## THE LAST GREAT PROBLEM

### WHAT DO UNSOLVABLE MATH PROBLEMS AND MOUNT EVEREST HAVE IN COMMON? THEY’RE BOTH “THERE.”

DISCUSSED: The Fermat Conjecture, Frozen Saps, The Riemann Hypothesis, Prime Obsession, The Music of the Primes, Mount Everest, Idealized Cognitive Models, Competitive Hot-Dog Eating, “We Guarantee to Maximise the Safety and Success of Your Adventure”

# (1.1) BECAUSE IT’S THERE

(1.1.1) In 1923, the New York Times asked George Mallory why he wanted to climb Everest. He told them,

(1.1.1.1) “Because it’s there.”

(1.1.1.2) “How do you know it’s there?”

Let alone,

(1.1.1.3) “What, exactly, is it?”

There are certain ontological issues mountain climbers are not called upon to address.

# (1.2) MAKING INFINITELY MANY ASSERTIONS AT ONCE

(1.2.1) Until 1995, the most famous open problem in mathematics was Fermat’s Last Theorem, or, as it ought to have been called, the Fermat Conjecture. The lawyer and number theorist Pierre de Fermat asserted in the seventeenth century that

(1.2.1.1) “There are no positive whole numbers a, b, c, and n, where n is at least three, such that an + bn = cn.”

Fermat jotted down (1.2.1.1) in the margins of his copy of Diophantus’s Arithmetica, with the remark

(1.2.1.2) “I have discovered a truly remarkable proof which this margin is too small to contain.”

It’s worth pausing, first of all, to appreciate what a serious achievement Fermat was claiming. To prove that an equation does have a solution is, in principle, easy business; you just exhibit the solution. But to show an equation has no solution is a different, and deeper, matter—it requires you to prove infinitely many assertions at once.

(1.2.1.1) says that for every one of the infinitely many choices of a, b, c, and n, the equation an + bn = cn does not hold. Such a statement can never be checked by a computer; even if you checked all values of a, b, c, and n less than a million, you wouldn’t know that a solution to the equation wasn’t lurking just a few steps past the end of your computation. Compare the following statement:

(1.2.1.3) “The sum of two odd numbers can never be an odd number.”

You can probably prove (1.2.1.3) by yourself. But (1.2.1.3) is deep in the same way (which is not to say as deep) as Fermat’s conjecture. By proving it, you’ve simultaneously made statements about every single odd number. You’re being a mathematician, not a computer.

(1.2.2) So you’re convinced that Fermat’s problem is hard. But why should you care? Or if you don’t care, why do so many other people? People who care enough about the problem that they work themselves ragged at it?

(1.2.2.1) “Nowhere does this book come right out and address the central question—Why would a normal person want to do this stuff?—head on; I circle the issue continually, poke at it from behind with a long stick now and then, but at no point do I jump right in the cage and wrestle with the beast directly….”

That’s Jon Krakauer, from the introduction to his 1990 collection Eiger Dreams, and he’s talking about mountain climbers, not mathematicians. But his “central question” is the same as ours. After all, what had Hillary and Norgay achieved when they stood at Everest’s tippy-top? The view they saw had already been seen plenty, from airplanes. No real knowledge was gained and nothing new learned about the human condition. Likewise, it’s not clear what makes the Fermat conjecture worth proving. There’s no nuclear reactor that will function or not according as (1.2.1.1) is or isn’t true.

Taking my cue from Krakauer, I’ll circle for a while.

(1.2.3) In 1993, Andrew Wiles announced a proof of the Fermat Conjecture which, after two years of checking and modification, was shown to be correct. Wiles had proved a special case of a conjecture made in the 1950s by Goro Shimura and Yutaka Taniyama, a conjecture which was known to imply Fermat thanks to the work of Ken Ribet, Barry Mazur, Gerhard Frey, and Jean-Pierre Serre. The Shimura-Taniyama conjecture, now proved in its entirety by a four-man team of mathematicians, is itself but a special case of a more general family of conjectures known as the Langlands Program: conjectures which are in fact so conjectural that even their precise statements remain matters of controversy and concern. And beyond that, one cannot doubt, will come other conjectures, of which the Langlands Program will be revealed as “the baby version.” In mathematics, there’s no highest peak.

