February 9, 1999

Genius or Gibberish? The Strange World of the Math Crank

By GEORGE JOHNSON

he letter, dated Christmas Day 1998 and addressed to a professor at the Niels Bohr Institute in Copenhagen, began portentously: "Nowadays, we seek to comprehend our comprehensions and call that comprehensiveness knowledge in the mistaken belief that as a science it is immortal. Such omniscience diffuses like Helium-3 into the penetralia mentis of omnipotent impotency within any God-head such that any caveat actor is saved. . . . "

Within a few sentences, the writer was holding forth on Heisenberg's Uncertainty Principle and "the concept of nothing" as the empty set, before launching into speculations involving number theory: "It's enough to make me conjecture that infinity's prime and Riemann's Zeta function accounts for fractional charge subatomically just for the Higg's boson with an involucral matrix of ogdoad parity as midwife!"

The letter was typed single-spaced with the tiniest of margins and embellished with hand-drawn diagrams and colored annotations. Copies were sent to a list that included the linguist Noam Chomsky, the physicists John Archibald Wheeler, David Deutsch and Stephen Hawking, and the mathematician John Casti.

"It has all the hallmarks of a crank," said Dr. Casti, who is affiliated with the Technical University of Vienna and the Santa Fe Institute in New Mexico. "It's amazing all the stuff you can get onto a single piece of paper."

But was it not just possible that couched in the obscure mix of mathematics, physics and Egyptian mysticism ("ogdoad parity" refers to four pairs of gods with names like Darkness, Absence and Endlessness), there lay an important insight?

Didn't two Cambridge University mathematicians dismiss the great self-taught Indian number theorist Srinvasa Ramanujan as a crackpot when he sent them long eccentric letters from India early in this century? Only their colleague G. H. Hardy had the foresight to recognize Ramanujan as a genius. And didn't the great German mathematician Carl Friedrich Gauss foolishly throw away unread a groundbreaking paper from his young Norwegian colleague Niels Heinrik Abel, calling it "another of those monstrosities"?

Dr. Casti was not too worried about the possibility. Though the stories of Ramanujan and Abel may linger in the backs of mathematicians' minds as they aim the latest unsolicited epistle toward the wastebasket, most become quickly jaded.

"After several hundred of these things you get into that mode," said Dr. Ian Stewart, a mathematician at Warwick University in England. "It has to do with your self-preservation.

"The writers of these letters range from pretty good amateur mathematicians who have made a mistake somewhere or skipped over an important step to people who are completely mad," he said. "You get very strange mail in 17 different fonts and 14 colors and with an idiosyncratic grammar." Many of the correspondents are intelligent, well-meaning, indefatigable souls who, in their untrained way, share the fascination mathematicians feel for the invisible world of numbers. And many are simply cranks.

The equipment necessary for discovering a subatomic particle in your own home costs too much and would never fit inside a basement laboratory or even a very large backyard.

But with nothing more than a pencil and paper, and maybe a compass and straightedge, even an amateur can explore the mathematical nether world, stumbling across important new truths.

That, anyway, is the dream or obsession that drives would-be Ramanujans all over the world to send mathematicians letter after letter crammed full of diagrams and equations promising answers to unsolved problems ranging from squaring the circle or proving (in a simpler way) Fermat's Last Theorem to revealing, through the wonders of mathematics, the meaning of life.

Physicists get their share of mail from amateurs attempting to reconcile quantum mechanics and general relativity or to show that Einstein was wrong. But the greater ease with which one can speculate about numbers has caused the mathematical crank to become enshrined in academic folklore. The phenomenon is even documented in a book called "Mathematical Cranks" (Mathematical Association of America, 1992), by Dr. Underwood Dudley, a mathematician at DePauw University in Indiana.

"I've been at this for a decade and still can't pin down exactly what it is that makes a crank a crank," said Dr. Dudley, who has met a few in person. "They are usually men, old men," he said. "All are humorless. None of them are fat," a characteristic he attributes to their obsessive personalities. "It's like obscenity -- you can tell a crank when you see one."

With recent films like "Good Will Hunting" and "Pi" giving mathematics a romantic sheen and popular new biographies romanticizing the lives of the mathematicians Paul Erdos and John Nash, the flow of crank mail will only increase, predicted Dr. John Allen Paulos, a mathematician at Temple University in Philadelphia.

Add millennial anxiety, including the Y2K problem, Dr. Paulos speculated, and the time he and his colleagues spend opening and discarding letters peppered with strange symbols and grand claims is bound to increase. "Popular mathematics used to be an oxymoron," he said, "but happily that's not true anymore." The celebrity, though, has its price.

"It's a real dilemma," said Dr. Reuben Hersh, professor emeritus of mathematics at the University of New Mexico in Albuquerque. "Who is going to stroll through pages and pages of stuff that is very hard to understand when you don't have to do it? From the point of view of these guys, we are arrogant, unwilling to reconsider ideas. And why shouldn't they expect a responsible scientist to look carefully at some new idea that might be important?

