Evariste Galois: A Study in Genius
ASSOCIATE PROFESSOR OF MATHEMATICS
PERHAPS the central purpose of our new undergraduate reading room in the Bradley Mathematics Center is to make available some of the evidence for the human side of mathematics. The impressions one gets from the formal course work at Dartmouth lead all too often to the popular conclusion that mathematics is a towering alabaster monument, ruthlessly eternal, erected to the memory of Truth by the hand of Reason. But mathematics is intensely human; it is the product of human passions, and is shaped by the human drama. Never to see this is never to understand its true character.
I should like to illustrate this thesis by recounting for you the story of Evariste Galois.* Galois was a young French mathematician of the early 19th century who carved into the granite pages of history a truly extraordinary legend. He, among all mathematicians, was perhaps the most nearly perfect expression of the spirit of genius, and his legend surely the most compelling romance of all.
The term genius should always be used with caution. Let us agree here to use it only to describe a truly extraordinary creative intellectual power which succeeds in breaking new ground in man's understanding of his universe. Under this definition, there are no more than a handful of men worthy of the term in each century. And Galois was one of them.
The facts are simple, and briefly described. Evariste Galois was born on-October 25, 1811, in the little village of Bourg-la-Reine, a suburb of Paris. His father was a relic of the 18th century, cultivated, intellectual, saturated with philosophy, a passionate hater of tyranny and lover of liberty. His father was later elected mayor of the town, and later still fell victim to its political forces. Evariste's mother came from a long line of distinguished jurists, and brought with her a thorough education in both Christianity and the classics, fused with a virile stoicism in an independent mind. She was generous to a fault, and original to a turn. She was to educate Evariste for the first eleven years of his life, and to provide for his subsequent years whatever strengths he had.
For this story I have drawn freely on the writings of E.T. Bell (Men of Mathematics,Simon and Schuster, 1937) and George Sarton (Evariste Galois, Osiris, Vol. 3, 1937, pp. 241-259).
Let me pause to set the scene. Some eighteen years before, Madame Defarge had sat knitting her fateful tapestry while the tumbrels rolled through the streets of Paris, the knife fell, and the heads of the proudest aristocracy in all history dropped from their shoulders. Soon victim, judge, and jury all gave way to a squat Corsican, another genius, who stamped his legend with cavalry boots into the face of Europe. By 1811 the career of this same Corsican had reached its zenith; and the next year he was to begin his last great adventure. By 1815 a badly battered crown was restored to the remnants of the royal family for another fifteen years. In 1819, when Evariste was only eight, the Supreme Court of a new nation across the sea decided the case which determined that Dartmouth would be the college that she is, instead of the university she might be.
For his first eleven years Galois lived at home under the sheltering hand of his mother. But in his twelfth year he entered the Lycee of Louis-le-Grand in Paris. The place was then a dismal horror. Barred and grilled, cold and gloomy, staffed by pedants and time-servers, and organized like a workhouse, it might have done justice to Oliver Twist. It was Galois's first school, and nearly his last.
We can trace his progress in Louis-le-Grand by quoting from his report cards. At first all went well, thanks to his mother's training, but as time passed and the dull, tedious, pedantic stupidities and injustices began to accumulate, and his attempts at independent thought and expression were rigorously suppressed, Galois grew bored and disgusted, and simply turned off. Listen to his teachers:
"This pupil, though a little queer in his manners, is very gentle and seems filled with innocence and good qualities. He never knows a lesson badly; either he has not learned it at all, or he knows it well."
"This pupil, except for the last fort- night during which he has worked a little, has done his class work only from fear of punishment. ... The queerness of his character keeps him aloof from his companions."
"His facility, in which one is supposed to believe but of which I have not yet witnessed a single proof, will lead him no-where: there is no trace in his tasks of anything but of queerness and negligence."
"Very bad conduct. . . . Does absolutely nothing for the class. The furor of mathematics possesses him. He is losing his time here and does nothing but torment his masters and get himself harassed with punishments."
This lamentable record documents the decline and fall of Galois the student. During this period he studied Latin and Greek, Rhetoric, and Mathematics, such as it was. His classwork was indifferent, or worse. But somehow he came across the writings of Lagrange, and then of Legendre, Gauss, and Abel, and he read them avidly, like novels. Somehow he discovered the fascination of real mathematics, and his genius, once aroused, was roaring to be let loose. And neither Galois nor his teachers knew how to cope with it.
By now it was 1828, and he was 16. His single compelling goal, important above all others, was to get out of Louis-le-Grand and into the famous Ecole Polytechnique, where contemporary mathematics was understood and appreciated, and contemporary politics flourished. He applied for entrance by examination, and failed. No doubt his preparation was spotty, and he tended to work everything in his head. But his failure is hardly a patch on that of his judges, who failed to see his tremendous talent. Galois was crushed. Here was a bitter blow to his fondest hopes.
Fortunately, at this point Galois fell at last into the hands of a teacher at Louis-le-Grand who could appreciate, however dimly, the kind of mind before him. Professor Richard knew mathe- matics, trained mathematicians, and encouraged Galois to study, think, and write. Galois was now seventeen and entering his greatest year. He mastered the works of his contemporaries, and laid the foundations for the monumental discoveries which bear his name. He wrote several original papers for publication or presentation before the Academy of Sciences. And he opened the doors upon whole new vistas of mathematics which are still under study today.
