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

captainbland 6d 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 6d 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 6d 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 6d 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 5d 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.
        karmakurtisaani 5d ago
        Firstly, I'm very happy we're having this conversation. It's so pointless, yet pedantic it warms my heart in the best way possible. Secondly, I stand by the statement that there is no merit to this joke. This is because the way it defines interesting is very hand-wavy. There are interesting and non-interesting problems, but by a sleigh of hand you can turn the non-interesting problems interesting, thus proving that basically everything in the universe is interesting. At least in the mathematically describable universe. When everything is interesting, nothing is interesting. So we can dismiss the proof as a silly joke. What mathematicians find interesting is a different story. However, we can almost certainly say there is only a finite number of problems mathematicians as physical beings can solve. If we have 200 mathematical symbols at our disposal, and we consider all strings of these symbols of length 1000,000, we have captured all the descriptions of problems that fit to 1M symbols. But that's a finite number. Going beyond that starts to be difficult for a human to grasp (if 1M is not too much already), so all mathematical problems that are solvable by a physical mathematician are in that set of strings. And that's not even saying anything about whether or not they're interesting..
- Smaug123 6d ago
  An interesting problem must have a description that fits in a brain, at least for now. Your description-length argument assumes arbitrarily large storage.
  - dcl 6d ago
    the smallest problem that cannot fit in a brain would be pretty interesting
    - Smaug123 6d ago
      Sorry, I assumed the inductive construction was implied; you can indeed describe properties of that particular interesting problem (though of course you can’t hold its definition in your head), so it goes in the list. Keep going. At some point you’ll hit problems where the process of constructing the problem doesn’t even fit in a brain, etc. There are at least countably many problems, but finitely many problems which any algorithm-which-fits-in-the-brain can describe given finitely many inputs-which-fit-in-the-brain. This isn’t an enormously important point - the actual question at issue is an empirical one, “in a steady state, can we produce interesting problems at a rate that exceeds our ability to solve them and integrate our understanding” or something like that - but I did rankle at a “trivial” proof which is invalid due to equivocating between multiple definitions of the word “interesting” (which should really take an object, “interesting to me” vs “interesting to something smarter than me”).
tim333 6d 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.

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 6d 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 6d 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 6d 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 5d 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.
      - karmakurtisaani 5d ago
        Firstly, I'm very happy we're having this conversation. It's so pointless, yet pedantic it warms my heart in the best way possible. Secondly, I stand by the statement that there is no merit to this joke. This is because the way it defines interesting is very hand-wavy. There are interesting and non-interesting problems, but by a sleigh of hand you can turn the non-interesting problems interesting, thus proving that basically everything in the universe is interesting. At least in the mathematically describable universe. When everything is interesting, nothing is interesting. So we can dismiss the proof as a silly joke. What mathematicians find interesting is a different story. However, we can almost certainly say there is only a finite number of problems mathematicians as physical beings can solve. If we have 200 mathematical symbols at our disposal, and we consider all strings of these symbols of length 1000,000, we have captured all the descriptions of problems that fit to 1M symbols. But that's a finite number. Going beyond that starts to be difficult for a human to grasp (if 1M is not too much already), so all mathematical problems that are solvable by a physical mathematician are in that set of strings. And that's not even saying anything about whether or not they're interesting..
Smaug123 6d ago
An interesting problem must have a description that fits in a brain, at least for now. Your description-length argument assumes arbitrarily large storage.
- dcl 6d ago
  the smallest problem that cannot fit in a brain would be pretty interesting
  - Smaug123 6d ago
    Sorry, I assumed the inductive construction was implied; you can indeed describe properties of that particular interesting problem (though of course you can’t hold its definition in your head), so it goes in the list. Keep going. At some point you’ll hit problems where the process of constructing the problem doesn’t even fit in a brain, etc. There are at least countably many problems, but finitely many problems which any algorithm-which-fits-in-the-brain can describe given finitely many inputs-which-fit-in-the-brain. This isn’t an enormously important point - the actual question at issue is an empirical one, “in a steady state, can we produce interesting problems at a rate that exceeds our ability to solve them and integrate our understanding” or something like that - but I did rankle at a “trivial” proof which is invalid due to equivocating between multiple definitions of the word “interesting” (which should really take an object, “interesting to me” vs “interesting to something smarter than me”).

tim333 6d 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.