What Wiles did is more important than proving a theorem; he introduced a new idea, and this is what brought him—so to speak—to the top. That he succeeded in proving Fermat’s conjecture is in a sense the least interesting part of his work. Indeed, much of the apparatus of the modern theory of numbers can be traced directly back to earlier, failed attempts on Fermat. The failures have been as influential and as deep as the success.

(1.2.4) Which seems to point up a big difference between mathematics and mountain climbing. In climbing, the point is to get to the top. In mathematics, the location of the top—even its existence—is never clear. (Cf. [1.1.1.2], [1.1.1.3])

But is this actually a difference? Krakauer profiles a climber named John Gill, who pioneered the modern art of bouldering. Boulderers don’t try to get to the tops of mountains; they climb small, complicated rocks. Krakauer writes:

(1.2.4.1) “Actually, to Gill’s mind, summits aren’t even very important. The real pleasure of ‘bouldering’ lies more in the doing than in attaining the goal. ‘The boulderer is concerned with form almost as much as with success,’ says Gill. ‘Bouldering isn’t really a sport. It’s a climbing activity with metaphysical, mystical, and philosophical overtones.’”

Gill is also, aptly, a mathematician. He sees the cognitive challenges posed by bouldering and mathematics as formally quite alike. It’s hard to know how seriously to take this opinion; we only get one brain each, after all, whose various functions have to pass through many of the same channels. One imagines, e.g., that professional chefs who race stock cars in their spare time hear echoed in the pit stop the hyperefficient racket of the kitchen at dinner rush. David Auburn, author of Proof (cf. [1.3.1]), thought the following remark of the number theorist G. H. Hardy was an excellent description of playwriting:

(1.2.4.2) “In a good proof, there is a very high degree of unexpectedness, combined with inevitability and economy. The argument takes so odd and surprising a form: the weapons used seem so childishly simple when compared with the far-reaching consequences; but here is no escape from the conclusions.”

And I, who write fiction and belletristic articles like this one when my job as a mathematician allows it, can also see the kinship between those two kinds of work; so much so, at the moment, that I’m borrowing for this article the numbered-paragraph style used for the mathematical publications of the Institut des Hautes Études Scientifiques.

Anyway, here’s Gill on bouldering and math:

(1.2.4.3) “One of the objectives for both is to achieve an interesting result—ideally an unexpected result—in an elegant fashion, with a smooth flow, using some unexpected simplicity. There is the question of style.” But beyond this, he adds, “to be a boulderer or a research mathematician you have to have this natural inclination to dig for something, a strong, completely inner motivation to be on the frontier, to discover things. The reward, in both activities, is almost-continual enlightenment, and that’s a great feeling.”

# (1.3) BOOKS UNDER REVIEW

(1.3.1) The resolution of the Fermat conjecture occasioned a wave of publicity, including at least four popular books, a television special, an off-off-Broadway musical, and an invitation for Wiles to pose in an advertisement for the Gap (he declined). Three years later, Sylvia Nasar’s biography of John Nash, A Beautiful Mind, appeared, followed quickly by a related movie, and David Auburn’s Pulitzer Prize-winning play, Proof, about a demented mathematician (or two) and a disputed theorem. Math, for the moment, is a public shorthand for difficulty, even maddening difficulty, as mountaineering might have been fifty years ago when Everest was climbed, or solo flying thirty years before that.

As a result, there’s a need for books, and when there’s a need for books, there’s a need (probably a somewhat lesser one) for topics. At the moment, especially after Grigori Perelman’s announcement of a proof of the Poincaré conjecture in topology, the most famous unsolved problem in mathematics is the Riemann Hypothesis. I’ll say here that the Riemann Hypothesis (RH, among pros) concerns the distribution of prime numbers and the behavior of a certain function, called zeta [or ζ], introduced by Bernhard Riemann in his 1859 paper, “On the number of primes less than a given quantity.” And that RH, like (1.2.1.1) and (1.2.1.3), has the vexing quality of encompassing infinitely many statements at once.

