It may be surprising to see emotional sensibility invoked à propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true aesthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility.
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?
I appreciate, as do many, that there is bad architecture, good architecture and great architecture just as there is bad, good, and great music or bad, good and great literature but neither my education, nor my experience nor, above all, my innate abilities allow me to distinguish with any certainty one from the other. Besides the boundaries are fluid and uncertain. With mathematics, my topic in this lecture, the world at large is less aware of these distinctions and, even among mathematicians, there are widely different perceptions of the merits of this or that achievement, this or that contribution.
Even when they are personally motivated 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 said
Harish-Chandra and Chevalley were certainly not alone in perceiving their goal as the revelation of God’s truths, which we might interpret as beauty, but mathematicians largely use a different criterion when evaluating the efforts of their colleagues. The degree of the difficulties to be overcome, thus of the effort and imagination necessary to the solution of a problem, is much more likely than aesthetic criteria to determine their esteem for the solution, and any theory that permits it. This is probably wise, since aesthetic criteria are seldom uniform and often difficult to apply. The search for beauty quickly lapses in less than stern hands into satisfaction with the meretricious.
The 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.
I [have] already mentioned the idea of mathematics as an art, a poetry of ideas. With that as a starting point, one would conclude that, in order for one to appreciate mathematics, to enjoy it, one needs a unique feeling for the intellectual elegance and beauty of ideas in a very special world of thought. It is not surprising that this can hardly be shared with nonmathematicians: Our poems are written in a highly specialized language, the mathematical language; although it is expressed in many of the more familiar languages, it is nevertheless unique and translatable into no other language; unfortunately, these poems can only be understood in the original. The resemblance to an art is clear. One must also have a certain education for the appreciation of music or painting, which is to say one must learn a certain language.
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
I’ve always loved mathematics. My earliest memory, which goes back to the beginning of elementary school, is of learning the multiplication table. When one loves to play, one tries to understand the reason. All my mathematics is like this, but a bit more complicated.
In his aforementioned essay Langlands wrote
Initially perhaps there is no gangue, not even problems, perhaps just a natural, evolutionary conditioned delight in elementary arithmetic – the manipulation of numbers, even of small numbers – or in basic geometric shapes – triangles, rectangles, regular polygons.
and then after reviewing the history of algebraic numbers:
Does mathematical beauty or pleasure require such an accumulation of concepts and detail? Does music? Does architecture? Does literature? The answer is certainly “no” in all cases. On the other hand, the answer to the question whether mathematical beauty or pleasure admits such an accumulation and whether the beauty achieved is then of a different nature is, in my view, “yes”. This response is open to dispute, as it would be for the other domains.
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:
How, in fact, does one decide which things in mathematics are important and which are not? Ultimately, the criteria have to be aesthetic ones. There are other values in mathematics, such as depth, generality, and utility. But these are not so much ends in themselves. Their significance would seem to rest on the values of the other things to which they relate. The ultimate values seem simply to be aesthetic; that is, artistic values such as one has in music or painting or any other art form.
At Harvard, a classmate said "Come with me to hear Professor Zariski's first lecture, even though we won't understand a word" and Oscar Zariski bewitched me. When he spoke the words 'algebraic variety,' there was a certain resonance in his voice that said distinctly that he was looking into a secret garden. I immediately wanted to be able to do this too. It led me to 25 yeras of stuggling to make this world tangilbe and visible. Especially, I became obsessed with a kind of passion flower in this garden, the moduli spaces of Riemann. I was always trying to find new angles from which I could see them better.
In his essay in Mathematicians: An Outer View of the Inner World, Don Zagier wrote
I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it's true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful. Some mathematicians find formulas and special cases less interesting and care only about understanding the deep underlying reasons. Of course that is the final goal, but the examples let you see things for a particular problem differently, and anyway it's good to have different approaches and different types of mathematicians.
In Recoltes et Semailles Alexander Grothendieck wrote about his subjective experience of his transition to algebraic geometry following a successful early career in analysis:
The year 1955 marked a critical departure in my work in mathematics: that of my passage from "analysis" to "geometry". I well recall the power of my emotional response (very subjective naturally); it was as if I'd fled the harsh arid steppes to find myself suddenly transported to a kind of "promised land" of superabundant richness, multiplying out to infinity wherever I placed my hand in it, either to search or to gather... This impression, of overwhelming riches has continued to be confirmed and grow in substance and depth down to the present day. (*)
(*) The phrase "superabundant richness" has this nuance: it refers to the situation in which the impressions and sensations raised in us through encounter with something whose splendor, grandeur or beauty are out of the ordinary, are so great as to totally submerge us, to the point that the urge to express whatever we are feeling is obliterated.
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:
I was in Steve Ullom's course on class field theory, finally understanding the statements of the theorems. I had been intrigued by class field theory ever since I encountered David Cox's book titled "Primes of the form x2 + ny2" in 2004 or so, but I was not able to form a mental picture of the subject from Cox's presentation.
I was initially drawn toward algebraic number theory primarily by its reputation rather than out of a love for the subject. By this I don't mean that I was motivated by careerism, but rather that I knew that the subject was a favorite of some of the greatest historical mathematicians and I had seen the Nova video on Fermat's Last Theorem while in high school in which the mathematicians interviewed (in particular Wiles, Mazur, Ribet and Shimura) seemed fascinated by the subject - I figured that if I stuck with it for long enough I would be so struck as well.
It took me a long time to come to a genuine appreciation of the subject. I don't think that this is uncommon - the early manifestations of the subject are somewhat obscure, and I don't think it an accident that it wasn't until the late 1800's that it became mainstream despite having a pedigree stretching back very far. And even today, few expositions highlight the essential points.
Anyway, in Steve Ullom's course I finally "got it" - both on a semantic level and why so many people might have been attracted to the subject. Sometime in November I revisited Silverman and Tate's Rational Points on Elliptic Curves and looked at the last chapter on complex multiplication. I knew that the theory of complex multiplication came highly recommended, Kronecker citing its development as his "dearest dream of youth" and Hilbert having said something like "The theory of complex multiplication of elliptic curves is not only the most beautiful part of mathematics but of all science." I had seen glimmerings of what made it interesting from Cox's book, but again, I had not been able to understand the subject from his exposition.
With the background of the course that I was taking, reading Silverman and Tate I was able to understand the instance of complex multiplication that they worked out and was totally bewitched to learn how the elliptic curve y2 = x3 + x organizes all finite abelian extensions of Q(i) in a very coherent way.
For the next several weeks I was in a plane of existence different both from what which I'm accustomed to and from that of the people surrounding me. Naturally it's difficult to describe such an experience in words and all the more so retrospectively. It was a state of great inner focus and tranquility. I was filled with a sense of limitless possibility. I was simultaneously able to acknowledge the problems in the human world around me while also not being discouraged by them in the least. There are certainly echoes of the Buddhist conception of enlightenment in my feeling. My state brought to mind a famous poem by the Chinese poet Li Bai:
Question and Answer in the Mountains
They ask me why I live in the green mountains.
I smile and don't reply; my heart's at ease.
Peach blossoms flow downstream, leaving no trace -
And there are other earths and skies than these