"You can't just say Ramanujan was a genius and these other guys were cranks," Dr. Hersh said. "With a superficial look, there is hardly any visible difference. There is not always a sharp line between eccentric mathematicians and intelligent but maybe obsessed amateurs."

Until recently much of the mail contained supposed proofs of Fermat's Last Theorem.

But since Dr. Andrew Wiles of Princeton University recently proved this famous puzzle and number theory, Dr. Paulos said, the focus has shifted to disproving Dr. Wiles.

Another favorite diversion is Goldbach's Conjecture, which holds that all even numbers are the sum of two primes. Though no one has found a counter example, this would-be theorem remains unproven, unless the solution has been crumpled up in a math department wastebasket somewhere.

Mathematicians are especially impatient with letters claiming to have solved one of the three classical problems of Greek mathematics: trisecting an angle (dividing it into three equal parts), doubling a cube, or squaring a circle (constructing a square with the exact same area as a circle) using only a compass and an unmarked straightedge. The reason for imposing these restrictions was to see how much mathematics could be derived from two basic concepts, the line and the circle.

Solving the three classic problems this way has been proved impossible. In 1882, for example, the German mathematician Ferdinand Lindemann established that trying to square the circle was hopeless. The reason is that pi, the ratio of the circumference of a circle to its diameter, is not only irrational (its infinitely long string of decimal places never repeats) but transcendental (it is not a solution of any polynomial equation with whole number coefficients).

Amateur attempts at squaring the circle generally amount to rounding off pi to a close approximation like 22/7 or 355/113. At best the result is a square that is very close but never exactly the size of the circle.

"These people don't know what impossible means," Dr. Dudley lamented. "They think it just means very hard."

Or the writers do not believe the impossibility proofs. "They start by saying they know there is a theorem that says it is impossible but they've done it anyway, so there must be something wrong with the theorem," Dr. Stewart said. "They are operating in a conceptual vacuum. It leads to strange things."

One of the rules mathematicians soon learn is never to answer a possible crank. The mail often comes with cover letters assuring that the work has been "endorsed" by a certain Harvard or Stanford professor, say, or even a member of the United States Senate.

The attached letters are usually no more than polite brushoffs: "Thank you for your interesting letter. . . ." A mathematician who makes a perfunctory reply can count on its being stapled to the next round of mailings.

Dr. Hersh replied to a letter from an amateur mathematician in India. "This guy wrote very well," he said, "in a good expository style. His penmanship was fine. What he said made enough sense that I thought I would try to explain and straighten him out."

Over several years, Dr.

Hersh realized that his correspondent had come to believe that the whole edifice of mathematics was about to crumble because of a dreadful mistake made centuries ago. Dr. Hersh finally wrote back in exasperation: "I've done all I can for you. I can't do anymore." The writer answered that he did not consider himself Dr. Hersh's student but rather his opponent in a debate.

Surprisingly, the rise of the Internet has not increased the amount of mathematical crank mail.

Most of the letters still come typewritten, often on what appear to be manual typewriters. "Cranks are always about one level of technology behind," Dr. Dudley said. Dr. Casti said he imagined some of his correspondents as "penniless guys in cold-water flats." They save paper and postage by single-spacing and often type on both sides of the page.

Sometimes, just sometimes, a mathematician finds that it pays to answer an unsolicited letter, one that does not have what Dr. Stewart calls "the strange fairy dusting of lunacy." He once received a clearly written letter from a man in China who believed he had trisected the angle. "I knew it had to be a fallacy, but it gave me a spur to try and see where it was wrong," Dr. Stewart said. "There were three points in the diagram that looked as if they were on a straight line, but actually were not." Dr. Stewart wrote back and received a reply thanking him for pointing out the error. "It is the only time I've had any success convincing someone like this that they were wrong," he said.

Several years ago Dr. Stewart heard from a man -- in India again -- who had found a new, simpler proof for an obscure, pointless theorem in number theory written by Ramanujan and a collaborator.

According to the Ramanujan-Nagell theorem, the only numbers one can square and add 7 to, yielding an answer that is a power of 2, are 1, 3, 5, 11 and 181. For example, squaring 3 and adding 7 gives 16, which is the fourth power (the square of the square) of 2.

Dr. Stewart was surprised to realize that the proof was correct, but it was badly typed on strange paper and cast in an idiosyncratic style that would have given any journal editor the impression that the writer was a crank. Dr. Stewart advised the writer to find an Indian number theorist who could teach him how to present a proper paper. Several years later the result was published, and soon after came another publication from the same man. "It is worth reading these things occasionally," Dr. Stewart said.

But only occasionally, Dr. Dudley advised. "There is always that chance of success, but it is so small," he said. "I've gone through enough reams of crank stuff to know that the probability is close to zero."