An OpenAI model has disproved a central conjecture in discrete geometry • Hacker News Board

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

#721

Post by vuquangchien » Thu, May 21, 2026, 6:31 AM UTC

al_borland wrote: post #554

miki123211 wrote: post #482

Earlier quoted context omitted.

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.

We don’t blindly trust the drug-sniffing dog. The dog gives a signal that it was trained to give, then humans understand what that signal means and verify the accuracy. Without the human understanding in the loop, the dog’s ability is of little value. Without a human in the loop and LLM could churn away spitting out results, some right, some wrong, and it would be of no consequence. Not much different than wild dogs…

The knowledge isn't of any use to us unless it is understandable to us => it seems that you have shifted the goalpost here. In the dog example, humans still don't understand how dogs sniff, but it is of use to us and thus is meaningful. The same for quantum effects - we don't understand how it works. We just guess that it works reliably and make use of it.

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

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Post by zamadatix » Thu, May 21, 2026, 6:32 AM UTC

no-name-here wrote: post #665

zamadatix wrote: post #624

Earlier quoted context omitted.

My only complaint is the claims always start spreading 6-12 months before the delivery. A little patience goes a long way in what's possible with AI and we all just have to wait and see what parts actually grow this next cycle or not. Guessing at it based on trend lines only leads to people getting excited when it matches their particular guess and ignoring it when it doesn't.

>> OpenAI said their models will have "PhD-Level Intelligence" > My only complaint is the claims always start spreading 6-12 months before the delivery. If delivering on such promises "always" occurs 6-12 months after the promise, is that pretty good?

Again, the promise isn't _always_ delivered, people just more often focus on when the particular result aligns with their view. When it is though, it's all too commonly 6-12 months later. Which is nice but a bit annoying - why not wait 6-12 months and claim when you can actually show it? Or just say that's where it might be soon instead of talking like it is now.

I generally like AI and use it plenty often, it does many things well and I'm curious to see how far it keeps going, but that doesn't mean I have to like overhyped marketing about it.

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

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Post by [deleted] » Thu, May 21, 2026, 6:34 AM UTC

mkl wrote: post #697

dyauspitr wrote: post #686

Earlier quoted context omitted.

> Only a rare few new theorems in mathematics nowadays have direct real world applicability. I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe. > If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative. I think the idea he…

> > Only a rare few new theorems in mathematics nowadays have direct real world applicability. > I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe. I am a mathematician. It is true. The key is we're talking about new theorems, and direct , current real world applicability. Some theorems that have no applicab…

[deleted]

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

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Post by black_knight » Thu, May 21, 2026, 6:36 AM UTC

hackinthebochs wrote: post #500

black_knight wrote: post #360

Earlier quoted context omitted.

Where does this mathematics exist before we discover it? I know of no realm where mathematical objects live except human minds. No, it seems clear to me that mathematics is a creation of our minds.

If it were merely a creation, there would be no reason for two independent mathematicians to land on the same creation given some directed effort. But of course we do see that. There is an objectivity to mathematics that must be accounted for. "Where" mathematics exists is in the abstract combinatorical space of an infinite repeating application of logical rules. This space doesn't exist in a substantive sense, but i…

If this space of possible structure is real, but seemingly immaterial, how does our matter brain access it?

I think we create mathematics as thought structure in our mind. We can agree on things when we create the same structures. But this structure did not exist prior to creation.

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

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Post by xyzzy123 » Thu, May 21, 2026, 6:37 AM UTC

shalmanese wrote: post #658

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…

> But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. AI is going to both help and hinder this process though. At the end of the day, mathematics is mostly a social process at this point. The goal is not raw number of…

Weird question, do you think AIs might prove a lot of theorems that are mainly useful to other AIs (i.e, make nearly no impact on the human culture of working mathematicians), which then get used to prove results that humans do actually care about?

It seems like if AIs can prove and index a huge number of (largely uninteresting to humans) things there might be sort of "parallel cultures"? Big results are most valuable to humans and AIs both (most context efficient!), but a very large number of less general but still non-obvious results might be an effective approach to solving problems?

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

#726

Post by ykl » Thu, May 21, 2026, 6:39 AM UTC

drdeca wrote: post #493

ykl wrote: post #446

Earlier quoted context omitted.

I think of most things you can get to by guess and checking as definitionally inside of the hull; most forms of guess and checking are you take some existing thing, randomize a bunch of its parameters, and see what you get. Whereas with something like relativity, there's not even a starting point that you can randomize and guess/check from the pre-existing knowledge space that will lead you to relativity. That's more…

I think your point about “you could randomly generate a sequence of words, which could in principle produce a text interpretable as expressing any particular expressible-as-a-sequence-of-words novel good idea” pretty much refutes the idea that guessing and checking can only result in things inside such a convex hull, unless said hull already contains everything. Of course, there’s a significant role to play by the “c…

This assumes that everything outside of the convex hull can already be described using existing language. If you need new language to describe what is outside of the convex hull, is this something an LLM can do?

I actually don't know the answer to that; my understanding is that LLMs by nature of what they are can't understand concepts that are independent of the existing language they are trained on, but I don't have enough in-depth nitty-gritty knowledge of like, core LLM implementation details and architecture and stuff to know if that understanding is correct or not.

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

#727

Post by teiferer » Thu, May 21, 2026, 6:43 AM UTC

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…

> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.

And by opening the door to LLM-generated results, you'll see greater and greater amounts without any hope of ever navigating this field again without machine help.

It's a little like a software project which more and more gets extended by a AI agents with less and less review by human software engineers and in the end the complexity and spaghetti design are so incomprehensible by humans that the maintenance requires an AI agent. The risk is that math as a whole (the field itself) will experience that effect.

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

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Post by kamaal » Thu, May 21, 2026, 6:47 AM UTC

bananaflag wrote: post #709

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.)

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.)

Well that's coming.

As a matter of fact more logic and structure to your work, the more easy it is for AI to conquer it. Due to this programming was the first thing that got solved, but pure sciences are next.

If what you do, and how you do can be written down on a piece of paper, then AI can do it.

I do believe programming getting solved will be double assault on these fields.

>>This is not snobbery

This is good for the species, what sense does it make to keep treating these fields like they are reserved for the top most intelligent micro percentage of humans? Getting LLM to these things gives some scale to these subjects and thats good.

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

#729

Post by black_knight » Thu, May 21, 2026, 6:48 AM UTC

necovek wrote: post #673

black_knight wrote: post #354

Earlier quoted context omitted.

It could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances! This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are di…

What frequently happens when we recombine axioms like that is that they end up leading to inconsistencies or contradictions. Do you know if this topos with every total function on real numbers is continuous has been constructed and proven to be a viable set of axioms? If so, I am curious about the source. My go to example still remains the one of hyperbolic geometry and axiom of parallel lines, so the more approachab…

Sure. These toposes are well known, and proven to be consistent (relative to set theory). For instance Hyland’s effective topos, or Johnstone’s topological topos. The ideas are that these toposes either require everything to be computable, or continuous in some greater sense.

There is also this blogpost by Amdrej Bauer, which can be seems as exploring how it is to be such such a topos: https://math.andrej.com/2006/03/27/sometimes-all-functions-a...

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

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Post by kamaal » Thu, May 21, 2026, 6:49 AM UTC

bananaflag wrote: post #661

necovek wrote: post #646

Earlier quoted context omitted.

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

They mean "new math" in the sense of more than a novel mathematical result, a new math paradigm or so.

Thats coming too.

Some times when you go some distance with a subject generates data for new ideas.

Once math gets done fast, newer ideas and paradigms also arrive.