Mentioned in

Beauty in Mathematics

5NancyLebovitz

0multifoliaterose

2realitygrill

1multifoliaterose

2ThomasR

-1Daniel_Burfoot

New Comment

I can easily believe that there's a lot of individual variation in which fields of mathematics seem most beautiful to which mathematicians, but is this also true of theorems in mathematicians' specialties?

Good question. I have some immediate reactions, but your question deserves a well-considered response, so I'll spend some time reflecting before attempting articulate my thoughts.

Upvoted for that lovely poem, though Li Bai is more aligned with Daoism than Buddhism.

I'm nowhere near being a mathematician, though I do like what little math I've been exposed to. I've always wondered what this 'beauty' meant. Some things seem elegant, clever, insightful - maybe distantly extrapolate-able to sublime or majestic - but 'beauty' eludes me. It doesn't help that mathematicians don't seem to agree on this sense either.

Is finding some relation 'remarkable' or 'intriguing' or 'mysterious' sort of like the baby versions of mathematical beauty?

Upvoted for that lovely poem, though Li Bai is more aligned with Daoism than Buddhism.

Glad that you enjoyed the poem.

I'm nowhere near being a mathematician, though I do like what little math I've been exposed to.

What have you seen before?

I've always wondered what this 'beauty' meant. Some things seem elegant, clever, insightful - maybe distantly extrapolate-able to sublime or majestic - but 'beauty' eludes me.

According to Luc Illusie's recent Reminiscences of Grothendieck and His School

Grothendieck had a very strong feeling for music. He liked Bach, and his most beloved pieces were the last quartets by Beethoven.

I was very interested to read this because I myself am strongly attracted to Bach and Beethoven's late quartets. I can only speak for myself, but the aesthetic appeal of math for me is similar to the aesthetic appeal of Bach and of the fugues variations by Beethoven and Brahms. The key for me is the build up of grand and extremely coherent structures through layering of simple motivic elements. I find the resulting juxtaposition between the simple things that are very close to our inborn instincts and the greatest human intellectual achievements to be very poignant. This seems to be in consonance with Zagier's quotation above.

A specific musical piece that has mentioned character for me is the third movement of Beethoven's Piano Sonata No. 30.

I'd be happy to say more if you have further questions.

Thanks for the nice article! Cox' book is really very beautifull on some of the most beautifull themes in mathematics! Here is the link to an old text on related issues. Conc. the comment below, I'd say that it does not relate to individual theorems or definitions, but to global ideas (which may allow several, different expressions, like Grothendieck's "Dessins d'Enfant").

Serious mathematicians are often drawn toward the subject and motivated by a powerful aesthetic response to mathematical stimuli. In his essay on Mathematical Creation, Henri Poincare wrote

The prevalence and extent of the feeling of mathematical beauty among mathematicians is not well known. In this article I'll describe some of the reasons for this and give examples of the phenomenon. I've excised many of the quotations in this article from the extensive collection of quotations compiled by my colleague Laurens Gunnarsen.

There's an inherent difficulty in discussing mathematical beauty which is that as in all artistic endeavors, aesthetic judgments are subjective and vary from person to person. As Robert Langlands said in his recent essay Is there beauty in mathematical theories?

Even when they are

personallymotivated by what they find beautiful, mathematicians tend to deemphasize beauty in professional discourse, preferring to rely on more objective criteria. Without such a practice, the risk of generalizing from one example and confusing one's own immediate aesthetic preferences with what's in the interest of the mathematical community and broader society would be significant. In the same essay Langlands saidThe asymmetry between personal motivations and professional discourse gives rise to the possibility that outside onlookers might misunderstand the motivations of mathematicians and consequently misunderstand the nature of mathematical practice.

Aside from this, another reason why outside onlookers are frequently mislead is the high barrier to entry to advanced mathematics. In his article Mathematics: art and science, Armand Borel wrote:

I think that Borel's statement about the inaccessibility of mathematics to non-mathematicians is too strong. For a counterpoint, in a reference to be added, Jean-Pierre Serre said

In his aforementioned essay Langlands wrote

and then after reviewing the history of algebraic numbers:

This can be viewed as a reconciliation of Borel's statement and Serre's statement.

I'll proceed to give some more specific examples of aesthetic reactions. In the spirit of the quotations from Langlands above, the remarks quoted below should be interpreted as expressions of personal preferences and experiences rather than as statements about the objective nature of reality. All the same, since human preferences are correlated, knowing about the personal preferences of others does provide useful information about what one might personally find attractive.

Furthermore, as Roger Penrose wrote in his article

The Role of Aesthetics in Pure and Applied Mathematical Research, the ultimate justification for pursuing mathematics for its own sake is aesthetic:In an autobiography, David Mumford wrote

In his essay in Mathematicians: An Outer View of the Inner World, Don Zagier wrote

In

Recoltes et SemaillesAlexander Grothendieck wrote about his subjective experience of his transition to algebraic geometry following a successful early career in analysis:On rare occasions I've been fortunate to experience the "superabundant richness" that Grothendieck describes in connection with mathematics. I've quoted a reflective piece that I wrote about a year ago about such an experience from November 2008: