Murdered People

"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..."

"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..."

"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."

"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 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."

"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."

"[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."

"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."

"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."

"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."

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

"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."

"[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."

"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."

"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."

"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."

"Probably almost anyone who has ever seriously attempted to solve differential equations by the will appreciate the labor inherent in any such heroic project as Wilczynski’s and agree with Galois that, whatever the nature of its unchallenged merits, the theory of groups does not afford a practicable method for solving equations. Galois of course was speaking of s, but his opinion, in the judgment of experts in the Lie theory, carries over to differential equations. Beyond a not very advanced stage of complexity, the calculations become prohibitive to even the most persevering obstinacy."

"[A]s noted by Lie in the grand summary (1893) of his lifework, the nineteenth century’s greatest effort in formal algebra—as distinguished from the more abstract, structural algebra originating with Galois—was absorbed in analysis."

"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.""

"Just as the algebraic theory characterizes the nature of the irrationalities required for the solution of a given algebraic equation, so does the structural theory of differential equations characterize and classify the functions defined by a system of differential equations. ...[t]he initial impulse for a structural theory came from Lie's transformation groups. In spite of Picard's deprecatory estimate of his own contribution, which, historically, inaugurated the project, as "only a very natural extension to an analytic problem of the extremely fruitful ideas introduced into algebra by Galois," the problem of devising a structural theory for differential equations was no facile exercise in principles already classic."

"Although... structural theories of a major division of analysis originated in the late nineteenth century, they are more in the spirit of the general analysis of the twentieth. Their primary objectives are to discover what can be done rather than to do it, and to give criteria for what cannot be done. ...[A]s in Abel’s proof that the general quintic is not solvable by radicals, a demonstration of impossibility definitely disposed of what might seem a reasonable problem. Once more the methodology of Abel and Galois made an outstanding contribution to the development of mathematics. In this connection it is interesting to recall Lie’s opinion that the pattern of nineteenth century mathematics was laid out by four men, Gauss, Cauchy, Abel, and Galois."

"With this highly abstract definition of a space in mind, we return once more to Klein’s program and its successors. It is interesting to observe the abstract identity between the following description of spacial structure and structure as described in connection with modern algebra, and further to note once more that the basic concepts originated with Galois. Two spaces are called equivalent or (simply) isomorphic if there is a one-one correspondence between the objects in the spaces which establishes a one-one correspondence between all the properties constituting the structures of the respective spaces. When this is applied to two spaces which are the same, there is thus defined what is called an of the space. It follows... from these definitions that all the automorphisms of a given space form a group."

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

"Another white man's trick! Let me go! Let me die fighting!!!!"

"Thus died one of the ablest and truest American Indians. His life was ideal; his record clean. He was never involved in any of the numerous massacres on the trail, but was a leader in practically every open fight. Such characters as those of Crazy Horse and Chief Joseph are not easily found among so-called civilized people. The reputation of great men is apt to be shadowed by questionable motives and policies, but here are two pure patriots, as worthy of honor as any who ever breathed God’s air in the wide spaces of a new world."

"“Another white man’s trick! Let me go! Let me die fighting!” cried Crazy Horse. He stopped and tried to free himself and draw his knife, but both arms were held fast by Little Big Man and the officer. While he struggled thus, a soldier thrust him through with his bayonet from behind. The wound was mortal, and he died in the course of that night, his old father singing the death song over him and afterward carrying away the body, which they said must not be further polluted by the touch of a white man. They hid it somewhere in the Bad Lands, his resting place to this day."

"When he reached the military camp, Little Big Man walked arm-in-arm with him, and his cousin and friend, Touch-the-Cloud, was just in advance. After they passed the sentinel, an officer approached them and walked on his other side. He was unarmed but for the knife which is carried for ordinary uses by women as well as men. Unsuspectingly he walked toward the guardhouse, when Touch-the-Cloud suddenly turned back exclaiming: “Cousin, they will put you in prison!”"

"The captain urged him to report at army headquarters to explain himself and correct false rumors, and on his giving consent, furnished him with a wagon and escort... He went of his own accord, either suspecting no treachery or determined to defy it."

"Under these circumstances Crazy Horse again showed his masterful spirit by holding these young men in check. He said to them in his quiet way: “It is well to be brave in the field of battle; it is cowardly to display bravery against one’s own tribesmen. These scouts have been compelled to do what they did; they are no better than servants of the white officers. I came here on a peaceful errand.”"

"His wife was critically ill at the time, and he decided to take her to her parents at Spotted Tail agency, whereupon his enemies circulated the story that he had fled, and a party of scouts was sent after him. They overtook him riding with his wife and one other but did not undertake to arrest him, and after he had left the sick woman with her people he went to call on Captain Lea, the agent for the Brules, accompanied by all the warriors of the Minneconwoju band. This volunteer escort made an imposing appearance on horseback, shouting and singing, and in the words of Captain Lea himself and the missionary, the Reverend Mr. Cleveland, the situation was extremely critical. Indeed, the scouts who had followed Crazy Horse from Red Cloud agency were advised not to show themselves, as some of the warriors had urged that they be taken out and horsewhipped publicly."

