Political Activists

"Both Abel and Galois died long before their time, Abel at the age of twenty-seven from tuberculosis induced by poverty, Galois at twenty-one of a pistol shot received in a meaningless duel. When Abel’s genius was recognized, he was subsidized by friends and the Norwegian government. By nature he was genial and optimistic. Galois spent a considerable part of his five or six productive years in a hopeless fight against the stupidities and malicious jealousy of teachers and the smug indifference of academicians. Not at first quarrelsome or perverse, he became both."

"[D]iscoverers of fundamental principles are not generally awarded much recompense. They often die misunderstood and unrewarded. But... the fame... of Galois ...is based upon the unlimited future. He well knew the pregnancy of his thoughts, yet they were even more far-reaching than he could possibly dream of."

"In his last year at the college, 1828-1829... [his] teacher of mathematics... wrote of him: "This student has a marked superiority over all his school-mates. ...He works only at the highest parts of mathematics." ...[O]ther teachers were less indulgent. For physics and chemistry, the note often repeated was: "Very absent-minded, no work whatever.""

"The simple of any two groups was defined in connection with the postulates for a group. Galois considered simply isomorphic groups as the same group which, abstractly, they are."

"[T]he soul of Galois will burn on throughout the ages and be a perpetual flame of inspiration."

"There are six major episodes to be observed, four of which will be described... The four are the definition by Gauss, E. E. Kummer.., and Dedekind of s; the restoration of the in s by Dedekind’s introduction of ideals; the definitive work of Galois on the solution of algebraic equations by radicals, and the theory of finite groups and the modern theory of fields that followed; the partial application of arithmetical concepts to certain linear algebras by R. Lipschitz.., A. Hurwitz.., L. E. Dickson.., Emmy Noether.., and others. All of these developments are closely interrelated."

"Galois... made the terminal contribution so far as are concerned, and subsequent reworkings of his initial theory have added nothing basically new to his criteria for solvability by radicals. Even the modernized presentation of the , as in the streamlined model of E. Artin... is a tribute to the mathematical creed of Galois, in its elimination of all superfluous machinery. For this modern release from algebraic calculation, the direct approach of A. E. ‘Emmy’ Noether... in the 1920’s was primarily responsible. Much of her mathematics was in the spirit of Galois. But his methods, sharpened and generalized by his successors, have transcended the problem for which they were invented, and have rejuvenated much of living pure mathematics."

"Since my mathematical youth, I have been under the spell of the classical theory of Galois. This charm has forced me to return to it again and again."

"It is... not true... that he proved and used the simplicity of s. He did not need to: he was much cleverer than that; his treatment of solubility of equations is... simpler and more elegant than... textbook tradition."

"The earliest recognitions of fields, but without explicit definition, appear to be in the researches of Abel (1828) and Galois (1830-1) on the solution of equations by radicals. The first formal lectures on the were those of Dedekind to two students in the early 1850's. Kronecker also at that time began his studies on abelian equations. It appears that the concept of a field passed into mathematics through the arithmetical works of Dedekind and Kronecker."

"[C]oncerning the vital residue of the theory of algebraic numbers... it, like the , can be traced to definite, highly special problems. Neither Galois nor the creators of the theory of s set out deliberately to revolutionize a mathematical technique; their comprehensive methods were invented to solve specific problems. Such appears to have been the usual path to abstractness, generality, and increased power. Some difficult problem... is taken as the point of departure without any conscious effort to create a comprehensive theory; repeated failures to achieve a solution by known procedures force the invention of new methods; and finally, the new methods, having been necessitated by a problem which appeared in the historical development, themselves pass into the main stream."

"Langlands and Grothendieck are both (at least) Giants by any measure, and both were consciously successors of Galois."

"[A]t the age of sixteen he believed that he had found a method of solving general equations of the fifth degree. ...[B]efore succeeding in proving the impossibility of such resolution, Abel had made the same mistake."

"Applications, or developments, of... extensions of number followed two main directions. ...The second, in the arithmetical spirit of Gauss, guided in part by the abstract algebraic outlook of Galois, led to a partial but extensive arithmetization of algebra."

"An excessive desire for conciseness was the cause of this fault which one must try to avoid when writing on the mysterious abstractions of pure Algebra. Clarity is indeed an absolute necessity. ... Galois too often neglected this precept."

