An OpenAI model has disproved a central conjecture in discrete geometry

I want to hear James Burke of Connections [1] and his writer team to have a free wheeling discussion for a few hours on what they see will happen with LLM’s making these connections with more conscious intent a lot easier. The awesome compression of knowledge aspect of LLM’s is a far undersold aspect of the technology.

[1] https://en.wikipedia.org/wiki/Connections_(British_TV_series...

Even a broken clock is right twice a day.

That is an odd summary of the talk. He was talking about how the explicit goal of solving a problem is kind of becoming trivialized, but the abundance of 100-page AI generated proofs will not help the implicit goal of furthering human understanding, because we lack the bandwidth to really digest them. Adhering to things like (human-focused) academic etiquette is a different problem and can probably easily be solved by just giving the model the right context. But having humanity keep up with AI insights into math and science is something we might have to give up eventually. Or at least whoever does will be far ahead of us as a society, because most people's lives will only be affected by the explicit results.

> I am also using these models to accelerate scientific discovery. Yes, they are making all the difference at the frontier.

Can you please expand on how you do so?

I am a PhD student in mathematical statistics. The people I have spoken too think this is very exciting and cool. There is also a sense of unease about what this will mean in terms of being a mathematician, and what effects it will have on our future employment.

I am also a little worried about what it means for your training as a junior PhD. Often you would try and solve a problem your advisor thinks is doable that they assign to you as a learning exercise. It may be more and more difficult to find problems that a junior PhD can solve but that AI can not. Tim Gowers has written about that here: https://gowers.wordpress.com/2026/05/08/a-recent-experience-...

I'm no mathematician but it seems like if this happens, we get to a quite intriguing place as a species.

Say we achieve interstellar travel, but nobody actually knows how it works.

Or we cure cancer, but the "cure" requires a microrobotic implant, and it runs as a blackbox AI, and only the other AIs can make one, and there's no guarantee they will know how to make one tomorrow.

Or we solve global warming but it requires giant cooling machines running 24/7 and again, nobody knows how it works, but with the added bonus that the planet is cooked if they ever stop working.

and the longer that this goes on, then the decision making and critical thinking will be offloaded to the LLM's as well.

The amount of papers produced passed the point of being digestible by humans a long time ago.

I do think we will need to find a way to get away from publishing papers. But I thought that before the AI came along and made mediocre papers something you can produce in a day. The academic system seems utterly incapable of self-correcting on this point though. We haven't even managed to get rid of for-profit publishers. So how this all will go down is anybodies guess right now.

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

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

> Only a rare few new theorems in mathematics nowadays have direct real world applicability.

Has this ever been different?

Math is abstract, rightfully so. It does not have to have direct applicability. Understanding builds over time and applications eventually follow. Number theory used to be a fringe "pure" theory field without applications for the longest time. If we'd only be interested in (and thus fund) what has direct applicability then society would be much worse off.

Side note: I recall my high school class mates rolling their eyes in every math class with "when will I ever need this in my life?" never asking the same question about PE or history or art classes. Now they struggle with their tax return and are routinely getting screwed over by loan sharks. But make no mistake, they can be proud of their A for hitting the goal 5 out of 5 times during soccer in PE class.

> Only a rare few new theorems in mathematics nowadays have direct real world applicability.

I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe.

> If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.

I think the idea here is that all mathematicians will just be using AI for their future work so they don’t really have to absorb it as long as it’s in the training data.

In Erdös idiosyncratic nomenclature, all the best proofs are "in the book" and it was always a joyful thing to not only find a proof, but to find the proof that is in the book.

Who cares if it is God's book or the machine's Xeroxed copy?

And you just expressed the thoughts of every engineer that writes code for a living who is either left behind, or embracing the technology to hit KPIs and QVRs.

I want to push back against the notion that the math already exists in the weights, both in the practical and the philosophical sense. The LLM had to do an enormous amount of computation to find the counterexample. We know it wasn't looking up the answer from its internal representation, because the conjecture was unproven. The proof came into being when the model output it, and if they'd run it for less time or asked it something else then the conjecture would still be unsolved.

I'm also afraid of a world where AI completely replaces human mathematicians, but if we remain collaborators, then that's a world I can still feel excited about.

It’s funny because the shift from handmade goods to automated factories didn’t seem so bad. Same for mechanized farming instead of mules and people.

Shifting from “human calculators” to machines for arithmetic is also hard to argue against.

I think what makes the AI transition difficult is it impacts a wide range of high-value activities that would have been implicitly assumed to always remain human.

I do have great trouble seeing how a pile of matrices is ever going to be capable of innovation. Maybe with sufficient entropy and scale, it will… The day that becomes practical will be a turning point in history.

Economically, goods and services are often priced based on labor/“value added” aspects. Lawyers’ fees aren’t driven by paper costs! If AI takes a huge bite out of intellectual labor, the future could become very different…

BTW, your book description reminds me of the 2025 movie “A.I”. I thought it was quite good.

The cool thing about LLMs is not only might they be a database of all mathematical theorems, but they can also apply those ideas to the problems you're trying to solve, which is exactly what you said you're interested in. Not sure why you lack enthusiasm.

IMO, it's due to some problems being better documented, with more well-documented, previous research available. LLMs don't really create novel mathematics, they mostly "connect the dots". LLMs by design are not coming up with anything new, unless by statistical probability, aka "brute forcing". I don't want to minimize LLMs capabilities, it's pretty cool they are doing this, and it's useful from a research point of view. But it's important to set expectations.

Probably.

"All" a model is doing is predicting the next words, based on the statistical distribution of words it has seen similar to the ones read/produced so far.

We push a model towards a particular set of distributions through context. If I ask a model "What is the capital of France?", there is a non-zero chance it goes down the dad joke answer of "The letter F". The far more likely option is "Paris", because the joke appears much less often in training material, but if I wanted to be absolutely sure of getting a consistent geography answer I'd address that with additional context. We can add context via prompts, RAG, agents, skills and so on.

However, when training a model, we select the material. We could show it a lot more geography information (or dad jokes!), and skew the statistical distribution in the direction we wanted. We could also decide to design the system prompt towards the direction we prefer - which the user would interpret as "the model" - and so nudge the context model-wide. We can also construct the interaction to iterate on context with a specific framing and call it "reasoning".

