Tao: Open math problems being non-renewably mined by AI(mathstodon.xyz) |
Tao: Open math problems being non-renewably mined by AI(mathstodon.xyz) |
> Pure mathematics is dead. Long live mathematics. I think all of interesting mathematics is applied mathematics in the end. Powerful AI means that the level at which we can do applied mathematics will be so much higher, though, and many more people will be able to be "mathematicians". The importance of pure mathematics is often argued for by citing examples of important applications that used pure mathematics invented a long time before the application became apparent. We can reverse this argument: by properly developing the mathematics our applications need, we surely will obtain all of interesting pure mathematics.
Perhaps the pace of applied mathematics would rise sharply, given cheap intelligence. And this* may end up being the forefront driving progress in mathematics.
*Or maybe a split between the human domain and the practical real world. Where the human domain might end up with a variation of a "No machine contributions" policy. Sorta like the recent gcc policy.
[Fig Tree] "By end of the story Gilgamesh is human; He is Us; We see the Deep; That's our gift as humans; We fight to understand something no god could ever could (emergence and richness from scarcity); The gods don't need to fear death so they don't have to live an impassionate life; It means nothing to them to run out of time; "Seeing the deep" is the apex of human potential, not a transgression into godhood; It's something that only humans can do."
He's not an AI Luddite by any means.
It’s like someone offers to build mag lev gym weights. It’s very cool that I can now lift the 500 pound weight with a finger. But what will I do when there’s no power and 500 pounds to lift?
Of course, cognition isn’t a single outcome problem like weight lifting. But we build cognition not wholly unlike how we build muscle: one needs resistance. Otherwise I’m not at all confident we “learn” in any depth.
Then an AI bot draws a false Ramanujan analogy, is corrected several times but keeps insisting.
At the end a meek comment that Tao helped in opening Pandora's box, without engagement or likes.
Modern academics have been conditioned to feel powerless and obedient. The proper way would be to ignore all AI proofs and not grant them access to journals, because OpenAI admitted themselves that plagiarism cannot be ruled out.
I have seen private correspondence between one mathematician working on Navier-Stokes and OpenAI that makes it sound like OpenAI deliberately scooped this Navier-Stokes result. The alleged correspondence also contained veiled threats if said mathematician went public.
Now that problem solving proved to be easy, mathematics will become even more interesting.
I would like to see the Clay Institute give zero recognition for formalizations without human-readable proofs. That would incentivize OpenAI to scram or create something that's actually useful.
> @tao curious to know what makes LLM fundamentally different compared to (other) automatic theorem provers?
It highlighted that "automatic" is a spectrum. And the norm is shifting toward "fully" (even if the process is chaotic).
Article arguing math is the next "human calculator".
That’s not universally true. Some conjectures are renamed after being proven. For example, Fermat’s Last Theorem is now sometimes called the Fermat-Wiles Theorem, the Taniyama-Shimura Conjecture is often referred to as the Modularity Theorem now, etc.
Yes. We already knew this. Are we actually surprised it's happening?
I guess we are.
The thing is, nobody has time for that. Look at Mochizuki's work. It takes years of hard labor by high-level mathematicians to come up with stuff like that, and years of hard labor on the part of other mathematicians to validate it. The low-hanging fruit in math has all been picked, AI or no AI, and Tao doesn't seem to acknowledge that.
The mathematics community needs better tools or they're out of business anyway. Now they're getting those tools... and bickering and complaining about it?
That's how I view modern math. There's the occassional discovery, like Linear Algebra, or the Fourier Transform which has basically allowed math to improve civilization to such a degree, that they deserve to be counted among the most important works of civilization.
Yet the best these weird math theories usually provide is proving something is fundamentally hard to compute, or distributed in some peculiar way, or maybe knock down some of those aforementioned problems into easier tiers.
This all ends up practically in cryptography. Hashes, symmetric asymmetric crypto, zero knowledge proofs etc. etc.
These are well and useful, but not that useful and the new ones often end up functionally identical to the old, from an utilitarian point of view.
It seems like new age requires a new form of publication instead of sticking to 17th century concept of a "paper".
Mochizuki's claimed proof of the ABC conjecture is not unintelligable; it has errors. There are no proofs written by humans that are not human-readable, because in order to come out of a human mind, the proof must have fit there originally.
The four color theorem states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. It was the first theorem proved with substantial computer assistance. The theorem was proved by showing there could not be a counterexample. The authors made a list of maps where if a minimal counterexample existed, it would be one of these maps. There were 1,834 maps in that list, and each one was checked by computer. You could turn each of those cases into a picture or paragraph, but the resulting artefact would not meet my definition of human-readable.
Human-readable does not just mean in English. Humans can only hold a few objects in their short-term memory at once, not hundreds. Though some proofs require significant background knowlege, any proof written by a human will respect the fundemental limits of the human mind. There are no proofs written by humans that are not human-readable, because in order to come out of a human mind, the proof must have fit there originally.
I suspect large lean proofs generated by LLMs do not respect the fundemental limits of the human mind. If no human can read and understand them, no one can extract value in the form of reusable techniques, additional problems, progress towards related problems, and everything else Tao mentioned. If LLM proof generators can be made to write proofs with the same value as humans, that would be great! OpenAI would be a celebrated collaborator if they created as much value as a human does.
Jokes aside, this seems like a pretty weird take. What's stopping mathematicians to make this renewable?
Why not spend some time and effort (presumably using AI) to pose new open problems that are fundamental in nature?
As a field, math could start crediting the person who comes up with a great question, rather than the AI brute forcing a LEAN proof.
It shifts power further from the worker toward those with capital.
Just because the length of the arm is longer with an AI org doesn't mean it's somehow fundamentally a different system.
The future is that if you don't use AI your work is a lot easer to reach against someone else who has it.
That dude hand writing code with punchcards can be lapped by a 20yo with python, what's different?
The solution is also different to theirs? Literally no evidence of plagarism?
In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?
Struggling to understand how a solution to a millennium problem like this isn't a net positive. Presumably Open AI employs mathematicians in these efforts anyway. And I can think of far worse uses of the AI compute resources.
Time to consider re-training to become a nurse, electrician, auto mechanic, or a plumber.
This is not at all what he said? I'm not sure how you got this from anything he wrote, actually.
You need to solve the second to get a PhD. For good reason. The second is way harder than the first. And now AI is making second way obsolete. It's not the end of the world, but it is the end of how things have been done for centuries.
I think the reasonable ethical question to consider when a company has an incentive to prove its value, and learns of the state of the art of a flashy research topic being close to a solution, and then throws loads of resources at trying to solve it first. I agree that "AI" didn't "take" but it's not an incomprehensible shortcut of criticism of the aforementioned behavior, not that precision should be avoided here.
I agree though if they did plagarism it's really really bad for OpenAI. They would instantly have evaporated all remaining good will to gloat over a stolen discovery.
I think there's an important part which is that the nature of the beast implies there's effectively zero way to do that.
They have no idea what context, triggering this pathway, was from this guys reddit post 4 years ago.
I also assume that if any AI lab could actually do that it'd be an instant pissing context over who gives who credit the mostest. "This software engineer wrote a sick sorting algorthm everyone now uses, thank you random dude for that training contribution"
Maybe we will in the future, maybe we'll learn that the solution to this prize is from 3 peoples schizo rant on reddit a decade ago. Would be kinda sick actually.
"Early in the history of Multivac, it had become apparent that there was one big bottleneck: the questioning procedure. Multivac could answer the problems of humanity, all the problems, if -- if it were asked meaningful questions. But as knowledge accumulated at an ever-faster rate, it became ever more difficult to locate those meaningful questions."
[0] https://web.archive.org/web/20150118004835/http://www.sffaud...
This is already the case with most DevOps jobs with Claude Code. There are only that many finite issues that need fixing in production, even for very large systems.
So you don't need all that many people to run ops these days.
Mathematics has always been highly competitive.
Dudes straight up used to hoard solutions to equations and use them in math battles.
(Because they are my private RSA keys)
"But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed."
But yes; him reaping the benefits of himself having the idea first was part of it too; as far as I am aware.
