Emmy Noether – Mathematician Extraordinaire 303063809X, 9783030638092

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David E. Rowe

Emmy Noether Mathematician Extraordinaire

Emmy Noether – Mathematician Extraordinaire

David E. Rowe

Emmy Noether – Mathematician Extraordinaire

David E. Rowe Institut für Mathematik Johannes Gutenberg-Universität Mainz Mainz, Germany

ISBN 978-3-030-63809-2 ISBN 978-3-030-63810-8 (eBook) https://doi.org/10.1007/978-3-030-63810-8 Mathematics Subject Classification (2020): 01A60, 01A70, 01A72, 01A73 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Emmy Noether Papers, Bryn Mawr College Special Collections This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents Preface 1

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1 2 6 15 23 35

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Emmy Noether’s Long Struggle to Habilitate in Göttingen 2.1 Opportunities for Women in Göttingen, 1890–1914 . . . . . . . . . 2.2 Habilitation as the Last Hurdle . . . . . . . . . . . . . . . . . . . . 2.3 Noether’s Attempt to Habilitate . . . . . . . . . . . . . . . . . . .

39 39 44 48

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Emmy Noether’s Role in the Relativity Revolution 3.1 Einstein’s Road to General Relativity . . . . . 3.2 Hilbert’s Approach to Einstein’s Theory . . . . 3.3 Einstein reads Hilbert . . . . . . . . . . . . . . 3.4 Klein’s Interests in General Relativity . . . . . 3.5 Noether on Invariant Variational Problems . . . 3.6 On the Slow Reception of Noether’s Theorems

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Max 1.1 1.2 1.3 1.4 1.5

vii and Emmy Noether: Mathematics in Erlangen Max Noether’s early Career . . . . . . . . . . . Academic Antisemitism . . . . . . . . . . . . . Emmy Noether’s Uphill Climb . . . . . . . . . Classical vs. Modern Invariant Theory . . . . . Max Noether’s Career in Retrospect . . . . . .

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Noether’s Early Contributions to Modern Algebra 4.1 On the Rise of Abstract Algebra . . . . . . . . . . . . . . . 4.2 Noether’s Contributions to Abstract Ideal Theory . . . . . . 4.3 Noether’s Ideal Theory and the Theorem of Lasker-Noether 4.4 Van der Waerden in Göttingen . . . . . . . . . . . . . . . . 4.5 Pavel Alexandrov and Pavel Urysohn . . . . . . . . . . . . . 4.6 Brouwer and the Two Russians . . . . . . . . . . . . . . . . 4.7 Urysohn’s Tragic Death . . . . . . . . . . . . . . . . . . . . 4.8 Helping a Needy Friend . . . . . . . . . . . . . . . . . . . .

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Contents Noether’s International School in Modern Algebra 5.1 Mathematics at “The Klie” . . . . . . . . . . 5.2 The Takagi Connection . . . . . . . . . . . . 5.3 Bologna ICM and Semester in Moscow . . . . 5.4 Helmut Hasse and the Marburg Connection . 5.5 Takagi and Class Field Theory . . . . . . . . 5.6 Collaboration with Hasse and Brauer . . . . . 5.7 Noether’s “Wish List” for Favorite Foreigners 5.8 Paul Dubreil and the French Connection . . .

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121 121 125 128 134 137 139 145 150

Emmy Noether’s Triumphal Years 6.1 The Marburg “Schiefkongress” . . . . . . . . 6.2 Rockefeller and the IEB Program . . . . . . 6.3 Birth of the Brauer-Hasse-Noether Theorem 6.4 Olga Taussky and the Hilbert Edition . . . 6.5 From Vienna to Göttingen . . . . . . . . . . 6.6 Taussky on Hilbert’s 70th Birthday . . . . . 6.7 Zurich ICM in 1932 . . . . . . . . . . . . . .

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155 155 159 163 170 172 176 181

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Cast 7.1 7.2 7.3

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Emmy Noether in Bryn Mawr 8.1 Bryn Mawr College and Algebra in the United States . 8.2 Emmy Noether’s New Home . . . . . . . . . . . . . . . 8.3 Emmy’s Efforts on Behalf of Fritz Noether . . . . . . . 8.4 Last Visit in Göttingen . . . . . . . . . . . . . . . . . 8.5 Lecturer at Princeton’s Institute for Advanced Study .

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213 213 219 226 235 241

Memories and Legacies of Emmy Noether 9.1 Obituaries and Memorials . . . . . . . . . 9.2 Fate of Fritz Noether and his Family . . . 9.3 Hasse’s Sympathies for Hitlerism . . . . . 9.4 Noether and “Hebraic Algebra” . . . . . . 9.5 “German Algebra” in the United States . 9.6 On Noether’s Influence and Legacies . . . 9.7 Courant, Alexandrov, and Grete Hermann

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255 255 267 279 280 284 286 291

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Out of Her Country 189 Dark Clouds over Göttingen . . . . . . . . . . . . . . . . . . . . . . 189 First Wave of Dismissals . . . . . . . . . . . . . . . . . . . . . . . . 194 Hasse’s Campaign for Noether . . . . . . . . . . . . . . . . . . . . . 199

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Bibliography

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Name Index

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Preface Emmy Noether is today one of the most celebrated figures in the history of mathematics, universally recognized as a brilliant algebraist and familiar to many as an iconic personality for women in science. This recognition was long in coming, and even if she enjoyed a taste of it toward the end of her life, Noether would have probably felt puzzled and bemused by such boundless acclaim. That’s pure speculation, of course, but then Emmy Noether enjoyed that sort of thing, especially when it involved mathematical fantasies. Otherwise, this extroverted woman was a rather private person who, for whatever reasons, tended not to talk about her worries and concerns, including those that eventually related to her own deteriorating health. Much about her life, which ended all too abruptly, we will never know and can only simply imagine. This book arose out of a longstanding fascination with Emmy Noether’s unique personality, but it never would have been written except for recent circumstances and events. In 2019, Mechthild Koreuber organized a major conference at the Freie Universität Berlin in cooperation with the Berlin Mathematics Research Center MATH+ and the Max Planck Institute for the History of Science to commemorate the hundredth anniversary of Noether’s Habilitation in Göttingen. 1 This event, which took place on June 4, 1919, was not only a major milestone for Noether herself but also for women longing to pursue academic careers in Germany. As one of the highlights of the conference, the ensemble portraittheater Vienna presented the premiere performance of their play, “Mathematische Spaziergänge mit Emmy Noether” [Schüddekopf/Zieher 2019]. Both of us, as historians of mathematics, had been involved in its production, and we were delighted by the result. So the thought of adapting the script for an English-speaking audience occurred to us right away. It must be said, though, that we had no idea how Sandra Schüddekopf and Anita Zieher would manage to stage a play about a mathematician, one whose work even her peers found to be highly abstract. Nevertheless, they found a very elegant way to finesse that problem, and in a manner that would have appealed to Emmy Noether, whose personality shines through despite the handicap that most of her audience has absolutely no idea what she’s really talking about. Noether was by no means a one-dimensional type who lived for mathematics and nothing else, and Anita Zieher truly brings her personality back to life on stage, now in the new adaptation of the original play, “Diving into Math with Emmy Noether” [Schüddekopf/Zieher 2020].2 Sensing that “Diving into Math” was an excellent vehicle for conveying the spirit of Emmy Noether’s life, Mechthild Koreuber and I soon got the idea of writing a book that would expand on some of its themes. This we hope to have achieved in Proving It Her Way: Emmy Noether, a Life in Mathematics 1 The interdisciplinary character of the conference – which brought together mathematicians, physicists, historians of science, gender researchers, and cultural historians – is reflected in the forthcoming conference volume [Koreuber 2021]. 2 For information or to book a performance, contact [email protected].

