I believe we're seeing a new kind of mathematics that will require completely new formats for publication, a bit similar to those used in experimental sciences. AI-powered mathematics should be fully reproducible, so it's the authors' responsibility to disclose the exact model type, inference settings/seeds and the full prompt history leading to the result. Of course that would ideally require open weights models.

It's not just about requiring to disclose AI use. AI-powered mathematics is a completely valid discipline that doesn't need to be shy, but it should develop its own publication culture.

Yeah I remember reading about something along the lines of Mathematics is now about the scaffolding around you find the problems/solutions not just the problems and solutions. For teaching purposes. This was before this ai craze

I don't think you want to bring cost into this argument.

Even if the cost was $1 mil for these 10 problems, that's maybe 10-20 math researchers for a year.

Do you really think that if you paid that to humans, they will deliver the same results?

Also, have there been examples of researchers not affiliated with OpenAI (or another LLM creator), who have done something similar?

Another question I have is whether or not OpenAI 'simply' hired capable combinatorics researchers to work on problems, and they have, and the use of the model is incidental / secondary to their work.

There are people who can’t grasp the universe without mandatory randomised controlled trial. Would tomorrow be a Sunday? Need an RCT for that boys!

My point here is to not snark. But there should be some level of self skepticism that doesn’t warrant an RCT theatre.

> therefore the $2000 number could be completely misleading, similar to P-value hacking by not disclosing the total experimental setup.

I don't think that comparison to p-hacking is fair. I mean not reporting price of all run is nothing like committing scientific fraud and fake results.

This is not at the top as it is actively flagged by people that can't psychologically cope with the advances of AI. Hacker News is no longer a web site of an elite.

I don't think that's it. Multiple or my friends from the target audience (academic mathematicians) admitted to scrolling past because the title made it sound like a review of last month's contributions, instead of 10 new ones.

It’s also very very divided (x companies, oss vs not and other interests)

What about AI research itself? Is OpenAI close to automating its human staff out of a job?

Honestly just a bit burnt out on posts like this

This is one of the most impactful mathematical publications in history by all accounts

I think we've now hit a point where 99.9% of the population gloss over these types of AI advancements because of human competence being insufficient

No human could have published this because it requires paradigm shifts (e. g. Section 5) in multiple mathematical domains. Mastering one of them to this degree is rare, mastering 3+ pretty much non existent for humans.

Never understood all this talk about moving goalposts - you understand that's how science works, right? We improve, we learn, we recalibrate our expectations based on what we've learned. If we never "moved the goalposts", we'd be stuck scoring the same goals over and over.

> The sooner people can be broken out of their denial

There is irony here

Its not denial that is the problem. It's apathy. The vast majority simply don't care. They can tell you everything there is to know about Kim Kardashian though.

This makes no sense. To believe this you have to think that the models are somehow being overfit explicitly on academic mathematics and it doesn’t carry over at all to more practical software engineering. I wouldn’t make that bet.

> we are also seeing incredible advances in software performance

Incredible?

> open ai announced like 15% improvement by fixing gpu kernel issue

That is... ordinary software optimization.

> novel advances in math

> we are seeing frontier level math breakthroughs (ie performance that would put it in the top 100 or 1000 mathematicians in the world if it were a human, meaning top .00001% or 800/8B)

i think you have misunderstanding of what mathematicians do

"Breakthrough research" can be defined (in the citation record) as research that both (1) becomes highly cited, and (2) brings together citation chains that were previously not showing up together.

Mundane incremental research is cobbled from existing citations that already appear nearby in the record.

Basically, innovative research is a measure of bridging thought and domains that were previously not bridged. It's quite concrete as a measure in the citation record.

So we can know pretty conclusively.

Puja Ohlhaver gave a talk on this[1], and ran some experiments (that I had the pleasure to support on)

[1]: https://www.youtube.com/watch?v=guLDNMAOn24

The ones I'm familiar with are big breakthroughs, but they are both counterexamples. Examples have an advantage in that once you have the example in hand and a sketch of the proof (which they have provided), then an expert can probably work out the details themselves.

The sofic groups question was the outstanding question about sofic groups. Almost everyone thought that non-sofic groups existed, and there were plausible candidates, but proving a group was non-sofic was out of reach. Now that we know how to do it once, we can probably do it a lot more.

The Connes rigidity conjecture I think people thought was false, but it was a provocative claim to make. The significance of conjectures is frequently not that the answer to the question is "yes", but that we don't know how to answer the question. And now, apparently, we do.

