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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?

#71

Post by suddenlybananas »

phantompeace wrote: post #49
suddenlybananas wrote: post #36

Earlier quoted context omitted.

Why are you so upset that someone is criticising LLMs that you call them schizophrenic? (I'd recommend refreshing your memory with this https://news.ycombinator.com/newsguidelines.html )

Nobody called anyone schizophrenic. They noted that the irrelevant link spam in a reply was reminiscent of schizophrenic posting. If you've spent any sort of time on places like 4chan, you'd understand. I don't think posting the HN guidelines is warranted here.

>Nobody called anyone schizophrenic. They noted that the irrelevant link spam in a reply was reminiscent of schizophrenic posting.

Come on, that's being incredibly pedantic. If I say, your comment reminds me of someone being an asshole, I'm just using a circumlocution to call you an asshole. No?

Re: What sort of maths are LLMs good at?

#72

Post by sdenton4 »

seanclayton wrote: post #68
scronkfinkle wrote: post #8

> 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. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think…

Would you trust a bridge built by it with zero human interference? A train? A plane? A skyscraper? What about fight in a war? Unless you can, I wouldn't consider it AGI, because you're actually trusting human intelligence to verify the bridge or train or plane or skyscraper is safe or that the robot is following orders. And even then, you're trusting human-influenced guardrails etc. I would consider it AGI when an AI…

Would you trust a bridge built by a single human? In reality, we have lots of guardrails to ensure that we don't screw up and kill a lot of people by deploying defective bridges (or cars to drive on them). Those guardrails often are written in blood, and still occasionally fail.

Re: What sort of maths are LLMs good at?

#73

Post by YeGoblynQueenne »

yorwba wrote: post #66
YeGoblynQueenne wrote: post #40

Disclaimer: I only scanned the article quickly; I might be re-stating something already in the article. We have just got some very strong evidence about the way in which LLM-based systems solve mathematical problems and this evidence supports what many have already suspected including myself. Here's what I'm talking about. On 10 August Anthropic released an article [1] claiming that: An unreleased research version of…

> We've had the ability to brute-force all of mathematics "eventually" In a way, yes. You can easily write a program that recursively enumerates all provable theorems in some order. But if you want a proof of a specific theorem, how do you find it in the list? You need to encode the theorem in a formal syntax first, and since mathematics is built on towers of definitions referencing other definitions, that alone is a…

>> In a way, yes. You can easily write a program that recursively enumerates all provable theorems in some order. But if you want a proof of a specific theorem, how do you find it in the list? You need to encode the theorem in a formal syntax first, and since mathematics is built on towers of definitions referencing other definitions, that alone is a significant amount of work before you can even write down what you want to prove.

My current framing of this is that the advantage of LLMs lies in their ability to generate the text of a proof without having derived each of its steps in order, like a theorem prover (automated or not) would have to. There's nothing forcing an LLM to derive conclusions from premises (or indeed making it at all capable to do that).

They don't have to understand what the proof they generate means, or to be able to tell whether it's true. In fact, they can't do either. But that's fine as long as it's possible to check the proof with an external verifier.

So most LLM-based proofs use the LLM as the generator and an external verifier as the tester, either a solver like Lean or a mathematician. That's the best of both worlds as far as generate-and-test goes. A powerful generator tied to a powerful tester.

EDIT: yeah, like this:

>> An example of a good division of labor is the SAT Attack on Tarski's High School Algebra Problem https://arxiv.org/abs/2608.08421 where they construct a formula with O(n⁴) variables and O(n⁶) clauses and use a SAT solver to show that it is unsatisfiable for n ≤ 11 but satisfiable for n = 12. Then they use an LLM to help them write a Lean proof that the SAT solver input is equivalent to the human-readable description of what they wanted to prove.

I'm not disergarding the fact that LLMs don't generate text completely at random. They generate likely text. I suspect that can make it more likely to generate the text of some correct proofs. But I have no idea how likely that "more likely" is or what proofs are those.

Re: What sort of maths are LLMs good at?

#74

Post by jerf »

najmbajwa123 wrote: post #46

From my experience, you have to be good at math to trust an LLM to do the math.

There is an old koan in the old hacker literature:

"A novice was trying to fix a broken Lisp machine by turning the power off and on.