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Which is still something completely different than some anonymous organisation keeping mathematical research secret because it is better for hype reasons. One is competition between individuals or groups within a field; the other is boring and sometimes borderline nihilistic generating of mathematical knowledge as an marketing asset.
I found this behavior against healthy science practices and only driven by ego. Unfortunately, I find this too often at work (working in academia). Most probably I'm too naive...
I read the Mastodon thread as more about the 'flattening' and 'rawness' of the proofs these systems and their operators are producing. I mean what is the cultural significance of a lean proof that is half a million lines long or something? And what tools can be extracted for further work from such a construction?
The late William Thurston wrote about the culture of mathematics in that sense.
It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle
Did you comment on the wrong thread? The original one makes more sense :)
(the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them).
FWIW, your post comes off as a middlebrow dismissal, surface level and not actually engaging with the substance of the comment. It's also just wrong. You claim "it’s also a misstatement of mathematical history", but don't specify which part. That there's "centuries of traditions of open science"? But your examples are from centuries (and millennia) ago, and there was never any claim that these traditions are universal.
But more fundamentally, competition doesn't mean you can't also have open science. And the very long, damaging events like the Leibniz/Newton feud are exactly what make many mathematicians work to maintain a spirit of collaboration and attribution even when they're competing on approaches.
Reads like nothing but historical context
Also, your third paragraph is highly ironic.
Should it be competitive? Is it more useful to be collaborative? How collaborative can it be when it's fundamentally competitive? Is it only fundamentally competitive because of some common 'quirks' of math types, or are there deeper forces pressuring it to be competitive?
These are all questions that I think are worth discussing, as is the note that the pendulum seems to be swinging away from cooperation in the face of competing for $trillion+ valuations (and a real enthusiasm for proving cool math stuff). The alternative, tweeting complaints on twitter without some context, is mostly a waste of space. I mentioned the history in hopes we could get informed complaints on twitter.
Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes attached.
"In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. "
I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.
Except way more nefarious than I expected
Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource."
The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and offer a prize to solve them.
It's a marketing game for them, any societal benefits are secondary. Winning a prize is going to get headlines and feed into the "AGI soon, machine replaces another career" narrative they crave unlike coming up with some (possibly) interesting problems.
For anyone facing an existential crisis on AI, your ace in the hole is your humanity. Only you have it, and only you will be the best judge of what is good and interesting (to a human at least).
Average HN Poster: [nervous sweating]
As far as I can tell, it's still not possible for an agent to reliably determine if a question is a good question. That means the test part of the loop cant be fulfilled.
If theft becomes more profitable than genuine creation, then nobody will create anything. Then there's nothing to steal, at which point all progress collapses
What is interesting is that LLM's do not directly violate copyright. The settlements we have seen are for how the works were acquired (that was a copyright violation) not the use of the works.
The vectors of a book, or a paper, are not the paper. They are, for all intents, facts about the work itself, and more generally writing. You can not copyright a fact.
It also means that the weights, the things that (mostly) matter can not be copyrighted either.
To block progress. Got it.
The awkward part about all of this is that we're about to enter an age of extreme enslavement at the hands of the major tech companies if we do not focus on distribution of hardware and research, so that everyone can participate in the abundance and automate their daily lives. If we're beholden to frontier labs because they have hoarded all of the cutting edge hardware and we're left with overpriced scraps, we're collectively screwed. They will ensure a false economy is maintained so they can clutch onto a permanent class hierarchy of haves and have-nots and remain the key global decision makers. Automating hardware manufacturing is irrelevant if the hardware is not being distributed fairly, and is weighted to real scarcity instead of artifical scarcity.
Take Louis Vuitton for example. They can mass-produce their products for pennies, but they're artificially scarce and incredibly expensive. Imagine if ALL clothing was the price of LV. Now imagine this applies to every single thing you can purchase (or rather, rent - if some of these "elite" get their way), because they've cooked the economy and swallowed all industry. That's where we are headed if distribution and decentralization is not a priority for the world and we let labs like Anthropic pull off their regulatory capture stunts.
I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.
Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.
The AI labs' approach to math is immature in a way they can't get away with in coding. In coding, they realize that a pile of code that technically works is not enough: they need the code output to be a foundation to build on, and they need their agents to work well with humans which means explaining things in a way that makes sense.
In math, their goal seems just to be to exploit mathematics' reputation as full of hard problems with a general population that can't tell a pile of Lean from a good proof. OpenAI pretty much said this work is just to show off at the end of the post. Anthropic said their FLT formalization is a research artifact they do not intend to clean up or improve in any way.
Besides uniting mathematicians in irritation at the labs, the other flaw with this strategy is that it ignores that organizing knowledge is part of intelligence, much like not just producing a mess that runs is part of programming. You can write a proof that uses algebraic geometry because someone organized what could have been a bunch of disparate ideas (or fragments of a Lean repo no one will read) into a toolbox where an expert can find the tool they need.
I hope they change tack. Perhaps instead of making an explicit strategy of taking the credit from mathematicians but doing little for actual understanding, they could let some math departments at their swarms or best models, ask for a bit of acknowledgement, and hopefully they approach it by trying to write good papers, simplify, etc. rather than just rushing for headlines. (Tao's post about digesting an LLM-generated proof https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the... is an interesting read for a sense of what he means by 'digestion'.)
On that last note, it's also important (Tao's also noted) for the mathematical community to properly value digestion and organization of results, so that given the incentives of mathematics and availability of new tools you end up with good papers and textbooks and so on, not just mathematicians taking the labs' current role of pushing incomprehensible-even-to-specialists proof code to repos.
My sense of the word 'share' is that it traditionally involves agency by all parties involved. There are a lot of words in English for describing taking things without permission and profiting thereby - words like piracy, banditry, and larceny.
> [I]t is now the identification of a promising problem which is the scarce and precious resource. We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential.
Sounds like they're going the way of the DoDo. better take that PhD, migrate to the new world and become a tuktuk driver.
> In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.
Presumably it is only a matter of time until these frontier models are used to create new interesting conjectures. I don't get Tao's line of reasoning.
I probably have delusional expectation of what a mathematician of his level should be talking about, but I expected from him a pure objective analysis on what to do with this new AI thing , what are its limitations, how it can improve the field and the creation of human knowledge, etc.
But within next 10 years as costs drop significantly and even more improvements are made, yes it is very likely that almost every single existing math problem will get a serious AI cracking done on it
If an AI solves a problem in an unenlightening way, then there's no reason for mathematicians to stop studying it. Pythagoream Theorem has hundreds of different proofs!
If an AI solves a problem in an enlightening way, mathematicians should study it and propose extensions.
Not sure I agree with this. AI generated proofs can still be analyzed and mined for useful insights. I suppose he's saying the process of banging our heads against the wall on a problem can itself yield useful insight? But what is stopping us from analyzing a proof after the fact. And if we can generate many different versions of a proof that should help us develop a much deeper understanding of the problem than we would have without being able to perceive the "proof landscape"...
What is going to happen is a complete revaluation of things like "finding a counter example to a famous problem". Even if someone finds a solution to a problem like this with pencil and paper, nobody will believe it, and they will assume that there was an AI involved.
Further, sitting and doing math with a pencil and paper will no longer be a reasonable strategy to build a reputation or career, beyond the benefit a mathematician gains to their own intuition and skill. People who work hard to build intuition and also use AI effectively will dominate the field.
In a world where everyone is using AI, the open problems that remain will be the ones that are AI resistant. This is no different that how things work now, mathematicians wait until they are fairly confident someone won't rapidly solve their problem before they start talking about it. They will do the same thing in the future, except in the future AI will be part of the toolset they use decide if they are ready to share yet or not.
Edit: Ok I believe I was generally right here, but I just read the details of what OpenAI did. They didn't solve a longstanding problem, they got tipped off to an approach a mathematician was using and would likely result in the solution very soon and they finished it first. If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.
Which I don't see a reason for Anthropic and "Open"AI not to, given their not so stellar track record with IP of individuals/entities-that-are-not-rich-enough ;)
This will be literally every field, sooner or later, at least until the human and their intuition is just slowing the AI down. There’s no scenario where humans without AI beat humans with AI in the long run, unless there are fields where the “alien intelligence” somehow hurts more than it helps (like artistic pursuits perhaps?)
