The epicycles model represents one arbitrary periodic function as the sum of multiple well understood periodic functions (uniform circular motion, i.e. complex exponentials). This is the same idea we use in a more systematic way today by way of the Fourier series, etc.

It's quite a good method for approximating any arbitrary smooth periodic function.

My best guess is that the originators of this model (perhaps as early as Hipparchus, but at any rate someone by the time of Ptolemy) didn't think there were literally circles being combined, but had a pretty clear idea that they were trying to make an approximate model to fit the data. A previous model, from Eudoxus, was also probably pretty accurate but more cumbersome to compute with, involving approximation of the visible planetary motion by the combination of uniform rotations of some imagined sphere(s) centered on the earth. (We don't know exactly because the books about it don't survive, so all we have are vague descriptions by other people.)

>If we were using LLMs to analyze astronomy, we might just get increasingly complicated epicycles and never realize that the Sun is the real center of the solar system

I think this is a fantastic point and this kind of complexifying can and will happen. And I think future of online debating and propaganda is going to complexify in a similar way.

I will say though, I think that can be controlled by building in principles that prefer competing theories on the grounds of cogency, and that cancel out competing theories by reducing them to the parts between them that are equivalent.

My hard disagree will be with this: I don't think at all that we would "never" have got to the theory of relativity. It reminds me a bit of the "embodied cognition" argument against simulated brains. That argument suggests you can't "just" simulate brains, because actual brains are the totality of their embodiment in bodies and environments. Regardless of whether you agree with that line of thinking (I don't), we can assume it's true, and change the target of our simulation to brains + bodies + environments. The problem is bigger, but still perfectly amenable to the same methods.

I think the same is true of theorizing about science and imposing metatheoretical constraints. I think you can optimize for healthy hypothesis production just like you can for first-order reasoning about data. It's definitely harder but not necessarily a difference in kind.

If Epicycles made good predictions, then they were not "completely wrong". Nor is putting the sun at the center "right" so long as the predictions are still good. It is a matter of taste, not correctness (it is, however, very good taste).

I think you might be underestimating how transformative an "advance search function" could be. If the search engine knows how to design and carry out experiments in order to synthesize new evidence that allows it to ensure that the answers it provides are well supported, then humans are essentially out of the "truth" part of science, leaving them only to advice on the "beauty" parts, which might be ok, but it's a pretty big shift.

The introduction of LLM in mathematics is no different than the introduction of automatic calculators. Probably the calculators (computers) where even more groundbreaking if you think so (something that was impossible to calculate or simulate it was now possible).

Just don't believe at all the shitty marketing that AI companies are creating to make you believe that they have found the Holy Grail

What you have to believe is that somehow the entire field of math was just leaving open problems on the ground that were actually easily solvable and were essentially free Fields Prizes, tickets to a lifetime of Fame and Academic superstardom. Orr LLMs are actually doing something novel.

Current models have undergone massive amounts of RL to "get shit done." There's incentive toward reasoning, but probably little toward elegance. Every engineer is familiar with the impulse to just hack things together.

But this need not be a permanent state of affairs. The model labs are probably already examining the elegance problem, since it is such a common complaint of researchers reading these machine generated proofs and papers. And I haven't seen anything disproving the notion that elegance can be RL'd.

BACON in the 1980’s showed using AI techniques to analyze geocentric astronomical observations, and it came up with the heliocentric model and Kepler’s laws on its own. So no, this very application has been demonstrated.

I don't think checking our work is a uniquely human trait. It's our uniquely human traits that made us willfully ignore the better model in favor of increasingly complicated epicycles.

> If we were using [humans] to analyze astronomy, we might just get increasingly complicated epicycles and never realize that the Sun is the real center of the solar system.

Of course. But we might have found out the sun is the center of the solar system much sooner. Who knows!

I feel like there's a lot of "humans are special" in this thread. There's a good chance we're not.

Precession of perihelion of Mercury was not the motivation of the discovery of general relativity theory. It's only an evidence that convinced why the theory ought be true.

it is perfectly valid to have earth as the center. you just get very complicated orbits. But it's just a reference shift. The sun at the center makes for much simpler equations. But it's arbitrary. you can still do the math from both perspectives

> It was completely wrong but the outputs were surprisingly close to reality.

that's not how physics works. epicycles weren't wrong. there is no wrong/right - modern scientists don't believe in platonism. physics is a collection of models which are empirically tested. if epicycles makes predictions to more decimal places than relativity then epicycles are "right" and relativity is "wrong".

now think about how this translates to LLMs...

Your experience mirrors so many managers' experience with engineering teams...

That's the point. When a human works on a problem they realize themselves "wait, i probably should re-architect now" - of course you can ask an LLM to do that, but at that point you already know yourself what you need, which defeats the point of LLM working on difficult problems that require insight automatically. A lot of these math problems are sessions over many hours. And of course you can also ask "Think about whether to re-architect at each step" and it will never do the right thing because the context it builds up for itself drives it into a specific solution space. It's literally trained to complete exactly that.

> You literally just have to ask it.

Does anyone have a good prompt for this - like if I’m adding a feature or fixing a bug and I want it to be open to more than just tacking on to the existing architecture?

why did you leave?

> You literally just have to ask it.

why doesnt it ask itself before proceeding?

I think everyone is missing the "...that nobody has thought of" part. You can have it switch known abstractions, but can it come up with one on it's own that doesn't exist in it's training data?

Thanks to improvements in the system prompts/harnesses/RL, the LLMs have gotten way better at questioning the premise and telling you to try something else. It's a huge difference from just 6 months ago.

Nowhere near the level of a suitably-bearded human, but they've gotten pretty good at stopping a lot of bad ideas.

They're very Declarative that way

Have you ever asked it to?

As a mathematician, I am looking forward to the day when nearly all of undergraduate mathematics and many of the lower level grad school math books are encoded in Lean (a proof checking language). Often I find that theorems are not stated precisely enough and I have trouble finding the exact statement of a theorem without digging through math books in my library. It would be nice if we put the physics and chemistry books into Lean also.

Simplifying all of math is another endeavor, but I imagine that you could have a bunch of LLMs trying to shorten existing Lean proofs.

there have been explicit attempts at this in the past. Nicholas Bourbaki was the pseudonym of a group of French mathematicians who had this goal mid 20th century. You can see e.g.

https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_math%C3%...

