April 18, 1988
The Joy of Math, or Fermat’s Revenge
For one brief shining moment, it appeared as if the 20th century had justified itself. The era of world wars, atom bombs, toxic waste, AIDS, Muzak and now, just to rub it in, a pending Bush-Dukakis race, had redeemed itself, it seemed. It had brought forth a miracle. Fermat’s last theorem had been solved.
Fermat’s last theorem is the world’s most famous unsolved mathematical puzzle. It owes its fame to its age — it was born about five years before Isaac Newton — and its simplicity. It consists of only one line. The Greeks had shown that there are whole numbers for which a2+b2=c2. One solution for Pythagoras’ theorem, for example, is 32+42=52. Pierre de Fermat conjectured that the Pythagorean equation doesn’t work for higher dimensions: for n greater than 2, an+bn=cn is impossible. It won’t work for n=3. (There are no integers for which a3+b3=c3.) Nor, theorized Fermat, for any higher power: for n=4 or n=5 and so on.
Then came the mischief. Fermat left the following marginal annotation: “I have discovered a truly remarkable proof [of this theorem], which this margin is too small to contain.” And which for more than three centuries the mind of man has been too dim to discern.
All these years mathematicians have given Fermat the benefit of the doubt: the consensus was that the last theorem was probably true, but that Fermat was mistaken in thinking or perverse in claiming that he had proved it. Its legend grew as it defied 15 generations of the world’s greatest mathematical minds. It became the Holy Grail of number theory. Then last month came news that a 38-year-old Japanese assistant professor had found the solution. Between the banal and the absurd that is the everyday, it seemed, something epic had happened.
Alas, it had not. Yoichi Miyaoka and his colleagues have been checking, and found fundamental if subtle problems deep in his proof. Miyaoka got a glimpse of the Grail, but no more. The disappointment is keen — the 20th century stands unredeemed — but it is mixed with a curious relief. “Next to a battle lost,” wrote Wellington, “the greatest misery is a battle gained.” Easy for him to say. (He won.) Still, there is wisdom in Wellington and comfort too. Solving Fermat would have meant losing him. With Miyaoka’s miss, Fermat — bemused, beguiling, daring posterity to best him — endures.
And mathematics gains. Miyaoka’s assault on Fermat is a reminder, an enactment of the romance that is mathematics. Math has a bad name these days. In the popular mind, it has become either a syndrome (math anxiety is an affliction to be treated like fear of flying) or a mere skill. We think of a math whiz as someone who can do in his head what a calculator can do on silicon. But that is not math. That is accounting. Real math is not crunching numbers but contemplating them and the mystery of their connections. For Gauss, “higher arithmetic” was an “inexhaustible store of interesting truths” about the magical relationship between sovereign numbers. Real math is about whether Fermat was right.
Does it matter? It is the pride of political thought that ideas have consequences. Mathematics, to its glory, is ideas without consequences. “A mathematician,” says Paul Erdös, one of its greatest living practitioners and one of the most eccentric, “is a machine for turning coffee into theorems.” Mathematicians do not like to admit that, because when they do, their grant money dries up — it is hard to export theorems — and they are suspected of just playing around, which of course they are.
Politicians and journalists need to believe that everything ultimately has a use and an application. So when a solution for something like Fermat’s last theorem is announced, one hears that the proof may have some benefit in the fields of, say, cryptography and computers. Mathematicians and their sympathizers, at a loss to justify their existence, will be heard to say, as a last resort, that doing mathematics is useful because “it sharpens the mind.”
Sharpens the mind? For what? For figuring polling results or fathoming Fellini movies or fixing shuttle boosters? We have our means and ends reversed. What could be more important than divining the Absolute? “God made the integers,” said a 19th century mathematician. “All the rest is the work of man.” That work is mathematics, and that it should have to justify itself by its applications, as a tool for making the mundane or improving the ephemeral, is an affront not just to mathematics but to the creature that invented it.
What higher calling can there be than searching for useless and beautiful truths? Number theory is as beautiful and no more useless than mastery of the balance beam or the well-thrown forward pass. And our culture expends enormous sums on those exercises without asking what higher end they serve.
Moreover, of all such exercises, mathematics is the most sublime. It is the metaphysics of modern man. It operates very close to religion, which is why numerology is important to so many faiths and why a sense of the transcendent is so keenly developed in many mathematicians, even the most irreligious. Erdös, an agnostic, likes to speak of God’s having a Book that contains the most elegant, most perfect mathematical proofs. Erdös’ highest compliment, reports Paul Hoffman in the Atlantic, is that a proof is “straight from the Book.” Says Erdös: “You don’t have to believe in God, but you should believe in the Book.”
In one of Borges’ short stories, a celestial librarian spends his entire life vainly searching for a similar volume, the divine “total book” that will explain the mystery of the universe. Then, realizing that such joy is destined not to be his, he expresses the touching hope that it may at least be someone else’s: “I pray to the unknown Gods that a man — just one, even though it were thousands of years ago! — may have examined and read it. If honor and wisdom and happiness are not for me, let them be for others.”
For a couple of days it seemed that honor and wisdom and happiness were Miyaoka’s. A mirage, it turns out. Yet someday Fermat’s last theorem will be solved. You and I will not understand that perfect proof any more than we understand Miyaoka’s version. Nonetheless, the thought that someone, somewhere, someday, will be allowed a look at Fermat’s page in the Book is for me, for now, joy enough.
Note from Joe: This article predates the now-famous proof of Fermat’s Last Theorem by Andrew Wiles, but the points made by Krauthammer about mathematics remain true, and this remains my all-time favorite magazine essay.