1. I give you a proof, you tell me if it's correct
2. I give you a theorem, you give me a correct proof
3. I give you nothing, you give me a theorem
1. is largely solved by modern LLMs and they took a big step toward 2. today with the Navier-Stokes proof. But they're definitely not there yet. It's unclear what progress is being made toward 3. for the time being that remains the realm of humans.
Once that is cracked, you throw more compute at it and practically every industry will collapse on a long enough horizon - with digital industries going first. Anything that requires physical hardware will require time for the machines to bootstrap, but that'll get there too.
Although, there are a few human-centric industries that will survive, for example: prostitution. Maintaining it's edge as the world's oldest and most enduring profession.
Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career of the professional that takes the challenge.
A lot. In fields where knowledge is incrementally building on previous work the reason the whole field hasn't collapsed from the replication crisis is that usually the results that are really high impact are replicated in as an initial step in new research building on it. It's almost never the focus of the paper but you'll often find a quick mention in methods/supplemental of some previous work that was verified to be valid by a replication of a key technique etc. you'll have crisis where old tools are found to be problematic and findings end up revisited etc. Plus fields like clinical research where there's an awful lot of focus on replicating findings using staged clinical trials with increasing statistical power to determine if new interventions work - that's driven by regulatory requirements grounded in good science and a lot of people make careers in just that.
I don’t think this is true, especially for novel or unexpected results. I suppose it depends on what you mean by scientifically, and there is a debate in the philosophy of science about what the value of research even is, but a successful replication does not result in substantial updates to one’s beliefs in the way new research does. And if the goal of science is to change our beliefs and bring them closer to what is “real”, successful replications can’t be as valuable as the initial research almost by definition.
If you examine the consequences of the inversion of the peak, you realise the need for a balanced perspective.
1. proof that there is a solution
2. a solution that you can work backwards from to build understanding
So, both of these pieces of info would be at least marginally useful for advancing human knowledge.
This is only true in the most trivial sense. A solution is a solution, sure... but how do you know it's a solution, and not an incoherent jumble of words? A human has to review and vouch for it.
Just because the AI gives you an arxiv-worthy PDF, or a Lean proof which compiles, doesn't mean it proves what the AI says it does. The AI could give you the same PDF/Lean code and says it proves the opposite, how would anyone know the difference?
You can't advance human understanding unless you produce things that humans can understand.
If Fermat had a book with larger margins, would Weil have devoted so much time to proving the Taniyama-Shimura conjecture? No one can say.
Yes, that is Tao's premise, I'm just not sure I buy it. Suppose an oracle existed which could answer any question truthfully. Let's ignore the mechanics of this for now, but it could say things like "the Riemann hypothesis is False" or whatever and we would take it as gospel.
Does this mean that we wouldn't have mathematicians or physicists or computer scientists or biologists anymore? I genuinely don't think so.
Is there value lost in them working on problems that don't have solutions instead of problems that do?
As far as researchers sharing their data/notes with the AI hyperscalars looks like that would be coming to an end wihth a mor guild-like structure going forward to prevent their progress being fron-run by the AI labs.
Removing this measure suddenly means that academic mathematic norms need to adapt rapidly, and, even more importantly, intrinsic motivation for many mathematicians needs to change rapidly. That is understandably a sea change for the current mathematics community.
- taking all the books in the exchange (which is technically allowed, no rules about how many books you can take)
- destructively scanning them (still technically allowed, no rules about what you do with the book)
- and not leaving any new books in their place (which is frowned upon, but there's no rule that says you have to replace books you take)
Is the bummer that we got a solution we didn’t want - that navier-stokes is not always applicable or something, but we hoped it was?
prize is not just monetary
You'd be more sure if you read the tweets.
Tao's point is very simple.
1. Working on problems that AI solvers can solve is a waste of human time.
2. We have no idea which problems can be solved by AI solvers...
3. ...Because the AI labs are keeping their negative results secret, and don't tell us which problems they've tried and failed to solve, and why they've failed to solve them (or succeeded at solving others).
There are additional points surrounding it, but that is the thrust of his argument. His issue is not the existence of AI, but the anti-scientific secrecy in how it is used to solve problems. All the incentives around its current use result in closed, uncollaborative work - which while very attractive to a vulture capitalist, is anathema to scientists.
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He also posits that having a solution to a problem is a small part of the value of solving a problem. What the AI labs are doing is the equivalent of a student turning in their homework, which has 100% of the right answers, but with none of the 'show your work' steps. Those steps are a critical artifact for doing mathematics, because the process of solving a difficult problem teaches us things about other problems.
In my work as a graphics programmer I often find that I look at a problem and will immediately see how to solve it, more or less. But the devil is in the details and often nothing works unless you get every detail right. So you spend a lot of time coming up with complex solutions, then boiling them down to simpler versions. In the end you often end up with a fix which is short, simple, and seems obvious. But it gets a lot of subtle details just right and avoids countless potential issues you wouldn't know if you hadn't failed a lot getting there.
And that is actually how you learn and master the craft.
Now, imagine you describe how you sort of solve it to a machine and it spits out the simple, correct implementation and you nod approvingly, never knowing all the ways it could have gone wrong. If this is how mathematics - or programming - is done from now on, no one will actually master their craft. I definitely see why this would worry someone whose career is built on mastery of the craft and a legacy meant to teach the next generation.
The (new) lack of need to redo solutions to already solved things opens up new craft we've not yet conceived.
We shouldn't reimplement solutions and call it self-education; humans have always stood on the shoulders of giants. These tools contain those giants (copyright notwithstanding) and let you receive solutions to solved things; if your mindset is in order you can leverage that to ignore the solved bits and do even more ambitious stuff.
I get kinda freaked out when I feel like all the AI "utopianists" haven't taken the next logical step of thinking about what society looks like when humans are subpar in every domain (and you may argue this won't happen, though I'm becoming more and more a believer that it will, but my point is the utopianists believe that this absolutely will happen, and that it's also a wonderful thing). How motivated do you think folks will be to do the hard cognitive work to focus on things like math problems when there is a good chance AI will do it better?
So, no, my skills have not become subpar, but have only become stronger because of the presence of AI.
I'd be quite happy as a citizen of The Culture.
But, on a more serious note, or more realistic, I find it vaguely insulting that you think that so many of us are just so shallow we'll not figure out how to live meaningful lives in an AI dominated future. The world is already meaningless. Some people live in the US, get tech jobs and live lives of almost indescribable luxury compared to someone born in South Sudan. There are beggars on the streets of most African countries, who are deformed of malnutrition and suffering in pain. There are children who will never know true happiness, and die in misery.
It's all fucking senseless and nasty, and as someone who originated from such a poor country it is infuriating to see spoiled developed world complaints sometimes.
We live, suffer, die. Some suffer less, some a lot more. We find out own meaning somehow, in this brief weird experience that we perceive as conscious life.
If an AI can bring a few billion 3rd world people out of misery, I don't think I'm going to shed tears if that means you have to somehow "find meaning" outside of a high paying tech job.
Disclosure: I did a theoretical physics PhD, but got admitted to quite a few top math programs back when I was applying to math and physics programs simultaneously. If you asked me whether I’d do a PhD today I’d say why bother.
PhDs from poorer overseas do try to get related jobs here, mainly to be able to get a permanent resident visa.
John Naur said the same of coding eons ago [1]. That code is only as healthy as the human mental model of its maintainers. Code is not the artifact, it is the intuitions and insight and mental models of its human maintainers. For a project to sustain, it is vital that the human mental models are kept alive.
[1] Programming as Theory Building https://pages.cs.wisc.edu/~remzi/Naur.pdf
I think the deeper point Terrence is making is simply that the information revolution which is the culmination of the mathematics revolution has less low-hanging fruit today. That was true ever since Claude Shannon
- taking all the books in the exchange (which is technically allowed)
- destructively scanning them (still technically allowed)
- and not leaving any new books in their place (which is frowned upon, but there's no rule that says you have to replace books you take)
The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point.
Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.
Here's an excerpt of the video where he explains his idea: https://www.youtube.com/watch?v=svl_1upFpQo
My summary: Science is like a hike. You have a general sense of what direction you want to go. You explore things along the way. You might make wrong turns but might also discover something new and interesting. While on your journey, you might spot new mountains or waterfalls or other vistas that look like they might be good to explore.
In contrast, AI is like taking a helicopter to your destination. Yes you got there quickly, but you missed a lot on the way.
Related, Tao later talks about the pipeline. It used to be that seminal proofs were rare, so there was lots of time for the community to review them, process them, share them, and summarize them in textbooks. Nowadays, AI helps a lot with generating more proofs, but not so much with the other parts of the pipeline.
Tao also explains the math pipeline issue in his talk at the International Congress of Mathematicians, also shared on HN last month: https://news.ycombinator.com/item?id=49056620
My overall take: Tao is very thoughtful and insightful about progress in AI and math, and about what the tradeoffs are. We're seeing similar problems in computer science (my field), where conferences are now getting over 10x the number of submitted papers over just a few years ago (not an exaggeration). It's straining the research community in ability to review, understand, and present the papers.
Sure there is: problems that require knowledge that simply doesn't exist yet. Until "AI" turns into general purpose robots that can develop new tools to explore the world, it is, in fact, pretty damned limited in what it can do without human help. The world is vast. Math is small.
Biology is replete with examples. Computers "solve" protein folding [1], and midwits immediately leap to conclusions that drug development will also quickly fall. But we literally have no idea how most of biology works, and simply getting to the starting line for drug development problems is often 95% of the battle. Come talk to me when you've done a million experiments to find the fundamental knowledge that unlocks the pathway(s) we didn't know about that makes a drug discovery program possible in the first place [2].
I am not pessimistic about humans running out of challenges. We'll just declare one class of problems "done" [3], and move on to the next frontier, as we always have. The problem with AI doomers is that they lack imagination that extends beyond computers, or perhaps more accurately, are so sophomoric in their thinking that they skip over the hard parts of any problem they don't fully understand. This stuff reminds me of the endless smartypants whinging about the end of human intelligence when chess machines started beating grandmasters. Chess was never really that great a measurement of human intellectual capacity, and we found new things to do with our big monkey brains.
[1] They did not solve protein folding, except in the minds of people who don't fully understand the problem.
[2] ...and invented new machinery to make the experiments possible in the first place.
[3] ...and we'll likely be wrong about that.
Obtaining the knowledge is a process of trial and error, bruteforce, observation, search, etc. Humans follow that same process. Machines can do that too. If they discover or are taught the same heuristics humans use, and a computational capacity greater than that of all humans combined, they will outpace us in this endeavour.
> Until "AI" turns into general purpose robots that can develop new tools to explore the world, it is, in fact, pretty damned limited in what it can do without human help.
For sure, but this isn't a question of if, but when. LLMs are already rapidly accelerating the pace of every scientific field, so advancements in technology will start to compound.
> Biology is replete with examples. Computers "solve" protein folding [1], and midwits immediately leap to conclusions that drug development will also quickly fall. But we literally have no idea how most of biology works, and simply getting to the starting line for drug development problems is often 95% of the battle. Come talk to me when you've done a million experiments to find the fundamental knowledge that unlocks the pathway(s) we didn't know about that makes a drug discovery program possible in the first place [2].
I don't disagree with you. The amount of complexity in biology is incredible and due to its unpredictable squishy and noisy nature and the advanced machinery needed to analyze it with precision at the microscopic scale, vast amounts of data accumulation are still required to discover the useful patterns here, and the search space will be enormous. Still, it will be done. Here is some recent movement in that direction: https://www.anthropic.com/news/model-hardware-standard-resea...
> I am not pessimistic about humans running out of challenges. We'll just declare one class of problems "done" [3], and move on to the next frontier, as we always have.
This is fair, you're probably right about that. I think they'll be challenges for the AI to solve though, not humans as a collective. Hopefully we have something to contribute to that process, other than just expressing our desires.
> The problem with AI doomers is that they lack imagination that extends beyond computers, or perhaps more accurately, are so sophomoric in their thinking that they skip over the hard parts of any problem they don't fully understand. This stuff reminds me of the endless smartypants whinging about the end of human intelligence when chess machines started beating grandmasters. Chess was never really that great a measurement of human intellectual capacity, and we found new things to do with our big monkey brains.
I'm not an AI doomer, I despise the AI doomer movement and the incessant fearmongering, it's incredibly unhealthy and its only purpose is to perpetuate panic in the aim of maintaining and establishing power, and preventing others from obtaining power that threatens the status quo. What cultish idealogical humans will do with this technology if they're left unchecked with staggering power and no one to stop them is certainly a big concern though.
The AI companies have already bought up the world’s entire supply of hardware. There won’t be any more.
I studied math, but I am not a mathematician, so I think I have a slightly different perspective on this than Tao overall. This is certainly the definitive end of an era in mathematics, but I think he's wrong that insightful new open problems are truly non-renewable. They might be non-renewable by humans at the rate at which they are being closed, but I see no reason why AI systems could not also discover insightful new open problems. In fact, once we have Riemann-capable AI mathematicians, I'd personally love to see what the next Riemann hypothesis is, which even these AI systems cannot solve with any amount of available compute.
I think we're about to find that, on the spectrum of mathematical intelligence, the best human mathematicians were only a fraction of a percent forward from the very beginning, and there's a vast universe of mathematical depth that's beyond our ability to imagine or work on directly in any way. We're used to feeling like we're able to directly perceive the Platonic realm, but we're almost certainly going to discover that our own minds, even when joined together over centuries of deliberation, can only interact with a tiny little shadow within it.
Actually, I have to admit I don’t really know what math is. With physics we suspect there’s a universe, and when we study physics we’re improving our description of the behavior of that universe, right? The universe exists whether or not we know how it works.
Eventually, as you suggest, maybe we’ll hit math that won’t fit in anybody’s head at all. What is the nature of mathematics that doesn’t fit in any human’s head? Does it even exist in some sense?
The only way to renew it is to make a new problem that is so hard systems and humans will be unable to solve it for the next ten years. And, in the spirit of trees, the best time to plant a tree is twenty years ago, the next best is today: we do need to start posing some hard math problems and deciding if they are interesting merely because there are challenging or because of something else (eg busy beaver problems are arbitrarily hard, but does solving them imply anything other than "another busy beaver problem was solved"?)
Now, I think AI will solve some, but we'll find out that some are just either unsolvable or wildly huge that nothing is solving them any time soon.
And a whole lot of these problems have been around quite some time, when even knowing how to do advanced math meant you were a landed gentry or someone of high wealth. If those problems fall, they fall. They aren't pets we keep around forever. And new problems will crop up over time for both AI and men to scratch their brains over.
In another comment below, I likened this to clear-cutting a forest. Growing the forest takes a lifetime; destroying it could happen in the next few months.
of course thrrr are tons of problems once you remove this social consensus based filter. if i’m not mistaken Ramanujan left a book of dozens of unproven theorems, for one quick example. i don’t think that that has opened up dozens of fields of mathematical research.
augmented Hilbert's problems of 1900.
Surely mathematicians are creative enough to ask new questions?
If not, then the next set of challenges will be to find questions to ask!
There's an interesting commentary about this: https://mathstodon.xyz/@tao/117237320796901560
> famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest.
Web search turns up Gauss's comment, with a bit more nuance: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of." (https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quot...)
The other problem that Tao identifies is that math has typically been an unusually open subject in many respects. This openness may not work if big AI labs can afford to throw $X million at a problem to scoop you if the rumor gets around that you think you have something promising. Hence, less collaboration, and less chance of identifying these exciting new problems, infinite though they may be.
That is exactly what Tao is explaining in that tweet.
TLDR: Open Problems are infinite, but those which are at the boundary of easy and hard problems and are interesting are far more scarce
For other folks, the post: https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
The transcript: https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...
It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.
