To the “LLMs just interpolate their training data” crowd:
Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules.
I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.
> 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.
What's your basis for assuming LLM is capable of doing this?
I honestly don't know personally either way. Based on my limited understanding of how LLMs work, I don't see them be making the next great song or next great book and based on that reasoning I'm betting that it probably wont be able to do whatever next "Descartes, Newton, Leibnitz, Gauss, Euler, Ramanujan, Galois" are going to do.
Of course AI as a wider field comes up with something more powerful than LLM that would be different.
Because by definition LLMs are permutation machines, not creativity machines. (My premise, which you may disagree with, is that creativity/imagination/artistry is not merely permutation.)
LLMs by themselves are not able to but you are missing a piece here.
LLMs are prompted by humans and the right query may make it think/behave in a way to create a novel solution.
Then there's a third factor now with Agentic AI system loops with LLMs. Where it can research, try, experiment in its own loop that's tied to the real world for feedback.
Agentic + LLM + Initial Human Prompter by definition can have it experiment outside of its domain of expertise.
So that's extending the "LLM can't create novel ideas" but I don't think anyone can disagree the three elements above are enough ingredients for an AI to come up with novel ideas.
This "new math" might be a recombination of things that we already know - or an obvious pattern that emerges if you take a look at things from a far enough distance - or something that can be brute-forced into existence. All things LLMs are perfectly capable of.
In the end, creativity has always been a combination of chance and the application of known patterns in new contexts.
> This "new math" might be a recombination of things that we already know
If you know anything about the invention of new math (analytic geometry, Calculus, etc.), you'd know how untrue this is. In fact, Calculus was extremely hand-wavy and without rigorous underpinnings until the mid 1800s. Again: more art than science.
I prefer to think of it as they’re interpolation machines not extrapolation machines. They can project within the space they’re trained in, and what they produce may not be in their training corpus, but it must be implied by it. I don’t know if this is sufficient to make them too weak to create original “ideas” of this sort, but I think it is sufficient to make them incapable of original thought vs a very complex to evaluate expected thought.
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.
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.
One can argue that mathematical facts are discovered, but the tools that allow us to find, express them and prove them, are mostly invented. This goes up to the axioms, that we can deliberately choose and craft.
Regardless of which, both Newton and Leibniz imprint in their findings a 'voice' and understanding different from each other and that of an LLM (for now?)
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.
Yes they do…? Who cares if they just predict the next token? The outcome is that they can invent new abstractions. You could claim that the invention of this new idea is a combination of an LLM and a harness, but that combination can solve logic puzzles and invent abstractions. If a really large spinning wheel could invent proofs that were previously unsolved, that would be a wildly amazing spinning wheel. I view LLMs similarly. It is just fancy autocomplete, but look what we can do with it!
Said differently, what is prediction but composition projected forward through time/ideas?
I'm not sure what the point of this exercise is. My prompt to ChatGPT: "Create a new English word with a reasonably sounding definition. That word must not come up in a Google search." Two attempts did come up in a search, the third was "Thaleniq (noun)". Definition: The brief feeling that a conversation has permanently changed your opinion of someone, even if nothing dramatic was said. Nothing in Google. There, a new word, not sure it proves or disproves anything. Or is it time to move the goal posts?
It's even more interesting if they just "predict the next token" because that would mean that all knowledge is obvious, even if no human knows it yet, because it has to be if the predictor is just writing an average of all possible words.
This means that those compositions are also obvious because they will be predicted by just outputing the next word.
For anyone using LLMs heavily for coding, this shouldn't be too surprising. It was just a matter of time.
Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool at a narrow objective and see what sticks, then build on that or combine improvements.
Reading the article, that seems exactly how the discovery was made, an LLM used a "surprising connection" to go beyond the expected result. But the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.
wow, that was indeed a brilliant essay. i particularly liked the framing that "solving a big conjecture was a cryptographic proof that you had come up with a genuine conceptual innovation".
> 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.
One could hardly ask for a task better suited for LLMs than producing math in Lean. Running a restaurant is so much fuzzier, from the definition of what it even means to the relation of inputs to outputs and evaluating success.
I just visited a McDonald's for the first time in a while. The self-order kiosk UI is quite bad. I think this is evidence in favor of the idea that an incompetent AI will soon be incompetently running a McDonald's.
