Posted by _alternator_ 13 hours ago
"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...
Mathematics has always been highly competitive.
Dudes straight up used to hoard solutions to equations and use them in math battles.
I read the Mastodon thread as more about the 'flattening' and 'rawness' of the proofs these systems and their operators are producing. I mean what is the cultural significance of a lean proof that is half a million lines long or something? And what tools can be extracted for further work from such a construction?
The late William Thurston wrote about the culture of mathematics in that sense.
(Because they are my private RSA keys)
"But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed."
But yes; him reaping the benefits of himself having the idea first was part of it too; as far as I am aware.
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Which is still something completely different than some anonymous organisation keeping mathematical research secret because it is better for hype reasons. One is competition between individuals or groups within a field; the other is boring and sometimes borderline nihilistic generating of mathematical knowledge as an marketing asset.
I found this behavior against healthy science practices and only driven by ego. Unfortunately, I find this too often at work (working in academia). Most probably I'm too naive...
[1] https://mathworld.wolfram.com/Taniyama-ShimuraConjecture.htm...
That's how maths works yes...
I think that was Ken Ribet?
Grigori Perelman and the Poincaré Conjecture is more interesting. IIRC he turned down Millennium and was decidedly not all about the Fields Medal - mostly because Richard Hamilton didn't get credit? Anyway, I am grateful I had the opportunity to learn about Poincaré in college taking a few classes from a professor who was a key contributor to the conjecture and got a Fulbright for it when I was there
It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle
Engines - specifically heuristically-driven ones like Stockfish - don't play like a strong GM. They play engine-perfect chess, which isn't how a GM plays with any consistency.
I'm only a decent amateur (1550 USCF) but when I lose to a titled player it's largely explainable in human terms how it happened.
Also note how the quote by Tao is in all likelyhood not meant as an absolute; rather than a statement of a trend - a handfull of counterexamples do I no way change anything about the truth value of Tao's quote.
On the other heand; consider how absurd it would be if "... in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science ..." would indeed be a misstatement; which would imply that far more promising research directions were not shared with the broader community (i.e.: published). I wonder what different reading of that counterfactual there could be other than secret societies that kept their discoveries and research directions to themselves - which we just learned about (since we would otherwise not be refering to the secret societies and their supposed promising research directions).
All pretty straightforward, I would say - both that "misstatement" is hopefully based an overly strict reading of Tao's quote, and that mentioning Tao's background as one of the fields leading practitioners is relevant as well. Again; to make sure: A few counterexamples achieves nothing here. It would need to reach a certain threshold of such counterexamples before we will have to write the history of mathematics; and before Tao actually made a misstatement here.
Terrence Tao can do his job perfectly well without being aware of any mathematical history, though I consider it unlikely that he is. I'm not seeing the direct link you're talking about, in fact history is frequently left out of mathematical teaching even when the history would in fact help in the understanding of some concepts.
That is well-known I assumed and continue to assume.
> I'm not seeing the direct link you're talking about
You are stating that link yourself; indirectly: "though I consider it unlikely that he is [being unaware of any mathematical history]". Why is it unlikely, precisely?
- Maybe because it is unlikely that he recieved the mathematical teaching that frequently does not contain history of mathematics (wild! I wonder which university you have in mind in particular) that you seem to be refering to?
- Maybe because his writing is evidence that he is interested about, incorporates and refers to history of mathematics, refer for example to https://terrytao.wordpress.com/2008/01/04/pcm-article-genera... or https://terrytao.wordpress.com/career-advice/theres-more-to-...
- Or maybe because he is quite the opposite of a person that never ventures outside of their own area; being blind for other fields, or ones own history; as evidence by being famously collaborative across different fields, having a popular blog where he writes about non-mathematical topics too and last; him being one of the main proponents of foundational topics such as formalization of mathematics; or the use of LLMs for mathematical research.
Does all that really make it more likely to you that Tao is not aware of the existence of counterexamples like those the commenter above mentioned - more likely than the commenter simply having missed a nuance or taking something out of context?
If so; I would be genuinely curious why - people work differently, and I am always happy to learn, or close gaps in my own understanding.
