> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question.
The other crucial part to this is the ability to actually encode and test the theorem (via Lean).
Otherwise, we would be swarmed with a billion lines of theorems that no one will be able to ever understand and verify anyway.
This includes a proof of Barnette's Conjecture, which is one of the graph theory conjectures that I tried attacking with SOTA models a few months ago. I like it because it is easy to understand with a basic knowledge of graph theory. I spent quite a bit of time on it and failed. Their proof looks approachable at first glance.
Presumably it's mainly the better model, I don't see much evidence of a particularly advanced harness based on the reasoning traces that they provided.
Unique Games Conjecture [0] is a seminal conjecture in Complexity Theory, and is an underlying assumption for many, many inapproximability results. A valid proof is a big deal!
I also don't think there was general consensus on which way this would resolve prior to this (or is that a little out dated?) unlike some of the other major problem resolutions. I heard rumors that there would be a big result in TCS and speculation it would be UGC that or P neq PSPACE but I'm still a bit shocked.
I'm really glad that OpenAI is formalizing these things, because I'm not convinced that their current internal frontier models are particularly good at writing down their thoughts in English. From the (probably awesome) Unique Games Conjecture/Theorem paper, the first two sentences of section 1.1 start to define the problem:
> A Unique Games instance has a finite vertex set, a finite alphabet K, and a nonempty list of oriented constraints e = (u_e,v_e,π_e), where π_e is a permutation of K. A labeling a satisfies e when a(v_e) = π_e(a(u_e)).
I'm sorry, what? I admit it's been quite a few years since I've thought about the Unique Games Conjecture, and I never dug that deeply, but this part is very, very elementary graph theory and notation. So let's unpack it.
1. e is maybe a name of a list.
2. The elements of that list are tuples, where each tuple is (a vertex, a vertex, a permutation). So e indexes into the list and u_e is the source vertex for the e-th constraint in the list called e. Thanks.
3. a is a labeling. I'm fairly confident that, by "a labeling", they mean that e is a function from vertices to colors, where the colors are the elements of k.
4. That vertex coloring a satisfies the list e, when, for, um, an index e into e, a(v_e) = π_e(a(u_e)). But this isn't for all e, it's for some e, and the goal is to count them.
So maybe e isn't a list? Maybe e is a constraint that is represented as a tuple, so e = (u_e,v_e,π_e) and u, v, and π aren't sequences at all but are, in fact, the trivial unpacking functions that unpack the pieces of the tuple.
Reading this stuff is pointlessly painful, and it's extremely easy to make mistakes when being sloppy like this.
If this were my paper, or if I were trying to train a model to write math, I'd want something like:
A Unique Games instance has a finite vertex set V, a finite edge set E = (V × V), a finite alphabet K of possible vertex colors, and a nonempty list of oriented constraints. Let Π be the set of permutations of V. Each constraint e is a tuple in E × E × Π, where we write u_e ∈ E for the first element, v_e ∈ E for the second element and π_e ∈ Π for the third.
A vertex coloring a : V → K satisfies e when a(v_e) = π_e(a(u_e)).
As a TCS/scheduling person, this one is definitely of lesser importance than UGC, but it has been an open problem since the book of Garey and Johnson in 1979:
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]
Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:
Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
As an AI "doomer" can I ask the non-doomer people here how you interpret the significance of results like these, and what kind of progress you expect to see in the next 1-5 years?
Like do you see the technology plateauing at the current level, do you expect progress will continue but only in mathematics, I'm interested to know why others are not concerned?
Can I ask you back, what your concern is here? It'll get so good so as to desire to hurt us or is it a misalignment event that you think will lead to disaster?
Or is it simply that you feel bad for Mathematicians.
I see it as “if this can be represented in tokens it can be trained in and ‘solved’”. I don’t think there will be a plateau, but there might be issues with how effectively we can represent some things in a tokenized form and still be efficient.
I'm not sure how AI solving math problems is related to "doom", perhaps you could expand on that? To me (a "non-doomer"), it seems like an overall positive.
From a few preprints I've checked, this is not reliant on just extrapolating existing theories but actually shows novel/surprising approaches. Very few people globally could come up with something like this, even when given time and ressources.
So in other words, since deep learning is algorithmic research, we are now in the RSI era.
> Thus at most one informative i. So cheater chooses arbitrary g_{v_i}, on exact duplicated input matches and passes, independent of actual satisfiability!
No idea what it's so excited about, but it's cute that it "is." I for one welcome having access to a math buddy 24/7 that's way above my level but also always "willing" to talk at where I'm at.
This is significant progress and released without all the drama. Some very important progress in Reinmann, Hodge and unique games theorem. Point the repo to your agent and ask for the significance! In a way this is probably 50-100 years of math progress by humans
Not to pick on you specifically, but as someone who spends a lot of time unproductively reading AI math discourse it's truly shocking how incapable all the supposed math enthusiasts are of spelling Riemann.
