In the Mountains of Mathematics

“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 to have infinitely many rational solutions (that is, solutions that are whole numbers or fractions); another is y² = x³ − x, which has only finitely many solutions. Though elementary in appearance, elliptic curves turn out to have a deep structure that makes them endlessly interesting.

Harris has spent much of his research career in Paris, and it shows: his book is full of Gallic intellectual playfulness, plus references to figures like Pierre Bourdieu, Issey Miyake, and Catherine Millet (“the sexual Stakhanovite”), and mention of the endless round of Parisian champagne receptions where “mathematical notes are compared for the first glass or two, after which conversation reverts to university politics and gossip.” It is rambling, sardonic (the term “fuck-you money” appears in the index), and witty. It contains fascinating literary digressions, such as an analysis of the occult mathematical structure of Thomas Pynchon’s novels, and lovely little interludes on elementary math, inspired by Harris’s gallant attempt to explain number theory to a British actress at a Manhattan dinner party.

Starting with the simple definition of a prime number, he builds, bit by bit, to an explanation of the aforementioned Birch-Swinnerton-Dyer conjecture—which, at a press conference given in Paris in the year 2000 by an international group of leading mathematicians, was declared to be one of seven “Millennium Prize Problems” whose resolution would be rewarded by a million-dollar prize. The book takes an intimate look at the deepest developments in contemporary mathematics, especially the visionary work of the recently deceased Alexander Grothendieck. And it successfully conveys what Harris calls the “pathos” of the mathematician’s calling.

Harris is rudely skeptical of the usual justifications for pure mathematics: that it is beautiful, true, or even much good, at least in the utilitarian “Golden Goose” sense. “It is not only dishonest but also self-defeating to pretend that research in pure mathematics is motivated by potential applications,” he writes. He notes that public-key cryptography, by making the world safe for Amazon, has destroyed the corner bookstore (in America, not France, where online retailers are prohibited by law from offering free shipping on discounted books). And he displays an olympian scorn for the sudden popularity of “finance mathematics,” which offers a path to derivative-fueled wealth on Wall Street:

A colleague boasted that Columbia’s mathematical finance program was underwriting the lavish daily spreads of fresh fruit, cheese, and chocolate brownies, when other departments, including mine in Paris, were lucky to offer a few teabags and a handful of cookies to calorie-starved graduate students.

Even at France’s elite École Polytechnique, 70 percent of the mathematics students today aspire to a career in finance.

Nor is Harris impressed with the claim, voiced by Hardy and so many others, that pure mathematics is justified by its beauty. When mathematicians talk about beauty, he tells us, what they really mean is pleasure. “Outside this relaxed field, it’s considered poor form to admit that we are motivated by pleasure,” Harris writes. “Aesthetics is a way of reconciling this motivation with the ‘lofty habit of mind.’”

Why should society pay for a small group of people to exercise their creative powers on something they enjoy? “If a government minister asked me that question,” Harris writes,

I could claim that mathematicians, like other academics, are needed in the universities to teach a specific population of students the skills needed for the development of a technological society and to keep a somewhat broader population of students occupied with courses that serve to crush the dreams of superfluous applicants to particularly desirable professions (as freshman calculus used to be a formal requirement to enter medical school in the United States).

Although physicians don’t really need calculus, Harris at least concedes that engineers, economists, and inventory managers couldn’t get by without a fair amount of math, even if it is trivial math by his lights.

Finally, there is the presumed value of mathematical truth. Since the ancient Greeks, mathematics has been taken as a paradigm of knowledge: certain, timeless, necessary. But knowledge of what? Do the truths discovered by mathematics describe an eternal and otherworldly realm of objects—perfect circles and so forth—that exist quite independently of the mathematicians who contemplate them? Or are mathematical objects actually human constructions, existing only in our minds? Or, more radically still, could it be that pure mathematics doesn’t really describe any objects at all, that it is just an elaborate game of formal symbols played with pencil and paper?

The question of what mathematics is really about is one that continues to vex philosophers, but it does not much worry Harris. Philosophers who concern themselves with the problems of mathematical existence and truth, he claims, typically pay little attention to what mathematicians actually do. He invidiously contrasts what he calls “philosophy of Mathematics” (with a capital M)—“a purely hypothetical subject invented by philosophers”—with “philosophy of (small-m) mathematics,” which takes as its starting point not a priori questions about epistemology and ontology, but rather the activity of working mathematicians.