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[Rowe/Koreuber 2020], which aims to provide an overall picture of her life, but with only minimal attention paid to her mathematics. The present book has a similar structure and purpose, namely, to illuminate Noether’s life by offering a full-blooded picture of her role in shaping the mathematical activity of her day and, as it happened, well beyond.3 In places, however, it is more technically demanding. Thus, Chapter 3, “On Emmy Noether’s Role in the Relativity Revolution,” deals with a topic omitted from the smaller volume, since to appreciate what she accomplished in that context requires familiarity with Einstein’s theory of general relativity. Likewise, the final Chapter 9, “Memories and Legacies of Emmy Noether,” which reaches well beyond the events of her life, was left out of the shorter book. Elsewhere, as well, the reader will find many brief discussions of Noether’s mathematics and related matters strewn throughout the pages of this volume. For those who wish to learn still more, this book contains numerous references to the works listed in its extensive bibliography, including all of Noether’s own publications from her Collected Papers [Noether 1983]. Emmy Noether was not a particularly prolific mathematician, and while a few of her papers are now classics, most have long since been forgotten. Noether’s fame and influence had much to with those well-known publications, of course, but one cannot really begin to grasp her importance merely by studying these published works. This would be to overlook her activities as a collaborator and critic, not to mention her role as referee for the journal Mathematische Annalen. Most of all, though, Noether’s influence flowed through her role as leader of a dynamic new mathematical school, one in which she taught younger mathematicians how to exploit the new concepts and methods she promoted in her lectures and published work. Her approach aimed to strip mathematical objects down to their bare essentials in order to recognize deeper underlying relationships among them. Doing so, however, meant learning to think about mathematics on a higher abstract plane. Noether’s enthusiasm was infectious, at least for those who entered her circle and caught the abstract algebra bug. By the mid-1920s, she was already riding a wave of modern methods that would eventually reshape major branches of mathematical knowledge. Noether lived during the pre-Bourbaki era, a time when modern forms of collaboration were only emerging. André Weil, the unofficial leader of the group that wrote under the pseudonym Nicolas Bourbaki, remembered the atmosphere in Noether’s Göttingen circle during the mid-1920s as very different from the one he encountered when talking with those in Richard Courant’s group, from whom he learned very little. Nearly every time he got into a conversation with one of the latter’s students, the exchange would end rather abruptly with a remark like, “sorry, I have to go write a chapter for Courant’s book” [Weil 1992, 51]. This

3 Both books, it should be noted, draw heavily on the information concerning Noether’s school as well as the interpretation of its impact found in [Koreuber 2015].

