Why Beauty is Truth Read online

  The attitude may seem rather negative, but time and time again it has proved its worth. The underlying philosophy is that most mathematical problems are too difficult for anyone to solve. So when somebody manages to solve something that has baffled all predecessors, merely celebrating the great solution is not enough. Either the solver got lucky (mathematicians do not believe in that sort of luck) or some special reason made the solution possible. And if it proves possible to understand the reason . . . why, lots of other problems might yield to similar methods.

  So while Abel was polishing off the specific question, “Can every quintic be solved?” and getting a clear “no,” an even deeper thinker was wrestling with a far more general issue: which equations can be solved by radicals, and which cannot? To be fair, Abel had begun to think along those lines, and might have found the answer if tuberculosis had spared him.

  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 imp
ortant 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.

  Galois, not surprisingly, was furious. He became convinced that what had happened was a conspiracy of mediocre minds to stifle the efforts of genius; he quickly found a scapegoat, the oppressive Bourbon regime. And he wanted to play a role in its destruction.

  Six years earlier, in 1824, King Charles X had come to the throne of France, following Louis XVIII, but he was far from popular. The liberal opposition did well in the 1827 elections and even better in 1830, gaining a majority. Charles, facing the imminent prospect of forced abdication, attempted a coup; on 25 July he issued a proclamation suspending freedom of the press. He misread the mood of the people, who promptly rose in revolt, and after three days a compromise was reached: Charles was replaced as king by the duke of Orléans, Louis-Philippe.

  The students of the École Polytechnique, the university Galois had hoped to attend, played a significant role in these events, demonstrating on the streets of Paris. And where was the arch antimonarchist Galois during this fateful period? Locked away inside the École Préparatoire along with his fellow students. The Director, Guigniault, had decided to play safe.

  Galois was so incensed at being denied his place in history that he wrote a blistering attack on Guigniault in the Gazette des Écoles:

  The letter which M. Guigniault placed in the lycée yesterday, on the account of one of the articles in your journal, seemed to me most improper. I had thought that you would welcome eagerly any way of exposing this man.

  Here are the facts, which can be vouched for by forty-six students.

  On the morning of 28 July, when several students of the École Normale wanted to join in the struggle, M. Guigniault told them, twice, that he had the power to call the police to restore order in the school. The police on 28 July!

  The same day, M. Guigniault told us with his usual pedantry: “There are many brave men fighting on both sides. If I were a soldier, I would not know which to decide. Which to sacrifice, liberty or LEGITIMACY?”

  There is the man who next day covered his hat with an enormous tricolor cockade [a symbol of the republicans]. There are our liberal doctrines!

  The editor published the letter but removed the author’s name from it. The director promptly expelled Galois for publishing an anonymous letter.

  Galois retaliated by joining the Artillery of the National Guard, a paramilitary organization that was a hotbed of republicanism. On 21 December 1830, this unit, very probably including Galois, was stationed in the vicinity of the Louvre. Four ex-ministers had gone on trial, and the public mood was ugly: they wanted the men executed, and were prepared to riot if they were not. But just before the verdict was announced, the Artillery of the National Guard was withdrawn and replaced by the regular National Guard, together with other soldiers who were loyal to the King. The verdict of a jail sentence was announced, the riot failed to materialize, and ten days later, Louis-Philippe disbanded the Artillery of the National Guard as a security risk. Galois was having no more success as a revolutionary than he had had as a mathematician.

  Practical issues now became more urgent than politics: he needed to make a living. Galois set himself up as a private mathematics tutor, and forty students signed up for a course of advanced algebra. We know that Galois was not a good written expositor, and it’s reasonable to guess that his teaching was no better. Probably his classes were laced with political commentary; almost certainly they were too difficult for ordinary mortals. At any rate, the enrollment rapidly dwindled.

  Galois had still not given up on his mathematical career, and he submitted yet a third version of his work to the Academy, entitled On the Conditions of Solvability of Equations by Radicals. With Cauchy having fled Paris, the referees were Siméon Poisson and Lacroix. When two months passed without any response, Galois wrote to ask what was happening. No one replied.

&nb
sp; By the spring of 1831, Galois was behaving ever more erratically. On 18 April the mathematician Sophie Germain, who had greatly impressed Gauss when she first began her research in 1804, wrote a letter about Galois to Guillaume Libri: “They say he will go completely mad, and I fear this is true.” Never the most stable person, he was now verging on full-blooded paranoia.

  That month, the authorities arrested nineteen members of the Artillery because of the events at the Louvre and put them on trial for sedition, but the jury acquitted the men. The Artillery held a celebration on 9 May in which about two hundred Republicans assembled for a banquet at the restaurant Vendanges des Bourgogne. Every one of them wanted to see Louis-Philippe overthrown. The novelist Alexandre Dumas, who was present, wrote, “It would be difficult to find in all Paris, two hundred persons more hostile to the government than those to be found reunited at five o’clock in the afternoon in the long hall on the ground floor above the garden.” As the event became more and more riotous, Galois was seen with a glass in one hand and a dagger in the other. The participants interpreted this gesture as a threat to the king, approved wholeheartedly, and ended up dancing in the streets.

  The next morning, Galois was arrested at his mother’s house—which suggests that there had been a police spy at the banquet—and charged with threatening the king’s life. For once he seems to have learned some political sense, because at his trial he admitted everything, with one modification: he claimed that he had proposed a toast to Louis-Philippe, and had gestured with the dagger while adding the words, “if he turns traitor.” He lamented that these vital words had been drowned in the uproar.

  Galois made it clear, however, that he did expect Louis-Philippe to betray the people of France. When the prosecutor asked whether the accused could “believe this abandonment of legality on the part of the king,” Galois responded, “He will soon turn traitor if he has not done so already.” Pushed further, he left no doubt as to his meaning: “The trend in government can make one suppose that Louis-Philippe will betray one day if he hasn’t already.” Despite this, the jury acquitted him. Perhaps they felt as he did.