The Ramanujan Machine: Using algorithms to discover new mathematics • Hacker News Board

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#21

Post by nurettin » Thu, Jul 04, 2019, 7:19 PM UTC

amelius wrote: post #9

nurettin wrote: post #5

It seems aptly named because Ramanujan also dealt with infinite series. One of his most famous equations is the ramanujan summation [1] where he derives that the sum of infinite series 1 + 2 + 3 + 4 + ... = - 1 / 12 [1] https://en.wikipedia.org/wiki/Ramanujan_summation

I still don't understand why this derivation is such an achievement. If anything, it seems like dirty hacking, but this time done by a mathematician.

I read somewhere that the result is confirmed in applied physics where it somehow relates to the casimir force, but I'm not a physicist.

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#22

Post by OscarCunningham » Thu, Jul 04, 2019, 8:08 PM UTC

m3kw9 wrote: post #13

Zenst wrote: post #12

I was pondering a few weeks back about the prospects of using ML/AI to find new ways to factor primes and if their is any sequence in primes and how to calculate them in a way that you input N and it will produce the Nth prime. That would be something.

Can primes be estimated?

The nth prime is about nlog(n).

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#23

Post by carapace » Thu, Jul 04, 2019, 8:52 PM UTC

Zenst wrote: post #12

I was pondering a few weeks back about the prospects of using ML/AI to find new ways to factor primes and if their is any sequence in primes and how to calculate them in a way that you input N and it will produce the Nth prime. That would be something.

There's something called "Deep Algebra" (no links handy sorry) where they are trying to parse LaTeX source of math papers to extract equations and figure out new stuff using ML.

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#24

Post by fzort » Thu, Jul 04, 2019, 9:10 PM UTC

Zenst wrote: post #12

I was pondering a few weeks back about the prospects of using ML/AI to find new ways to factor primes and if their is any sequence in primes and how to calculate them in a way that you input N and it will produce the Nth prime. That would be something.

> you input N and it will produce the Nth prime

Look for Willans' formula (there's a nice explanation in the book "Hacker's Delight").

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#25

Post by eximius » Thu, Jul 04, 2019, 10:48 PM UTC

nurettin wrote: post #5

It seems aptly named because Ramanujan also dealt with infinite series. One of his most famous equations is the ramanujan summation [1] where he derives that the sum of infinite series 1 + 2 + 3 + 4 + ... = - 1 / 12 [1] https://en.wikipedia.org/wiki/Ramanujan_summation

Except he didn't. He created a way of assigning surrogate values to forms that otherwise wouldn't have one. To quote from the Wikipedia article you linked:

> assigning a value to divergent infinite series

Every time someone claims that series is EQUAL to -1/12, someone loses faith in math.

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#26

Post by mathandpoop » Thu, Jul 04, 2019, 11:06 PM UTC

nurettin wrote: post #21

amelius wrote: post #9

Earlier quoted context omitted.

I still don't understand why this derivation is such an achievement. If anything, it seems like dirty hacking, but this time done by a mathematician.

I read somewhere that the result is confirmed in applied physics where it somehow relates to the casimir force, but I'm not a physicist.

It's a theorem of analytic number theory. It's just a special value of the ζ-function. People misunderstand how analytic continuation works and therefore (wrongly) interpret ζ(-1) as the divergent series 1+2+3+... (The correct interpretation is as the unique analytic continuation of the holomorphic function $\sum_{n\geq 1}\frac{1}{n^s}$ defined on the half-plane $\Re(s)>1$ to $\mathbb{C}\setminus\{1\}$.)

It's actually a pretty simple consequence of the functional equation for ζ and a few special values of Γ. That in turn comes from a theta-function identity which can be proven using Poisson's summation formula.

What I'm trying to say is that it's legit maths, that has been distorted due to the shock value of writing the equation "1+2+3+... = -1/12".

If you're looking for a reference, go to Davenport's "Multiplicative Number Theory". It's short, self-contained, and extremely well-written. Serre's "A Course in Arithmetic" should also work.

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#27

Post by mathandpoop » Thu, Jul 04, 2019, 11:27 PM UTC

"New mathematics" is a bit of an overstatement. There is a lot more to maths than continued fractions.

It bugs me a bit that the authors write (I would love for anyone affiliated with the project to talk to me about this) "Any new conjecture, proof, or algorithm suggested will be named after you.". No offense, but there are very few mathematicians out there with that kind of a world view.

Seems a bit like a high school project without proper guidance from a mathematician.

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#28

Post by beagle3 » Sun, Jul 07, 2019, 8:21 AM UTC

thom wrote: post #2

Seems slightly less rich than Doug Lenat's Automated Mathematician (which in turn led to Eurisko which was one of the earliest genuinely interesting AI systems): https://en.wikipedia.org/wiki/Automated_Mathematician

The story of Eurisko doesn't add up. So advanced for its time, not replicated for 30 years (or at all) and Lenat lost the source code and was not interested in doing anything similar again, with the next version (Cyc, IIRC) not getting anywhere close?

I would guess there was a lot more manual handholding that was not documented.

Re: The Ramanujan Machine: Using algorithms to discover new mathematics

#29

Post by thom » Mon, Jul 08, 2019, 4:19 PM UTC

beagle3 wrote: post #28

thom wrote: post #2

Seems slightly less rich than Doug Lenat's Automated Mathematician (which in turn led to Eurisko which was one of the earliest genuinely interesting AI systems): https://en.wikipedia.org/wiki/Automated_Mathematician

The story of Eurisko doesn't add up. So advanced for its time, not replicated for 30 years (or at all) and Lenat lost the source code and was not interested in doing anything similar again, with the next version (Cyc, IIRC) not getting anywhere close? I would guess there was a lot more manual handholding that was not documented.

Yes, I mostly agree here. But even with that manual handholding, it's no different from current, mostly-supervised AI approaches. Even if you think of Eurisko as a very clever optimiser when posed problems in a certain way, it clearly delivered some interesting results. Also, you _can_ find the source code for large parts of Eurisko if you look hard enough, I believe.

Cyc never seemed in the least bit interesting to me, tbh. Even today several "we taught our AI common sense!" articles have hit HN, and it's still not _really_ true.