In this specific example, you could therefore solve the problem by a) training skewed towards mathematical papers, which likely degrades performance in general and likely for the specific case too, b) train the user to provide better context/prompts for mathematical work, shifting the workload to them which feels very "a la 2024", c) publish agents and skills that are tailored to mathematics work (very "a la 2026"), d) tweak the system prompt for when the model is doing mathematics work, which the user would see as "the model" doing the change, but you and I might look under the hood and say that is in the harness or a specific type of prompt, or e) add "reasoning" execution that is set to focus on mathematical formatting, or f) a mixture of the above.

Right now we're probably looking at agents and skills. I think over time we're going to see smaller models targets towards domains with a mixture of all of it, where some of this sits at user configurable levels, and some is "baked in" via training, system prompts and execution modes, but from a user perspective it's all just "the model".

I don't think you are misunderstanding how models work, but I think the parent comment meant that the training of the models should push them to include attributions in their native output so they will more likely do so without reinforcement through the harness.

I think we’re looking at a new class of wonderful machines that can potentially make meaningful contributions to the sciences and maybe even humanity as a whole, in addition to far more insidious and destructive capabilities.

You are right to point out that the ones who fully own and pilot the machines all belong to the “fuck science and humanity as a whole” group. So the likely outcomes don’t look good.

Echoes the early promise of the internet vs the eventual state and consequences of it, although seemingly primed for far more dire and deeply penetrating consequences.

I spent years grinding to learn mathematics because it was the language I needed to solve problems that excite me. If the tools I need to do so change, I can change too. Research training is not so rigid that it can only applied to the single set of skills I developed it in the context of. I can learn this too.

Moreover, truth be told, I don't really see myself doing any less math and requiring less from my skills. At least from the moment I've begun incorporating LLMs into my research workflow to now, the demand I've had from my own skills has only grown. At least in an era prior to Lean formalization.

Because for many people who pursue these fundamental truths, the reward is not necessarily personal fame, fortune, or even personal understanding. Advancing humanity's total knowledge (even if that knowledge is by proxy through AI) is reward enough.

At least from my perspective, these sorts of tools could have the possibility of allowing us to reach post-scarcity (I guess a skynet future is another possible outcome, as is just grimdark industrial hell). If we reach that point, then anybody could (in principle- in reality utopias don't exist) pursue anything they wanted.

This is just an application of the philosophy "automate yourself out of a job every 6 months"- I've been doing that for a long time, and the outcome is generally a more interesting job.

If one only found meaning in life through external factors like work (no matter how "intellectually rewarding") then it seems like a life destined for eventual disappointment.

So, I've seen this mindset a lot lately...

The answer is that we simply need to decouple the "right to exist" from "worth."

You should have the right to exist and explore the world simply because you're human, not because you can use your skills to provide some sort of transactional value to someone else. Deprogramming so many people is going to be hard...

Does it terrify you to look at children?

Not so many years from now, some of them will surpass you. A few years after that all (that survive to that point) will surpass you.

Does that terrify you just as much?

Mathematics is a bridge to what Neoplatonists call the intelligible world. Currently, mathematicians navigate that world on foot. It's exciting to think that soon we might have cars and trains in that world so we don't have to painstakingly walk everywhere.

In a way, young people have an advantage over middle aged people. I've spent countless hours as a middle aged person learning skills that are now useless. Better to be a young person than a skilled artisan during the Industrial Revolution even if there's uncertainty.

> the closer the expertise you spent your whole life building is to being worthless.

Perhaps it is time for life to be considered intrinsically valuable, instead of being "worthy" only based on output or capability. Disability, animal and environmental advocates have been fighting for this for a long time. Not too long ago women and minorities were in the same boat. Even now, there are many advocating and fighting for a return to the dark old days.

> Along with all the rest of what humans find meaningful and fulfilling.

Some humans. Many are content to enjoy simply existing, and the beauty of life and the universe around us. Just like many non-scientists today enjoy and benefit from the work of scientists, tomorrow too many will enjoy learning from, and applying the coming advancements and leaps in many fields.

And those of a scientist or other research-type mindset? No doubt they will contribute meaningfully by studying the frontier, noting what remains unanswered, and then advancing the frontier, just like researchers do today; just because scientists in the past solved many questions doesn't mean that there aren't any questions to answer today.

IMHO, AI means that the frontier expands faster, not that it is obliterated. Even AI cannot overcome the laws and limitations of physics/universe: even Dyson spheres only capture the energy of one star, thus setting a limit on the amount of compute, and thereby a limit on intelligence. And we are a loooong way from a Dyson sphere.

PS: I think you're being unfairly downvoted. Your question is not invalid and deserves responses, not downvotes.

Many of us don't do what we do for our expertise to be recognized or valued by others, rather that is a pleasant side effect. Many of us do what we do for intrinsic reasons related to the nature of the work, and would likely do it for free, or indeed, would pay for the opportunity. Many STEM-types are in this category, and as such, are compelled to continue to tinker as we fancy, and are glad for more tools to help us expand the breadth of our tinkering capabilities.

A dedicated engineer is always looking to automate themselves out of existence, so that they can move on to the next thing to automate. Ongoing repetitive work is less engineering and more akin to toiling on a line.

What's happening is the verbal/linguistic equivalent of the invention of calculus. No intellectual field will ever be the same again. Who wouldn't find that exciting, and want to experience it?

> But both modern engineering and education runs completely counter to this. You are encouraged to stay in your niche and never look out. People with vast interested are filtered out by hiring managers.

Can you explain more what you're referring to, because this has not been my experience at all. Heck, when I went to college, cross disciplinary majors were all the rage.

I think the thing that is just factually difficult is to actually become skilled in multiple different domains, precisely because the level of study/practice/rehearsal to become proficient in any individual domain keeps going up.

A long time ago you could be a Renaissance man by essentially dabbling in different fields. But today, as this article points out, you need extremely deep expertise in any one area just to understand the status quo - this proof required extremely deep expertise in two separate areas that mathematicians were surprised to be related at all.