It has many benefits, and many people appreciate the books. It also has many downsides. For example, they started publishing in 1939. As part of this, they needed to work through the basis that most other mathematical objects are defined in terms of (they used sets).

Unfortunately for them, contemporaneously with their work, other mathematicians were beginning to define mathematical objects (categories) that can be an alternative basis for mathematics, which many modern expositions prefer to sets. So, their approach either

1. needed a massive "refactoring", or

2. would be hopelessly dated.

They ended up going with the approach that is now dated. It may sound peculiar that mathematics can be "dated". But it very much can. The mathematics community can go through many different styles for how to explain/collect mathematical understanding. Different styles can have different benefits, and be easier/harder for different subfields. A simplified/unified perspective will necessarily privilege certain perspectives.

It is analogous to how you might want there to be a simplified/unified (set of) libraries for programming. Perhaps that everyone uses. This sounds nice, and many programming languages do this with their standard libraries. But these always make concessions! I'll speak about Rust's, as I'm most familiar

1. fallible allocation or infallible allocation?

2. C-style strings or (ptr, len) strings?

3. Should interfaces take as input &mut [T] references, or should they take as input T in an "owned" way (this would make compatibility with io_uring easier)

for each, it is not that one answer is right. Both can be argued for. You have to choose one. The one choice may not be satisfactory for every practitioner though.

Math simplified? The "basic" math (say less than graduate math) is already simplified and minimized and refined. Yet most people have trouble understanding and mastering it.

I mean he’s basically quoting what Wolfram (a man famed for his humility) said on a panel just after these dropped.

The amount of money they're lately ploughing into proving math theorems is inconsistent with how societies and markets have priced pure mathematics. The entire US federal budget for math research is something like $100M annually. A single college football coach can already earn 10 percent of that.

Pretty much the only enterprise that historically pays some mathematicians handsomely is quant finance, but those people are actually compensated not for proving theorems but rather for statistical modeling and programming skills. And even that industry is so technologically driven these days that pure research mathematicians no longer hold a clear edge over strong programmers with undergrad level probability and statistics at their fingertips.

I don't think it will be the case, maths have real utility. I found something interesting at the intersection of combinatorics and information geometry. To be quite frank I don't understand what I'm doing. And yet, when I ask ChatGPT to use the framework we're developing to write an algorithm, it turns out it has quasi-parity with the state of the art. I have to measure absolute perfs to decide which one is better – theirs, not mine. Ok. Time to keep improving on what I have. And this implies dropping the code and going back to the blackboard doing more super abstract math that are way out of my league.

Yeah chess is a good example. DeepMind came for publicity with AlphaZero. Arranged a match with Stockfish with rigged rules to make AlphaZero look better than it really was (it was amazing but the match wasn't fair) and then just published some games and went home.

I was bitter about that back in the day as I hoped for more answers, more matches, more "truth" about chess being shown. Soon after that community project Leela Chess Zero was started and not only surpassed original AlphaZero but added few hundred ELO points over it. Then the combination of NN and classical engines happened with NNUE and current Stockfish is again a few hundred ELO points stronger.

Today we pretty much know the truth in chess for all practical purposes. Human analysts/preparation experts focus on finding interesting path and opponent profiling (what is the most unpleasant for the opponent to face). They don't look for truth anymore. The game is doing great, it's more popular than it ever was.

> I wonder if top labs will soon abandon math progress like they did go and chess.

I definitely think that this is marketing, just "with good side effects". My doubt is when they will be able to move to "marketing with better side effects", that is, research with more concrete outcomes (health, materials etc.).

Problem is, that type of research is much harder. Some doubt that progress in such areas will be quick (https://www.noahpinion.blog/p/wheres-the-intelligence-explos...).

This misses the raw advantage of a good proof. It makes conceptualization simpler. In some ways math is like a hash list of of theorems. This list makes it simpler to prove other calculations, and will always be useful, to both humans and AI models. I can see two new directions 1 - the creation of specialist theorem models; that can answer questions efficiently about one topic and 2 - we probably need to incentivize and codify ownership of theorems; charging a proportion of the compute saved by using them. Ultimately enabling mathematicians to be paid our true market value!

I disagree. Firstly, people in AI likely care about math on a personal level. Secondly math is useful. Playing go or chess is basically a party trick. Being useful gives it staying power.

But, I do think you are right that there will be some level of moving on. The spotlight is currently on maths and that won't last. It will move to some other area where there is more impact to be had. So while they might shift gears and put less focus on math, it will always be there as part of the portfolio.

I think this is an interesting and good theory. They've probably eked out the large majority of the PR benefit at this point, so whether they continue in this vein will tell us a lot about their motivations for this work.

To take this to the next step, what happened after deep mind pretty much solved Go is that they started looking for the next set of things that hadn't been done yet. It does strike me as very likely that this will follow that same path.

Interesting path forwards, and probably partly true, but there are some important distinctions:

Go was a specialized application. All the math results come as a side effect of reading the whole internet, and it will keep reading the whole internet. It will keep practicing thinking questions. Actually, math might be one of the best ways to keep them contemplating and measure their contemplation abilities, so math will always stay in the loop.

Also, math might not be useful just for humanity, but also for AI, so the system might actively benefit from new math results itself. (Not sure if any of the recent proofs qualify, but future work might.)

I had the same idea recently. You've solved all the famous conjectures (all formulated by humans because humans found them interesting), what next? I doubt "AI formulated a math conjecture that nobody else cares about and immediately solved it" will produce that much hype. The actually interesting thing is indeed how mathematicians themselves will use these AI models going forward and how that will shape mathematics of the future.

I think there’s a venue where they start focusing on introducing hypotheses where the model currently can’t solve it, or maybe this is already happening?

Being able to present useful novel ideas would likely generate a lot of press, for a while. I don’t know how this would look since I’m useless at math, but Im sure there are plenty of unknown problems with massive implications, that once formulated can be solved.

AlphaGo and AlphaZero weren't generalized models. Math capability will presumably keep improving along with the other general capabilities, even if there wasn't a special RL focus for math itself.

They already proved the point

>Merely solving not yet solved problem might not be it anymore.

Never has been.

See: Mochizuki's "proof" of the ABC conjecture [1].

In short, mathematics is an art of storytelling as much as anything else.

