People From Paris

"I can accept no responsibility for any changes in your existence, miraculous or otherwise. You must take responsibility for your own life. You must have your own life because if you haven’t had at least that, what have you had?"

"Calamitous collapse is better than mediocre defeat!"

"Art is like a fart for the soul. Better out than in."

"Women are much more honourable than men."

"For we may remark generally of our mathematical researches, that these auxiliary quantities, these long and difficult calculations into which we are often drawn, are almost always proofs that we have not in the beginning considered the objects themselves so thoroughly and directly as their nature requires, since all is abridged and simplified, as soon as we place ourselves in a right point of view."

"I am the conscience of the 21st Century."

"Never fall for someone with a body to diet for."

"Everybody should be entitled to one lie, one failing, one infidelity."

"Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulations of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?""

"I am not here concerned with intent, but with scientific standards, especially the ability to tell the difference between a fact, an opinion, a hypothesis, and a hole in the ground."

"I don't like the nonsense that passes for rational discourse so often in our society. I am very much bothered by the inaccuracies, ambiguities, code words, slogans, catch phrases, public relation devices, sweeping generalizations, and stereotypes, which are used (consciously or otherwise) to influence people."

"Historically, Jean d’Alembert precedes Augustin-Louis Cauchy. However, in the context of functional equations, it seems more natural to consider his contributions after Cauchy. Jean d’Alembert was a man of many names. The illegitimate son of an army officer, Louis-Camus Destouches, and a writer, Claudine Guérin de Tencin, he was born in Paris in 1717, while his father was abroad. Shortly after his birth, his mother abandoned him at the church of Saint-Jean-le-Rond. Following tradition, he was named Jean le Rond after the church, and placed in an orphanage. Upon the return of his father, he was removed from the orphanage, and placed with Mme. Rousseau, the wife of a glazier. Although Destouches continued to support his son financially, he chose not to publicly acknowledge his son. In 1738, Jean le Rond entered law school, where he was registered under the name Daremberg. He later changed this name to d’Alembert."

"Rien n'est plus incontestable que l'existence de nos sensations; ..."

"D'Alembert was always surrounded by controversy. … he was the lightning rod which drew sparks from all the foes of the philosophes. … Unfortunately he carried this... pugnacity into his scientific research and once he had entered a controversy, he argued his cause with vigour and stubbornness. He closed his mind to the possibility that he might be wrong..."

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined nontentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given a new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920."

"Even before Galois coined the term 'group,' A. L. Cauchy... made (1815) extensive investigations in what are now called s, and discovered some of the simpler basic theorems."

"The earliest discussion from the standpoint of groups of the (modular) equations arising in the division of s was by Galois."

"[A]ny mathematician today must be impressed by the apparent permanence of the ideas introduced by Abel... and Galois.., and the profound difference between their approach to mathematics and that of their predecessors including, in some respects, Gauss... To these young men, perhaps more than to any other two mathematicians, can be traced the pursuit of generality which distinguishes the mathematics of the recent period, beginning with Gauss in 1801, from that of the middle period. They initiated... the deliberate search for inclusive methods and comprehensive theories. Their forerunners in the middle period were Descartes with his general method in geometry; Newton and Leibniz with the differential and integral calculus created to attack the mathematics of continuity by a uniform procedure; and Lagrange, with his universal method in mechanics. Their contemporary in recent mathematics was Gauss, who in his arithmetic sought to unify much of the uncorrelated work of the leading arithmeticians from Fermat to Euler, Lagrange, and Legendre. Both Abel and Galois acknowledged their indebtedness to the theory of cyclotomy created by Gauss; and although they went far beyond him in their own algebra (Abel in analysis also), it is at least conceivable that neither Abel nor Galois would have chosen the road he followed had it not been for the hints in the Gaussian theory of binomial equations."

"Lagrange did not explicitly recognize groups. Nevertheless, he obtained equivalents for some of the simpler properties of s. For example, one of his results, in modern terminology, states that the order of a of a divides the order of the group. Normal (self-conjugate, invariant) subgroups, basic in the theory of algebraic equations and in that of group structure, were introduced by Galois, who also invented the term 'group.'"

"Both Abel and Galois were indebted to Lagrange in their own, profounder work on s. ...The unique importance of Abel’s proof is that it inspired Galois to seek a deeper source of solvability, which he found in the theorem that an is solvable by radicals its group, for the field of its s, is solvable."

"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."

"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."

"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."

"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"."

"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."

"[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."

"[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 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."

"[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."

"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.]"

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

"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.]"

"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.]"

"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."

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

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

"[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."

"Le vrai point d'honneur [d'un scientifique] n'est pas d'être toujours dans le vrai. Il est d'oser, de proposer des idées neuves, et ensuite de les vérifier."

"How to console oneself for having exhausted in one month the greatest source of happiness which is in man — of having exhausted it without happiness, without hope, being certain that one has drained it for life? Oh! come and preach peace after that! Come and ask men who suffer to take pity upon what is! Pity, never! Hatred, that is all. He who does not feel it deeply, this hatred of the present, cannot really have in him the love of the future. ...I like to doubt your cruel prophecy when you say that I shall not work any more. But I admit that it is not without likelihood. To be a savant, I should need to be that alone. My heart has revolted against my head. I do not add as you. do: It is a pity."

"The final lines are not mine: they come from an experiment on soft matter, after Boudin… An English translation might run like this:"

"... un auteur ne nuit jamais tant à ses lecteurs que quand il dissimule une difficulté."

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

"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]"

"A dense film of a conventional surfactant is quite impermeable. On the other hand, a dense film of Janus grains always has some interstices between the grains, and allows for chemical exchange between the two sides; "the skin can breathe". This may possibly be of some practical interest."

"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."

"Congruences were responsible for one theory of far more than merely arithmetical interest. The notation for a congruence suggests the introduction of appropriate 'imaginaries' to supply the congruence with roots equal in number to the degree of the congruence when there is a deficiency of real roots. As in the corresponding algebraic problem, it is not obvious that imaginaries can be introduced consistently. That they can, was first proved in 1830 by Galois, who invented the required 'numbers,' since called Galois imaginaries, for the solution of any irreducible congruence F(x) = 0 mod p , where p is prime. He thus obtained a generalization of Fermat’s theorem, and laid the foundation of the theory of s. ...Galois was eighteen when he invented his imaginaries."

"Benjamin Franklin performed a beautiful experiment using surfactants; on a pond at Clapham Common, he poured a small amount of oleic acid, a natural surfactant which tends to form a dense film at the water-air interface. He measured the volume required to cover all the pond. Knowing the area, he then knew the height of the film, something like three nanometers in our current units. This was to my knowledge the first measurement of the size of molecules. In our days, when we are spoilt with exceedingly complex toys, such as nuclear reactors or synchrotron sources, I particularly like to describe experiments of this Franklin style to my students. Surfactants allow us to protect a water surface, and to generate these beautiful soap bubbles, which are the delight of our children."

"Galois... had nothing of the topologist about him..."

"Galois, who was Lie's idol, indirectly inspired the application of continuous groups to differential equations. In a letter of 1874 to A. Mayer, Lie observes that "In the theory of algebraic equations before Galois only these questions were proposed: Is an equation solvable by radicals, and how is it to be solved? Since Galois, among other questions proposed is this: How is an equation to be solved by radicals in the simplest way possible? I believe the time is come to make a similar progress in differential equations.""