Explaining anything deeper than that is a book-length job, one which three new books, all of which appeared in April 2003, take as part of their brief. All three aim to explain the meaning of RH to lay readers, and all—taking after Krakauer’s best-selling Everest memoir Into Thin Air—interleave history and technical tidbits with blow-by-blow accounts of the contemporary, as-yet-unsuccessful attempts on the Riemann Hypothesis: the latest highest summit.

(1.3.2) The oldest of these is Karl Sabbagh’s The Riemann Hypothesis, which was published last year in Britain under the off-puttingly goofy title Dr. Riemann’s Zeros. Sabbagh’s book makes the admirable attempt to start with first principles—he includes, for instance, a lengthy excursus on why negative times negative equals positive—and to lead his readers all the way to the vanguard of current research.

But Sabbagh is hampered throughout by a posture of alienation from the mathematicians, his subjects. “The interior life of mathematicians is a closed world to the rest of us,” he begins one chapter; and of the number theorists Hardy, Littlewood, and Ramanujan he says:

(1.3.2.1) “They give the impression of effortless superiority when you read their finished works, especially for those of us for whom any mathematical expression more complicated than 9 x 7 = 63 can induce head-scratching and even revulsion.”

This business is, I think, meant to reassure the math-phobic. Instead one gets the the impression that Sabbagh just doesn’t like math very much. One imagines touring Rome with a guide who keeps saying, “Strange as it may seem, there are people who really care about this creepy old junk! Maybe you should too!” Sabbagh’s apparent distaste for his subject matter manifests itself in a steady sequence of errors. At one point, he describes Riemann’s zeta function as “entirely unfamiliar to 100 percent of the educated population,” with a helpful footnote reminding us that, in mathematical usage,

(1.3.2.2) “100 percent doesn’t always mean ‘all.’”

There’s nothing wrong in (1.3.2.2), and it’s worth giving an example. It would be right to say “100 percent of positive whole numbers are greater than 10,” because the set of positive whole numbers not greater than 10 is a finite subset of the infinite set of positive whole numbers; it seems fair to say that a finite set cannot make up 1 percent, or 0.5 percent, or 0.01 percent, or any positive percentage of an infinite set. If we are to assign a percentage at all, we must say the finite set is 0 percent of the infinite set, which means that the whole numbers greater than 10 comprise 100 percent of the positive whole numbers—without, of course, being all of those numbers.

On the other hand, the whole numbers greater than 10 don’t make up 100 percent of the whole numbers between 1 and 1000. They make up 99 percent. The fact is, 100 percent of a finite set (like “the educated population”) is the whole set, just as you might think; it’s really only in the study of infinite sets that ordinary English and math talk start to diverge. As far as I can tell, Sabbagh has missed this distinction entirely and thinks mathematicians say “100 percent” when they mean “a percentage pretty close to 100 percent.” This kind of sloppiness is, unfortunately, characteristic of his book. The Riemann Hypothesis is safe only for those people who know enough math to skip over the mistakes; and such people would do better to learn about the math from a more advanced treatment.

(1.3.3) John Derbyshire’s Prime Obsession is an altogether better effort. Derbyshire is very nearly Sabbagh’s opposite; he demonstrates a seemingly bottomless enthusiasm for mathematics, dancing cheerfully away from his main subject to discuss the meaning of infinity divided by infinity, the divergence of the harmonic series, the Möbius function, the p-adic numbers, and dozens more topics just as exotic to those outside the fold. He speaks of these things as a zealous college junior might to an enraptured freshman; said freshman only takes in half of what she hears, but is carried along nonetheless by the excitement, even as she wishes, from time to time, that he knew when to stop.