The problems from CS (CVP and circuit complexity) are very important problems that have been worked on by top researchers for 30-40 years. Some of these researchers include Turing Award winners. A solution to them would be a best-paper award at many top CS conferences.

> However, I am worried that the language they are using in this blog post is exaggerating for the sake of marketing

Your worry.... is because they used the word advanced? For marketing? The word is used very appropriately here. There were PhD's who spent a big part of their career tackling these problems.

> The results were achieved by an internal version of Astra, our next major model.

Why would you factor in salary unless they had to baby it through. You would only count the hours for setting up the harness and prompt and checking the result.

Training the model is going to be amortized over other uses.

Given that OpenAI pays their employees with stock surely a breathtaking number, but not a very meaningful number now that the infrastructure is in place and the models are trained. AI could never get better and it would still be incredibly disruptive.

> The cost of generating the proofs for all 10 of these breakthroughs combined was under $2,000 at Sol API prices.

https://x.com/polynoamial/status/2083470822258467194

Not much point to pure math being kept secret, in all honesty. There isn't really industrial value, its only purpose (to them) is showing off their model's capabilities. More realistically they'll just stop paying for it.

Edit: Oh, are you suggesting they just use it to privately improve their models? I imagine a few more correct proofs would have a very marginal benefit, if any. Also, they'll probably just get extracted, meaning it still gets out but OpenAI doesn't get to fancily announce it themselves.

What does this even mean lol. These are not solved questions. The solution never existed.

For OpenAI, research is marketing. I’m sure they’ve got plenty of budget for that.

Presumably it's a rounding error compared to their full output, and they're making sure they have enough compute set aside for research by limiting public models. The more datacenters they build the less they have to limit them.

RL training can use all of them - idk what needed means.

I love how people come up with creative ideas to prove the bubble. This one is even more ridiculous - that OpenAI had spare compute to advance mathematics proves that data centres will not be needed. WHAT.

If anything it proves more data centres are needed. That's literally the only reasonable conclusion from this news.

well, a bug in the Lean kernel was discovered last week by way of an LLM tricking itself and its handler into believing it had found a non-constructive proof of the existence of a nontrivial Collatz cycle, see https://infosec.exchange/@0xabad1dea/117002106099986943 and https://lipn.info/@mevenlennonbertrand/116997917683191056

No, the correctness isn't for the "inside the Lean proofs", but for the translation of "human language math" and its formal Lean variant.

I'm not an expert at it myself, but my understanding is there are numerous ways to "cheat" in a Lean proof (via `sorry` and similar). They're taking responsibility for fully verifying that none of these cheats were used (and that the theorem statements themselves were all correctly formalized.)

A friend’s PhD advisor has been chasing non-sofic groups for 25 years (and was shown a preprint of the results by openai to verify them). He believed a solution would be Fields-worthy

This was not a problem that was for sale

This isn't really a productive way to think about these things, IMO. It's quite possible it would take hundreds of years for any specific group of PhDs to solve them. Or one individual PhD could have the correct flash of insight and solve it in a month. There's absolutely no way to predict this, besides trying to gauge the apparent simplicity of the proof or counterexample (which is likely to be misleading). Until someone actually runs an experiment like this it's not a viable metric.

The old way of establishing career credibility is being destroyed, for better or worse. Accomplishments that used to be career-defining are hard to distinguish from AI, and correlate more with access to compute. Think about Bill Gates's math paper he wrote in college. That kind of thing is gone now as a path to credibility. There's still competitions and grades, but the diversity of paths is going away. Maybe new ones will open up. This is a competitive advantage for old people who have credible pre-2025 accomplishments they can point to.

I find this whole way of looking at things weird. Did maths exist just to entertain and employ mathematicians? Surely maths is, like, useful? Not immediately, not predictably, but in the long run? In which case, whether mathematicians feel bad about it is mostly irrelevant - it's like complaining about the railway because it may put coaching inns out of business.

Given that we were nowhere near this state even two years ago, I think it’s a question of velocity more so than just distance.

Do mathematicians have the right to say "no AI PRs please, the volume is too much" just like how some open source maintainers do it? I guess they feel a loss of control, there is no way to turn the hose off.

Thinking of this a little bit with the perspective of every new proof as a burden, dumped for review by actual mathematicians.

Every time someone makes a comparison to chess I die inside. Chess is a spectator sport primarily funded by a few eccentric billionaires. Players artificially constrain themselves in timed environments knowing that they will never be able to produce better moves than a smartphone because a select few people find it interesting. Only ~30 top professionals actually make enough money to have a full career playing chess, maybe a few hundred more can sustain a meager lifestyle with coaching gigs. I shudder to imagine what will happen to the tens of thousands of non-Fields medalist caliber mathematicians if math goes the way of chess. Perhaps Terence Tao and a few other famous mathematicians will be funded by Peter Thiel to report on how well humanity can keep up with the machines? How do you expect any mathematician to be optimistic about this comparison.

