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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

#521

Post by naasking »

isotypic wrote: post #463
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…

I cannot quite share your enthusiasm. The clearest analogy that I can think of to try to explain why I feel this way is that it seems there will eventually be a phantom textbook of all of mathematics contained in the weights of an LLM; every definition, every proof, etc; and the role of a mathematician is going to be reduced towards reading certain parts of this phantom textbook (read: prompting an LLM to generate a…

The cool thing about LLMs is not only might they be a database of all mathematical theorems, but they can also apply those ideas to the problems you're trying to solve, which is exactly what you said you're interested in. Not sure why you lack enthusiasm.

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

#522

Post by lg5689 »

zeofig wrote: post #397
johnfn wrote: post #349

Earlier quoted context omitted.

Have you updated your priors after this announcement? If not, why not?

I don't have enough information about the announcement for it to mean much to me. I don't know much about this field of maths. I don't know how many mathematicians were actively working on this problem. It could be zero, which would indicate it's not really that interesting. The article gushes about how it's a Very Important Problem, but it's not even mentioned on https://en.wikipedia.org/wiki/List_of_conjectures_by_…

The problem was pretty well known, and had many human attempts. There's some room to argue that the right humans hadn't attempted it, as the solution used advanced methods from another field of math. But imho, whereas many prior AI victories could be explained by not enough human attention, there is no such excuse in this case, and one should acknowledge this is a notable achievement.

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

#523

Post by vatsachak »

trostaft wrote: post #243
vatsachak wrote: post #193

Earlier quoted context omitted.

Yeah, but I wouldn't be surprised if they train the model on verification assisted by Lean.

Arguing similarly to how stockfish, the chess engine, trains I would not be surprised if this is more common in the future. I don't know if they use any proof verification tools during their reinforcement learning procedure right now, as far as I know they've been focusing more on COT based strategies (w/o Lean). But I'm hardly an LLM expert, I don't know.

They most definitely threw in rl with formal verification somewhere between GPT 4 and now. The models are better at not hallucinating. I don't think their IMO team are only show ponies...

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

#524

Post by libraryofbabel »

getnormality wrote: post #468
libraryofbabel wrote: post #448

This HN thread depressed me. I’m still thinking about why. Look past the press-releasey gushing from OpenAI and there are all sorts of interesting and subtle questions here about the role for LLMs in mathematical research. I urge folks to click through to the accompanying comments from mathematicians published alongside the result. There is a really interesting discussion going on. I particularly recommend Tim Gowers…

Yes, this and every internet forum will still be doing this two years hence. Your life will be better if you take to heart this famous passage from Nietzsche: I do not want to wage war against what is ugly. I do not want to accuse; I do not even want to accuse those who accuse. Looking away shall be my only negation.

> Looking away shall be my only negation.

I’ve been thinking of building myself my own frontend to HN that makes it impossible to view comments, for this reason. Yet sometimes there are still really interesting discussions and it’s hard to let go of what for me feels like the last social media I want to be part of.

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

#525

Post by JacobAsmuth »

fromMars wrote: post #520

Seems rather depressing to me but maybe I am a Luddite.

Exactly. I would rather we let these discoveries stay hidden for a while longer such that human ingenuity may untangle them from the coils of reality. A machine? A mechanical man? Deins to produce something as pure as mathematics without the divine fervor of the ineffable spirit of Man?! It's just not what God wants.

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

#526

Post by retardkiller »

cman1444 wrote: post #488
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

Because for many people who pursue these fundamental truths, the reward is not necessarily personal fame, fortune, or even personal understanding. Advancing humanity's total knowledge (even if that knowledge is by proxy through AI) is reward enough.

[dead]

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

#527

Post by vatsachak »

KalMann wrote: post #261
vatsachak wrote: post #11

As I have stated before, AI will win a fields medal before it can manage a McDonald's A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation. LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.

I think your analogy is good but I don't believe modern LLMs use Lean or any lean-like structure in their proofs. At least recent open source ones like DeepSeek can do advanced math without it (maybe the most cutting edge ones are doing it I can't say).

They are most likely using them in training. I doubt their IMO team are show ponies

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

#528

Post by retardkiller »

ted_dunning wrote: post #507
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

Does it terrify you to look at children? Not so many years from now, some of them will surpass you. A few years after that all (that survive to that point) will surpass you. Does that terrify you just as much?

[dead]

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

#529

Post by vatsachak »

red75prime wrote: post #368
vatsachak wrote: post #209

Earlier quoted context omitted.

My claim is that LLMs waste a lot of time training on all available data. Math is a sequence of formal rules applied to construct a proof tree. Therefore an AI trained on these rules could be far more efficient, and search far deeper into proof space

It has been tried. Lenat's Automated Mathematician, for example. The problem is that the system succumbs to combinatorial explosion, not knowing which directions are interesting/promising/productive. LLMs seem to pick up some kind of intuition from the data they are fed. The generated data might not have the needed "human touch" or whatever it is.

It might just be that we didn't have enough compute till now. StockFish definitely has superior intuition

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

#530

Post by mathgradthrow »

cman1444 wrote: post #488
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

Because for many people who pursue these fundamental truths, the reward is not necessarily personal fame, fortune, or even personal understanding. Advancing humanity's total knowledge (even if that knowledge is by proxy through AI) is reward enough.

I think when your work is no longer required, you will probably come to regret this sentiment, not that it matters.
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