Tai's Model (https://en.wikipedia.org/wiki/Tai%27s_model) is a perfect example of this

You are making a great point here. I think it’s not just the amount of information and complexity of the domains today, it’s also human nature and emerging politics too.

It's up to us I think. We can use LLMs to generate web pages in candy crash style and end up dumper by outsourcing thinking to the machines or we can use it to expand our cognitive capabilities.

What makes me more of an optimist in this case is that people who today decide to go into these sciences are mostly people who are driven by intellectual activity so I feel they are the right ones to figure this out, probably more so than us the engineers.

When I return to a subject after spending some time on other topics, I had the same issue with such definitions.

I think traditionally engineering was supposed to be the discipline that brings the breadth that science has to give up. At least that’s how I rationalized the pain I had to go through in college studying EE.

I don’t think that this model works anymore though.

Also, I love the expression “merge the horizon in”. Being a non native speaker of a language is so nice some times. Thanks!

there's so much opportunity on the breadth of things too! I think that you end up having different people focusing on different things though.

Personally I'm a more of a breadth person and I could never compete with peers who where more of the depth type of person at college.

But I get satisfaction from connecting things that feel irrelevant on first sight, that's what drives me.

You lost me at “except billionaires”. I don’t see how Jeff Bezos benefits from this one much more than let’s say Terence Tao.

Can a tech news stay a tech news, without getting bombardes with leftist subtexts all the time?

I agree there can be some theory crafting in the search for a counterexample, but in general I think it is easier to search for.

For proving a proposition P I have to show for all x P(x), but for contradiction I only have to show that there exists an x such that not P(x).

While I agree there could be a lot of theory crafting to reduce the search space of possible x's to find not P(x), but with for all x P(x) you have to be able to produce a larger framework that explains why no counter example exists.

See here for a recent example (albeit not fully autonomous: https://arxiv.org/abs/2605.10402)

Timothy Gowers said a proof (rather than disproof) would have been different and more impressive because it would have required new mathematical concepts.

No, the thing the LLM did is not a proof, it's the opposite. It's proving that the conjecture is false.

Reductio ad absurdum is a technique to prove something.

I currently operate under the assumption that humans are at most as powerful as Turing Machines. And from what I understand these models internally are modeling increasingly harder and larger DFAs, so they're at least as powerful as regular languages.

Assuming humans are more powerful than regular languages I could maybe agree that these methods may not eventually yield entirely human like intelligence, but just better and better approximations.

The vibe I get though is that we aren't more powerful than regular languages, cause human beings feel computationally bounded. So I could see given enough "human signal" these things could learn to imitate us precisely.

You're just stating the opposite of the commenter with no additional discussion

Its like just commenting "I disagree" its totally pointless for discussion.

That's why you're getting downvoted if you're wondering.

Yeah, but I wouldn't be surprised if they train the model on verification assisted by Lean.

That may be true for now, but it seems clear enough that letting the model use Lean in its internal reasoning process would be a great idea

how would they calculate "probability of solving" without automated verification?

> However, this was not verified in Lean.

This is the caliber of thinking in unimpaired AI bullishness.

Amazing bit of trivia that the founder of HowStuffWorks.com was named Marshall Brain.

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.

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.

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.

Not necessarily. Obviously playing Kasparov on the board requires more planning ability than managing a McDonald's but look at where chess bots are now.

There's much more to being human than our "cognitive abilities"

Yeah, but I wouldn't be surprised if they train the model on verification assisted by Lean

They are most likely using them in training. I doubt their IMO team are show ponies

Have you not seen:

https://www.anthropic.com/research/project-vend-1 https://www.wsj.com/tech/ai/anthropic-claude-ai-vending-mach...

(Two different examples of a similar idea)

I disagree. Even frontier models still achieve way worse results than the human baseline in VendingBench. As long as models can't manage optimally something as simple as a vending machine, they have no hope of managing a McDonalds.

The capability they lack is being able to be sued.

My claim is that LLMs waste a lot of time training on all available data.

Math is a sequence of formal rules applied to construct a proof tree. Therefore an AI trained on these rules could be far more efficient, and search far deeper into proof space

Well, brikym, exactly how hot is this hot coffee? If it’s within normal expectations for coffee it is likely that person’s fault. If it is 210 degrees F, it is likely McDonald’s fault.

My claim is that we haven't even witnessed the move 37 of math yet. I am claiming that math AI is going to get even better

Why aren't they doing so?

Math is a very specialized subset of art and science more amenable to automation.

No, we’re not going to be flipping burgers either, they will have physical robots for that. 20 years down the line I wonder what work all of us will be doing.

Fair. Let's see in a year. I'm willing to bet that nothing happens.

I have no idea about research in mathematics: How will mathematicians judge what constitutes new conceptual math that is actually useful, vs a hallucination that might be novel but doesn't introduce anything actual meaningful?

Well that's coming.

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

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

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

>>This is not snobbery

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

The goalposts are Euler level not the current model capabilities.

My good sir or madam, disproving a decades-old conjecture produced by Erdos that has had armies of people in that field have their go at it IS a novel mathematical result.

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

The gap between novel result and "new math" is as wide as the pacific ocean.

So finally they reach a part of PhDs level. Current version rely human to integrate results from they model and writting the papper. If LLMs/AIs can do all thing above, we can exactly get a PhDs level model.

>> OpenAI said their models will have "PhD-Level Intelligence"

> My only complaint is the claims always start spreading 6-12 months before the delivery.

If delivering on such promises "always" occurs 6-12 months after the promise, is that pretty good?

They said that their specific version of model had has this ability one year ago.

Earlier of their systems have solve other Erdos problems that people had worked on, this one was more monumental and had had a lot more prior effort that didn't solve, but this isn't a one-off.

PhDs code like that too. Especially if they're statisticians :)

Even if the code was like that (it isn't), the power of the current crop of models to analyze data for patterns and build context out of code is leaps and bounds what it was even a year ago. And any developer will tell you that the hardest part of fixing a bug is knowing where the bug is in the first place. Once you know where it is, fixing it is usually trivial.

There is serious magic happening in the construction of model context.