Derbyshire’s fondness for higher math extends to its practitioners; indeed, he seems seldom to have met a mathematician he didn’t like. Hardy was a “handsome and charming man,” Landau “a gifted and enthusiastic teacher.” Gauss “truly was a gentleman,” and Hadamard and de la Vallée Poussin, not to be outdone, were “perfect gentlemen.” David Hilbert—who put the Riemann Hypothesis on his famous 1900 list of twenty-three challenges for mathematicians of the twentieth century—was “a keen dancer and a popular lecturer.” Even Oswald Teichmüller, an ardent Nazi, gets a not unsympathetic treatment. Perhaps the grandest praise is heaped on the character of the great Swiss mathematician Leonhard Euler:

(1.3.3.1) “Another reason I find Euler so attractive is that, without being striking or eccentric or interesting in any particular way, he was a very admirable human being. When you read about his life you get a strong impression of serenity and inner strength…. Underneath it all was a rock-solid religious faith. Euler had been raised a Calvinist and never wavered in his belief… We are told that while living in Berlin, ‘he assembled the whole of his family every evening, and read a chapter of the Bible, which he accompanied with an exhortation.’ This, while attending a court at which, according to Macaulay, ‘the absurdity of all religions known among men was the chief topic of conversation.’ Hardworking, pious, stoical, devoted to his family, plain-living, and plain-spoken—no wonder Frederick [the Great] didn’t like him.”

And Derbyshire doesn’t like Frederick:

(1.3.3.2) “Frederick wanted his court to be a sort of salon, full of brilliant people saying brilliant things to each other… Further, Frederick was a manipulative egotist who, while in principle wishing to surround himself with geniuses, in practice preferred second-raters who would flatter him. Setting aside a few luminaries like Voltaire and Euler, the general intellectual level at Frederick’s court was probably less than scintillating.”

Here one can’t help recalling the author’s day job as a columnist for the National Review. Is it too much to see in his godly, no-nonsense Euler a hint of the Republican vision of George W. Bush? Or in the louche confines of Frederick’s court the reviled administration of Bush’s predecessor?

Prime Obsession is built on a clever chassis; the even chapters advance the story of Riemann and the questions he left behind, while the odd ones do the heavy mathematical lifting. (Derbyshire is probably drawing here from Douglas Hofstadter’s still-unequaled work of mathematical exposition, Gödel, Escher, Bach, which interleaved discussions of the limits of formal logic with musical scores and whimsical dialogues between talking animals.) The alternation allows readers to work through as many or as few computations as they like, while leaving Derbyshire room to do what Sabbagh does not: tell us what, precisely, Riemann hypothesized.

(1.3.4) The only working mathematician to give this microgenre a try is Marcus du Sautoy, a group theorist at Oxford whose useful volume on analytic pro-p groups I have close by my hand as I write. The Music of the Primes is the most mathematically sophisticated of the three books discussed here and, for most readers, the best. Du Sautoy leads the reader with a sure hand through many of the highlights of contemporary mathematics, attending to both the famous (Gödel’s Incompleteness theorem, the stories of Erdös and of Ramanujan, the surprising application of number theory to cryptography) and the undeservedly obscure. His chapter on the role of the abstract notion of “computation” is particularly nice, deftly combining the math celebrities Gödel, Alan Turing, and Georg Cantor (the subject of David Foster Wallace’s new book, Everything and More) with a crisp summary of the work of Julia Robinson and Yuri Matijasevich, who settled another of Hilbert’s problems by proving the following remarkable fact: There cannot exist a computer program that, given an equation, reliably determines whether or not the equation has a solution in whole numbers.

(1.3.4.1) (I was going to say Robinson and Matijasevich’s work merited a book of its own, until I learned that the book, Julia: A Life in Mathematics, already exists. The editor is the mathematical biographer Constance Reid who, it turns out, is Robinson’s sister.)

What all this has to do with Riemann is another question. Du Sautoy’s book is not so much a book about the Riemann Hypothesis, but a book about the main themes of twentieth-century mathematics, with RH as an organizing principle to which the story periodically returns.

Du Sautoy, unlike Derbyshire, draws a scrim over technicalities and formulas to avoid shocking the modesty of nonmathematicians; the graph of a function is here called a “landscape,” every entity with resonant frequencies is a “drum,” and the method of modular arithmetic is “Gauss’s clock calculator.” There’s nothing wrong with metaphor, and du Sautoy’s are well-chosen and -deployed. But inevitably The Music of the Primes feels a bit distant from its subject. It’s a guided tour in an air-conditioned bus. Readers who want unmediated contact with the integrals, complex numbers, and infinite series that Riemann bequeathed us should choose Derbyshire’s book. For everyone else, the bus is comfy, and there’s plenty to see.

