Inventors

"[A]bstract algebra, as a conscious discipline, starts with Noether's 1921 paper "Ideal Theory in Rings.""

"One of the major problems of algebra as it is practiced in today's schools is the lack of mathematical, pedagogical, and psychological connection between these two kinds of algebra—between the pre- and post-Noether views of the subject."

"Her thesis ends with a table of the complete system of covariant forms for a given ternary quartic consisting of not less than 331 forms in symbolic representation. It is an awe-inspiring piece of work; but today I am afraid we should be inclined to rank it among those achievements with regard to which Gordan himself once said when asked about the use of the theory of invariants: "Oh, it is very useful indeed; one can write many theses about it.""

"The computation of algebraic invariants did not end with Hilbert's work. Emmy Noether... did a doctoral thesis in 1907 "On Complete Systems of Invariants for Ternary Biquadratic Forms." She also gave a complete system of covariant forms for a ternary quartic, 331 in all. In 1910 she extended Gordan's result to n variables. The subsequent history of algebraic invariant theory belongs to modern abstract algebra. ...From 1911 to 1919 Emmy Noether produced many papers on finite bases for various cases using Hilbert's technique and her own. In the subsequent twentieth-century development the abstract algebraic viewpoint dominated. As complained in his text on invariant theory, there was lack of concern for specific problems and only abstract methods were pursued."

"The theory of rings and ideals was put on a more systematic and axiomatic basis by Emmy Noether, one of the few great women mathematicians... Many results on rings and ideals were already known... but by properly formulating the abstract notions she was able to subsume these results under the abstract theory. Thus she reexpressed Hilbert's basic theorem... as follows: A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis. In this reforumulation she made the theory of invariants a part of abstract algebra."

"The first "modern" text in algebra, van der Waerden's Modern Algebra, which appeared in 1931, was heavily influenced by Emmy Noether. It is an enlightening exercise to compare this work with algebra books of just a few decades earlier to see the profound influence that she had on our present conception of algebra. Nevertheless, even Noether realized that one needs to be familiar with a wide variety of concrete examples from all parts of mathematics before one can understand the value of the generalizations she was able to make."

"Her strength lay in her ability to operate abstractly with concepts. It was not necessary for her to allow herself to be led to new results on the leading strings of known concrete examples. ...[S]he was sometimes but incompletely cognizant of the specific details of the more interesting applications of her general theories. She possessed a most vivid imagination, with the aid of which she could visualize remote connections; she constantly strove toward unification. In this she sought out the essentials in the known facts, brought them into order by means of appropriate general concepts, espied the vantage point from which the whole could best be surveyed, cleansed the object under consideration of superfluous dross, and thereby won through to so simple and distinct a form that the venture into new territory could be undertaken with the greatest prospect of success. ...She possessed a strong drive toward axiomatic purity. All should be accomplished within the frame and with the aid of the intrinsic properties of the structure under investigation; nothing should be brought from without, and only invariant processes should be applied. ...This can be carried too far, however ..."

"[I]t surely is not much of an exaggeration to call her the mother of modern algebra."

"Another change in the formulation of basic combinatorial properties, made... 1923 to 1930 by a number of men and possibly suggested by Emmy Noether, was to recast the theory of chains, cycles, and bounding cycles into the language of group theory."

"It is queer that a formalist like Gordan was the mathematician from whom her mathematical orbit set out; a greater contrast is hardly imaginable than between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics that abhorred all calculation and operated in a much thinner air of abstraction than Hilbert, the young lion, ever dared."

"In the judgement of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day generation of younger mathematicians."

"demonstrates that wherever there is symmetry in nature, there is also a conservation law, and vice versa. In other words, the symmetries of space and time are not only linked with conservation of energy, momentum, and angular momentum, but each implies the other. Conservation laws are necessary consequences of symmetries, and symmetries necessarily entail conservation laws. The simplicity, power, and depth of Noether's theorem only slowly became apparent. Today, it is an indispensable part of the groundwork of modern physics... [with] over a dozen important conservation laws and their associated symmetries..."

