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When can infer somewhat from this section in the FrontierMath paper [1]: "Assessment of FrontierMath difficulty All four mathematicians characterized the rese

When can infer somewhat from this section in the FrontierMath paper [1]: "Assessment of FrontierMath difficulty All four mathematicians characterized the research problems in the FrontierMath benchmark as exceptionally challenging, noting that the most difficult questions require deep domain expertise and significant time investment. For example, referring to a selection of several questions from the dataset, Tao remarked, "These are extremely challenging. I think that in the near term basically the only way to solve them, short of having a real domain expert in the area, is by a combination of a semi-expert like a graduate student in a related field, maybe paired with some combination of a modern AI and lots of other algebra packages...” However, some mathematicians pointed out that the numerical format of the questions feels somewhat contrived. Borcherds, in particular, mentioned that the benchmark problems “aren’t quite the same as coming up with original proofs.” It sounds like it will be able to crack some hard math problems, but not actually do mathematics. Which makes sense to me, given how these oN models are being trained. Synthetic data is bound to be in some (more or less obvious) way contrieved, since it's not like we can just generate tons of new/natural/original mathematical results to train the model on. [1] https://arxiv.org/pdf/2411.04872 https://arxiv.org/pdf/2411.04872

versteegen 2y ago
> It sounds like it will be able to crack some hard math problems, but not actually do mathematics. What, do you say that because format of the FrontierMath problems is a bit contrived? I think I must misunderstanding you; a benchmark can't rule out something it doesn't test. And saying solving these problems requiring "deep domain expertise" isn't real mathematics sure sounds like a No True Scotsman argument. Why shouldn't o3 be able to produce novel mathematical proofs? It's got plenty of maths textbooks and papers to train on, the same thing mathematicians train on.
- pegasus 2y ago
  Those proofs are not by far enough - which is why the raw frontier models don't do that well on this test. The oN models are trained on synthetic data: so generated mathematical theorems and associated proofs. If we could generate legitimately new math we would already have accomplished the hard task, so the generated theorems are going to feel contrieved. I.e. they won't be theorems that ever make it into a future math book. Doesn't mean they won't be useful to mathematicians, they will, just like computers in general are useful in doing maths today.
  - versteegen 2y ago
    Think about what happens when a student solves some math exercises using some maths they don't fully understand, or some example they contrived to test what they read. They might make some false starts, work through some steps, make reference to other ideas that they do understand, and finally find a path to the solution. This is actually incredibly analogous to LLM-generated synthetic training data. And nobody's going to print those crossed-out scrawls in a textbox either! But then the student learns from the chain-of-reasoning which arrived at the result. They learn what steps they should have followed one after the other so they can apply it effortlessly next time, what mathematical tool A on problem type B results in, and so forth. They distill some insights from it. Well LLMs can also learn many of these same things. They certainly don't learn as well or as much as humans (just look at their appalling data requirements), but my point is synthetic training data really does work. So expect training compute used on frontier models to keep scaling.
    - isotypic 2y ago
      > and finally find a path to the solution. But how does the student, or in your case the LLM, know that it actually has the solution? For students, this is done by: a grader grading the homework, asking the professor at OH, working on problems with other peers who crosscheck as you go. I see no reason why this LLM produced synthetic data, without this correction factor, would not devolve into a mess of incorrect, maybe even not-even-wrong style "proofs". And then how can training on this yield anything?
      - cevi 2y ago
        When you get good enough at mathematics, you can tell if your proofs are correct or not without asking a TA to grade them for you. Most mathematicians reach this level before they finish undergrad (a rare few reach it before they finish high school). While AI hasn't reached this level yet, there is no fundamental barrier stopping it from happening - and for now, researchers can use formal proof-checking software like Lean, Coq, or Isabelle to act as a grader. (In principle, it should be also be possible to get good enough at philosophy to avoid devolving into a mess of incoherence while reasoning about concepts like "knowledge", "consciousness", and "morality". I suspect some humans have achieved that, but it seems rather difficult to tell...)
- pegasus 2y ago
  Look at the sample problems and it will be obvious. For example, look at all the arbitrary constants here: "Let an for n ∈ Z be the sequence of integers satisfying the recurrence formula1 an = (1.981 × 1011)an−1 + (3.549 × 1011)an−2 − (4.277 × 1011)an−3 + (3.706 × 108 )an−4 with initial conditions ai = i for 0 ≤ i ≤ 3. Find the smallest prime p ≡ 4 mod 7 for which the function Z → Z given by n 7→ an can be extended to a continuous function on Zp."