Here, Harris is being a little unfair. He fails to remark that the standard competing positions in the philosophy of mathematics were originally staked out not by philosophers, but by mathematicians—indeed, some of the greatest of the last century. It was David Hilbert—a “supergiant,” in Harris’s estimation—who originated “formalism,” which views higher mathematics as a game played with formal symbols. Henri Poincaré (another “supergiant”), Hermann Weyl, and L.E.J. Brouwer were behind “intuitionism,” according to which numbers and other mathematical objects are mind-dependent constructions. Bertrand Russell and Alfred North Whitehead took the position known as “logicism,” endeavoring to show in their massive Principia Mathematica that mathematics was really logic in disguise. And “platonism”—the idea that mathematics describes a perfect and eternal realm of mind-independent objects, like Plato’s world of Forms—was championed by Kurt Gödel.

All of these mathematical figures were passionately engaged in what Harris slights as philosophy of Mathematics-with-a-capital-M. The debate among them and their partisans was fierce in the 1920s, often spilling over into personal animus. And no wonder: mathematics at the time was undergoing a “crisis” that had resulted from a series of confidence-shaking developments, like the emergence of non-Euclidean geometries and the discovery of paradoxes in set theory. If the old ideal of certainty was to be salvaged, it was felt, mathematics had to be put on a new and secure foundation. At issue was the very way mathematics would be practiced: what types of proof would be accepted as valid, what uses of infinity would be permitted.

For reasons both technical and philosophical, none of the competing foundational programs of the early twentieth century proved satisfactory. (Gödel’s “incompleteness theorems,” in particular, created insuperable problems both for Hilbert’s formalism and for Russell and Whitehead’s logicism: they showed—roughly speaking—that the rules of Hilbert’s mathematical “game” could never be proved consistent, and that a logical system like that of Russell and Whitehead could never capture all mathematical truths.) The issues of mathematical existence and truth remain unresolved, and philosophers have continued to grapple with them, if inconclusively—as witness the frank title that Hilary Putnam gave to a 1979 paper: “Philosophy of Mathematics: Why Nothing Works.”

To Harris, this looks a bit vieux jeu. The sense of crisis in the profession, so acute less than a century ago, has receded; the old difficulties have been patched up or papered over. If you ask a contemporary mathematician to declare a philosophical party affiliation, the joke goes, you’ll hear “platonist” on weekdays and “formalist” on Sundays: that is, when they’re working at mathematics, mathematicians regard it as being about a mind-independent reality; but when they’re in a reflective mood, many claim to believe that it’s just a meaningless game played with formal symbols.

Today, as Harris observes, paradigm shifts in mathematics have less to do with “crisis” and more to do with finding superior methods. It used to be thought, for example, that all mathematics could be constructed out of sets. Starting with the simple idea of one thing being a member of another, set theory shows how structures of seemingly limitless complexity—number systems, geometrical spaces, a never-ending hierarchy of infinities—can be built up out of the most modest materials. The number zero, for example, can be defined as the “empty set”: that is, the set that has no members at all. The number one can then be defined as the set that contains one element—zero and nothing else. Two, in turn, can be defined as the set that contains zero and one—and so on, with the set for each subsequent number containing the sets for all the previous numbers. Numbers, instead of being taken as basic, can thus be viewed as pure sets of increasingly intricate structure.

In the 1930s, a cabal of brilliant young Paris mathematicians, including André Weil, resolved to make the house of mathematics more secure by rebuilding it on the logical foundation of set theory. The project, under the collective nom de guerre “Bourbaki,” went on for decades, resulting in one fat treatise after another. Among its consequences, crazily enough, was the advent of the “new math” education reforms back in the 1960s, which so befuddled American schoolchildren and their parents by replacing intuitive talk of numbers with the alien jargon of sets.

Physicists talk about finding the “theory of everything”; well, set theory is so sweeping in its generality that it might appear to be (as Harris quips) “the theory of theories of everything.” It certainly appeared that way to the members of Bourbaki. Yet a few decades after their program got underway, the extraordinary Alexander Grothendieck came into their midst and transcended it. In doing so, he created a new style of pure mathematics that proved as fruitful as it was dizzyingly abstract. Long before his death last November at the age of eighty-six in a remote hamlet in the Pyrenees, Grothendieck had come to be regarded as the greatest mathematician of the last half-century. As Harris observes, he likely qualifies as the “most romantic,” too: “his life story begs for fictional treatment.”