"A keen mind and infectious enthusiasm for mathematical research made Emmy Noether an effective teacher. Her classroom technique, like her thinking, was strongly conceptual. Rather than simply lecturing, she conducted discussion sessions in which she would explore a topic with her students. ...Outstanding mathematicians often make their greatest contributions early in their careers. Emmy Noether was an exception: she began to produce her most powerful and creative work around the age of 40. ...She never attained the top rank of full professor, although she contributed so much to making Göttingen the premier mathematical center in Europe—many would say in the world. When the Nazis seized power in 1932, one of their first acts was to deprive non-Aryan[s]... of their positions. ...For a time Emmy Noether continued to meet informally with students and colleagues, inviting groups to her apartment... In the meantime, efforts were being made on her behalf... and she secured a temporary position at , a new college for women near Philadelphia."

"s were old acquaintances from classical physics. ... asserts that any continuous symmetry leads to a conservation law. It is rather intuitive... After all, symmetry reflects invariance under a transformation, and therefore there must exist a quantity that remains invariant or, in other words, that is conserved. For instance, a circle is invariant under rotations about its centre. ...Hence, the symmetry of a circle is associated with the conservation of distance ...The power of Noether's theorem was to show that this intuitive concept is valid for any continuous symmetry ...from Noether's theorem we discover that the conservation of electric charge is the consequence of the special rotational symmetry of QED... [acting upon] an abstract space defined by the quantum fields."

"With the appearance of Einstein's general theory of relativity, Hilbert turned to that subject, which also occupied his colleague Felix Klein. Interestingly, the most lasting mathematical contribution out of this effort came from an algebraist who had recently engaged in studies of differential invariants. This was Emmy Noether... the daughter of the algebraic geometer , whom Hilbert and Klein brought to Göttingen to assist them in research. Her results were published in 1918; best known as ""..."

"Following ['s] work, Emmy Noether, in 1921, transferred s for ideals in algebraic number fields to those for ideals in arbitrary rings. ...Noether and her students made other major contributions to ring theory before she turned to a treatment of finite group representations from an ideal-theoretic point of view. ...Chain conditions had been used since the days of Hölder and Dedekind but were brought to the fore in the 1921 paper [above]. Through Noether's influence... algebraic notions were linked to topology in the work of and ..."

"I do not see that the sex of the candidate is an argument against her admission as Privatdozent [teaching assistant]. After all, we are a university and not a bathing establishment."

"She continually advised her students to read and re-read Dedekind's works, in which she saw an inexhaustible source of inspiration. When praised for her own innovations, she used to repeat: "Es steht alles schon bei Dedekind.""

"The third great epoch in the extension of arithmetic is that of the twentieth century after 1910. To anticipate, the introduction of general methods into , beginning in the first decade of the twentieth century, prepared that vast field of mathematics, first opened up by Hamilton and Grassman in the 1840s, for partial arithmetization in the second and third decades of the century. In 1910, E. Steinitz... proceeding from, and partly generalizing, Kronecker's theory (1881) of "algebraic magnitudes," made a fundamental contribution to the modern theory of (commutative) fields. His work was one of the strongest impulses to the abstract algebra of the 1920s and 1930s, with its accompanying generalized arithmetic. The outstanding figure in the later phase of this development is usually considered to be Emmy Noether... who, with her numerous pupils, laid down the broad foundations of the modern abstract theory of ideals, also a great deal more in the domain of modern algebra. The application of this work to the 'integers' of linear s affords the ultimate extension up to 1940 of common arithmetic."

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined non-tentatively 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 and the theory of s 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 new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920."

"The third and last exception to general sterility connects the arithmetic of forms with that other major outgrowth of ancient diophantine analysis, the Gaussian concept of congruence. Dickson in 1907 began the congruencial theory of forms, in which the coefficients of the forms are either natural integers reduced modulo p, p prime, or elements of a Galois field. The linear transformations in the theory, corresponding to those in the classical problem of equivalence, were similarly reduced, and hence modular invariants and covariants were defineable. By 1923 the theory was practically worked out, except for two central difficulties, by Dickson and his pupils. Simplified derivations for some of the results were given (1926) by E. Noether by an application of her methods in abstract algebra."

"Dedekind's concern with algebra goes back to the 1850s, when he attended Dirichlet's lectures on number theory... and pursued intensive studies of . ...[H]e developed an abstract treatment of elementary group theory at that time. After Dirichlet's death, Dedekind was charged with publishing Dirichlet's lectures on number theory. In appendices he presented... his ideal theory... The most axiomatic approach [1894]... was the one that especially influenced Emmy Noether and her school of algebraists in the 1920s."

