What is the most beautiful equation?
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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?
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.
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?
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.
And again, thanks for the replies - this has made a bleak Tuesday fun.