versteegen 2y ago

> It sounds like it will be able to crack some hard math problems, but not actually do mathematics. What, do you say that because format of the FrontierMath problems is a bit contrived? I think I must misunderstanding you; a benchmark can't rule out something it doesn't test. And saying solving these problems requiring "deep domain expertise" isn't real mathematics sure sounds like a No True Scotsman argument. Why shouldn't o3 be able to produce novel mathematical proofs? It's got plenty of maths textbooks and papers to train on, the same thing mathematicians train on.

pegasus 2y ago
Those proofs are not by far enough - which is why the raw frontier models don't do that well on this test. The oN models are trained on synthetic data: so generated mathematical theorems and associated proofs. If we could generate legitimately new math we would already have accomplished the hard task, so the generated theorems are going to feel contrieved. I.e. they won't be theorems that ever make it into a future math book. Doesn't mean they won't be useful to mathematicians, they will, just like computers in general are useful in doing maths today.
- versteegen 2y ago
  Think about what happens when a student solves some math exercises using some maths they don't fully understand, or some example they contrived to test what they read. They might make some false starts, work through some steps, make reference to other ideas that they do understand, and finally find a path to the solution. This is actually incredibly analogous to LLM-generated synthetic training data. And nobody's going to print those crossed-out scrawls in a textbox either! But then the student learns from the chain-of-reasoning which arrived at the result. They learn what steps they should have followed one after the other so they can apply it effortlessly next time, what mathematical tool A on problem type B results in, and so forth. They distill some insights from it. Well LLMs can also learn many of these same things. They certainly don't learn as well or as much as humans (just look at their appalling data requirements), but my point is synthetic training data really does work. So expect training compute used on frontier models to keep scaling.
  - isotypic 2y ago
    > and finally find a path to the solution. But how does the student, or in your case the LLM, know that it actually has the solution? For students, this is done by: a grader grading the homework, asking the professor at OH, working on problems with other peers who crosscheck as you go. I see no reason why this LLM produced synthetic data, without this correction factor, would not devolve into a mess of incorrect, maybe even not-even-wrong style "proofs". And then how can training on this yield anything?
    - cevi 2y ago
      When you get good enough at mathematics, you can tell if your proofs are correct or not without asking a TA to grade them for you. Most mathematicians reach this level before they finish undergrad (a rare few reach it before they finish high school). While AI hasn't reached this level yet, there is no fundamental barrier stopping it from happening - and for now, researchers can use formal proof-checking software like Lean, Coq, or Isabelle to act as a grader. (In principle, it should be also be possible to get good enough at philosophy to avoid devolving into a mess of incoherence while reasoning about concepts like "knowledge", "consciousness", and "morality". I suspect some humans have achieved that, but it seems rather difficult to tell...)
pegasus 2y ago
Look at the sample problems and it will be obvious. For example, look at all the arbitrary constants here: "Let an for n ∈ Z be the sequence of integers satisfying the recurrence formula1 an = (1.981 × 1011)an−1 + (3.549 × 1011)an−2 − (4.277 × 1011)an−3 + (3.706 × 108 )an−4 with initial conditions ai = i for 0 ≤ i ≤ 3. Find the smallest prime p ≡ 4 mod 7 for which the function Z → Z given by n 7→ an can be extended to a continuous function on Zp."