The chess analogy is awful. If you simply want to know the answer to a chess problem, give it to the engine. Chess only lives on because it's a competition between humans to test their skill (just like bicycles, cars, trains didn't eliminate foot races) ... the computer is largely factored out, but not entirely -- people train with the computer, use it to check whether they played correctly, ... and they cheat. A lot. Thus there are more and more sophisticated mechanisms to detect and prevent cheating.

If you translate that to math, then all you get is math competitions, not math as a career. Of course the translation isn't nearly exact ... there's a lot more room for professional mathematicians because the math space is far more vast than the chess space and can't generally be cranked out mechanically (we have proof).

P.S. The response is nonsense ... I explained exactly why it's awful (others have too) and the response doesn't in any way refute the explanation ... rather it offers up a ridiculous strawman.

As in chess and go and also coding for the past ~year there are two groups of people: the disappointed and the enthusiastic. The disappointed are sad that they lost their advantage and that the craft they honed for years or decades has rapidly lost its value; the enthusiastic are excited about the future and what computers can bring to their domain and how it will evolve. I’m a bit of both if it comes to programming, more enthusiastic than disappointed, but also more than a bit terrified about the pace of it all. I imagine that’s how Kasparov felt back then, that’s how Lee Sedol felt and now that’s how Terry Tao feels.

The most disappointed folks will simply drop out, but the enthusiastic ones will keep going and with luck make up for the ones who decided to quit. Chess and go certainly went this way.

It will not happen with existing LLM techniques.

I’m personally hoping for the next big AI gangbanger to be theoretical physics. Boy does that field need a good reshuffle. I think when any novel mathematical theory can be done by AI you’ll see simultaneously theoretical physics getting wrecked as hard as pure math is. At that point we might see new physics or paradigm shifting technology emerging.

There's no clean line between a collection of theorems and a theory.

Solving math problems doesn’t require the cooperation of rival factions.

Yes, but all of those are orders of magnitude harder than math.

What's stopping the non-profits that exist today from just putting more money into tokens to get the solutions they want?

Seriously?

We already know how to solve all of these issues. What we lack is collective political will.

Almost like superintelligence solves these problems...

The Maxwell conjecture was a conjecture in theoretical physics (though not a particularly important one)

It's just another tool that can help solve problems. It doesn't know _what_ problems to solve. It turns out that a lot of old problems are now low hanging fruit for these new models. In terms of 'expanding the frontier', we've just discovered dynamite and can now blast our way through mountains. The bottom of the ocean or space are still as hard to reach as ever.

No bud, it’s just the beginning!

Ambient 0: Math for Airports

You shouldn't be getting downvoted for something that a majority of pure math and art enthusiasts believe to be true. The truth is many of these entrepreneurs and VCs are obsessed with AI not for money or human progress, but because it makes them feel closer to being a "god" rather than a mere mortal. Much of it (especially AI art) is out of spite for human creativity, which is done by mortals with limitations.

I asked ChatGPT and it told me these aren’t not important /s

Sorry, OpenAI's take is correct here. If you're not convinced, here is how they prompted LLM: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98... [0]

A slightly smarter highschooler could write these. I could write these. It's clear as day that the LLM, not the human, did the heavy lift. It'd be ridiculous to give full credit to whoever wrote the prompt.

[0]: Not one of the proofs in the linked article, but from OpenAI too.

Your brain also is physical. Electrochemical gradients flow between physical molecular constructs. Isn't it just chemistry? Do you attribute it to physics or some whole-is-greater-than-the-parts idea?

A better analogy would be a manufactured object, say 3d printed for simplicity. The 3d printer is given an input, and an object manifests itself after some time. We say that the creator of the object is the person turning on the machine, sending the data, and collecting the object. Not the machine itself.

> Do you attribute the build to the tool? The "system's contribution" is helped by many other things all the way down to chips, datacenters and power generation. If the authorship requires attributing to a tool, then it should happen all the way down.

When the tool is a 3D printer, or any CNC system really, you bet I attribute a build to it.

I could also attribute the operator; there is no contradiction, it's a free choice, just like saying "I am in Berlin" does not contradict "I am in Germany".

> AI has no self-awareness

What is your mechanistic model of self awareness that yields this conclusion?

> It's a tool

Does your model suggest that tools can't have self awareness?

Very marketable nerd snipes indeed.