"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."
I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession.
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.
> An AI-generated solution always provides ... proof that there is a solution
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.
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.
Humanity is very biased for the culmination of work, considering everything that comes before and after busywork for the lower masses.
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.
In mathematics, finding novel proofs of a given result is often valuable; it may be a shorter proof (demonstrating better/expanded understanding of the problem) or a translation of the problem into a new domain, setting up more cross-domain advances.
>Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
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.
From a pure statistical perspective the first scientific paper shouldn’t update your beliefs as much as the independent replication study.
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.
> Mathematicians will be less likely to work on a problem if there is a solution
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.
I think his point is that AI is not creating new problems. It may solve "the Riemann hypothesis" but may completely fail to posit a "Mythos hypothesis" which is vital to advance the field. In fact, achieving the former may make the latter even harder because it will disincentivize production of human mathematics which has till now been the only source of "interesting" problems.
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?
The sphere of human comprehensible mathematics is finite. Once everything is solve it is not necessary to advance the field. The recurring error her is to say ai is not the product of human effort but another agent. Ai is human. Ai may well be speeding up human comprehension of math to its limits in which case there is no further need to advance the field and mathematicians might need to get a job. Why is this a bad thing?
But this oracle doesn't just say true / false. It also gives a proof. That makes it much less exciting (not to mention beneficial for your career) to find another one (or even worse, the same one).
I mean, the oracle doesn't really seem so hypothetical right now. And clearly it's going to drastically change these fields, and mathematics, particularly pure mathematics, must change most of all in order to adapt to the existance of a math oracle (or something close to it).
Current career structure of mathematicians works partially by looking at whether they have solved novel and interesting problems, or at least done theory-building that can help solve such problems. Many mathematicians are also motivated by being the world's first to solve such problems
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.
I didn't realize that open math problems were a finite resource.
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.
They aren't, but the problem is that open problems tend to emerge when people are working on other problems. If fewer people are spending time deeply thinking about current problems since a handful of labs are solving them with AI without an eye towards understanding and only on verification, the pool of open problems won't be continuously growing. There is a fear that there will be a chilling effect on the community if people are disincentivized from trying to solve deep problems or study them for understanding as opposed to simply focusing on verification. It's more of a social and community problem than a fundamental problem with mathematics itself becoming "completed".
So we can let the ai generate some math problems based on the solutions found? Other fields (computer science, physics, ...) can generate math problems too.
There's an infinite number of possible math problems, but the things that make these open problems worthwhile is they're interesting to people who have worked in related areas.
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.
This kinda reminds me of the guys who decided to industrialize digging up dinosaur fossils, in order to feed the dinosaur fossil collector market. They were amazed that paleontologists were so "inefficient" at finding and digging up dinosaur fossils.
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.
These open problem solutions often reveal tighter bounds on prior conjectures. Even if the solutions produced are far from elegant and only machine verifiable, we do learn new information. But I agree that just like writing prose and code, brainstorming frontier math proofs is a perishable skill
It's easy to come up with new open problems. It's hard to come up with new open problems that seem to teach us something fundamentally new about the world. Our current batch of problems went through a complex selection process over decades (or centuries) based not purely on difficulty but also on perceived insightfulness.
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.
I haven’t been following the AI proof stuff very closely, but the impression I got was that these models are producing massive Lean programs that prove the statement one way or another, but are quite difficult to fully understand.
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?
I'm surprised nobody has stated the obvious: a hard math problem that has been open for ten years (because many serious people have given it serious thought and been unable to make significant progress) is, in fact, nonrenewable.
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"?)
I think "close to completion" is not the right framing. Creating good open problems was an achievement because these problems often sit at the edge of known techniques, and solutions require inventing "new math". It's hard to find these problems, and they take decades to mature as they withstand scrutiny by many people.
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.
his whole point is that specifically problems that have been held as important by consensus in the field are a finite resource. obvious example being the Clay millennium prize problems. seems like they function to shape the direction of future research into useful directions. which is to say, the process of developing a solution itself generates more useful problems.
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.