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“publish or perish” mentality predominated in Courant’s circle, whereas Emmy Noether felt no such urgency to rush her work into print.4 Weil recalled conversations with Pavel Alexandrov in Noether’s cramped little attic apartment. Its ceiling was so angular that Edmund Landau – who lived in a veritable palace by comparison – wondered out loud whether Euler’s polyhedral formula still applied to her living room. Here and elsewhere, Weil saw how Emmy Noether good-naturedly played the role of mother hen and guardian angel, constantly clucking away in the midst of a group from which van der Waerden and Grell stood out. Her courses would have been more useful had they been less chaotic, but nevertheless it was in this setting, and in conversations with her entourage, that I was initiated into what was beginning to be called “modern algebra” and, more specifically, into the theory of ideals in polynomial rings. [Weil 1992, 51] Many who heard Noether’s lectures reacted similarly, like the young Carl Ludwig Siegel, who remembered them as badly prepared. In one of her courses, which ended at 1 o’clock, he scribbled in the margin of a notebook: “It’s 12:50, thank God!” [Dick 1970/1981, 1981: 37]. Siegel much preferred lectures in the style of his mentor Landau, who prided himself on presenting polished lectures already nearly ripe for publication. Landau’s teaching style, which Siegel largely emulated, aimed to convey the formalized end products of mathematical activity. For Noether, on the other hand, the excitement came when she was still searching, groping along halfway in the dark. She gradually developed a teaching style in which oral communication, dialogue, and collaboration dominated. Doing mathematics meant, for her, engaging with all facets of the process, and in this way she came to embody the oral component in Göttingen’s vibrant mathematical culture.5 Her approach, however, should by no means be understood as one that neglected the importance of formal rigor in published communications. Indeed, the relative sparsity of her own published work reflects the fact that she always resisted putting less than perfect texts into print. Moreover, Noether’s letters and postcards – in particular those she sent to Helmut Hasse, published in [Lemmermeyer/Roquette 2006] – reveal very clearly that she always upheld the highest standards for mathematical publications. “Pauca sed matura” (few but ripe), the famous watchword of Carl Friedrich Gauss, applies just as well to Emmy Noether. Yet Gauss, who was anything but generous when it came to communicating his unripe ideas with others, stands in this respect in complete opposition to Noether, whose success and influence had much to do with her un4 Weil was especially struck by the radically different atmosphere in Frankfurt, where Max Dehn and Carl Ludwig Siegel cultivated mathematics as an art form, in conscious opposition to the factory-like production facilities in Courant’s Göttingen [Weil 1992, 52–53]. 5 On the importance of the oral dimension in Göttingen, see [Rowe 2004]. For an analysis of Noether’s oral style as part of her conceptual approach to mathematics, see [Koreuber 2015, Kap. 2].

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selfish generosity. Indeed, despite her unorthodox teaching style, she made her greatest impact as a teacher and through her influence as leader of what came to be called the “Noether school” (or “Noether community” or sometimes “Noether family”). For Emmy Noether, just as for Emil Artin, mathematicians were in the first instance artists, not scientists.6 This rather contentious position was apparently a favorite topic of friendly disputes between Emmy and her brother Fritz, a leading applied mathematician who worked on problems like modeling turbulence in continuum mechanics. Their father, Max Noether, was a different type of mathematician still, as will be seen in Chapter 1, which deals with their years together in Erlangen. Mathematical talent ran through the Noether family, leading the Göttingen number-theorist Edmund Landau to liken Emmy Noether’s kinfolk with a coordinate system in which she occupied the origin [Dick 1970/1981, 1981: 95]. Had Landau lived longer, he would have needed to imagine a new coordinate axis for Fritz Noether’s younger son Gottfried, who became a leading authority in the field of non-parametric statistics (see Section 9.2). Labels are often misleading, and in the case of mathematical elites like the Noethers lumping them together as “mathematicians” simply overlooks the wide range of different intellectual pursuits in which these researchers were engaged. In the case of Emmy Noether, this is a crucially important point; her way of thinking about mathematics using abstract concepts, rather than concrete objects, was by no means new. She became, however, the foremost exponent of this approach to mathematical theorizing, which she promoted in a radical manner, a style quite unlike that of any other contemporary figure. She and Artin also both believed that all truly deep mathematical truths must be beautiful. One can only begin to understand what that means, of course, by delving deeply into the mathematical world they shared, as many famous figures who came after them did. Quite apart from her accomplishments as a mathematician, Emmy Noether was also a singularly remarkable representative of that famous group of German Jewish intellectuals who fled from Nazi Germany. How this tragedy unfolded in Göttingen is described in Chapter 7, which provides a fairly detailed account of the events that led to the destruction of its star-studded mathematical faculty. Much has been written about “Hitler’s gift” to the Western democracies and how Weimar’s exiled intellectuals enriched cultural life in the United States. Emmy Noether’s name would surely have appeared in many more of those studies had she only lived longer. Instead, her tragic and wholly unanticipated death in April 1935 prevented her from importing her distinctive style for doing abstract algebra to the United States, even though others, in particular Artin and Richard Brauer, promoted similar ideas soon afterward. Still, we can easily imagine how that story might have unfolded at Bryn Mawr College, but especially at Princeton’s Institute for Advanced Study. Her “girls” at 6 On the broader context of earlier debates over the status of mathematics as art or science, see [Rowe 2018a, 401–411].

