In the Mountains of Mathematics

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Kurt Gödel and Albert Einstein, Princeton, New Jersey, 1954

“The science of pure mathematics…may claim to be the most original creation of the human spirit.” So declared the philosopher (and lapsed mathematician) Alfred North Whitehead. Strange, then, that the practitioners of this “science” still feel the need to justify their vocation—not to mention the funding that the rest of society grants them to pursue it. Note that Whitehead said “pure” mathematics. He wasn’t talking about the “applied” variety: the kind that is cultivated for its usefulness to the empirical sciences, or for commercial purposes. (The latter is sometimes disparagingly referred to as “industrial mathematics.”) “Pure” mathematics is indifferent to such concerns. Its deepest problems arise out of its own inward-looking mysteries.

From time to time, of course, research in pure mathematics does turn out to have applications. The theoretical goose lays a golden egg. It was to this potential for unexpectedly useful by-products that Abraham Flexner, the founder of the Institute for Advanced Study in Princeton, called attention in a 1939 article in Harper’s titled “The Usefulness of Useless Knowledge.” But the “Golden Goose argument” (as historian Steven Shapin has dubbed it) is not one that much appeals to pure mathematicians. The British mathematician G.H. Hardy, for one, was positively contemptuous of the idea that “real” mathematics should be expected to have any practical importance.

In his 1940 book A Mathematician’s Apology—justly hailed by David Foster Wallace as “the most lucid English prose work ever on math”—Hardy argued that the point of mathematics was the same as the point of art: the creation of intrinsic beauty. He reveled in what he presumed to be the utter uselessness of his own specialty, the theory of numbers. No doubt Hardy, who died in 1947, would be distressed to learn that his “pure” number theory has been pressed into impure service as the basis for public-key cryptography, which allows customers to send encrypted credit card information to an online store without any exchange of secret cryptographic keys, thus making trillions of dollars worth of e-commerce possible; and that his work in a branch of mathematics called functional analysis proved fundamental to the notorious Black-Scholes equation, used on Wall Street to price financial derivatives.

The irony of pure mathematics begetting crass commercialism is not lost on Michael Harris, whose Mathematics Without Apologies irreverently echoes Hardy’s classic title. Harris is a distinguished middle-aged American mathematician who works in the gloriously pure stratosphere where algebra, geometry, and number theory meet. “The guiding problem for the first part of my career,” he writes, was “the Conjecture of Birch and Swinnerton-Dyer,” which “concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite.” One such elliptic curve is y2 = x3 − 25x, which can be shown…



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