19th-century-german-mathematicians

"I was amazed at how this outstanding connoisseur of Indian knowledge in the field of exact sciences showed himself to be so captive to Cantor’s authority."

"For Egypt, in my opinion, the one fact of experience was sufficient that there was a right-angled triangle 4, 3, 5. Then Pythagoras took the matter geometrically in hand and proved the theorem named after him by ascending in ancient Greek manner from individual case to individual case, finally to the quite general theorem. (Cantor 1905: 70)."

"[…] should it […] boil down to the fact that individual parts of the Sulbasutras are relatively modern interpolations?"

"Therefore, we do not doubt for a moment that Pythagoras’ stay in Egypt, that the lessons he received from the priests there, were among the things that were common truth."

"So much appears certain to us, that Pythagoras could have been in Babylon."

"Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)"

"Pythagoras remarked, we think, that 9+16=25. When he made this under all circumstances interesting remark, he already knew, no matter from which source, the experienced fact that a right angle is constructed by assuming the measure numbers 3, 4, 5 for the lengths of the two legs and for the distance of the end points of the same. We have pointed out that the Egyptians and the Babylonians perhaps possessed the same knowledge that the Chinese certainly had (Cantor 1880: 153)"

"Pythagoras, so we tried to prove, certainly acquired mathematical knowledge in Egypt, perhaps in the Euphrates countries. The former can be seen from the explicit traditions, as well as from the Egyptian character of some geometrical developments, the latter from the Babylonian seeming number differences of the Pythagoreans. The sum of geometrical knowledge, which was made accessible by Pythagoras and his school to the Greeks before the year 400, is not quite a minor one [...] It included the Pythagorean theorem […]. (Cantor 1880: 158)"

"It is hardly conceivable that these writers [Aryabhata, Brahmagupta, Bhaskaracharya etc.], should have been unacquainted with the Greek sources, which were available to Varahamihira [the earliest of these writers]; it is even less conceivable that they [the Indian writers], acquainted with the same, did not let themselves be influenced by them [the Greek sources]. Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works."

"But instead of Cantor admitting the independence or at least the relative independence of the Indians, i.e. the independence of their geometry from the Greeks, he expresses himself extremely tortuous, even he did not give up his Heron hypothesis by hiding it behind the doubtful question at the end, whether there are not at the end in the Sulbasutras relatively modern insertions."

"Cantor concludes that Indian geometry and Greek geometry, especially of Heron, are related; and the only question is, Who borrowed from whom? He expresses the opinion that the Indians were, in geometry, the pupils of the Greeks."

"The first writer to attribute this proposition to Pythagoras is Vitruvius, hardly a reliable witness. From then on, this account became more widespread, but always in connection with the famous hecatomb that Pythagoras is said to have offered in celebration of discovering the proposition—an anecdote that severely undermines the credibility of the entire story. This sacrifice is incompatible with the strict prohibition of all bloody sacrifices, which writers of the same period, indeed often the very same ones who elsewhere recount the hecatomb, have handed down to us from the Pythagorean ritual laws. Cicero himself took offense at this anecdote, and in the latest Neopythagorean tradition, the bloody sacrifice is replaced by that of an ox formed from flour. For this reason, the hecatomb is not only incongruous with the Pythagoreans, but also with the Pythagoreans themselves. Proclus, an insightful writer, expresses himself remarkably vaguely: ‘When we listen to those who want to tell old stories, we find that they trace this theorem back to Pythagoras.’ As this shows, he too was unaware of any reliable source."

"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 ..."

"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.""

"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..."

"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."

"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."

"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."

"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."

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

"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."

"Group theory is the mathematical language of symmetry, and it... seems to play a fundamental role in the very structure of nature. ...In the midst of the fomenting of the new twentieth century physics was the... life of the greatest female mathematician who ever lived, Emmy Noether. ...At Göttingen, Noether achieved fame for her research into the fundamental structure of mathematics. However, she stepped briefly into the realm of theoretical physics... is a profound statement, perhaps running as deeply into the fabric of our psyche as the famous theorem of Pythagoras. Noether's theorem directly connects symmetry to physics, and vice versa. It frames our modern concepts about nature and rules modern scientific methodology. ...For scientists it is the guiding light to unraveling nature's mysteries, as they delve into the innermost fabric of matter ...To this task scientists apply ...the great s ...Emmy Noether's work interweaves our understanding of nature—through physics and mathematics—with the beauty and harmony that surrounds us... Noether's theorem provides a natural centerpiece for any discussion that unifies physics and mathematics, such as in the teaching of these... in a way that enlivens them both."

"She started by examining continuous symmetries. These are symmetries under transformations that can be varied continuously, such as rotations (where the angle can be changed continuously). The result... was stunning. She showed that to every continuous symmetry of the laws of physics there corresponds a conservative law and vice versa. In particular, the familiar symmetry of the laws under translations corresponds to conservation of momentum, the symmetry with respect to the passage of time (the fact that the laws do not change with time) gives us , and the symmetry under rotations produces conservation of angular momentum. ... fused together symmetries and conservation laws—these two giant pillars of physics are actually nothing but different facets of the same fundamental property."

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

"In crediting Emmy Noether with her share in this transformation of mathematics, most biographers have followed Hermann Weyl's analysis... noting that it falls in three periods, of which the first, lasting until about 1919, was one of "relative dependence," whereas the other two were characterized by the algebraic work for which she is remembered. ...[D]ifficulties arise in drawing a sharp distinction between... "relatively dependent" and the rest, however. One can find examples of originality in her early work, and many instances of dependence in her later period... the exclusion of "dependent" work from consideration makes it impossible to study any process of conceptual change. ...The work that was most influential was done when she was in her forties; The "Noether school" of those who collaborated with her in attempting to make algebra the tool and foundation of all mathematics consists of individuals who knew her only in the last decades of her life. In short, her historic influence in effecting conceptual change is based on the events in the last decade of her life. Her stature as a creative mathematician is better understood if we examine her mathematical career in its entirety, however. Only then can we appreciate to what extent Emmy Noether's work fits Poincare's famous description of mathematical creativity..."

"In clarifying conservation law issues for the coupled matter-field systems of relativistic gravitation, Emmy Noether helped David Hilbert, Felix Klein, and Albert Einstein put the finishing touches on the general theory of relativity in 1915. ...Because of the central role of conservation laws, one could argue that Noether's Theorem offers a strategic unifying principle for most if not all of physics."

"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."

"We may assume that Emmy Noether studied, like Weyl, all of Hilbert's papers, at least those which were concerned with algebra or arithmetic. In particular she would have read the paper ["Über die Theorie der algebraischen Formen" (1890)] where Hilbert proved that every ideal in a polynomial ring is finitely generated; in her famous later paper ["Idealtheorie in Ringbereichen" (1921)] she considered arbitrary rings with this property, which today are called "s". ...Hilbert's ' too was... studied; it was the standard text which every young mathematician of that time read... to learn algebraic number theory. ...Steinitz' great paper "Algebraische Theorie der Körper"...marks the start of abstract field theory... [and] is often mentioned in her later publications, as the basis for her abstract viewpoint of algebra."

"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."

"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."

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

"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)."

"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 ..."

"Emmy Noether's creative power was directed quite generally towards the clarification of mathematical structures and concepts through abstraction, which means leaving all unnecessary entities and properties aside and concentrating on the essentials. Her basic work in this direction can be subsumed under algebra, but her methods eventually penetrated all mathematical fields, including number theory and topology."

"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."

"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."

"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."

"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."