I just don't understand how it happens. If they had ever taken an intro to real analysis class they would learn to spell his name. If they were just parroting what an AI told them... shouldn't they still just say his name? An individual could just be dyslexic or mistaken but it seems to be a substantial volume. I guess they just don't care enough about it to commit the correct name to memory, only remembering the "pattern" of the name and filling in the spelling via guesswork?
My point is it is crazy to make public claims about how important or not important a mathematical result is when you can't spell Riemann. Yes, it technically doesn't matter, but it betrays a damning lack of familiarity with introductory mathematics.
The mathematicians don't make this mistake, I assure you. I doubt there is a math professor on planet earth who would spell Riemann as Reinmann. In fact, I imagine no one who has ever heard the name pronounced would do so.
Math is the tool humans use to compress knowledge. So until we can comprehend it there really isn't much progress. Math theorems are tautologies, the truth of which are not dependent on proofs and proofs are erasable, at least classically. But the AI progress is exciting and AI proofs are a gold mine for humans (at least non domain experts) to explore.
What makes you think no one can comprehend this? It has been less than an hour since it dropped and there is already a ton of online chatter from people explaining the results, pointing out their favorites and more. Some of it is happening on this very thread.
I didn't say that. I am responding to "In a way this is probably 50-100 years of math progress by humans." I am actually very excited about AI proof and I am working overtime in my own way to try to comprehend as much as I can.
He spent years formalizing his sphere packing theorem because the proof (human produced) was already beyond the ability of peer review. Now his formalization effort likely can be easily reproduced by a model. However one should read his experience about what a formal proof is: often the problem is the statement not the proof. The example he gave is the Jordan curve theorem. It's actually quite challenging to formalize the concept of a planar curve (there are space filling curves). So it is not necessary that someone can look at a formal statement and say aha it is about a planar curve, unlike FLT where there is not much problem in recognizing what the statement is about.
Math is far more than that. If you can solve prime factorization for example, suddenly you can listen and interfere with almost every private conversation on the internet.
We are not far away from the moment where these models will be restricted, and sharing the results will be done more carefully.
This doesn't contradict what I said. But I do appreciate the fact AI can produce side effects not just humans. I made it sound like only human knowledges matter. That's too narrow.
Another way to state this: math theorems are like programs without side effects; it is immaterial whether a program without side effects is ever run. We study math for the side effects: it changes how we organize our thoughts.
>So until we can comprehend it there really isn't much progress.
Not really? We are at a point if an AI today can solve it, it can be stepping stone of understanding something deeper to tomorrows AI and it continues. Sort of like our limitations doesn't matter. Obviously there are many scenarios in this recursive loop but saying it isn't much progress is not how I view this as
Academics have been treating it that way because they had no other choice, and its been a waste of everyone’s time and often times taxpayer resources
Look at that, taxpayer funding was cut and a private sector solution came in just the nick of time, far accelerating the holding patterns we’ve been in for decades
Humanity doesn’t need all iterations towards the blueprints, the blueprint is good enough, we all stand on the shoulders of giants
Right. Before all the AI disruption, pure Math traditionally welcomed anyone who wanted to study its esoteric proofs, right? I remember all the excitement of the average Math enthusiast casually reading Wiles' proof over coffee.
Bottom line is, the relevant people can still understand the generated proofs. The disorienting part is they are a little slower than they'd like, but they'll get there.
AI is very helpful with understanding AI proofs. Agent swarms produce messy proofs overall but locally they are excellent and can teach anyone who wants to study them. No one controls math (in a material way funders do control an aspect of practicing math). Still, theorems are already true before we prove them. The difference a proof makes is whether it convinces the reader.
It fascinates me that there's something like this in something as solid and rigid like matrix multiplication. What causes something so rigid to break apart and "leak" at very large scale? Why does the "optimization" appear to be very, very small? Why does galactic algorithm exists? I can't imagine long division suddenly breaking apart after a billion digit, the structure seems very stable? I have heard before that matrix multiplication is apparently optimize-able at very, very large scale.
Does anyone have an intuition to what causes it? What happens at these large scale (or very small)?
Then 'n' means kind of different things for sorting vs. multiplication though. For example for sorting we assume constant time comparison, which doesn't make sense inputs of O(n) bits
These results are wild. Several individual findings are crazy good and use mostly unexplored methods (the improvement over Riemann for example)... I'm pretty sure some of these results would have been Fields-worthy.
But having so many of them at once? Damn. We really live in the future.
In previous math sharing there was speculation about stealing human researcher's results or progress, via prompt inputs from those researchers, and sharing that as their own result. They're adjusting their process, and it seems ok.
After how poorly OpenAI and Anthropic handled the previous cases, I approve of the more measured and cautious approach this time.
We cannot have them rushing to publish amidst tons of confusion, rumors of threats/scooping and outright plagiarism of existing work (by failing to cite said work).
If they're going to participate as scientists in these more rigorous fields, they're going to have to match that level of rigor, not lower it to the disastrous low that ML research publication is at.
