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.

[1]: https://hex.ooo/library/nine_billion_names_of_god.html

I don't see why that should be a problem, as we already know the answer is 42 in any case.

The mathematical community was very competitive in its early years, but in the last 70 to 100 years, it has been generally less competitive and very collegial. The community was in a good place, and progress has been very good. In a few cases when competitiveness was ramped up, it lead to bad behaviour and destructive fights. Few would like to return to those competitive years.

The story of the cubic equations is another great example: https://en.wikipedia.org/wiki/Cubic_equation

Dudes straight up used to hoard solutions to equations and use them in math battles.

My computer contains the prime factorization of probably several dozen (if not more) large integers, and I refuse to share them with anyone!

(Because they are my private RSA keys)

Also Andrew Wiles working in secret for 7 years out of fear of someone scooping him.

Yeah but a highly productive last two decades of math research from https://en.wikipedia.org/wiki/Polymath_Project has come from collaboration.

You miss the point. Humans don't mind competing with others. I love competition, but I don't want to compete with you and your machine. I love to play chess, I don't care if you are grand master, whoop my ass. But not if you are going to pair up with stockfish. I don't even care if you are a newbie that started playing yesterday with an ELO rating of 900. If I wanted to play the damn computer I'll do it myself. Likewise, mathematicians will not mind sharing and competing with other fellows, but if another has a billion dollars worth of GPU and you don't? Then you best be carefully what you say.

   It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle

Surprised to see someone on HN arguing against open science. Seems like the opposite of the lessons we should learn from Newton and Gauss, actually, hoarding results for decades at the expense of progress.

(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.

this is not untrue but it's a pendulum swinging back to ancient times man

Between people.

Doesn't this seem to be where all domains are headed?

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?

How is this any different from people in any field that are impacted by AI and lose the utility of their skills and endeavors over the past decades? Are we saying that we're running out of problems to solve because of AI and hence it should be stopped? I am not underestimating the importance of the collective knowledge of the mathematics community and the role of mathematics as the enablers of other sciences, but opposing meaningful progress in that discipline or any for that matter feels counter intuitive. I would rather have the mathematics community start collaborating closely with the this newly evolving and powerful tool to expedite humanity's progress.

Seems to be it, though Im not particularly concerned about this problem personally.

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.

> it's only a matter of hardware and scale at this point.

The AI companies have already bought up the world’s entire supply of hardware. There won’t be any more.

Humanity is very biased for the culmination of work, considering everything that comes before and after busywork for the lower masses.

Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication?

If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career of the professional that takes the challenge.

An AI-generated solution always provides two pieces of info:

    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.

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.

Mathematicians will be less likely to work on a problem if there is a solution - even an incomprehensible one.

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.

Why was there a prize attached to this problem then? What does humanity get out of this being proved?

Current career structure of mathematicians works partially by looking at whether they have solved novel and interesting problems, or at least done theory-building that can help solve such problems. Many mathematicians are also motivated by being the world's first to solve such problems

Removing this measure suddenly means that academic mathematic norms need to adapt rapidly, and, even more importantly, intrinsic motivation for many mathematicians needs to change rapidly. That is understandably a sea change for the current mathematics community.

> 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 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.

---

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.

What theft? LLM output has never been copyrightable.

Nobody owns math and it is ridiculous to suggest someone should.

> why enabling mass theft is so incredibly damaging to society. This is literally why we need a functional copyright system

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.

This is literally why we need a functional copyright system

To block progress. Got it.

Not so dissimilar to my first comment on this site (which I got piled on): https://news.ycombinator.com/item?id=48959395

Except way more nefarious than I expected

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.

> ask challenging questions

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.

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).

If AI can generate questions and then answer them, what are the people for?

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".

It's easy to come up with new open problems. It's hard to come up with new open problems that seem to teach us something fundamentally new about the world. Our current batch of problems went through a complex selection process over decades (or centuries) based not purely on difficulty but also on perceived insightfulness.