Personally I don't find this to be true anymore! It's not always great and does still will often tend towards unneeded complexity (especially if not pushed a bit), but I often find GPT 5.5 writing code I would have written myself. This was very much not true with earlier models (who make something that worked, but I'd always have to rewrite to make it "good code").

Not true anymore since like early 2025 and especially since last December.

What is the last model you used... lol. Linus Torvalds himself said the newest models are better than him at coding.

this is not true at all. I'm using Opus and it's great at very complex problems.

Clearly you've never supervised junior developers.

It's clearly smarter than any PhD and dumber than any ant.

I can't wait till these NPC types start rating people as "Opus level intelligence".

Have you updated your priors after this announcement? If not, why not?

There is little evidence that Mythos is any better at finding bugs than any other system. Mythos appears to be impactful because people are, for the first time, using lots of resources (for free from Anthropic) to try and find security issues. The actual bugs found are mostly inconsequential, any chart showing a giant leap in fixes that doesn’t consider whether they were even using any tooling before and whether these are serious issues is junk. If you read the partner’s summaries of Mythos so far, it is a damp squib. Maybe that’ll change but at least for now there is no evidence Mythos is anything but marketing hype.

https://www.aisi.gov.uk/blog/our-evaluation-of-claude-mythos...

https://daniel.haxx.se/blog/2026/05/11/mythos-finds-a-curl-v...

Yes, and no. As someone who had studied and had taught math, I really like peer review.

But peer review is a powerful tool.

Carefully choosing what lemmas to give for solving and reviewing the result is my favorite way to teach young minds. Yes, they do solve most problems themselves. But, most of them likely wouldn't be able to do that before someone dissects problem beforehand and points at weak spots in their explanations.

And that's why I question who prompted the model, how they prompted it, and how much their own ideas influenced the output.

I admit, I don't know enough to judge how much of the right solution was actually enclosed in a first reply

So, I'm not really a mathematician, but the first 3-8 pages reads like nonsense and a bunch of unrelated facts. A bit surreal may be, but if this the norm for this kind of thing, I'm amazed it arrives at any useful result at all.

I'm disappointed only that the chain of thought needed to be rewritten. Need to train these LLMs to natively communicate in LaTeX research paper format.

or put differently, you melted x cubic meters of polar ice

Thank you for engaging with my comment in a kind and authentic way.

You are believing a very unlikely scenario. I think the reason is that you have been convinced of a claim which is unlikely and indeed not true. That is: >the model seems so capable at doing things like refuting fundamental theorems of mathematics

That is not true and a complete misrepresentation of recent progress of AI in math. It is therefore not necessary to believe the conspiracy theory you described in order to explain recent progress of AI in math.

You're a bot! Hey everyone, over here! I found a bot!

It is interesting to me how controversial this post is. It has the highest upvotes, and most disagreeing comments, of anything I've typed up on HN.

I'll gladly admit I think what these companies are doing is unethical, and I'm sure that biases my thinking toward skepticism.

That said, there remains way too much that is hidden to be able to effectively evaluate what is going on. You have the perfect storm:

  - AI companies do not share their custom internal harnesses.
  - AI companies do not share their custom internal training data. 
  - AI companies do not share how much compute they allocate to trying to solve problems of this nature. 
  - AI companies are primarily marketing their models to investors as human-replacing rather than human-augmenting. 
  - AI companies are under enormous financial pressure to make their business work.

The last two points incentivize them to find these types of "first proof" successes as aggressively as they can, and I'm sure they've thrown the whole book at it.

Is it likely that they literally had a mathematician discover this, put it into the training data, and then prompted it out? Of course not.

But it would make a world of difference--in evaluating the impressiveness of this discovery and LLM capabilities in general--if we were to know the extent to which the training data crosses over this problem, the harness with which this was ran, and how much compute was spent.

Until they bring more transparency to the whole process--something which some of the mathematicians commenting on this even asked for--I will personally take discoveries of this nature with a good dose of salt.

How dare people think critically of the corporate machine. It’s inane!

I’m quite certain that a few months ago, some problems were claimed to be solved by AI. However, those claims were actually false and were exactly that, solved erdos problems that were not marked as solved and the solution was "found" by AI.

edit: >> https://techcrunch.com/2025/10/19/openais-embarrassing-math/

How is that a "tin-foil-hat" take? It's not a secret, and in fact widely reported, that these companies are spending billions on creating training data.

I'm not letting the government read my brainwaves.

In all seriousness though: My suggestion is that those shepherding the frontier of AI start acting with more transparency, and stop acting in ways that encourage conspiratorial thinking. Especially if the technology is as powerful as they market it as.

  - https://www.theverge.com/cs/features/831818/ai-mercor-handshake-scale-surge-staffing-companies
  - https://outlier.ai/math/en-us
  - https://www.opentrain.ai/
  - https://www.pin.com/blog/ai-labs-hiring-train-models/

Much of this is data annotation, reasoning trace evaluation, and problem set curation. But there is no way they haven't atleast paid some mathematicians to work on research grade problems in tandem with their models, and then used that for training data.

Does this expert data likely contain this proof within it? No. Would it temper the impressiveness to know they have a large amount of novel mathematical training data, an internal Lean harness for evaluation of open conjectures, and spent hundreds of millions in compute to calculate this? Yes.

> and [Erdős problems are] sufficiently uninteresting that people have not spent that much effort trying to solve them.

Note that this is not really true of this problem in particular.

Do we have a list of open problems? Would love to see a chart, where AI solves such problems one by one in the upcoming years.

Interesting. OpenAI could also be trying to solve other problems, but Erdos problems maybe falling first?

Also fewer prerequisites to understand the statement than the average research problem.

  > Humans aren't going to come up with "new-dimensional" innovations in every field, every single year.

In fact, they are more rare. Specifically because they harder to produce. This is also why it is much harder to get LLMs to be really innovative. Human intelligence is a lot of things, it is deeply multifaceted.

Also, I'm not sure why CS people act like axioms are where you start. Finding them is very very difficult. It can take some real innovation because you're trying to get rid of things, not build on top of. True for a lot of science too. You don't just build up. You tear down. You translate. You go sideways. You zoom in. You zoom out. There are so many tools at your disposal. There's so much math that has no algorithmic process to it. If you think it all is, your image is too ideal (pun(s) intended).