Writing a boring story, much less an incomprehensible one is worth exactly nothing, and doesn't matter.

Is it cool that the machine can connect the dots that people put out there in a form that happens to be easily ingestible?

Yeah.

Can the machine answer why give a fuck about a result like that?

Nope. That work has already been put in by people.

The most obvious impact of this BS is that interesting problems with potential to be solved, which were already a scarce resource, will no longer be shared.

It's already become a huge issue without AI being involved, due to the rat race nature of academia.

There's even an industry term for when someone gets a wind of another person's progress on such a problem, and solves it before them - it's called "getting scooped"[2].

It doesn't mean that they got access to the results. It can be as simple as e.g. an experienced mathematician hearing about a problem that a PhD student is working on for their thesis, and scooping it.

Honestly, as a mathematician-turned-software-engineer, I see and hear both sides, and both sides are lying:

* The AI companies are lying that what the LLMs produce is "mathematics". Writing a proof, even a correct proof, is not what mathematics is about. The Fundamental Theorem of Algebra has been "proven" incorrectly four times IIRC, but it was mathematics nonetheless.

Mochizuki's "proof", on the other hand, may be correct, but over a decade later, it's still not really mathematics, because transfer of insight is a fundamental component of it, and insight is a thing that the AI, presently at least, doesn't do, no matter how many proofs it produces.

Proof automation has been a field for a long time, for that matter; hyperscaling it doesn't change the nature of things.

The lie is also that a correct proof is the final point. The Four-Color Theorem[3] was proven in 1976, yet people were still proving it 20 years later, trying to reduce the number of cases to leave to the machine, even as we got it super-duper-verified by theorem-proving software in 2005[4].

The writing was on the wall since computers were invented, so why are mathematicians making Pikachu faces now?

That's where the other lie comes in.

* Mathematicians are lying to everyone else by suggesting that the technology is the problem, or that the issue is with how the technology impacts people, and not with the COMPLETELY, ABSURDLY FUCKED UP ORGANIZATIONAL STRUCTURE OF ACADEMIA, and I can't scream it in bold enough caps lock.

There's a reason websites like "100 reasons to NOT go to grad school"[5] exist, and all who've gone through the masochistic 5+ year journey can attest that actually, the reality is worse.

Closer to the point, there's a reason Grothendieck burned most of his work and fucked off from mathematics in the 90s, and Perelman did the same, in effect, in 2000s[6], while putting a million dollar weight to his statement that MATH SOCIETY IS FUCKED.

To all the mathematicians running in circles screaming AaaaAaaAAAaa, Perelman can now say: told you so.

See, after we discussed what math is about (insight transfer), let's do a peculiar little observation that this is exactly what's rewarded THE LEAST in the field.

"Teaching jobs" are a punishment for those who don't do good enough, and doing good means cranking out proofs in a publish-or-perish race, emitting just enough for others to verify correctness, but not too much lest you don't have enough for the next year's publication quota (or, if you are a mastodon, enough for your grad students/postdocs to work on).

The industry where the incentive structure creates and rewards scooping as a concept turned out to be vulnerable to scooping at scale.

O the humanity!

'No Way to Prevent This,' Says Only Industry Where This Regularly Happens.

So the lie to everyone else is that problem isn't 100% a social one in the mathematics community. Not even mathematics.

The way we do math doesn't have to change, and it won't. We've had theorem provers for decades, remember?

What will, I hope, change - what has to change, as Perelman cried out loud (and paid $1M for people to hear him - without listening, it appears) -

- is the way mathematicians treat each other.

That's to say, what gets rewarded, what gets penalized; the incentive structure, and so on.

In Google-speak: the real problem here is that the perf is fucked up, and what counted as impact before can now be automated, threatening the entire hierarchy of resource allocation (in particular, headcount).

This is not only NOT a tech problem (which won't be solved by changing the behavior of AI companies), it's not a problem with the way we do math or adapting to the progress either.

It's the ROTTEN INCENTIVE STRUCTURE OF ACADEMIA that is showing its cracks, and mathematics is where it's apparent the most at this moment because, as Vladimir Arnold told us, mathematics is a branch of physics in which experiments are cheap[7] - and more pertinently, could be performed by LLMs without human involvement.

Unlike the biotech lab where someone still has to wash the beakers, mathematics doesn't have that moat. Hence the conundrum.

Will the mathematical community adapt and fix its publish-or-perish problem, or will it simply become a cult, like it used to be[8]?

Given that a colleague of mine wrote that satire years before LLM proofs were on anyone's radar, I'm not having much hope.

Stay tuned.

[1] https://www.quantamagazine.org/titans-of-mathematics-clash-o...

[2] https://quantixed.org/2018/03/19/scoop-some-practical-advice...

[3] https://en.wikipedia.org/wiki/Four_color_theorem

[4] https://en.wikipedia.org/wiki/Proof_assistant

[5] https://100rsns.blogspot.com

[6] https://www.nbcnews.com/id/wbna38039068

[7] https://archive-dsweb.siam.org/The-Magazine/All-Issues/vi-ar...

[8] https://www.mcsweeneys.net/articles/an-open-letter-to-the-ma...

> assumptions which will probably not hold in the very near future [...] One is that AI will continue hallucinating in a manner that is not easy to verify

Hold up, that's an even bigger assumption in the opposite direction, and I don't see anything to support it.

At least in terms LLMs getting all the "AI" hype these days, there is no structural/mathematical reason to believe they won't continue to have the same problem they've always had of generating plausible text over rational text, and I don't think anybody even has a clear idea how it could eventually be accomplished.

I've seen "then the magic singularity occurs and somehow it solves the problem for itself", but I would classify that more as mysticism than engineering.

> One is that AI will continue hallucinating in a manner that is not easy to verify

It is an old saw at this point, but what an LLM does still cannot be divided into hallucination and non-hallucination. This is literally an anthropomorphism trap.

Layers and layers of application-specific verification can reduce the risks inherent to LLMs, to a really remarkable degree, but nothing about what these tools are suggests that this problem will go away; it will just bubble up again somewhere else.

Putting hallucination aside, LLM "theory of mind" has gotten worse over time. I feel it peaked in Opus 3 and Sonnet 3.5, GPT 4 and then GPT 4.5 for OpenAI. Since then, even with Opus 5.5, phrasing has needed careful crafting, in order that it not be taken too literally. OpenAI models suffer from this much more than Anthropic models but Claudes have backslid over time too.