Out of curiosity, what issue did you have with the McDonald’s self-order kiosk? I actually think McDonald’s has the best kiosk I’ve ever encountered. The little animation that plays when you add an item to your cart is a little annoying (but I think they’ve sped that up). But otherwise, it’s everything I’d want. It shows you all the items, tells you every ingredient, and lets you add or remove ingredients. I have a better experience ordering through the kiosk than I do talking to a cashier.
Casual reminder that the author's proposed solution to the labor-automation dystopia is to invent a second identity-verification dystopia. Also casual reminder that the author wanted the death penalty to anyone over the age of 65.
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.
Assuming you can still sue McDonalds I am not sure if this is a problem in the robotic llm case. I'm also trying to imagine a case where you would want to sue the llm and not the company. Given robots/llm don't have free will I'm not sure the problem with qualified immunity making police unaccountable applies.
There already exist a lot of similar conventions in corporate law. Generally, a main advantage of incorporation is protecting the people making the decisions from personal lawsuits.
McDonald's are franchises - you generally want to sue the local owner or threaten them in addition to the holding company.
That only requires someone own the ai managed McDonald's though. so long as they can't avoid responsibility by pointing to the AI I don't see why you couldn't sue them.
>Police officers are human. In the United States in the vast majority of cases you can't sue the police, only the community responsible for them.
Police are a monopoly; nobody has a choice about which police company to use. McDonalds are not a monopoly, and many customers would prefer to eat at competitors run by entities that could be sued or jailed if they did anything particularly egregious.
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.
I think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture.
I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true.
In my eye's proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by grounding it in a larger theory while with the counterexample the model has to just perform a more advanced form of search to find the correct construction.
Obviously this search is impressive not naive and requires many steps along the way to prove connections to the counterexample, but instead of developing new deep mathematics the model is still just connecting existing ideas.
Not to discount this monumental achievement. I think we're really getting somewhere! To me, and this is just vibes based, I think the models aren't far from being able to theory craft in such a way that they could prove more complicated conjectures that require developing new mathematics. I think that's just a matter of having them able to work on longer and longer time horizons.
Is there a reason why we only hear of Erdos problems being solved? I would imagine there are a myriad of other unsolved problems in math, but every single ChatGPT "breakthrough in math" I come across on r/singularity and r/accelerate are Erdos problems.
No, Erdos problems were accepted as sort of a benchmark. There's a bunch of reasons they're favorable for this task:
1. They have a wide range of difficulties.
2. They were curated (Erdos didn't know at first glance how to solve them).
3. Humans already took the time to organize, formally state, add metadata to them.
4. There's a lot of them.
If you go around looking for a mathematics benchmark it's hard to do better than that.
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.
The summarized chain of thought for this task (linked in the blogpost) is 125 pages. That's an insane scale of reasoning, quite akin to what Anthropic has been teasing with Mythos.
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.
Gemini is like someone with short-term memory loss; after the first response, it forgets everything. That being said, I have checked multiple model and gemini can sometime give accurate answer.
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.
Not to dismiss the AI but the important part is that you still need someone able to recognize these solutions in the first place. A lot of things were just hidden in plain sight before AI but no one noticed or didn't have the framework either in maths or any other field they're specialized in to recognize those feats.
Would be interesting to know what kind of preparatory work actually went into this - how long did it take to construct an input that produced a real result, and how much input did they get from actual mathematicians to guide refining it
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.
One heuristic I used during my masters degree research thesis was to look for the seminal people or papers in a field by using google scholar to find the most cited research papers and then reading everything else by that author / looking at the paper's references for others. You often only need to go back 3-4 papers to find some really seminal/foundational stuff.
Yeah, that's actually how I discovered Leslie Lamport like ten years ago. I was looking for papers on distributed consensus, and it's hard not to come across Paxos when doing that. It turns out that he has oodles of really great papers across a lot of different cool things in computer science and I feel like I understand a lot more about this space because of it.
It doesn't hurt that Lamport is exceptionally good at explaining things in plain language compared to a lot of other computer scientists.
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's a very heavy hammer. I used it in the way you describe and after double-checking noticed some crucial details were missed and certain facts were subtly misrepresented.
But I agree with you, especially in areas where they have a lot of training data, they can be very useful and save tons of time.
I don't think there's a substitute for reading the source material. You have to read the actual paper that's cited. You have to read the code that's being sourced/generated. But used as a reasoning search engine, it's a huge enabler. I mean so much of research literally is reasoning through piles of existing research. There's probably a large amount of good research (especially the kind that don't easily get grant funding) that can "easily" shake out through existing literature that humans just haven't been able to synthesize correctly.
