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Formalization of the Solution to the Hopf Problem

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Re: Formalization of the Solution to the Hopf Problem

#11

Post by tancop »

elil17 wrote: post #10
medler wrote: post #9

Earlier quoted context omitted.

We pretty much crossed this bridge in 1976 with the proof of the four-color theorem: https://en.wikipedia.org/wiki/Four_Color_Theorem

The process of automating mathematics: We understand the proof (most math from all of history) -> We understand how the proof was made (computer-assisted proofs like the four-color theorem) -> We have to trust the computer's explanation for how the proof was made (some LLM proofs)

Is that really a problem? The only risk in a Lean proof is the LLM finding a bug in multiple kernels (2 with the standard setup) that would let it add a new theorem without legitimately proving it. If you have reasoning traces or send a second model to look at the proof you can easily detect that and file a bug report.

For the rest you only need to check that the formal problem statement matches the actual problem and the proof is not using extra axioms. That's easy to check manually or with a simple script.

Re: Formalization of the Solution to the Hopf Problem

#12

Post by tux3 »

tancop wrote: post #11
elil17 wrote: post #10

Earlier quoted context omitted.

The process of automating mathematics: We understand the proof (most math from all of history) -> We understand how the proof was made (computer-assisted proofs like the four-color theorem) -> We have to trust the computer's explanation for how the proof was made (some LLM proofs)

Is that really a problem? The only risk in a Lean proof is the LLM finding a bug in multiple kernels (2 with the standard setup) that would let it add a new theorem without legitimately proving it. If you have reasoning traces or send a second model to look at the proof you can easily detect that and file a bug report. For the rest you only need to check that the formal problem statement matches the actual problem an…

Many mathematicians care about understanding the method and the tools used more than the result. LLMs have a large advantage in memory and reading speed, you can feed them the entire internet and published literature at training time, and then make them check dozens of papers in a few minutes at runtime. They might not be very creative, but they've seen that obscure 1990's theorem in the training data that happened to use an idea that might just work again.

Reading the LLM's 12 MB lean output is a bit like trying to maintain a codebase that has been fully vibe-coded as a single giant file, by hand. Even if it's possible in principle, no one is going to enjoy that.

Re: Formalization of the Solution to the Hopf Problem

#14

Post by GPerson »

QuesnayJr wrote: post #2

I was thinking about trying this exact problem with AI. I missed that it had already been solved. It's not that surprising that someone else already tried it. What's surprising is that open problems get solved so quickly now that it's impossible to keep up with them all.

Unless you have access to the same compute limits as the guy at Anthropic, and the internal model they use, you probably would just have spent a lot of tokens without success. So you can take solace in that.

Re: Formalization of the Solution to the Hopf Problem

#15

Post by bananaflag »

QuesnayJr wrote: post #2

I was thinking about trying this exact problem with AI. I missed that it had already been solved. It's not that surprising that someone else already tried it. What's surprising is that open problems get solved so quickly now that it's impossible to keep up with them all.

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