With deep dismay Emmy Noether’s friends living in America learned about her sudden passing away on Sunday, April 14. She seemed to have got well over an operation for tumor; we thought her to be on the way to convalescence when an unexpected complication led her suddenly on downward the path to her death within a few hours. She was such a paragon of vitality, she stood on the earth so firm and healthy with a certain sturdy humor and courage for life, that nobody was prepared for this eventuality. She was at the summit of her mathematical creative power; her far-reaching imagination and her technical abilities accumulated by continued experience, had come to a perfect balance; she had eagerly set to work on new problems. And now suddenly — the end, her voice silenced, her work abruptly broken off.
Down, down, down into the darkness of the grave
Gently they go, the beautiful, the tender, the kind;
Quietly they go, the intelligent, the witty, the brave.
I know. But I do not approve. And I am not resigned.
A mood of defiance similar to that expressed in this “Dirge without music” by Edna St. Vincent Millay, mingles with our mourning in the present hour when we are gathered to commemorate our friend, her life and work and personality.
I am not able to tell much about the outward story of her life; far from her home and those places where she lived and worked in the continuity of decades, the necessary information could not be secured. She was born the 23rd of March, 1882, in the small South German university town of Erlangen. Her father was, himself a great mathematician who played an important rôle in the development of the theory of algebraic functions as the chief representative of the algebraic-geometric school. He had come to the University of Erlangen as a professor of mathematics in 1875, and stayed there until his death in 1921. Besides Emmy there grew up in the house her brother Fritz, younger by two and a half years. He turned to applied mathematics in later years, was until recently professor at the Technische Hochschule in Breslau, and by the same fate that ended Emmy’s career in Göttingen is now driven off to the Research Institute for Mathematics and Mechanics and in Tomsk, Siberia. The Noether family is a striking example of the hereditary nature of the mathematical talent, the most shining illustration of which is the Basle Huguenot dynasty of the Bernoullis.
Side by side with Noether in Erlangen was his close and intimate friend, also a mathematician, who had come to Erlangen shortly before, in 1874, and he, too, remained associated with that university until his death in 1912. Emmy wrote her doctor’s thesis under him in 1907. Besides her father, Gordan must have been well-nigh one of the most familiar figures in Emmy’s early life, first as a friend of the house, later as a mathematician also; she kept a profound reverence for him though her own mathematical taste soon developed in quite a different direction. I remember that his picture decorated the wall of her study in Göttingen. These two men, the father and Gordan, determined the atmosphere in which she grew up. The father was — such as the impression I gather from his papers and even more from the many obituary biographies he wrote for the Mathematische Annalen — a very intelligent, warm-hearted, harmonious man of many-sided interests and sterling education. This scientific kinship of father and daughter — who became in a certain sense his successor in algebra, but stands beside him independent in her fundamental attitude and in her problems — is something extremely beautiful and gratifying.
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, the young lion, ever dared.
It is not quite easy to evoke before an American audience a true picture of that state of German life in which Emmy Noether grew up in Erlangen; maybe the present generation in Germany is still more remote from it. The great stability of burgher life was in her case accentuated by the fact that Noether (and Gordan too) were settled at one university for so long as uninterrupted period. One may dare to add that the time of the primary proper impulses of their production was gone, though they undoubtedly continued to be productive mathematicians; in this regard, too, the atmosphere around her was certainly tinged by a quiet uniformity. Moreover, there belongs to the picture the high standing, and the great solidity in the recognition of spiritual values; based on a solid education, a deep and genuine active interest in the higher achievements of intellectual culture, and on a well-developed faculty of enjoying them. There must have prevailed in the Noether home a particularly warm and companionable family life. Emmy Noether herself was, if I may say so, warm like a loaf of bread. There irradiated from her a broad, comforting, vital warmth. Our generation accuses that time of lacking all moral sincerity, of hiding behind its comfort and bourgeois peacefulness, and of ignoring the profound creative and terrible forces that really shape man’s destiny; moreover of shutting its eyes to the contrast between the spirit of true Christianity which was confessed, and the private and public life as it was actually lived. Nietzsche arose in Germany as a great awakener. It is hardly possible to exaggerate the significance which Nietzsche (whom by the way Noether once met in the Engadin) had in Germany for the thorough change in the moral and mental atmosphere. I think he was fundamentally right — and yet one should not deny that in wide circles in Germany, as with the Noethers, the esteem in which the spiritual goods were held, the intellectual culture, good-heartedness, and human warmth were thoroughly genuine — notwithstanding their sentimentality, their Wagnerianism, and their plush sofas.