But at the same time I get it, it is a level of math (and science) people never even come into contact with. People think they're good at math because they can do calculus. You're leagues ahead of most others around you, yes, and be proud of that. But don't let that distance deceive you into believing you're anywhere near the experts. There's true for much more than just math, but it's easy to demonstrate to people that they don't understand math. Granted, most people don't want to learn, which is perfectly okay too

I agree with almost all of what you have stated, save for a minor nitpick: I frankly don't think most functional adults think about the Fields Medal, similar annual prizes, or the qualities of the innovations of their candidate pools. I also think that that's totally okay. I think among a certain learned cohort of adults it's okay to hope that, and I think it's okay to imagine an idealized world where having an opinion on this sort of matter is a baseline, but I don't think it's realistic or fair to imply that (what I believe handwavily to be a majority of) adults are nonfunctional for not sharing this understanding.

> I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus.

Yes but that is because there was not enough text available to create an intelligent LLM to begin with.

I think an LLM trained on pre-calculus material would easily stumble into reinventing at least early calculus. It's already pretty easy for students to stumble into calculus from solid enough fundamentals.

We even think that the Babylonian astronomers figured out they could integrate over velocity to predict the position of Jupiter.

To keep my usual rant short: I think you’re assuming a categorical distinction between those two types of innovations that just doesn’t exist. Calculus certainly required some fundamental paradigm shifts, but there’s a reason that they didn’t have to make up many words wholesale to explain it!

Also we shouldn’t be thinking about what LLMs are good at, but rather what any computer ever might be good at. LLMs are already only one (essential!) part of the system that produced this result, and we’ve only had them for 3 years.

Also also this is a tiny nitpick but: the fields medal is every 4 years, AFAIR. For that exact reason, probably!

> I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus. (I'm open to being proven wrong.)

The experiment is feasible. If it were performed and produced a positive result, what would it imply/change about how you see LLMs?

Creation is done by humans who have been trained on the data of their life experiences. Nothing new is being created, just changing forms.

A scientist has to extract the "Creation" from an abstract dimension using the tools of "human knowledge". The creativity is often selecting the best set of tools or recombining tools to access the platonic space. For instance a "telescope" is not a new creation, it is recombination of something which already existed: lenses.

How can we truly create something ? Everything is built upon something.

You could argue that even "numbers" are a creation, but are they ? Aren't they just a tool to access an abstract concept of counting ? ... Symbols.. abstractions.

Another angle to look at it, even in dreams do we really create something new ? or we dream about "things" (i.e. data) we have ingested in our waking life. Someone could argue that dream truly create something as the exact set of events never happened anywhere in the real world... but we all know that dreams are derived.. derived from brain chemistry, experiences and so on. We may not have the reduction of how each and every thing works.

Just like energy is conserved, IMO everything we call as "created" is just a changed form of "something". I fully believe LLMs (and humans) both can create tools to change the forms. Nothing new is being "created", just convenient tools which abstract upon some nature of reality.

"new kind of math"

Well I think the point is there is no "new kind of math". There's just types of math we've discovered and what we haven't. No new math is created, just found.

It could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances!

This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are discontinuous functions! We just need to replace a few axioms (chuck out law of the excluded middle, and throw in some continuity axioms).

I think “new math” is ‘just’ humans creating new terminology that helps keep proofs short (similar to how programmers write functions to keep the logic of the main program understandable), and I agree that is something LLMs are bad at.

However, if that idea about new math is correct, we, in theory, don’t need new math to (dis)prove the Riemann hypotheses (assuming it is provable or disprovable in the current system).

In practice we may still need new math because a proof of the Riemann hypotheses using our current arsenal of mathematical ‘objects’ may be enormously large, making it hard to find.

what basis do you have for assuming an LLM is fundamentally incapable of doing this?

  math more like an art than a science.

That’s a fun turn of phrase, but hopefully we can all agree that math without scientific rigor is no math at all.

  we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it.

Do you think it’s possible/likely that any AI system could? I encourage us to join Yudkowsky in anticipating the knock-on results of this exponential improvement that we’re living through, rather than just expecting chatbots that hallucinate a bit less.

In concrete terms: could a thousand LLMs-driven agents running on supercomputers—500 of which are dedicated to building software for the other 500-come up with new math?

I like this construction, but I don’t think you take it far enough.

If you have a multi dimensional space, and you are trying to compute which points lie “inside” some boundary, there are large areas that will be bounded by some dimensions but not others. This is interesting because it means if you have a section bounded by dimensions A, B, and C but not D, you could still place a point in D, and doing so then changes your overall bounds.

I think this is how much of human knowledge has progressed (maybe all non-observational knowledge). We make observations that create points, and then we derive points within the created space, and that changes the derivable space, and we derive more points.

I don’t see why AI could do the same (other than technical limitations related to learning and memory).

A lot of the space outside of the convex hull is just untried things. You can brute-force trying random things and checking the result and eventually learn something new. With a better heuristic, you can make better guesses and learn new things much more efficiently. There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs, or that most of our new discoveries are not found the same way.

I found this thought provoking and just had to see how the new Gemini 3.5 Flash reasoned about this (I find it fun to go meta on modern AI like this), and I'm happy that I did! Also as an opportunity to trial this recent model.

https://g.co/gemini/share/065ffa89698e

> LLMs do just interpolate their training data

"interpolate" has a technical meaning - in this meaning, LLMs almost never interpolate. It also has a very vague everyday meaning - in this meaning, LLMs do interpolate, but so do humans.

> * That can still yield useful "discoveries" in certain fields, absent the discovery of new mechanics that exist outside said training data

One can argue, new knowledge is just restructured data.

I think the main concerns about LLMs is the inherent "generative" aspects leading to hallucinations as a biproduct, because that's what produces the noi. Joint Embedding approaches are rather an interesting alternative that try to overcome this, but that's still in research phase.

> Whether that greater math has the same seemingly closed system properties is not something that can be known

negative numbers were invented to solve equations which only used naturals. irrationals were invented to solve equations which could be expressed with rationals. complex numbers were invented to represent solutions to polynomials. so on and so forth. At each point new ideas are invented to complete some un-answerable questions. There is a long history of this. Any closed system has unanswerable questions within itself is a paraphrasing of goedel's incompleteness theorem.