I think it's generally a good thing that OpenAI is noticing when their projects are harming human communities, and deciding to respect their norms, especially when their math discoveries do not have immediate application and build on the thousands of years of that community's work.
Does this imply that it was a one shot prompt with ChatGPT Pro style models (i.e. best-of-N), rather than the agent swarm approach that was used for Navier-Stokes?
I'd really like some clarity on what that means. E.g. DeepMind has 'cheated' with this in the past, claiming that AlphaZero only took 4 hours to reach super-human chess levels while conveniently leaving out the fact that it was 4 hours x 5000+ TPUs. Sure it's impressive that it only took 4 hours wall-clock but it's very misleading as to cost.
Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?
I think they should put human names on the papers as someone who has reviewed the result, even if just a preliminary review. (I’m assuming they didn’t just pipe their model output directly to the internet and these had some amount of review?)
I appreciate you acknowledging the politics behind this. OpenAI does not intend to be a software company for long, they intend to be scientific infrastructure. They'll want to be faucet from which pours embryos, orbital calculation, geothermal/substructural rating, and etc.
Let's hypothetically say I'm a PHD student who is half way through my studies and I have a halfway written version of one of these "preprints" - what do I do?
Seems like a lot of PHD students are doing to have to pivot the entire structure of their PHD studies? Or just produce something which is already written by OpenAI?
Interesting. If you don't mind, could you please share a little bit about that ? You already finished your dissertation ?
It was about the works of Doudna and Charpentier ?
No, back in the late 90s and early 00s, people were trying to engineer custom nucleases and transcription factors, my work was on doing molecular dynamics simulations to optimize TF sequence specificity (similar to engineered zinc fingers) for gene therapy. I wrote up my dissertation and published it in 2001, and then went off to find enough compute, IO, and smart people to make it happen (https://research.google/blog/groundbreaking-simulations-by-g...).
My approach would require custom engineering for every different sequence we'd want to target. With CRISPR, you just "program" the system with a guide sequence, you don't need to do massive engineering to solve a protein design problem.
This happens all the time, even without AI. Other researchers or PhD students can publish the same results before you. I say that based on my experience during my PhD.
In other words, you've already taken a gambit with the first half of your PhD, now take a second gambit, praying that you have something to publish by the end of your PhD.
This has always been a challenge for PhD students and researchers, it's just far more likely to occur now it seems.
Getting scooped doesn't feel good, but it's a signal you've been thinking about things other people care about.
I think eventually companies won't get as much stuff that's usable for marketing, so they'll stop investing so much into ai for math, so eventually cheap and poor graduate students will be able to do relevant work again without worrying about getting scooped by a company with a million GPUs :-/
That would be unfortunate but the world does not owe you anything and is not going to stop for you. Which is also a valuable thing to learn in your 20s (ideally earlier).
Precisely what all NLP researchers and the ML community at large did in the last few years: embrace the frontier and realize that attention is all you need.
GitHub is a significantly worse place to store important results than Arxiv. Of course, slop does not belong on Arxiv, buy slop should also not get published.
I'm a software engineering/biology/ML guy who loves when clever math ideas get turned into real solutions (https://en.wikipedia.org/wiki/Compressed_sensing). I am curious if any of the results have immediate applications in any kind of engineering or science.
It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.
The Advisory Group states in its recommendations [1]:
"We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."
To me, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
To me, the issue is that the models are proprietary which are only accessible to a few people in 2 digits. It's not about progress but access.
Maybe, but it still seems like an excuse. OpenAI has a proprietary model capable of solving these problems and is willing to share the results with the mathematical community. So basically, the ask is to just not use the model and leave the problems unsolved?
It all hinges on the definition of “progress”. The debate of the past month is all about questioning whether formally proving outstanding unproven theorems without human understanding constitutes progress. This is quite different from solving diseases.
I don't think this is a reasonable take at all. Most work on maths has no real benefit other than to further human understanding of maths - it's more like an art. Nothing is gained from OpenAI solving all these problems but taking jobs from mathematicians. Other than advertising for OpenAI at least.
It could not be more different from having AI work on disease research etc.
I fear that professional mathematics will wither, and there will be nobody left to digest the AI results of the future, leaving us unable to challenge the AI.
The trope you're drawing a parallel with has a (to me) compelling counter though: there being a cure for every disease wouldn't stop people from getting sick.
A lot of these seem to be proving conjectures rather than finding counterexamples, a lot of people used that to claim that these models are not really smart/creative etc.
I wonder if the Lean compiler can change it's license so that a for-profit corporation can only use it if it pays, say, a few hundred billion dollars. This is the correct path.
With so many results in so many different areas no way they even remotely spot checked well enough.
Prediction: one of these is wrong and this (publicity stunt) will backfire.
Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
> Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.
As Kevin Buzzard recently said:
> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question.
The other crucial part to this is the ability to actually encode and test the theorem (via Lean). Otherwise, we would be swarmed with a billion lines of theorems that no one will be able to ever understand and verify anyway.
