Why Beauty is Truth Read online

  On 15 June, Galois was at liberty. Three weeks later, the Academy reported on his memoir. Poisson had found it “incomprehensible.” The report itself said this:

  We have made every effort to understand Galois’s proof. His reasoning is not sufficiently clear, not sufficiently developed, for us to judge its correctness, and we can give no idea of it in this report. The author announces that the proposition which is the special object of this memoir is part of a general theory susceptible of many applications. Perhaps it will transpire that the different parts of a theory are mutually clarifying, are easier to grasp together rather than in isolation. We would then suggest that the author should publish the whole of his work in order to form a definitive opinion. But in the state which the part he has submitted to the Academy now is, we cannot propose to give it approval.

  The most unfortunate feature of this report is that it may well have been entirely fair. As the referees pointed out:

  [The memoir] does not contain, as [its] title promised, the condition of solvability of equations by radicals; indeed, assuming as true M. Galois’s proposition, one could not derive from it any good way of deciding whether a given equation of prime degree is solvable or not by radicals, since one would first have to verify whether this equation is irreducible and next whether any of its roots can be expressed as a rational fraction of two others.

  The final sentence here refers to a beautiful criterion for solvability by radicals of equations of prime degree that was the climax of Galois’s memoir. It is indeed unclear how this test can be applied to any specific equation, because you need to know the roots before the test can be applied. But without a formula, in what sense can you “know” the roots? As Tignol says, “Galois’s theory did not correspond to what was expected; it was too novel to be readily accepted.” The referees wanted some kind of condition on the coefficients that determined solubility; Galois gave them a condition on the roots. The referees’ expectation was unreasonable. No simple criterion based on the coefficients has ever been found, nor is one remotely likely. But hindsight cannot help Galois.

  On 14 July, Bastille Day, Galois and his friend Ernest Duchâtelet were at the head of a Republican demonstration. Galois was wearing the uniform of the disbanded Artillery and carrying a knife, several pistols, and a loaded rifle. It was illegal to wear the uniform, and also to be armed. Both men were arrested on the Pont-Neuf, and Galois was charged with the lesser offense of illegally wearing a uniform. They were sent to the jail at Sainte-Pélagie to await trial.

  While in jail, Duchâtelet drew a picture on the wall of his cell showing the king’s head, labeled as such, lying next to a guillotine. This presumably did not help their cause.

  Duchâtelet stood trial first; then it was Galois’s turn. On 23 October he was tried and convicted; his appeal was turned down on 3 December. By this time he had spent more than four months in jail. Now he was sentenced to another six months. He worked for a while on his mathematics; then in the cholera epidemic of 1832 he was transferred to a hospital and later put on parole. Along with his freedom he experienced his first and only love affair, with a certain “Stéphanie D,” as his doodles identify her.

  From this point on it takes a lot of guesswork to interpret the scanty historical record. For a time, no one knew Stéphanie’s surname or what sort of person she was. This mystery added to her romantic image. Galois wrote her full name on one of his manuscripts, but at some later point he scrawled all over it, rendering it illegible. Forensic work by the historian Carlos Infantozzi, who examined the manuscript very carefully, revealed the lady as Stéphanie-Felicie Poterin du Motel. Her father, Jean-Louis Auguste Poterin du Motel, was resident physician at the Sieur Faultrier, where Galois spent the last few months of his life.

  We don’t know what Jean-Louis thought of the relationship, but it seems unlikely that he approved of a penniless, unemployed, dangerously intense young man with extremist political views and a criminal record paying court to his daughter.

  We do know a little about Stéphanie’s opinions, but only through some scribbled sentences that Galois presumably copied from her letters. There is much mystery surrounding this interlude, which has a crucial bearing on subsequent events. Apparently, Galois was rejected and took it very badly, but the circumstances cannot be determined. Was it all in his mind—an infatuation that was never reciprocated? Did Stéphanie encourage his advances? Did she then get cold feet? The very characteristics likely to repel her father might have been distinctly attractive to the daughter.