This means when writing documentation, tutorials or commit messages, their output is often a garbled jumble. Assuming shared context, using invented terminology without explaining, leaking conversational states due to improper epistemic boundaries and failing to model the reader. This all usually leads to their freely generated explanations being terrible. Getting good explanations requires chaining questions that force them to line things up properly, which is not easy the less you know. These failures as something LLMs naturally struggle with make sense, given the nature of attention and RL with weak signals from human data.

Math is not merely a collection of proofs, it's a way of understanding. A proof presented in a manner that cannot be incorporated remains useless. It does not make it's way to physics like Riemannian geometry and matrix math did. This is no less true when done by humans too.

Your hallucination conclusion, checking if a proof is one, is exactly the counterproductive cost.

Most of us cannot verify that the claims in the OpenAI lore dump are in fact all correct. It will take tons of work from experts to do this. It took subject expert mathematicians to identify the discrepancy and disconnect in the Navier Stokes proofs, for example. LLMs will struggle to make use of their own proofs or turn them into knowledge that accumulates over time.

The act of proving is often more valuable than the proof itself. Human constraints and limitations force us to invent tools and abstractions that a 100,000 x 1M context swarm can bypass. The tradeoff from that AI swarm advantage is work that doesn't usually lend itself to being built upon. It's like doing all the side quests and reading all the books of an RPG versus min maxing a straight path with a guide. We might try to identify new abstractions, but the fact that we don't get access to CoT and that much of it will be illegible means mining LLM traces for what human mathematicians produce naturally will be a tedious chore.

> but it is very productive in terms of hundreds of proofs being produced that had previously consumed uncountable hours of the brightest minds

You’re making the following assumptions:

1. the exercise of struggling to find proofs was not productive, but this is precisely how new techniques in math were produced. Brute forcing solutions doesn’t lend itself to the creation of much new mathematics (except maybe the exercise of developing verifiable proofs)

2. the point of doing mathematics is to be “productive” in the first place. This is silly. Many people get into mathematics because of the beauty of understanding, for example.

> One is that AI will continue hallucinating in a manner that is not easy to verify, second is that AI will not be enhanced to produced more simplified amd robust outputs, and third that a human will be required to do that.

There is literally not a single shred of evidence to indicate either of your supposed eventualities. The core technology of an LLM is sampling from a distribution so there is literally no way to make it deterministically robust (only probabilistically).

Even if you somehow have a 100% correct AI, it's not useful unless we can understand and internalise (and communicate) its results.

AI cannot explain chess moves it comes up with in an elegant way. What makes you think it will be able to do so for math?

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Did you even RTFA? His argument absolutely doesn't make any assumptions about hallucinations, implicit or not. It's you who assumes Tao must have surely been complaining about hallucinations or some such. You've not addressed any of his arguments and moreover ask questions his post answers.

Here's a longer article which goes into a bit more of the details: https://terrytao.wordpress.com/2026/10/05/the-future-of-math...

Expression of tech-faith is not intellectually honest argument.

Where does this "will probably not hold in the very near future" come from? People correctly warn about extrapolating current things onto the future, but then just throw some vague "probabilities" without providing any argument why their "probably" is somehow more grounded than others.

In TFA I read (scroll up from the link anchor) this was explained: OpenAI are not giving talks - because they can't answer any questions about the model's work. There is little follow-up activity - the actual elaboration of human understanding of the new ideas is stifled since the problem is solved.

But you are right, this is not OpenAI's "fault". The problem is - as others have said recently - that many people in mathematics want recognition for solving open questions more than they want the answers to the open questions. Everything about the economics and social environment of Mathematics will have to change.

I think that this is exactly the same split we see in software: there are those who mainly enjoy the craft aspect of building software, and are uninterested in the product or business they are supporting. Others are primarily interested in the production of useful software or building a platform or company.

I've always been in both camps myself. When it became obvious that AI was going to destroy the craft aspect - at least two years before it actually could do so - I became very discouraged, even depressed. But once it was actually good at building software, I became very excited about all the stuff I could now build. Sadly, I think a lot of people in our field have never had something they really wanted to build.

I'm in the same boat, I guess I had this naive idea that unsolved math problems would mean something if they were solved. But it seems like a lot of them at least were more thought experiments than anything else.

Perhaps mathematics was never about proving things? I know it sounds like moving the goal post, and it certainly was what motivated mathematicians on a day-to-day basis, but bear with me for a moment. I think that beyond being a creative activity that humans enjoy, math was about building new tools and systems of thought. Axiomizing things we take for granted, logic, linear algebra and calculus (on which modern LLMs rely so heavily) are such examples. People chose to participate in this field because they found it enjoyable and satisfying in some way. As a side effect, society reaped the benefits every couple of centuries. Will harvesting open problems with AI ever give us these things, or will we just be left laundry pile of Lean formalizations?

Mathematics is less about the end product and having a healthy community of people to actually understand the proofs and eventually apply them.

That community won't exist as many people simply won't even enter the field because it's reprehensible and contemptible, not to mention boring, just to read machine-generated proofs and verify them.

Collaborating on, or at least working on unsolved problems is what motivates most people.

AI is like a cheat code in a video game. You get to the end faster but fewer people want to play if the cheat code is always on. You can't turn it off either because the very challenge is to do something unique.

Its pure cope from the math community. One year ago if god came down and asked any of these math guys if they would want to see the proof of the Riemann Hypothesis they would all say yes. Now that the AI math god is here and handing out proofs they are all like 'not like that'!

What does it mean to “genuinely” contribute to mathematics? Because the definition you use is load-bearing ;) and it might be different from that of others.

Because they published a ton of slop papers with terrible English, impossible for humans (even experts) to understand, full of non-standard terms, invented jargon etc.

So effectively, stuff got proved, but people don't really understand how, so it's mostly fucking useless and done for OpenAI's marketing team, while also pissing off the maths world at large.

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I’m not deep in to math but the op tweets make sense to me. In that it’s not just the final proof that mattered, but the mind and understanding of the person who arrived at the answer. An LLM dumping the answer can’t elaborate on it, can’t tell the story of how they got there, etc. But it also deprives someone else of that achievement and learning.

You know it's valid. You're not working on incorrect assumptions. Surely there's value in that?