Let me try to sketch the highlights of his work. Galois addressed himself to the general problem of solving algebraic equations. Solutions for various special cases were already known. Formulas for the solutions of equations of the second degree had been found by Arabian mathematicians in the middle ages, and for those of the third and fourth degree by the Italians in the fifteenth century. Concentrated and determined efforts by leading mathematicians since then had failed to find a formula for the solutions of equations of the fifth degree, and finally in 1826 young Abel succeeded in proving that no such formula could ever be found. Galois read the work of Abel, and he proceeded to investigate the general case. And here he discovered the key. He found that the relevant properties of the solutions of any algebraic equation of any degree could be determined from the symmetries of the number system which they generate. He learned to calculate these symmetries, and from them to decide which equations can be solved by formulas and which cannot. He learned how to obtain the formulas when they exist, and what could be done when they do not. And his methods possess a generality, lucidity, and elegance almost unrivalled in the highest reaches of human thought.
We owe to Galois, first of all, our present understanding of algebraic equations and their solutions. Moreover, many other problems of diverse kinds can be rephrased in terms of algebraic equations. Among them are the three puzzles of classical geometry: the problems of the trisection of an angle, the squaring of a circle, and the duplication of a cube. Galois' methods showed, after a search of more than two thousand years, that the general solutions of these three problems cannot be constructed by the classical methods of ruler and compass alone.
We also owe to Galois our appreciation for the value of the study of abstract structures, and particularly the study of symmetries. Here is a very powerful technique, completely modern in spirit, which today finds applications all along the frontiers of mathematics, and the physical sciences as well. And finally, we owe to Galois our understanding of the limitations of our tools. We now know that there are equations whose solutions cannot be obtained by formulas. Similarly, we now know that there are statements whose truth cannot be decided by logic. This last fact, one of the monuments of the twentieth century, proved in 1930 by Kurt Godel, is generically descended from the first. In a very real sense Galois was the first mathematician of our times.
THUS did a nineteenth century French schoolboy turn the course of human thought. Surely now he would find the recognition he needed so badly! But Galois' guardian angel slept on, and formidable forces gathered in opposition. His first paper was published in March of 1829, but his major efforts, one after the other, were submitted to the Academy of Sciences, and unaccountably lost. He applied again for entrance to the Ecole Polytechnique, and again he failed. Men not fit to sharpen his pencils sat in judgment over him and turned him down. At this point his father, hounded beyond endurance by his political enemies, committed suicide. Perhaps this was the last straw. Such an accumulation of events might well have broken a strong man. Galois, driven by his demon, alienated from his peers, scorned by his masters, rejected by his establishment, his best efforts wasted, his career blighted, his father dead, sickened, embittered and desperate, took the only course open to him - he joined the New Left.
By now he had managed to graduate from the Lycee, and had entered the Ecole Normal to prepare for a career in teaching. But now he abandoned all other interests and flung himself into the politics of the coming revolution, goaded by the evidences of social injustice he saw on all sides. "Genius," he said, "is condemned by a malicious social organization to an eternal denial of justice in favor of fawning mediocrity." He was not alone. That summer his impatient countrymen had had enough of Bourbon stupidity at last, and began ripping up the paving stones of the Parisian streets to build the barricades of 1830. Galois was overjoyed. When the director prudently locked in the would-be participants behind the high walls of the Ecole Normal, Galois led the rebellion, and published a scathing rebuke, more notable for its language than its reason, in the Gazette desScales. For this effort, of course, he was expelled.
The rest is almost inevitable. He first hired out as a private tutor in advanced mathematics, but could find no students. He then joined a radical wing of the National Guard, and devoted all his energies to revolutionary politics. In May of 1831 he was arrested for insulting the new King, jailed, tried, and acquitted. In July he was rearrested on general principles, jailed without charge, and held for two months. He was then convicted of wearing an illegal uniform and held for six more. In March of 1832 he was paroled, and finally set free in May.
Four days later he was stopped on the street, provoked, and challenged to a duel. Accounts vary as to the cause. Some say a woman ("quelque coquettede bas étage") whom he had met earlier on parole was involved. Others maintain that the quarrel was purely political. In any case Galois did not seek it, and tried every other means of accommodation without success.
Fearing that he was doomed, Galois sat up all that night composing an outline and summary of his mathematical discoveries. In the fleeting hours he sketched his scientific testament, writing against time a record of his principal results and their possible extensions. Again and again he broke off to scribble in the margin "I have not time, I have not time," before passing on to the next frantically scrawled outline. What he wrote in those last desperate hours before dawn will keep generations of mathematicians busy for hundreds of years.
At an early hour the next morning, Galois met his adversary and was shot in the abdomen. He was taken to Cochin Hospital where he lingered through the day but died the next morning. "Preserve my memory," he had written, "since fate has not given me life enough for my country to know my name." He was not yet twenty-one.
Prof. Reese T. Prosser
PROFESSOR PROSSER'S ARTICLE is based on the talk he gave January 9 on the occasion of the dedication of the Mirkil Reading Room for undergraduates in the Bradley Mathematics Center.