# (2.1) GREAT PROBLEMS

(2.1.1) It’s an undeniable fact that books about mathematicians and books about mountain climbing often draw from the same rhetorical bag. The words “amazing,” “difficult,” and “brutal” are frequent players. Climbers call each hardest-yet ascent a “last great problem,” a phrase which could be (but isn’t) a commonplace in mathematics, too. Derbyshire quotes Mallory (1.1.1.1) and compares the Prime Number Theorem to Everest, while Sabbagh and du Sautoy reserve this comparison for the Riemann Hypothesis itself (which du Sautoy, slightly off-puttingly, repeatedly calls “Mount Riemann”). Taniyama (see (1.2.3)) described mathematical accomplishment this way:

(2.1.1.1) “It may be said, we are allowed in the course of progress to climb to a certain height in order to look back at our tracks, and then to take a view of our destination.”

Jon Krakauer says the prospect of death on the mountain is “as abstract a notion as non-Euclidean geometry.” There’s the mathematician-climber Gill, of course, who says

(2.1.1.2) “Even though one activity is almost completely cerebral and the other is mainly physical, there is something common to bouldering and mathematical research. I think it has something to do with pattern recognition, a natural instinct to analyze a pattern.”

When Hardy, who tackled RH in the first decades of the 1900s, gave a list of six goals for the New Year, “prove the Riemann Hypothesis” was first. “Be the first man at the top of Mount Everest” was fourth. (Not that Hardy was serious about Everest; “Be proclaimed first president of the U.S.S.R. of Britain and Germany,” was on the list, too.)

(2.1.2) Derbyshire dives straight at the analogy we’re dancing around in his counterposition of Riemann and Karl Weierstrass:

(2.1.2.1) “Reading Weierstrass’s papers is like watching a rock climber. Every step is firmly anchored in proof before the next step is taken…. Certainly the precise logical progression of Weierstrass’s work, with every least fact carefully justified before proceeding to the next, and no appeals to geometrical intuition at all, is representative of the logical mathematician.

Riemann is at the other pole. If Weierstrass is a rock climber, inching his way methodically up the cliff face, Riemann is a trapeze artist, launching himself boldly into space in the confidence—which to the observer often seems dangerously misplaced—that when he arrives at his destination in the middle of the sky, there will be something there for him to grab.”

Derbyshire slightly misrepresents both mathematics and rock climbing here. Poincaré, in his essay, “Intuition and Logic in Mathematics,” makes the same distinction between the logical Weierstrass and the intuitive Riemann; but then, “seized with scruples,” he concludes that Weierstrass must also have relied on intuition, albeit of a more formal, less sensual kind than Riemann’s. And modern rock climbing, thanks, in part, to John Gill’s innovations, depends on dynamic moves, where the climber releases one hold before achieving the next, relying on momentum to make it to the next ledge—more like Riemann than Weierstrass.

But the distinction between Weierstrass and Riemann is a real one, and—more to our point—the analogy between climbing and mathematics, even as extended by Derbyshire, doesn’t begin to strain.

Mathematicians spend most of their working lives in offices and libraries, held at a comfortable temperature; so why, in these books, and in mathematicians’ own words, does the alpine keep popping up?

# (2.2) THE CHALLENGE

(2.2.1) Let’s start with a ruthless oversimplification of a piece of cognitive science.

According to some linguists, there are only so many things we know how to know about. We know about mothers, for instance; we’re probably built to know that. I’m going to borrow a term from George Lakoff (without, I hope, doing too much violence to his work) and say we carry around a skeletal description of “motherness” called an idealized cognitive model (ICM); this ICM might contain the facts that mothers provide nourishment, that mothers are there before you are, and that mothers are people. Other concepts in the world, such as “home country,” fit the ICM to a certain extent; your home country, while not a person, is a source of nourishment and a predecessor. And so, for convenience, we put “home country” into the “mother” box by means of a metaphor; the fact that we know how to think about mothers makes it quicker and easier to think about home countries. Since the fit with the ICM isn’t perfect, we may draw some wrong conclusions—but that’s the price we pay for the marvelous ability to simplify the world to the point where we can talk about it, and not just gape at the incredible particularness of everything in sight.