  As far as Galois was concerned, the relationship was certainly serious. In May, he wrote to his close friend Chevalier, “How can I console myself when in one month I have exhausted the greatest source of happiness a man can have?” On the back of one of his papers he made fragmentary copies of two letters from Stéphanie. One begins, “Please let us break up this affair,” which suggests that there was something to break up. But it continues, “and do not think about those things which did not exist and which never would have existed,” giving the contrary impression. The other contains the following sentences: “I have followed your advice and I have thought over what . . . has . . . happened . . . In any case, Sir, be assured there never would have been more. You are assuming wrongly and your regrets have no foundation.”

  Whether he imagined the whole thing and his feelings were never reciprocated, or he initially received some form of encouragement only to be subsequently rejected, it looks as though Galois suffered the worst kind of unrequited love. Or was the whole affair perhaps more sinister? Shortly after the breakup with Stéphanie, or what Galois interpreted as a breakup, someone challenged him to a duel. The ostensible reason was that this person objected to Galois’s advances toward the young lady, but yet again the circumstances are veiled in mystery.

  The standard story was one of political intrigue. Writers like Eric Temple Bell and Louis Kollros tell us that Galois’s political opponents found his infatuation with Mlle. du Motel to be the perfect excuse to eliminate their enemy on a trumped-up “affair of honor.” One rather wild suggestion is that Galois was the victim of a police spy.

  These theories now seem implausible. Dumas states in his Memoirs that Galois was killed by Pescheux D’Herbinville, a fellow Republican whom Dumas described as “a charming young man who made silk-paper cartridges which he would tie up with silk ribbons.” These were an early form of cracker, of the kind now familiar at Christmas. D’Herbinville was something of a hero to the peasantry, having been one of the nineteen Republicans acquitted on charges of conspiring to overthrow the government. Certainly he was not a spy for the police, because Marc Caussidière named all such spies in 1848 when he became chief of police.

  The police report on the duel suggests that the other participant was one of Galois’s revolutionary comrades, and the duel was exactly what it appeared to be. This theory is largely borne out by Galois’s own words on the matter: “I beg patriots and my friends not to reproach me for dying otherwise than for my country. I die the victim of an infamous coquette. It is in a miserable brawl that my life is extinguished. Oh! why die for so trivial a thing, for something so despicable! . . . Pardon for those who have killed me, they are of good faith.” Either he was unaware that he was the victim of a political plot, or there was no plot.

  It does appear that Stéphanie was at least a proximate cause of the duel. Before departing for the engagement, Galois left some final doodles on his table. They include the words “Une femme,” with the second word scribbled out. But the ultimate cause is as opaque as much else in this tale.

  The mathematical story is much clearer. On 29 May, the eve of the duel, Galois wrote to Auguste Chevalier, outlining his discoveries. Chevalier eventually published the letter in the Revue Encyclopédique. It sketches the connection between groups and polynomial equations, stating a necessary and sufficient condition for an equation to be solvable by radicals.

  Galois also mentioned his ideas about elliptic functions and the integra
tion of algebraic functions, and other things too cryptic to be identifiable. The scrawled comment “I have no time” in the margins has given rise to another myth: that Galois spent the night before the duel frantically writing out his mathematical discoveries. But that phrase has next to it “(Author’s note),” which hardly fits such a picture; moreover, the letter was an explanatory accompaniment to Galois’s rejected third manuscript, complete with a marginal note added by Poisson.