If you think AI-generated Lean proofs are unreadable, imagine Opus 5 generating informal proofs.
I think OP is saying Lean does indeed help.
This includes a proof of Barnette's Conjecture, which is one of the graph theory conjectures that I tried attacking with SOTA models a few months ago. I like it because it is easy to understand with a basic knowledge of graph theory. I spent quite a bit of time on it and failed. Their proof looks approachable at first glance.
https://github.com/openai/math/blob/main/preprints/Paired-st...
Any idea what made OpenAI successful where you weren’t?
Presumably it's mainly the better model, I don't see much evidence of a particularly advanced harness based on the reasoning traces that they provided.
Trillions of dollars might be a bit of an advantage.
Their internal model is allegedly like 4x as capable as the publicly available ones
They ingested all of his sessions with their SOTA models from a few months ago. ;)
A quick check shows that this list claims to fully solve 90 of the top 500 open problems in math (https://proofatlas.ai/open-problems/).
The highest ranked would be:
| 22 | Hilbert’s tenth problem over ℚ |
| 29 | Unique Games |
| 31 | Anderson-model extended states |
| 37 | Spacetime Penrose inequality |
| 48 | Nonexistence of Landau–Siegel zeros |
| 52 | Baum–Connes |
| 78 | Abundance |
| 80 | Hadwiger |
| 87 | Bose–Einstein condensation |
| 92 | Two-dimensional entanglement area law |
Dear lord that website is laggy
Why didn't they just give mathematicians access, so they could at least understand and write up the results in publishable form?
They're all here https://github.com/openai/math/tree/main/preprints
I think they meant "access to the model" rather than the results.
Unique Games Conjecture [0] is a seminal conjecture in Complexity Theory, and is an underlying assumption for many, many inapproximability results. A valid proof is a big deal!
[0] https://en.wikipedia.org/wiki/Unique_games_conjecture [1] https://github.com/openai/math/blob/main/preprints/The-Uniqu...
I also don't think there was general consensus on which way this would resolve prior to this (or is that a little out dated?) unlike some of the other major problem resolutions. I heard rumors that there would be a big result in TCS and speculation it would be UGC that or P neq PSPACE but I'm still a bit shocked.
I'm really glad that OpenAI is formalizing these things, because I'm not convinced that their current internal frontier models are particularly good at writing down their thoughts in English. From the (probably awesome) Unique Games Conjecture/Theorem paper, the first two sentences of section 1.1 start to define the problem:
> A Unique Games instance has a finite vertex set, a finite alphabet K, and a nonempty list of oriented constraints e = (u_e,v_e,π_e), where π_e is a permutation of K. A labeling a satisfies e when a(v_e) = π_e(a(u_e)).
I'm sorry, what? I admit it's been quite a few years since I've thought about the Unique Games Conjecture, and I never dug that deeply, but this part is very, very elementary graph theory and notation. So let's unpack it.
1. e is maybe a name of a list.
2. The elements of that list are tuples, where each tuple is (a vertex, a vertex, a permutation). So e indexes into the list and u_e is the source vertex for the e-th constraint in the list called e. Thanks.
3. a is a labeling. I'm fairly confident that, by "a labeling", they mean that e is a function from vertices to colors, where the colors are the elements of k.
4. That vertex coloring a satisfies the list e, when, for, um, an index e into e, a(v_e) = π_e(a(u_e)). But this isn't for all e, it's for some e, and the goal is to count them.
So maybe e isn't a list? Maybe e is a constraint that is represented as a tuple, so e = (u_e,v_e,π_e) and u, v, and π aren't sequences at all but are, in fact, the trivial unpacking functions that unpack the pieces of the tuple.
Reading this stuff is pointlessly painful, and it's extremely easy to make mistakes when being sloppy like this.
If this were my paper, or if I were trying to train a model to write math, I'd want something like:
A Unique Games instance has a finite vertex set V, a finite edge set E = (V × V), a finite alphabet K of possible vertex colors, and a nonempty list of oriented constraints. Let Π be the set of permutations of V. Each constraint e is a tuple in E × E × Π, where we write u_e ∈ E for the first element, v_e ∈ E for the second element and π_e ∈ Π for the third.
A vertex coloring a : V → K satisfies e when a(v_e) = π_e(a(u_e)).
Yeah, that's one of the big things of TCS. I think I see that as bigger than that Millenium Prize problem.
This was the biggest highlight for me as well. Astounding...
As a TCS/scheduling person, this one is definitely of lesser importance than UGC, but it has been an open problem since the book of Garey and Johnson in 1979:
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]
Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:
Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...
As an AI "doomer" can I ask the non-doomer people here how you interpret the significance of results like these, and what kind of progress you expect to see in the next 1-5 years?
Like do you see the technology plateauing at the current level, do you expect progress will continue but only in mathematics, I'm interested to know why others are not concerned?
Can I ask you back, what your concern is here? It'll get so good so as to desire to hurt us or is it a misalignment event that you think will lead to disaster?
