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.
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.
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.
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.
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.
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"
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.
https://arxiv.org/abs/2308.11829