An OpenAI model has disproved a central conjecture in discrete geometry

> I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.

Most discoveries are indeed implied from axioms, but every now and then, new mathematics is (for lack of a better word) "created"—and you have people like Descartes, Newton, Leibnitz, Gauss, Euler, Ramanujan, Galois, etc. that treat math more like an art than a science.

For example, many belive that to sovle the Riemann Hypothesis, we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it.

It’s easy to see that LLMs don’t merely recombine their training data. Claude can program in Arc, a mostly dead language. It can also make use of new language constructs. So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.

See the longstanding debate on whether new math is "invented" or "discovered". Most mathematicians I knew thought it's discovered.

There was a project long long ago where every piece of knowledge known was cross polinated with every other piece of knowledge, creating a new and unique piece of knowledge, and it was intended to use that machine to invalidate the patent process - obviously everything had therefore been invented.

But that's now how new frontiers are conquered - there's a great deal of existing knowledge that is leveraged upon to get us into a position where we think we can succeed, yes, but there's also the recognition that there is knowledge we don't yet have that needs to be acquired in order for us to truly succeed.

THAT is where we (as humans) have excelled - we've taken natural processes, discovered their attributes and properties, and then understood how they can be applied to other domains.

Take fire, for example, it was in nature for billions of years before we as a species understood that it needed air, fuel, and heat in order for it to exist at all, and we then leveraged that knowledge into controlling fire - creating, growing, reducing, destroying it.

LLMs have ZERO ability (at this moment) to interact with, and discover on their own, those facts, nor does it appear to know how to leverage them.

I feel this is the case whenever I "problem solve". I'm not really being creative, I'm pruning a graph of a conceptual space that already exists. The more possibilities I see, the easier it is to run more towards an optimal route between the nodes, but I didn't "create" those nodes or edges, they are just causal inevitabilities.

Pretty much everything that appears novel in life is derivative of other works or concepts.

You can watch a rock roll down a hill and derive the concept for the wheel.

Seems pretty self evident to me

I think you are conflating composition and prediction. LLMs don't compose higher abstractions from the "axioms, symbols and rules", they simply predict the next token, like a really large spinning wheel.

"LLMs just interpolate their training data"

Cracks me up.

What exactly do we think that human brains do?

There is a long and interesting recent essay on that topic by a mathematician: https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...

> A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation.

However, this was not verified in Lean. This was purely plain language in and out. I think, in many ways, this is a quite exciting demonstration of exactly the opposite of the point you're making. Verification comes in when you want to offload checking proofs to computers as well. As it stands, this proof was hand-verified by a group of mathematicians in the field.

I disagree. It will be able to perform work deserving if a fields medal before it is capable of running a McDonalds. I think it will be running a McDonalds well before either of those things happen, and a fields medal long after both have happened.

The proof is not written in Lean, though. It’s written in English and requires validation by human experts to confirm that it’s not gibberish.

> we'll see more specialized math AI resembling StockFish soon

Heuristically weighted directed graphs? Wow amazing I'm sure nobody has done that before.

> manage a McDonald's

Dystopia vibes from the fictional "Manna" people-management system. [0]

[0] https://en.wikipedia.org/wiki/Manna_(novel)

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.

AI is already too old for that.

our local AI models are already capable of running McDonalds.

the only thing keeping the mcdonalds from happening will be political, likewise the same with fields medal

Enough body cameras and audio recordings of each job function should be enough the build the world model for operating a fast food franchise. Training on academic publications wouldn't yield the same effect.

Erdos problems are easier to state, thus they make a great benchmark for the first year of AI mathematics.

Afaik this is because there is a community and database around them.

It's not just Erdos problems - https://news.ycombinator.com/item?id=48213189

They're just famous because Erdos was a great mathematician, kinda like the Hilbert problems a century earlier.

It's a large set of problems that are both interesting and difficult, but not seen as foundational enough or important enough that they have already had sustained attention on them by mathematicians for decades or centuries, and so they might actually be solvable by an LLM.

OpenAI specifically targeted Academia a lot and gave out a lot of free/unlimited usage to top academics and universities/researchers.

They also offer grants you can apply for as a researcher. I'm sure other labs may have this too but I believe OpenAI was first to this.

Hasn't AlphaFold been used to make real discoveries for a few years now?

From my limited testing, Gemini can dig out hard to find information given you detail your prompt enough.

Given that Google is the "web indexing company", finding hard to find things is natural for their models, and this is the only way I need these models for.

If I can't find it for a week digging the internet, I give it a colossal prompt, and it digs out what I'm looking for.

Gemini seems better trained for learning and I think Google has made a more deliberate effort to optimize for pedagoical best practices. (E.g. tutoring, formative feedback, cognitive load optimization)

As far as academic research is concerned (e.g. this threads topic), I can't say.

