3 ms·

I would be extremely surprised if something as elegant, terse, and useful as the Fourier Transform had been missed by human mathematicians up until now. All exp

I would be extremely surprised if something as elegant, terse, and useful as the Fourier Transform had been missed by human mathematicians up until now. All expressible theorems are enumerable, after all (if we limit ourselves to a finite alphabet). It seems likely that any new theorems are long, highly complex and esoteric, regardless of human or machine origin.

fn-mote 2mo ago
1. The computer is going to struggle to recognize elegance. I’m not sure it’s relevant at this point (but who knows). 2. The statement about proofs is just way wrong. It doesn’t sound like you are familiar enough with them. This isn’t exactly what you implied, but witness the very short disproof of the Jacobean Conjecture.
- gowld 2mo ago
  > the very short disproof of the Jacobean Conjecture. That's a counterexample (finding a needle in a haystack), not an elegant proof. Proving the conjecture would be elegant, if it were true but somehow still resisted proof nearly as much as the conjecture did because the conjecture was false.
- woopsn 2mo ago
  Big leaps in insight I don't think are received as elegant, generally. Fourier's contemporaries actually thought he was wrong (which he was), about both the series and heat equations, and even with the whole thing worked out it's not really straightforward. It fits into integral transforms/linear operators but in a way more complicated than can explained in those terms alone. Plus truncation error is commonly unbounded! Complex numbers and "fluxions" were also objectionable on their face although now one can frame them more or less elegantly.
Smaug123 2mo ago
Shannon was 1948, one-way crypto in 1978, univalence/HoTT something like 2007; I would be surprised if there weren’t simple new fundamental primitives out there! One problem is that some great advances are from viewing complex objects in a simple way, which take a lot of characters to define in formal logic but which are “simple” in platonic maths-space.

fn-mote 2mo ago

1. The computer is going to struggle to recognize elegance. I’m not sure it’s relevant at this point (but who knows). 2. The statement about proofs is just way wrong. It doesn’t sound like you are familiar enough with them. This isn’t exactly what you implied, but witness the very short disproof of the Jacobean Conjecture.

gowld 2mo ago
> the very short disproof of the Jacobean Conjecture. That's a counterexample (finding a needle in a haystack), not an elegant proof. Proving the conjecture would be elegant, if it were true but somehow still resisted proof nearly as much as the conjecture did because the conjecture was false.
woopsn 2mo ago
Big leaps in insight I don't think are received as elegant, generally. Fourier's contemporaries actually thought he was wrong (which he was), about both the series and heat equations, and even with the whole thing worked out it's not really straightforward. It fits into integral transforms/linear operators but in a way more complicated than can explained in those terms alone. Plus truncation error is commonly unbounded! Complex numbers and "fluxions" were also objectionable on their face although now one can frame them more or less elegantly.

Smaug123 2mo ago

Shannon was 1948, one-way crypto in 1978, univalence/HoTT something like 2007; I would be surprised if there weren’t simple new fundamental primitives out there! One problem is that some great advances are from viewing complex objects in a simple way, which take a lot of characters to define in formal logic but which are “simple” in platonic maths-space.