Physicists From Germany

"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."

"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)."

"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..."

"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."

"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."

"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 ..."

"[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."

"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 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 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."

"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 ""..."

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

"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."

"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."

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

"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.]"

"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."

"The members of the Vienna Circle (Moritz Schlick, Rudolf Carnap, , Hans Hahn, , Fritz Waismann, Kurt Godel, Otto Neurath and others) are working out a ‘Logical Empiricism’. Following Mach and Poincare, but above all Russell and Wittgenstein, all the sciences are treated uniformly. Carnap’s Logischer Aufbau der Welt (1928) shows in which direction future systematic work will move. Wittgenstein’s Tractatus Logico- Philosophicus (1921) clarified, among other things, the position of logic and mathematics; besides the statements that make additions to what is meaningful, there are the ‘tautologies’ that show us which transformations are possible within language. By its syntax the language of science excludes anything that is meaningless from the very beginning."

"Schlick ( [Wende] p.8 ) interprets Wittgenstein's position as follows: philosophy "is that activity by which the meaning of propositions is established or discovered" ; it is a question of "what the propositions actually mean. The content, soul, and spirit of science naturally consist in what is ultimately meant by its sentences; the philosophical activity of rendering significant is thus the alpha and omega of all scientific knowledge"."

"The Vienna Circle was a discussion group of philosophically interested specialists who came together in 1923 and from 1925 to 1936 met regularly once a week in an institute of Vienna University. These gatherings were conducted by Moritz Schlick, the physicist and philosopher who was appointed professor of the philosophy of inductive sciences in 1922. Over the years, members included Hans Hahn, Otto Neurath, Philipp Frank, Viktor Kraft, Herbert Feigl, Friedrich Waismann, Rudolf Carnap, Kurt Godel, Karl Menger, Bela Juhos and others. There was no conscious aim of radically revising traditional views on the task and place of philosophy, but the members were on the whole well aware that current findings of research into the foundations of logic, mathematics and the natural sciences had important philosophic consequences. Among subjects for discussion were Wittgenstein's Tractatus, the possibility of reducing all concepts of science to what is directly given in experience, the setting up of a criterion of meaningfulness for non-logical utterances, the character of the basic propositions of empirical science, and the devising of a meta-language for the syntactic analysis of scientific language systems."

"Philosophy is not a system of propositions, and not a science."

"Philosophy... is that activity by which the meaning of propositions is established or discovered. Philosophy elucidates propositions, science verifies them. In the latter we are concerned with the truth of statements, but in the former with what they actually mean."

"The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other ; it is rather a word denoting a kind of conceptual construction, as, e.g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas."

"Grassmann's first publication of his new system was made in 1844 in a book entitled "Die Lineale Ausdehnungslehre Ein Neuer Zweig der Mathematik." His novel and fruitful ideas were however presented in a somewhat abstruse and unusual form, with the result, as the author himself states in the preface to the second edition issued in 1878, that scarcely any notice was taken of the book by Mathematicians. He was finally convinced that it would be necessary to treat the subject in an entirely different manner in order to gain the attention of the mathematical world. Accordingly in 1862 he published "Die Ausdehnungslehre vollständig und in strenger Form bearbeitet," in which the treatment is algebraic... Since that time his great work has been more fully appreciated, but not even yet, in the opinion of the writer, at its real value."

"The exchange theorem... is sometimes called the Steinitz exchange theorem after Ernst Steinitz... The result was first proved Hermann Günther Graßmann..."

"Some of the groundbreaking work in the treatment of n-dimensional geometry—was carried out by Hermann Günther Grassmann. ...Grassmann was responsible for the creation of an abstract science of "spaces," inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as Ausdehnungslehre... Grassmann's suggestion that BA = -AB violates one of the sacrosanct laws of arithmetic... Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions."

"As the great generality of Grassmann's processes—all results being obtained for n-dimensional space—has been one of the main hindrances to the general cultivation of his system, it has been thought best to restrict the discussion to space of two and three dimensions."

"The wonderful and comprehensive system of Multiple Algebra invented by Hermann Grassmann, and called by him the Ausdehnungslehre or Theory of Extension, though long neglected by the mathematicians even of Germany, is at the present time coming to be more and more appreciated and studied. In order that this system, with its intrinsic naturalness, and adaptability to all the purposes of Geometry and Mechanics, should be generally introduced to the knowledge of the coming generation of English-speaking mathematicians, it is very necessary that a text-book should be provided, suitable for use in colleges and universities, through which students may become acquainted with the principles of the subject and its applications."

"I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results."

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept."

"The concept of centroid as sum led me to examine Möbius' Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and... the only point of contact which my new system of analysis had with the one that was already known."

"A work on tidal theory... led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagrange's work."

"It was natural that Grassmann chose to introduce his system, not by means of a paper, but rather by means of a long and complicated book. ...such ideas as Grassmann's form of the scaler (dot) and vector (cross) products... have counterparts in modern vector analysis."

"As I was reading the extract from your paper in the geometric sum and difference... I was struck by the marvelous similarity between your results and those discoveries which I made even as early as 1832... I conceived the first idea of the geometric sum and difference of two or more lines and also of the geometric product of two or three lines in that year (1832). This idea is in all ways identical to that presented in your paper. But since I was for a long time occupied with entirely different pursuits, I could not develop this idea. It was only in 1839 that I was led back to that idea and pursued this geometrical analysis up to the point where it ought to be applicable to all mechanics. It was possible for me to apply this method of analysis to the theory of tides, and in this I was astounded by the simplicity of the calculations resulting from this method."

"From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all."

"The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line. This then was the first step which led to a new branch of mathematics... I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea."

"While I was pursuing the concept of geometrical product, as this idea was established by my father... I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third... and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC.] From this harmony I came to see a whole new area of analysis was opening up which could lead to important results."

"It is clear... that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution."

"Geometry can in no way be viewed... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry."

"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative. Zero can never be a unit."

"One may say without great exaggeration that Grassmann invented linear algebra and, with none at all, that he showed how properly to apply it to geometry. ...He ...anticipated in its most important aspects Peano's treatment of the natural numbers, published 28 years later. ...A feature of Grassmann's work, far in advance of the times, is the tendency towards the use of the implicit definition. ...The definition of a linear space (or vector space) came into mathematics, in the sense of becoming widely known, around 1920, when Hermann Weyl and others published formal definitions. ...Grassmann did not put down a formal definition—again, the language was not available—but there is no doubt that he had the concept."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of descriptive geometry by Monge in 1795. The quaternions of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"[W]e may apply the theorems concerning cyclical processes to all thermo-dynamic machines, and thereby arrive at conclusions... independent of the nature of the processes executed by the several machines."

"The s... may be divided into two classes: those which the atoms of a body exert upon each other... which depend... upon the nature of the body, and those which arise from the foreign influences to which the body may be exposed. According to these two classes of forces... I have divided the work done by heat into interior and exterior work."

"If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat. 1. The energy of the universe is constant. 2. The entropy of the universe tends to a maximum."