
What is the most beautiful equation? - DanielRibeiro
http://www.quora.com/Mathematics/What-is-the-most-beautiful-equation?srid=hZ
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bartonfink
Not to troll, but why do people consider math "beautiful?" I hardly ever use
the word, so I might be missing a nuance of meaning, but when I have used it
it's been to refer to something physical and never to an idea.

Elegance, I can understand - it's a lot of fun to see a complex problem get
reduced to something simpler by a twist of logic. I can understand calling
something compelling as well if the logic is especially clear and concise. But
what do people find "beautiful" about math?

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RiderOfGiraffes
What is "beauty"? Define that, and I can help you. Until you can define it,
it's hard to give meaning to the question.

Warning: embarrassing stream-of-consciousness ahead ...

There are some parts of math that I find genuinely beautiful. They evoke an
emotional reaction, I want to work on them, I want to consider them, I want to
give time to them. To me, the word "beautiful" comes closest to how I think of
them.

But I've done lots of math, and have a great deal of experience to bring to
bear. People with less experience of different varieties and forms of music
usually can't appreciate more "classical" pieces in the same way as people who
have immersed themselves in music for decades. Similarly, without being an
experienced programmer it's hard to tell hacked together code from elegantly
written code.

Similarly, without enough experience, discerning quality in math can be hard.

Finally, math is morethan solving problems. Some of it is, yes, but some of it
is building theories. I remember the first time I really grokked some obscure
parts of low dimensional topology. Suddenly things fell into place, and what
had until then been just valid proofs suddenly became obvious truths.

It's like proving that sqrt(2) is irrational. You can follow the equations,
you can believe the proof, but there's more going on. It's the difference
between knowing that it's true, and seeing that it can be no other way.

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bartonfink
Hey, Rider -

Thanks for the reply. I agree that this is sort of a fluffy subject, but I'm
genuinely curious and figured here on HN I'd be able to get some interesting
opinions on the subject.

I use the term 'beauty' in a more limited sense than you are. The two phrases
that come to mind are 'pleasing to the eye' and 'pleasing to the ear', and
neither of those has anything substantial to do with math. When you say that
there are things you want to work on and consider, the word I would use would
be "compelling." I have ?'s I've chewed on in my own head for over a decade,
and they keep coming back, but I wouldn't call them "beautiful". Is that what
you mean?

You bring up an interesting point that things can be "acquired tastes", so to
speak, and that I might simply be math blind. I don't have substantial math
background compared to most of the people here. My official math education
stopped at differential equations, and I picked up what I needed throughout
grad school but it was very piecemeal. When I look at things like Cantor's
diagonalization, for example, I don't see an obvious truth - I see a trick of
logic with interesting applications. I can follow the logic, but at no point
do I feel like I'm doing anything more than moving symbols around in an
internally consistent manner. Just out of curiosity, what's your background
with math?

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RiderOfGiraffes
I use beauty as "Pleasing to the perception" which doesn't limit it to sight,
sound, touch, taste, or any other physical sense.

To take your example, Cantor's diagonalisation - after a time - takes on a
life of its own. Treating math as simply pushing symbols around makes it
devoid of meaning, and it's meaning that makes it possible to do. Creating a
proof requires more than exploring all the ways of pushing symbols around -
you need a sense of "rightness" to guide you. Similarly, writing a program is
more than banging out something that's syntactically correct. You form it from
your internal sense of what's right.

For reference, I did a PhD in 1983-88 specialising in Graph Theory and
Combinatorics. Since then, in addition to my day job, I've given around 100
presentations a year on what math is really about. Patterns, predictions,
testing, proofs, structure, and why things work.

Whay are all primes of the form 4k+1 equal to the sum of two squares? You can
puch symbols around all you like, but finding a proof requires seeing
structure, form, pattern and "truths" underneath the simple statements.
Following someone else's proof is usually unenlightening.

There really is beauty _and_ elegance there.

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bartonfink
Thank you very much, Rider - this is very interesting. I hate to take up more
of your time, but do you mind if I shoot you an e-mail at your solipsys
account to continue this if you have the time or interest? I'm not sure an HN
thread is the best way to maintain what seems to be a dialogue instead of a
discussion.

And again, thanks for the replies - this has made a bleak Tuesday fun.

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RiderOfGiraffes
Feel free. No time, responses will be sporadic, but happy to engage as able.

