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The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if

The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. This subjective attitude turns mathematics into nothing more than number-poetry. That would reduce mathematics to something very pathetic. Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from! For many theorems, I can still recall which exposition was the one that "sank in" for me (often not the first one!) a decade after grad school. In my mind, this makes current LLMs unfit to deserve any credit at all -- they cannot give credit to others, so they and their owners deserve no credit themselves. OpenAI's LLM took the last tiny step, but not any of the important ones.

traject_ 13d ago
I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.
- octoberfranklin 13d ago
  What you describe is Tao's rebuttal of #2. I fully agree with him there. It's his rejection of #1 that makes me sad.
  - traject_ 13d ago
    If you mean > What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics. Then that is not necessarily subjective either if an AI can produce an actually intelligible proof. The problem is that as mathematicians with PDE expertise have mentioned on Twitter the actual solution seems to devolve into an unreadable mess focusing on irrelevant details after a more readable first few pages in the proof. If it wasn't a Lean compiled proof and presented as a human artifact, it would be hard to assess if the deviser of the solution had any actual understanding of the solution.
    - grey-area 13d ago
      Well perhaps we should use occam’s razor here. If it doesn’t seem like the lllm understands the proof, it probably doesn’t. Just because it generated something that works doesn’t mean it understands it, and the evidence from its proof is that it does not, which is not very surprising given the technology we’re talking about, which generates likely phrases based on a corpus and training.
  - auggierose 13d ago
    it is not his rejection, this is a guest article written by two other people
deleted 13d ago
[deleted]
Pulcinella 13d ago
Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from! In my mind, this makes it unfit to deserve any credit at all -- it cannot give credit to others, so it deserves no credit itself. It took the last tiny step, but certainly not any of the important ones. Yes I fear a lot of doom and gloom around AI is unearned and only really serves to prop up the valuation of AI companies. It's still very much unclear how much work OpenAI actually did versus just copying the nearly complete homework of someone 5 minutes earlier.
- deleted 13d ago
  [deleted]
adalacelove 13d ago
Grigori Perelman rejected the Fields Medal and the Clay prize among other things for what he considered unjust decisions and lack of ethics and proper attribution to other mathematicians. I find it surprising that he hasn't been mentioned yet, given it involves another Clay prize (ironically another lack of attribution, I guess). There is a long tradition of mental problems among great mathematicians, but maybe this was not the case, or maybe in a Lovecraftian way the observation of deep truths has a terrible toll.
fwlr 13d ago
Conversely, I think mathematics should retain this aspect of “number-poetry”. Consider a mathematical pursuit that lacks number-poetry but retains other features such as attribution, calculation, and puzzle-solving: I think of competitions like the largest prime number or the furthest digit of pi. Do these not feel in some sense trivial, more suited to IFLS Facebook posts than arxiv preprints?
robotpepi 13d ago
> This subjective attitude turns mathematics into nothing more than number-poetry. I don't think you understood the point. The point applies to all of basic science. You of course want some explanation supporting the raw answer, so you can use that insight in other contexts.

traject_ 13d ago

I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.

octoberfranklin 13d ago
What you describe is Tao's rebuttal of #2. I fully agree with him there. It's his rejection of #1 that makes me sad.
- traject_ 13d ago
  If you mean > What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics. Then that is not necessarily subjective either if an AI can produce an actually intelligible proof. The problem is that as mathematicians with PDE expertise have mentioned on Twitter the actual solution seems to devolve into an unreadable mess focusing on irrelevant details after a more readable first few pages in the proof. If it wasn't a Lean compiled proof and presented as a human artifact, it would be hard to assess if the deviser of the solution had any actual understanding of the solution.
  - grey-area 13d ago
    Well perhaps we should use occam’s razor here. If it doesn’t seem like the lllm understands the proof, it probably doesn’t. Just because it generated something that works doesn’t mean it understands it, and the evidence from its proof is that it does not, which is not very surprising given the technology we’re talking about, which generates likely phrases based on a corpus and training.
- auggierose 13d ago
  it is not his rejection, this is a guest article written by two other people

deleted 13d ago

[deleted]

Pulcinella 13d ago

Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from! In my mind, this makes it unfit to deserve any credit at all -- it cannot give credit to others, so it deserves no credit itself. It took the last tiny step, but certainly not any of the important ones. Yes I fear a lot of doom and gloom around AI is unearned and only really serves to prop up the valuation of AI companies. It's still very much unclear how much work OpenAI actually did versus just copying the nearly complete homework of someone 5 minutes earlier.

deleted 13d ago
[deleted]

adalacelove 13d ago

Grigori Perelman rejected the Fields Medal and the Clay prize among other things for what he considered unjust decisions and lack of ethics and proper attribution to other mathematicians. I find it surprising that he hasn't been mentioned yet, given it involves another Clay prize (ironically another lack of attribution, I guess). There is a long tradition of mental problems among great mathematicians, but maybe this was not the case, or maybe in a Lovecraftian way the observation of deep truths has a terrible toll.

fwlr 13d ago

Conversely, I think mathematics should retain this aspect of “number-poetry”. Consider a mathematical pursuit that lacks number-poetry but retains other features such as attribution, calculation, and puzzle-solving: I think of competitions like the largest prime number or the furthest digit of pi. Do these not feel in some sense trivial, more suited to IFLS Facebook posts than arxiv preprints?

robotpepi 13d ago

> This subjective attitude turns mathematics into nothing more than number-poetry. I don't think you understood the point. The point applies to all of basic science. You of course want some explanation supporting the raw answer, so you can use that insight in other contexts.