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When reading mathematical books written for a general audience, or even searching questions on this site, the adjective beautiful is often used to describe mathematics. My question is whether there has been scholarly work on the semantics of the word beautiful as used in this sense.

More precisely, what do mathematicians find beautiful, and why do they choose this word to describe math? Have any authors focused on how the idea of "beauty" in mathematics may have changed over time, or how mathematicians may find different ideas beautiful depending on their social/cultural influences?

To clarify, I am not looking for examples of why mathematics is "beautiful". I am also not looking for quotes or aphorisms from famous mathematicians about the beauty of math. My personal opinion is that mathematicians often use beautiful when they could instead choose words such as simple, elegant, or clever to describe proofs or theorems. I am interested in why they choose to use 'beauty', and the implications of this choice in mathematics exposition. References that investigate the success (or failure!) of efforts to show mathematical beauty in education would also be welcomed.

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And not only "beautiful". I have a friend who often uses the adjective "sexy" when speaking of mathematics... –  Daniel Robert-Nicoud Dec 29 '13 at 19:51
@DanielRobert-Nicoud : That's commonplace, isn't it? –  Michael Hardy Dec 30 '13 at 3:50
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4 Answers

There is an essay by Gian-Carlo Rota, Professor of Applied Mathematics and Philosophy at MIT, titled "The Phenomenology of Mathematical Beauty", which appears as Chapter X in his book Indiscrete Thoughts (not to be confused with one of his other books, Discrete Thoughts). I am inclined to disagree with his bottom-line conclusion, but I agree with his explanation that beauty and elegance are two quite different things. His bottom line: "Mathematical beauty is the expression mathematicians have introduced in order to obliquely admit the phenomenon of enlightenment while avoiding acknowledgment of the fuzziness of the phenomenon." I am not convinced of that.

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I don't know if this is what you're looking for, but this is a recent study correlating mathematical beauty with other forms of beauty in the experience of mathematicians' brains in fMRI scanners. Here's a summary by Scientific American.

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I'm sure there have been pop novels and such written on the subject, but I doubt it has been considered noteworthy of scholarly attention.

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Why would it not be considered worthy of scholarly attention? That certain things in mathematics are strikingly beautiful is frequently mentioned in scholarly writing. But actually analyzing the phenomenon has probably not been done much at all or we'd have heard a lot more about it. –  Michael Hardy Dec 29 '13 at 22:51
Yes, but beautiful-ness isn't a measurable thing. It can't be studied. There may be works, but it's unlikely a professor or someone reputable spent time on such a book or article. –  TheVolatileChemist Dec 30 '13 at 21:56
@TheVolatileChemist What do you meant it can't be studied? There are hundreds of people currently living whose life work is studying beauty. It is one of the major branches of philosophy called aesthetics. –  Matt Dec 31 '13 at 15:31
@TheVolatileChemist : Professors and reputable people write opinion articles on politics and on lots of things that cannot be measured all the time. –  Michael Hardy Dec 31 '13 at 17:56
Not Math professors. –  TheVolatileChemist Dec 31 '13 at 22:49
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