The raw facts are astounding enough. Alexander Grothendieck was born in Berlin in 1928 to parents who were both active anarchists. His father, a Russian Jew, took part in the 1905 uprising against the tsarist regime and the 1917 revolution. He escaped imprisonment under the Bolsheviks; clashed with Nazi thugs on the streets of Berlin; fought on the Republican side in the Spanish civil war (as did Grothendieck’s mother); and was deported, after the fall of France, from Paris to Auschwitz to be murdered.

His mother, a gentile from Hamburg, raised Grothendieck in the south of France. There the boy showed talent both for numbers and for boxing. After the war, he made his way to Paris to study mathematics under the great Henri Cartan. Following early teaching stints in São Paulo, Kansas, and Harvard, Grothendieck was invited in 1958 to join the Institut des Hautes Études Scientifiques, which had just been founded by a private businessman outside Paris in the woods of Bois-Marie. There Grothendieck spent the next dozen years astonishing his elite colleagues and younger disciples as he recreated the landscape of higher mathematics.

Grothendieck was a physically imposing man, shaven-headed and handsome, as charismatic as he was austere. A staunch pacifist and antimilitarist, he refused to go to Moscow in 1966 (where the International Congress of Mathematics was being held) to accept the Fields Medal, the highest honor in mathematics. He did, however, make a trip the next year to North Vietnam, where he lectured on pure mathematics in the jungle to students who had been evacuated from Hanoi to escape the American bombing. He remained (by choice) stateless most of his life, had three children by a woman he married and two more out of wedlock, founded the radical ecology group Survivre et Vivre, and once got arrested for knocking down a couple of gendarmes at a political demonstration in Avignon.

Owing to his unyielding and sometimes paranoiac sense of integrity, Grothendieck ended up alienating himself from the French mathematical establishment. In the early 1990s he vanished into the Pyrenees—where, it was reported by the handful of admirers who managed to track him down, he spent his remaining years subsisting on dandelion soup and meditating on how a malign metaphysical force was destroying the divine harmony of the world, possibly by slightly altering the speed of light. The local villagers were said to look after him.

Grothendieck’s vision of mathematics led him to develop a new language—it might even be called an “ideology”—in which hitherto unimaginable ideas could be expressed. He was the first, Harris observes (breaking into emphatic boldface), “to be guided by the principle that knowing a mathematical object is tantamount to knowing its relations to all other objects of the same kind.” In other words, if you want to know the real nature of a mathematical object, don’t look inside of it, but see how it plays with its peers.

Such a peer group of mathematical objects is called, in a deliberate nod to Aristotle and Kant, a “category.” One category might consist of abstract surfaces. These surfaces play together, in the sense that there are natural ways of going back and forth between them that respect their general form. For example, if two surfaces have the same number of holes—like a donut and a coffee mug—one surface can, mathematically, be smoothly transformed into the other.

Another category might consist of all the different algebraic systems that have an operation akin to multiplication; these algebras too play together, in the sense that there are natural ways of going back and forth between them that respect their common multiplicative structure. Such structure-preserving back-and-forth relations among objects in the same category are called “morphisms,” or sometimes—to stress their abstract nature—“arrows.” They determine the overall shape of the play within a category.

And here’s where it gets interesting: the play in one category—say, the category of surfaces—might be subtly mimicked by the play in another—say, the category of algebras. The two categories themselves can be seen to play together: there is a natural way of going back and forth between them, called a “functor.” Armed with such a functor, one can reason quite generally about both categories, without getting bogged down in the particular details of each. It might also be noticed that, since categories play with one another, they themselves form a category: the category of categories.

Category theory was invented in the 1940s by Saunders Mac Lane of the University of Chicago and Samuel Eilenberg of Columbia. At first it was regarded dubiously by many mathematicians, earning the nickname “abstract nonsense.” How could such a rarefied approach to mathematics, in which nearly all its classical content seemed to be drained away, result in anything but sterility? Yet Grothendieck made it sing. Between 1958 and 1970, he used category theory to create novel structures of unexampled richness. Since then, the heady abstractions of category theory have also become useful in theoretical physics, computer science, logic, and philosophy.^*

The project undertaken by Grothendieck was one that began with Descartes: the unification of geometry and algebra. These have been likened to the yin and yang of mathematics: geometry is space, algebra is time; geometry is like painting, algebra is like music; and so on. Less fancifully, geometry is about form, algebra is about structure—in particular, the structure that lurks within equations. And as Descartes showed with his invention of “Cartesian coordinates,” equations can describe forms: x² + y² = 1, for example, describes a circle of radius 1. So algebra and geometry turn out to be intimately related, exchanging what André Weil called “subtle caresses.”