I studied math, but I am not a mathematician, so I think I have a slightly different perspective on this than Tao overall. This is certainly the definitive end of an era in mathematics, but I think he's wrong that insightful new open problems are truly non-renewable. They might be non-renewable by humans at the rate at which they are being closed, but I see no reason why AI systems could not also discover insightful new open problems. In fact, once we have Riemann-capable AI mathematicians, I'd personally love to see what the next Riemann hypothesis is, which even these AI systems cannot solve with any amount of available compute.

I think we're about to find that, on the spectrum of mathematical intelligence, the best human mathematicians were only a fraction of a percent forward from the very beginning, and there's a vast universe of mathematical depth that's beyond our ability to imagine or work on directly in any way. We're used to feeling like we're able to directly perceive the Platonic realm, but we're almost certainly going to discover that our own minds, even when joined together over centuries of deliberation, can only interact with a tiny little shadow within it.

I'm surprised nobody has stated the obvious: a hard math problem that has been open for ten years (because many serious people have given it serious thought and been unable to make significant progress) is, in fact, nonrenewable.

The only way to renew it is to make a new problem that is so hard systems and humans will be unable to solve it for the next ten years. And, in the spirit of trees, the best time to plant a tree is twenty years ago, the next best is today: we do need to start posing some hard math problems and deciding if they are interesting merely because there are challenging or because of something else (eg busy beaver problems are arbitrarily hard, but does solving them imply anything other than "another busy beaver problem was solved"?)

I think "close to completion" is not the right framing. Creating good open problems was an achievement because these problems often sit at the edge of known techniques, and solutions require inventing "new math". It's hard to find these problems, and they take decades to mature as they withstand scrutiny by many people.

In another comment below, I likened this to clear-cutting a forest. Growing the forest takes a lifetime; destroying it could happen in the next few months.

his whole point is that specifically problems that have been held as important by consensus in the field are a finite resource. obvious example being the Clay millennium prize problems. seems like they function to shape the direction of future research into useful directions. which is to say, the process of developing a solution itself generates more useful problems.

of course thrrr are tons of problems once you remove this social consensus based filter. if i’m not mistaken Ramanujan left a book of dozens of unproven theorems, for one quick example. i don’t think that that has opened up dozens of fields of mathematical research.

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.

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.

> I didn't realize that open math problems were a finite resource.

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

He addresses your point in the first paragraph.

> I didn't realize that open math problems were a finite resource.

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...)

You can indeed generate many nonsensical problems. Generating ones which require interesting and non-trivial mathematics is much more difficult.

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.

Besides Sendov’s Conjecture, Terence Tao has also shared his AI-assisted digestion of the counterexample to the Jacobian conjecture. In simple words, digestion just means full human understanding of the results. It could come from understanding the result from a different perspective, or perturbing the result and seeing what breaks.

>We’re basically sharing everything we have with these companies/AI systems.

My sense of the word 'share' is that it traditionally involves agency by all parties involved. There are a lot of words in English for describing taking things without permission and profiting thereby - words like piracy, banditry, and larceny.

If math is solved then move to an area that’s not. Why do fields need “protecting” from ai?

> How can you protect a field that's under this much pressure in the next 6 months?

Sounds like they're going the way of the DoDo. better take that PhD, migrate to the new world and become a tuktuk driver.

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.

It's possible but the approaches these tools take are usually verbose and strange. Think about it like anything else llms do. Even when the picture is right and there are only 5 digits on each hand all the textures are off and so is the lighting and postures. Or in code, the code is always way larger then it needs to be and tightened up strangely with weird loose ends. Or in writing weird idioms, words, structure, and a weaselly way to turn 3 sentences into 8 paragraphs.

People usually use these tools in math and science to find an answer. Then often they will work it back using more sane or human pathways. So it's shareable or even beautiful.

Knowing the answer has value. But, often in math the best thing was how someone got there.