I want to propose an ICM of “the challenge,” whose rules look like this:

(2.2.1.1) The challenge is difficult.

(2.2.1.2) The challenge is dangerous.

(2.2.1.3) It is unambiguous whether you’ve succeeded in the challenge.

(2.2.1.5) The challenge is undertaken for its own sake.

The Riemann Hypothesis and Mount Everest are challenges; challenges, I think, that fit these rules extremely well: prototypical challenges. As such, they’re natural targets of metaphor, from other challenges, and—as we’ve seen—from each other.

Most challenges don’t fit these rules so well. Writing a great poem satisfies (2.2.1.5) and (2.2.1.1) but fails (2.2.1.2) and (2.2.1.3). Winning the World Series meets (2.2.1.3) but fails (2.2.1.5). Being a good parent satisfies (2.2.1.1) and (2.2.1.4) but fails the rest.

(2.2.2) When you tell a story, you foreground the features that fit the model; you overdetermine the metaphor between the story in front of your face and the idealized Challenge in the back of your mind. You can hardly open a page of one of Krakauer’s books (or a book about the Riemann Hypothesis) without finding reference to the difficulty of the problem faced, the all-or-nothing nature of success, and the moral brawn necessary for whoever would surmount the challenge.

Here Krakauer gestures at (2.2.1.2) and (2.2.1.3) in adjacent sentences:

(2.2.2.1) “Achieving the summit of a mountain was tangible, immutable, concrete. The incumbent hazards lent the activity a seriousness of purpose that was sorely missing from the rest of my life.”

There’s a touch of both (2.2.1.4) and (2.2.1.5) in the “seriousness of purpose.” And, of course, (2.2.1.5) is the whole point of “Because it’s there.” Derbyshire describes an breakthrough of Carl Ludwig Siegel with a handoff from (2.2.1.1) to (2.2.1.4):

“Several researchers had investigated them [notes left behind by Riemann], but all had been defeated by the fragmentary and disorganized style of Riemann’s jottings, or else they lacked the mathematical skills needed to understand them.

“Siegel was made of sterner stuff.”

The geometer Alain Connes, quoted by du Sautoy, packs (2.2.1.1), (2.2.1.2), and (2.2.1.4), not to mention the ubiquitous identification of math with mountain-climbing, into a single paragraph:

(2.2.2.2) “For me, mathematics has always been the greatest school of humility. Mathematics is mainly valuable because of the immensely difficult problems which are like the Himalayas of mathematics. To reach the peak will be extremely difficult and we might even have to pay the price.”

And Eric Shipton speaks to (2.2.1.1):

(2.2.2.3) “No, it is not remarkable that Everest did not yield to the first few attempts; indeed, it would have been very surprising and not a little sad if it had…. We had forgotten that the mountain still holds the master card, that it will grant success only in its own good time. Why else does mountaineering retain its deep fascination?”

Why else? Maybe it’s the danger, as Krakauer writes:

(2.2.2.4) “Those who participate in this hazardous pastime do so not in spite of the unforgiving stakes, but precisely because of them.”

(2.2.3) But solving math problems isn’t dangerous! True: But in popular accounts, math is about as hazardous to the mind as a solo ascent of K2 is to the body. A Beautiful Mind and Proof both feature deranged mathematicians as main characters; in the movie Pi and the novel Presumed Innocent, the crazy mathematicians are explicitly made so by their mathematical frustrations. Du Sautoy tells the story of the great geometer Alexander Grothendieck, who retreated into seclusion and became a shepherd after revolutionizing the field in the 1960s, remarking: “The sheer number of hours Grothendieck had spent exploring at the edges of the world of mathematics left him unable to chart his way home.”

