Devlin's Angle

August 1996

Of Men, Mathematics, and Myths

With the Unabomber now firmly implanted in the public psyche as "a mathematician", the caricature of the mathematician as the misunderstood, antisocial loner has received a considerable boost. That much of mathematics these days is done by two or more mathematicians working in collaboration is overlooked when such caricatures are being bandied about. Also forgotten is that fact that practically all present-day mathematicians earn their daily crust by working alongside dozens of other academics at a college or university. Even for those mathematicians who spend most of their time engaged in highly esoteric, pure research, interaction with colleagues is an essential ingredient. The myth of the mathematical loner--the misunderstood genius who cannot relate to his (this mythical creature is always male) colleagues--is surely belied by the eagerness with which the contemporary mathematician heads for the airport before the red ink on the last exam script is barely dry, jetting off to meet old acquaintances and catch up on the latest gossip. But so what?

The idea of the mathematical genius dropping out of society to contemplate abstract, Platonic worlds has a nice Hollywood ring to it, but it's hardly the truth. For all the press coverage given to Andrew Wiles' New Jersey attic, the British-born solver of Fermat's Last Theorem has a wife, a teaching position at Princeton, and a drawer full of airline-ticket stubs. In the case of the alleged Unabomber, the fact is that this societal misfit gave up mathematics to pursue his life of isolation. But who cares? Please don't spoil a good story.

Still and all, it's hard to condemn journalists for milking a popular conception when mathematicians themselves create and perpetuate romantic myths. Perhaps the most famous mathematical fairy story is the tale of Evariste Galois, the man who gave the world group theory.

Killed in a duel at the tender age of twenty, the final hours of this undoubted mathematical genius were described in the following breathless prose by the mathematician Eric Temple Bell in his book "Men of Mathematics", first published in 1937:

"All night long he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death he saw could overtake him. Time after time he broke off to scribble in the margin "I have not time; I have not time," and passed on to the next frantically scrawled outline. What he wrote in those last desperate hours before the dawn will keep generations of mathematicians busy for hundreds of years. He had found, once and for all, the true solution of a riddle which had tormented mathematicians for centuries: under what circumstances can an equation be solved?"

It's a great story. The stuff of legends. Which one of us, when we were starting out on our mathematical careers, on hearing the story for the first time--or perhaps reading Bell's words for ourselves--did not find our emotions stirred? The more so if we took into account the tragic details of the short life that preceded the young Frenchman's violent end. At least, the details as described to us by Bell, who tells us how this young genius suffered the frustrations of his work being shunned by a mathematical establishment that was too stupid to see true genius when it was presented to them.

The facts, sadly to say, do not support this romantic picture--not that anything as mundane as hard fact will stop a good story once it gets going. The facts in this case have been around for a long time. They were thrust firmly in front of the mathematical community in 1982, when Tony Rothman published an article in the February issue of the American Mathematical Monthly entitled "Genius and Biographers: The Fictionalization of Evariste Galois". An expanded version of the article appears as Chapter 6 in Rothman's 1989 book "Science a la Mode", published by Princeton University Press, on which I am basing most of this column.

What then, is the true story of young Galois?

He was born on October 25, 1811, in Bourg-la-Reine, near Paris. His parents were well-educated. He started his formal schooling in October 1823, at first doing well in all subjects. In February 1827 he took his first real mathematics course, studying the work of Legendre on geometry and of Lagrange on algebra. The immediate passion with which he embraced mathematics soon led to problems with his other subjects, and the beginning of a paranoid antagonism toward educational authority that was to be characteristic of the rest of his academic life.

Galois' paranoia was given a further boost when, in 1828, he took the examination for entry to the prestigious Ecole Polytechnique a year early and failed. That his failure was due to lack of adequate preparation, and not an absence of talent, is clear from the fact that only the next year, in April 1829, he published his first paper, on continued fractions. But that paper was little more than a diversion from his major work--the analysis of the solvability of equations that would lead to what we now call Galois Theory. On May 25 and June 1 of 1829, Galois sent his results to the Academy Francais for publication. Cauchy was appointed referee.