"The Galois theory of equations itself was the concluding episode in about three centuries of effort to penetrate the arithmetical nature of the roots of algebraic equations."

"Whoever, if anybody, was responsible for the colossal waste represented by these two premature deaths, it seems probable that mathematics was needlessly deprived of the natural successsors of Gauss. What Abel and Galois might have accomplished in a normal lifetime cannot be even conjectured. ...Early maturity and sustained productivity are the rule, not the exception, for the greatest mathematicians. It may be true that the most original ideas come early; but it takes time to work them out."

"It is perhaps less well known that [Galois] had also, without any possible doubt, discovered the essentials of the theory of abelian integrals, as Riemann would develop it 25 years later. By what route did he arrive at these conclusions? The fragments of calculations in Analysis found among his papers do not seem to permit much of an answer to that question, but there is room to imagine that he must have been very close to the idea of the Riemann surface associated with an algebraic function, and that such an idea must also be fundamental in his investigations into what he calls the "théorie de l'ambiguïté"."

"He was still a mere boy, yet within these short years he had accomplished enough to prove indubitably that he was one of the greatest mathematicians of all times."

"Of all these influences, two in particular are germane... the development of ; and the infiltration of the ideas of Abel and Galois into algebra as a whole. The of equations was acknowledged by both Dedekind and Kronecker to be the inspiration for their own general and semi-arithmetical approach to algebra. Two of the basic concepts of the Galois theory, domains of rationality, or fields, and groups, were the point of departure. ...Common algebra is the most familiar example of a field."

"The first formulas for the computation of the s of the roots of an equation seem to have been worked out by Newton, although Girard (1629) had given, without proof, a formula for a power of the sum, and Cardan (1545) had made a slight beginning in the theory. In the 18th century Lagrange (1768) and Waring (1770, 1782) made several valuable contributions to the subject, but the first tables, reaching to the tenth degree, appeared in 1809 in the Meyer Hirsch Aufgabensammlung. In Cauchy's celebrated memoir on determinants (1812) the subject began to assume new prominence, and both he and Gauss (1816) made numerous and important additions to the theory. It is, however, since the discoveries by Galois that the subject has become one of great significance."

"Gauss took as the subject for his doctor’s dissertation (1799) a proof of the : an algebraic equation has a root of the form a + bi, a , b real. ...After Girard’s conjecture, there had been attempts at proof, including essays by D’Alembert (1746) and Euler (1749). All were faulty, as were the first and fourth attempts (1799) by Gauss. ...[T]he fundamental theorem in its classic form, as proved in the theory of functions of a complex variable, is no longer regarded as belonging to algebra. It is supplanted in modern algebra by a statement which is almost a triviality. The basic ideas of the modern treatment go back to Galois.., Dedekind.., and Kronecker.., not to Gauss."

"Nous avons transcrit textuellement la démonstration que nous avons donnée de ce lemme dans un mémoire présenté en 1830. Nous y joignons 154 IV The First Memoir comme document historique le note suivante qu’a cru ^devoir^ y apposer M. Poisson. On jugera. Note de l’auteur [We have faithfully transcribed the proof of this lemma that we have given in a memoir presented in 1830. We append as a historical document the following note which Mr Poisson believed he should add. Posterity will judge. Note by the author]"

"[In the first case, the problem is solving equations, including solving an algebraic equation with an unknown variable. In the second case, the problem is integrating differential equations. Galois was the first to recognize with absolute clarity how extraordinarily important the concepts of substitution group and invariant of a discontinuous group are for dealing with problems of the first kind.] Im ersten Falle hat man das Problem der Auflösung der Gleichungen, unter Anderm der Auflösung einer algebraischen Gleichung mit einer Unbekanuten. Im zweiten Falle hat man das Problem der Integration von Differentialgleichungen. Galois war der erste, der vollkommen klar erkannte, wie ausserordentlich wichtig die Begriffe Substiutionengruppe und Invariante einer discontinuirlichen Gruppe für die Behandlung von Problemen jener ersten Art sind."

