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.
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 to prove or disprove. 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.
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.
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.
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.
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.
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.
> The AI was able to race its car very well, but it took a lot of risks that a human player probably would not.
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.
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!
> 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 problems in fluid dynamics. IIUC, this particular proof is understandable by human mathematicians, but if it weren't, it might as well be 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.
Think of it like software going from programmers understanding every instruction, knowing where every byte of memory was being used and why, and using this knowledge to build optimised systems
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
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.
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.)
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.
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.
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.
Well maybe its time to pivot from mathematics, and science as whole from personal attribution to being about progress of the field? Maybe your contribution to humanity as a mathematician is to find the right meaningful question to ask, and not to stamp your name on some fact?
Yes. But this is hard for mathematicians to stomach, because like everyone else, deep down in a place where they don't like to talk about at parties, they have egos and a sense of purpose based in part on demonstrating mastery of a technically difficult field, as well as social connections based on their participation in it, and taking all that away from them probably feels like a kind of death.
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?
I totally understand and agree, I just feel like with the progress that we have in automating informational work, you are going to have an exponential amount of these "deaths" as new fields where humans can provide any kind of value get more and more short-lived.
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.
At some point from your description there will be no need for human workers at all. The real question is what happens as we continue to advance in that direction and we eliminate mathematicians, and then more broadly scientists and engineers, tech workers, and all knowledge workers. People can’t “doing what you like doing” if it doesn’t support them being able to make a living.
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).
This is the direction of experimental particle physics and observational astronomy, where the budgetary scale at which progress occurs means we now fund these efforts at a societal level. These fields have graduated beyond "tabletop science".
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.
Pure math is practiced mostly for the intellectual thrills and peer 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. 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.
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 human pursuits. 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? Etc, etc. The incentives are there to strip-mine almost every domain.
There's plenty of people on HN who think it's nothing new, ignoring the huge quantitative shift; or 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 and 100 of them won't be able to name any living mathematician. A few of them might know Linus, though.
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.
Seems like a natural way to conclude that if people lose interest in the field its not a useful field anymore. Most people lost interest in horse and carriage as well, you know?
Anyone feeling sad about this is clutching pearls for egocentric reasons of feeling good and important
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.
Is there an analogy here to the phenomenon that senior {engineers, designers, PMs} are now able to be insanely productive with AI, but it's also very hard to train junior folks to develop the sense of judgment that senior folks have?
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.
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.
> 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.
I might be wrong, but making an assumption that you could learn to read the mathematical output of the AI long before you could write a solution yourself. But hey, what do I know, I'm not a mathemagition.
Not an expert by any means but the assumption here as I understand it is that the arxiv worthy PDF would not be acceptable or meaningful for impossible to understand proofs. And the lean proof would be meaningless unless the specific expression being proven is human understandable as the direct translation of the question the human is asking in formal form. So proving the negation is not a thing but if you make a subtle mistake in translating the statement you want to prove then obviously the QI is going to be proving the wrong thing. And otherwise you're relying on the correctness of lean as a system and on identifying/preventing if the proof is adversarially exploiting bugs in lean to falsely prove things.
> 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.
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.
This is definitely true in an information theory sense: having more knowledge is always better than less knowledge. However, it may not be true in math as a social human endeavor, and having answers without interesting paths to get there may not expand human mathematics in the same way.
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.
This is my question too. If we are all just going “well that sucks” after AI solves this problem, why did anyone care about the problem being solved in the first place?
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?
I think the Navier Stokes problem kind of illustrates what he’s highlighting. I think most people even before AI expected that this would resolve in the negative and that you could get finite time blow up. There wasn’t really ever going to be a situation where the resolution to this question, or really any of the other Millenium Prize problems as far as I know, gives some kind of immediate massive practical feedback.
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 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.
Honestly, the attitude of the math community is a bit cringe and increasingly I think some of the elite/mystical aura is fading. Rather than a rich fertile jungle where AI can barely chomp through a fraction of the luscious terrain, one gets the sense it's a desert and all the oases are running dry.
Because the mathematicians consulted 25 or so years ago believed their solutions would lead to the greatest amount of interesting new maths to explore, and because they had been validated as being hard by being attempted and not solved for a long time.
More of a 'its the journey' rather than the destination type of thing.Since the insights , quirks, tricks and procedures gained along the way allows insights intoother at that moment unknown problem/domains in the future.
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.
I wonder if it would be possible for researchers and scientists to submit their papers to an organization which would then collect them, submit them for peer review by other experts in the field, and then release them in periodical form to individuals and organizations who pay a subscription fee in order to read them?
Why would society fund mathematicians if they decided to become a guild that hides secrets? They could pursue that as a hobby, but they’d end up like the coders who refuse to use LLMs - rapidly becoming irrelevant and a bit sad from an outsider’s perspective.
Ah but heres the thing , society/gov expects mathematicians to be productive and tries to measure that by awards/publications/citations gained. Within a guild ope or secret they could possibly use a local LLM (even if slow) to accelerate their collective output.While ensuring their credit/publication/citations remain intact rather than with the AI labs taking a lions share of that.