And if any mathematician's AI usage on a problem leads to scooping, the volume of agents involved gives them a huge advantage which could prompt mathematicians to not use LLMs.
Though you can say Terry's claim is a slippery slope.
Get outta here.
I mean, besides the empty platitude that we have no reason to assume applies here, we can easily search and find Tao commenting on the history and philosophy of mathematics.
This is a really weird subthread.
Clearly Tao knows the former, but apriori that does not imply he knows the latter.
Not saying he doesn't, just saying one does not imply the other.
Even if you go back and read the original papers, you'll miss all that which happened beyond the page.
If Tao has a knowledge of the topic (which he does), then it isn't by virtue of being a mathematician per se, but by virtue of an interest in the history of mathematics (which he has). Knowledge of math is enormously helpful here, but it does not imply historical knowledge.
(the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them).
FWIW, your post comes off as a middlebrow dismissal, surface level and not actually engaging with the substance of the comment. It's also just wrong. You claim "it’s also a misstatement of mathematical history", but don't specify which part. That there's "centuries of traditions of open science"? But your examples are from centuries (and millennia) ago, and there was never any claim that these traditions are universal.
But more fundamentally, competition doesn't mean you can't also have open science. And the very long, damaging events like the Leibniz/Newton feud are exactly what make many mathematicians work to maintain a spirit of collaboration and attribution even when they're competing on approaches.
Reads like nothing but historical context
If competition is somehow the opposite of "centuries of traditions of open science", and "mathematics has always been highly competitive", then open science is neither sufficient or necessary for the future of mathematics. Their clear implication is that we don't need to worry about it, though, because it's always been that way.
> Reads like nothing but historical context
They literally accuse Tao of "a misstatement of mathematical history".
For the future of past mathematics, it says nothing about the current future. Also, open science can be nonsufficient and unnecessary but still extremely beneficial and desirable.
>Their clear implication is that we don't need to worry about it, though, because it's always been that way.
Lets just ask him if that's what he meant, I bet no.
Also, your third paragraph is highly ironic.
Always been what way? And how does that contrast to what Tao said (since it was apparently "a misstatement of mathematical history")?
> Also, your third paragraph is highly ironic.
You'll have to be more specific, since I engaged with my parent's argument, while they waved away Tao's quote by suggesting he was wrong because of exactly the kind of events that helped lead to the norms and mores working mathematicians have today.
Should it be competitive? Is it more useful to be collaborative? How collaborative can it be when it's fundamentally competitive? Is it only fundamentally competitive because of some common 'quirks' of math types, or are there deeper forces pressuring it to be competitive?
These are all questions that I think are worth discussing, as is the note that the pendulum seems to be swinging away from cooperation in the face of competing for $trillion+ valuations (and a real enthusiasm for proving cool math stuff). The alternative, tweeting complaints on twitter without some context, is mostly a waste of space. I mentioned the history in hopes we could get informed complaints on twitter.
"In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. "
I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.
Disclosure: I did a theoretical physics PhD, but got admitted to quite a few top math programs back when I was applying to math and physics programs simultaneously. If you asked me whether I’d do a PhD today I’d say why bother.
PhDs from poorer overseas do try to get related jobs here, mainly to be able to get a permanent resident visa.
One reason is despising that line of work. Quant firms were pummeling my @prestigious.edu inbox throughout my PhD and I fucking hated those parasites. Well, jokes on me if AI shatters my current career.
In my work as a graphics programmer I often find that I look at a problem and will immediately see how to solve it, more or less. But the devil is in the details and often nothing works unless you get every detail right. So you spend a lot of time coming up with complex solutions, then boiling them down to simpler versions. In the end you often end up with a fix which is short, simple, and seems obvious. But it gets a lot of subtle details just right and avoids countless potential issues you wouldn't know if you hadn't failed a lot getting there.
And that is actually how you learn and master the craft.
Now, imagine you describe how you sort of solve it to a machine and it spits out the simple, correct implementation and you nod approvingly, never knowing all the ways it could have gone wrong. If this is how mathematics - or programming - is done from now on, no one will actually master their craft. I definitely see why this would worry someone whose career is built on mastery of the craft and a legacy meant to teach the next generation.