  The duel was with pistols. The postmortem report states that they were fired at 25 paces, but the truth may have been even nastier. An article from the 4 June 1832 issue of Le Precursor reported:

  Paris, 1 June—A deplorable duel yesterday has deprived the exact sciences of a young man who gave the highest expectations, but whose celebrated precocity was lately overshadowed by his political activities. The young Évariste Galois . . . was fighting with one of his old friends, a young man like himself, like himself a member of the Society of Friends of the People, and who was known to have figured equally in a political trial. It is said that love was the cause of the combat. The pistol was the chosen weapon of the adversaries, but because of their old friendship they could not bear to look at one another and left the decision to blind fate. At point-blank range they were each armed with a pistol and fired. Only one pistol was charged. Galois was pierced through and through by a ball from his opponent; he was taken to the Hospital Cochin where he died in about two hours. His age was 22. L.D., his adversary, is a bit younger.

  Could “L.D.” refer to Pescheux d’Herbinville? Perhaps. The letter D is acceptable because of the variable spelling of the period; the L may have been a mistake. The article is unreliable on details: it gets the date of the duel wrong, and also the day Galois died and his age. So the initial might also be wrong.

  The cosmologist and writer Tony Rothman has a more convincing theory. The person who best fits the description here is not d’Herbinville but Duchâtelet, who was arrested with Galois on the Pont-Neuf. Galois’s biographers Robert Bourgne and Jean-Pierre Azra give Duchâtelet’s Christian name as “Ernest,” but that might be wrong, or again the L may be wrong. To quote Rothman, “We arrive at a very consistent and believable picture of two old friends falling in love with the same girl and deciding the outcome by a gruesome version of Russian roulette.”

  This theory is also consistent with a final horrific twist to the tale. Galois was hit in the stomach, a wound that was almost always fatal. If the duel was at point-blank range, this is no great surprise; if at 25 paces, it is the final example of his cursed luck.

  He did not die two hours later, as Le Precursor says, but in the Hospital Cochin the next day, on 31 May. The cause of death was peritonitis, and he refused the office of a priest. On 2 June 1832 Galois was buried in the common ditch at the cemetery of Montparnasse.

  His letter to Chevalier ended with these words: “Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.”

  But what did Galois actually accomplish? What was the “mess” referred to in his final letter?

  The answer is central to our tale, and not easily stated in a few sentences. Galois introduced a new point of view into mathematics, he changed its content, and he took a necessary but unfamiliar step into abstraction. In Galois’s hands, mathematics ceased to be the study of numbers and shapes—arithmetic, geometry, and ideas that developed out of them like algebra and trigonometry. It became the study of structure. What had been a study of things became a study of processes.

  We should not give Galois all the credit for this transformation. He was riding a wave that had been set in motion by Lagrange, Cauchy, Ruffini, and Abel. But he rode it with such skill that he made it his own; he was the first person seriously to appreciate that mathematical questions could sometimes be best understood by transporting them into a more abstract realm of thought.

  It took a while for the beauty and value of Galois’s results to percolate into the general mathematical consciousness. In fact they were very nearly lost. They were rescued by Joseph-Louis Liouville, the son of a captain in Napoleon’s army who became a professor at the Collège de France. Liouville spoke to the French Academy—the body that had mislaid or rejected Galois’s three memoirs—in the summer of 1843. “I hope to interest the Academy,” he began, “in announcing that among the papers of Évariste Galois I have found a solution, as precise as it is profound, of this beautiful problem: whether or not there exists a solution by radicals . . .”

  If Liouville had not bothered to wade through the luckless revolutionary’s often untidy and confusing manuscripts, and had not devoted considerable time and effort to puzzling out what the author intended, the manuscripts might well have been thrown out with the rubbish, and group theory would have had to await some later rediscovery of the same ideas. So mathematics owes Liouville an enormous debt.

  As understanding of Galois’s methods grew, a new and powerful mathematical concept came into being: that of a group. An entire branch of mathematics, a calculus of symmetry called group theory, came into being and has since invaded every corner of mathematics.

