3 ms·

Algorithm-assisted discovery of an intrinsic order among mathematical constants

85 points by BerislavLopac 3y ago

YeGoblynQueenne 3y ago
Original on arxiv: https://arxiv.org/abs/2308.11829 https://arxiv.org/abs/2308.11829
- miguelmurca 3y ago
  This should be the link. As is, currently, you get a PDF viewer that does not allow zoom (without zooming surrounding UI), and with a weird comments integration.
henrydark 3y ago
Still waiting to see the first new irrationality proof with these ideas, wishing you lots of good luck!
ackbar03 3y ago
Can someone tldr this please?
- pringk02 3y ago
  Some people involved in this have a YouTube explainer here: https://www.youtube.com/watch?v=Uk04gfIt8yM https://www.youtube.com/watch?v=Uk04gfIt8yM
- contravariant 3y ago
  There are a lot of continued fraction formulas for mathematical constants of the form: c = a(0) + a(1) / (b(1) + a(2) / ( .... Here a and b are polynomials (this looks a bit more sensible in their paper, I don't really know how to denote continued fractions in ASCII). If you stop this fraction at a certain point you get a fraction like p_n / q_n. In general the value of p_n and q_n grow extremely rapidly. Their conjecture is that if you look at this fraction in its lowest terms then p_n and q_n grow way less rapidly (exponential vs. a power of a factorial), but only if the limit is a mathematical constant. Obviously this can't hold for all mathematical constants, but as a rule of thumb it seems to work well enough to identify when two polynomials a and b correspond to some mathematical constant or a combination of mathematical constants.
  - Garlef 3y ago
    And what is a "mathematical constant" in this context? Not in terms of examples but rather: Is this something that can be defined in a meaningful way? Remembering that one joke about inductive proofs ("there are no uninteresting natural numbers"), my first impression would be that this is a bit of a void concept. Regarding the joke: "there are no uninteresting natural numbers because if there were any, there would also be a smallest uninteresting number, making this number interesting, which is a contradiction"
    - jjgreen 3y ago
      I'd say not, mathematical constants are usually irrational (often transcendental) numbers which tend to crop-up consistently in different contexts, so they're culturally (and so ill-) defined. Such concepts are not uncommon in mathematics, "canonical" springs to mind.
nonrandomstring 3y ago
This sounds big. Can some mathematicians please weigh in.
- kevinventullo 3y ago
  Seems to be in the spirit of the “Ramanujan Machine” from a few years ago, which… well here is one prominent mathematician’s take: https://www.galoisrepresentations.com/2019/07/17/the-ramanujan-machine-is-an-intellectual-fraud/ https://www.galoisrepresentations.com/2019/07/17/the-ramanuj... Until they prove a new irrationality result I personally won’t be paying much attention.
  - nonrandomstring 3y ago
    Thanks. This is a fascinating rabbit hole in itself regardless of the merit of the claims, because I was unaware of how rapidly advancing experimental mathematics is banging heads with traditional work. I see the problem, which goes deep into issues of ML/AI, that interesting results without "proofs of understanding" are "Idiotic" in the original Greek sense - that they stand alone and separate from the wider corpus - leaving someone else the work of "connecting them up", as it were. Am I even half right?
    - kevinventullo 3y ago
      To be honest, we’re not even to that point. There have been no major mathematical proofs to come out of ML/AI, regardless of understanding. I say “major” because, sure, you can prove something “new” in a constrained framework that lends itself to different types of search techniques, but this is sort of akin to writing down the “undiscovered” mathematical statement that X*Y=Z, where X and Y are some super large numbers whose product has never been explicitly computed. So, if an ML technique came out and proved the Riemann hypothesis or that zeta(5) was irrational, even if the proof was totally inscrutable, that would be absolutely groundbreaking. At the risk of hyperbole, potentially the most important result of my lifetime; not because of the result per se, but because it would signal a shift in how all of mathematics is done.

85 points by BerislavLopac 3y ago

YeGoblynQueenne 3y ago

Original on arxiv: https://arxiv.org/abs/2308.11829 https://arxiv.org/abs/2308.11829

miguelmurca 3y ago
This should be the link. As is, currently, you get a PDF viewer that does not allow zoom (without zooming surrounding UI), and with a weird comments integration.

henrydark 3y ago

Still waiting to see the first new irrationality proof with these ideas, wishing you lots of good luck!

ackbar03 3y ago

Can someone tldr this please?

pringk02 3y ago
Some people involved in this have a YouTube explainer here: https://www.youtube.com/watch?v=Uk04gfIt8yM https://www.youtube.com/watch?v=Uk04gfIt8yM
contravariant 3y ago
There are a lot of continued fraction formulas for mathematical constants of the form: c = a(0) + a(1) / (b(1) + a(2) / ( .... Here a and b are polynomials (this looks a bit more sensible in their paper, I don't really know how to denote continued fractions in ASCII). If you stop this fraction at a certain point you get a fraction like p_n / q_n. In general the value of p_n and q_n grow extremely rapidly. Their conjecture is that if you look at this fraction in its lowest terms then p_n and q_n grow way less rapidly (exponential vs. a power of a factorial), but only if the limit is a mathematical constant. Obviously this can't hold for all mathematical constants, but as a rule of thumb it seems to work well enough to identify when two polynomials a and b correspond to some mathematical constant or a combination of mathematical constants.
- Garlef 3y ago
  And what is a "mathematical constant" in this context? Not in terms of examples but rather: Is this something that can be defined in a meaningful way? Remembering that one joke about inductive proofs ("there are no uninteresting natural numbers"), my first impression would be that this is a bit of a void concept. Regarding the joke: "there are no uninteresting natural numbers because if there were any, there would also be a smallest uninteresting number, making this number interesting, which is a contradiction"
  - jjgreen 3y ago
    I'd say not, mathematical constants are usually irrational (often transcendental) numbers which tend to crop-up consistently in different contexts, so they're culturally (and so ill-) defined. Such concepts are not uncommon in mathematics, "canonical" springs to mind.

nonrandomstring 3y ago

This sounds big. Can some mathematicians please weigh in.

kevinventullo 3y ago
Seems to be in the spirit of the “Ramanujan Machine” from a few years ago, which… well here is one prominent mathematician’s take: https://www.galoisrepresentations.com/2019/07/17/the-ramanujan-machine-is-an-intellectual-fraud/ https://www.galoisrepresentations.com/2019/07/17/the-ramanuj... Until they prove a new irrationality result I personally won’t be paying much attention.
- nonrandomstring 3y ago
  Thanks. This is a fascinating rabbit hole in itself regardless of the merit of the claims, because I was unaware of how rapidly advancing experimental mathematics is banging heads with traditional work. I see the problem, which goes deep into issues of ML/AI, that interesting results without "proofs of understanding" are "Idiotic" in the original Greek sense - that they stand alone and separate from the wider corpus - leaving someone else the work of "connecting them up", as it were. Am I even half right?
  - kevinventullo 3y ago
    To be honest, we’re not even to that point. There have been no major mathematical proofs to come out of ML/AI, regardless of understanding. I say “major” because, sure, you can prove something “new” in a constrained framework that lends itself to different types of search techniques, but this is sort of akin to writing down the “undiscovered” mathematical statement that X*Y=Z, where X and Y are some super large numbers whose product has never been explicitly computed. So, if an ML technique came out and proved the Riemann hypothesis or that zeta(5) was irrational, even if the proof was totally inscrutable, that would be absolutely groundbreaking. At the risk of hyperbole, potentially the most important result of my lifetime; not because of the result per se, but because it would signal a shift in how all of mathematics is done.