I get kinda freaked out when I feel like all the AI "utopianists" haven't taken the next logical step of thinking about what society looks like when humans are subpar in every domain (and you may argue this won't happen, though I'm becoming more and more a believer that it will, but my point is the utopianists believe that this absolutely will happen, and that it's also a wonderful thing). How motivated do you think folks will be to do the hard cognitive work to focus on things like math problems when there is a good chance AI will do it better?
So, no, my skills have not become subpar, but have only become stronger because of the presence of AI.
> when AI can do most of the work
You're not really responding to the core hypothetical of his comment
To me it reads as that: for utopians, you may benefit from LLMs now, but they'll still surpass you later, what then ?
Not necessarily. The "something superior about human intelligence" may have dependencies that "these AI solutions" are able to eliminate, such as the motivation to refine intellectual talent to a high level. Basically, AI could kick the ladder out from under human intelligence but be incapable of actually surpassing it in important ways, enabling a burst of advancement that's also a dead end. Sort of like https://en.wikipedia.org/wiki/The_Road_Not_Taken_(short_stor....
So the AI could be inferior but there's still no useful work for humans, because the environment doesn't allow them to work up to that level anymore.
This is kinda feeling a bit like SBF's coin flip bet: https://www.businessinsider.com/sam-bankman-fried-coin-flip-.... Achieve human-superior AGI this generation or humanity stagnates.
The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point.
Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.
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.
Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?
If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career of the professional that takes the challenge.
A lot. In fields where knowledge is incrementally building on previous work the reason the whole field hasn't collapsed from the replication crisis is that usually the results that are really high impact are replicated in as an initial step in new research building on it. It's almost never the focus of the paper but you'll often find a quick mention in methods/supplemental of some previous work that was verified to be valid by a replication of a key technique etc. you'll have crisis where old tools are found to be problematic and findings end up revisited etc. Plus fields like clinical research where there's an awful lot of focus on replicating findings using staged clinical trials with increasing statistical power to determine if new interventions work - that's driven by regulatory requirements grounded in good science and a lot of people make careers in just that.
I don’t think this is true, especially for novel or unexpected results. I suppose it depends on what you mean by scientifically, and there is a debate in the philosophy of science about what the value of research even is, but a successful replication does not result in substantial updates to one’s beliefs in the way new research does. And if the goal of science is to change our beliefs and bring them closer to what is “real”, successful replications can’t be as valuable as the initial research almost by definition.
People don’t behave this way, but a high percentage of all papers have known flaws and that goes up even higher when you consider unknown flaws. Replication doesn’t own its own solve the underlying issue, but independent replication removes a huge range of potential issues on top of providing more information.
That alleged superconductor from a few years ago - everybody kind of held their breath and waited for the reproduction.
You have created a fraud machine. Why? With no answer checking then why not make up the most fraudulent crap you can get away with?
Examples: A huge portion of recent non-reproducable science papers.
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Your thinking, along with everybody that's doing this rat race is causing the pumping out of papers with questionable data, but very little to ensure we are actually making correct science.
If you examine the consequences of the inversion of the peak, you realise the need for a balanced perspective.
1. proof that there is a solution
2. a solution that you can work backwards from to build understanding
So, both of these pieces of info would be at least marginally useful for advancing human knowledge.
This is only true in the most trivial sense. A solution is a solution, sure... but how do you know it's a solution, and not an incoherent jumble of words? A human has to review and vouch for it.
Just because the AI gives you an arxiv-worthy PDF, or a Lean proof which compiles, doesn't mean it proves what the AI says it does. The AI could give you the same PDF/Lean code and says it proves the opposite, how would anyone know the difference?
You can't advance human understanding unless you produce things that humans can understand.
> 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.
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.
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]
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?
You're either being intentionally obtuse, or unintentionally ignorant.
It's similar to Mochizuki claiming to have proved the ABC conjecture, with a proof depending on ideas developed over a large number of obscure papers, that required mathematicians to spend a lot of time before they felt they understood it well enough to point out flaws.
