
Two Forms of Mathematical Beauty - theafh
https://www.quantamagazine.org/how-is-math-beautiful-20200616/
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Jonnston
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.

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

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kirse
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.

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iambrj
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.”

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bluetomcat
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.

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JadeNB
> 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.)

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bluetomcat
> 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.

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thanatropism
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.

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

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

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kkylin
This article reminds me of this:

[https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf](https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf)

I especially like the Atiyah quote.

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magneticnorth
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.

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kkylin
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."

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Koshkin
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.)

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

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CalmStorm
> 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.