Emmy Noether, as a young girl, took part in the housework, dusted and cooked and went to dances, and it seems her life would have been that of an ordinary woman had it not happened that just about that time it became possible in Germany for a girl to enter on a scientific career without meeting any too marked resistance. There was nothing rebellious in her nature; she was willing to accept conditions as they were. But now she became a mathematician. Her dependence on Gordan did not last long; he was important as a starting point, but was not of lasting scientific influence upon her. Nevertheless the Erlangen mathematical air may have been responsible for making her into an algebraist. Gordan retired in 1910; he was followed first by, and the next year by . 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.
Already in Erlangen about 1913 Emmy lectured occasionally, substituting for her father when he was taken ill. She must have been to Göttingen about that time, too, but I suppose only on a visit with her brother Fritz. At least I remember him much better than her from my time as a Göttinger Privatdozent, 1910–1913. During the war, in 1916, Emmy came to Göttingen for good; it was due toand direct influence that she stayed. Hilbert at that time was over head and ears in the general theory of relativity, and for Klein, too, the theory of relativity and its connection with his old ideas of the Erlangen program brought the last flareup of his mathematical interests and mathematical production. The second volume of his history of mathematics in the nineteenth century bears witness thereof. To both Hilbert and Klein Emmy was welcome as she 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.
Still during the war, Hilbert tried to push through Emmy Noether’s “Habilitation” in the Philosophical Faculty in Göttingen. He failed due to the resistance of the philologists and historians. It is a well-known anecdote that Hilbert supported her application by declaring at the faculty meeting, “I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university and not a bathing establishment.” Probably he provoked the adversaries even more by that remark. Nevertheless, she was able to give lectures in Göttingen, that were announced under Hilbert’s name. But in 1919, after the end of the War and the proclamation of the German Republic had changed the conditions, her Habilitation became possible. In 1922 there followed her nomination as a “nicht-beamteter ausserordentlicher Professor”; this was a mere title carrying no obligations and no salary. She was, however, entrusted with a “Lehrauftrag” for algebra, which carried a modest remuneration. During the wild times after the Revolution of 1918, she did not keep aloof from the political excitement, she sided more or less with the Social Democrats; without being actually in party life she participated intensely in the discussion of the political and social problems of the day. One of her first pupils, Grete Hermann, belonged to Nelson’s philosophic-political circle in Göttingen. It is hardly imaginable nowadays how willing the young generation in Germany was at that time for a fresh start, to try to build up Germany, Europe society in general, on the foundations of reason, humaneness and justice. But alas! the mood among the academic youth soon enough veered around; in the struggles that shook Germany during the following years and which took on the form of civil war here and there, we find them mostly on the side of the reactionary and nationalistic forces. In later years Emmy Noether took no part in matters political. She always remained, however, a convinced pacifist, a stand which she held very important and serious.