I tend to disagree with OP, for the same reason.

It's unclear what more could be expected than releasing the presumably already verified results and write-ups for each problem. Should they run a mathematics school too?

Then the comment goes on to argue AI labs were not interested in actually advancing mathematics, and that investments into AI were manically excessive.

IMO none of this follows and demand is there to justify the investments.

The comment then goes further to argue that AI labs were putting too much effort into pretending there was exponential progress rather than actually making progress.

The factual basis for this claim seems to be that OpenAI released math results and write-ups, and it's not even clear what more they could do on that topic.

That's a very negative opinion.

They should not solve the math problems. It’s simply a net negative for the world.

Look at how actual researchers are currently using the tools: they'll generally use the LLMs to create slop papers, sometimes supported by auto-formalizations, just like OpenAI does. Then they will go through the lengthy process of digesting the results, turning the often incomprehensible and poorly organised outputs into something that humans can understand and build upon, they will then give seminars on the results, further helping with dissemination. Doing so still requires expertise, and probably will for a good while.

So yes, that is exactly what they should do. Alternatively, if they are too lazy or incompetent to put in the effort themselves, do what AGMAI proposed and fund a third party to help out.

> This does not affect people building new theories.

Actually, the problem is, somehow the skill of building new theories in math is directly tied to slaving hard over a problem. It's the very experience of slaving away that actually somehow causes ideas to form. Pretty much all mathematicians understand this. Yes, senior mathematicians now can form some new theories, but what about junior ones who will have very little experience in working hard on a problem by hand?

Of course, they could work on the problem by hand anyway, but they won't because no one will pay them when a machine can do it.

My grunt work is different than doing grunt work for a company to fix their problems so that they can make more money.

OpenAI has kicked things over. Dealing with that is the math community's problem. OpenAI had absolutely no obligation to defer to that community's deficiencies. If this has upset their reward system and historical traditions, tough luck.

AI is fucking shit up for humans without providing any benefits.

It’s causing unemployment but no real gains unless you have a lot of equity and can benefit from lower costs from fewer people working.

I think in a ideal world your right.

I think a problem is that math seems like a deeply toxic, ego driven domain.

I think he argued that e.g because the navier stokes millennium problem ist considered solved now, you won't get any recognition for being the first human to solve.(How would you even proof you solved it yourself and not just regurgitated the ai proof?)

And since recognition is the main objective, noone would spend time on dissecting the proof, and perhaps finding some unique approach to solving the problem, that could be transferred to other open issues.

And therefore the problem is now "poisoned". Since it's assumed to be solved noone will research it, and the potential revelations won't be found

I think it's quite clear the mathematics panel didn't read most of the papers with the scrutiny it would take to publish it, if only because the lean certificates and the informal proofs are not 100% the same.

And if it's hard to understand (which seems to be the most common reaction) it's not exactly devoid of noise either

There was an opportunity for people to work with the AI to produce a proof, now it almost feels they're working against it.

Etic is a factor of productivity, and the larger the contextual window is the weightier it becomes. That’s even integrated within the paperclip parabola.

If the focus in placed on maximizing some easily measurable output on a narrow perspective, situation is unlikely going to match a sweet spot of holistic equilibrium which is maximizing harmony and happiness through humanity as a whole.

How to make sure it doesn't evolve into some sort of Library of Babel of science?

And what use of that?

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The problem is that one a person writes a 60 page proof in theory that person has spent an inordinate amount of time on the proof and can answer questions, describe some insight, etc etc.

If a random person is given a 60 page proof to digest and not the author, those hidden insights that _aren't_ in the paper might be completely inaccessible. Maybe the AI will "just" be able to provide the insights. Maybe. But pedagogy is tricky work, and despite these AIs being able to do all this fancy math we can't get them to write good cover letters yet, so....

Ultimately we might be left with just a bunch of intellectually unsatisfying proofs. This means way less drive to simplify the proofs or rework them.

End result: we generate a layer of "less efficient" mathematics, that won't get built upon. We will not actually have any shoulders upon which to stand.

> I'm not sure there's any evidence to suggest exploring and refining results is outside their capabilities

OP didn’t suggest that.

The bar has been raised. Everyone has to meet it now. An inelegant solution squatted onto the internet doesn’t count as discovery per se, even if it’s impressive.

But what should they do? They got all these proofs, should they just have sat on them?

Honestly, that's a weakness shared by a lot of human-generated science as well.

Pretty tenuous connection to blame mathematicians for everything bad created in the world, that happened to use math.

That's not the hard part. The hard part still requires humans (and/or _maybe_ a bunch of tokens) and a lot of work.

And then what?

Proofs are valuable but I believe the mathematical community values understanding more. Proofs were previously a great way to develop understanding. Now, less so.

Not an expert, but explaining the proof and it being independently verifiable as important as just putting a paper of it. Reminds me of 1000 page proof of Goldbach’s conjecture that a Mathematician reached sometime back. He was told plainly that no one is going to invest time in verifying the proof because there’s a good chance there’s an error somewhere in between.

Proofs of open problems are valuable because we are assuming that proving the problem requires some new method or infrastructure in math to prove it. Basically proving open problems isn't actually useful if it doesn't develop new tooling for mathematics, which can help us create new open problems, solve other ones etc.

I recommend RTFA, it explains exactly that: why proofs should not be considered the goalpost, and why dumping all these AI-generated proofs might be an overall negative for mathematics as a whole.

This looks indeed like a public relations campaign, it would be helpful to the conversation to contribute your opinion, in a polite way.

I think that's debatable. Anthropic's interpretability research suggest models do have self-introspection ability, at least in the "J-Space": https://transformer-circuits.pub/2026/workspace/ ; and this private working/'introspection' space is distinct and distinguishable from the tokens they output (CoT tokens are output too).

Bold claim. What data is informing it?

The reasoning traces were traditionally keept secret by legacy mathematicians as well.

"When the architect completes a fine building, he removes the scaffolding." - Carl Friedrich Gauss

I think this happens all the time actually, but it's often smaller scale. Like someone writes a neat architecture to do X at their company, and later someone else walks in, looks at this code and sees it's now super easy to do Y, which then turns out to have immense user/business value.

> I don’t think I’ve heard of elegant code leading to discovery on its own.

