Algorithm-assisted discovery of an intrinsic order among mathematical constants
fermatslibrary.com
fermatslibrary.com
Until they prove a new irrationality result I personally won’t be paying much attention.
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?
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.
just to share what I was thinking while out walking this morning.
I doubt very much we'd see much come out of ML in the capacity of exhaustive search or pattern seeking vis a vis "large models", because maths is to language as intergalactic space is to a teacup.
Doesn't feel like there's a generative potential for stumbling upon "proofy-like" things the way it works with natural language.
But I imagine AI in a few years being able to train on a great corpus of known proofs. We show it how to do constructive, deconstructive, contradiction, induction, equivalence...
And hopefully what it gives us is a meta insight into the nature of proofs themselves, perhaps by finding new techniques, or revealing refutations... that sort of thing.
Those would be tools useful to mathematicians, but I think the BIG finds will still be by made humans who think for 10,000 hours about a problem.
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.
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"