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This result is some evidence that AI will not solve RH soon. If there were any easy solution hiding in plain sight then it probably would have found it. Solvin

This result is some evidence that AI will not solve RH soon. If there were any easy solution hiding in plain sight then it probably would have found it. Solving RH likely requires AI that is substantially more creative. But we haven't even solved the creativity problem for writing let alone mathematics. I believe that transformers are a trillion dollar local optimum that we will find it very hard to escape. Let's wait for the models to produce a good novel first.

jetrink 2mo ago
I would expect to see LLMs that are creative in math before any that are creative in writing. Creativity is more easily specified in math and the solutions can be formally verified. There's no good way to classify creative writing. Many truly great works are overlooked by experts and the public until decades later. Many derivative works are commercially successful.
- zarzavat 2mo ago
  LLMs, both in writing and mathematics seem to only be capable of coming up with texts that are inside the distribution of the training data. With writing it's just more obvious. LLMs don't write with personality. They don't create new and exciting worlds on their own. Everything they output feels derivative. In mathematics you see the same effect. They are very good at finding results that humans missed, taking advantage of their broad knowledge and tireless work ethic. But just as they have been unable to create new literary worlds, they also have so far been unable to create new mathematics. I believe this lack of creativity is intrinsic to how these models are architected and trained. We want models that produce these in-distribution outputs because those types of models are more economically valuable. Nobody wants a coding agent with spontaneity, we want models that predictably and obediently solve problems - and that's what we got.
  - deleted 2mo ago
    [deleted]
  - goekjclo 2mo ago
    Man the cope on HN is something else. Neighbour, 99.9(9)% of professional mathematicians don't go around creating new mathematics. The goalposts have been moved so far that we're now at "AI isn't Grothendieck yet".
- polanyer 2mo ago
  Not to mention it is all subjective taste. There is nothing objective beyond consensus that a "great" work of creative writing is "great" no matter how great a consensus thinks it is. It is really the same with any art form.
redox99 2mo ago
> This result is some evidence that AI will not solve RH soon. If there were any easy solution hiding in plain sight then it probably would have found it. There's no way you can conclude that. Yes, "Fable 2" or whatever this was probably won't. But we can't know what Fable 3/4/5/etc will be able to do. If anything, if we have 1 or 2 more years of progress like the last 12 months, which have been insane, I'd say LLMs are likely to solve it.
- HappyPanacea 2mo ago
  You underestimate the difficulty of RH, there are far easier problems broadly related to RH (density hypothesis, Artin's holomorphy conjecture, Stark conjectures) which are still aren't solved.
  - redox99 2mo ago
    I think verifiable tasks like math will soon be like Chess. Machines way beyond human intelligence.
    - nbulka 2mo ago
      Check out the work of Godel Turing and Chaitin!
- handle584 2mo ago
  The keyword is soon, so OP simply meant current generation of LLMs are not likely to prove RH, judging from the performance shown in the paper. Why? Because such explicit numerical improvements are not that interesting, which is best summed up in a review paper [0] of our efforts on RH spanning a century and a half, > The pathetic attempts to enlarge the ridiculous zero free region in the critical strip is a perfect example of what brute force can do without fully exploiting fundamental arithmetic aspects of the problem. (italic added) For outsiders, zero free region is another angle abundant with numerical improvements but no groundbreaking insights. For percentage people stop at ~40% because there is no need to proceed further, just like we are not interested in computing the googol-th digit of pi although in principle we could. The groundbreaking results are like Selberg's that goes from zero to 0.01% (actually it is left unspecified, all we know is the percentage is positive), or Zhang's twin prime bound from infinity to 70,000,000. After this leap the pure numerical difference between 0.01%, 40%, 67% or even 100% is not substantial, and that's partly why Selberg did not even bother to compute it. Also RH will not follow from 100%, because in mathematics 100% does not mean all. On the other hand, it is also wrong to dismiss such results all together. Riemann already know the real part of all zeros are bewteen zero and one, and RH says they equal 1/2. If someone or some LLM proved they are all less than 0.99, well this would be huge, and I'd bet they would easily get a Fields and be remembered forever. Innocent looking results could have drastically different technical depth behind them. Alas math ppl tend to agree that RH will not be proved one bit at a time. The fundamental arithmetic aspects, once found out, will likely knock out not only RH but all the other L-functions in one go. [0]: https://arxiv.org/abs/1707.01770 https://arxiv.org/abs/1707.01770

jetrink 2mo ago

I would expect to see LLMs that are creative in math before any that are creative in writing. Creativity is more easily specified in math and the solutions can be formally verified. There's no good way to classify creative writing. Many truly great works are overlooked by experts and the public until decades later. Many derivative works are commercially successful.

