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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

#881

Post by recitedropper »

fergie wrote: post #746
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…

> 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 original ideas. Really? Any references to read more?

  - https://www.theverge.com/cs/features/831818/ai-mercor-handshake-scale-surge-staffing-companies
  - https://outlier.ai/math/en-us
  - https://www.opentrain.ai/
  - https://www.pin.com/blog/ai-labs-hiring-train-models/
Much of this is data annotation, reasoning trace evaluation, and problem set curation. But there is no way they haven't atleast paid some mathematicians to work on research grade problems in tandem with their models, and then used that for training data.

Does this expert data likely contain this proof within it? No. Would it temper the impressiveness to know they have a large amount of novel mathematical training data, an internal Lean harness for evaluation of open conjectures, and spent hundreds of millions in compute to calculate this? Yes.

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

#882

Post by jeremyjh »

ricardobayes wrote: post #775
bandrami wrote: post #693

Earlier quoted context omitted.

I am curious if LLMs are better at some kinds of problems than others. IIRC this and another big recent one were cases of the LLM producing a counterexample to a conjecture.

IMO, it's due to some problems being better documented, with more well-documented, previous research available. LLMs don't really create novel mathematics, they mostly "connect the dots". LLMs by design are not coming up with anything new, unless by statistical probability, aka "brute forcing". I don't want to minimize LLMs capabilities, it's pretty cool they are doing this, and it's useful from a research point of v…

> LLMs don't really create novel mathematics, they mostly "connect the dots".

That is not what the mathematicians are saying. I don't have the knowledge to evaluate this myself, but a number of mathematicians - for example, in the SP - are saying it goes further than that - they really do introduce novel ideas. Of course everything is based on and inspired by some previous work, but that is true of all human mathematics as well.

LLMs that have been trained through reinforcement learning on mathematics are NOT simply token predictors. Only base models can be accurately described that way. They have learned how to do mathematics. They have learned to do coding. Its really amazing we're three years into instruct models and such a large part of Hacker News still does not understand the most basic facts about this field.

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

#883

Post by [deleted] »

tedbradley wrote: post #691
necovek wrote: post #646

Earlier quoted context omitted.

PhDs used to mean publishing a novel mathematical result: when has that changed?

My good sir or madam, disproving a decades-old conjecture produced by Erdos that has had armies of people in that field have their go at it IS a novel mathematical result.

[deleted]

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

#884

Post by jeremyjh »

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.

Stockfish did not teach itself to play chess. You are probably thinking of Leela Chess Zero - an open re-implementation of AlphaZero - both were given nothing but the rules of chess and a board and played millions of games against themselves until they were the strongest engine available at the time.

Stockfish's neural net evaluation model was trained on millions of its positions with its own original algorithmic evaluation function (entirely developed by humans) and search tree. The result was a much smaller model than Leela's that requires little computation (not even a GPU), paired with its already extremely efficient search/pruning algorithms that made it stronger than Leela in competitive play. Leela's evaluation function is much stronger (at one ply it has an ELO of around 2300, Stockfish is probably closer to 1800), but it requires vastly more resources and those are always bounded in a match.

Humans haven't learned as much new information about chess from Stockfish as we have from Leela.

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

#886

Post by ontouchstart »

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 dug through the chain-of-thought publication and did actually find (a few of) them. If people working on these LLMs are reading, it's very important to me that these are contained in the actual model output.

This is a very important point, especially when the output is from a non-deterministic random walk with some unknown probability distribution.

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

#887

Post by RedCinnabar »

toilet wrote: post #795
bananaflag wrote: post #709

Earlier quoted context omitted.

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.)

I have no idea about research in mathematics: How will mathematicians judge what constitutes new conceptual math that is actually useful, vs a hallucination that might be novel but doesn't introduce anything actual meaningful?

It’s basically up to the domain experts. What I found interesting in mathematical optimization/combinatorics (my fields of interest) when an AI proved some major results some time ago was probably dismissed as a boring fact by someone else. What OP is mentioning is just their personal preference and doesn’t reflect the actual opinion of the mathematical world.

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

#888

Post by TrackerFF »

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.

As others have written, Erdős was a lifelong curator of mathematical problems, from high-school level problems to the types that will land you a Fields medal. Like the Collatz conjecture.

Most new math problems appear in other papers, doctoral dissertations, etc. Usually you'll find them in the "future work" / "future research" section.

So obviously in order to present and formalize these problems, you either need the author(s) to do it, or some reader. At this level of math, there are many extremely niche fields, where the papers might only be read by a small amount of people.

In short, it is a visibility problem.

But, I figure, there's some potential use in AI models to extract and present these problems, which would make them available to a larger audience.

That is exactly what Erdős did. His life revolved around math, and seeking mathematical questions.

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

#889

Post by RedCinnabar »

dawnerd wrote: post #299
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.)

Yet it still codes like a junior developer that memorized all of stack overflow.

Not true anymore since like early 2025 and especially since last December.

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

#890

Post by NiloCK »

pocksuppet wrote: post #845
Nesco wrote: post #714

Earlier quoted context omitted.

You lost me at “except billionaires”. I don’t see how Jeff Bezos benefits from this one much more than let’s say Terence Tao. Can a tech news stay a tech news, without getting bombardes with leftist subtexts all the time?

Billionaires are going to benefit from AI at the expense of everyone else. That's not leftist ideology, that's just a fact. That's happened with every technology that's ever been created. It happened with the industrial revolution. Why would it be different this time?

What do you think of things like the changes to infant mortality or life expectancy between the industrial revolution and present day?

EG, my own oldest child needed a surgery at birth that would have been logistically impossible even 50 years ago. I'd say that she and I have benefited enormously, despite not being billionaires.

edit: I solemnly swear that the sibling comment with the strikingly similar "impossible 50 years ago" claim is a pure coincidence and that I at least am not a bot campaign. Haha.

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