Every software design pattern came from elegant code. People wrote code, summarize code, learnt from code, and taught code. That's discovery

> I don’t think I’ve heard of elegant code leading to discovery on its own.

I’ve not heard of it either, but code is an abstraction of math, so I don’t see why this couldn’t theoretically happen.

Anecdotally, I’ve started spending time advancing my math skills beyond the early college level I stopped at and I’ve frequently found I already know concepts of more advanced math - I just didn’t know what they were called or how to apply them to an equation on paper, but I’ve been using them for years and intrinsically grasped the underlying academics.

> elegant code

Usually it's the opposite. "That's in prod? And it works? It shouldn't work and I thought it was doing something else. Why does it work?"

But it has. Think of design patterns and other programming paradigms (logic programming, functional programming, etc).

you must not be familiar with haskell

I mean if we're being honest, most of CS studying (as practiced) was a waste of time anyway.

The S fell short in actual reality for the most part, as it was merely a hiring requirement. A hiring requirement that didn't even make sense, because the skillset of academic CS only marginally overlaps with the skillset one wants to hire for.

Material engineering at least for the most part has actual real-world applications where one can push humanity further. CS (as practiced, not necessarily the idea of real CS but the CS we got due to it being used as a hiring filter) for the most part is just self-referential spinning with mostly unclear results.

There is real impressive work being done in that field, of course, but I'd argue that the majority of it over the last decade or so at least was just performative nonsense.

Maybe by again allocating new resources to other fields, what hides under the label CS can become more pure actual CS again. I think that would also be a much less miserable experience for everyone involved.

Well i mean, the whole idea of discovery is that you're not sure what you will discover until you do

> What new things are even left to discover in software?

The will to implement the stuff we learned, instead of letting dark patterns and churn for the sake of churn get the better of that :P

Byte-matching has been a side-effect of gaining an understanding of how the software works. With this you are skipping the understanding and you will not get people like Kaze Emanuar on YouTube who have dedicated a lot of time to building upon this understanding to create something better.

I don't think it's motivating to solve a black box by having AI generate another black box if what you want is to understand how the thing worked.

If your goal was to do something for the games you love and understand them on a deeper level, then just handing that off to an LLM doesn't really give you the same sense of accomplishment.

I am advocating for effective communications, it is not as if knowledge once understood remains difficult. To communicate and convey understanding is good, and to withhold understanding is tyranny.

communicating results clearly is just as important for specialists/experts. A reaction by many mathematicians to the OpenAI dump (and many LLM breakthroughs) is simultaneously

1. if this is true, it is interesting, and

2. the exposition of this is a huge pile of slop.

This requires mathematicians to have to clean up the slop for it to be useful. It has happened for most LLM-generated proofs in the last few months. Anthropic explicitly payed two top complexity theorists to do it for the 3SUM and APSP breakthrough.

They didn't do this because those complexity theorists are advocating for a tyranny of the illiterate lmao

If a coal miner or car mechanic would say something like this, they would be called a luddite by the same science people.

Its different when your own job is on the line.

When human manual arts were being automated away it was supposed to be not only acceptable but any complain and you were told you were a progress blocking luddite.

Now that mental labor is getting automated, the response to automation is very different.

The amount developers saying how "now we can focus on the bigger picture" rather than the code are just delusional. The bigger picture has always been a bigger problem than writing the code. It's the part that engineering stands for in "software engineering". If you couldn't code something well or weren't involved in bigger picture decisions in the past, you're just gonna engineer big picture spaghetti.

This is exactly why half the conversation conversations about AI eventually become a critique of capitalism. AI isn’t for fun or joy, it’s explicitly for business use. For replacing people. For extracting more out of each worker. That’s all the major companies (OAI, Anthropic, etc) are offering. None of this is designed to give us any more freedom or time to pursue the things we are passionate about.

I feel like we’re circling back to that 2010s energy of “everyone can be an entrepreneur.” Now it’s “everyone can build software”

He grew up in Australia, didn't he? He was the one doing the replacing I suppose.

Pretty much every published paper I see now is a combination of Chinese and Indian immigrant authors. Given enough time they all become American. I don’t see what the problem is.

He is a imigrant.

> even if they like doing it as a day to day job, they'd probably be doing something else if they had absolute freedom over their time.

Not the mathematicians I know. They’d happily drop the academic admin stuff, but they’d absolutely keep doing mathematics in much the same way.

Based on the research mathematicians I have known, I'd disagree with 'they'd probably be doing something else if they had absolute freedom over their time' for at least a large number of them, although it probably varies from individual to individual.

Some call that "entertainment", or "a hobby".

Similar things are happening in the competitive programming world.

Quite a lot of people are not happy they aren't elite anymore, and many have spent years to decades to arrive here.

Simply put you invest years of your life to establish a kind of distinction over others, and that goes away. That hurts.

But its not something surprising. Most of these competitive programming problems were actually English languages puzzles, because you couldn't dial up the mathematical difficulty anymore making it a fields medal problem. And in most cases in simple language weren't even that hard to begin with, and you could look up solutions to these problems in an hour of Google searching.

I'm not totally against "math as art" but I don't think that formulation goes deep enough towards what's really going on with mathematics. There is something deeper where I lean more towards the philosophers who have said that mathematics is essentially ontology.

Since the Greeks we've had the idea that "Being and thinking are one," or that Being (in the sense of all of existence as such) has some essential unity with thought, and therefore can be thought, and expressed or submitted to the logos or reason. Being is in some sense fundamentally intelligible, and mathematics is the most developed, exacting, and articulate expression of Being.

Logic was understood in this older sense up to roughly the the mid to late 19th century. This is why a work like Hegel's Science of Logic begins not with syllogisms or propositions but with Being and Nothing. But this was forgotten after logic was mathematized by the English around the time of Russell, and its connection to ontology was gradually overshadowed by a focus on epistemology (still, it should be remembered, originally as a means of getting back to ontology).

There may be truth in art, but it's always haunted by its own historicity or contingency, which is to say untruth. Mathematics seems on the contrary the only really timeless, absolute thing we have. Part of what makes it captivating is stumbling on a construction or concept or proposition or theorem that simply must be, independent of us.

The AIs are certainly now more than automatic theorem provers, mechanically traversing some space of true propositions. They are able to push things forward and connect seemingly disparate domains to get to a proof, but to my mind it remains to be seen how well they will be able to form new concepts and definitions.

