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Two Forms of Mathematical Beauty

quantamagazine.org

72 pointstheafh20 comments

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 clever punchline is. A series of facts are setup, and then your viewpoint is suddenly shifted forcing you to recontextualize those facts and see something new. A lot of “elegant” proofs are of this flavor.

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

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

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.

I guess Category Theory then is mathematical beauty industrialized.

It's all beautiful if you know how to look. Fully explaining a blade of grass to its core would have more engineering, biological, physical, and mathematical knowledge than all of mankind's best web startups, businesses, and intellectual efforts combined.

The beautiful irony of the once popular "god of the gaps" argument is that the gaps are continuing to widen toward infinitude each passing day. Each passing day we discover that "knowing" one thing reveals 9 more things we do not. How arrogant it would be to miss the awe-inspiring beauty, consistency, and self-sustaining processes that are everywhere, from mathematics to the physical and beyond.

Reminds me of Russell's quote - “Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”

Mathematics, at its core, is about tangible and easily perceptible stuff like counting things and measuring space. Through the introduction of layers of notational and conceptual abstractions, many dependencies can be discovered and many claims can be made. They are just "there", but we are seeing them through the lens of our own man-made abstractions.

> Mathematics, at its core, is about tangible and easily perceptible stuff like counting things and measuring space.

I think that it depends on what you mean by 'core'. This is certainly the historical core of mathematics—where things started, and so around which all later developments have accreted—and I suspect it characterises a large part of most 'users'' interactions with mathematics, but I think that there are many mathematicians who would not describe your characterisation as the core of what they do professionally.

(It happens that I can't substantiate that even by a flimsy appeal to my own work, because there is a reasonable sense in which counting things is at the heart of my work (even though it's not combinatorics); but there are other fields that I think don't have that sort of connection informing their everyday work, even though it is of course always there historically.)

> but I think that there are many mathematicians who would not describe your characterisation as the core of what they do professionally.

Those mathematicians are certainly doing something much more intellectually-challenging than counting things and measuring space, but I would argue that those basic activities represent the basic problems upon which most of the low-level math abstractions are built. "Serious" math is about operating at much higher abstraction levels, but it is not disconnected from those low-level foundations.

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 explains why Rayleigh coefficients have that name, and links practical statistics/ML concerns (low-rank matrix approximation) to light and refraction.

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

Surely that entire comment would be improved by deleting the pointless "Eh." at the beginning?

That entire comment would be improved by deleting the pointless "Surely" at the beginning?

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.

Thanks! I was feeling lazy. The context of the quote is a discussion of whether one solves problems to understand mathematics, or builds general theory (my paraphrasing) to be able to solve specific problems. I read Atiyah's response as "neither."

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.)

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

> 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.

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