Pure math is practiced mostly for the intellectual thrills and recognition among a very small group of peers. There's little else to it. You don't become rich, you don't become a celebrity. You teach students, write papers, and probably know most other people who work in the same subfield as you. Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.

If you take that away and turn math into a less fulfilling pursuit where you mostly try to make sense of the output of an LLM, and it's "Astra's theorem #18398" and not "John Doe's last theorem", I'd wager that far fewer people will have any interest in the field.

This is really not unique to math, by the way. AI is undermining a lot of creative work. Why blog when you have much better odds of making it to the top of HN with autogenerated blog-slop? Why write books when many nonfiction categories on Amazon are now dominated by AI? The list goes on.

There's plenty of people on HN who think it's nothing new, ignoring the huge change in scale. And those who think this is good because there's no inherent value to human creativity if we can get the same content faster and for less. I disagree.

Well no because it works by joining together existing novel insights.

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?

The "ecosystem" is dead. Tao should be thinking about what will replace it. I don't understand why he's taking this tack.

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.

I don't think we can confidently say, yet, that LLMs can't discover those connections. We just haven't explored them yet, because 99.99% of the prestige is locked up in proving hard results, which also happen to be easier to assess objectively (thanks to automated proof checkers), and so that's where all the effort has thus far been exerted.

Probably an AI-written Lean proof is very different to how a human would write it, and some may say it's more like mathy neuralese. For sure it works but it is not human-friendly and needs to be transformed into something more readable and digestible to be able to extract insights from it.

Not that different from when trying to read an out-of-control vibe coded codebases, or an sloppy AI long email that someone may send you at 9 AM.

Because the problem has almost no value unto itself. The clay statement of navier stokes is not relevant to how CFD is done in practice.

It's about what is non verifiable versus verifiable. The same way it produces "slop" code (which, if you give it test cases, will be 100% correct), it also produces "slop" math.

Code that serves a business function, it's ok if its slop. Math that serves directly a business function also can be slop.

But most open problems are not directly for a particular usecase. People agree widely to attack it due to the perceived possibility of encountering useful mathematical objects along the way, that will then expand the world's mathematical toolset. This is not something that you can easily express in a verifier, and is thus something that is hard to force an LLM system to do.

You are right in that understanding it retrospectively is possible, but that is not going to be as useful as the desired "elegant" objects that expand and unify mathematics. You can't represent these concepts in verifiers.

Again, if you let AI rip at something like say "beat shannon capacity" and suppose it comes up with MIMO as paulraj did, great! It's useful and you can retrospectively understand it, say by expanding shannon to multiple dimensions, as foschini and telatar did. But most math problems are not in that category.

The question then is, if AI is really good at this type of math, how much of the existing mathematical community+process is necessary? I think it will still be necessary, just maybe in fewer cases. Wherever the primary purpose of the math is in a domain and that domain has a verifiable target, we can directly optimise it to that verifiable target in-domain rather than reach for the mathematical community. How well will this work? We'll see. It's not clear if it's even possible to represent most problems this way.

Perhaps reading https://www.math.toronto.edu/mccann/199/thurston.pdf will help.

The point of mathematics is not to prove results. It is to build conceptual thinking about mathematics. Important problems are important because in order to solve them we have to build concepts tying different things together.

We're not searching for answers. We're searching for insights. Trying to understand the problem causes us to draw the connections and find those insights.

AI gives us answers. But it doesn't help us build those insights. AI has a complete mastery of existing human insights. But doesn't build new ones from its own experience. In a real way, it does not find the opportunity to really learn.

So it tackles problems and either solves them or not. If solved, we now have an answer. If not, it's too hard for humans.

Not a mathematician.

The issue I see with a handed-over proof is tunnel-vision: you explore only the understanding of the proof.

Without a proof, your exploration branches out much further, in directions that could seem fruitless, but may uncover new understandings that are now "hidden" because the handed-over proof drastically lowered the incentives to find them.

- If you have only a fuzzy idea of how to get to your travel destination, wrong turns and alternate routes may reveal sights and places you'd never have encountered without that wandering.

