Strangely often, I’m thinking about beauty in mathematics. For example, Euclid’s proof of an infinity of primes is a thing of beauty. Or consider the patterns Stanislaw Ulam discovered primes make among the numbers. Or patterns in nature that have mathematical roots . Or … well, you get the picture. And yet, what does a word like “beauty" really mean in mathematics? What are mathematicians getting at when they use the word?
Whatever it is, it’s always been clear that they think it’s important in some deep way. That is, mathematicians think a certain kind of mathematical beauty is fundamental to their work. When they find it, they know they have something profound.
The great G.H. Hardy, Srinivasa Ramanujan’s mentor at Cambridge University, grappled with this notion in his famous book, A Mathematician’s Apology: “A mathematician, like a painter or a poet, is a maker of patterns. …[They] must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: There is no permanent place in the world for ugly mathematics…"
“It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind—we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it."
So yes, we all generally recognize and even agree about beauty, among humans, a piece of music, a poem, a landscape. But in numbers and mathematical themes? We may not consciously acknowledge beauty there. Yet often the things we find beauty in have mathematics underlying them. Like the stripes you see on a seashore, or on a tiger’s body. Like the spirals in a sunflower, or the way one small floret of cauliflower resembles the whole thing. Like the perfect arc a basketball makes as it swishes home without a sound, or the devastating swing a fast bowler generates.
Hardy again: “[E]very chess-player can recognize and appreciate a ‘beautiful’ game or problem. Yet a chess problem is simply an exercise in pure mathematics, and everyone who calls a problem ‘beautiful’ is applauding mathematical beauty."
All of which is a prelude to suggest how beautiful mathematics can be, and that it is a preoccupation for its practitioners. And some of its practitioners strive to share this beauty with the rest of us. One, a good friend, spent a morning playing pieces of music for me, trying to persuade me that certain phrases were particularly beautiful and made me happier with each hearing, and that I felt happier because of the precise, exquisite mathematical relationship between notes in those phrases. I’m not sure I felt happier, nor that I heard the beauty — but I understood how much it mattered to him.
So my friend was not exactly persuasive. But two more practitioners have just published a study suggesting the rest of us can indeed appreciate aesthetics in mathematics.
Stefan Steinerberger is an assistant professor of mathematics at Yale University. Teaching one day, he found an analogy for a proof: That it was like a “really good Schubert sonata". (I used to play one on the piano, so believe me: Schubert wrote some gorgeous sonatas). To his surprise, this resonated with several students. For mathematics students at Yale, he found, “also do a statistically impressive amount of music". This unexpected connection prompted Steinerberger to get in touch with a psychologist, Samuel G.B. Johnson, whose interests lay in how people reason. Together, they devised experiments to test whether laypeople — not just mathematicians — could see beauty in mathematics just like they see it in Schubert, or in art. In fact, because plenty of people already know about and acknowledge strands common to music and mathematics, they especially wanted to test math alongside art.
How would you test such a hypothesis? Steinberger and Johnson chose four relatively simple mathematical proofs that can be explained easily to non-mathematicians.
One, for example, is the sum of the infinite series:
1/2 + 1/4 + 1/8 + … = 1.
There’s a remarkably intuitive way of showing this. Take a square sheet of paper one foot on each side; its area is, of course, 1 sq. ft. Slice down the middle to produce two rectangles, each 1/2 sq. ft in area. Lay one piece aside and cut the other down the middle to produce two squares, each 1/4 sq. ft in area. Put one of those next to the 1/2 sq. ft rectangle (thus 1/2 + 1/4) and cut the other down the middle to produce two rectangles, each 1/8 sq. ft in area. Put one of those with the previous two shapes (thus 1/2 + 1/4 + 1/8) and … you get the idea. You will rack up a pile of alternating squares and rectangles whose combined area gets ever closer to 1 sq. ft. (Naturally, a diagram would be helpful. Still, try it).
Three more proofs with this one; and to go with them, Steinberger and Johnson chose four landscape paintings and four sonatas. They divided the participants into three groups. The first was given the proofs and the paintings, and asked to match proof to painting, according to how similar they were, aesthetically. They were not primed with any prior opinion, just asked to match them for beauty in any way that occurred to them. The second did the same with the proofs and sonatas. Finally, they asked the third set to individually score each proof and painting using nine criteria, and assign it an overall beauty score. (Those criteria, incidentally, were inspired by six that Hardy speaks of in A Mathematician’s Apology.) The nine are: seriousness, universality, profundity, novelty, clarity, simplicity, elegance, intricacy and sophistication.
The results of the experiments were surprising and satisfying (Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas, Cognition, August 2019).
The first two groups tended to match proof to painting, or proof to sonata, in similar ways. This suggests that there was a common and consistent idea of aesthetics across participants, and that it applied to mathematics. The third group generally used just elegance, clarity and profundity, from the nine given criteria, to decide the beauty of their proofs and paintings. A high elegance score generally indicated a high beauty score, and the participants also seemed to agree on which proofs and paintings were elegant.
But even better, there were threads in common between the first and third groups. The first group matched the aesthetics in painting and proof just by intuition. The third used those nine criteria. Yet there was still broad agreement between the groups. In other words, not only was there a match between beauty in art and beauty in mathematics, there was even a common understanding of beauty itself.
Naturally there are limitations to such a study, or at least questions to ask. Landscapes appeal to many of us, so what if we use paintings of other subjects? What if instead of sonatas we use the blues, or Bappi Lahiri, or Boney M? What if participants cannot understand even apparently simple mathematics? Would any of this change the findings? Perhaps. Yet I can’t help thinking Steinberger and Johnson are on to something here. They write in their abstract: “We argue that these results shed light on broader issues in how and why humans have aesthetic experiences of abstract ideas."
That’s worth pursuing further, I’d say. “The beauty of mathematics," the late Fields Medal winner
Miryam Mirzakhani once said, “only shows itself to more patient followers."
As true beauty should.
What Steinberger and Johnson used:
Proofs: the infinite series above; Gauss’ famous sum of integers; the “pigeonhole" principle; and a particular geometric proof.
Sonatas: Schubert’s Moment Musical in F Minor; Bach’s Fugue from Toccata in E Minor; Beethoven’s Diabelli Variations; and Shostakovich’s Prelude in D-flat Major.
Landscape paintings: Albert Bierstadt’s “Looking Down Yosemite Valley, California" and “A Storm in the Rocky Mountains, Mt. Rosalie"; John Constable’s “The Hay Wain"; and Frederic Edwin Church’s “The Heart of the Andes".
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun