Tao: Open math problems being non-renewably mined by AI

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

Maybe the solution is pretty inscrutable, but it's almost always better than nothing.

So, both of these pieces of info would be at least marginally useful for advancing human knowledge.

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.

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

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

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".

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"?)

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 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.

He addresses your point in the first paragraph.

> 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

> 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...)

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.

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).

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

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.

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

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.

> Well, we name conjectures after the conjecturer not the (dis)prover

That’s not universally true. Some conjectures are renamed after being proven. For example, Fermat’s Last Theorem is now sometimes called the Fermat-Wiles Theorem, the Taniyama-Shimura Conjecture is often referred to as the Modularity Theorem now, etc.

Actually there will because if you've actually ever spent time doing math, you only really get to that level of truly understanding and appreciating the stories if you've actually done the hard work yourself, which in turn will be economically infeasible due to AI. The analogy is interesting but incomplete and misleading.

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)