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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

#471

Post by doginasuit »

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…

I find it understandable, it is common to evaluate human intelligence vs AI as a zero-sum competition, because that is how employers typically understand it and LM providers market it. AI proving itself moves the needle in an uncomfortable direction for all of us without very robust job security.

> I wonder if we’ll still be doing this two years hence.

It is going to take some time for people to recognize that AI has a very different set of competencies that compliments human intelligence rather well. It is unlikely to eclipse human intelligence at scale, and the companies betting on that will fall behind. That is when the conversation will start to shift.

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

#472

Post by nathan_compton »

kelseyfrog wrote: post #278
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…

> I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus. (I'm open to being proven wrong.) The experiment is feasible. If it were performed and produced a positive result, what would it imply/change about how you see LLMs?

I don't think its really feasible - there just isn't enough training data before calculus. I would guess all the mathematical and philosophical texts available to Newton and Leibniz would fit on a CD-ROM with loads of space to spare.

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

#473

Post by slashdave »

paulddraper wrote: post #150
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…

"LLMs just interpolate their training data" Cracks me up. What exactly do we think that human brains do?

You have to define what you mean by "interpolate". The mechanisms that LLM use are not mysterious, and they are not the same as used by humans.

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

#474

Post by dap »

sigmoid10 wrote: post #49
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.

Managing a McDonalds is a question of integration and modalities at this point. I don't think anyone still doubts that these models lack the reasoning capability or world knowledge needed for the job. So it's less of a fundamental technical problem and more of a process engineering issue.

Have you not seen:

https://www.anthropic.com/research/project-vend-1 https://www.wsj.com/tech/ai/anthropic-claude-ai-vending-mach...

(Two different examples of a similar idea)

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

#475

Post by Lost-Futures »

missyougowers wrote: post #348
ks2048 wrote: post #133

Timothy Gowers' tweet about this: "If you are a mathematician, then you may want to make sure you are sitting down before reading futher.". woah.

Unfortunately Gowers has taken Tao's lead on this one. Gowers has one of my favourite video series about how he approaches a problem he is unfamiliar with: https://www.youtube.com/watch?v=byjhpzEoXFs It is disheartening to see him jump into this GenAI puffery. I hope these GenAI labs are paying Tao handsomely for legitimizing their slop, but more likely he's feeling pressure from his University to promote and work wi…

Ngl, this sounds like a defensive coping mechanism

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

#476

Post by bdamm »

doginasuit wrote: post #471
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…

I find it understandable, it is common to evaluate human intelligence vs AI as a zero-sum competition, because that is how employers typically understand it and LM providers market it. AI proving itself moves the needle in an uncomfortable direction for all of us without very robust job security. > I wonder if we’ll still be doing this two years hence. It is going to take some time for people to recognize that AI has…

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 collar workers no longer understand how to participate in whatever the economy is becoming. Because it is starting to look like old money is coming around, and soon we will all be serfs to the creature comforts of those who have money now, upward mobility will be a thing of the past, and a small ruling elite over the vast subservient majority will form, reorganizing societies to more resemble middle ages lordship rather than what emerged in the 50's and 60's following WWII.

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

#477

Post by missyougowers »

horhay wrote: post #431
missyougowers wrote: post #348

Earlier quoted context omitted.

Unfortunately Gowers has taken Tao's lead on this one. Gowers has one of my favourite video series about how he approaches a problem he is unfamiliar with: https://www.youtube.com/watch?v=byjhpzEoXFs It is disheartening to see him jump into this GenAI puffery. I hope these GenAI labs are paying Tao handsomely for legitimizing their slop, but more likely he's feeling pressure from his University to promote and work wi…

I'm not sure your characterization of Tao is accurate lol. In that companion paper, only Gowers seems to extensively show no pragmatism in the implications of this accomplishment. Even the younger math experts in that paper were a lot more cautious with their statements. Tao seems to follow that same tune most of the time even though he uses AI for first-pass inspections of solutions brought to his attention.

Tao was absent from the formal verification circles until GenAI orgs saw formal verification as a way to legitimize their obscene existence, and since has been making the rounds on the podcast bro circuit pumping up these GenAI orgs.

His university is deeply entrenched with the GenAI org that released this result both with having alumni on staff, integrating their tools into the school's processes and curriculum, and paying for lots of grants. (I understand Tao is absent from this specific announcement, perhaps because it found its solution without utilizing formal verification tooling)

Is it unreasonable to assume he's feeling pressure to do so?

Gowers similarly appeared largely uninterested in this current crop of GenAI until some months ago when he announced a 9M$ fund to develop "AI for Maths" and since then his social media has included GenAI promotion.

Now he is being asked about this result and his first sentence is:

> I do not have the background in algebraic number theory to make a detailed assessment of the disproof of Erdős’s unit-distance conjecture, so instead I shall make some tentative comments about what it tells us about the current capabilities of AI.

Why did this GenAI org reach out to mathematicians outside of the discipline that this result addresses?

Why did they respond?!

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

#478

Post by wayeq »

estetlinus wrote: post #217
zozbot234 wrote: post #38

The summarized chain of thought for this task (linked in the blogpost) is 125 pages. That's an insane scale of reasoning, quite akin to what Anthropic has been teasing with Mythos.

Today I generated the equivalent of two LOTR books just to fix three missing rows in my SQL models (and open a PR), so +1

or put differently, you melted x cubic meters of polar ice

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

#479

Post by ted_dunning »

red75prime wrote: post #345
skybrian wrote: post #91

Earlier quoted context omitted.

Any design already exists as a possibility, so it could be said to be both invented and discovered, depending on how you look at it.

On the other hand, it is proven that if you need to count things, the only thing you can discover/invent is the natural numbers.

Really?!

Care to cite a reference to that proof?

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

#480

Post by missyougowers »

cm2012 wrote: post #455
missyougowers wrote: post #348

Earlier quoted context omitted.

Unfortunately Gowers has taken Tao's lead on this one. Gowers has one of my favourite video series about how he approaches a problem he is unfamiliar with: https://www.youtube.com/watch?v=byjhpzEoXFs It is disheartening to see him jump into this GenAI puffery. I hope these GenAI labs are paying Tao handsomely for legitimizing their slop, but more likely he's feeling pressure from his University to promote and work wi…

If seems like you have an axe to grind about AI capabilities that is making you think irrationally

This is a popular HNism.

Focusing solely on "capabilities" is the irrational thinking.

Asbestos is the most "capable" material where extreme thermal, chemical and electrical resistance is required.

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