Tuesday 30 July 2019

The aesthetic beauty of math

In 1939, as the buildup to war in Europe intensified, a brilliant French mathematician named André Weil made a plan to emigrate to the U.S. He was thirty-three and didn’t want to serve in the army; his life’s purpose was math, he felt, not soldiering. His escape turned out to be more difficult than he anticipated, in part because, as he would write in his memoir, “the Americans, who so warmly welcome those who do not need them, are much less hospitable to those who happen to be at their mercy”—as we’ve gone on to prove repeatedly since then.

He was vacationing in Finland when the war broke out, and he tried to lay low in Helsinki but was arrested and returned to France, where he sat in jail during the spring of 1940, awaiting trial for desertion. While there, he took some consolation from the fact that jail allowed him to work undisturbed, as well as to read novels and write letters, in particular letters to his sister, Simone Weil, who was also remarkably talented, a philosopher and spiritual thinker.

Though her brother’s incarceration infuriated her, Simone saw an opportunity. His work in advanced mathematics was to her, as it would be to most of us, esoteric. Since you have some spare time on your hands, she wrote to him, why don’t you explain to me exactly what it is you do?

There wouldn’t be any point, he replied. Trying to explain my work to a non-mathematician, he wrote, would be like trying to explain a symphony to someone who can’t hear. Later he would rely on another metaphor, calling math “art in a hard material.”

Mathematics is an artistic endeavor, his words suggest. Yet Simone was skeptical. What kind of art? What is the material? Even poets have language, but your work seems to rely on sheer abstraction, she wrote her brother.

That math is an art, that one of its signature qualities is its beauty—these are ideas that continue to be articulated by mathematicians, even as non-mathematicians may wonder, as Simone did, what that could possibly mean. I myself become wary when a mathematician or scientist speaks about the beauty of her discipline, since it can seem vague and high-handed, if not wrong.

In the same year that André Weil spent months in jail, British mathematician G. H. Hardy penned what is perhaps still the most eloquent attempt to give non-mathematicians a sense of math’s aesthetic appeal, in the form of a book-length essay called A Mathematician’s Apology. As with the letters between the Weil siblings, it was the war that occasioned and shaped Hardy’s book, prompting him to argue that math has an intrinsic value unrelated to any military uses. His Apology is a stylish work and a wistful one. Hardy, then in his sixties, felt that he was past his prime and that writing about mathematics—as opposed to doing mathematics—was symptomatic of his decline.

“A mathematician, like a painter or a poet, is a maker of patterns,” he wrote. “If his patterns are more permanent than theirs, it is because they are made with ideas.” Hardy went on to characterize what makes a mathematical idea worthy: a certain generality, a certain depth, unexpectedness combined with inevitability and economy.

My own pursuit of math ended in college, but this rings as true to me, and it could just as readily apply to a great poem. Elegant is a word often used to commend a good mathematical proof. It is a construction that can seem like a kind of magic trick without sleight of hand, in which nothing has been hidden, each step building up another layer of a black hat that turns out to contain, in the end, a rabbit.

Philosophers can argue whether beauty is the property of an object or lies in our perceptions of it; Hardy would have it both ways. The best mathematics is eternal, he maintained, and like the best literature, it will “continue to cause intense emotional satisfaction to thousands of people after thousands of years.” Recent research in neuroscience has lent support to this idea of “emotional satisfaction.” A few years ago, a neurobiologist in London, Semir Zeki, performed fMRI scans of mathematicians while they contemplated equations they’d rated as beautiful, and the region of their brains that lit up has been associated in other studies with perceptions of visual and musical beauty. (Contemplating equations they found less inspiring, on the other hand, did not activate that part of the mathematicians’ brains.) In the brain, a mathematician’s affective response to math is similar to, or maybe the same as, the way in which we respond to beauty in the arts.

And there’s another sense in which math could be considered beautiful. In addition to the aesthetic appeal of a particular equation or a proof, there’s a kind of cumulative marvelousness to math, to its landscape of ideas. Here is an elaborate model world, in which the more you explore, the more fantastic it gets. “ ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one,” Hardy wrote.

As for the Weils, Simone wouldn’t let her brother off the hook. After he told her he couldn’t explain his work, she urged him, in her next missive, to just please try. The letter he finally wrote, in which he did attempt to present his area of study to her, is still cited by mathematicians—not for its breakneck review of the history of number theory, which gave Simone headaches, but for its description of the process of mathematical discovery. Progress in mathematics, he wrote to her, is often made by working out analogies between one subject area and another. “Nothing is more fecund than these slightly adulterous relationships” between analogous theories, he wrote; “nothing gives greater pleasure to the connoisseur.”

It’s worth noting that Simone didn’t doubt the value of mathematics, but she suspected that in the hands of her brother and his contemporaries, mathematical research had grown too abstract. For her, ancient Greek geometry was the epitome of mathematical thinking and of a piece with other ancient Greek achievements. In that culture, she believed, art and math and science were all bridges between the human and the divine, beauty a means of access to grace. Mathematics, she would write, “is first, before all, a sort of mystical poem composed by God himself.”

Eventually Simone and André both made it to America. André would enjoy a long and illustrious career, while Simone would travel back across the ocean, to England, where she would die in 1943 after an illness left her unable to eat. In a memoir, published in 1992, André would remember a vacation in the mountains his family took when he and Simone were young: “My sister had this vacation in mind when she later wrote that contemplating a mountain landscape had once and for all impressed the notion of purity upon her soul,” he wrote. “I was left with a totally different impression. Seeing the shafts of sunlight criss-cross in the distance at sundown gave me the idea of composing on several planes simultaneously.”

As with the mountains, so with mathematics—the beauty of the discipline, to her, resided in its mystical connection to the divine, while to him it lay in the marvel of its connections, its adulterous relationships. Pressed to say why we deem something beautiful, we might all come up with something different. The very idea of beauty might slip away as we try to articulate it, and yet we would still know it was there.

(Source: The Paris Review)

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