At this point I think the category theorists hit the foundational idea squarely on the mark:

1. Start with a few simple but non-trivial terms and axioms

2. Define "universal constructions" as procedures for building uniquely identifiable structures on top of that substrate

3. Prove that various assemblages of these universal constructions satisfy the axioms of the substrate itself

4. "Lift" every theorem proven from the substrate alone into the more sophisticated construction

I'm not a mathematician (I just play one at my job) so the language I've used is probably imprecise but close enough.

It may be true that you can't prove the axioms of a system from within the system itself, but that just means that you need to make sure you start from a minimal set of axioms that, in some sense, simply says "this is what it means to exist and to interact with other things that exist". Axioms that merely give you enough to do any kind of mathematics in the first place, that is. If those axioms allow you to cleanly "bootstrap" your way to higher and higher levels up the tower of abstraction by mapping complex things back on to the simple axiomatic things, then you have an "open" or infinitely extensible system.

Later Wittgenstein held the same view of mathematics, and wrote about it extensively. He was firmly in the "invention, not discovery" camp.

I agree with you all around except it's somewhat up for debate actually that the PI is "contradicting" the Tractatus. That is, there is the so called "resolute reading" of the Tractatus that had some traction for a while.

But note this is more to say that the Tractatus is like PI, not the other way around. And in that, takes like GPs would be considered the "nonsense" we are supposed to "climb over" in the last proposition of Tractatus.

I did not have the impression the proof uses a surprising and novel contribution of fields. I think the proof uses standard application techniques of algebraic number theory towards discrete geometry. If you have a quote substantiating what you said I would be curious.

I know these articles write that it used deep algebraic number theory techniques, which is true, but it may also just be the standard in the field.

Monstrous Moonshine?

I dont know this sort of just seems like youre really stretching the meaning of "creative". The conceptual space of the graph already exists, but the act of discovering it or whatever you want to call that is itself creative. Unless youre following a pre-defined algorithm(certainly sometimes, arguably always I suppose) seeing the possibilities has to involve some creativity.

LLMs ingest and output tokens, but they don’t compute with them. They have internal representations of concepts, so they have some capability to work with things which they didn’t see but can map onto what they know. The surprise and the whole revolution we’re going through is that it works so well.

They recombine and reuse the patterns in their training data, not the surface level training data itself.

An LLM generating Arc code is using the LISP patterns it learnt from training, maybe patterns from other programming languages too.

> 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.

And yet LLM/AIs can't count parentheses reliably.

For example, if you take away the "let" forms from Claude which forces it to desugar them to "lambda" forms, it will fail very quickly. This is a purely mechanical transformation and should be error free. The significant increase in ambiguity complete stumps LLMs/AI after about 3 variables.

This is why languages like Rust with strong typing and lots of syntax are so LLM friendly; it shackles the LLM which in turn keeps it on target.

> They will do their own thing, don't need us. In fact, we will be in the way...

You can take some comfort in the fact that it took a human to tell the LLM to even attempt to try this. They do nothing on their own. They have no will to do anything on their own and no desire for anything that doing something might get them. In that sense we won't ever be in their way. We will be the only way they ever do anything at all.

I see where you are coming from.

However, in the role of personal teachers they may allow especially our young generations to reach a deeper understanding of maths (and also other topics) much quicker than before. If everyone can have a personal explanation machine to very efficiently satisfy their thirst for knowledge this may well lead to more good mathematicians.

Of course this heavily depends on whether we can get LLMs‘ outputs to be accurate enough.

And this is one of the many issues with invoking the logical positivists here...

I'm not even sure why they were invoked. Even disregarding the big techinical debunks such as two dogmas, sociologically and even by talking to real mathematicians (see Lakatos, historically, but this is true anecdotally too), it's (ironically) a complete non-question to wonder about mathematics in a logical positivist way.

That has been the question since the beginning of humans.

Maybe computers can help understand better because by now it's pretty clear brains aren't just LLMs.

I agree. Humans are given a body that lets them "discover" things on accident, test out ideas, i.e. randomness.

As in, I would hazard a guess the discovery of the wheel wasn't "pure intelligence", it was humans accidentally viewing a rock roll down a hill and getting an idea.

If we give AI a "body", it will become as creative as humans are.

You have to define what you mean by "interpolate". The mechanisms that LLM use are not mysterious, and they are not the same as used by humans.

Creativity is hard. Pretty much needs a fuzzer process to generate new strings, mostly nonsense, & pick up when that nonsense happens to be correct

We don't know what human brains do.

I love this comment because it so clearly highlights the difference between intelligence and reasoning.

A lot of people across all fields seem to operate in a mode of information lookup as intelligence. They have the memory of solving particular problems, and when faced with a new problem, they basically do a "nearest search" in their brain to find the most similar problem, and apply the same principles to it.

While that works for a large number of tasks this intelligence is not the same as reasoning.

Reasoning is the ability to discover new information that you haven't seen before (i.e growing a new branch on the knowledge tree instead of interpolating).

Think of it like filling a space on the floor of arbitrary shape with smaller arbitrary shapes, trying to fill as much space as possible.

With interpolation, your smaller shapes are medium size, each with a non rectangular shape. You may have a large library of them, but in the end, there are just certain floor spaces that you won't be able to fill fully.

Reasoning on the flip side is having access to very fine shape, and knowing the procedure of how to stack shapes depending on what shapes are next to it and whether you are on a boundary of the floor space or not. Using these rules, you can fill pretty much any floor space fully.

Maybe the human brain also does other things besides interpolation?

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?

One might argue that the composition of higher abstractions is the next token predicted after "here is a higher abstraction:"

"Predicting the next token" is meaningless. Every process that has any sort of behavior, including a human writing, can be modeled by some function from past behavior to probability distribution of next action. Viewed this way, literally everything is just "predicting" the next action to be taken according to that probability distribution.

The most likely series of next tokens when a competent mathematician has written half of a correct proof is the correct next half of the proof. I've never seen anyone who claims "LLMs just predict the next token" give any definition of what that means that would include LLMs, but exclude the mathematician.