"At this juncture General Crook proclaimed Spotted Tail, who had rendered much valuable service to the army, head chief of the Sioux, which was resented by many. The attention paid Crazy Horse was offensive to Spotted Tail and the Indian scouts, who planned a conspiracy against him. They reported to General Crook that the young chief would murder him at the next council, and stampede the Sioux into another war. He was urged not to attend the council and did not, but sent another officer to represent him. Meanwhile the friends of Crazy Horse discovered the plot and told him of it. His reply was, “Only cowards are murderers.”"

"For some time he held out, but the rapid disappearance of the buffalo, their only means of support, probably weighed with him more than any other influence. In July, 1877, he was finally prevailed upon to come in to Fort Robinson, Nebraska, with several thousand Indians...on the distinct understanding that the government would hear and adjust their grievances."

"What drew me to those two people was that both of them wanted peace and insisted on working toward peace. Both of them believed in life. Nijinski believed that his body was god, and it was. Crazy Horse knew that god was in everything, and it was. They had a knowledge we all need to have."

"Crazy Horse dreamed and went into the world where there is nothing but the spirits of all things. That is the real world that is behind this one, and everything we see here is something like a shadow from that one. He was on his horse in that world, and the horse and himself on it and the trees and the grass and the stones and everything were made of spirit, and nothing was hard, and everything seemed to float. His horse was standing still there, and yet it danced around like a horse made only of shadow, and that is how he got his name, which does not mean that his horse was crazy or wild, but that in his vision it danced around in that queer way. It was this vision that gave him his great power, for when he went into a fight, he had only to think of that world to be in it again, so that he could go through anything and not be hurt. Until he was killed at the Soldiers' Town on White River, he was wounded only twice, once by accident and both times by some one of his own people when he was not expecting trouble and was not thinking; never by an enemy. He was fifteen years old when he was wounded by accident; and the other time was when he was a young man and another man was jealous of him because the man's wife liked Crazy Horse. They used to say that he carried a sacred stone with him, like one he had seen in some vision, and that when he was in danger, the stone always got heavy and protected him somehow. That, they used to say, was the reason that no horse he ever rode lasted very long. I do not know about this; maybe people only thought it; but it is a fact that he never kept one horse long. They wore out. I think it was only the power of his great vision that made him great."

"Upon suffering beyond suffering; the Red Nation shall rise again and it shall be a blessing for a sick world. A world filled with broken promises, selfishness and separations. A world longing for light again. I see a time of seven generations when all the colors of mankind will gather under the sacred Tree of Life and the whole Earth will become one circle again. In that day there will be those among the Lakota who will carry knowledge and understanding of unity among all living things, and the young white ones will come to those of my people and ask for this wisdom. I salute the light within your eyes where the whole universe dwells. For when you are at that center within you and I am that place within me, we shall be as one."

"Hokahey! Today is a good day to die."

"A very great vision is needed, and the man who has it must follow it as the eagle seeks the deepest blue of the sky."

"My friend, I do not blame you for this. Had I listened to you this trouble would not have happened to me. I was not hostile to the white men. Sometimes my young men would attack the Indians who were their enemies and took their ponies. They did it in return. We had buffalo for food, and their hides for clothing and for our tepees. We preferred hunting to a life of idleness on the reservation, where we were driven against our will. At times we did not get enough to eat and we were not allowed to leave the reservation to hunt. We preferred our own way of living. We were no expense to the government. All we wanted was peace and to be left alone. Soldiers were sent out in the winter, they destroyed our villages. The "Long Hair" [Custer] came in the same way. They say we massacred him, but he would have done the same thing to us had we not defended ourselves and fought to the last. Our first impulse was to escape with our squaws and papooses, but we were so hemmed in that we had to fight. After that I went up on the Tongue River with a few of my people and lived in peace. But the government would not let me alone. Finally, I came back to the Red Cloud Agency. Yet, I was not allowed to remain quiet. I was tired of fighting. I went to the Spotted Tail Agency and asked that chief and his agent to let me live there in peace. I came here with the agent [Lee] to talk with the Big White Chief but was not given a chance. They tried to confine me. I tried to escape, and a soldier ran his bayonet into me. I have spoken."

"One does not sell the earth upon which the people walk."

"My lands are where my dead lie buried."

"It's so long ago now. Even I don't know the truth. If I had ever known it, I have long forgotten it. All I know is that I took the blame as a good CO should have been and was punished accordingly."

"History is always written by the victor and histories of the vanquished belong to a shrinking circle of those who were there."

"When seeing today the defendants on the dock, don’t believe them to be the old Combat Group Peiper. All my old friends and comrades have gone before! These people who plead for mitigating circumstances are only the negative selection! The real outfit is waiting for me in Valhalla!"

"My boys may charge me with all they want. The main thing is, it helps them. They are not evil and no criminals. They are the products of total war, grown up on the streets of scattered towns without any education! The only thing [they] knew was to handle weapons for the Dream of [the] Reich. They were young people with a hot heart and the desire to win or to die, according to the word: right or wrong—my country!"

"I admit willingly that after the Normandy battles, my unit was composed of young fanatic soldiers. Many of them had lost their parents, or brothers and sisters in the bombardments. Some had seen for themselves at Cologne where thousands of bodies were crushed after the terrorist raids. Their hatred of the enemy was such that I admit that I could not always control them. At Malmédy, there were, no doubt, some excesses."

"I was a Nazi and I remain one...The Germany of today is no longer a great nation, it has become a province of Europe."