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Bryn Mawr (graduate students and post-docs) were among the first to sink their teeth into “German algebra” by reading B.L. van der Waerden’s new textbook Moderne Alegbra [van der Waerden 1930/31] and by studying the Ausarbeitung of Helmut Hasse’s Marburg lectures on class field theory. Noether taught them in English, but with such large dosages of German jargon that this became part of their natural vocabulary, and so they spoke to each other about this “new math” in a kind of pidgin German. Noether’s favorite pupil at Bryn Mawr, Ruth Stauffer, later recalled how “it was very easy for us to simply accept the German technical terms and to think about the concepts behind the terminology. Thus from the beginning we discussed our ideas and our difficulties in a strange language composed of some German and some English” [Quinn et al. 1983, 142]. Although Emmy Noether is justly famous as the “mother of modern algebra,” it is important to understand what she meant by “doing algebra.” Her vision of its role in mathematics did not seek to erect clear disciplinary boundaries setting algebraic investigations off from those in other fields. On the contrary, her work was closely tied to an older trend that aimed to algebraicize other fields, from complex functions and number theory to topology, i.e. major parts of all mathematical knowledge. In this respect, she took inspiration from earlier studies by Richard Dedekind, Heinrich Weber, Ernst Steinitz, and David Hilbert. Moreover, she clearly identified with words she once cited from Leopold Kronecker’s 1861 inaugural address, when he was inducted into the Berlin Academy: “algebra is not actually a discipline in itself but rather the foundation and tool of all mathematics.” 7 Not that many of Noether’s contemporaries shared this view; far from it. Nor should we imagine that Noether meant this literally; she was well aware of the vast fields in analysis and applied mathematics that lie well beyond the realm of even her all-embracing view of algebraic research. Still, she was without doubt the leading spokesperson of her generation for this position, one that many of her contemporaries found extreme. One of Noether’s closest collaborators, Helmut Hasse, clearly recognized the import of Noether’s message, but he also sensed the need to spread the word. In a lecture on “The modern algebraic Method,” he made this mission clear: The aim of my talk is to promote modern algebra among non-specialists instead of preaching to the choir. It is not my intention to lure anyone from his field of specialization to become an algebraist. I see my task, rather, as laying the groundwork for a favorable understanding of modern algebra, helping to establish its methods – insofar as they are of general importance, and integrating these methods into the common knowledge of contemporary mathematicians. [Hasse 1930, 22] Noether’s former Göttingen colleague, Hermann Weyl, on the other hand, had deep misgivings, though not so much with regard to abstract algebra per se. What concerned Weyl was the general trend toward abstraction in mathematical research, 7 [Noether 1932c]; the relevance of this citation is discussed in [Koreuber 2015, 225] as well as in [Merzbach 1983, 161].