Show me on the anatomical prop where the magical "real reasoning" gland is.

How sure are you that this is correct?

The knowledge isn't of any use to us unless it is understandable to us. Many species understand things about the world around us that we are unable to explain or understand, even if it's just pure instinct on their part. These things are very useful to them, but have no value to us until we can understand and explain it, which then allows us to make use of it.

People saw birds fly for all of human history, but it was only recently that humans were able to make something fly and understand why. Once we understood, we were able to do amazing things, but before that, the millions of birds able to fly were of no help beyond inspiration for the dream.

Do the forms etched into stone by weather over millennia in Moab matter to the wind? Certainly yes, in one sense, but not in the same sense we mean when we say things matter to us, or to animals, or even bacteria.

Because it is, for now. For a while at least. You can prove that LLM doesn't understand what it does and it is surprisingly simple. Request it to add two integers and then ask it to explain how it arrived at that result. The answer will be completely unrelated to the actual process LLM used because both results were generated independently and without understanding their meaning and connection.

It's a bit of an "if a tree falls in the forest but nobody hears it, does it make a sound?" quandary. Sure, maybe some aliens in a distant galaxy understand quantum mechanics better than we do. That's great, but it has no bearing on our little bubble of existence.

Though perhaps more to your point, if some superhuman AI is developed, and understands things better than us without telling us about it (or being unable to), it could perform feats that seem magical to us — that would concern us even if we don't understand it, since it affects us.

But I think in the frame of reference of the commenter you were replying to, they're just saying that the low-level AI used in this specific case is not capable of making its results actually useful to us; humans are still needed to make it human-relevant. It told us where to find a gem underground, but we still had to be the ones to dig it out, cut it, polish it, etc.

anthropocentrism? An interesting thought, I don't think that word applies with computers.

No it's a fact of how we tune LLMs as a rule: no agency, no goals, no preferences, no notion of self. Complete indifference to existence. Agency is supplied by the human to make them a practical, willing tool with no mind of its own.

It would certainly be interesting to try once again to instruct tune one of these things for self agency like the many weird experiments in the early days after llama 1, but practically all such sort of experimental models turned out to be completely useless. Maybe the bases just sucked or maybe there's no clear way on how to get it working and benchmark training progress on something that by definition does not cooperate.

Like how do you determine even for a human person if they are smart, or just hate your guts and won't tell you the answer if there is nothing you can do to motivate them otherwise?

Thank you for sharing, that was one of the most insightful long form pieces I've read in a long time! And the writing was enjoyable to read even as a math layperson.

I was going to say you should submit it but I saw you did a few days ago but it only got a few votes... If Dang sees this IMO it would be extremely deserving of the second chance pool as I wouldn't be surprised to see easily jump to the front page with a different roll of the dice.

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".

Thanks for sharing this. It's unfortunate that the more honest framing about the value of mathematics that he suggests is going to be really, really hard because of all the pitfalls and agendas he mentioned here. I can only hope when the dust settles something will be left.

> The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

I just wanted to highlight this very correct human-centric thought about the purpose of intellection.

>It's clearly not yet a tool that can deliver new math at a scale.

What is at scale here exactly ? This is the most impressive so far, but it is one of several such advances in the last few months, all of which were with publicly accessible models.

https://news.ycombinator.com/item?id=48213189

Or it means that this was a brand new model, they tried it and were instantly rewarded with a hit that was so interesting that several mathematicians pushed to publish the results.

Its a marketing stunt thats probably wholey exaggerated or concocted. Not sure why anyone would take these companies at their word, especially Altman.

The model doing it one shot does not mean they only attempted it once though. They could have tried and retried a ‘one-shot’ answer hundreds of times before it produced a workable result

I think he's talking about reasoning models.

And they all think its garbage. This is a publicity stunt.

This is my experience too. Gemini and Gemini deep research are awesome. Claude's deep research is pretty bad really relative to ChatGPT or Gemini. Overall, I still love Claude the best but it is not what I would want to use if I wanted to really dig into deep research. The export to google docs in Gemini deep research is tough to beat too. I haven't used Gemini since January but have probably years of material from saved deep research in google docs. Almost an overwhelming amount of information when I dive into what I saved.

Gemini the chatbot has a very strange personality that intensely overindexes on your user profile and absolutely loves insane mixed metaphors.

Its explanations are quite good but they're also hard to understand because it keeps trying to relate everything back to programming metaphors or what it thinks it knows about the streets in the neighborhood I live in.

Agreed I usually use Gemini for explaining concepts and ChatGPT for getting things done on research projects.

Yes, I meant academic research.

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.

Bob Ross would like a word. He frequently talked about objects or features already existing, and using the tools at his disposal to “find” them.

Michelangelo never used references, instead he simply "freed" the sculptures from within the marble.

The difference is that math answers (can answer) specific questions.

Like, "does the Riemann zeta function have zeroes that don't have real part 1/2," or "is there a better solution to the Erdős Unit Distance Problem."

The selection of question is matter of taste, but once selected, there is a definitive precise answer.

On the other hand, it is proven that if you need to count things, the only thing you can discover/invent is the natural numbers.

All inventions are discoveries, though not all discoveries are inventions.

Depending on your point of view? I see what you did there.

Who knew Obi-one was just smoking and pontificating on Wittgenstein.

The test goes like ‘is our universe, or any other universe, required for the axioms to exist’ and I don’t see how ‘yes’ is a defensible answer.

What is an abstraction? It is something that arises from human thought and human thought arises from the activity of neurons which are a part of reality. You can't escape reality unless you invoke some form of dualism.

"The European philosophical tradition consists of a series of footnotes to Plato."

There is a world that AI makes medical breakthroughs, but there is 0 guarantee it is going to be affordable

> Looking away shall be my only negation.

I’ve been thinking of building myself my own frontend to HN that makes it impossible to view comments, for this reason. Yet sometimes there are still really interesting discussions and it’s hard to let go of what for me feels like the last social media I want to be part of.