- If your GPS directs you straight to your travel destination, you are now where you wanted to be but missed out on the exploration. This is the sort of consequences the AI math proofs have.

STEM research thrives on that side exploration and unearthing unexpected things along the way. James Burke's famous documentary Connections spends the middle episodes talking about the unexpected directions that exploration has taken science. It's very hard to credibly make the case that this sort of meandering exploration is not valuable.

I think you're misunderstanding the point of math problems. Mathematics is as much a process as it is a result. This is why even from early on, relatively rudimentary mathematics questions you are graded by your capacity to correctly achieve the desired process to the answer than getting the answer correct. The risk here is that AI generated proofs removes the process part of mathematics, where actually interesting concepts live (because then you can apply novel concepts to other unsolved problems and then thereby unlock new concepts that way...) Sure you can kind of try to reverse-engineer it but you lose the entire intuition and "we tried applying it in X, Y, Z ways and it didn't work" intuition, because even the non-working process can teach you about how not to apply the working process to novel problem spaces.

Basically: Tasting a delicious soup doesn't tell you how to layer the flavors, but if you want to be a good chef, you better be learning flavors more than you learn dishes!

He's saying that in such a scenario, almost all of the value is located in the analysis and just dumping the proof has "negligible or even negative value". (The negative value would occur in the cases where the proof doesn't contain enough information to reconstruct what insights would have led a person to it.)

While AI companies have almost infinite money, they still don’t want to blow million dollar budgets on problems if there isn’t high likelihood that it will be successful.

But within next 10 years as costs drop significantly and even more improvements are made, yes it is very likely that almost every single existing math problem will get a serious AI cracking done on it

A "massive amount of AI-powered effort" costs a ton of money. What's the return on investment for these frontier labs? Do headline-grabbing successes in mathematics translate to expected *profitability* in disciplines with more immediately quantifiable economic value.

It would be trivially easy to convert Lean into English, so I’m not sure what the human-readable criteria gets you. There are also human written proofs that are considered not human-readable by most of the mathematics community (ABC conjecture).

OpenAI has already said they aren't going to claim the $1,000,000. If this proof claim holds up, then the Millennium prizes will be 2 for 2 for rejections of the prize money for valid solutions. Maybe that will be the precedent for other AI labs as well.

it's 7

> 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.

The future sole owners of half of San Francisco’s mansions thank you for your enlightened stance.

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.

> If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.

Which I don't see a reason for Anthropic and "Open"AI not to, given their not so stellar track record with IP of individuals/entities-that-are-not-rich-enough ;)

People who work hard to build intuition and also use AI effectively will dominate the field.

This will be literally every field, sooner or later, at least until the human and their intuition is just slowing the AI down. There’s no scenario where humans without AI beat humans with AI in the long run, unless there are fields where the “alien intelligence” somehow hurts more than it helps (like artistic pursuits perhaps?)

I would expect this for service providers that provide their "free" services but the mathematicians said they paid OpenAI to help generate this result. This is a clear conflict of interest. Similar to inside trading. Or the same lawyer being hired from two opposing sides which is a big no no. There are rules and laws that already deal with this in other industries and I expect that there will have to be regulation developed for cases where there is a conflict between AI customers and AI provider interest.

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)

The difference is no one writes punchcards so the comparison is inapt. It is more like supposing AI eats everything humans attempt to do. They will take your work as you are working on it, any interaction at all. It is meta-plagiarism, on another level.

What? No, it's nothing like that. It means that if you have a critical insight and you share it out loud, somebody with no particular mathematical inclination can just spend lots of money to steal the credit from you.

It shifts power further from the worker toward those with capital.

If there are no incentives for mathematicians to work on and share progress in tough problems because they will get scooped, then they will end up quitting and in net we will see less progress overall.

> In other words, the "right" people need to solve it: the mathematicians who made it their job and not the people working to push AI models forward?

This is not at all what he said? I'm not sure how you got this from anything he wrote, actually.