{"id":138201,"date":"2019-07-22T11:16:49","date_gmt":"2019-07-22T15:16:49","guid":{"rendered":"https:\/\/www.theparisreview.org\/blog\/?p=138201"},"modified":"2019-07-23T10:06:46","modified_gmt":"2019-07-23T14:06:46","slug":"the-aesthetic-beauty-of-math","status":"publish","type":"post","link":"https:\/\/www.theparisreview.org\/blog\/2019\/07\/22\/the-aesthetic-beauty-of-math\/","title":{"rendered":"The Aesthetic Beauty of Math"},"content":{"rendered":"

\"\"<\/a><\/p>\n

In 1939, as the buildup to war in Europe intensified, a brilliant French mathematician named Andr\u00e9 Weil made a plan to emigrate to the U.S. He was thirty-three and didn\u2019t want to serve in the army; his life\u2019s 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, \u201cthe Americans, who so warmly welcome those who do not need them, are much less hospitable to those who happen to be at their mercy\u201d\u2014as we\u2019ve gone on to prove repeatedly since then.<\/p>\n

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.<\/p>\n

Though her brother\u2019s 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\u2019t you explain to me exactly what it is you do?<\/p>\n

There wouldn\u2019t 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\u2019t hear. Later he would rely on another metaphor, calling math \u201cart in a hard material.\u201d<\/p>\n

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.<\/p>\n

That math is an art, that one of its signature qualities is its beauty\u2014these 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.<\/p>\n

<\/p>\n

In the same year that Andr\u00e9 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\u2019s aesthetic appeal, in the form of a book-length essay called A Mathematician\u2019s Apology<\/em>. As with the letters between the Weil siblings, it was the war that occasioned and shaped Hardy\u2019s book, prompting him to argue that math has an intrinsic value unrelated to any military uses. His Apology<\/em> 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\u2014as opposed to doing mathematics\u2014was symptomatic of his decline.<\/p>\n

\u201cA mathematician, like a painter or a poet, is a maker of patterns,\u201d he wrote. \u201cIf his patterns are more permanent than theirs, it is because they are made with ideas<\/em>.\u201d Hardy went on to characterize what makes a mathematical idea worthy: a certain generality, a certain depth, unexpectedness combined with inevitability and economy.<\/p>\n

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 <\/em>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.<\/p>\n

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 \u201ccontinue to cause intense emotional satisfaction to thousands of people after thousands of years.\u201d Recent research in neuroscience has lent support to this idea of \u201cemotional satisfaction.\u201d A few years ago, a neurobiologist in London, Semir Zeki, performed fMRI scans of mathematicians while they contemplated equations they\u2019d 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\u2019 brains.) In the brain, a mathematician\u2019s affective response to math is similar to, or maybe the same as, the way in which we respond to beauty in the arts.<\/p>\n

And there\u2019s another sense in which math could be considered beautiful. In addition to the aesthetic appeal of a particular equation or a proof, there\u2019s 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. \u201c \u2018Imaginary\u2019 universes are so much more beautiful than this stupidly constructed \u2018real\u2019 one,\u201d Hardy wrote.<\/p>\n

As for the Weils, Simone wouldn\u2019t let her brother off the hook. After he told her he couldn\u2019t 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\u2014not 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. \u201cNothing is more fecund than these slightly adulterous relationships\u201d between analogous theories, he wrote; \u201cnothing gives greater pleasure to the connoisseur.\u201d<\/p>\n

It\u2019s worth noting that Simone didn\u2019t 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, \u201cis first, before all, a sort of mystical poem composed by God himself.\u201d<\/p>\n

Eventually Simone and Andr\u00e9 both made it to America. Andr\u00e9 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, Andr\u00e9 would remember a vacation in the mountains his family took when he and Simone were young: \u201cMy 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,\u201d he wrote. \u201cI 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.\u201d<\/p>\n

As with the mountains, so with mathematics\u2014the 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.<\/p>\n

Karen Olsson\u00a0is the author of the novels\u00a0<\/em>Waterloo, which was a runner-up for the 2006 PEN\/Hemingway Award for First Fiction, and\u00a0<\/em>All the Houses. Her most recent book,\u00a0<\/em>The Weil Conjectures, is out this month.\u00a0<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

The best mathematics, like the best literature, will continue to cause intense emotional satisfaction to thousands of people after thousands of years.<\/p>\n","protected":false},"author":1805,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[419],"tags":[],"class_list":["post-138201","post","type-post","status-publish","format-standard","hentry","category-arts-culture"],"acf":[],"yoast_head":"\nThe Aesthetic Beauty of Math by Karen Olsson<\/title>\n<meta name=\"description\" content=\"July 22, 2019 \u2013 The best mathematics, like the best literature, will continue to cause intense emotional satisfaction to thousands of people after thousands of years.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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