Imagine the controversy surrounding Cantor, for example, but put an AI in the place of Cantor. If an AI proposed something like the (infinite) hierarchy of infinity, would we have accepted it? What would the intuitionism debates have looked like? Would they even have taken place? And aside from that, has it actually been shown conclusively that an AI could propose such a thing?

There are lots of attempts right now to recover a humanism for mathematics, or restore man's pride of place with respect to it, but maybe we don't need to worry about that. Tao's attempts to preserve the mathematical community, while allowing for practices to change through the crisis may look like a kind of rearguard action, but seems reasonable to me and not really dependent on any kind of humanism. It's a way to avoid the question for now while things play out (and not conservative/reactionary like Scholze and others), which may be exactly what we need, because after all, perhaps we still don't understand why we do mathematics, what it's really for, and what our relation is to it. Whether it's enough to preserve funding is another issue.

I don't think they prefer it, but the price is a deciding factor.

Most people would be happier with La Marzocco coffe machines which costs thousands of dollars, but if you get an OK shot with a 100 USD DeLongi, then the choice is clear for majority of the population.

The dynamics of software are vastly different from manufactured goods though. Software can be distributed at essentially zero cost so the disposable crap analogy breaks down.

Like given the choice, the vast majority of people would prefer one quality game like Minecraft, LoL, or Fortnite, vs. thousands of one-shot generated games, and looking at user playtime this is exactly what we see. If anything AI is just going to entrench these pre-AI franchises even more.

I don't think this is really true for games, a lot of games have enormous resources thrown at their development and marketing and they fall flat because they simply aren't very fun. Then we have a million slop games that sell 3 copies on steam, and in practice only the most unique/fun/addictive games can break through.

Well, that’s the thing, isn’t it? They somehow invented a way of solving an open problem without advancing the field. That’s something that mathematicians had never even thought was possible. And while the mathematicians were beginning the great conversation to re-examine their fundamental understanding of the pursuit of math in light of this new phenomenon, OpenAI decided they would do it again 372 more times.

What a disingenuous, willfully ignorant take. None of these AI companies have been acting in good faith or in any way other than grotesque self-aggrandizement to the expense of everyone else.

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For deeper complexity, elegance may turn out to be a hindrance. Better or worse may turn out to be descriptive of taste and nothing else. On the other hand, elegant alternatives might be brought into reach by this first messy contact with new ideas. Who can say at this stage?

Are you suggesting that Tao, whose banner image on his blog is a quote from On Proof and Progress, has not read Thurston's essay?

> The difference in reactions by the maths community should probably be spit up into those who have read and understood On Proof and Progress and actually thought deeply about why they do mathematics, and those who haven't.

I think David Bessis (in an essay from April) has the best thoughts on this since Thurston:

https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...

Also discussed here: https://news.ycombinator.com/item?id=48758048

Stupid, but could be fun: grants are given for how amusing your proposed prompts are.

Everyone should have a portfolio of prompts that are indecipherable by other humans but when fed to a frontier model, produces shocking one paragraph english version of a 10000 line lean proof

Obfuscated prompt grant contest

Plus requiring attribution in papers.

It’s just as unlikely that this will cease. hiring a professor at a top school because they were an excellent math teacher… It won’t happen.

I don't think it's smart at all to position yourself in bucket 1.

There's a reason there are maybe a few hundred professional marathon runners in the world vs tens of thousands of professional mathematicians. Bucket 1 is basically an "amusement for the upper classes" type of deal. Any field that goes in that direction would have to shrink down massively.

It also devalues the field in my opinion from something really profound with actual impact in the world to a somewhat vain leisure activity. (basically going back to gentleman scientists) But I know other people would see it exactly the opposite way.

Ah but your buckets are based on your assumptions about the topic.

You say marathons are just for fun, but prior to the wheel it was the only way to get around. (Other than horses in some places)

So it’s not that running is immune to automation, it’s that we’re already post-automation and that only people doing it for fun are left.

I think your second bucket would need to be changed, as it's not really undecided for some and Tao in fact argues (in a way) that there is no need for the field to decide. There is a need for results, but there is also a need for humans to fully understand why the results are the way they are and how they were obtained. This doesn't replace the human in the loop it's more of a cooperative effort.

I'd end up with the buckets

a) Fully human

b) Hybrid human-AI

c) Fully AI

Seems like neat buckets but you may want to find one example for the first which can actually be done by an llm as running is not its strong suit afaik

I am not so sure that's true. Even if AI intelligence were to plateau at today's levels, there are still many gains to be had in speeding up today's intelligences. ASICs with burned in weights could become economical to invest in as they would retain usefulness longer than 18 months, and I feel we have only scratched the surface on possible usecases of local AI and what it means for almost any technology product or interface.

It couldn't make every human understand. But it could make Tao and other mathematicians understand. You're not going to become knowledgeable at everything suddenly, even though I doubt what you say. Many months of 1-1 tutoring and hard work from a student with a top mathematician would make you understand a lot. Just not sit down and read it first day.

No, you're right. I clearly missed reading the room :D But I'm glad how Terence Tao sees it. I was relying him to do that for the NS results as well.

I didn't follow through to the PhD, but I spent few years building up to understanding of Fluid Dynamics and Functional Analysis to come close to NS. It's intriguing that it's "solved", but what interests me is then "what do we learn from it" and what lies beyond in non-linearity.

In the 10 years I've spent away from academia, I still cherish what Math taught me best: looking at equivalences and I still feel the kick that I surely wouldn't want an Agent to do on my behalf. NS was never the point. And who can't see that I can only feel that they missed out.

Maybe they should, you know. Maybe OpenAI is exactly working towards so that we need to do that :)

It's fascinating how people like to push any statement to its limits, because it circles around to absurdity and they believe they've made a point. Moderation seems to be chasing you, but you clearly are faster!

You're just moving the problem around. Instead of OpenAI and Anthropic dumping proofs, you'd instead have someone (maybe an aspiring graduate student, maybe a hobbyist, maybe an established mathematician who has disdain for existing process) dump these proofs en masse personally.

I imagine that would take months at least. Meanwhile, only a selected few get to have access to what might be some of the most significant developments in mathematics. That seems incredibly unfair.

We should think about rejecting this AI future.

