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People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? Every pr

People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post. Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free). So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight. BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.

itishappy 7d ago
It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.
- captainbland 7d ago
  I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.
  - itishappy 7d ago
    I would argue that novel and/or valuable results are not necessarily interesting! I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm. I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.
- n4r9 7d ago
  Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.
  - karmakurtisaani 7d ago
    Why is x interesting? Just because it has a minimal description in P\I? That makes it interesting in strictly technical sense only.
    - renyicircle 6d ago
      I think that's a variation on the interesting numbers paradox joke. Statement: All numbers are interesting. Proof: Assume by contradiction that there's a non-empty set of uninteresting numbers. Then that set contains the smallest uninteresting number. That property makes it interesting.
      - karmakurtisaani 6d ago
        Yeah, I've seen this before as well. I guess I've just become old and grumpy and can't appreciate jokes like these anymore. Also taking jokes seriously is peak HN so.. Edit: actually, forget about the above. I just find it very annoying when people dismiss good conversations with not-so-good jokes.
        renyicircle 6d ago
        I mean, there's some truth to that joke in this case, so I wouldn't agree that it's dismissing. The point of it is that what matters is how we define "interesting", because by the joke's definition in particular, there can't be uninteresting problems. That seems to be a very subjective concept that can also change with time. Mathematics is so specialized that there can be 3 people in the world who find one specific problem interesting. If they solve it, they'll move on to something else. I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.
- tim333 7d ago
  Interesting is a bit in the eye of the beholder. Some people probably find maths boring full stop, some probably find all of it interesting.
- anon291 6d ago
  That's a good question that can be answered by methods in mathematics.
unsupp0rted 6d ago
There probably is a finite amount of math to be done The universe is bounded by rules, as far as we can tell, and not a lot of them
- logicchains 6d ago
  >The universe is bounded by rules That's physics. Not all math is physics.
  - unsupp0rted 6d ago
    The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes. At those levels math and physics are the same bound: the bound of things the universe allows to be conceived of inside it. It’s possible the math our monkey brains + sand can ever conceive of in this universe is a low and accessible amount.
    - mathgeek 6d ago
      > The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes. When you have a moment, please reference some proofs supporting this.
      - unsupp0rted 6d ago
        There are lots of thoughts you’re not biologically capable of thinking. That means the list of thoughts you’re capable of thinking is finite. That means the amount of math you’re capable of discovering, even if you lived forever, is finite. That’s true of all humans and all constructs. So however much math there is to discover, that’s the finite subset you’ll ever have access to. It’s hard to prove what thoughts no human and no construct is capable of generating, but surely there are some, and it’s possible or even likely some of those are math-related.
nicce 6d ago
> Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post. I guess Terence's main point has been all the time that if we let AI solve all these existing problems, we don't notice the new ones and then there is stagnation.

itishappy 7d ago

It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.

captainbland 7d ago
I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.
- itishappy 7d ago
  I would argue that novel and/or valuable results are not necessarily interesting! I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm. I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.
n4r9 7d ago
Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.
- karmakurtisaani 7d ago
  Why is x interesting? Just because it has a minimal description in P\I? That makes it interesting in strictly technical sense only.
  - renyicircle 6d ago
    I think that's a variation on the interesting numbers paradox joke. Statement: All numbers are interesting. Proof: Assume by contradiction that there's a non-empty set of uninteresting numbers. Then that set contains the smallest uninteresting number. That property makes it interesting.
    - karmakurtisaani 6d ago
      Yeah, I've seen this before as well. I guess I've just become old and grumpy and can't appreciate jokes like these anymore. Also taking jokes seriously is peak HN so.. Edit: actually, forget about the above. I just find it very annoying when people dismiss good conversations with not-so-good jokes.
      - renyicircle 6d ago
        I mean, there's some truth to that joke in this case, so I wouldn't agree that it's dismissing. The point of it is that what matters is how we define "interesting", because by the joke's definition in particular, there can't be uninteresting problems. That seems to be a very subjective concept that can also change with time. Mathematics is so specialized that there can be 3 people in the world who find one specific problem interesting. If they solve it, they'll move on to something else. I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.
tim333 7d ago
Interesting is a bit in the eye of the beholder. Some people probably find maths boring full stop, some probably find all of it interesting.
anon291 6d ago
That's a good question that can be answered by methods in mathematics.

unsupp0rted 6d ago

There probably is a finite amount of math to be done The universe is bounded by rules, as far as we can tell, and not a lot of them

logicchains 6d ago
>The universe is bounded by rules That's physics. Not all math is physics.
- unsupp0rted 6d ago
  The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes. At those levels math and physics are the same bound: the bound of things the universe allows to be conceived of inside it. It’s possible the math our monkey brains + sand can ever conceive of in this universe is a low and accessible amount.
  - mathgeek 6d ago
    > The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes. When you have a moment, please reference some proofs supporting this.
    - unsupp0rted 6d ago
      There are lots of thoughts you’re not biologically capable of thinking. That means the list of thoughts you’re capable of thinking is finite. That means the amount of math you’re capable of discovering, even if you lived forever, is finite. That’s true of all humans and all constructs. So however much math there is to discover, that’s the finite subset you’ll ever have access to. It’s hard to prove what thoughts no human and no construct is capable of generating, but surely there are some, and it’s possible or even likely some of those are math-related.

nicce 6d ago

> Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post. I guess Terence's main point has been all the time that if we let AI solve all these existing problems, we don't notice the new ones and then there is stagnation.