Is this something that can be made explainable to someone without any of the relevant background, or is this one of those things where all that background is needed to understand it? Because I have no idea what's going on here, but would like to.
I would have thought a triangular grid works better than a grid of squares. You get ~3n links vs ~2n for the square grid. Curious what the AI came up with.
Knowing OpenAI, the solution's probably being withheld as a trade secret, lest it fall victim to distillation attacks (i.e. exactly the same shit they did to the open Internet).
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.
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...
"The proof came from a general-purpose reasoning model, not a system built specifically to solve math problems or this problem in particular, and represents an important milestone for the math and AI communities."
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.
It seems plausible given that people have been using off the shelf 5.5 xhigh to decent success with some erdos problems. There is likely still some scaffolding around it though (like parallel sampling or separate verifier step) since it's not clear if you can just "one shot" problems like this.
"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
While the result is impressive, this blog post is extremely disappointing.
- It does not show an example of the new best solution, nor explain why they couldn't show an example (e.g. if the proof was not constructive)
- It does not even explain the previous best solution. The diagram of the rescaled unit grid doesn't indicate what the "points" are beyond the normal non-scaled unit grid. I have no idea what to take away from it.
- It's description of the new proof just cites some terms of art with no effort made to actually explain the result.
If this post were not on the OpenAI blog, I would assume it was slop. I understand advanced pure mathematics is complicated, but it is entirely possible to explain complicated topics to non-experts.
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
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.
Every time I interact even with OpenAI's pro model, I am forced to come to the conclusion that anything outside the domain of specific technical problems is almost completely hopeless outside of a simple enhanced search and summary engine.
For example, these machines, if scaling intellect so fiercely that they are solving bespoke mathematics problems, should be able to generate mundane insights or unique conjectures far below the level of intellect required for highly advanced mathematics - and they simply do not.
Ask a model to give you the rundown and theory on a specific pharmacological substance, for example. It will cite the textbook and meta-analyses it pulls, but be completely incapable of any bespoke thinking on the topic. A random person pursuing a bachelor's in chemistry can do this.
Anything at all outside of the absolute facts, even the faintest conjecture, feels completely outside of their reach.
I dunno, I'm skeptical without proof. I've had the MAX+ plan for a while and I'm sorry, the quality between GPT vs Claude is night and day difference. Claude understands. GPT stumbles over every request I give it.
Except its not a proof. It's an existential proof of what? Projecting points and loosing density? Nah, it's wrong. At least with Edros you could solve f(x) or not solve it (inf). You can not with this. All they did was balance a really fancy quadratic equation. The projection from C^f to R² doesn't demonstrate geometric injectivity, so nⱼ = |X| isn't established, and the bound collapses.
Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules.
I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.
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.
I honestly don't know personally either way. Based on my limited understanding of how LLMs work, I don't see them be making the next great song or next great book and based on that reasoning I'm betting that it probably wont be able to do whatever next "Descartes, Newton, Leibnitz, Gauss, Euler, Ramanujan, Galois" are going to do.
Of course AI as a wider field comes up with something more powerful than LLM that would be different.
because I have no basis for assuming an LLM is fundamentally capable of doing this.
LLMs are prompted by humans and the right query may make it think/behave in a way to create a novel solution.
Then there's a third factor now with Agentic AI system loops with LLMs. Where it can research, try, experiment in its own loop that's tied to the real world for feedback.
Agentic + LLM + Initial Human Prompter by definition can have it experiment outside of its domain of expertise.
So that's extending the "LLM can't create novel ideas" but I don't think anyone can disagree the three elements above are enough ingredients for an AI to come up with novel ideas.
In the end, creativity has always been a combination of chance and the application of known patterns in new contexts.
If you know anything about the invention of new math (analytic geometry, Calculus, etc.), you'd know how untrue this is. In fact, Calculus was extremely hand-wavy and without rigorous underpinnings until the mid 1800s. Again: more art than science.
Who knew Obi-one was just smoking and pontificating on Wittgenstein.
You can watch a rock roll down a hill and derive the concept for the wheel.
Seems pretty self evident to me
Cracks me up.
What exactly do we think that human brains do?
Said differently, what is prediction but composition projected forward through time/ideas?
This means that those compositions are also obvious because they will be predicted by just outputing the next word.
Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool at a narrow objective and see what sticks, then build on that or combine improvements.
