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An OpenAI model has disproved a central conjecture in discrete geometry

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Re: An OpenAI model has disproved a central conjecture in discrete geometry

#701

Post by colordrops »

trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

Maybe I'm misunderstanding how these models work, but isn't it more the responsibility of the harness and its prompts rather than the model itself to make sure that a result is generated with explicit sources?

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#703

Post by justinnk »

School-Cotton wrote: post #421
justinnk wrote: post #274

Earlier quoted context omitted.

I see where you are coming from. However, in the role of personal teachers they may allow especially our young generations to reach a deeper understanding of maths (and also other topics) much quicker than before. If everyone can have a personal explanation machine to very efficiently satisfy their thirst for knowledge this may well lead to more good mathematicians. Of course this heavily depends on whether we can ge…

Something that can instantly tell you the answer to every math question will make people worse at math, not better. Building "mathematical maturity", skill, and understanding requires struggle.

Totally agree that it requires struggle and I did not say you should use it to just get the solution. What I think one can do is use it more like a personalized textbook which you can ask any question. It can also provide you with problems just at the right level for you and judge the solution. Now, of course for many students it is tempting to just get the solution if provided with the means but they can be taught to use an LLM in a didactically useful way.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#704

Post by edanm »

stevefan1999 wrote: post #542
mooreat wrote: post #164

I think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture. I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true. In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by…

That's proof by contradiction: https://en.wikipedia.org/wiki/Reductio_ad_absurdum

No, the thing the LLM did is not a proof, it's the opposite. It's proving that the conjecture is false.

Reductio ad absurdum is a technique to prove something.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#705

Post by golol »

JacobAsmuth wrote: post #531
recitedropper wrote: post #244

This is impressive, no question. Without knowing all this model has been trained on though, it is pretty hard to ascertain the extent to which it arrived to this "on its own". The entire AI industry has been (not so secretly) paying a lot of experts in many fields to generate large amounts of novel training data. Novel training data that isn't found anywhere else--they hoard it--and which could actually contain origi…

Exactly. Maybe OpenAI paid mathematicians to keep this discovery quiet, then added their proof to the training data, then manipulated a second team into prompting for this question such that the model could regurgitate the solution. This would plausibly explain why the model seems so capable at doing things like refuting fundamental theorems of mathematics while in things like competitive programming, biology, and ph…

You are believing a very unlikely scenario. I think the reason is that you have been convinced of a claim which is unlikely and indeed not true. That is: >the model seems so capable at doing things like refuting fundamental theorems of mathematics

That is not true and a complete misrepresentation of recent progress of AI in math. It is therefore not necessary to believe the conspiracy theory you described in order to explain recent progress of AI in math.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#706

Post by [deleted] »

golol wrote: post #696
raincole wrote: post #280

I like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)

No it is not Leibniz/Euler/Galois. More like writing good papers that contribute to the broader understanding of a theory. I think if one evaluated a mathematicians research output and it consisted of mostly the kinds of problems AI has solved so far, it would give the impression that this person is somehow very good at picking accessible problems to target, but has not made a larger impact on the field.

[deleted]

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#707

Post by edanm »

jltsiren wrote: post #210
0x5FC3 wrote: post #30

Is there a reason why we only hear of Erdos problems being solved? I would imagine there are a myriad of other unsolved problems in math, but every single ChatGPT "breakthrough in math" I come across on r/singularity and r/accelerate are Erdos problems.

Erdős problems form a substantial fraction of all mathematical problems that have been explicitly stated but not solved; are sufficiently famous that people care about them; and are sufficiently uninteresting that people have not spent that much effort trying to solve them. Solving problems people have already stated is a niche activity in mathematical research. More often, people study something they find interestin…

> and [Erdős problems are] sufficiently uninteresting that people have not spent that much effort trying to solve them.

Note that this is not really true of this problem in particular.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#708

Post by golol »

hammock wrote: post #213
m-hodges wrote: post #40

To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…

Recombining existing material is exactly right, and in this case LLMs were uniquely positioned to make the connection quicker than any group of humans. The proof relies on extremely deep algebraic number theory machinery applied to a combinatorial geometry problem. Two humans expert enough in either of those totally separate domains would have to spend a LONG time teaching each other what they know before they would…

I did not have the impression the proof uses a surprising and novel contribution of fields. I think the proof uses standard application techniques of algebraic number theory towards discrete geometry. If you have a quote substantiating what you said I would be curious.

I know these articles write that it used deep algebraic number theory techniques, which is true, but it may also just be the standard in the field.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#709

Post by bananaflag »

raincole wrote: post #280

I like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)

As a mathematician, new, conceptual math is when I'll become interested in reading LLM output.

I appreciate very much the work done so far, but this sort of asymptotic/quantitative result didn't interest me much even when it was done by humans.

(This is not snobbery, just a personal preference.)

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#710

Post by golol »

black_knight wrote: post #354
dvt wrote: post #61

Earlier quoted context omitted.

> I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. Most discoveries are indeed implied from axioms, but every now and then, new mathematics is (for lack of a better word) "created"—and you have people like Descartes, Newton, Leibniz, Gauss, Euler, Ramanujan, Galois, etc. that treat math more like an art than a science. For example, many belive that to…

It could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances! This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are di…

I think based on the class of problem that RH is an independence result is not something that "really happens".
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