"The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her—in published papers, in lectures, and in personal influence on her contemporaries."

"Emmy Noether herself was... warm like a loaf of bread. There irradiated from her a broad, comforting, vital warmth."

"Emmy Noether introduced the notion of a representation space— a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformations reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition. Her point of view stresses the essential fact about a simple algebra, namely, that it has only one type of irreducible space and that it is faithfully represented by its operation on this space. 's statement that the simple algebra is a total matrix algebra over a quasifield is now more understandable. It simply means that all transformations of this space which are linear with respect to a certain quasifield are produced by the algebra. This treatment of algebras may be found in 's '. Recently it has been discovered that this last described treatment of simple algebras is capable of generalization to a far wider class of rings."

"My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously."

"Ich habe das symbolische Rechnen mit Stumpf und Stil verlernt. I have completely forgotten the symbolic calculus."

"If one proves the equality of two numbers a and b by showing first that a \leqq b and then that a \geqq b, it is unfair; one should instead show that they are really equal by disclosing the inner ground for their equality."

"Wissenschaftliche Anregung verdanke ich wesentlich dem persönlichen mathematischen Verkehr in Erlangen und in Göttingen. Vor allem bin ich Herrn E. Fischer zu Dank verpflichtet, der mir den entscheidenden Anstoẞ zu der Beschäftigung mit abstrakter Algebra in arithmetischer Auffassung gab, was für all meine späteren Arbeiten bestimmend blieb. I obtained scientific guidance and stimulation mainly through personal mathematical contacts in Erlangen and in Göttingen. Above all I am indebted to Mr. E. Fischer from whom I received the decisive impulse to study abstract algebra from an arithmetical viewpoint, and this remained the governing idea for all my later work."

"A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis."

"Es steht alles schon bei Dedekind. [It is already all in Dedekind.]"

"[Noether] taught us to think in terms of simple and general algebraic concepts—homomorphic mappings, groups and rings with operators, ideals—and not in cumbersome algebraic computations; and she thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some complicated special situation."

"Her dependence on Gordan did not last long; he was important as a starting point, but was not of lasting scientific influence... Gordan retired in 1910; he was followed first by , and the next year by Ernst Fischer. Fischer’s field was algebra again, in particular the theory of elimination and of invariants. He exerted upon Emmy Noether, I believe, a more penetrating influence than Gordan did. Under his direction the transition from Gordan’s formal standpoint to the Hilbert method of approach was accomplished. She refers in her papers at this time again and again to conversations with Fischer. This epoch extends until about 1919."

"During the war, in 1916, Emmy came to Göttingen for good; it was due to Hilbert’s and Klein’s direct influence that she stayed. Hilbert at that time was over head and ears in the general theory of relativity, and for Klein, too... [S]he was able to help them with her invariant theoretic knowledge. For two of the most significant sides of the general relativity theory she gave at that time the genuine and universal mathematical formulation: First, the reduction of the problem of differential invariants to a purely algebraic one by use of "normal coordinates"; second, the identities between the left sides of Euler's equations of a problem of variation which occur when the (multiple) integral is invariant with respect to a group of transformations involving arbitrary functions (identities that contain the conservation theorem of energy and momentum in the case of invariance with respect to arbitrary transformation of the four world coordinates)."

"I wish there was a knob on the TV so you could turn up the intelligence. They got one marked “brightness” but it don’t work, does it?"

"I find television to be very educating. Every time somebody turns on the set, I go in the other room and read a book."

"They say that ninety percent of TV is junk. But, ninety percent of everything is junk."

"Phil saw television as a marvelous teaching tool. There would be no excuse of illiteracy. Parents could learn along with their children. News and sporting events could be seen as they were happening. Symphonies would mean more when one could see the musicians as they played, and movies would be seen in our own living rooms. He said there would be a time when we would be able to see and learn about people in other lands. If we understood them better, differences could be settled around conference tables, without going to war."

"If it weren’t for Philo T. Farnsworth, inventor of television, we’d still be eating frozen radio dinners."