If AI solves all famous open problems and the non-famous ones, too, without advances in the readability of their output, there'll still be some work to do to digest and rearrange the proofs for human consumption. During that process, the mathematician may well get some new ideas...
In the case of mathematicians, I think not as researchers. What would a research mathematician do? I don't think there would be any reason to try to gain insight from proofs that AI made for the sake of understanding. I don't see what that would achieve besides just retaining extremely niche knowledge (which AI or the oracle already does). The whole point of having that knowledge was to build toward novel work which the AI/oracle does. Also, the time spent and difficulty understanding them could be very high but with no payoff besides just understanding them because the AI/oracle would be used to solve all the problems anyway.
Is there value lost in them working on problems that don't have solutions instead of problems that do?
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 give us tools that allow us to create lower bounds on a lot more problems we do care about and perhaps some bigger insight onto what makes a problem intrinsically hard or easy. These long term considerations are helpful but are definitely more vague. The remarkable part is that AI is separating the part about proving theorems and the “free” insight you get.
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.
You'd be more sure if you read the tweets.
Tao's point is very simple.
1. Working on problems that AI solvers can solve is a waste of human time.
2. We have no idea which problems can be solved by AI solvers...
3. ...Because the AI labs are keeping their negative results secret, and don't tell us which problems they've tried and failed to solve, and why they've failed to solve them (or succeeded at solving others).
There are additional points surrounding it, but that is the thrust of his argument. His issue is not the existence of AI, but the anti-scientific secrecy in how it is used to solve problems. All the incentives around its current use result in closed, uncollaborative work - which while very attractive to a vulture capitalist, is anathema to scientists.
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He also posits that having a solution to a problem is a small part of the value of solving a problem. What the AI labs are doing is the equivalent of a student turning in their homework, which has 100% of the right answers, but with none of the 'show your work' steps. Those steps are a critical artifact for doing mathematics, because the process of solving a difficult problem teaches us things about other problems.
The awkward part about all of this is that we're about to enter an age of extreme enslavement at the hands of the major tech companies if we do not focus on distribution of hardware and research, so that everyone can participate in the abundance and automate their daily lives. If we're beholden to frontier labs because they have hoarded all of the cutting edge hardware and we're left with overpriced scraps, we're collectively screwed. They will ensure a false economy is maintained so they can clutch onto a permanent class hierarchy of haves and have-nots and remain the key global decision makers. Automating hardware manufacturing is irrelevant if the hardware is not being distributed fairly, and is weighted to real scarcity instead of artifical scarcity.
Take Louis Vuitton for example. They can mass-produce their products for pennies, but they're artificially scarce and incredibly expensive. Imagine if ALL clothing was the price of LV. Now imagine this applies to every single thing you can purchase (or rather, rent - if some of these "elite" get their way), because they've cooked the economy and swallowed all industry. That's where we are headed if distribution and decentralization is not a priority for the world and we let labs like Anthropic pull off their regulatory capture stunts.
Sure there is: problems that require knowledge that simply doesn't exist yet. Until "AI" turns into general purpose robots that can develop new tools to explore the world, it is, in fact, pretty damned limited in what it can do without human help. The world is vast. Math is small.
Biology is replete with examples. Computers "solve" protein folding [1], and midwits immediately leap to conclusions that drug development will also quickly fall. But we literally have no idea how most of biology works, and simply getting to the starting line for drug development problems is often 95% of the battle. Come talk to me when you've done a million experiments to find the fundamental knowledge that unlocks the pathway(s) we didn't know about that makes a drug discovery program possible in the first place [2].
I am not pessimistic about humans running out of challenges. We'll just declare one class of problems "done" [3], and move on to the next frontier, as we always have. The problem with AI doomers is that they lack imagination that extends beyond computers, or perhaps more accurately, are so sophomoric in their thinking that they skip over the hard parts of any problem they don't fully understand. This stuff reminds me of the endless smartypants whinging about the end of human intelligence when chess machines started beating grandmasters. Chess was never really that great a measurement of human intellectual capacity, and we found new things to do with our big monkey brains.
[1] They did not solve protein folding, except in the minds of people who don't fully understand the problem.
[2] ...and invented new machinery to make the experiments possible in the first place.