This is wildly shortsighted view of mathematics. Historically many of the subfields of math which are presently most valuable were considered useless for decades or centuries. Number theory, non-Euclidean geometry, group theory, and Boolean algebra, to name a few.

If you only judge by practical applications you eliminate vast swaths of human endeavor.

The main issue here is that people that are able to "Strong disagree" is shrinking with every model release, because it requires skills. So it will get more difficult to get funding.

You may say "hey, before AI people payed for math salaries even though they didnt understand the math or the economic outcome". But the issue is there are now "mathematitians" trying to convince not to fund.

In this new reality, you will get "mathematitians" trying to convince that only AI maths matter. And on the other side someone speaking about "understanding", "taste", "community". And the people deciding to fund dont have the skills to differentiate. So they will fund the AI boosters with a higher probability.

Thats how the job of professional mathematitian dissapears. By being replaced by something that on the surface looks similar, but its just an ugly copy.

Yes, I am projecting out into the future, not talking about the current state. I said "might be the first".

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It was the understanding for some, and the result for others. Math is going to bifurcate along those lines. Many are the builder type who use math for a purpose. To accelerate an algorithm, to improve the numerics or convergence of a computation, to verify statements about real things, to use it in angineering applications, etc etc.

In my view, over the last century, math has turned into an intellectual analogue of extreme bodybuilding competitions. A navel gazing runaway optimization in making useless stuff just to demonstrate cleverness. That's fine, why not. But society has no obligation to fund that, just as it doesn't fund other extreme hobbies. Ideally if we ever get something like UBI, math can be still their hobby.

> What are these "greater things" that people can dedicate time to, when clearly even classically cerebral activities like mathematics can be automated away.

Their applications?

For example I'm not a mathematician but I love thinking about weird "useless" shit like how math might be like for aliens? Are numbers as fundamental as we assume? i.e. humans developed math for "arithmetic" first, then latched geometry etc on top of that. We took ages to admit zero and negative numbers.. what if an alien species develops math for "navigation" first, and starts out with complex numbers right away!?

> What happens when humanity becomes obsolete?

There's an infinity out there to explore.

> And what happens when the machines that cause this obsolescence are controlled by a tiny amount of people, who suddenly don't need the rest of us?

That's a social problem we needed to tackle more than 100 years before AI or even computers appeared.

Apparently the minutes hand was added to clock to keep time in factories, for the benefit of the factory owners, not the workers — something I learned from this 1991 show from the BBC with Terry Jones: "So This Is Progress" https://www.youtube.com/watch?v=-Em96NVxO9Q

> and now that skill is not paying the bills. ... you cannot put your life on pause to learn something new

Yes, but that's a social issue, external to technology but exacerbated by every new technology, AI or not:

UBI should be a thing: let people work on what they find fulfilling, instead of having to work to survive.

AI could help design a system for UBI that everyone agrees with, since it's so good at maths and shit now.

This problem HAS to be tackled. Removing/slowing AI will only kick it further down the road, not eliminate it.

I know that is the whole point. But i may dream of being as tall as a house, but it does not matter how hard i dream, it wont happen.

In my experience, every new model release allows me to go further, although every time i see every time the limitations, and i identify where i add value. And this value gets bigger every time.

But on the other hand, every model release reduces the amount of people that are able to value this "added value", because it requires more skills.

So we have the paradox that the added value i can bring on top gets bigger and bigger, but the perception of the economic value for the majority of the population gets smaller.

If anything he is very pro-AI and far from a naysayer. Labelling anyone who doesn't isn't Linkedin style "stoked" "excited" about AI is not helpful.

This is a technology not like prior technologies. Are we okay if the technology discourages a whole generation of Mathematicians? If the technology leads to 10x fewer mathematicians -- what impact does that have on the field? These are the questions Tao is asking. And I don't think he himself claims to have all the answers, he just doesn't wanna see the math _community_ die.

Certainly not implying that! He's the one with the deferred judgement and the wisdom.

Yann LeCun's criticisms are at least balanced with an alternative approach he has teams actively working on and showing good progress in areas LLMs are weak.

The less said about Gary Marcus the better.

Gary Marcus has has a long career now of shifting goalposts

Nobody determined the usefulness of the internal combustion engine or the aeroplane after the fact. They were goals specifically worked towards. For every useful discovery that comes from this hobbyist approach there are far more that aren’t, and those would have been discovered during a goal orientated research program.

However, with AI it actually may become so cheap that the scattergun random approach becomes more viable rather than less. It’s when human time and resources are scarce that you need to optimise. The hobbyist approach may therefore ironically continue, but without the hobbyists.

> usefulness can only be determined after the fact.

Thats great! So in other words it should be perfectly fine for AI to solve all of the supposedly useless math problems so that it might possibly be useful later. No need to worry about the mathematicians hobbies here.

> Publicly funded maths research should continue, even if some taxpayers feel it's a waste of money.

Note, I think the debate is mainly over what research should be funded, not whether any research should be funded.

Actually yes, I think that AI will end up in a place where additional human effort, even at the highest level of suggesting what directions to look in, will become a rounding error compared to what the AI will achieve by itself.

So as measured by utility, I absolutely believe we don’t need humans doing science into the future. I’m sure people will continue doing it, but not for utility, for enjoyment - as a hobby. Probably we’ll all end up as dedicated hobbyists.

Haha. That’s almost verbatim what the guy said. Here, let me grab that for you:

> things that I do and that my colleagues do, this kind of like curiosity-driven, you know, applied math, computational physics type research has always been justified by, I would argue, intentionally blurring the line between what I would call, you know, science as product versus science as process

> science as product is very kind of clear-cut. It’s, you know, things like, you know, cure cancer, solve nuclear fusion, generate, you know, clean energy.

> And then there’s science as process, which is kind of the curiosity-driven stuff about, you know, like, “I want to understand protein folding,” or, “I want to understand, you know, turbulence,” or, “I want to understand quantum gravity,” or something. And broadly speaking, we have tended to justify the latter by kind of laundering it through the former

And the examples he gives are actually the more defensible ones, he talks about a friend of his working on some abstract algebra under the false guise of cryptography research later.

And it’s not just him saying it, this is simply true. He should be lauded for admitting it publicly, this is the only way any progress is made. At least, it used to be. Now it’ll be AI instead.

It's already unbelievably abstract as it is...

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