zarzavat 2mo ago
LLMs, both in writing and mathematics seem to only be capable of coming up with texts that are inside the distribution of the training data. With writing it's just more obvious. LLMs don't write with personality. They don't create new and exciting worlds on their own. Everything they output feels derivative. In mathematics you see the same effect. They are very good at finding results that humans missed, taking advantage of their broad knowledge and tireless work ethic. But just as they have been unable to create new literary worlds, they also have so far been unable to create new mathematics. I believe this lack of creativity is intrinsic to how these models are architected and trained. We want models that produce these in-distribution outputs because those types of models are more economically valuable. Nobody wants a coding agent with spontaneity, we want models that predictably and obediently solve problems - and that's what we got.
- deleted 2mo ago
  [deleted]
- goekjclo 2mo ago
  Man the cope on HN is something else. Neighbour, 99.9(9)% of professional mathematicians don't go around creating new mathematics. The goalposts have been moved so far that we're now at "AI isn't Grothendieck yet".
polanyer 2mo ago
Not to mention it is all subjective taste. There is nothing objective beyond consensus that a "great" work of creative writing is "great" no matter how great a consensus thinks it is. It is really the same with any art form.

redox99 2mo ago

> This result is some evidence that AI will not solve RH soon. If there were any easy solution hiding in plain sight then it probably would have found it. There's no way you can conclude that. Yes, "Fable 2" or whatever this was probably won't. But we can't know what Fable 3/4/5/etc will be able to do. If anything, if we have 1 or 2 more years of progress like the last 12 months, which have been insane, I'd say LLMs are likely to solve it.

HappyPanacea 2mo ago
You underestimate the difficulty of RH, there are far easier problems broadly related to RH (density hypothesis, Artin's holomorphy conjecture, Stark conjectures) which are still aren't solved.
- redox99 2mo ago
  I think verifiable tasks like math will soon be like Chess. Machines way beyond human intelligence.
  - nbulka 2mo ago
    Check out the work of Godel Turing and Chaitin!
handle584 2mo ago
The keyword is soon, so OP simply meant current generation of LLMs are not likely to prove RH, judging from the performance shown in the paper. Why? Because such explicit numerical improvements are not that interesting, which is best summed up in a review paper [0] of our efforts on RH spanning a century and a half, > The pathetic attempts to enlarge the ridiculous zero free region in the critical strip is a perfect example of what brute force can do without fully exploiting fundamental arithmetic aspects of the problem. (italic added) For outsiders, zero free region is another angle abundant with numerical improvements but no groundbreaking insights. For percentage people stop at ~40% because there is no need to proceed further, just like we are not interested in computing the googol-th digit of pi although in principle we could. The groundbreaking results are like Selberg's that goes from zero to 0.01% (actually it is left unspecified, all we know is the percentage is positive), or Zhang's twin prime bound from infinity to 70,000,000. After this leap the pure numerical difference between 0.01%, 40%, 67% or even 100% is not substantial, and that's partly why Selberg did not even bother to compute it. Also RH will not follow from 100%, because in mathematics 100% does not mean all. On the other hand, it is also wrong to dismiss such results all together. Riemann already know the real part of all zeros are bewteen zero and one, and RH says they equal 1/2. If someone or some LLM proved they are all less than 0.99, well this would be huge, and I'd bet they would easily get a Fields and be remembered forever. Innocent looking results could have drastically different technical depth behind them. Alas math ppl tend to agree that RH will not be proved one bit at a time. The fundamental arithmetic aspects, once found out, will likely knock out not only RH but all the other L-functions in one go. [0]: https://arxiv.org/abs/1707.01770 https://arxiv.org/abs/1707.01770