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]
AI companies don't share the dead ends and only sometimes a bit of the process toward success so people don't understand what was curious along the way.
> and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge.
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 nature in the secrecy of 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.
Let's say for the sake of argument that an LLM finds a counter example to the Reimann hypothesis. Wouldn't that just create a bunch of opportunities for understanding and explaining _why_ there is a counter example?
Rephrasing the argument made in the post, math was like a "take a book, leave a book" exchange where you solved a problem, and in the process you discover more problems so the pool of interesting open problems is renewed. What OpenAI and the other labs are doing is akin to
- 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)
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
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.
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)
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.
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.
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.
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 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 think part of mathematics is taking things that don't fit in our head and giving them human abstractions so they can.
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.
The models produce both Lean code for formal verification and a traditional-style narrative proof. Like the general long-form output of frontier models, the math papers produced appear to be generally correct technically, but written in an ungraceful and sometimes hard-to-follow style, so they are often polished by a human mathematician as of today.
I think math is compressible structure. That's why we care about something like the Riemann hypothesis but, to use Tao's example, we really couldn't care less about computing the 10^10^10th digit of pi. The first compresses a vast amount of information about the primes, while the second decompresses information that we've already compressed (a few lines of code can define every digit of pi).
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.
What you are saying implies that by some technique that hasn't been discovered yet, we can make the models to have the capabilities of extrapolate the information they are trained on and also interpret that what they are extrapolating are Riemann-capable hypothesis.
I do believe it will accelerate the discovery of that "vast universe of mathematical depth that's beyond our ability" but at the cost of removing the "fun part" of solving the problems. Not sure if the community is willing to do that.
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
My more cynical take is that it’s press release generation intended solely to message “behold, we have built the most capable machine humans have ever known: please line the dump trucks full of money up for us now.” Bulldozing an intellectual forest into oblivion is an acceptable cost for them if their only goal is huge piles of cash.
> 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.
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.
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"?)
Eh, if AI quickly solves most of our mathematics problems that are solvable then it might be time for us to hang up our hat as our little monkey brains aren't very good at this stuff.
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.
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).
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.
That should be the easiest part of the loop. Frontier models are good judges of human preference and taste and should be excellent at prioritizing proposed questions. The generation part is much harder.
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?
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.
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.
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..
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.
Perhaps his takes are evidence of just usual human fear to new things. I see he relies quite much on the "community" or "social" aspects of the discussion.
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.
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.
> 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.
"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."
You may recall though that whole problem with '42' as the answer was the lack of knowledge of the question. The Earth was created as an attempt to provide the question, but was unfortunately demolished to make way for a hyperspace bypass just before the question was resolved.
> You may recall though that whole problem with '42' as the answer was the lack of knowledge of the question. The Earth was created as an attempt to provide the question, but was unfortunately demolished to make way for a hyperspace bypass just before the question was resolved.
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?"
I'd say that a more appropriate reference from that time would be "The Nine Billion Names of God" by Arthur C. Clarke [1], which actually deals with the finiteness of the list of problems that a machine successfully exhausts.
>>which actually deals with the finiteness of the list of problems that a machine successfully exhausts.
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.
If we're being anecdotal, I haven't had a single outage of anything affect me at all this year, but have had many in previous years. Often the thing blamed publicly has been incorrect too.
Well you see we need to migrate system A to B and X to Y and here's another new integration to look at and we need to finally get off of this EOL distro and... It'll never stop.
I am kind of saddened by technology in general that we need LLMs to deal with it in the first place.
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.
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[ 0.17 ms ] story [ 16.2 ms ] threadThe 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 to prove or disprove. 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?
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]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.
Article arguing math is the next "human calculator".
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.
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.
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.
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!
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 problems in fluid dynamics. IIUC, this particular proof is understandable by human mathematicians, but if it weren't, it might as well be 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.
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.
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.
(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.
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.
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).
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.
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 human pursuits. 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? Etc, etc. The incentives are there to strip-mine almost every domain.
There's plenty of people on HN who think it's nothing new, ignoring the huge quantitative shift; or 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 and 100 of them won't 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%...
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.
> 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.
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.
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.
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.
> 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.
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.
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?
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 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.
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.
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]
[1] https://abakcus.com/articles/nicolas-bourbaki
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 nature in the secrecy of 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.
prize is not just monetary
- 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)
[1] https://mathstodon.xyz/@tao/117219101339291693
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.
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.
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.
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!
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.
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.
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?
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.
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.
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.
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
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.
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]
My humanity is not paying my bills.
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 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.
Yes. We already knew this. Are we actually surprised it's happening?
I guess we are.
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.
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.
> 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.
"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...
[1]: https://www.imdb.com/title/tt0708807
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?"
[1]: https://hex.ooo/library/nine_billion_names_of_god.html
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.
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.
https://www.youtube.com/watch?v=dcolM6W5Odc