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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
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@DanielRobert-Nicoud : That's commonplace, isn't it? –  Michael Hardy Dec 30 '13 at 3:50

5 Answers 5

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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There seem to be something written, googling brings http://www.jstor.org/discover/10.2307/40247796?uid=3738744&uid=2&uid=4&sid=21103271738653

and

http://eric.ed.gov/?id=ED501123

This article contains references you can fol,low :

https://en.wikipedia.org/wiki/Mathematical_beauty

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There's a very recent book by Ulianov Montano called Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics. Here's the first paragraph of the blurb:

This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a view advanced by James McAllister, the assertion is that beauty in science does not confine itself to anecdotes or personal idiosyncrasies, but rather that it had played a role in shaping the development of science. Mathematicians often evaluate certain pieces of mathematics using words like beautiful, elegant, or even ugly. Such evaluations are prevalent, however, rigorous investigation of them, of mathematical beauty, is much less common. The volume integrates the basic elements of aesthetics, as it has been developed over the last 200 years, with recent findings in neuropsychology as well as a good knowledge of mathematics.

Here's the Table of Contents:

Introduction.-

Part 1. Antecedents.-
Chapter 1. On Non-literal Approaches.-
Chapter 2. Beautiful, Literally.-
Chapter 3. Ugly, Literally.-
Chapter 4. Problems of the Aesthetic Induction.-
Chapter 5. Naturalizing the Aesthetic Induction.-

Part 2. An Aesthetics of Mathematics.-
Chapter 6. Introduction to a Naturalistic Aesthetic Theory.-
Chapter 7. Aesthetic Experience.-
Chapter 8. Aesthetic Value.-
Chapter 9. Aesthetic Judgement I: Concept.-
Chapter 10. Aesthetic Judgement II: Functions.-
Chapter 11. Mathematical Aesthetic Judgements.-

Part 3. Applications.-
Chapter 12. Case Analysis I: Beauty.-
Chapter 13. Case Analysis II: Elegance.-
Chapter 14. Case Analysis III: Ugliness, Revisited.-
Chapter 15. Issues of Mathematical Beauty, Revisited. ​

Here's a link to Springer's page on the book.

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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
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@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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