqua 1y ago
    When I asked experts on formal proofs a year ago, their intuition was that there isn't enough formal proofs out there for LLMs to be very good at the syntax. It's, as far as I know, quite hard to teach an LLM things it doesn't know.
    - UltraSane 1y ago
      He is right and it doesn't matter because you can instantly tell if the proof the LLM generates is true or not.
      - deleted 1y ago
        [deleted]
      - kmill 1y ago
        I see people on Zulip using Copilot to write Lean proofs, and they have some success, but the quality is really bad right now, creating long, unmaintainable proofs. New users get stuck, thinking they're 90% of the way to the end, but really the whole thing probably should be scrapped and they should start over. 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.
ethan_smith 1y ago
Formalizing Wiles' proof requires translating hundreds of pages of sophisticated mathematics with implicit reasoning steps into a precise logical framework, which is fundamentally different from the pattern-matching AI uses to solve competition problems.
- adastra22 1y ago
  That's not how state of the art models work.
  - yorwba 1y ago
    All three claims of gold medal performance on IMO 2025 that I'm aware of solved the first 5 problems, that were designed to be solvable by application of standard techniques, but got stumped on the sixth problem that was a bit more unusual. So it does seem like state-of-the-art models solve competition problems by recognizing which kind of problem it is and applying a corresponding solution template. Which is not too different from human competitors exploiting common question patterns, but humans seem to be able to degrade more gracefully by falling back to a more explorative mode when none of the standard tricks seem to apply.
jhanschoo 1y ago
In addition to other comments, see https://xenaproject.wordpress.com/2024/12/11/fermats-last-theorem-how-its-going/ https://xenaproject.wordpress.com/2024/12/11/fermats-last-th... 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.
- NooneAtAll3 1y ago
  > and my understanding of Maria Ines’ talk is that these issues have now been sorted out
  - jhanschoo 1y ago
    I think you are trying to say that this matter has since been resolved and so presumably the whole informal proof somehow resides in literature. I suppose that to the first point you may be right (I'm guessing that it's since been made available), but to the second point I think you are overconfident that similar gaps do not exist. I admit that my original comment was inaccurate, as it seems to suggest that the gap still exists.
kmill 1y ago
My understanding is that the proof doesn't exist in written form in its entirety. 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.

tripplyons 1y ago

You're either underestimating the length of the proof, or overestimating the length of tasks that models can currently accomplish.

amelius 1y ago
The blueprint is a step-by-step outline.
- tripplyons 1y ago
  If the goal is to formalize the proof, you would need more than an outline.

rcxdude 1y ago

This is a significantly harder problem than winning gold in IOM. A large part of it is figuring out how to represent some of the relevant ideas in Lean at all.

Davidzheng 1y ago
I disagree. I think it's a sequence of a huge number of modular moderately hard tasks each much easier than a hard IMO question.
- markusde 1y ago
  IMO problems are stated in and solved by math a high schooler could understand. Getting the definitions right (one of the harder parts of mechanizing proofs IME) is a different beast altogether.

seanwilson 1y ago

"The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students", so not to undermine it but it's below university or graduate level. 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.

amelius 1y ago
Yes but the hard work (coming up with a human-readable proof) has already been done.
- rcxdude 1y ago
  No, some of the harder work has been done. Translating human-readable proofs into machine-readable ones is also very hard work and an area of active research.
- seanwilson 1y ago
  Human readable (informal) proofs are full of gaps that all have to be traced back to axioms e.g. gaps that rely on shared intuition, background knowledge and other informal proofs. 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.
  - bee_rider 1y ago
    I’m almost certain this is ignorance on my part, but it seems like this would mean the proof is… possibly wrong? I mean if there are gaps and other informal proofs in there? But I thought it was a widely celebrated result.

voxl 1y ago

Proof assistant code is high reliability, there is no room to fudge it. This is perhaps the one place where you can really see how bad LLMs are when you care about reliability.

adastra22 1y ago
Why? Coding assistants tend to do even better in contexts where they have tools like type checkers and linters to verify their implementations. This area seems actually uniquely well suited to LLM usage.
- rocqua 1y ago
  When I asked experts on formal proofs a year ago, their intuition was that there isn't enough formal proofs out there for LLMs to be very good at the syntax. It's, as far as I know, quite hard to teach an LLM things it doesn't know.
  - UltraSane 1y ago
    He is right and it doesn't matter because you can instantly tell if the proof the LLM generates is true or not.
    - deleted 1y ago
      [deleted]
    - kmill 1y ago
      I see people on Zulip using Copilot to write Lean proofs, and they have some success, but the quality is really bad right now, creating long, unmaintainable proofs. New users get stuck, thinking they're 90% of the way to the end, but really the whole thing probably should be scrapped and they should start over. 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.

ethan_smith 1y ago

Formalizing Wiles' proof requires translating hundreds of pages of sophisticated mathematics with implicit reasoning steps into a precise logical framework, which is fundamentally different from the pattern-matching AI uses to solve competition problems.

adastra22 1y ago
That's not how state of the art models work.
- yorwba 1y ago
  All three claims of gold medal performance on IMO 2025 that I'm aware of solved the first 5 problems, that were designed to be solvable by application of standard techniques, but got stumped on the sixth problem that was a bit more unusual. So it does seem like state-of-the-art models solve competition problems by recognizing which kind of problem it is and applying a corresponding solution template. Which is not too different from human competitors exploiting common question patterns, but humans seem to be able to degrade more gracefully by falling back to a more explorative mode when none of the standard tricks seem to apply.

jhanschoo 1y ago

In addition to other comments, see https://xenaproject.wordpress.com/2024/12/11/fermats-last-theorem-how-its-going/ https://xenaproject.wordpress.com/2024/12/11/fermats-last-th... 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.

NooneAtAll3 1y ago
> and my understanding of Maria Ines’ talk is that these issues have now been sorted out
- jhanschoo 1y ago
  I think you are trying to say that this matter has since been resolved and so presumably the whole informal proof somehow resides in literature. I suppose that to the first point you may be right (I'm guessing that it's since been made available), but to the second point I think you are overconfident that similar gaps do not exist. I admit that my original comment was inaccurate, as it seems to suggest that the gap still exists.

kmill 1y ago

My understanding is that the proof doesn't exist in written form in its entirety. 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.