Relevant, interesting problems that we have some immediate hope of making genuine work on might be, if not finite, quite difficult to produce. And it's also plausible that AI will not do as good a job of producing these as it does at solving them.
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.
Deforestation might be a better metaphor than mining. Logging is renewable if for each tree you chop down you plant several more. AI companies are operating "in a non-renewable fashion" by chopping down trees without planing seeds. Open problems are a renewable resource, but only if harvested sustainably.
> 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...)
I think you can't have read the thread.
The whole point is that there is no end of mathematics, an infinite sea; but the constitution of an 'open math problem' is a delicate piece of mathematical thought, at any moment a small supply of drinking water developed by finitely many human being.
Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them?
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.
The incentives are massively skewed towards the AI labs investing their massive amounts of compute into being the first to solve an outstanding problem.
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.
I think the problem with AI asking questions is that it will ask questions that are interesting to it but not necessarily us. AI, as a model, will never be a perfect copy of a human. It will always be a simulation, and thus to some extent, will ask questions that humans find irrelevant and solve problems that humans find irrelevant.
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).
Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.
Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of Navier-Stokes will probably take a similarly long time. In the mean time, it looks like all open problems will be solved (or proved that they can't be solved).
It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.
This reminds me of the time an AI was taught how to play a racing sim game (Gran Turismo if I remember correctly). The AI was able to race its car very well, but it took a lot of risks that a human player probably would not. A human player might be able to copy the approach the AI took, but they would probably crash.
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.
So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before?
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!
Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor.
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.
there's a book I read "The Practice Effect" such that technology becomes super advanced based on using something, it gets better and better, but the people regress and become more like a medieval society as they just care that using things improves them.
> 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?
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 [...] mathematicians are effectively useless?
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.
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.
Mathematicians provide two complementary services bundled together.
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.
Perhaps an AI could! Today they do not, because the people driving them understand constructing the proof rather than understanding the proof to be "the problem".
(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.)
It's possible but the approaches these tools take are usually verbose and strange. Think about it like anything else llms do. Even when the picture is right and there are only 5 digits on each hand all the textures are off and so is the lighting and postures. Or in code, the code is always way larger then it needs to be and tightened up strangely with weird loose ends. Or in writing weird idioms, words, structure, and a weaselly way to turn 3 sentences into 8 paragraphs.
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.
Pure math is practiced mostly for the intellectual thrills and recognition among a very small group of peers. There's little else to it. You don't become rich, you don't become a celebrity. You teach students, write papers, and probably know most other people who work in the same subfield as you. Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.
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.
> Now, Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.
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.
I'm going to charitably assume that you forgot to include quotes around the names in your query, because that's absolutely not what Google Trends shows.
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.
Seems like in current cultural and economic context, short term extraction is what we’re going to do
> 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.
This series of posts by Terry Tao is a direct response to the Navier-Stokes results (multiple results!) from the last 24 hours. The question is what is left after the levelling of mathematics, in all its senses, occurs? How can you protect a field that's under this much pressure in the next 6 months?
> [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.
yes but the tweets don't really address why the field needs to be protected instead of adapting and evolving.
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?
The reason is because the entirety of society, historically, has been based upon humans using their differential skills to further it, which in turn promotes societal cohesion. If most human endeavours are solved, then we will enter a period of abundance that paradoxically will erode the glue holding society together. In short, endless abundance of solutions and ideas cannot coexist with a healthy society. Only those who are priveleged and have a naive belief in a Star Trek utopia think otherwise.
>endless abundance of solutions and ideas cannot coexist with a healthy society.
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.
This far into history I don't think many are going to buy a "the next technology is to be the undoing of society itself" pitch until after society is already undone by something. They're as easy to make and hard to concretely evaluate ahead of time as the utopian predictions while offering little in the way of practical approach to preventing the same result from occurring anyways.
Aren't we in a similar position to what chess went through in the 2000s when Deep Fritz came out, and a desktop PC was able to defeat a reigning World Chess Champion? Did chess players just give up and stop playing? No, they didn't. They used these new chess engines to become better players. Computer programmers and mathematicians will probably go through something analogous.
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.