Going back to chess, I think the situation is similar where you can’t expect an amateur player to get better by trying to play like a strong engine. I think even professional chess players mainly use engines to prepare or memorize variations that are counterintuitive for their opponent. In other words, getting into situations that look wild, but that part of one player’s preparation.
I’m not sure how it is in math, but in chess, it seems like top players can play just like engines when they are in “normal” positions, so that is where I get a bit confused as to where the direction of insight is coming from because it’s been my view that AI is able to make leaps that we would never think of taking and I’m not sure that anyone could actually learn how to do that on their own unless they were willing to keep failing over and over.
Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?
If this is V0.5 of AGI/ASI then by V1 the only system that will be understanding any of this is the AI itself. If AI creates a new field of mathematics month 1, then solutions to new problems in month 2, then another field of mathematics on top of that at month 3 there's no human who will ever keep up with that.
Or the alternative is a flattening of abilities, the AI cannot proceed further than the collective intelligence of humans and in that case this is correct. We'd be in a future where nobody wants to work in a field with an AI dominating it and when AI hits the limit of no useful training data input we'd have this giant gap of nobody know wtf it's done for years and nobody willing to figure it out and advance it.
Ooo here's a dytopian story: - AI gets better at everything humans do - humans stop trying - AI cannot improve anymore than its input data + human support - AI slowly degrades itself (model collapse) for decades, it slowly hallucinates little by little until its hallucinating entire scientific fields losing quality over time - there's a mass population of people in the future who never learned to do anything and now have to relearn and figure out the equivalent of 100k years of AI work in order to prevent its slow degredation while all the systems they've come to rely on start failing around them. The AI has solved every problem but every real solution is saturated with 1000 false ones. - humanity starts from scratch?
I love the idea of an archive of every solution to every problem existing but it's impossible to figure out the correct one. Infinite library like!
People usually use these tools in math and science to find an answer. Then often they will work it back using more sane or human pathways. So it's shareable or even beautiful.
Knowing the answer has value. But, often in math the best thing was how someone got there.
If you take that away and turn math into a less fulfilling pursuit where you mostly try to make sense of the output of an LLM, and it's "Astra's theorem #18398" and not "John Doe's last theorem", I'd wager that far fewer people will have any interest in the field.
This is really not unique to math, by the way. AI is undermining a lot of creative work. Why blog when you have much better odds of making it to the top of HN with autogenerated blog-slop? Why write books when many nonfiction categories on Amazon are now dominated by AI? The list goes on.
There's plenty of people on HN who think it's nothing new, ignoring the huge change in scale. And those who think this is good because there's no inherent value to human creativity if we can get the same content faster and for less. I disagree.
This is an absurd thing to say. Hacker news is not the only place that knows about the most famous mathematician in the world. Glancing at Google Trends he seems to be roughly as famous as Linus Torvalds. Not exactly a household name but by no means obscure.
The point of mathematics is not to prove results. It is to build conceptual thinking about mathematics. Important problems are important because in order to solve them we have to build concepts tying different things together.
We're not searching for answers. We're searching for insights. Trying to understand the problem causes us to draw the connections and find those insights.
AI gives us answers. But it doesn't help us build those insights. AI has a complete mastery of existing human insights. But doesn't build new ones from its own experience. In a real way, it does not find the opportunity to really learn.
So it tackles problems and either solves them or not. If solved, we now have an answer. If not, it's too hard for humans.
The issue I see with a handed-over proof is tunnel-vision: you explore only the understanding of the proof.
Without a proof, your exploration branches out much further, in directions that could seem fruitless, but may uncover new understandings that are now "hidden" because the handed-over proof drastically lowered the incentives to find them.
Basically: Tasting a delicious soup doesn't tell you how to layer the flavors, but if you want to be a good chef, you better be learning flavors more than you learn dishes!
- If your GPS directs you straight to your travel destination, you are now where you wanted to be but missed out on the exploration. This is the sort of consequences the AI math proofs have.
STEM research thrives on that side exploration and unearthing unexpected things along the way. James Burke's famous documentary Connections spends the middle episodes talking about the unexpected directions that exploration has taken science. It's very hard to credibly make the case that this sort of meandering exploration is not valuable.
Not that different from when trying to read an out-of-control vibe coded codebases, or an sloppy AI long email that someone may send you at 9 AM.
It's about what is non verifiable versus verifiable. The same way it produces "slop" code (which, if you give it test cases, will be 100% correct), it also produces "slop" math.
Code that serves a business function, it's ok if its slop. Math that serves directly a business function also can be slop.
But most open problems are not directly for a particular usecase. People agree widely to attack it due to the perceived possibility of encountering useful mathematical objects along the way, that will then expand the world's mathematical toolset. This is not something that you can easily express in a verifier, and is thus something that is hard to force an LLM system to do.
You are right in that understanding it retrospectively is possible, but that is not going to be as useful as the desired "elegant" objects that expand and unify mathematics. You can't represent these concepts in verifiers.
Again, if you let AI rip at something like say "beat shannon capacity" and suppose it comes up with MIMO as paulraj did, great! It's useful and you can retrospectively understand it, say by expanding shannon to multiple dimensions, as foschini and telatar did. But most math problems are not in that category.
The question then is, if AI is really good at this type of math, how much of the existing mathematical community+process is necessary? I think it will still be necessary, just maybe in fewer cases. Wherever the primary purpose of the math is in a domain and that domain has a verifiable target, we can directly optimise it to that verifiable target in-domain rather than reach for the mathematical community. How well will this work? We'll see. It's not clear if it's even possible to represent most problems this way.
Both of these are simply incorrect - serving a business function means it's valuable to that function.
In the second book of the book series, "The Restaurant at the End of the Universe", the question gets revealed: "What do you get if you multiply six by nine?"
My humanity is not paying my bills.
That's how maths works yes...
[1] https://mathworld.wolfram.com/Taniyama-ShimuraConjecture.htm...
I think that was Ken Ribet?
Grigori Perelman and the Poincaré Conjecture is more interesting. IIRC he turned down Millennium and was decidedly not all about the Fields Medal - mostly because Richard Hamilton didn't get credit? Anyway, I am grateful I had the opportunity to learn about Poincaré in college taking a few classes from a professor who was a key contributor to the conjecture and got a Fulbright for it when I was there
Always been what way? And how does that contrast to what Tao said (since it was apparently "a misstatement of mathematical history")?
> Also, your third paragraph is highly ironic.
You'll have to be more specific, since I engaged with my parent's argument, while they waved away Tao's quote by suggesting he was wrong because of exactly the kind of events that helped lead to the norms and mores working mathematicians have today.
And to reply to the sibling since I hit my comment limit and I'm going to probably forget about this conversation until tomorrow:
But our situation before 2023 was one in which we had an endless abundance of solutions and ideas. I understand that AI can generate bad ideas faster than we can discern them, but we already have tried and true mechanisms to filter good ideas from bad (e.g. the scientific process), why can't they be adapted?
If competition is somehow the opposite of "centuries of traditions of open science", and "mathematics has always been highly competitive", then open science is neither sufficient or necessary for the future of mathematics. Their clear implication is that we don't need to worry about it, though, because it's always been that way.
> Reads like nothing but historical context
They literally accuse Tao of "a misstatement of mathematical history".
For the future of past mathematics, it says nothing about the current future. Also, open science can be nonsufficient and unnecessary but still extremely beneficial and desirable.
>Their clear implication is that we don't need to worry about it, though, because it's always been that way.
Lets just ask him if that's what he meant, I bet no.
This really cuts to the heart of the problem with AI. Not only does AI undermine the monetary economy, it undermines the intellectual economy. What is humanity without the need for collaboration for survival or for intellectual progress, ultimately providing the impetus to build something greater as a result? I don't know, and I'm not looking forward to finding out.
A society can live just fine in a period of abundance. The societies that we currently have on earth do make it questionable of 'we' can right now. I mean I see people posting stuff like "I'd rather burn it all to the ground rather than see one cent more tax" kind of stuff when they have millions. That kind of person doesn't want more people uplifted and it takes away from their idea of being special.
>naive belief in a Star Trek
The naive ones don't read into ST lore to know it comes after WWIII.
