Not many of us fulfill our childhood dreams and see our names entered into history books as a result.
So it’s little wonder that mathematician Andrew Wiles smiled broadly as he sat in his office at Princeton University, recounting just such a double coup.
“I first encountered Fermat’s Last Theorem as a child,” said Wiles, 40. “That was a long time ago. Perhaps I was 10. So I’ve forgotten in just which book. But it related that, for centuries, mathematicians had been seeking a solution to that puzzle.”
Wiles wasn’t the only youngster of his generation to be so fascinated. But he had a leg up, which may account for the early age at which he encountered a classic problem inadvertently bequeathed to all mathematicians by the 17th Century Frenchman Pierre de Fermat (pronounced fer-MA). Wiles, whose father was a professor of theology at Oxford University in England, grew up surrounded by books and ideas.
Fermat’s problem used to be printed in math textbooks as a kind of intellectual goad, showing students that the subject was far from finished and that fame, if not fortune, awaited whoever finally solved it.
“Fermat’s Last Theorem has been perennially intriguing,” Wiles said, “precisely because it so long remained unsolved, though it can be so easily stated.”
Indeed, it’s a masterpiece of simplicity, especially if set alongside comparable scientific problems. Short of physicists, who can comprehend the question running through Einstein’s mind when he tackled the Theory of Relativity?
But the only prerequisite for understanding the problem that Fermat set is a smattering of elementary algebra. (See accompanying box.)
So like Wiles, lots of youngsters have joined professional mathematicians in having a go at the puzzle. Both were probably still scratching their heads over it when Wiles recently shocked a meeting of mathematicians at Cambridge University by stepping up to a blackboard and scribbling a solution to Fermat’s Last Theorem.
With that, he vaulted into the front ranks of mathematicians, not just of this generation but of all ages.
Even several weeks later, Wiles can’t stop grinning. A smile of sheer joy never once left his face during a two-hour interview in which he recalled the years of hard work preceding his moment of glory.
“I just found it irresistible,” Wiles said, explaining why he kept up a quest other mathematicians abandoned as beyond them. His Princeton colleague, John Conway, himself a mathematician of note, notes that there are only four or five mathematicians on Earth-and he’s not one-even capable of checking Wiles’ proof for errors.
Wiles’ smile could also be inspired by the unique nature of his accomplishment. In virtually every other field of endeavor, today’s champion is tomorrow’s has-been. The 4-minute mile and the 15-foot pole vault once seemed impossible, but who recalls the athletes who first broke those barriers, now that others routinely surpass their marks?
Even scientific breakthroughs have a limited lifespan. Nobel Prize winners in physics or chemistry know that their achievements may become outdated because of subsequent discoveries. But mathematics is a forever thing, noted Robert Zimmer, chairman of the University of Chicago’s math department. The permanence of a mathematical discovery, he noted, is guaranteed by the inviolability of the logic upon which it rests.
“Assuming he’s got it correct, Wiles’ solution of the Fermat theorem will always be his,” Zimmer said. “Others may come along to tidy up his proof a bit. But they’ll never take it away from him. That’s the nature and the beauty of mathematics.”
The Uncertainty Principle
The nature of their enterprise also means that mathematicians talk about their work differently than other scholars do. A physicist is schooled in the Uncertainty Principle, an axiom of the science that roughly says the phenomena studied are inevitably distorted by attempts to measure them. So a physicist’s imagination is humbled by a sense that, though he employs powerful scientific tools, his discipline can only sketch an approximation of reality.
By contrast, though armed only with a piece of chalk and his logical faculties, a mathematician feels that when he writes the solution to a problem on his blackboard, there remain no intellectual veils between his equations and truth.
“You’re trying to trick Nature into revealing one of her secrets to you,” Wiles said. “We assume that Nature has left clues around, here and there.”
The particular secret of Nature that intrigued Wiles, noted his Princeton colleague Conway, was first glimpsed by the ancient Babylonians perhaps as far back as the second millennium B.C.
“There survive from the beginnings of civilization clay tablets,” Conway said, “inscribed with sets of what we call Pythagorean triplets, such as the numbers 3, 4 and 5.”
He explained that some Babylonian mathematician must have noticed that squares of such numbers have a unique relationship. Notice: 3 squared (or 3 x 3) (equals) 9; 4 squared (equals) 16; and 5 squared (equals) 25-which is the same number you get by adding together the squares of the first two numbers (9 + 16).
Centuries later, the Greek philosopher Pythagoras (569-500 B.C.) recognized that such sets of numbers have a geometric spinoff, namely that if you draw two lines perpendicular to each other, and make one 3 units long and the other 4 units long, then connect their endpoints, you will have created a triangle whose third side is 5 units long.
That discovery passed into math textbooks, where high school students still learn it as the Pythagorean theorem. In the 17th Century, Fermat encountered it while reading the works of another ancient Greek mathematician, Diophantus, leading him to consider what mathematicians call the general case.
What happens, Fermat wondered, if you write the equation xn + yn (equals) zn, where x, y, z and n is each a whole number other than zero?
For instance, what happens if you cube numbers (multiply them by themselves twice) or raise them to the 4th power (multiply them by themselves three times), etc.? Fermat concluded that in such cases the general equation wouldn’t hold: That is, you can’t cube a number, add it to the cube of another number and wind up with the cube of a third whole number. Likewise for numbers raised to the 4th power, etc.