"The modern in general is commonly said to date from Abel and Galois. The latter's posthumous (1846) memoir... established the theory... To him is due the discovery that to each equation there corresponds a group of substitutions (the "group of the equation") in which are reflected its essential characteristics. Galois' early death left without sufficient demonstration several important propositions, a gap which has since been filled."

"Three new approaches to number, in 1801 and in the 1830's, were to hint at the general concept of mathematical structure and reveal unsuspected horizons... That of 1801 was the concept of congruence, introduced by Gauss [age 24] in... the Disquisitiones arithmeticae... To this and the revolutionary work (1830-2) of E. Galois... in the theory of algebraic equations can be traced the partial execution of L. Kronecker’s... revolutionary program in the 1880's for basing all mathematics on the s."

"It angered Galois sufficiently that he wrote directly below it:"

"It is painful to think that a few rays of generosity from the heart of his elders might have saved this boy or... sweetened his life."

"Galois... accomplished his task and... few men will... accomplish more. He has conquered the purest kind of immortality. As he wrote to his friends: "I take with me to the grave a conscience free from lie, free from patriot's blood". How many of the conventional heroes of history, how many of the kings, captains and statesmen could say the same?"

"Poisson, reading Galois' First Memoir, found the proof of Lemma III insufficient, and wrote in pencil the following comment."

"We must... indicate the involuntary participation of E. Galois... and N. H. Abel... in the development of Kronecker’s Pythagoreanism. Galois... adhered to no such creed. Nor did Abel. But it was in the attempt to understand and elucidate the of equations, left (1832)... in a... fragmentary and unapproachable condition, that Kronecker acquired some of his skill. Both Kronecker and Dedekind, two of the founders (E. E. Kummer... being a third) of the theory of algebraic numbers, were inspired partly by their scrutiny of the Galois theory to begin their... revolutionary work in algebra and arithmetic. ...Galois and Abel mark the beginning of one modern approach to algebra. The transition from highly finished individual theorems to abstract and widely inclusive theories..."

"La démonstration de ce lemme n’est pas suffisante; mais il est vrai d’après le No . 100 du mémoire de Lagrange, Berlin, 1771. [The proof of this lemma is insufficient; but it is true according to No . 100 of the memoir by Lagrange, Berlin, 1771.]"

"[We will not dwell further on the theory of transformation groups of a linear equation. We believe we have sufficiently demonstrated... the value of this theory, which is simply the natural extension to a question of analysis of the fruitful ideas introduced into algebra by Galois] Nous n'insisterons pas davantage sur la théorie des groupes de transformations d'une équation linéaire. Nous pensons avoir suffisamment montré... l'intérèt de cette théorie, qui n'est que l'extension bien naturelle à une question d'Analyse des idées si fécondes introduites en Algèbre par Galois."

"Although the Greek use of s in the case of greatest common measure was well known in the Middle Ages, the modern theory of the subject may be said to have begun with Bombelli (1572). ...The first great memoir on the subject was Euler's De fractionibus continuis (1737), and in this work the foundation for the modern theory was laid. Among other interesting cases Euler developed e as a continued fraction, thus: e = 2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{1+...}}}}} Of the later contributors to the theory, special mention should be made of Lagrange (1767) and Galois."

"Il parait après cela qu'il n'y a aucun fruit à tirer de la solution que nous proposons. [It seems there is no fruit to be drawn from the solution we offer.]"

"Four days later two men challenged him to a duel. ...According to Evariste's younger brother the duel was not fair. Evariste, weak as he was, had to deal with two ruffians hired to murder him."

"The last letter addressed to... Auguste Chevalier, was a... scientific testament. Its seven pages, hastily written... contain a summary of the discoveries which he had been unable to develop. This statement is so concise and... full that its significance could be understood only gradually... It proves the depth of his insight, for it anticipates discoveries of a much later date. At the end of the letter, after requesting his friend to publish it and to ask Jacobi or Gauss to pronounce upon it, he added: "After that, I hope some people will find it profitable to unravel this mess. Je t'embrasse avec effusion." ...[T]he greatest mathematicians of the century have found it very profitable ...to clear up Galois' ideas."

"Preserve my memory, since fate has not given me life enough for the country to know my name."

"[This] science is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it."

"[G]enius possessing... a mere boy, a fragile little body divided within itself by disproportionate forces, an undeveloped mind crushed mercilessly between the exaltation of scientific discovery and the exaltation of sentiment."