In the modest position of a “nicht-beamteter ausserordentlicher Professor” she worked in Göttingen until 1933, during the last years in the beautiful new Mathematical Institute that had risen in Göttingen chiefly byenergy and the generous financial help of the Rockefeller Foundation. I have a vivid recollection of her when I was in Göttingen as visiting professor in the winter semester of 1926–1927, and lectured on representations of continuous groups. She was in the audience; for just at that time the hypercomplex number systems and their representations had caught her interest and I remember many discussions when I walked home after the lectures, with her and , who was in Göttingen as a Rockefeller Fellow, through the cold, dirty, rain-wet streets of Göttingen. When I was called permanently to Göttingen in 1930, I earnestly tried to obtain from the Ministerium a better position for her because I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as mathematician in many respects. I did not succeed, nor did an attempt to push through her election as a member of the Göttinger Gesellschaft der Wissenschaften. Tradition, prejudice, external considerations, weighted the balance against her scientific merits and scientific greatness, by that time denied by no one. In my Göttingen years, 1930–1933, she was without doubt the strongest center of mathematical activity there, considering both the fertility of her scientific research program and her influence upon a large circle of pupils.
Her development into that great independent master whom we admire today was relatively slow. Such a later maturing is a rare phenomenon in mathematics; in most cases the great creative impulses lie in early youth. [e1] which seems to mark the decisive turning point. It is here for the first time that Emmy Noether appears whom we all know, and who changed the face of algebra by her work., like Emmy Noether, is one of the few great exceptions. Not until 1920, thirteen years after her promotion, appeared in the Mathematische Zeitschrift that paper of her written with , “Über Moduln in nicht-kommutativen Bereichen, insbesondere aus Differential-und Differenzen-Ausdrüken,”
Not less characteristic for Emmy was her collaboration with another, in this case with Schmeidler. I suppose that Schmeidler gave as much as he received in this coöperation. In later years, however, Emmy Noether frequently acted as the true originator; she was most generous in sharing her ideas with others. She had many pupils, and one of the chief methods of her research was to expound her ideas in a still unfinished state in lectures, and then discuss them with her pupils. Sometimes she lectured on the same subject one semester after another, the whole subject taking on a better ordered and more unified shape every time, and gaining of course in the substance of results. It is obvious that this method sometimes put enormous demands upon her audience. In general, her lecturing was certainly not good in technical respects. For that she was too erratic and she cared too little for a nice and well arranged form. And yet she was an inspired teacher; he who was capable of adjusting himself entirely to her, could learn very much from her. Her significance for algebra cannot be read entirely from her own papers; she had great stimulating power and many of her suggestions took final shape only in the works of her pupils or co-workers. And one cannot read the scope of her accomplishments from the individual results of her papers alone: she originated above all a new and epoch-making style of thinking in algebra.
She lived in close communion with her pupils; she loved them, and took interest in their personal affairs. They formed a somewhat noisy and stormy family, “the Noether boys” as we called them in Göttingen.
In the spring of 1933 the storm of the National Revolution broke over Germany. The Göttinger Mathematisch-Naturwissenschaftliche Fakultät, for the building up and consolidation of which Klein and Hilbert had worked for decades, was struck at its roots. After an interregnum of one day by Neugebauer, I had to take over the direction of the Mathematical Institute. But Emmy Noether, as well as many others, was prohibited from participation in all academic activities, and finally her venia legendi, as well as her “Lehrauftrag” and the salary going with it, were withdrawn. A stormy time of struggle like this one we spent in Göttingen in the summer of 1933 draws people closer together; thus I have got a particularly vivid recollection of these months. Emmy Noether, her courage, her frankness, her unconcern about her own fate, her conciliatory spirit, were, in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace. It was attempted, of course, to influence the Ministerium and other responsible and irresponsible but powerful bodies so that her position might be saved. I suppose there could hardly have been in any other case such a pile of enthusiastic testimonials filed with the Ministerium as was sent in on her behalf. At that time we really fought; there was still hope left that the worst could be warded off. It was in vain. Franck, Born, Courant, Landau, Emmy Noether, Neugebauer, Bernays and others — scholars the university had before been proud of — had to go because the possibility of working was taken away from them. Göttingen scattered into the four winds! This fate brought Emmy Noether to Bryn Mawr, and the short time she taught here and as guest at our Institute for Advanced Study in Princeton is still too fresh in our memory to need to be spoken of. She harbored no grudge against Göttingen and her fatherland for what they had done to her. She broke no friendship on account of political dissension. Even last summer she returned to Göttingen, and lived and worked there as though all things were as before. She was sincerely glad that Hasse was endeavoring with success to rebuild the old, honorable and proud mathematical tradition of Göttingen even in the changed political circumstances. But she had adjusted herself with perfect ease to her new American surroundings, and her girl students here were as near to her heart as the Noether boys had been in Göttingen. She was happy at Bryn Mawr; and indeed perhaps never before in her life had she received so many signs of respect, sympathy, friendship, as were bestowed upon her during her last one and a half years at Bryn Mawr. Now we stand at her grave.