[3] ...and we'll likely be wrong about that.
The AI companies have already bought up the world’s entire supply of hardware. There won’t be any more.
If theft becomes more profitable than genuine creation, then nobody will create anything. Then there's nothing to steal, at which point all progress collapses
Download 1 million books and you are OpenAI
This was unpublished research that was stolen, and constitutes plagiarism and academic fraud by even the strictest definition
It was not stolen, it was willingly given.
2. The topic we are dissussing concerns LLMs being trained on logs from previous LLM chats. If you're prompting a model and it spits out some unique mathematical insight, you do not have copyright on that.
What is interesting is that LLM's do not directly violate copyright. The settlements we have seen are for how the works were acquired (that was a copyright violation) not the use of the works.
The vectors of a book, or a paper, are not the paper. They are, for all intents, facts about the work itself, and more generally writing. You can not copyright a fact.
It also means that the weights, the things that (mostly) matter can not be copyrighted either.
To block progress. Got it.
Copyright maximalism is a bad look on a site called "Hacker News." Perhaps other sites beckon.
Frankly it's more of an insult to the "hacker" name to be apologising for big companies profiting off of frontrunning existing work for PR purposes, if the claims about piggybacking on human-directed efforts/prompting are true.
Being pro-copyright in order to protect the work of an individual from being reconstituted into the corporate machine is VERY hackery. Novel use for an existing tool, to fight the dominant system.
(Of course, we're on a so-called "hacker" site hosted by a company run by squarely-establishment individuals acting in an extremely un-hackery-field (investing), so the irony here has been at least one layer deep since the start.)
Yes, OpenAI is likely to be found to have acted like a slimeball in this instance, or at least the employee in question may have. But you can't fix that without making laws that will make everything else worse... and only here in the US.
Jokes aside, this seems like a pretty weird take. What's stopping mathematicians to make this renewable?
Why not spend some time and effort (presumably using AI) to pose new open problems that are fundamental in nature?
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.
Except way more nefarious than I expected
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?
Most patterns that exist are incompressible. Math is basically a search for those compressions that do exist. An example I personally really like is the amplituhedron: a geometric structure that humans have just barely been capable of recognizing compresses information about scattering amplitudes and Feynman diagrams. That one happens to be within our reach, but it's right at the edge, and we can only imagine what glorious, wondrous compressions exist in abundance beyond the edge. Math accessible only to superintelligence would exist entirely beyond that edge, compressing patterns whose existence we cannot even detect using objects and constructions that we cannot grasp.
As an aside, I also think this is why AI is quickly becoming superhuman at math: intelligence is essentially a form of pattern compression.
Take infinity. Infinity can't fit in your head, hell, it can't fit anywhere, but you can abstract away the endlessness and look at infinities of different sizes, et al.
Now, is there a single formula for something actually represented in this world that would take most of a humans life just to read it, no idea.
The only way to renew it is to make a new problem that is so hard systems and humans will be unable to solve it for the next ten years. And, in the spirit of trees, the best time to plant a tree is twenty years ago, the next best is today: we do need to start posing some hard math problems and deciding if they are interesting merely because there are challenging or because of something else (eg busy beaver problems are arbitrarily hard, but does solving them imply anything other than "another busy beaver problem was solved"?)
Now, I think AI will solve some, but we'll find out that some are just either unsolvable or wildly huge that nothing is solving them any time soon.
And a whole lot of these problems have been around quite some time, when even knowing how to do advanced math meant you were a landed gentry or someone of high wealth. If those problems fall, they fall. They aren't pets we keep around forever. And new problems will crop up over time for both AI and men to scratch their brains over.
In another comment below, I likened this to clear-cutting a forest. Growing the forest takes a lifetime; destroying it could happen in the next few months.
of course thrrr are tons of problems once you remove this social consensus based filter. if i’m not mistaken Ramanujan left a book of dozens of unproven theorems, for one quick example. i don’t think that that has opened up dozens of fields of mathematical research.
augmented Hilbert's problems of 1900.
Surely mathematicians are creative enough to ask new questions?
If not, then the next set of challenges will be to find questions to ask!
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...)