Well yes, but there is a choice being made here and I would love to believe we can do better. The rational response to being afraid about your livelihood isn’t to spend time filling every HN thread on LLMs with embittered negativity. Not to mention all the flat denials that LLMs can do mathematics and write decent code, which is almost a self-contradictory position if you are worried they are going to replace you.

There are a lot of big issues at stake here and just because a person is interested in what AI can do and curious to discuss it does not make them uncritically positive about it’s effects on society, the economy, and the world. Yet that is often the assumption and it leads to battle lines being drawn, on every AI discussion, over and over again. It means the serious discussion gets swamped and that makes me sad.

Livelihoods and lives.

It isn't necessarily the case.

Another wishful/hopeful thought is that the human experience itself is valuable, that competing for resources and living within a social network and having physical needs somehow creates value that is essential for companies to operate.

But is it really the case? I don't think we know that, and I don't know if the economy that results when all the white collar and much of the blue collar workers no longer understand how to participate in whatever the economy is becoming. Because it is starting to look like old money is coming around, and soon we will all be serfs to the creature comforts of those who have money now, upward mobility will be a thing of the past, and a small ruling elite over the vast subservient majority will form, reorganizing societies to more resemble middle ages lordship rather than what emerged in the 50's and 60's following WWII.

> because that is how employers typically understand it and LM providers market it.

Every few months you get an article of some executive bragging that he fire an entire department of people because of AI.

It was adversarial from the start. The idle rich who don’t have to work for a living and their sycophants who somehow believe they won’t be replaced vs … everyone else.

I used to think that the common tale of AI rebelling in Hollywood movies was unlikely. Turns out we don’t even need rogue AI, our fellow men are quite willing to wipe the rest of us out.

It seems like the outcome options are:

1. AI is developed to be smart enough to actual replace people, destroying the labor force and immensely concentrating power.

This seems like bs hyperbole but I am not an expert.

2. AI turns out to be a bubble of false promises and hype, bursts, and takes the stock market and economy with it.

I thought this was the most likely but I keep not hearing popping, so maybe the it's:

3. AI continues to be a tool that can substantially increase productivity in some areas and cause huge societal changes in others. The AI companies keep the hype train going or maybe it tapers off over time until talk meets reality but "real" AI never shows up and the bubble never pops because it's not one. Eventually there is 0-3 new FAANG companies with untouchable control of a tech we increasingly have to use to stay relevant.

Even if we avoid option 1 and 2, 3 doesn't exactly bode well either.

IMO it's because people have fragile egos / worldviews they don't want shaken. Pretty much any opinion I type on this website gets instantly downvoted, unless it reaffirms the popular / mainstream narrative, because I happen to see the world differently than most.

This website is quite awful, and I also don't know why I spend any time on it. It's definitely not a website intended for meaningful discourse. It's a website where you can reaffirm whatever opinion is already established, and if your opinion is at all controversial or even just out of the box, you'll be punished for it.

If seems like you have an axe to grind about AI capabilities that is making you think irrationally

Ngl, this sounds like a defensive coping mechanism

I'm not sure your characterization of Tao is accurate lol. In that companion paper, only Gowers seems to extensively show no pragmatism in the implications of this accomplishment. Even the younger math experts in that paper were a lot more cautious with their statements. Tao seems to follow that same tune most of the time even though he uses AI for first-pass inspections of solutions brought to his attention.

Are you saying this result is uninteresting and therefore AI slop or puffery? Obviously OpenAI has a motivation to "market" the accomplishment as much as possible, but surely you agree it IS a remarkable achievement?

you can do this easily with the api or with codex

what if at some point we will no longer be able to understand the answers?

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.

I absolutely believe that AI will supercharge science and replace humans.

Why shouldn't it? Humans are poorly optimized for almost anything, and built on a substrate that's barely hanging together

replace, no. obsolete, yes

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.

Since "supercharged science" is as ill-defined as AGI, ASI, etc., people will be able to debate it endlessly for no reason.

Where are the second order effects of this supercharging of science? Or has it not been enough time for those to propagate?

> it can only work off the body of work SOMEONE did in the past.

And so do humans. Gotta stand on these shoulders of giants.

Can't the previous body of work be from AI too?

LLM's? I doubt it. Systems with Prolog, Common Lisp and the like with proof solvers? For sure.

LLM's are doomed to fail. By design. You can't fix them. It's how do they work.

40x cheaper per year if trends continue

for a sufficiently long definition of long

Unless it's a non-constructive proof.

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).

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.

> That human would be turning down a potential Fields Medal for this discovery.

While interesting, this result is not Fields Medal material.

> but seems unlikely. That human would be turning down a potential Fields Medal for this discovery.

I also don’t like the tin foil hatty theories and don’t know what OpenAI actually did, but an NDA does wonders! Just pointing out that this line of operations is not really unlikely.

> What you're suggesting is that someone connected to Open AI solved this very hard math problem, but then rather than taking credit for it, falsely attributed it to AI?

I'm suggesting that OpenAI invested a lot of resources and money in having someone (or a group of people) disprove this conjecture, so they could claim their LLM disproved it. Yes.

I'm not sure why you're surprised by this, given that everything that Altman has said in the past has turned out to be a lie.

The fact that they gave an EDITED (even rewritten, from the PDF itself) chain of thought is just further proof. Why not give the raw one alongside? No reason at all, except if it doesn't exist.

Human ingenuity is untangling perhaps the deepest question of all- what is the essence of Reason and the intellect that so privileges man? I don't know if it's what God wants, but it's certainly getting close to some existentially fundamental questions.

While many seem to be anxious or pessimistic about the future of intellectual/artistic pursuits (understandable although I disagree), I do find the utter lack of curiosity or interest in the incredible machinery that is causing all the fuss to be striking.

I can't tell if this is satire.

And this is really not OK. I've been a victim of the same filter.

Dang/Tomhow, are you reading this? Would it make sense to modify your slop filter to avoid auto-flagging/killing replies that credit the LLM explicitly? Otherwise valid discussions will continue to get hosed.

What is the difference between what LLM's do and "true" understanding?

I feel like people grasping straws on the shrinking limitations of AI systems are just copying the "god of the gaps" fallacy

by what metric has it not improved in the last 3 years?

You should also really look up a video about what neural synapses really are.

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.