Or is it simply that you feel bad for Mathematicians.
Misuse of extremely capable models, misalignment during RL are both very large risks as capabilities grow imo
I see it as “if this can be represented in tokens it can be trained in and ‘solved’”. I don’t think there will be a plateau, but there might be issues with how effectively we can represent some things in a tokenized form and still be efficient.
I'm not sure how AI solving math problems is related to "doom", perhaps you could expand on that? To me (a "non-doomer"), it seems like an overall positive.
Try using AI for your work, whatever you do. You will quickly understand the limitations.
From a few preprints I've checked, this is not reliant on just extrapolating existing theories but actually shows novel/surprising approaches. Very few people globally could come up with something like this, even when given time and ressources.
So in other words, since deep learning is algorithmic research, we are now in the RSI era.
> this is not reliant on just extrapolating existing theories but actually shows novel/surprising approaches
"Surprising" is a, well, surprisingly high bar to clear, and requires thorough understanding of the paper. ("Novel" is tautological.)
How did you determine this in 1 hour? Are you a researcher in multiple of these areas?
Can you give an example, or explain more how you came to this conclusion?
They can replace anyone but they can't replace everyone.
It’s fascinating to read through the reasoning traces: https://github.com/openai/math/tree/main/reasoning_traces
Look at one of their examples of an initial prompt: https://github.com/openai/math/blob/main/reasoning_traces/re...
> Look at one of their examples of an initial prompt
Interesting that its only an excerpt. I wonder what else they include but didn't share.
> Thus at most one informative i. So cheater chooses arbitrary g_{v_i}, on exact duplicated input matches and passes, independent of actual satisfiability!
No idea what it's so excited about, but it's cute that it "is." I for one welcome having access to a math buddy 24/7 that's way above my level but also always "willing" to talk at where I'm at.
This is significant progress and released without all the drama. Some very important progress in Reinmann, Hodge and unique games theorem. Point the repo to your agent and ask for the significance! In a way this is probably 50-100 years of math progress by humans
Not to pick on you specifically, but as someone who spends a lot of time unproductively reading AI math discourse it's truly shocking how incapable all the supposed math enthusiasts are of spelling Riemann.
“How many Ns in Riemann?”
It's because he isn't a mathematician: he's an OpenAI shill. Check his comment history
I just did a quick search on this and apparently the misspellings are German surnames as well:
https://en.wikipedia.org/wiki/Reimann
https://en.wikipedia.org/wiki/Reinmann
I just don't understand how it happens. If they had ever taken an intro to real analysis class they would learn to spell his name. If they were just parroting what an AI told them... shouldn't they still just say his name? An individual could just be dyslexic or mistaken but it seems to be a substantial volume. I guess they just don't care enough about it to commit the correct name to memory, only remembering the "pattern" of the name and filling in the spelling via guesswork?
Maybe they skipped straight to Lebeg integrals.
People just don’t spell that seriously buddy, especially when it is so immaterial to the point.
My point is it is crazy to make public claims about how important or not important a mathematical result is when you can't spell Riemann. Yes, it technically doesn't matter, but it betrays a damning lack of familiarity with introductory mathematics.
You’re (as Claude would say) absolutely right, and I suspect the original commenter has no idea what they’re talking about.
Yes sure but they are different surnames and pronounced differently.
I didn't mean to oppose OP's point, I just found it interesting as a non-German speaker!
They're mathematicians, not linguists :)
The mathematicians don't make this mistake, I assure you. I doubt there is a math professor on planet earth who would spell Riemann as Reinmann. In fact, I imagine no one who has ever heard the name pronounced would do so.
Uh oh, no true scottsman....
There was A LOT of drama about this release.
Math is the tool humans use to compress knowledge. So until we can comprehend it there really isn't much progress. Math theorems are tautologies, the truth of which are not dependent on proofs and proofs are erasable, at least classically. But the AI progress is exciting and AI proofs are a gold mine for humans (at least non domain experts) to explore.
What makes you think no one can comprehend this? It has been less than an hour since it dropped and there is already a ton of online chatter from people explaining the results, pointing out their favorites and more. Some of it is happening on this very thread.
I didn't say that. I am responding to "In a way this is probably 50-100 years of math progress by humans." I am actually very excited about AI proof and I am working overtime in my own way to try to comprehend as much as I can.
I think it would be helpful to people who want to understand what a formalized proof is to read Thomas Hales on this: https://www.math.stonybrook.edu/~bishop/classes/math536.S24/...
He spent years formalizing his sphere packing theorem because the proof (human produced) was already beyond the ability of peer review. Now his formalization effort likely can be easily reproduced by a model. However one should read his experience about what a formal proof is: often the problem is the statement not the proof. The example he gave is the Jordan curve theorem. It's actually quite challenging to formalize the concept of a planar curve (there are space filling curves). So it is not necessary that someone can look at a formal statement and say aha it is about a planar curve, unlike FLT where there is not much problem in recognizing what the statement is about.