The dangerous ones do.
Copyright maximalism is a bad look on a site called "Hacker News." Perhaps other sites beckon.
Frankly it's more of an insult to the "hacker" name to be apologising for big companies profiting off of frontrunning existing work for PR purposes, if the claims about piggybacking on human-directed efforts/prompting are true.
Being pro-copyright in order to protect the work of an individual from being reconstituted into the corporate machine is VERY hackery. Novel use for an existing tool, to fight the dominant system.
(Of course, we're on a so-called "hacker" site hosted by a company run by squarely-establishment individuals acting in an extremely un-hackery-field (investing), so the irony here has been at least one layer deep since the start.)
Copyright is not a hacker thing because it restricts peoples freedom to share and hack on anything they see. Now abusing a bad system to do something good, as you said, would be hackery. But that's not the same thing as supporting copyright as it is right now or wanting to expand it. It needs to be in the spirit of "we turn their own tools against them". The ultimate goal is still eliminating IP laws as far as they limit individuals free speech.
It would be different if it was an asymmetric system. Protect creators from companies trying to make money off their work, but not the companies from users taking their profits. That would mean switching to a form of copyright that only applies to commercial use and abolishing work for hire / any form of rights assignment as opposed to a license. That's one idea I can get behind.
Yes, OpenAI is likely to be found to have acted like a slimeball in this instance, or at least the employee in question may have. But you can't fix that without making laws that will make everything else worse... and only here in the US.
Engines - specifically heuristically-driven ones like Stockfish - don't play like a strong GM. They play engine-perfect chess, which isn't how a GM plays with any consistency.
I'm only a decent amateur (1550 USCF) but when I lose to a titled player it's largely explainable in human terms how it happened.
Think along the lines of the Nicolas Bourbaki persona/collective : " was a collective pseudonym chosen in 1934 by a group of young French mathematicians. None of them carried the name alone; all of them carried it together. And under that name, they launched the most ambitious mathematical publishing project of the twentieth century: a series of texts rebuilding modern mathematics from scratch, on entirely axiomatic foundations."[1]
The hope with many of these problems in math is that in trying to prove that, we get some additional insight into why it blew up that could be applied elsewhere to more general PDEs that cannot be easily controlled.
I think the observation from Tao and many others is that when humans solved these problems, the additional insights into intuition and theory building came for free since humans can give expository on what they found hard or what was their own intuition. This is much more difficult or tedious to extract from an AI model. Even when people did have access to the chain of thought, it wasn’t always very helpful to figure out what was the exact thing that made it all click. This is even more difficult how that the CoT are hidden but I would think the sort of difficulty of extracting the key ideas for a human might be worse now with more advanced models.
There’s a long term aspect to this too where we have historically used these problems as markers for the other parts of mathematics but if AI can solve it all, then suddenly this signal is not very meaningful.
Maybe to bring it closer to home. If an oracle just gave you P \neq NP, then this would be generally uninteresting since this was already expected. There’s a deeper question of why that needs to be answered. However, one would hope that creating such a separation would give us tools that allow us to create lower bounds on a lot more problems we do care about and perhaps some bigger insight onto what makes a problem intrinsically hard or easy. These long term considerations are helpful but are definitely more vague. The remarkable part is that AI is separating the part about proving theorems and the “free” insight you get.
There is a parallel with autonomous vehicles in real life. On northbound 1 in SF going through GG Park, the left turn lane onto Crossover Drive is always backed up. Waymos often do a very late merge into that turn lane in order to jump the queue and save time. With 360 degree sensing they can do this safely in real time but it feels too risky for most humans to attempt.
If solving any problem is possible with ai, the job of mathematician is finding new problems, not trying to do what a machine can do better.
As for training, most of it is already done by solving known problems, and no one was suggesting to forget analytical geometry so that solving school geometry problems becomes cutting edge research.
Not necessarily. The "something superior about human intelligence" may have dependencies that "these AI solutions" are able to eliminate, such as the motivation to refine intellectual talent to a high level. Basically, AI could kick the ladder out from under human intelligence but be incapable of actually surpassing it in important ways, enabling a burst of advancement that's also a dead end. Sort of like https://en.wikipedia.org/wiki/The_Road_Not_Taken_(short_stor....
So the AI could be inferior but there's still no useful work for humans, because the environment doesn't allow them to work up to that level anymore.
This is kinda feeling a bit like SBF's coin flip bet: https://www.businessinsider.com/sam-bankman-fried-coin-flip-.... Achieve human-superior AGI this generation or humanity stagnates.
I meant what you're saying. That it's OK if it's slop if it serves a business function.
Edited
Most patterns that exist are incompressible. Math is basically a search for those compressions that do exist. An example I personally really like is the amplituhedron: a geometric structure that humans have just barely been capable of recognizing compresses information about scattering amplitudes and Feynman diagrams. That one happens to be within our reach, but it's right at the edge, and we can only imagine what glorious, wondrous compressions exist in abundance beyond the edge. Math accessible only to superintelligence would exist entirely beyond that edge, compressing patterns whose existence we cannot even detect using objects and constructions that we cannot grasp.
As an aside, I also think this is why AI is quickly becoming superhuman at math: intelligence is essentially a form of pattern compression.
Take infinity. Infinity can't fit in your head, hell, it can't fit anywhere, but you can abstract away the endlessness and look at infinities of different sizes, et al.
Now, is there a single formula for something actually represented in this world that would take most of a humans life just to read it, no idea.
Download 1 million books and you are OpenAI
2. The topic we are dissussing concerns LLMs being trained on logs from previous LLM chats. If you're prompting a model and it spits out some unique mathematical insight, you do not have copyright on that.
People don’t behave this way, but a high percentage of all papers have known flaws and that goes up even higher when you consider unknown flaws. Replication doesn’t own its own solve the underlying issue, but independent replication removes a huge range of potential issues on top of providing more information.
That alleged superconductor from a few years ago - everybody kind of held their breath and waited for the reproduction.
You have created a fraud machine. Why? With no answer checking then why not make up the most fraudulent crap you can get away with?
Examples: A huge portion of recent non-reproducable science papers.
---
Your thinking, along with everybody that's doing this rat race is causing the pumping out of papers with questionable data, but very little to ensure we are actually making correct science.
Sure, you can reduce the 99.9% of research labor that matters to "a process of trial and error, bruteforce, observation, search, etc.", but that's like saying that nuclear fusion is only a few technical details away from implementation. We already know the theory!
The part where you're closest to being correct is "trial and error" -- it would be great if a robot existed that could do any experiment, tirelessly, with the mechanical fidelity, intelligence and creativity of a human. That robot does not exist. Moreover, the fundamental techniques to do the kinds of observation necessary to unlock the parts of science we don't know about do not exist. They must be invented. So now we have two problems. The problems are recursive and interlocking.
Biology and chemistry are the sciences I know best, so I will use those examples -- every major breakthrough of the last 50 years has involved invention of some fundamental new mode of observation, such as crystallography, NMR, mass spec, electron microscopy, various kinds of light microscopy, DNA sequencing, PCR, etc. Someone invents some innovative technique, and a wave of progress happens. Expert practitioners in in the lab are probably the second rate-limiting step, but the part that LLMs can do -- taking data and turning it into hypotheses -- is the part that carries the least value. Any postdoc has enough ideas to keep a lab going forever.
The thing you linked about Anthropic creating a "robot standard" for operation of lab tools is great for Anthropic, but that's about all. There's tons of lab automation tooling already. Having LLMs run the microscope is maybe a cool automation technique if you have the kinds of experiments that benefit from it, but those are rare, and they're still ultimately limited by people doing the upstream work.
AI will certainly help people be more efficient at their current scientific jobs, make better methodology more universal, etc., but suggesting that it will replace actual scientists is just science fiction.
Technology is not made in a vacuum though, it is a process of incremental advancements, often in parallel, over multiple industries. If LLMs are watching the worlds scientific and hardware progress on all fronts and compute is dedicated to exploring combinations of new ideas and ranking them on estimated practicality toward outstanding problems or current goals that humans have, I see no reason why they won't be able to come up with creative new technology. The issue with cutting edge technology is that it's expensive and time consuming to validate. If AI can develop sufficient simulations, it could be validated digitally.
