Two Forms of Mathematical Beauty • Hacker News Board

Re: Two Forms of Mathematical Beauty

#11

Post by mcherm » Tue, Jun 16, 2020, 3:20 PM UTC

Jonnston wrote: post #2

I think most mathematical facts described as beautiful fall into one of the two categories of “consequences of definitions” and “shocking connections”. The first happens when the structure of your terms is lined up in just the right way as to make a proof feel automatic and clear, every piece follows right from the previous one in a natural way. The second one is rarer imo, and is enjoyable almost in the same way a c…

In my opinion, your comment contains far more insight than the original article.

Re: Two Forms of Mathematical Beauty

#12

Post by kingkawn » Tue, Jun 16, 2020, 3:59 PM UTC

Reduction to a binary is only useful for producing comedically inaccurate generalizations

Re: Two Forms of Mathematical Beauty

#13

Post by Koshkin » Tue, Jun 16, 2020, 4:25 PM UTC

Jonnston wrote: post #2

I think most mathematical facts described as beautiful fall into one of the two categories of “consequences of definitions” and “shocking connections”. The first happens when the structure of your terms is lined up in just the right way as to make a proof feel automatic and clear, every piece follows right from the previous one in a natural way. The second one is rarer imo, and is enjoyable almost in the same way a c…

I guess Category Theory then is mathematical beauty industrialized.

Re: Two Forms of Mathematical Beauty

#14

Post by CalmStorm » Tue, Jun 16, 2020, 4:34 PM UTC

> Trying to appreciate mathematics without understanding its inner workings is like reading a description of Beethoven’s Fifth Symphony instead of hearing it.

I personally feel J.S. Bach would be a better metaphor here.

Re: Two Forms of Mathematical Beauty

#15

Post by enugu » Tue, Jun 16, 2020, 4:36 PM UTC

thanatropism wrote: post #5

Eh. There's two kinds of mathematical beauty (maybe more, I'm making this up): concepts and proofs. The other day I saw a proof of the minimax principle (about eigenvalues maximizing the Rayleigh coefficient) that used a variational problem over eigenfunctions. This is fairly "ugly" mathematics conceptwise, and there are simpler standard proofs, but this one made the top of my head pop out like that emoji. It explain…

Thanks for the comment, was an interesting lookup and I learnt something. Have to figure how it connects to optics/ML.

Re: Two Forms of Mathematical Beauty

#16

Post by Koshkin » Tue, Jun 16, 2020, 4:37 PM UTC

I forget who of the great men said it but it's stuck in my head that compared to the ellipse ("the general") the circle ("the particular") looks like an idiot's smile. (I guess that's one way of looking at the comparative value of various objects of modern mathematical research.)

Re: Two Forms of Mathematical Beauty

#17

Post by rogue7 » Tue, Jun 16, 2020, 5:04 PM UTC

Jonnston wrote: post #2

I think most mathematical facts described as beautiful fall into one of the two categories of “consequences of definitions” and “shocking connections”. The first happens when the structure of your terms is lined up in just the right way as to make a proof feel automatic and clear, every piece follows right from the previous one in a natural way. The second one is rarer imo, and is enjoyable almost in the same way a c…

Proofs from THE BOOK compiles such results in a nice way. Springer has made it available [1] for free these days.

[1]: https://link.springer.com/book/10.1007%2F978-3-662-57265-8

Re: Two Forms of Mathematical Beauty

#18

Post by kkylin » Tue, Jun 16, 2020, 5:19 PM UTC

Jonnston wrote: post #2

I think most mathematical facts described as beautiful fall into one of the two categories of “consequences of definitions” and “shocking connections”. The first happens when the structure of your terms is lined up in just the right way as to make a proof feel automatic and clear, every piece follows right from the previous one in a natural way. The second one is rarer imo, and is enjoyable almost in the same way a c…

I remember reading somewhere (one of Gian-Carlo Rota's essays?) that one of the most powerful words in mathematics is "but." Can't recall the exact source, unfortunately.

Re: Two Forms of Mathematical Beauty

#19

Post by kkylin » Tue, Jun 16, 2020, 5:21 PM UTC

This article reminds me of this:

https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

I especially like the Atiyah quote.

Re: Two Forms of Mathematical Beauty

#20

Post by magneticnorth » Tue, Jun 16, 2020, 5:51 PM UTC

kkylin wrote: post #19

This article reminds me of this: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf I especially like the Atiyah quote.

For those like me who have bookmarked this to read later but are curious what the Atiyah quote is:

MINIO: How do you select a problem to study?

ATIYAH: I think that presupposes an answer. I don’t think that’s the way I work at all. Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.