I think the mathematicians on X are all using GPT 5.5 Pro

OpenAI models seem to have been trained on a lot of auto-generated theorem proving data; GPT 5.5 is really good at writing Lean.

A simpler explanation is that more people are using ChatGPT

Erdos, or the model?

shame we won’t see any of these medical breakthroughs when we all lose our jobs and thus our healthcare

Breakthroughs in pure mathematics aren't scientific though. They say us nothing about the world, and they are not useful.

They only proved that one exists; computing the actual construction is non-obvious (the naive way to construct it is computationally infeasible).

This only a proof that a field with more connections is possible, not what it looks like.

I’m very out of my depth, but the structure of the proof seems to follow a pattern similar to a proof by contradiction. Where you’d say for example “assume for the sake of contradiction that the previously known limit is the highest possible” then prove that if that statement is true you get some impossible result.

They have a "before" picture but not an "after"!

I'm not a scientist but I like to LARP as one in my free time, and I have found ChatGPT/Claude extremely useful for research, and I'd say I'd go as far as to say it supercharged it for me.

When I'm learning about a new subject, I'll ask Claude to give me five papers that are relevant to what I'm learning about. Often three of the papers are either irrelevant or kind of shit, but that leaves 2/5 of them that are actually useful. Then from those papers, I'll ask Claude to give me a "dependency graph" by recursing on the citations, and then I start bottom-up.

This was game-changing for me. Reading advanced papers can be really hard for a variety of reasons, but one big one can simply be because you don't know the terminology and vernacular that the paper writers are using. Sometimes you can reasonably infer it from context, but sometimes I infer incorrectly, or simply have to skip over a section because I don't understand it. By working from the "lowest common denominator" of papers first, it generally makes the entire process easier.

I was already doing this to some extent prior to LLMs, as in I would get to a spot I didn't really understand, jump to a relevant citation, and recurse until I got to an understanding, but that was kind of a pain in the ass, so having a nice pretty graph for me makes it considerably easier for me to read and understand more papers.

I absolutely believe that AI will supercharge science.

I do not believe it will replace humans.

Isn’t that a joke? It already has supercharged science

Not only it supercharged science it supercharges scientist. Research on any narrow topic is a different world now. Agents can read 50 papers for you and tell you what's where. This was impossible with pure text search. Looking up non-trivial stuff and having complex things explained to you is also amazing. I mean they don't even have to be complex, but can be for adjacent field where these are basics for the other field but happen to be useful in yours. The list goes on. It's a hammer you need to watch your fingers, it's not good at cutting wood, but it's definitely worth having.

It will notice things that humans may have missed. That said - it can only work off the body of work SOMEONE did in the past.

To be strict, Math is not Science.

But AI is supercharging Math like there is no tomorrow.

The grid of squares actually gets > Cn for any C. (More in fact… C can grow like n^a/loglog(n).) The AI proved > n^{1 + b} for some small b > 0, which a human (Will Sawin) has now proved can be about b = 0.014. The grid can be rescaled so the edges are not necessarily length 1, but other pairs will have length 1; that is necessary to get more than 2n unit distances.

Yes, not providing visualization of the solution seems criminal.

Both 3n and 2n are linear, the broken conjecture is that you can't do better than linear.

Sadly math professors aren't very expensive. Academics are paid terrible wages. Until recently, tenure was the carrot at the end of a grueling education. But now that tenure positions are getting rarer (well, tenure positions aren't growing vs the number of aspiring academics is), they continue to be cheap highly educated labor.

it will only get cheaper in the long run

There's pages of comments from like 8 mathematicians in the attached pdf

The blog post links a pdf that OpenAI put together of nine mathematicians that commented on the proof. Each is quite brief and written in accessible terms (or more accessible terms, at least). https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...

"This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves." - Noam Brown (OpenAI reasoning researcher) on X

Yeah, I remember it was one of my biggest disappointments with LLMs.

Indeed, it's a pity. While many advanced math problems are highly abstract or convoluted to explain to a layman audience, this one in particular is about points in a 2D plane and distances. A drawing would have been nice.

apparently the proof is not constructive in the sense of not giving an easy to compute recipe for generating a set of points that you can plot on a 2d plane

all reasoning is .. well problem reasoning. restricting black-box AIs to specific human-defined domains because we believe that's better is such a human-ist thing to do.

I trust openAI's marketing team 100%

People thought neural networks were just an interesting thought exercise a few decades ago and not for practical ML problems, and look what happened since then.

How/why should we know this, it does not explain the process in the text?

Weird thing to say about a report which literally has the attached mathematical proof.