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a tendency he felt could easily lead to artificiality and superficial results. He mentioned these concerns in his famous memorial address [Weyl 1935], delivered at Bryn Mawr College. In this respect, Hermann Weyl was far closer to his former mentor, David Hilbert, than he was to Emmy Noether, much as he admired her brilliance.8 Beyond these matters, some of Weyl’s remarks reflect a rather condescending view of Emmy Noether and her family. This attitude seems particularly striking when he writes about her background and life in Erlangen, a time and place Weyl could only imagine. He portrayed the Noethers as impressive intellectuals, even drawing a parallel between their family, with its three distinguished mathematicians, and the Bernoullis, whose Huguenot ancestors fled Spanish repression in the Netherlands to settle in Basel. But he also saw the Noether family as representatives of a shallow bourgeois society, with “their sentimentality, their Wagnerism, and their plush sofas” [Weyl 1935, 430]. Perhaps Weyl also felt unnerved by Emmy’s apparent inability to grasp evil in the world. She had lived her whole life as a fully integrated German Jew, which meant of course that antisemitism was no stranger to her, but when the barbarians came to power and threatened to sweep away everything she loved, she reacted not only with restraint but with an almost super-human equanimity. Those lonely months during the spring of 1933 – the time when they were last together in Göttingen – no doubt profoundly shaped Weyl’s view of her, yet his opinion seems to have wavered between two extremes: Emmy was either a tower of moral strength or she was simply naive. Although Hermann Weyl knew Wilhelmian Germany exceedingly well, he may never have met Max Noether. He did know Fritz Noether, who came to Göttingen as a post-doctoral researcher when Weyl was teaching there as a Privatdozent. As for Emmy, he could easily have met her at the annual meetings of the German Mathematical Society, but he probably only got to know her well in 1927 when he spent a semester as a guest professor in Göttingen. Notwithstanding the importance of his testimony, Weyl’s personal opinions need not be taken as authoritative, particularly since his impressions in 1935 were colored by the turbulent events he experienced during the previous two years. Moreover, he did not have access to most of the contemporary documentary sources that form the basis for the present study and which inform its interpretation of Emmy Noether’s social and intellectual background. This book also consciously avoids certain stereotypic themes found in much of the secondary literature dealing woth Noether. Many standard studies of women in the history of mathematics have chosen to follow Weyl’s lead by comparing Noether with the internationally renowned Russian mathematician Sofia Kovalevskaya, who also appears in several places below as well. These comparisons, to be sure, rarely have anything to do with serious interest in what these women accomplished as mathematicians. Very often, they are coupled together as two trailblazers in a field then totally dominated by men, even though neither really saw herself in such a role. Talk of glass ceilings, after 8 For

an analysis of Weyl’s scientific work and views, see [Scholz 2001].

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all, was yet to come, whereas gender roles in that era were exceedingly constrained. Earlier commentators usually could not get past the notion that a “lady mathematician” was a freak of nature, a view clearly supported by the scarce number of these creatures then walking the earth. How that has changed! Contemporary opinions of Emmy Noether – and these were quite mixed – clearly have considerable importance for understanding the context in which she lived. Even more important – especially for the present undertaking – are those sources that tell us how she thought about herself and the world around her, and especially how she expressed those thoughts. Others occasionally compared her as a mathematician with Richard Dedekind (no one writing about her mathematics would have imagined a comparison with Sofia Kovalevskaya), but she quite rightly said about herself “I always went my own way.” 9 Hermann Weyl’s account of Emmy Noether’s intellectual development has, in one sense, been very influential. Many subsequent commentators have, in fact, adopted his tripartite division of her career:10 (1) the period as a post-doc, 1907– 1919, followed by (2) her work on the general theory of ideals, 1920–1926, and then (3) her contributions to non-commutative algebras with applications to commutative number fields, 1927–1935 [Weyl 1935, 439]. This periodization is certainly apt and even quite useful to a point, but it can also easily lead to quite misleading impressions. Those who have adopted it have tended to underplay the significance of the first period, while overlooking some of the threads that ran through all three phases of Noether’s career.11 Emmy Noether was nearly forty years old when she began publishing the papers on modern algebra that made her famous. By the mid-1920s, she had become the leader of an international school that would soon thereafter exert a deep and lasting influence on mathematical research. All her most familiar and significant work was thus undertaken during the latter two periods, when many of her ideas and findings quickly propagated through the network of the Noether school. Little wonder, then, that this success story has completely dominated nearly all the accounts of Noether’s life. Moreover, as Uta Merzbach has stressed, one of the great ironies behind her success was that part of it stemmed from never having gained a regular professorial appointment at Göttingen. This “allowed her to organize her algebraic research as single-mindedly as she did, to display that generosity to her followers to which van der Waerden, Alexandroff, and others have given eloquent testimony, and to engage so fully in the editing of Dedekind’s 9 See

the opening of Chapter 2 in Proving it Her Way: Emmy Noether, a Life in Mathematics. obituary of Hilbert was somewhat similar; there he discerned that the master’s work fell into five periods [Weyl 1944, 4: 135]. 11 Pavel Alexandrov fully appreciated the importance of Noether’s early work on finiteness results, but wrote that she herself was partly responsible for the fact that this work had been unjustly neglected, since she “considered those results to have been a diversion from the main path of her research, which had been the creation of a general, abstract algebra” [Alexandroff 1935, 2]. Since Alexandrov and Urysohn pioneered the theory of general compact spaces (in which every open covering has a finite subcovering), one can easily imagine their affinity for finiteness results in algebra. 10 Weyl’s