"Knight[, one of the principle designers of the Lisp machine], seeing what the student was doing, spoke sternly: 'You cannot fix a machine by just power-cycling it with no understanding of what is going wrong.'

"Knight turned the machine off and on.

"The machine worked."

I feel like AI is manifesting this even more concretely. I don't feel like I'm guiding the AI super intensely as I work on it with software engineering. I'd have a hard time pointing you at where in the prompt my decades of experience are manifesting. But I definitely can have better results, even with a less frontier-level AI, than people who don't know the same amount of stuff.

Terence Tao also released some unedited transcripts of some of his conversations with AI, and many people observed that while many mathematicians may have been able to formulate the initial question, very few people could have given the same feedback to the AI.

Perhaps someday AI will eliminate the need for competence to use it properly. But that day is not today. And to be honest, that tech is probably not LLMs, no matter how large they get. Some other breakthrough will be necessary to truly eliminate the human element. Those psychopathic elites making plans to turn Earth into one of the Spacer worlds from Asimov's works with a small elite population supported entirely with robots take notes... it's not possible yet.

Re: What sort of maths are LLMs good at?

#75

Post by jerf »

Given coding agent's demonstrated difficulties with concurrent code, even relatively simple concurrent code, it would be interesting to see how they do with temporal logic. I don't know enough to throw AI at the problems in that space but I wonder if they wouldn't crash and burn on it.

(I haven't had the opportunity to throw a current-gen frontier model at a concurrent problem because I haven't had one to try out lately. The best concurrency is no concurrency and the second-best concurrency is the "web request" model where many web requests are nominally running concurrently but they are otherwise fully isolated from each other and not trying to communicate at all. So maybe they're better, but I feel like if they were a lot better somebody would have noted that in a place I'd have seen by now.)

Re: What sort of maths are LLMs good at?

#77

Post by scronkfinkle »

root-parent wrote: post #44
scronkfinkle wrote: post #8

> 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. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think…

>> To me, this is more or less what I would think "Artificial General Intelligence" is So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621 "Our testing shows humans can solve 100% of the environments, in contrast to frontier AI systems which, as of March 2026, score below 1%." Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent. https://ww…

> So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621

I don't, we originally had the turing test which was designed to determine human intelligence by its ability to imitate us with natural dialogue, but we've since defeated that. I stated "to me" because it's my personal opinion on a definition whose goalpost will probably never stop being moved.

> Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent.

EQP doesn't have the 3 criteria I outlined, which were different than "solving a math problem"

Re: What sort of maths are LLMs good at?

#78

Post by parhamn »

> If they were, then their big speed advantage over us would mean that there would be much more of a flood of results.

Is this true right now? Just recently Jarred Sumner tweeted [1] that he managed to make some progress on the Riemann hypothesis while on a jog. Managed to get somewhere by encouraging the llm to “keep going” and “believe in yourself”.

This raised a few questions for me. Had no one at Anthropic thought to try this earlier? It's an interesting footnote that a software engineer there pursued this. How many people in the world can actually verify a proof? How many would we need to sit around and do the right incantations to get a proof out of it? How many would we need to verify and give those proofs value and meaning? What happens when there are more proofs than verifiers? How many will be around in 100 years?

I think it just turns out that a lot this stuff is more socially useful than anything else. The 10 proofs drop came and went in the daily news cycle. Perhaps math is already in it's chess like "for fun" period. I am interested in when we find a very high real-world utility breakthrough math/physics, some space where we've already poured our best human resources at it.

[1] https://x.com/jarredsumner/status/2086869681785500011?s=20

Re: What sort of maths are LLMs good at?

#79

Post by greenmoonx »

philipwhiuk wrote: post #67
pinkmoonx wrote: post #3

[dead]

> the human brain unconsciously does calculus and linear algebra We're not unconciously doing calculus and linear algebra. If you're arguing you 'do calculus' to predict how to catch a ball, I'm sorry but it's not supported by the data.

[dead]

Re: What sort of maths are LLMs good at?

#80

Post by [deleted] »

greenmoonx wrote: post #79
philipwhiuk wrote: post #67

Earlier quoted context omitted.

> the human brain unconsciously does calculus and linear algebra We're not unconciously doing calculus and linear algebra. If you're arguing you 'do calculus' to predict how to catch a ball, I'm sorry but it's not supported by the data.

[dead]

[deleted]
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