4 ms·

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

by jfengel 17d ago

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.

pitchlatte 17d ago
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.
- gowld 17d ago
  > the Clay millennium prize problems 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!
  - dgellow 17d ago
    Did you read Tao‘s tweets? That’s what he addresses
_alternator_ 17d ago
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.
nilkn 17d ago
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.
- bee_rider 17d ago
  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?
  - pixl97 17d ago
    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.
porcoda 17d ago
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".
- hkalbasi 17d ago
  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.
  - mlyle 17d ago
    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.
    - jordanb 17d ago
      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.
gowld 17d ago
> I didn't realize that open math problems were a finite resource. There's an interesting commentary about this: https://mathstodon.xyz/@tao/117237320796901560 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/quotations/ https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quot...)
applicative 17d ago
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.
cool_dude85 17d ago
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.
agnishom 17d ago
> 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
mellosouls 17d ago
He addresses your point in the first paragraph.
pvillano 17d ago
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.
wrsh07 17d ago
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"?)
- pixl97 17d ago
  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.
Ar-Curunir 17d ago
You can indeed generate many nonsensical problems. Generating ones which require interesting and non-trivial mathematics is much more difficult.

by jfengel 17d ago

pitchlatte 17d ago

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.

gowld 17d ago
> the Clay millennium prize problems 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!
- dgellow 17d ago
  Did you read Tao‘s tweets? That’s what he addresses

_alternator_ 17d ago

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.

nilkn 17d ago

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.

bee_rider 17d ago
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?
- pixl97 17d ago
  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.

porcoda 17d ago

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".

hkalbasi 17d ago
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.
- mlyle 17d ago
  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.
  - jordanb 17d ago
    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.

gowld 17d ago

> I didn't realize that open math problems were a finite resource. There's an interesting commentary about this: https://mathstodon.xyz/@tao/117237320796901560 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/quotations/ https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/quot...)

applicative 17d ago

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.

cool_dude85 17d ago

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.

agnishom 17d ago

> 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

mellosouls 17d ago

He addresses your point in the first paragraph.

pvillano 17d ago

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.

wrsh07 17d ago

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"?)

pixl97 17d ago
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.

Ar-Curunir 17d ago

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