Reading the article, that seems exactly how the discovery was made, an LLM used a "surprising connection" to go beyond the expected result. But the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.
A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation.
LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.
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.
Dystopia vibes from the fictional "Manna" people-management system. [0]
[0] https://en.wikipedia.org/wiki/Manna_(novel)
https://en.wikipedia.org/wiki/Qualified_immunity
Assuming you can still sue McDonalds I am not sure if this is a problem in the robotic llm case. I'm also trying to imagine a case where you would want to sue the llm and not the company. Given robots/llm don't have free will I'm not sure the problem with qualified immunity making police unaccountable applies.
There already exist a lot of similar conventions in corporate law. Generally, a main advantage of incorporation is protecting the people making the decisions from personal lawsuits.
That only requires someone own the ai managed McDonald's though. so long as they can't avoid responsibility by pointing to the AI I don't see why you couldn't sue them.
Police are a monopoly; nobody has a choice about which police company to use. McDonalds are not a monopoly, and many customers would prefer to eat at competitors run by entities that could be sued or jailed if they did anything particularly egregious.
I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true.
In my eye's proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by grounding it in a larger theory while with the counterexample the model has to just perform a more advanced form of search to find the correct construction.
Obviously this search is impressive not naive and requires many steps along the way to prove connections to the counterexample, but instead of developing new deep mathematics the model is still just connecting existing ideas.
Not to discount this monumental achievement. I think we're really getting somewhere! To me, and this is just vibes based, I think the models aren't far from being able to theory craft in such a way that they could prove more complicated conjectures that require developing new mathematics. I think that's just a matter of having them able to work on longer and longer time horizons.
1. They have a wide range of difficulties. 2. They were curated (Erdos didn't know at first glance how to solve them). 3. Humans already took the time to organize, formally state, add metadata to them. 4. There's a lot of them.
If you go around looking for a mathematics benchmark it's hard to do better than that.
For those in academics, is OpenAI the vendor of choice?
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.
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.
As far as academic research is concerned (e.g. this threads topic), I can't say.
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.
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.
It doesn't hurt that Lamport is exceptionally good at explaining things in plain language compared to a lot of other computer scientists.
I do not believe it will replace humans.
Why shouldn't it? Humans are poorly optimized for almost anything, and built on a substrate that's barely hanging together
(That's the first time I use that expression on HN.)
But I agree with you, especially in areas where they have a lot of training data, they can be very useful and save tons of time.
And so do humans. Gotta stand on these shoulders of giants.
But AI is supercharging Math like there is no tomorrow.
woah.
1. Erdos 1196, GPT-5.4 Pro - https://www.scientificamerican.com/article/amateur-armed-wit...
There are a couple of other Erdos wins, but this was the most impressive, prior to the thread in question. And it's completely unsupervised.
Solution - https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
2. Single-minus gluon tree amplitudes are nonzero , GPT-5.2 https://openai.com/index/new-result-theoretical-physics/
3. Frontier Math Open Problem, GPT-5.4 Pro and others - https://epoch.ai/frontiermath/open-problems/ramsey-hypergrap...
4. GPT-5.5 Pro - https://gowers.wordpress.com/2026/05/08/a-recent-experience-...
5. Claude's Cycles, Claude Opus 4.6 - https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cyc...
- It does not show an example of the new best solution, nor explain why they couldn't show an example (e.g. if the proof was not constructive)
- It does not even explain the previous best solution. The diagram of the rescaled unit grid doesn't indicate what the "points" are beyond the normal non-scaled unit grid. I have no idea what to take away from it.
- It's description of the new proof just cites some terms of art with no effort made to actually explain the result.
If this post were not on the OpenAI blog, I would assume it was slop. I understand advanced pure mathematics is complicated, but it is entirely possible to explain complicated topics to non-experts.
can we please put these ground breaking AIs to work on actual problems humans have?
For example, these machines, if scaling intellect so fiercely that they are solving bespoke mathematics problems, should be able to generate mundane insights or unique conjectures far below the level of intellect required for highly advanced mathematics - and they simply do not.
Ask a model to give you the rundown and theory on a specific pharmacological substance, for example. It will cite the textbook and meta-analyses it pulls, but be completely incapable of any bespoke thinking on the topic. A random person pursuing a bachelor's in chemistry can do this.
Anything at all outside of the absolute facts, even the faintest conjecture, feels completely outside of their reach.
Who else disproved this longstanding conjecture before the model did so, since obviously it must have been in the training data since before?