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Learning more about Claude's mathematical capabilities

anthropic.com

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Re: Learning more about Claude's mathematical capabilities

#171

Post by afdbcreid »

jcfrei wrote: post #151
bryan0 wrote: post #120

I'm not sure what's crazier: AI improving a lower bound on RH, or AI improving a lower bound on RH and it not even making the front page of HN.

That's how quickly your perception shifts when you are inside of a real, fundamental technological shift. If some similar computer program figured out the same improved bound just 6 years ago there would have been numerous blog posts, experts interviews and NYTimes, WaPo op-eds about some computer working on the Riemann hypothesis. Now that we are in the middle of it it's going to be a footnote at best.

True, however: if some human would improve the lower bound that would still end as some few posts on mathematics magazines, even if it would be no less impressive.

The general public is not interested in the mathematical details, it is interested in the fact this is unique. And it is no longer unique.

Re: Learning more about Claude's mathematical capabilities

#172

Post by evenhash »

johnnienaked wrote: post #142
sosodev wrote: post #69

I wonder why we have yet to see more systematic exploration of Math. Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop. We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little…

I wouldn't be surprised if half these proofs turn out to be well crafted hallucinations, barring of course the ones actually verified in Lean

Being “verified in Lean” doesn’t magically solve the problem of hallucinations unfortunately.

It just shifts the work from

> reading the (natural language) proof and confirming it has no errors

to

> reading the Lean code and confirming it correctly encodes the theorem

For example here is a statement of the Pythagorean theorem in Lean:

theorem EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ : P) : dist p₁ p₃ * dist p₁ p₃ = dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ angle p₁ p₂ p₃ = Real.pi / 2

This is just one possible way of formalizing it and it depends on other definitions, wherein you also need to understand the assumptions they make, etc.

Answering the question of “whether proving this theorem in Lean proves the Pythagorean theorem” thus requires expert judgement as well as domain knowledge of Lean’s libraries.

So if the AI says “this theorem is true, here is the proof in Lean” it’s still possible that it’s not correct, even if the Lean code compiles. The result will still be in question until a human expert reviews it.

Re: Learning more about Claude's mathematical capabilities

#173

Post by JuanFAngel »

mkl wrote: post #159
simonw wrote: post #63

Earlier quoted context omitted.

Brute-force compute hasn't been an option for most of the history of science.

It still isn't an option for most problems, either because it would take far more computing power than is available or because the number of things to compute is infinite. In both cases, new theory is what can succeed instead. Developing new theory by brute force can be attempted, but will almost always run into the same problems. Using AI to develop new theory is using heuristics rather than brute force.

But it does raise the bar for what is brute-force solvable by a lot, which is very significant

Re: Learning more about Claude's mathematical capabilities

#175

Post by jrflo »

laszlojamf wrote: post #144

60 subagents for a day and a half. I guess he's not on the same plan I am on.

The price wasn't that ridiculous IMO for the quality of the discovery, it generated 31M output tokens which is ~$1500 in API cost if it was on Fable. A new lower bound on the biggest unsolved problem in mathematics for less than a couple day's pay of a prestigious mathematics professor is kind of insane. Obviously this discovery doesn't change anything as it's just moving a lower bound, but that's something many mathematicians have tried and failed to do.

Re: Learning more about Claude's mathematical capabilities

#176

Post by fragmede »

bwfan123 wrote: post #56
simonw wrote: post #43

> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress. I remain delighted at how absurd our current timeline has become.

> Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1 If there is anything to learn from the history of science, it is that breakthroughs happen via better or new theory and not by brute-force compute [1]. [1] https://arxiv.org/pdf/2607.27794

Whether or not Thomas Edison actually said this, the saying goes genius is 1% inspiration and 99% perspiration, and that predates computers by decades.

Re: Learning more about Claude's mathematical capabilities

#177

Post by goekjclo »

zarzavat wrote: post #129
jetrink wrote: post #77

Earlier quoted context omitted.

I would expect to see LLMs that are creative in math before any that are creative in writing. Creativity is more easily specified in math and the solutions can be formally verified. There's no good way to classify creative writing. Many truly great works are overlooked by experts and the public until decades later. Many derivative works are commercially successful.

LLMs, both in writing and mathematics seem to only be capable of coming up with texts that are inside the distribution of the training data. With writing it's just more obvious. LLMs don't write with personality. They don't create new and exciting worlds on their own. Everything they output feels derivative. In mathematics you see the same effect. They are very good at finding results that humans missed, taking advan…

Man the cope on HN is something else. Neighbour, 99.9(9)% of professional mathematicians don't go around creating new mathematics. The goalposts have been moved so far that we're now at "AI isn't Grothendieck yet".

Re: Learning more about Claude's mathematical capabilities

#178

Post by fragmede »

simonw wrote: post #43

> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress. I remain delighted at how absurd our current timeline has become.

I don't if it's better or worse than the time I had Claude say it couldn't do something so I prompted it "yes you can", and it obliged. Absurd either way!

Re: Learning more about Claude's mathematical capabilities

#179

Post by gessha »

sosodev wrote: post #69

I wonder why we have yet to see more systematic exploration of Math. Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop. We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little…

> I wonder why we have yet to see more systematic exploration of Math.

> 60 Claude subagents

Re: Learning more about Claude's mathematical capabilities

#180

Post by Kotlopou »

rockmeamedee wrote: post #147

It's very very important to note that there was an existing 2025 arxiv preprint that had a >66% proof assuming some weak condition, and this result removes that weak condition. It didn't do this whole 41.6->67.2% jump by itself, humans had done most of the work and it came in at the end and found a way to remove the condition. Impressive, but not as massively impressive as when it sounds like it did the jump by itsel…

Could you share a link to this paper? A quick search only found reports about this Claude result and a bunch of crank proofs of RH.
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