Surely you don't believe that this isn't around the corner, given the scale we are now seeing? For example dedicating 80,000 agents over 88 hours to write a proof for Navier-Stokes.
Maybe this isn't a wildly creative exercise or you consider math "small" in comparison, but I think it's quite clear to see that this isn't a question of whether it's possible but rather a question of how long it will take to get there.
There's no shortage of talent being dedicated to this pursuit, either. For example: https://finance.yahoo.com/technology/ai/articles/deepmind-ch... - The company aims to accelerate scientific and engineering breakthroughs by building autonomous systems that handle entire research cycles.
They're good to give to new mathematicians, and they're good to help humans understand the shape of the problem space and relative difficulty with the tools we have.
Cheesing these problems with LLMs gets rid of both the training benefit and our ability to create good related problems. There's an aesthetic part of this, too, that LLMs do not capture.
But from paleontologists' perspective, they go out looking for dinosaur fossils when they have questions that digging up a fossil may answer. The metric they're focusing on isn't tons of fossil mined out of the ground, it's a developing understanding of extinct life.
Ask 1000 different individuals if Terrence Tao rings a bell. If 5% or less can answer you who Tao is, it is safe to say that Tao is obscure.
I'd be very surprised if you can find over 50 individuals, out of the 1000, who can tell you who Terrence Tao is. Even big names like Euler or Gauss would surprise me.
Stop 100 people on the street in NYC or Berlin or Tokyo and I bet none of them will be able to name any living mathematician. A few of them might know Linus, though.
[0]: https://trends.google.com/explore?geo=US&q=%2Fm%2F047mjr%2C%...
Were early in this cycle, we will learn to do more, and exercise our new capabilities more fluently, which in turn will create more skilled practitioners
Consider the abacus, calculator, computer, etc, each of these enhanced mathematicians’ capabilities and thus outputs.
For 3000 years mathematics has only been a "tabletop science". Even big programs like the classification of finite simple groups have been comprised of small teams chipping away at different (publishable) parts of an overall program.
This latest Navier-Stokes advance cost something like $22m in tokens, already well beyond what a mathematician's research grant can fund. As the easy open problems get mined, the cost of frontier progress will continue to climb. Some part of mathematics as a field will need to transition from tabletop science to big science: Coordinated top-down programs addressing high-priority objectives.
TBD is what the role of individual mathematicians will look like in a "big science" paradigm, but we could look to experimental high energy physics for ideas. For all practical purposes, AI converts math from a theoretical field into an experimental/observational one.
The situation is not that different from John Henry competing against the machine. The question is really: Do people deserve to be allowed to continue doing what they have always done, when doing it is no longer necessary to advance the greater good?
At some point in my suggestion the machine will ask better questions than you, and that will be pointless as well, and you keep doing what you like doing, or you move on to something new. But if you keep tying your value to outcome and recognition instead of process you are going to have some incredibly depressing years ahead, and every time will just be as hard to stomach because of your ego.
It sounds like it could be a pretty horrible world for the majority of people, especially if the AI overlords are only concerned about themselves (be it a small number of people who control the AIs, or the AIs themselves becoming independent entities that prioritize their own lives).
It's a bit baffling to me that folks think that the reason so many live in poverty now is a resource problem, and that an abundance of resources from AI would solve this problem. It is certainly a resource allocation problem, so I don't understand why folks believe that AI or our AI overlords will decide that everyone gets 40 acres and a mule.
After all, its not like some benevolent super-AGI will be deciding who the spoils get divided amongst, they will remain tools to perpetuate and intensify every incumbent power imbalance.
Maybe glance back at your comment history and find you mostly are complaining about very developed world things and investigate why there might be a disconnect between these two aspects of you?
> If an AI can bring a few billion 3rd world people out of misery, I don't think I'm going to shed tears if that means you have to somehow "find meaning" outside of a high paying tech job.
Well, this is begging the question (how awful are people for worrying about the future when the future is going to be so great?), but you're also missing that there's more to humanity than surviving or not, and there always has been. There's no reason to reduce this to a dichotomy.
It is rather consensus that by now,
- too many potential weaknesses for RSA are known (in particular in how it is used in standardized cryptosystems [1]),
- it is a bad idea if too many "free parameters" of the cryptosystem are decided by the practical implementation (in RSA: the decision which primes to use to generate the key pair) instead of these parameters being part of the standard
Thus the advice to drop RSA is in my opinion sound.
But otherwise: many people indeed have the suspicion that the panic that a sufficiently powerful quantum computer might get developed in the next year is indeed used to push novel "post quantum algorithms" which
- have not been analyzed as thoroughly as older algorithms, and thus might contain weaknesses,
- were covertly developed by some three letter agency, and contain backdoors.
In other words: your suspicion might not be unfounded - this just does not imply that RSA does not have its risks and should thus arguably indeed be abandoned.
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[1] Just to give one example: in the past, RSA was often used together with the padding scheme PKCS#1 v1.5, see [2]
[2] https://en.wikipedia.org/w/index.php?title=PKCS_1&oldid=1342...
> You can't advance human understanding unless you produce things that humans can understand.
And you can't advance human understating unless you maintain that understanding.
I can see a version of the junior software engineer problem here: AI wrecks the problems that could train and motivate the next generation mathematicians, so students abandon the field because there's no place for them. The senior mathematicians who can review/vouch/prompt for AI output like Tao retire and die. Then there's no more math that anyone can understand and no more open problems for it to solve.
And that's probably happening already. I've read articles about AI performing the journeyman work that mathematicians cut their teeth on, rendering years of work obsolete, and derailing the careers that work was meant to start.
One reason is despising that line of work. Quant firms were pummeling my @prestigious.edu inbox throughout my PhD and I fucking hated those parasites. Well, jokes on me if AI shatters my current career.
Btw a former classmate Caroline Ellison who went to Jane Street eventually landed in prison, lol.
And it's not just github, everything seems to be down on a daily basis right now.
It used to be that you compiled an executable, copied it onto the target computer and it worked. Then came servers, where you had to open some ports, set up some init scripts, maybe a db connection string.
This has evolved into a fractal of awfulness of a myriad of options and problems, each of which more elaborate and specifically limited than what came before, so you can't really use any one of them, but a combination, which is bound to interact weirdly, giving you some unique problems.
LLMs are great at solving these kinds of problems, but these problems should not exist in the first place, and they just waste everyone's time.
When it comes to actual meaningful product questions (and not just 'how/where does this code run'), LLMs are basically next to hopeless.
Since LLMs are very good with dealing with the BS, I hope they'll be useful in making people recognize and do away with BS altogether, and we can go back to the simple olden days.
>It used to be that you compiled an executable, copied it onto the target computer and it worked.
Java had been marketed last century as a 'write once, run anywhere' language. Tells you a lot about how portable executables in other languages were.
This was unpublished research that was stolen, and constitutes plagiarism and academic fraud by even the strictest definition
It was not stolen, it was willingly given.
It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even more useful to have mathematicians available who already understand that specific math.
The random engineer looking at a funny problem 10 years later now has the literal author of the math to talk to about it and implement it.
I have never even spoken to a world class mathematician and now I can have them design with me?
How is this not better in almost everyway?
If said human is kicked to the street with thousands of other homeless people that can't get jobs because AI then robots replaced them, then those fast math problems sound like a pretty bad trade off.
Now, if there's some future where AI leads to abundance and we can all live off UBI, well, probably a worthwhile trade.
The biggest issue I see is the more controversial people leading the AI race at the moment are not the kind of people I'd hand kids safety scissors much less the future of the human race.
Most modern mathematical problems are sufficiently abstract that their proofs or disproofs have no direct application. There's no problem you can fix or invention you can build based solely on OpenAI's construction, because analytic solutions to the Navier-Stokes equations are not used for practical purposes in fluid dynamics. The problems and their proofs are only interesting to the degree that they help us better understand how the math works.
IIUC the Navier-Stokes proof is understandable by human beings, but if it weren't it would be no more useful than a proof that 3 dimensional florg-complete entry seams have no durdle-nodes.
