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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

#481

Post by bdamm »

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.

Even if the code was like that (it isn't), the power of the current crop of models to analyze data for patterns and build context out of code is leaps and bounds what it was even a year ago. And any developer will tell you that the hardest part of fixing a bug is knowing where the bug is in the first place. Once you know where it is, fixing it is usually trivial.

There is serious magic happening in the construction of model context.

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

#482

Post by miki123211 »

al_borland wrote: post #406
daishi55 wrote: post #202

Earlier quoted context omitted.

> the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept. Isn't this just anthropocentrism? Why is understanding only valid if a human does it? Why is knowledge only for humans? If another species resolved the contradictions be…

The knowledge isn't of any use to us unless it is understandable to us. Many species understand things about the world around us that we are unable to explain or understand, even if it's just pure instinct on their part. These things are very useful to them, but have no value to us until we can understand and explain it, which then allows us to make use of it. People saw birds fly for all of human history, but it was…

This is not true.

We use drug-sniffing and guide dogs in a way similar to how we use LLMs. We don't really understand them at a fundamental level, we can't make electronic dog noses (otherwise we'd dispense with the silliness and just install drug detectors instead), but dogs are useful, so we use them.

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

#483

Post by missyougowers »

aroman wrote: post #408
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…

Are you saying this result is uninteresting and therefore AI slop or puffery? Obviously OpenAI has a motivation to "market" the accomplishment as much as possible, but surely you agree it IS a remarkable achievement?

I'll let the mathematicians in the field determine the level of "interest" in this result, but saying "you may want to make sure you are sitting down" is pure puffery.

> has a motivation to "market" the accomplishment as much as possible

I am so sick of HN promoting unethical behaviour as virtuous due to it's financialization worship at the foot of "valuations".

> but surely you agree it IS a remarkable achievement?

If you could define the bounds of "remarkable" I could answer this question.

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

#484

Post by class3shock »

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 seems like the outcome options are:

1. AI is developed to be smart enough to actual replace people, destroying the labor force and immensely concentrating power.

This seems like bs hyperbole but I am not an expert.

2. AI turns out to be a bubble of false promises and hype, bursts, and takes the stock market and economy with it.

I thought this was the most likely but I keep not hearing popping, so maybe the it's:

3. AI continues to be a tool that can substantially increase productivity in some areas and cause huge societal changes in others. The AI companies keep the hype train going or maybe it tapers off over time until talk meets reality but "real" AI never shows up and the bubble never pops because it's not one. Eventually there is 0-3 new FAANG companies with untouchable control of a tech we increasingly have to use to stay relevant.

Even if we avoid option 1 and 2, 3 doesn't exactly bode well either.

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

#485

Post by Chamix »

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.

I note that (though summarized), this is ~100k tokens. Anyone who routinely works with Codex (or any agentic harness really) can tell you how trivial it is to eat up 100k tokens doing complex work. I've personally had plenty of codex 5.5 xhigh sessions where just the pure chain of thought token count in a single turn exceeds 200k (and I assume doesn't go further only due to compaction meta-guidance; the harness will push the model to stay under 256k per turn/thinking block) .

I think the more interesting question is how many tokens were spent all told; the most interesting graph in the article imo is the success rate by log test-time compute: how many tokens are being spent on the right of the graph to hit a winning CoT/solution like this >50% of the time?

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

#486

Post by parineum »

pear01 wrote: post #220
logicchains wrote: post #159

Earlier quoted context omitted.

>Police officers are human. In the United States in the vast majority of cases you can't sue the police, only the community responsible for them. Police are a monopoly; nobody has a choice about which police company to use. McDonalds are not a monopoly, and many customers would prefer to eat at competitors run by entities that could be sued or jailed if they did anything particularly egregious.

You are missing the point. The point is you can still sue the McDonalds. With the police there is a human intuition to also want to sue the officer, given the officer is a human being who has free will and thus made a choice to violate your rights. The same intuition applies if you walk into McDonald's and a person there mistreats you. You want that person held responsible. But the LLM is not a person. What is there…

> given the officer is a human being who has free will and thus made a choice to violate your rights.

The purpose of qualified immunity is for when an officer does something that turns out to be illegal but they were both told to by their superiors and did not think it was in violation at the time.

An officer making a choice to violate your rights would not be eligible for qualified immunity.

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

#487

Post by llbbdd »

autoexec wrote: post #437
llbbdd wrote: post #402

Earlier quoted context omitted.

I come back to something like this idea when I consider the distinction being made that LLMs can only combine and interpolate between points in their training material. I could write a brute-force program that just used an English dictionary to produce every possible one-billion-gazillion word permutation of the words within, with no respect for rules of language, and chances are there would be some provable, testabl…

If we managed to create very fast monkeys with typewriters and software that can review their output quickly enough that we end up with a result that's worth reading we'd still have people insisting that we've created intelligence. The monkeys however remain monkeys.

I think intelligence is an orthogonal, mostly philosophical question aside from whether these tools can produce novel, useful output vs purely recombinant output.

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

#488

Post by cman1444 »

School-Cotton wrote: post #361
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…

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.

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

#489

Post by signatoremo »

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_…

You don't have enough knowledge to dismiss them, but you still laugh? For?

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

#490

Post by horhay »

missyougowers wrote: post #477
horhay wrote: post #431

Earlier quoted context omitted.

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…

I think the intention of this paper is to build some type of culture of "math generalists" that don't quite exist in today's academia. The thing is, is that a good half of the people in that paper were actually very pragmatic on the implications of such a success and present questions in terms of the measurability of the difficulty of the problem and the generalizability of the solution provided for other questions. Gowers in particular offers no resistance and in fact resorts to the theatrics of "being the bearer of bad news" on Twitter for some reason.

As with Tao, he's always been a measured optimist even before the tools were consistently usable for his work. And even still nowadays, he adds stipulations to his statements on the successes of AI. Yes, he's part of Math Inc. now and is in close contact with Google Deepmind for some projects but his interest lies in using the tools today. Gowers has been hypothesizing on the future of math in the tone he has taken now ever since o3/GPT5. There's no comparison between the two who should attract more scrutiny.

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