According to the Bell account, and the popular legend that Bell in large part inspired, Cauchy lost the papers. But as Rothman indicates, there is evidence in Cauchy's own hand that he not only read Galois' work, he was sufficiently impressed by it to offer to present it at a meeting of the Academy in January 1830. (Illness prevented Cauchy from attending the meeting on the day in question. He did not take the opportunity to present the work at a subsequent meeting, but it seems likely that this was because he encouraged Galois to turn his notes into a submission for the Grand Prize in Mathematics instead, for which the submission deadline was March 1.)

In 1829, Galois once again tried to gain entrance to the Ecole Polytechnique, but again failed. This time, the problem seemed to be more one of a rapidly growing academic arrogance rather than lack of preparedness. He reportedly told the examiner that he would not answer his questions because they were trivial. He might well have been right, but the result was predictable. Galois ended up enrolling at the Ecole Normale.

In 1830, Galois sent in his submission for the Grand Prize. As secretary for mathematics and physics at the Academy, Fourier handled the submission. The referees were to be Lacroix, Poisson, Legendre, and Poinsot. That such a high caliber review team was involved puts paid to another aspect of the Galois legend, one started by the paranoid Galois himself, that Fourier "buried" his paper. It was however the case that when Fourier died in May 1830, no one could find Galois' submission, and he did miss out on the prize.

Prize or no prize, however, during the same year, 1830, Galois carried out the research that would give immortality to his name. That year, he published three papers. Their titles were "An analysis of a memoir on the algebraic resolution of equations", "Notes on the resolution of numerical equations", and "On the theory of numbers". In these three papers he developed what we now call Galois Theory. In all three papers he used the word "group" to mean a "group of permutations". So much for that famous last night of feverish writing, when the mathematical world came within a few scribbled pages of never having group theory!

Whether or not his feeling of rejection by the mathematical community spurred him on, at the same time he was developing his mathematical theory, Galois' long standing interest in radical politics intensified. On May 10, 1831, he was arrested for making a seditious remark at a banquet the day before, but when his trial came on June 15 he was acquitted. A month later he was arrested again, this time for publicly wearing the banned uniform of the Artillery Guard, an act of political defiance that can only have been done in order to provoke arrest and the six months of prison that followed.

And so to that tragic end. It seems clear that the deadly duel was over a woman. However, the popular accounts give us two alternatives. One version has the dispute a straightforward quarrel between two young men vying for the same female. The other has the whole affair politically motivated, with the woman a prostitute hired to maneuver Galois to his untimely death. Purveyors of both versions usually add mystery to the drama by claiming that the identity of the woman in question is not known.

She was Stephanie-Felice Poterin du Motel, a woman from a good and respected family.

What does seem in doubt--oddly enough given the amount of evidence for most other aspects of Galois' life--is who was the other duelist. Rothman thinks it was Galois' revolutionary friend Duchatelet. However, in his memoirs, Galois' contemporary Alexander Dumas gives the name of the duelist as Pescheux d'Herbinville. At last, we do seem to have a bit of mystery! But whoever the other man was, it seems likely that Galois entered into the final duel with some kind of (romantic?) death wish.

The night before the duel Galois did indeed do a lot of writing. In particular, in a long letter to his friend Chevalier, he outlined his mathematical work. However, though it was still not accepted by the Academy, it had pretty well all been published, and Cauchy for one had clearly recognized its importance. At one point in his letter, Galois remarked that a published proof was missing a key detail, adding (just that once) that "I have not the time" [to give the details].

And so the world lost a great mathematical talent. Of that there is no doubt. At the same time, the seeds were set for the creation of a great legend. But of that legend, there is not just considerable doubt, in most respects there is concrete evidence to show that the story is just plain false.

But the truth, as I remarked once already, is hardly likely to stop a good story. No more than we can hope to stop journalists from describing the alleged Unabomber as "a mathematician".

-Keith Devlin

Devlin's Angle is updated at the start of each month.

Keith Devlin ([email protected]) is the editor of FOCUS, the news magazine of the MAA. He is the Dean of Science at Saint Mary's College of California, and the author of Mathematics: The Science of Patterns, published by W. H. Freeman in 1994.