Why Beauty is Truth (23 page)

The person who was to change the course of mathematics and science was Évariste Galois, and his life story is one of the most dramatic, and also the most tragic, in the history of mathematics. His magnificent discoveries were very nearly lost altogether.

If Galois had not been born, or if his work had really been lost, someone would no doubt have made the same discoveries eventually. Many mathematicians had voyaged across the same intellectual territory, missing the great discovery by a whisker. In some alternative universe, someone with Galois's gifts and insights (perhaps a Niels Abel who avoided tuberculosis for a few more years) would eventually have penetrated the same circle of ideas. But in this universe, it was Galois.

He was born on 25 October 1811, in Bourg-la-Reine, in those days a small village on the outskirts of Paris. Now it is a suburb in the département Hau-de-Seine, at the intersection of the N20 and the D60 highways. The D60 is now named Avenue Galois. In 1792, the village of Bourg-la-Reine had been renamed Bourg-l'Égalité, a name that reflected the era's political turmoil and its ideology: “Queen Town” had given way to “Equality Town.” In 1812, the name reverted to Bourg-la-Reine, but revolution was still in the air.

The father, Nicolas-Gabriel Galois, was a republican and leader of the village Liberal Party—Liberté in the town of Egalité—whose main policy was the abolition of the monarchy. When, in a fudged compromise of 1814, King Louis XVIII was returned to the throne, Nicolas-Gabriel became the town mayor, which cannot have been a comfortable office for someone of his political leanings.

The mother, Adelaide-Marie, was born to the Démante family. Her father was a jurisconsult, a paralegal expert whose job was to offer opinions about legal cases. Adelaide-Marie was a fluent reader of Latin and passed her classical education on to her son.

For his first twelve years, Évariste remained at home, educated by his mother. He was offered a place at the college of Reims when he was ten, but his mother seems to have thought it too early for him to leave home. But in October 1823, he started attending the Collège de Louis-le-Grand, a preparatory school. Soon after Évariste arrived, the students refused to chant in the school chapel, and the young Galois saw at first hand the fate of would-be revolutionaries: a hundred pupils were promptly expelled. Unfortunately for mathematics, the lesson did not deter him.

For his first two years he was awarded first prize in Latin, but then he became bored. In consequence, the school insisted that he repeat his classes to improve his performance, but of course this made him even more bored, and things went from bad to worse. What saved Galois from the slippery slope to oblivion was mathematics, a subject with enough intellectual content to retain his interest. And not just any mathematics: Galois went straight to the classics: Legendre's
Elements of Geometry.
It was a bit like a modern physics student starting out by reading the technical papers of Einstein. But in mathematics there is a kind of threshold effect, an intellectual tipping point. If a student can just get over the first few humps, negotiate the notational peculiarities of the subject, and grasp that the best way to make progress is to
understand
the ideas, not just learn them by rote, he or she can sail off merrily down the highway, heading for ever more abstruse and challenging ideas, while an only slightly duller student gets stuck at the geometry of isosceles triangles.

Just how hard Galois had to work to understand Legendre's seminal work is open to dispute, but in any case it did not daunt him. He started to read the technical papers of Lagrange and Abel; not surprisingly, his later work concentrated on their areas of interest, in particular the theory of equations. Equations were possibly the only things that really grabbed
Galois's attention. His ordinary schoolwork suffered in proportion to his devotion to the works of the mathematical greats.

At school, Galois was untidy, a habit he never lost. He baffled his teachers by solving problems in his head instead of “showing his work.” This is a fetish of mathematics teachers that afflicts many a talented youngster today. Imagine what would happen to a budding young footballer if every time he scored a goal, the coach demanded that he write out the exact sequence of tactical steps he followed, or else the goal would be invalid. There was no such sequence. The player saw an opening and put the ball where anyone who understood the game would know it had to go.

So it is with able young mathematicians.

Ambition led Galois to aim high: he wanted to continue his studies at one of the most prestigious institutions in France, the École Polytechnique, the breeding ground of French mathematics. But he ignored the advice of his mathematics teacher, who tried to make the young man work in a systematic manner, show his work, and generally make it possible for the examiners to follow his reasoning. Fatally underprepared and overconfident, Évariste took the entrance examination—and failed.

Twenty years later, an influential French mathematician named Orly Terquem, who edited a prestigious journal, offered an explanation for Galois's failure: “A candidate of superior intelligence is lost with an examiner of inferior intelligence. Because they do not understand me,
I
am a barbarian.” A modern commentator, more aware of the need for communication skills, would temper that criticism with the observation that a student of superior intelligence has to make allowances for those less able. Galois did not help his case by being uncompromising.

So Galois remained at Louis-le-Grand, where he had a rare piece of good fortune. A teacher named Louis-Paul Richard recognized the young man's talent, and Galois enrolled in an advanced mathematics course under Richard's tuition. Richard formed the opinion that Galois was so talented that he should be admitted to the École Polytechnique without being examined. Very likely, Richard had an idea of what would happen if Galois were to take the examination. There is no evidence that Richard ever explained his view to the École Polytechnique. If he did, they took no notice.