The publicity sheet for Philibert Schogt’s The Wild Numbers, a slight but enjoyable novel about a failing number theorist, puts it plainly: “The line between genius and madness is a thin one.” Is it really? It doesn’t matter. The story in these books and movies is always the same story, the one summed up by David Foster Wallace in his essay “Rhetoric and the Math Melodrama” (Science, 22 December 2000):

(2.2.3.1) “The Math Melodrama’s own allegorical template appears to be more classically Tragic, its hero a kind of Prometheus-Icarus figure whose high-altitude genius is also hubris and Fatal Flaw. If this sounds a bit grandiose, well, it is; but it’s also a fair description of the way Math Melodramas characterize the project of pure math—as nothing less than the mortal quest for Divine Truth.”

(2.2.3.1.1) Note the “high-altitude.”

The story has to be this way, though most mathematicians die at ease in their beds, no madder than average. Without the danger, the theorem is not the Challenge; and if the theorem is not the Challenge, its story is not the story we intended to sit in our low-altitude chairs and read.

(2.2.4) Rule (2.2.1.3) distorts the picture too—it’s the reason Edmund Hillary is world famous and John Gill is not, though Gill, by any standard, has influenced the craft of climbing more. Likewise, it’s why three books about the Riemann Hypothesis came out this spring, and none about Grothendieck, who didn’t solve a Hilbert problem, but who contributed to mathematics the foundational and now indispensable insight that arithmetic and geometry were, at bottom, two aspects of one formerly unimagined subject.

But there’s nothing that stands out more, when you read a lot of math books and mountain books, than the unceasing fascination with virtue, and with the heavy moral weather through which the mountaineers and their desk-bound cousins, the mathematicians, decide to struggle.

(2.2.5) The moral requirements for climbing mountains, if the literature is to be believed, are so stringent that it’s hard to imagine how most people get up a flight of stairs. One must be cautious, but also daring; humble, but also a dreamer; indomitable, but certainly not stubborn. Finally and most importantly, one must, following (2.2.1.5), climb for the right reasons: to realize a dream, to find oneself, or for the glory—not to prove a point, and certainly not for the money. One of Krakauer’s subjects, Alan Burgess, ascribes the death of two skilled mountaineers to their bad motivations: One needed to pay off a loan, another was trying to forget a failed romance. “It’s ’ard enough making the right decisions at ’igh altitude,” Burgess says, “without having that kind of pressure clouding your judgment.” (Transliterated Cockney Krakauer’s.)

Edmund Hillary, who delivers the best distillation of (2.2.1.4) and (2.2.1.5) I could find:

(2.2.5.1) “Modern developments in machinery and equipment have produced major changes in the technique of exploration…. But despite all this I firmly believe that in the end it is the man himself that counts. When the going gets tough and things go wrong the same qualities are needed to win through as they were in the past—qualities of courage, resourcefulness, the ability to put up with discomfort and hardship, and the enthusiasm to hold tight to an ideal and to see it through with doggedness and determination.

“The explorers of the past were great men and we should honour them. But let us not forget that their spirit still lives on. It is still not hard to find a man who will adventure for the sake of a dream or one who will search, for the pleasure of searching, and not for what he may find.”

Into Thin Air, Krakauer’s best-selling 1997 account of a disaster that killed nine climbers on Everest, is a moral parable from start to finish. When Ngawang Topche, a Sherpa climber, falls ill, the other Sherpas blame an unmarried couple whose sexual exertions on the mountain have angered the local goddess. Krakauer seems to take a similar view: The mountain is a tragedy machine, punishing good people for the smallest and most momentary moral lapse. Many of the victims were members of a \$65,000-per-head guided expedition, which advertised itself thus:

(2.2.5.2) “So you have a thirst for adventure!… Most of us never dare act on our dreams and scarcely venture to share them or admit to great inner yearnings.… We will not drag you up a mountain—you will have to work hard—but we guarantee to maximise the safety and success of your adventure. For those who dare to face their dreams, the experience offers something special beyond the power of words to describe. We invite you to climb your mountain with us.”

This is the language register you use to sell time-shares in San Diego, not the one in which Hillary promotes courage, resourcefulness, and enthusiasm. In the event, of course, neither success nor safety were guaranteed. One gets the sense Krakauer believes, though he’s too smart to state it as fact, that the kind of people who demand such guarantees belong as close to sea level as possible. They’re the kind, whatever their other qualities, who make mistakes and die; they give up too easily and die, or they press on too stubbornly and die, or they’re briefly careless, and they die.