Unless your argument is that mathematicians are effectively useless?
I am assuming that's not your point though.
It's always been a bit bizarre that this isn't the case. Mathematicians are almost always working on problems that there is no good reason to expect to have utility in the real world... problems they selected because of their elegance or whatever... yet there is a strong historical trend of their work having huge importance after the fact. Sometimes in fields that weren't even invented yet at the time of the work.
There's something to be said for the idea that disrupting a system that is working well for no apparent reason is a bad idea.
(I suppose it's possible that in some distant AI future there might be no value in people understanding theoretical math, but I'm pretty skeptical of that; to me it seems like the same error as thinking nobody needs to understand multiplication because you can ask the computer to solve any multiplication problem.)
The tech got so advanced people kind of used the advanced tech, but looked pretty much like a how medieval society would. Even their politics.
FWIW this is my understanding of his argument and I am not a mathematician.
As Tao points out, merely suggesting new open questions isn't really sufficient. Part of what gives these problems their fame is their notoriety, their difficulty, the fact that many prodigious mathematicians have spent an evening or week or month or several years studying it.
It wouldn't be as interesting if it had just been solved by the fifth random mathematician who considered it
Notably, gardening a new field of study in math is somewhat nontrivial. You have to introduce the field, illustrate some relevance or connections, and then - and this is key - not solve all of the low-hanging fruit yourself! Because you need somebody else to become an expert in that particular field.
The analog in programming is: if a large company merely open sources a product that's decent but not great and in a language nobody wants to maintain, but they don't commit to maintaining it themselves.
Suddenly there's a bit of a vacuum because in order to provide something of value, you either need to:
1. Implement something more complete than was initially open sourced
2. Or maintain something in a horrendous language while incrementally improving it and keeping it relevant
3. Or rewrite it into a tolerable and maintainable modern language.
What the large company has done is create a vacuum in the tool space where you now require extreme motivation to get someone else to step in.
Note that in this scenario, in 2026, it's actually not such a big deal. I think several recent models could happily translate it into a more maintainable language themselves or happily maintain it in the original crufty one. And so the question is: which parts of this analogy are true in math, too?
You're either being intentionally obtuse, or unintentionally ignorant.
In the case of mathematicians, I think not as researchers. What would a research mathematician do? I don't think there would be any reason to try to gain insight from proofs that AI made for the sake of understanding. I don't see what that would achieve besides just retaining extremely niche knowledge (which AI or the oracle already does). The whole point of having that knowledge was to build toward novel work which the AI/oracle does. Also, the time spent and difficulty understanding them could be very high but with no payoff besides just understanding them because the AI/oracle would be used to solve all the problems anyway.
It's similar to Mochizuki claiming to have proved the ABC conjecture, with a proof depending on ideas developed over a large number of obscure papers, that required mathematicians to spend a lot of time before they felt they understood it well enough to point out flaws.
If AI solves all famous open problems and the non-famous ones, too, without advances in the readability of their output, there'll still be some work to do to digest and rearrange the proofs for human consumption. During that process, the mathematician may well get some new ideas...
Now I wonder if someone could port his proof to Lean
For all your bombast, do take a moment to note that no mathematician has actually said openai has not produced a correct solution. Should make you think.
To make an unrelated anology, the presence of Lebron James definitely dampens my ability to contribute meaningfully to basketball. Even with all my practice, I cannot achieve his level. But if he didn't exist, and the best 90% of basketball players were eliminated, then I might be able to meaningfully contribute. Of course, I can still play basketball as a hobby, but you were talking about making meaningful contributions on hard novel problems at the frontier of knowledge and ability. That frontier is moving away from humans and will soon be the domain of electronic experts.
> when AI can do most of the work
You're not really responding to the core hypothetical of his comment
To me it reads as that: for utopians, you may benefit from LLMs now, but they'll still surpass you later, what then ?
I don't blame them, I also believed that until very recently. I thought there was something unique and special about sentience and consciousness, but my thinking has changed in that I don't think machines need to develop the sentience we experience to still surpass us cognitively along all dimensions.
But iff you really examine what grounds your beliefs, and then consider that AI very well could surpass humans even in domains like curiosity and real creativity (at several orders of magnitude faster), it starts to look like a very, very uncomfortable future for humans IMO.
I saw this quote on Twitter:
> Best outcome: we become pets
> Worst outcome: we become cows
These AI things are machines which are completely under our control. Even if they become more intelligent than us, that doesn't mean that it has authority over us. There are plenty of very intelligent humans which work under the authority of much less intelligent humans. Many engineers are smarter than their CEOs.
What makes you think that just because an AI is more intelligent that it will not be completely subject to our authority? It's just a tool in our hands.
Like what?
Because it would be like 100 chimps trying to make a million humans subject to their authority.
Geoffrey Hinton, considered "The Godfather of AI" for his pioneering work on neural networks, and who now says he somewhat regrets his life work due to the existential risks, has said similar things, e.g. "How many examples do you know of a more intelligent thing being controlled by a less intelligent thing? There are very few examples"
* (1) Mathematics, the true things known by humans
* Mathematics, the things that are true
* (3) Mathematics, the institution which gets funding and manages resources to expand and maintain 1
AI agents can discover more true things, but that doesn't necessarily expand 1 and it might undermine 3
If someone is worried about 3 and you're talking about 2, then you are talking past each other
(4) Mathematics, the community which is a living system that decides what is interesting, constructs shared frameworks, transfers ideas between domains, develops taste, teaches new mathematicians, and continually emphasizes what counts as important mathematics, especially in it's overall value to humanity.
This is the level in which theory building, simplification, integration and applications are built on. Contributing meaningfully here requires much more than generating solutions - it requires direction and restraint.
Tao seems to be saying that this direction and restraint is the scarce resource that drives mathematical progress, not problem-solving ability. And AI is not only insufficient at it, but results generated by AI are destroying human ability to excercise this resource.
This seems similar to another problem AI sucks at - drafting legal agreements. Despite being great at evaluating, interpreting, and comparing legal agreements it fails amazingly at drafting them. This is because what you don't say/do is vastly more important than what you do say/do.
And AI is great saying and doing things, it's the selection based on implied values that it struggles with.
There are many ways to stop progress and productivity
If AI can perfectly replicate their work but faster and better then what?
SWE have nobody crying for them as they've been massively disrupted.
1. Proving theorems - what AI can apparently replicate faster and better.
2. Creating definitions and new theorems from those definitions to prove, selecting which of the possible statements to work on. I.e. developing the "language" of mathematics. So far there is no evidence that LLM can do this at all well. And there's some reason to think that mathematicians won't be as good at this if they aren't also doing the first part.
The value to society only comes when they do both "well", and it's 2 which is really the black magic where we don't understand why they've been so useful to us.
So I totally agree if AI also cannot do the second part better than a person.
Honestly though, I wouldn't want to take that bet. I never thought that the first thing AI would become super human AGI like is math.
You ask me 10years ago and I'd think the opposite. I think we all would have said we'd have super human HR employees before a super human mathematician.
But here we are.
During the process of optimising and understanding the programmer might learn something new or have some kind of "aha" moment of insight that might lead them down a new path of study where fantastic new technologies and capabilities can be realised
Fast forward to 2026
Most web pages take several seconds to load
Applications crash often for no apparent reason
A vast majority of programmers have no idea what their applications are actually really even doing anymore, so they stack bloat on top of bloat and if something breaks, well I guess that's someone elses problem cos I have no idea what's going on anymore
There's something to be said about levels of abstraction being useful, but abstracting away understanding of the task itself is not the path to generating useful knowledge or applications for humanity
We might be gaining the "what" but we are losing the "why" and the "how" and these are generally fundamentally more important
The answer is 42 but what is the question?
The gatekeeping in math academia is extremely unfair, or should I say objectively fair but personally unfair. I won’t cry crocodile tears.
Because there's lots and lots of money in that and there's not in funding pure math. It sounds like your problem is with the people holding the purse strings.
The universe is either Euclidean or not. If it were, how can theorems on non-Euclidean geometry be true?