A mathematical tease
Beyond the second power, no solutions to the general equation can be found, Fermat announced. He did so on the margins of his copy of Diophantus, adding that he lacked space to write out a full proof of his theorem.
“Fermat wasn’t trying to tantalize other mathematicians,” Conway said. “In his day, there weren’t scientific journals where proofs could be published.”
In fact, Fermat, who was one of the founders of modern mathematics, left several other important theorems in similarly abbreviated fashion. Subsequent generations of mathematicians were able to reconstruct proofs for all of them except one, which thus came to be known as Fermat’s Last Theorem.
For 300 years, that missing proof haunted his successors.
“Mathematicians were able to prove why the theorem holds for certain cases of n,” Conway said. “Computers have been programmed to try tens of thousands of combinations of numbers and have never produced a single exception to Fermat’s rule.”
But mathematicians aren’t content with computer-generated results because a computer might produce an exception if asked to try more numbers. Physicists live with that ambiguity, prepared to adjust their formulas when an anomalous computer printout requires them to. “But a mathematician,” Conway said, “wants to be able to say to the computer: `You can stop looking. I’ve proved you’ll never find it.’ “
So over the centuries, thousands of mathematicians, professional and amateur, have sought purely mathematical proofs for Fermat’s Last Theorem. Scientific societies in France and Germany offered prize money for a correct solution.
Why?
Some mathematicians point to practical derivatives of such seemingly abstract research. They’ll note, for example, that some dead-end approaches to Fermat’s problem proved important to the development of cryptography, giving the CIA a vested interest in advanced math.
In truth, a mathematician is no more motivated by such mundane concerns than a crossword-puzzle fan is interested in increasing his reading skills. A mathematician is simply in love with numbers.
Conway recalled that as a youngster he memorized the value of pi (the ratio of the circumferance of a circle to its diameter) to a thousand decimal places. When he and his wife, who is also a mathematician, were courting, they’d take walks through their native English countryside, taking turns reciting that sequence to each other.
“Once we stopped off someplace where an American graduate student heard us,” Conway said. “He joined in, right at the proper digit.”
Branching out
Wiles had to set aside his fascination when he entered graduate school at Cambridge in the 1970s. By then it was apparent that traditional approaches to Fermat’s problem, through what’s known as number theory, were going nowhere. So he went into a different branch of mathematics.
“I began to study elliptical curves,” Wiles said. “At the time, I didn’t see any connections to Fermat’s theorem.”
Regularly, though, mathematics goes through periods when seemingly unrelated fields come together, notes the U. of C.’s Zimmer. In fact, mathematical creativity seems to involve the ability to see connections where others see differences.
“Because mathematical proofs are deductive, it seems like mathematicians push ahead on a narrow front,” Zimmer said. “Actually, the great ones solve problems by absorbing developments in fields on either side of the question they’re working on.”
Seven years ago, Kenneth Ribet, a University of California mathematician, demonstrated that if a seemingly unrelated mathematical proposition, called the Taniyama conjecture, could be proved, Fermat’s theorem would follow automatically. “Conjecture” is mathematician shorthand for “a fascinating but as-yet-unproved idea.” The Taniyama conjecture involved elliptical curves.
“The very day I heard about Ribet’s (linking of Taniyama and Fermat), I set to work,” Wiles said. “It tied together the two things I enjoyed most, elliptical curves and Fermat’s theorem.”
His decision to try an end-run on Fermat, Wiles recalled, required his withdrawal from professional life, beyond teaching his classes at Princeton. Partly, that was because proving the Taniyama conjecture involved difficult constructions that few other mathematicians could follow, leaving him virtually no colleagues to talk to. But also, to do otherwise would be like a pitcher announcing he was aiming for a no-hitter.
“You can’t tell people you’re working on Fermat’s problem,” Wiles said.
After almost seven years of secret labor, Wiles felt he had his proof in hand, except for one special case that continued to bedevil him. Then about two months ago, he happened across another mathematician’s paper on an unrelated topic.
“I read one sentence in that paper,” Wiles said, “and instantly saw that it gave me the tool I needed.”
A dramatic unveiling
He presented his proof of Fermat’s theorem with a bit of theatricality in keeping with the tantalizing history of the problem. Wiles simply signed up to lecture on three successive days at a small gathering of mathematicians at Cambridge, not even hinting that he would deal with Fermat.
By the end of the second day, some members of the audience thought he might be headed in that direction. Yet Wiles said nothing until the end of his final lecture, when he announced his proof of the Taniyama conjecture. Almost as an afterthought, he added that that meant Fermat’s Last Theorem finally had a proof too.
“I enjoy unraveling that which seems unravelable,” Wiles said, looking back on the experience. “But I think I won’t instantly pick up something that is as long term next time.”
XN + YN (EQUALS) ZN A PROVEN CHALLENGE
Consider this simple equation: xn + yn (equals) zn
When n equals 2, it’s not hard to substitute numbers for x, y and z that make the equation work. For example, we can make x (equals) 3, y (equals) 4 and z (equals) 5. Then the equation becomes: 3 squared + 4 squared (equals) 5 squared, or 9 + 16 (equals) 25. Other sets of numbers also work.
But the French mathematician Pierre de Fermat announced in the 17th Century that once n becomes greater than 2, no whole numbers will satisfy the equation.
Try to find some; you won’t (zero doesn’t count).
Why?
Until Andrew Wiles of Princeton University recently announced his proof of Fermat’s assertion, known as Fermat’s Last Theorem, even the greatest mathematicians in history couldn’t say.