Math is far more than that. If you can solve prime factorization for example, suddenly you can listen and interfere with almost every private conversation on the internet.
We are not far away from the moment where these models will be restricted, and sharing the results will be done more carefully.
This doesn't contradict what I said. But I do appreciate the fact AI can produce side effects not just humans. I made it sound like only human knowledges matter. That's too narrow.
Another way to state this: math theorems are like programs without side effects; it is immaterial whether a program without side effects is ever run. We study math for the side effects: it changes how we organize our thoughts.
>So until we can comprehend it there really isn't much progress.
Not really? We are at a point if an AI today can solve it, it can be stepping stone of understanding something deeper to tomorrows AI and it continues. Sort of like our limitations doesn't matter. Obviously there are many scenarios in this recursive loop but saying it isn't much progress is not how I view this as
If it changes how we think then yes it has an effect.
Academics have been treating it that way because they had no other choice, and its been a waste of everyone’s time and often times taxpayer resources
Look at that, taxpayer funding was cut and a private sector solution came in just the nick of time, far accelerating the holding patterns we’ve been in for decades
Humanity doesn’t need all iterations towards the blueprints, the blueprint is good enough, we all stand on the shoulders of giants
> until we can comprehend it there really isn't much progress
Who is "we" here exactly?
Whoever wants to study the result.
> Whoever wants to study the result.
Right. Before all the AI disruption, pure Math traditionally welcomed anyone who wanted to study its esoteric proofs, right? I remember all the excitement of the average Math enthusiast casually reading Wiles' proof over coffee.
Bottom line is, the relevant people can still understand the generated proofs. The disorienting part is they are a little slower than they'd like, but they'll get there.
AI is very helpful with understanding AI proofs. Agent swarms produce messy proofs overall but locally they are excellent and can teach anyone who wants to study them. No one controls math (in a material way funders do control an aspect of practicing math). Still, theorems are already true before we prove them. The difference a proof makes is whether it convinces the reader.
> Math theorems are tautologies
Proven math theorems are tautologies.
Flt was no less a tautology before it was proved. We just weren't sure about it. Proofs only change us, not math.
True.
From their github: "The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking." That's pretty crazy.
Some of these are interesting ngl.
109. Integer multiplication below n log n
Surprising that this is possible.
158. The Euclidean plane cannot be colored with five colors.
Only 6 and 7 remain!
376. Universal computation in forced Navier–Stokes flows.
Morning coffee proven turing complete
Integer multiplication is very unexpected, I think most people believed in the n log n lower bound!
> We give a deterministic algorithm that multiplies two n-bit integers in O(n (log n)^(1−κ)) worst- case time, with κ = 2^(−182).
LMAO, I don't think I ever saw such a small number in a CS result.
It fascinates me that there's something like this in something as solid and rigid like matrix multiplication. What causes something so rigid to break apart and "leak" at very large scale? Why does the "optimization" appear to be very, very small? Why does galactic algorithm exists? I can't imagine long division suddenly breaking apart after a billion digit, the structure seems very stable? I have heard before that matrix multiplication is apparently optimize-able at very, very large scale.
Does anyone have an intuition to what causes it? What happens at these large scale (or very small)?
Yeah its ridiculously small, but any improvement on n log n is wild.
Like there is somehow redundancy in a fourier transform that makes it sub Linearithmic?
Which low and behold ->
130. Fourier transforms below n log n.
They also separately give algorithm for Fourier transform over complex number faster than O(n log n)
I am fully braced for it to be a https://en.wikipedia.org/wiki/Galactic_algorithm
Very surprising result though! Multiplication is easier than sorting.
Then 'n' means kind of different things for sorting vs. multiplication though. For example for sorting we assume constant time comparison, which doesn't make sense inputs of O(n) bits
It would be absolutely unbelievable if such an improvement were practical.
https://github.com/openai/math
These results are wild. Several individual findings are crazy good and use mostly unexplored methods (the improvement over Riemann for example)... I'm pretty sure some of these results would have been Fields-worthy.
But having so many of them at once? Damn. We really live in the future.
Extremely unfortunate that gate keeping got to the point where they felt the need to ask for permission to share math.
In previous math sharing there was speculation about stealing human researcher's results or progress, via prompt inputs from those researchers, and sharing that as their own result. They're adjusting their process, and it seems ok.
After how poorly OpenAI and Anthropic handled the previous cases, I approve of the more measured and cautious approach this time.
We cannot have them rushing to publish amidst tons of confusion, rumors of threats/scooping and outright plagiarism of existing work (by failing to cite said work).
If they're going to participate as scientists in these more rigorous fields, they're going to have to match that level of rigor, not lower it to the disastrous low that ML research publication is at.
Just to be very clear: they aren't asking for permission, they are framing it that way because of the bad press.
There's no gatekeeping here!
