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Comment by m3kw9 | Hacker News Reader

Just waiting on what Terrance Tao thinks of the new o3 intern

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

> 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.

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.

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.

> 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?

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...)

> 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.

This is simply not true - you can get a very good sense of when your argument is correct, yes. But having graded for (graduate, even!) courses, even advanced students make mistakes. It's not limited to students, either; tons of textbooks have significant errata, and its not as if no retraction in math has ever been issued.

These get corrected by talking with other people - if you have an LLM spew out this synthetic chain-of-reasoning data, you probably get at least some wrong proofs, and if you blindly try to scale with this I would expect it to collapse.

Even tying into a proof-checker seems non-trivial to me. If you work purely in the proof-checker, you never say anything wrong - but the presentations in proof checking language is very different from textual ones, so I would anticipate issues of the LLM leveraging knowledge from, say, textbooks in its proofs. You might also run into issues of the AI playing a game against the compiler rather than building understanding (you see elements of this in the proofs produced by AlphaProof). And if you start mixing natural language and proof checkers, you've just kicked the verification can up the road a bit, since you need some way of ensuring the natural language actually matches the statements being shown by the proof checker.

I don't think these are insurmountable challenges, but I also don't think its as simple as the "generate synthetic data and scale harder" approach the parent comment thinks. Perhaps I'm wrong - time will tell.

The error rate of human mathematical work is not zero, but it does go down exponentially with the amount of time that the mathematician spends carefully thinking about the problem. Mistakes tend to be the result of typos, time pressure, or laziness - showing your work to others and having them check it over does help (it's one of the reasons we have peer review!), but is absolutely not necessary.

If the error rate is low enough - and by simply spending a constant factor more time finding and correcting errors, you can get it below one in a million - then you do get a virtuous feedback loop even without tying in a proof-checker. That's how humans have progressed, after all. While you are right to say that the proof-checker approach certainly is not trivial, it is currently much easier than you would expect - modern LLMs are surprisingly good at converting math written in English directly to math formalized in Lean.

I do think that LLMs will struggle to learn to catch their mistakes for a while. This is mostly because the art of catching mistakes on your own is not taught well (often it is not taught at all), and the data sets that modern LLMs train on probably contain very, very few examples of people applying this art.

A tangent: how do human mathematicians reliably manage to catch mistakes in proofs? Going line-by-line through a proof and checking that each line logically follows from the previous lines is what many people believe we do, but it is actually a method of last resort - something we only do if we are completely lost and have given up on concretely understanding what is going on. What we really do is build up a collection of concrete examples and counterexamples within a given subject, and watch how the steps of the proof play out in each of these test cases. This is why humans tend to become much less reliable at catching mistakes when they leave their field of expertise - they haven't yet built up the necessary library of examples to allow them to properly interact with the proofs, and must resort to reading line by line.

It does work, for in-distribution detection/generalization. Which is avery useful tool for a mathematician (basically, a calculator on steroids), but it's not at all what a good mathematician ultimately does. Discovering new maths is by definition stepping out of distribution and into the unknown. There, mathematical intuition is much more important, and that's not something we understand and even less can model today.

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."

Why does it matter what Terrance Tao thinks? Genuinely I find this obsession with what some specific math researcher thinks about x topic bizarre.

What do you think of it?

Tao is a Fields Medal winner and widely regarded as one of the most brilliant mathematics of his generation. He also often tests these models in good faith.

Why wouldn't someone wish to know what he thought of this new advance?

Because who cares? There are any number of mathematicians out there, why put so much value on one "thought leader"? I just find HNs obsession with knowing what one particular person thinks odd. It's a consistent reoccurrence.

There are not many mathematicians on Tao's level to put it mildly. And evidently, lots of people care lol. Yes people are interested in what an expert has to say about a potential advancement in their field. This shouldn't be hard to grasp.

Ok. I'm fine agreeing to disagree.

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