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Terry Tao @teorth.bsky.social: Indiscriminate brute-force application of AI to mathematics is depleting the field of good questions without renewing this resource or providing commensurate value. OpenAI: We hear you. Here's a .zip file with 377 hasty AI solutions to important open problems.

nytimes.comOpenAI Releases Findings on 377 Math Problems, Further Roiling Field (Gift Article)After a previous solution angered mathematicians, the company characterized the new release as being more responsive to concerns about A.I. disrupting research.

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I'm not a mathematician, but I was of the understanding that the value in proofs is more what we learn about problems and novel tools we can apply to other problems moreso than the proving in most cases

It's not clear to me... do we all just trust LEAN as a model to verify proofs to unsolved problems? How much of this is even verifiably correct? And then of course it's not useful if it can't be explained and understood.

"Famously, Lean, like many proof assistants, defines division so that 1/0 = 0 and subtraction on the natural numbers so that 2-3 =0." 🙃 How is this a thing people use? It must be useful in some cases right? Or is it literally just complete garbage and the people using it don't read anyways?

My sense overall is that it has been good for doublechecking that proofs are solid, *but* the increasing reliance on computer program A telling you the output of computer program B is correct has called attention to the peculiarities and shortcomings of program A.

Some parts of Lean may appear odd at first, but it is extremely useful in math. Experts use it, including famous ones like Terence Tao. A proof in Lean is considered a valid logical proof by mathematicians, uncontroversially. (That does not mean Lean does not have bugs, like any software! It does)

We all trust Lean. It's a mechanist proof system. Therefore, the only thing you need to check is that the thing it proves is actually the same as the thing you care about. That does not leave much room for funny business.

The idea with Lean is if you trust that you’ve correctly expressed the statement of the problem in Lean then you can trust Lean to tell you if it’s proved or not. With N-S a few weeks ago, they used a preexisting translation that people already agreed was right.

From a paper uploaded to Arvix yesterday: "We provide several examples of mistranslations of NL statements...resulting in mismatches between NL proofs and their lean "verification." These include Openai's announced Navier-Stokes proof."

Yes, but it's also true that we (as in “the mathematical community”) were unable to come up with an answer as to how OpenAI should release these results so as to mitigate the harm:

Gro-Tsen@gro-tsen.bsky.social

I agree about OpenAI's goals, but we should also acknowledge that we (the mathematical community) haven't been able to put forward a credible answer to how they should have released the results. They put us in a double bind where we neither want the results withheld nor published.

This is the best criticism of "AI does math": AI solving math problems doesn't teach humans new approaches to advancing the field. However...counterpoint...what if we don't need humans to advance the field anymore?

I mean, you have to define “the field”—but if the field involves human understanding of mathematics, I don’t see how you cut the humans out. If it doesn’t, and it’s just machines manipulating symbols that people don’t understand, I don’t see what the point is.

Well, in the simplest terms, is it useful if Kuncklebunkle's Theorem is proved true, regardless of whether humans understand the proof? So, no understanding, but now we have a new tool in the toolbelt.

They didn't even provide a Lean proof for the Hilbert problem they claimed that they solved. But they did provide a disclaimer.

BTW, I'm working on an AI generated Lean-free 100,000 page proof that P ≠ NP AND that OpenAI will be profitable next year. Technical report due soon.

Solutions that could actually be very convincing bullshit because these are not pre-formalised problems, the LLM was tasked with doing the formalising, but the LLM cannot walk you through its thinking, because that's not thinking, it's just more tokens.

Those automated solution finders for problems where a human mathematician can pre-formalise it: fine. But asking the LLM to do the formalisation as well is vibe mathing. Analogous to when vibe coders think the LLM understands the codebase.