I think it's generally a good thing that OpenAI is noticing when their projects are harming human communities, and deciding to respect their norms, especially when their math discoveries do not have immediate application and build on the thousands of years of that community's work.
> do not have immediate application
How do you know? Seems statistically unlikely with 720 problems, most of them well known
From the GitHub description: “On average, each result used 3 hours of ChatGPT Pro thinking compute”
Oh cool, we will all now have a math genius on our computer.
It was using their internal math model, so not yet for us
I used future tense. It was implied this will be available.
Well, on their computers. But you can rent them for a price.
That estimate is obviously going to conveniently ignore all the failed runs.
Does this imply that it was a one shot prompt with ChatGPT Pro style models (i.e. best-of-N), rather than the agent swarm approach that was used for Navier-Stokes?
It doesn't imply that, it's just measuring the amount of compute.
I'd really like some clarity on what that means. E.g. DeepMind has 'cheated' with this in the past, claiming that AlphaZero only took 4 hours to reach super-human chess levels while conveniently leaving out the fact that it was 4 hours x 5000+ TPUs. Sure it's impressive that it only took 4 hours wall-clock but it's very misleading as to cost.
Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?
Depends what your metrics are. If you suddenly found a way to have 9 women make a baby in one month that is huge.
I'm not denying that, but I'd still like to know what that cost.
They did say that. "3 hours of ChatGPT Pro thinking compute"
Yes, what does that mean?
>OpenAI has 'cheated' with this in the past, claiming that AlphaZero [...]
That doesn't sound right
Oops, edited.
Actual results: https://github.com/openai/math/blob/main/overview.pdf
HTML version, https://github.com/openai/math/blob/main/CONTENTS.md
I think they should put human names on the papers as someone who has reviewed the result, even if just a preliminary review. (I’m assuming they didn’t just pipe their model output directly to the internet and these had some amount of review?)
There are competitive reasons that they don't want to share all the people on the team.
Presumably they don't because they're training the audience (us) to trust the machine, not its verifiers, even if they were included.
I appreciate you acknowledging the politics behind this. OpenAI does not intend to be a software company for long, they intend to be scientific infrastructure. They'll want to be faucet from which pours embryos, orbital calculation, geothermal/substructural rating, and etc.
Long term this will be the only reasonable author list: Chad G. Peter {1}, Mat H. Lean {2}.
1: Author 2: Verifier
/s
holy fucking shit
Let's hypothetically say I'm a PHD student who is half way through my studies and I have a halfway written version of one of these "preprints" - what do I do?
Seems like a lot of PHD students are doing to have to pivot the entire structure of their PHD studies? Or just produce something which is already written by OpenAI?
Let me give you some perspective: my entire phd was made obsolete by CRISPR. It was a wonderful thing.
Interesting. If you don't mind, could you please share a little bit about that ? You already finished your dissertation ? It was about the works of Doudna and Charpentier ?
No, back in the late 90s and early 00s, people were trying to engineer custom nucleases and transcription factors, my work was on doing molecular dynamics simulations to optimize TF sequence specificity (similar to engineered zinc fingers) for gene therapy. I wrote up my dissertation and published it in 2001, and then went off to find enough compute, IO, and smart people to make it happen (https://research.google/blog/groundbreaking-simulations-by-g...).
My approach would require custom engineering for every different sequence we'd want to target. With CRISPR, you just "program" the system with a guide sequence, you don't need to do massive engineering to solve a protein design problem.
This happens all the time, even without AI. Other researchers or PhD students can publish the same results before you. I say that based on my experience during my PhD.
Use whatever is published as the new base for your research. Use AI tools to help you going forward.
In other words, you've already taken a gambit with the first half of your PhD, now take a second gambit, praying that you have something to publish by the end of your PhD.
It has to feel awful to be in this position.
Could be worse, imagine having years of experience in a profession these things can now handle by themselves.
:)
This has always been a challenge for PhD students and researchers, it's just far more likely to occur now it seems. Getting scooped doesn't feel good, but it's a signal you've been thinking about things other people care about.
That’s always been a thing, it’s called “getting scooped”
Perhaps consider switching to a more applied field. Experiments in the physical realm hold value, especially if you document the process.
I think eventually companies won't get as much stuff that's usable for marketing, so they'll stop investing so much into ai for math, so eventually cheap and poor graduate students will be able to do relevant work again without worrying about getting scooped by a company with a million GPUs :-/
Don't worry, if you use openAI and get lucky they might offer to share credit with you for your work.
It could still be interesting if your approach to the problem was different to theirs.
Don’t paste your research into these AI because they will train on it and then scoop you.
I think that happened after word of the project reached OpenAI and they allocated resources to it.
That would be unfortunate but the world does not owe you anything and is not going to stop for you. Which is also a valuable thing to learn in your 20s (ideally earlier).
Look forward to being obsolete I guess
> what do I do?
Precisely what all NLP researchers and the ML community at large did in the last few years: embrace the frontier and realize that attention is all you need.
Yes, and?