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works and selected correspondence” [Merzbach 1983, 169–170]. A similar view was expressed by Emmy Noether’s first biographer, Auguste Dick, who wrote that she preferred the position of Dozentin because it gave her freedom. As an Ordinarius “she would have been obliged to teach basic courses and exercises for which she was not well suited. Much of her time would have been absorbed by preparations for classes, and her own research would have suffered” [Dick 1970/1981, 1981: 72–73]. Most of Noether’s publications from the first period, on the other hand, received little attention during her lifetime. This applies even to her famous paper “Invariant Variational Problems” [Noether 1918b], which today is perhaps her best-known single work. As documented in [Kosmann-Schwarzbach 2006/2011], this paper was rarely ever cited, much less carefully read, until many years after Noether’s death. No doubt Weyl’s periodization of her research interests offers a useful schematic, so long as we are not misled into thinking that Emmy Noether’s earlier work had little to do with her publications from the 1920s. As Merzbach noted, a great deal of her work had clearly identifiable classical roots: Her deep knowledge of the literature and her ability to recognize and bring to the fore those concepts that would prove most fruitful prepared her . . . to undertake her grand synthesis. If one examines her work after 1910, one finds continual growth, but little change in methodological pattern. [Merzbach 1983, 169]12 This should come as no surprise if we remember that Emmy Noether had a thorough knowledge of the mathematical literature of her time; she was also well-versed in major works from the latter half of the nineteenth century. As noted in [Koreuber 2015, 5], Weyl’s tripartite framework is highly problematic if one hopes to gain a deeper understanding of Emmy Noether’s intellectual growth. To gain a more balanced picture requires recognizing, first of all, the critical importance of the first period in her career. Those years form part of the larger context taken up in Chapter 1, which deals with her life in the mid-size university city of Erlangen, where she grew up as the daughter of the eminent mathematician Max Noether. The Noether family – Max and Ida and their four children – were members of the local Jewish community, which numbered around 200 persons during Emmy’s childhood, less than 2% of the city’s population. More important still, all of them were recent arrivals, as before 1861 Jews were not permitted to live within the city limits. Max Noether and his older colleague Paul Gordan, who was also Jewish by birth, were the only mathematicians on the faculty, a highly unusual situation, especially given the small number of university professorships that existed throughout Germany. Young women had virtually no chances of even studying at a university, let alone dreaming of a teaching career at one of these institutions. That Emmy Noether dreamed of such a life at an early age probably cannot be documented, but clearly she did, and the fact that she longed to follow 12 A

more recent study that argues for a similar view is [McLarty 2017].