"The duel took place on the 30th in the early morning, and he was grievously wounded by a shot in the abdomen. He was found by a peasant who transported him at 9:30 to the Hôpital Cochin. His younger brother... came and stayed with him, and as he was crying, Evariste tried to console him, saying: "Do not cry. I need all my courage to die at twenty." ...[H]e refused the assistance of a priest. ...[H]e breathed his last ...the following morning."

"Dès le commencement de ce siècle, l'algorithme avait atteint un degré de complication tel que tout progrès était devenu impossible par ce moyen, sans l'élégance que les géomètres modernes ont su imprimer à leurs recherches et au moyen de laquelle l'esprit saisit promptement et d'un seul coup un grand nombre d'opérations. Il est évident que l'élégance si vantée et à si juste titre n'a pas d'autre but. Du fait bien constaté que les efforts des géomètres les plus avancés ont pour objet l'élégance on peut donc conclure avec certitude qu'il devient de plus en plus nécessaire d'embrasser plusieurs opérations à la fois, parce que l'esprit n'a plus le temps de s'arrêter aux détails. ... Sauter à pieds joints sur les calculs, grouper les opérations, les classer suivant leurs difficultés et non suivant leurs formes; telle est, suivant moi, la mission des géomètres futurs; telle est la voie où je suis entré dans cet ouvrage."

"Galois's introduction of imaginary roots of congruences has not only led to an important extension of the theory of numbers, but has given rise to wide generalizations of theorems which had been obtained in subjects like linear congruence groups by applying the ordinary theory of numbers."

"Galois was already trying to... enter the ... as early as 1828 — but failed. This failure was very bitter to him... he considered it as unfair... [but] his extra knowledge could not compensate for his deficiencies... The next year he published his first paper, and sent his first communication to the Academie des Sciences... lost through Cauchy's negligence. This embittered Galois even more. A second failure to enter Polytechnique seemed to... climax... his misfortune..."

"When genius... explodes suddenly, at the beginning and not at the end of life, or when we are at a loss to explain its... genesis, we can but feel that we are in the sacred presence of something vastly superior to talent. ...[I]t is not necessary to introduce any mystical idea, but it is one's duty to acknowledge the mystery. ...Galois' fateful existence helps one to understand Lowell's saying: "Talent is that which is in a man's power, genius is that in whose power man is." If Galois had been simply a mathematician of considerable ability, his life would have been far less tragic, for he could have used his mathematical talent for his own advancement and happiness; instead... the furor of mathematics—as one of his teachers said—possessed him and he had no alternative but absolute surrender to his destiny."

"Younger than Gauss by thirty-four years, and dying twenty-three years before him, Galois... seems more modern... Gauss terminated his investigations on the nature of the solutions of with the binomial equationGalois grasped and solved (1830) the general problem, proving, among other things, necessary and sufficient conditions for the solution by radicals of any algebraic equation. Mathematics after Gauss, and partly during his own lifetime, became more general and more abstract... Interest in special problems sharply declined if there was a general problem including the special instances to be attacked... [i.e.,] mathematics after Gauss turned to the construction of inclusive theories and general methods which, theoretically... implied... detailed solutions of infinities of special problems. In this sense Galois was more modern..."

"Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !"

"Mais je n’ai pas le temps et mes idées ne sont pas encore bien développées sur ce terrain qui est immense. [But I don't have the time and my ideas are not yet well developed on this immense terrain.]"

"He was now a prisoner on parole and took advantage... to carry on an intrigue with a woman... probably not... reputable ("une coquette de bas etage," says Raspail) ."

"Gauss, the penniless son of a day laborer, was educated by society as represented by the Duke of Brunswick. Today he would be educated at public expense... An Abel, no doubt, would be sent by the municipal health authorities to a sanitarium, where he might recover. A Galois... would find himself at outs with respectability, or in the protective custody of the police on some trumped-up charge... or in a concentration camp. For there is but little evidence that teachers are less helpless in the disturbing presence of a mind of the very highest intelligence than they were... or that the guardians of law and order are less nervous than they were when they sentenced Galois to... jail on a legal technicality. Aesop’s fable of the peacock and the crows has an element of permanence... you are different from us; get out or be plucked."