So happy this is shared on GitHub rather than some gatekeeping paid journal. Truly a new age for science.
Most mathematical results are shared on Arxiv. Journals add peer review.
End of an age for journals?
GitHub is a significantly worse place to store important results than Arxiv. Of course, slop does not belong on Arxiv, buy slop should also not get published.
Unique games conjecture and matmul <= 2.25. What the hell.
FFT BELOW NLOGN
I'm a software engineering/biology/ML guy who loves when clever math ideas get turned into real solutions (https://en.wikipedia.org/wiki/Compressed_sensing). I am curious if any of the results have immediate applications in any kind of engineering or science.
It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.
I find the claimed matrix multiply result (w<= 2.25) shocking. I hope it holds up.
You can read the papers here: https://hub.valency.io/collections/openai-math
The Advisory Group states in its recommendations [1]:
"We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."
To me, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?
[1] https://agmai.org/general-sep29/
Why did you leave out the entire quote?
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
To me, the issue is that the models are proprietary which are only accessible to a few people in 2 digits. It's not about progress but access.
Maybe, but it still seems like an excuse. OpenAI has a proprietary model capable of solving these problems and is willing to share the results with the mathematical community. So basically, the ask is to just not use the model and leave the problems unsolved?
The ask is to let mathematicians outside of OpenAI use it, ie. at least wait until the model is released.
Which mathematicians though? Only fields medalists? Grad students? Any hobbyist wanting access?
These models are too expensive for broad access unfortunately.
ChatGPT pro is accessible to literally anyone who has a job and lives in a developed country.
It all hinges on the definition of “progress”. The debate of the past month is all about questioning whether formally proving outstanding unproven theorems without human understanding constitutes progress. This is quite different from solving diseases.
I don't think this is a reasonable take at all. Most work on maths has no real benefit other than to further human understanding of maths - it's more like an art. Nothing is gained from OpenAI solving all these problems but taking jobs from mathematicians. Other than advertising for OpenAI at least. It could not be more different from having AI work on disease research etc.
> Most work on maths has no real benefit other than to further human understanding of maths - it's more like an art.
It's been a while since I was reminded of this xkcd: https://xkcd.com/435/
The rest of that document makes a pretty compelling case for why this is a bad practice
I fear that professional mathematics will wither, and there will be nobody left to digest the AI results of the future, leaving us unable to challenge the AI.
Comical ask
The trope you're drawing a parallel with has a (to me) compelling counter though: there being a cure for every disease wouldn't stop people from getting sick.
Not so for maths.
The latest posts from Terry Tao on Mastodon effectively ask for an AI to explain its results to human mathematicians.
> "I believe that AI can contribute positively in all of these directions [NB: exposition, community building, new directions of study]"
https://mathstodon.xyz/@tao/117395269325940185
Advocating purely for progress and not humanitarian value is how we'll all get enslaved.
...but we're not talking about diseases. Publishing an AI-generated Navier-Stokes solution does not save lives. (And, in fact, it harms some.)
Glad to seem them changing their tact with the whole NS debacle. Hopefully we can all focus on the results now rather than the surrounding drama.
A lot of these seem to be proving conjectures rather than finding counterexamples, a lot of people used that to claim that these models are not really smart/creative etc.
That copium didn't last for what, three months?
Don't worry, more copium will be delivered. TBF so far it's still only solved the easiest of the millennium problems.
I wonder if the Lean compiler can change it's license so that a for-profit corporation can only use it if it pays, say, a few hundred billion dollars. This is the correct path.
will this make the math for building data centers work?
Curious if Sébastien Bubeck still work at OpenAI? He came out quite dirty after Navier-Stokes drama
This is a pretty hilarious thing to read juxtaposed with AGMAI's requests.
Basically "Here you go, have fun with this, fuck all your demands, by the way we're gonna be releasing the model stay tuned!"
Have any real mathematicians working on these problems reviewed any of these and determined if they are just gobbledegook or not?
The ones with lean proofs could still be formulated incorrectly
The stochastic parrots have predicted the next token once again.
wow
Can someone knowledgeable about the subject outline the most significant portions of the results?
With so many results in so many different areas no way they even remotely spot checked well enough.
Prediction: one of these is wrong and this (publicity stunt) will backfire.
Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
You have not read their readme:
> Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.
i have and i'm exactly saying that if it comes to pass one of them is wrong it's going to backfire. ie yes that's my exact point/bet.
What does a backfire look like? It's ok to be wrong in the science/math world.
of course in science/math it is but it's not okay if you're a business selling supercalifragilisticexpialidocious infallible intelligence.
Do they claim that's the case? I don't think they do.
does company A making product B claim that the product is robust and consistent? is this a serious question?
Good to see they're engaging with the mathematical community, even if they had to be publicly shamed into doing so.
This is the opposite of what the mathematical community were asking for. I say this as someone who strongly approves of this approach.
I don't know, this does not feel like the message hit OpenAI where it needed to hit, if this is their primary response.
You want them to stop doing math research?