It shall not be forgotten what America did during these last two stressful years for Emmy Noether and for German science in general.
If this sketch of her life is to be followed by a short synopsis of her work and her human and scientific personality, I must attempt to draw in a few strokes the scene of her work: the world of algebra.
Emmy Noether’s scientific production seems to me to fall into three clearly distinct epochs: (1) the period of relative dependence, 1907–1919; (2) the investigations grouped around the general theory of ideals, 1920–26; (3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics, from 1927 on. The first epoch was described in the sketch of her life. I should now like to say a few words about the second epoch, the epoch of the general theory of ideals.
I must forego giving a picture of the content of these profound investigations. Instead, I had better try to close with a short general estimate of Emmy Noether as a mathematician and as a personality.
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. This had the disadvantage, however, that she 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.
Emmy Noether was a zealous collaborator in the editing of the Mathematische Annalen. That this work was never explicitly recognized may have caused her some pain.
It was only too easy for those who met her for the first time, or had no feeling for her creative power, to consider her queer and to make fun at her expense. She was heavy of build and loud of voice, and it was often not easy for one to get the floor in competition with her. She preached mightily, and not as the scribes. She was a rough and simple soul, but her heart was in the right place. Her frankness was never offensive in the least degree. In everyday life she was most unassuming and utterly unselfish; she had a kind and friendly nature. Nevertheless she enjoyed the recognition paid her; she could answer with a bashful smile like a young girl to whom one had whispered a compliment. No one could contend that the Graces had stood by her cradle; but if we in Göttingen often chaffingly referred to her as “der Noether” (with the masculine article), it was also done with a respectful recognition of her power as a creative thinker who seemed to have broken through the barrier of sex. She possessed a rare humor and sense of sociability; a tea in her apartments could be most pleasurable. But she was a one-sided who was thrown out of balance by the overweight of her mathematical talent. Essential aspects of human life remained undeveloped in her, among them, I suppose, the erotic, which, if we are to believe the poets, is for many of us the strongest source of emotions, raptures, desires, and sorrows, and conflicts. Thus she sometimes gave the impression of an unwieldy child, but she was a kind-hearted and courageous being, ready to help, and capable of the deepest loyalty and affection. And of all I have known, she was certainly one of the happiest.
Comparison with the other woman mathematician of world renown,, suggests itself. Sonya had certainly the more complete personality, but was also of a much less happy nature. In order to pursue her studies Sonya had to defy the opposition of her parents, and entered into a marriage in name only, although it did not quite remain so. Emmy Noether had, as I have already indicated, neither a rebellious nature nor Bohemian leanings. Sonya possessed feminine charm, instincts and vanity; social successes were by no means immaterial to her. She was a creature of tension and whimsy; mathematics made her unhappy, whereas Emmy found the greatest pleasure in her work. […] But Emmy Noether without doubt possessed by far the greater power, the greater scientific talent.
Indeed, two traits determined above all her nature: First, the native productive power of her mathematical genius. She was not clay-pressed by the artistic hands of God into a harmonious form, but rather a chunk of human primary rock into which He had blown His creative breath of life. Second, her heart knew no malice; she did not believe in evil — indeed it never entered her mind that it could play a rôle among men. This was never more forcefully apparent to me than in the last stormy summer, that of 1933, which we spent together in Göttingen. The memory of her work in science and of her personality among her fellows will not soon pass away. She was a great mathematician, the greatest, I firmly believe, that her sex has ever produced and a great woman.