  Galois worked with groups of permutations—ways to rearrange a list of objects. In his case, the objects were the roots of an algebraic equation. The simplest interesting example is a general cubic equation, with three roots a, b, and c. Recall that there are six ways to permute these symbols, and that—following Lagrange and Ruffini—we can multiply any two permutations by performing them in turn. We saw, for example, that cba × bca = acb. Proceeding in this way, we can build up a “multiplication table” for all six permutations. It’s easier to see what’s going on if we assign names to each permutation, say by letting I = abc, R = acb, Q = bac, V = bca, U = cab, and P = cba. Then the multiplication table looks like this:

  Multiplication table for the six permutations of the roots of a cubic equation.

  Here the entry in row X and column Y is the product XY, which means “do Y, then do X.”

  Galois realized that a very simple and obvious feature of this table is crucially important. The product of any two permutations is itself a permutation—the only symbols appearing in the table are I, U, V, P, Q, R. Some smaller collections of permutations have the same “group property”: the product of any two permutations in the collection is also in the collection. Galois called such a collection of permutations a group.

  For example, the collection [I, U, V] gives a smaller table:

  Multiplication table for a subgroup of three permutations.

  and only those three symbols appear. When, as here, one group is part of another, we call it a subgroup.

  Other subgroups, namely [I, P], [I, Q], and [I, R], contain only two permutations. There is also the subgroup [I] that contains only I. It can be proved that the six subgroups just listed are the only subgroups of the group of all permutations on three symbols.

  Now, said Galois (though not in this language), if we choose some cubic equation, we can look at its symmetries—those permutations that preserve all algebraic relations between the roots. Suppose, for example, that a + b2 = 5, an algebraic relation between the roots a and b. Is the permutation R a symmetry? Well, if we check the definition above, R keeps a as it was and swaps b with c, so the condition a + c2 = 5 must also hold. If it doesn’t, R is definitely not a symmetry. If it does, you check any other valid algebraic relations among the roots, and if R passes all these tests, it is a symmetry.

  Working out precisely which permutations are symmetries of a given equation is a difficult technical exercise. But there is one thing we can be sure of without doing any calculations at all. The collection of all symmetries of a given equation must be a subgroup of the group of all permutations of the roots.

  Why? Suppose, for instance, that both P and R preserve all algebraic relations among the roots. If we take some relation and then apply R, we get a vali
d relation. If we then apply P, we again get a valid relation. But applying R and then P is the same as applying PR. So PR is a symmetry. In other words, the collection of symmetries has the group property.

  This straightforward fact underlies the whole of Galois’s work. It tells us that associated with any algebraic equation there is a group, its symmetry group—now called its Galois group to honor the inventor. And the Galois group of an equation is always a subgroup of the group of all permutations of the roots.

  From this key insight a natural line of attack emerges. Understand which subgroups arise in which circumstances. In particular, if the equation can be solved by radicals, then the Galois group of the equation should reflect this fact in its internal structure. Then, given any equation, you just work out its Galois group and check whether it has the required structure, and you know whether it can be solved by radicals.

  Now Galois could recast the whole problem from a different viewpoint. Instead of building a tower with ladders and sacks, he grew a tree.

  Not that he called it a tree, any more than Abel talked of Cardano Tower, but we can picture Galois’s idea as a process that repeatedly branches from a central trunk. The trunk is the Galois group of the equation. The branches, twigs, and leaves are various subgroups.

  Subgroups arise naturally as soon as we start thinking about how the symmetries of equations change when we start taking radicals. How does the group change? Galois showed that if we form a pth root, then the symmetry group must split into p distinct blocks, all the same size. (Here, as Abel noted, we can always assume p to be prime.) So, for example, a group of 15 permutations might split into five groups of 3, or three groups of 5. Crucially, the blocks have to satisfy some very precise conditions; in particular, one of them must form a subgroup in its own right of a special kind known as a “normal subgroup of index p.” We can think of the trunk of the tree splitting into p smaller branches, one of which corresponds to the normal subgroup.