"The damned thing works! (telegram, on the first successful television broadcast)"

"There’s nothing on it worthwhile, and we’re not going to watch it in this household, and I don’t want it in your intellectual diet. (to his son, on television)"

"This has made it all worthwhile. (The live televised first step by Neil Armstrong on the moon.)"

"I was fortunate that my parents were bright, loving people who did everything they could to help me. I’m glad I was able to play baseball and soccer. I’m gratified that I was a smart boy who went to MIT. But looking back, I know there was anger and dislocation. I’m glad I’ve been able to overcome that, as many other immigrant kids have as well. But some haven’t. Some have dysfunctional and destructive families. Some are bitter. Some are blamers. Some are just crazy – call them sociopaths if you prefer — and their alienation has brought out the worst in them. Instead of filing a lawsuit when they felt wronged, as I have done, they bought guns. This is the nation we live in."

"Thirty-four thousand, four hundred and forty-eight total images: of which seventeen-thousand three-hundred and thirty-two ... three hundred and TWENTY two were duplicates, from seventeen-thouand one-hundred and twenty-six unique voters. This by the way, we'll get to it, was not reported in the report."

"Dr. Shiva’s team found a sudden surge in duplicate ballots between 11-04-2020 and 11-09-2020. There was no mention of duplicates in the Maricopa Canvass Report. 17,126 voters sent in two or more ballots (duplicates)."

"Adam set the bar so high for portraying the role of Batman, He was wonderful, spot on, with a twinkle in his eye. He had it all -- looks, charm, intelligence, I could go on and on. In conversation, he was very animated and once told me that Batman was the father that everyone wanted! I never thought of it that way! He had a great way of playing that 'tongue-in-cheek' nature in so much of the dialogue. I had a long, engaging conversation with him and his wife, Marcelle, about their life in Montana, If I had to describe him in a word or two, they would be 'stellar' and 'exemplar,' qualities that we want to encourage in ourselves and in young people."

"My kind of female power can’t be owned. I may feign enslavement, but I never let myself be unloved, even by myself, that would not do. I rarely argue - I consider. I wait till enough information is in and then the answer, answers itself. My imagination does tolerate empty space. Female power does not disagree with a compliment. Compliments shall be relished, like rich chocolate, like an inside caress. I make time to feel it, all the way up and down my body. Let the world please you. Use your female power. It was set up that way."

"The doctor's aim is to do good, even to our enemies, so much more to our friends, and my profession forbids us to do harm to our kindred, as it is instituted for the benefit and welfare of the human race, and God imposed on physicians the oath not to compose mortiferous remedies."

"I prayed to God to direct and lead me to the truth in writing this book. It grieves me to oppose and criticize the man Galen from whose sea of knowledge I have drawn much. Indeed, he is the Master and I am the disciple. Although this reverence and appreciation will and should not prevent me from doubting, as I did, what is erroneous in his theories. I imagine and feel deeply in my heart that Galen has chosen me to undertake this task, and if he were alive, he would have congratulated me on what I am doing. I say this because Galen's aim was to seek and find the truth and bring light out of darkness. I wish indeed he were alive to read what I have published."

"... In short, while I am writing the present book, I have written so far around 200 books and articles on different aspects of science, philosophy, theology, and hekmat(wisdom). ... I never entered the service of any king as a military man or a man of office, and if I ever did have a conversation with a king, it never went beyond my medical responsibility and advice. ... Those who have seen me know that I did not [go] into excess with eating, drinking or acting the wrong way. As to my interest in science, people know perfectly well and must have witnessed how I have devoted all my life to science since my youth. My patience and diligence in the pursuit of science has been such that on one special issue specifically I have written 20,000 pages (in small print), moreover I spent fifteen years of my life—night and day—writing the big collection entitled Al Hawi.It was during this time that I lost my eyesight, my hand became paralyzed, with the result that I am now deprived of reading and writing. Nonetheless, I've never given up, but kept on reading and writing with the help of others. I could make concessions with my opponents and admit some shortcomings, but I am most curious what they have to say about my scientific achievement. If they consider my approach incorrect, they could present their views and state their points clearly, so that I may study them, and if I determined their views to be right, I would admit it. However, if I disagreed, I would discuss the matter to prove my standpoint. If this is not the case, and they merely disagree with my approach and way of life, I would appreciate they only use my written knowledge and stop interfering with my behavior."