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What sort of maths are LLMs good at?

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Re: What sort of maths are LLMs good at?

#181

Post by antonvs »

n4r9 wrote: post #163
antonvs wrote: post #146

Earlier quoted context omitted.

> with hindsight come to seem beautiful and natural. …because they’ve been internalized. I wonder if the author has ever questioned where his notions of “beauty” and “natural” come from.

I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".

It seems surprising before you've learned and internalized it. It seems natural after that.

I think this is a case of people not being very introspective about their beliefs about their work.

Re: What sort of maths are LLMs good at?

#182

Post by n4r9 »

antonvs wrote: post #181
n4r9 wrote: post #163

Earlier quoted context omitted.

I'm sure that he has, and I think that his point is solid. Mathematical beauty often has something to do with the transition from "surprising" to "natural".

It seems surprising before you've learned and internalized it. It seems natural after that. I think this is a case of people not being very introspective about their beliefs about their work.

Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt.

Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely linked to ideas of beauty.

Re: What sort of maths are LLMs good at?

#183

Post by josteinhylin »

steinwinde wrote: post #24
laszlojamf wrote: post #13

Earlier quoted context omitted.

for somebody who's out of the loop: what's the fuss over the ABC conjecture?

This is a reference to Inter-Universal Teichmüller Theory. Its Wikipedia article gives a good overview ( https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCll... ). In maths lasting disagreements over a published "proof" are rare, but IUTT is an example of it. What the article misses: There is a more recent, ongoing effort to formalize the published proof in Lean under the name of "LANA" (e.g. see https://zen.…

[dead]

Re: What sort of maths are LLMs good at?

#184

Post by [deleted] »

wyager wrote: post #174
TuringTest wrote: post #134

Earlier quoted context omitted.

Yeah but the LLM can only handle proofs that hold inside its context window. Proofs for novel theories requiring thousands of pages with dozen millions of steps will need support from external tools to organize the full structure of the formal document; it cannot be done by the LLM inference process alone, which was my point. It would be like asking a mathematician to proof theorems without pen and paper; external to…

Every LLM harness released in the last year has handled this gracefully via delegation, structured memory systems, etc.

[deleted]

Re: What sort of maths are LLMs good at?

#185

Post by black_knight »

zahlman wrote: post #175
black_knight wrote: post #144

Earlier quoted context omitted.

Not sure how true this restrictions is once you have the agents hammering on at a big code base of formalised proofs. Proofs stretching thousands of pages are split into lemmas, grouped into sub theories. What I haven’t seen agents do yet is to develop new ideas for entire such theories. I have usually seen them bite into some existing idea and grinding out related results. But I am less sure than ever that they won’…

The thread title is "What sort of maths are LLMs good at?" The codebase you describe would be an external tool in GP's conception.

If I walked outside, I would also be an external tool. So, I don't.

Re: What sort of maths are LLMs good at?

#186

Post by antonvs »

n4r9 wrote: post #182
antonvs wrote: post #181

Earlier quoted context omitted.

It seems surprising before you've learned and internalized it. It seems natural after that. I think this is a case of people not being very introspective about their beliefs about their work.

Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt. Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely link…

I was referring to the Gowers quote you provided:

> A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural.

That last part, "new and surprising but that with hindsight come to seem beautiful and natural," is an almost perfect description of how processing fluency (https://en.wikipedia.org/wiki/Processing_fluency) or the mere-exposure effect (https://en.wikipedia.org/wiki/Mere-exposure_effect) works. There's plenty of other work along these lines.

Gian-Carlo Rota wrote about this more philosophically in "The Phenomenology of Mathematical Beauty" (1997). He argued (among other things) that the same proof can go from opaque to beautiful purely through familiarity.

My understanding is that no single person has "learned" finite simple group classification from top to bottom, so it's easy to see why it wouldn't fit this description - there's no-one in a position to say "with hindsight, it seems simple and natural." That makes it a poor example for your/Gowers' position.

Re: What sort of maths are LLMs good at?

#187

Post by n4r9 »

antonvs wrote: post #186
n4r9 wrote: post #182

Earlier quoted context omitted.

Are you claiming that anything you learn and internalise becomes natural? Because there are plenty of proofs (such as the classification of finite simple groups) that remain ugly and unnatural well after they're learnt. Gowers' claim is that there are particular areas of (or proofs in) mathematics that become especially natural and intuitive after internalising them. This seems uncontroversial to me, and closely link…

I was referring to the Gowers quote you provided: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. That last part, "new and surprising but that with hindsight come to seem beautiful and natural," i…

Gowers claims that some methods appear beautiful and natural in hindsight, whilst others do not. That suggests beauty is more than a simple matter of familiarity.

Re: What sort of maths are LLMs good at?

#188

Post by red75prime »

YeGoblynQueenne wrote: post #178
ozgung wrote: post #164

Earlier quoted context omitted.

>> In this case, by happy accident, what stuck was an improved bound of the zeroes of the zeta function etc. I’m not familiar with this specific example (or Mathematics) but I assume they keep intermediate results (python functions, lemmas, computations, intermediate proofs etc.) and formulate and explore adjacent ideas. Even with a failed attempt you can learn things. LLMs make a difference here because they can eva…

Thanks, but I can't see that in the article by Anthropic. AFAICT the "650 ideas" that were wrong were all generated independently of each other. I don't agree that LLMs can evaluate an experiment. There's nothing in LLM training that makes them capable of telling what is e.g. a correct hypothesis from an incorrect one. I know that is a common claim particularly encouraged by AI companies but whenever that claim has b…

> On the Self-Verification Limitations of Large Language Models on Reasoning and Planning Tasks

An empirical paper from 2024 that doesn't give a principled reason why LLMs will always be bad at self-verification.

Re: What sort of maths are LLMs good at?

#189

Post by YeGoblynQueenne »

red75prime wrote: post #188
YeGoblynQueenne wrote: post #178

Earlier quoted context omitted.

Thanks, but I can't see that in the article by Anthropic. AFAICT the "650 ideas" that were wrong were all generated independently of each other. I don't agree that LLMs can evaluate an experiment. There's nothing in LLM training that makes them capable of telling what is e.g. a correct hypothesis from an incorrect one. I know that is a common claim particularly encouraged by AI companies but whenever that claim has b…

> On the Self-Verification Limitations of Large Language Models on Reasoning and Planning Tasks An empirical paper from 2024 that doesn't give a principled reason why LLMs will always be bad at self-verification.

"An empirical paper that doesn't give a principled reason" is a great description for 99% of machine learning papers since the 1980's. I don't see anyone batting an eyelid e.g. that "Attention is All you Need" or "Language Models are Few Shot Learners" have no theoretical results.

Which is to say, I think your criticism is unfair and an attempt to avoid engaging with the arguments in the paper.

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