(2.2.5.3) Remark: They are also the kind who are sitting at home reading Krakauer’s best-selling book, imagining how they’d fare in alpine extremity—imagining, probably, that they’d heroically come through—therefore concluding, this test passed, that they know what they’re made of, and don’t need to climb mountains to find out, like the poor frozen saps on Everest. Don’t get me wrong—I think this is a noble way to read Krakauer’s book. I think it’s actually kind of the point.

The same could be said of popular math books, like (1.3.2), (1.3.3), (1.3.4). Mathematicians often falsely believe that a book is an accurate representation of mathematics precisely when the reader has, in the end, done some mathematics. No—just as a mountaineering book can be an accurate rendition of mountaineering even if the reader has not, by virtue of reading the book, climbed a mountain.

(2.2.6) If virtues help you climb mountains and prove theorems, then people who climb mountains and prove theorems are virtuous. Their story becomes a tablet on which you can chalk out whatever moral drama you have in mind, attaching your favorite virtues to the winners. We’ve already observed Derbyshire’s casting (1.3.4.1) of Leonhard Euler as a contemporary Republican, a characterization invisible in du Sautoy’s book, which emphasizes Euler’s romantic attachment to numbers and their affiliated mysteries.

Don George, writing in Salon.com on Edmund Hillary:

(2.2.6.1) “Hillary embodied the dash, the pluck, the stiff-upper-lip and what-the-hell, let’s-go-for-it aplomb the British Empire still aspired to.”

But Jan Morris, whose profile of Hillary in Time takes a more anti-imperial tack, says of Hillary and Norgay:

(2.2.6.2) “They were not, though, heroes of the old epic kind, tight of jaw and stiff of upper lip.”

On the matter of the lip Hillary himself is silent; he’s got national pride on his agenda.

(2.2.6.3) “In some ways I believe I epitomise the average New Zealander: I have modest abilities, I combine these with a good deal of determination, and I rather like to succeed.”

(2.2.7) But why mountain climbing, and why math? Why not poetry, why not urban planning, why not competitive hot-dog eating? Actually, the last case is the especially interesting one. I found speculating about a world in which in which competitive hot-dog eating serves the function that mountain-climbing does in ours to be a salutary mental exercise. After all, hot-dog eating on the scale necessary to win contests is difficult and dangerous; one either wins or one loses; and one can certainly postulate a world in which it was carried out without the prospect of monetary reward or brief fame it carries here.

The problem is, the moral tablet isn’t blank. You know what it feels like to stuff yourself with hot dogs. And you’d laugh if someone suggested that by means of that sensation you could take your moral measure. (Added in press: Indeed, a piece by Frederick Kaufman in the October 2003 Harper’s tries covering a hot-dog-eating contest in the high athletic manner, and ends up having no choice but to play the scene for laughs.) You can’t use the hot-dog champ the way Derbyshire uses Euler, or the way George and Morris use Hillary—or, for that matter, the way Hillary uses his younger self.

That’s why it’s hard for mathematicians to read the books of Sabbagh, Derbyshire, and du Sautoy the way they’re supposed to be read. For us, theorems are too close to home to make good dreams. When we’re at work we don’t feel our generosity, humility, virility, or good cheer helping us along. The virtue that helps us solve math problems is that of being good at math. For mountain climbers, I imagine the situation is much the same; though once the mountain climber steps off the mountain, and the mathematician away from her work, and books start getting written, all bets are off.

(2.2.8) Krakauer says, “Climbing was like life itself, only it was cast in much sharper relief.” But I don’t think that’s right. If climbing were like life, it wouldn’t be so interesting—it wouldn’t be so useful. That’s the thing about a mountain, or an unproved theorem, as Mallory pointed out. It’s there.

What’s it? It can be whatever you need it to be.

And where’s there? Right over here, where you can get a good look at it?

No. There.

Jordan Ellenberg is the author of The Grasshopper King (Coffee House Press, 2003), a novel about the worst poet in history and languages that are impossible to learn. He is an assistant professor of mathematics at Princeton University.

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