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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

#531

Post by JacobAsmuth »

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 physics it's merely only in the top 99.9%.

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

#532

Post by virgildotcodes »

davebren wrote: post #515
virgildotcodes wrote: post #502

Earlier quoted context omitted.

If you look at the life efforts and accomplishments of the ~100 billion humans who have ever lived, how many lifetimes would you discount as having "non-human intelligence" based on the lack of "novel" contributions to frontier of our species' scientific understanding according to the same high bar you apply to LLMs? Do you pass that bar yourself?

Ordinary humans do novel things all the time. Where do you think LLMs got all the training data that their responses come from?

You're not quite addressing the question. More and more of the training data is now synthetic.

To be very specific - what novel things did the majority of the ~8 bil humans on Earth do say, yesterday, that you wouldn't otherwise dismiss as non-intelligent rehashing of the same tired patterns they always inhabit were those same actions attributed to LLMs?

What I'm getting at is that I think you're falling into the trap of thinking of the rare geniuses of human history, and furthermore their rare moments of accomplishment (relative to the long span of their lifetimes filled mostly without these accomplishments) when you think of "human intelligence", which is of course far overstating what actual human intelligence is.

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

#533

Post by JacobAsmuth »

conception wrote: post #461
jryan49 wrote: post #457

Earlier quoted context omitted.

People are afraid for their livelihood. What do you expect?

Livelihoods and lives.

Exactly. And when one's life is threatened, what are we to do if not fight?

Fight! Fight! Fight!

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

#534

Post by dellamonica »

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_…

It is extremely well known. Lots of people have tried to solve it and it stood basically stuck for 80 years. It is getting harder every day to downplay these models.

Given its elementary nature (very easy to state), you can bet that a lot of very bright people have worked on it (I know of one MIT graduate who specialized in Geometry had a lot of interest in it).

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

#535

Post by JacobAsmuth »

lacewing wrote: post #326
dwroberts wrote: post #99

Would be interesting to know what kind of preparatory work actually went into this - how long did it take to construct an input that produced a real result, and how much input did they get from actual mathematicians to guide refining it

Why? It's clearly not yet a tool that can deliver new math at a scale. I say this because otherwise, the headline would be that they proved / disproved a hundred conjectures, not one. This is what happened with Mythos. You want to be the AI company that "solved" math, just like Anthropic got the headlines for "solving" (or breaking?) security. The fact they're announcing a single success story almost certainly means…

Or it means that this was a brand new model, they tried it and were instantly rewarded with a hit that was so interesting that several mathematicians pushed to publish the results.

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

#536

Post by libraryofbabel »

doginasuit wrote: post #496
bdamm wrote: post #476

Earlier quoted context omitted.

It isn't necessarily the case. Another wishful/hopeful thought is that the human experience itself is valuable, that competing for resources and living within a social network and having physical needs somehow creates value that is essential for companies to operate. But is it really the case? I don't think we know that, and I don't know if the economy that results when all the white collar and much of the blue colla…

We haven't seen a significant increase in the quality of LMM output since 2023 that hasn't been the result of throwing even more energy and compute at it. AI "reasoning" is just recursive iteration on their output, with diminishing improvement on each pass. It seems to be the reason why Mythos is not generally available, maybe a canary of sorts. If LLMs were improving significantly independent of scaling up compute r…

> We haven't seen a significant increase in the quality of LMM output since 2023 that hasn't been the result of throwing even more energy and compute at it.

This is completely false. Most of the dramatic improvements in LLM quality in the last two years were due to the application of new post-training methods, especially RLVR. It’s really interesting to read about (you should!) and it is the whole secret to why LLMs did not plateau in 2024 or 2025 like many people confidently predicted. Sure, RLVR requires compute to do, but this is not just throwing more compute at 2023 LLMs.

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

#537

Post by m4x »

bbor wrote: post #246
pseudocomposer wrote: post #222

Earlier quoted context omitted.

I'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year. I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz…

To keep my usual rant short: I think you’re assuming a categorical distinction between those two types of innovations that just doesn’t exist. Calculus certainly required some fundamental paradigm shifts, but there’s a reason that they didn’t have to make up many words wholesale to explain it! Also we shouldn’t be thinking about what LLMs are good at, but rather what any computer ever might be good at. LLMs are alrea…

I think your comment about inventing new words is an interesting one. One of the things that I believe limits our ability to discover new ideas is our ability to describe related concepts. For example, the reason we still can't have clear discussions on consciousness is probably partly due to the fact that the necessary concepts haven't been cemented in language. We need new language before we can describe consciousness.

I would guess LLMs are limited in their ability to be genuinely novel because they are trained on a fixed language. It makes research into the internal languages developed by LLMs during training all the more interesting.

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

#538

Post by tomjakubowski »

Terr_ wrote: post #67
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.

> manage a McDonald's Dystopia vibes from the fictional "Manna" management system [0] used at a hamburger franchise, which involved a lot of "reverse centaur" automation. > At any given moment Manna had a list of things that it needed to do. There were orders coming in from the cash registers, so Manna directed employees to prepare those meals. There were also toilets to be scrubbed on a regular basis, floors to mop,…

Amazing bit of trivia that the founder of HowStuffWorks.com was named Marshall Brain.

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

#539

Post by h4h4itsfunny »

JacobAsmuth wrote: post #525
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.

[dead]

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

#540

Post by pickleRick243 »

JacobAsmuth wrote: post #525
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.

Human ingenuity is untangling perhaps the deepest question of all- what is the essence of Reason and the intellect that so privileges man? I don't know if it's what God wants, but it's certainly getting close to some existentially fundamental questions.

While many seem to be anxious or pessimistic about the future of intellectual/artistic pursuits (understandable although I disagree), I do find the utter lack of curiosity or interest in the incredible machinery that is causing all the fuss to be striking.

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