In the 1940s, thanks to Weil’s insight, it became apparent that the dialectic between geometry and algebra was the key to resolving some of the most stubbornly enduring mysteries in mathematics. And it was Grothendieck’s labor that raised this dialectic to such a pitch of abstraction—one that was said to leave even the great Weil daunted—that a new understanding of these mysteries emerged. Grothendieck laid the groundwork for many of the greatest mathematical advances in recent decades, including the 1994 proof of Fermat’s Last Theorem—a magnificent intellectual achievement of zero practical or commercial interest.

Harris, whose own impressive work is much indebted to Grothendieck, is eloquent in praising his “ruthless… minimalism,” which extended to his contempt for money and his monkish wardrobe. I only wish that Harris had more conspicuously credited another figure in the modern transformation of mathematics, Emmy Noether. It was Noether, born in Bavaria in 1882, who largely created the abstract approach that inspired category theory. Yet as a woman in a male academic world, she was barred from holding a professorship in Göttingen, and the classicists and historians on the faculty even tried to block her from giving unpaid lectures—leading David Hilbert, the dean of German mathematics, to comment, “I see no reason why her sex should be an impediment to her appointment. After all, we are a university, not a bathhouse.” Noether, who was Jewish, fled to the United States when the Nazis took power, teaching at Bryn Mawr until her death from a sudden infection in 1935.

The intellectual habit of grappling with a problem by ascending to higher and higher levels of generality came naturally to Emmy Noether, and it was shared by Grothendieck, who said that he liked to solve a problem not by the “hammer-and-chisel method,” but by letting a sea of abstraction rise to “submerge and dissolve” it. In his vision, the familiar things dealt with by mathematicians, like equations, functions, and even geometrical points, were reborn as vastly more complex and versatile structures. The old things turned out to be mere shadows—or, as Grothendieck preferred to call them, “avatars”—of the new. (An avatar is originally an earthly manifestation of a Hindu god; as Harris notes, “a taste for Indian…metaphysics inflected [the] terminology” of many French mathematicians.)

Nor is this a one-off process. Each new abstraction is eventually revealed to be but an avatar of a still-higher abstraction. As Harris puts it, “the available concepts are interpreted as the avatars of the inaccessible concepts we are trying to grasp.” With the grasping of these new concepts, mathematics ascends a kind of “ladder” of increasing abstraction.

This, Harris says, is what philosophers should be paying attention to. “If you were to ask for a single characteristic of contemporary mathematics that cries out for philosophical analysis,” he writes, “I would advise you to practice climbing the categorical and avatar ladders in search of meaning, rather than searching for solid Foundations.” And what lies at the top of this ladder? Perhaps, Harris suggests with playful seriousness, there is One Big Theorem from which all of mathematics ultimately flows—“something on the order of samsāra = nirvana.” But since there are infinitely many rungs to climb, it is unattainable.

Here, then, is the pathos of mathematics. Unlike theoretical physics, which can aspire to a “final theory” that would account for all the forces and particles in the universe, pure mathematics must concede the futility of its own quest for ultimate truth. As Harris observes, “every veil lifted only reveals another veil.” The mathematician is doomed to what André Weil called an endless cycle of “knowledge and indifference.”

But it could be worse. Thanks to Gödel’s second incompleteness theorem—the one that says, roughly, that mathematics can never prove its own consistency—mathematicians can’t be fully confident that the axioms underlying their enterprise do not harbor an as-yet-undiscovered logical contradiction. This possibility is “extremely unsettling for any rational mind,” declared the Russian-born mathematician (and Fields medalist) Vladimir Voevodsky, in a speech on the occasion of the eightieth anniversary of the Institute for Advanced Study. Indeed, the discovery of such an inconsistency would be fatal to pure mathematics, at least as we know it today. The distinction between truth and falsehood would be breached, the ladder of avatars would come crashing down, and the One Big Theorem would take a truly terrible form: 0 = 1. Yet, oddly enough, e-commerce and financial derivatives would be left untouched.