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Comment by johnisom2001 | Hacker News Reader

johnisom2001

Don't worry, next month's internal model will be able to posit all the next big questions that matter.

How?

It might as well happen, similar to how AlphaGo was superseded by AlphaZero, at some point a model might produce better math if it's trained through self-play where it poses its own problems, instead of looking for open problems in literature.

What's the objective function or RL environment for "interesting conjecture"? Not saying it can't be done - I no longer have any specific task that I'm confident AI won't be able to do - but I don't see how. It feels to me like something that would require a qualitatively new approach.

> It feels to me like something that would require a qualitatively new approach

This sentiment has been a recurring theme throughout the history of the field.

Sure, but the question stands. The ability to evaluate some measure of success seems pretty fundamental to how we train models and iterate with them on tasks like theorem proving. What is that measure for mathematical conjecture generation? How do we evaluate success, either on a particular task for iteration (like we do by eg. setting an agent to produce a lean proof of a specific result) or on a large enough set of training data to learn a set of rules (like we do when eg. using an RL environment to train a model to generate source code that passes automated validity/correctness checks).

> but I don't see how

LLM's are trained on human knowledge and taste. They are actually pretty good at deciding if a conjecture would be found "interesting" by the mathematical community or not.

Note that I am saying LLM, and not chatbot or agent. But even a chatbot can often still reasonably rank a list of mathematical statements by vague properties like "interestingness".

How to RL this is a bit of an open question, but there are interesting conjectures of how to do it.

Can you expand on these interesting conjectures?

The fact that it's an open problem is the point I'm making. There's a very high degree of hubris right now, with people just assuming any open problem will be flattened by the AI steamroller soon. And sure, if that's what someone wants to believe that's up to them, but it's not a terribly interesting point of view to me. "What about X" "It'll be solved somehow", "What about Y" "It'll be solved somehow". Not exactly scintillating. If you know of any actual ideas on this I'd be interested to hear them.

Also any specifics on what you said about LLMs rating (preferably novel) mathematical claims for "interestingness" would be interesting.

I can only discuss published work, but take for instance this paper as one of the conjectured approaches: https://arxiv.org/abs/2603.20396

There is a general idea that beauty in mathematics is about being maximally compressing. Say I have a book with all formally correct logical statements. I could prove everything by truth table, or I can maximally compress my book with all proofs of all statements, and that will make my math beautiful. Because it forces you to reduce everything to a core of very general statements which are powerful compared to the length of the proof.

Math as some kind of condensed crystal from the sea of all possible logic.

There are other ideas of how to do it. The time has come now to just try a bunch and see which ones produce good results.

Interesting, I'll have a proper read of that. Sounds not unrelated to Kolmogorov complexity.

It may not be an interesting point of view, but it is based on the most fruitful philosophical position in history, plain-old empiricism.

Your statement that “I no longer have any specific task that I’m confident AI won’t be able to do” is founded on that.

That's not empiricism. It's the kind of extrapolation that predicts negative Germanies and 10 ton babies. (And it's not the reason I said that, either.)

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