> 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.
The way to tame a profit-maximizer is to make the most profitable choice the one that creates the most societal good.
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.
It would be trivially easy to convert Lean into English, so I’m not sure what the human-readable criteria gets you. There are also human written proofs that are considered not human-readable by most of the mathematics community (ABC conjecture).
My model of mathematical intelligence for a little while now has been 3 levels:
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.
What is this man talking about. You can speak physics into existance now, yet it still has to be proven with math. Until we are walking through worm holes and driving around in spaceships that travel in a warp drive could he even begin to say there is non-renewable. But even then..
Well, we name conjectures after the conjecturer not the (dis)prover so there is some incentive to be the guy who comes up with a hard problems. It is curious that we haven’t had something like this improve OR etc. problems. Perhaps not glorious enough.
> Well, we name conjectures after the conjecturer not the (dis)prover
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.
Actually there will because if you've actually ever spent time doing math, you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI. The analogy is interesting but incomplete and misleading.
I've actually spent time doing math and literally everyone I know who knows math learnt it by reproving things that people proved before them. I don't see why AI proving things makes this form of learning any more economically infeasible than the field of mathematics already is - and since it was apparently economically feasible before AI I expect it to stay that way.
There are two stages to learning research level math. First you learn to reprove other people work. This is relatively easy because the language, notion and prior explanations have already been optimized to help prove the next thing, and you know a solution exists. Then in your PhD you try to solve problems you don't really know a solution exists, and if it does, how long and complicated it is, what techniques will be used, etc.
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.
you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI
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?
I agree that the low-hanging fruit has largely been picked. But then I think we should acknowledge that and stop innovating, or just work less on new solutions and more on clarifying old ones, and start to work on degrowth rather than useless problem solving. Because I really don't feel that any of this is really necessary or beneficial for the human race in the long-term.
Plenty of new open problems will come from applications, and applications are what actually matters. Pure mathematicians are wrapped up in math for the sake of math which is a fun academic game but not something the rest of us should care about.
See also his previous thread from before the result was published (and before he knew it was coming [1]) on how a to this problem seemed increasingly likely to be solved by AI in a way that caused us to miss the insights that would traditionally be associated with solving it: https://mathstodon.xyz/@tao/117207849921390904
It's the same pipeline problem coders have been talking about; once AI does all the work, how are people going to get the experience necessary to take part productively?
There seems to be a sense wher mathematicians are gamifying math, but are frustrated that AI labs are better at gamifying math.
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.
The lack of reasoning traces in frontier model output is hurting science and progress... thats my take away from reading that, and why open source models are so critical and so needed, because they actually do expose the chain-of-thought reasoning traces recently missing from the frontier models (openAI, anthropic, etc). By encrypting and purposely hiding this important information from public inspection, it makes for a world where people lack the true understanding of how a problem gets solved.
This is just the age of slop mathematics, if it doesnt lead to our lives neing improved none of this matters. Math peeps are being nerd sniped by AI in the same way SWEs (the worst ones) got sniped by claude code. Building solutions to problems that dont matter for the sake of doing it just because you can.
You'll ultimately waste a ton of time and get lapped by people doing real world work that actually improves the lives of regular people.
alright, i'll bite: what real work did you put out there that has improved the lives of regular people in the past one year? (without the use of ai, needless to say)
"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...
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.
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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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"?)
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!
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
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...)
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).
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!
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?
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.
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.
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.
(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.)
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.
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.
https://trends.google.com/explore?geo=US&q=%2Fm%2F047mjr%2C%...
> 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]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.
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?
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.
Sounds like they're going the way of the DoDo. better take that PhD, migrate to the new world and become a tuktuk driver.
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.
> 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.
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.
https://www.youtube.com/watch?v=dcolM6W5Odc
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.
Yes. We already knew this. Are we actually surprised it's happening?
I guess we are.
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.
Article arguing math is the next "human calculator".
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.
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?
[1] https://mathstodon.xyz/@tao/117219101339291693
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.
You'll ultimately waste a ton of time and get lapped by people doing real world work that actually improves the lives of regular people.