For those who have learned something of higher mathematics, nothing could be more natural than to use the word “beautiful” in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic.
So powerful are these aesthetic impressions that one great mathematician, G.H. Hardy, declared that beauty, not usefulness, is the true justification for mathematics. To Hardy, mathematics was first and foremost a creative art. “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful,” he wrote in his classic 1940 book, A Mathematician’s Apology. “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
And what is the appropriate reaction when one is confronted by mathematical beauty? Pleasure, certainly; awe, perhaps. Thomas Jefferson wrote in his seventy-sixth year that contemplating the truths of mathematics helped him to “beguile the wearisomeness of declining life.” To Bertrand Russell—who rather melodramatically claimed, in his autobiography, that it was his desire to know more of mathematics that kept him from committing suicide—the beauty of mathematics was “cold and austere, like that of sculpture…sublimely pure, and capable of a stern perfection.” For others, mathematical beauty may evoke a distinctly warmer sensation. They might take their cue from Plato’s Symposium. In that dialogue, Socrates tells the guests assembled at a banquet how a priestess named Diotima initiated him into the mysteries of Eros—the Greek name for desire in all its forms.
One form of Eros is the sexual desire aroused by the physical beauty of a particular beloved person. That, according to Diotima, is the lowest form. With philosophical refinement, however, Eros can be made to ascend toward loftier and loftier objects. The penultimate of these—just short of the Platonic idea of beauty itself—is the perfect and timeless beauty discovered by the mathematical sciences. Such beauty evokes in those able to grasp it a desire to reproduce—not biologically, but intellectually, by begetting additional “gloriously beautiful ideas and theories.” For Diotima, and presumably for Plato as well, the fitting response to mathematical beauty is the form of Eros we call love.1
Edward Frenkel, a Russian mathematical prodigy who became a professor at Harvard at twenty-one and who now teaches at Berkeley, is an unabashed Platonist. Eros pervades his winsome new memoir, Love and Math. As a boy, he was hit by the beauty of mathematics like a coup de foudre. When, while still in his teens, he made a new mathematical discovery, it was “like the first kiss.” Even when his career hopes seemed blighted by Soviet anti-Semitism, he was sustained by the “passion and joy of doing mathematics.” And he…
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