By 1829, Galois had published his first research paper, a competent but pedestrian article on continued fractions. His unpublished work was more ambitious: he had been making fundamental contributions to the theory of equations. He wrote up some of his results and sent them to the French Academy of Sciences, for possible publication in their journal. Then, as now, any paper submitted for publication would be sent to a referee, an expert in the field concerned, who made recommendations about the novelty, value, and interest of the work. In this case the referee was Cauchy, then probably France's leading mathematician. Having already published in areas close to those involved in Galois's paper, he was a natural choice.

Unfortunately, he was also extremely busy. There is a prevalent myth that Cauchy lost the manuscript; some sources suggest that he threw it away in a fit of pique. The truth seems more prosaic. There is a letter from Cauchy to the Academy, dated 18 January 1830, in which he apologizes for not presenting a report on the work of “young Galoi,” explains that he was “indisposed at home,” and also mentions a memoir of his own.

This letter tells us several things. The first is that Cauchy had not thrown Galois's manuscript away but still had it six months after submission. The second is that Cauchy must have read the manuscript and decided that it was important enough to be worth drawing to the Academy's attention.

But when Cauchy turned up at the next meeting he presented only his own paper. What had happened to Galois's manuscript?

The French historian René Taton has argued that Cauchy was impressed by Galois's ideas—perhaps a little too impressed. So instead of reading the work to the Academy as originally intended, he advised Galois to write a more extensive and presumably much improved exposition of the theory, to be submitted for the Grand Prize in Mathematics, a major honor. There is no documentary evidence to confirm this claim, but we do know that in February 1830 Galois submitted just such a memoir for the Grand Prize.

We cannot know exactly what was in this document, but its general contents can be inferred from Galois's surviving writings. It is clear that history might have been very different if the far-reaching implications of his work had been fully appreciated. Instead, the manuscript just vanished.

One possible explanation appeared in 1831 in
The Globe
, a journal published by the Saint-Simonians, a neo-Christian socialist movement.
The
Globe
reported a court case in which Galois was accused of publicly threatening the life of the king, and suggested that “This memoir . . . deserved the prize, for it could resolve some difficulties that Lagrange had failed to do. Cauchy had conferred the highest praise on the author about this subject. And what happened? The memoir is lost and the prize is given without the participation of the young savant.”

The big problem here is to decide the factual basis of the article. Cauchy had fled the country in September 1830 to avoid the revolutionaries' anti-intellectual attentions, so the article cannot have been based on anything he had said. Instead, it looks as though the source was Galois himself. Galois had a close friend, Auguste Chevalier, who had invited him to join a Saint-Simonian commune. It seems likely that Chevalier was the reporter—Galois was otherwise engaged at the time, on trial for his life—and if so, the story must have come from Galois. Either he made it all up, or Cauchy had indeed praised his work.

Let us return to 1829. On the mathematical front, Galois was becoming increasingly frustrated by the apparent inability of the mathematical community to give him the recognition he craved. Then his personal life began to fall to pieces.

All was not well in the village of Bourg-la-Reine. The village mayor, Galois's father, Nicolas, became involved in a nasty political dispute, which enraged the village priest. The priest took the decidedly uncharitable step of circulating malicious comments about Nicolas's relatives and forging Nicolas's own signature on them. In despair, Nicolas committed suicide by suffocating himself.

This tragedy happened just a few days before Galois's final opportunity to pass the entrance examination for the École Polytechnique. It did not go well. Some accounts have Galois throwing the blackboard eraser into the examiner's face—it was probably a cloth, not a lump of wood, but even so, the examiner would not have been favorably impressed. In 1899, J. Bertrand provided some details that suggest that Galois was asked a question he had not anticipated, and lost his temper.

For whatever reason, Galois failed the entrance exam, and now he was in a bind. Having been utterly confident that he would pass—he really does seem to have been an arrogant young man—he had not bothered to prepare for the exams to enter the only alternative, the École Préparatoire.
Nowadays, this institution, renamed the École Normale, is more prestigious than the Polytechnique, but in those days it came a poor second. Galois hastily boned up on the necessary material, passed his mathematics and physics with flying colors, made a mess of his literature exam, and was accepted anyway. He obtained qualifications in both science and letters at the end of 1829.

As I mentioned, in February 1830 Galois submitted a memoir on the theory of equations to the Academy for the Grand Prize. The secretary, Joseph Fourier, took it home to give it the once-over. The ill-fortune that constantly dogged Galois's career struck again: Fourier promptly died, leaving the memoir unread. Worse, the manuscript could not be found among his papers. However, there were three other committee members in charge of the prize: Legendre, Sylvestre-François Lacroix, and Louis Poinsot. Maybe one of them lost it.