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in her father’s footsteps gives us the first key to understanding how such a thing could even happen. When Emmy Noether finally left Erlangen in 1915, she did so with the hope of joining the faculty in Göttingen, a plan supported by its two senior mathematicians, Felix Klein and David Hilbert. Their efforts, however, at first failed, and as recounted in Chapter 2, it took four long years before Noether was allowed to habilitate in Göttingen. This matter, which hinged entirely on the fundamental question of whether qualified women were entitled to become members of a university faculty, led to a dramatic clash of opinions within Göttingen’s highly polarized philosophical faculty. Indeed, the Noether affair was perhaps the most infamous in a series of running battles which would eventually lead to a complete cessation of relations between its two departments, comprised of natural scientists in one division, and humanists in the other. In 1922, the Ministry finally approved a proposal, put forth by the latter group (members of the historical-philological department), that called for the formation of two wholly distinct faculties. In that same year, the newly established faculty of mathematics and natural sciences appointed Emmy Noether as an honorary associate professor, a title normally bestowed only six years after habilitation. In recommending her for this honor, the faculty noted that she had been unjustly denied the right to habilitate in 1915. 13 During the war years, both Hilbert and Klein had become deeply immersed in mathematical problems connected with Albert Einstein’s novel approach to gravitation, the general theory of relativity. Working first with Hilbert and then with Klein, Noether ultimately unraveled one of the major mathematical mysteries that they and Einstein had struggled to solve, namely, the role of energy conservation in physical theories based on variational principles. Chapter 3 provides a fairly detailed account of Emmy Noether’s role in that particular phase of the relativity revolution. This story culminates with the publication of the “Noether theorem” (actually two theorems) in “Invariant Variational Problems” [Noether 1918b], a result nearly every physicist today is familiar with in some guise. The story of how she actually came to write that paper, however, has rarely been told and surely deserves to be better known, despite the technical complexities involved. Those who are unfamiliar with mathematical methods in general relativity can skip this chapter without losing the main threads that tie Emmy Noether’s first creative period with her work from the early 1920s. Noether’s most influential papers stem from her second period, when she made major contributions to ideal theory. She was almost 40 when she published “Ideal Theory in Ring Domains” [Noether 1921b], one of her most famous algebraic works. Here she introduced the general concept of rings satisfying the ascending chain condition, familiar today as Noetherian rings. Soon afterward, her reputation as a leading algebraist began to spread beyond Göttingen, leading to her fame as “der Noether.” To appreciate the importance of this work, described briefly in Chapter 4, one needs to understand its role in the general shift from classical to 13 Universitätsarchiv

Göttingen, Personalakte Emmy Noether, UAG.Math.Nat.Pers.9.

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modern algebra. This is easily illustrated by comparing Noether’s approach with Richard Dedekind’s earlier theory of ideals in number fields.14 Noether’s second major contribution to ideal theory was [Noether 1927a] (“Abstract Structure of Ideal Theory in Algebraic Number and Function Fields”). Here she followed Dedekind, who had proved a fundamental decomposition theorem for the ideals of a number ring. Noether was able to prove an analogous, but much more general theorem, valid for all commutative rings that satisfied the five axioms for a Dedekind ring – and vice versa – which means this theorem characterized Dedekind rings.15 In this theory, the prime ideals play the same role as the prime numbers in elementary number theory. So her theorem was a fundamental structure theorem for ideal theory – which is now understood as part of the broader discipline of commutative algebra.16 Noether’s other great achievement came in her earlier paper [Noether 1921b]. Here she was able to place Emanuel Lasker’s decomposition theorem for ideals in a ring of polynomials on a much broader and clearer basis. The building blocks in this case were the primary ideals introduced by Lasker, but instead of five axioms Noether essentially only needed one restriction, namely, that the ring does not contain an infinitely ascending chain of ideals. This property was not new, but Emmy Noether was the first to recognize its central importance. This is why rings that satisfy the ascending chain condition (acc) are today called Noetherian rings. She later made this acc condition the first of her five axioms in [Noether 1927a]. Noether’s structure theorem for polynomial rings was of great importance for the algebraization of algebraic geometry. Her father had proved a fundamental theorem for this discipline in 1871, which later served as the foundation for the work of the “Italian school.” However, his daughter took up earlier results of Hilbert and Lasker in order to lay the foundation for a new and far more general direction in algebraic geometry based on polynomial ideals. Yet even more important than these results were Noether’s methods, which clearly revealed the strength of her conceptual arguments compared with earlier more computational methods. Her goal throughout was to make everything as transparent as possible, and her most important works can still be read today with interest and understanding, a rare achievement in mathematics. After this brief excursion into Noether’s work on ideal theory, the focus in Chapters 4, 5, and 6 shifts to her relationships with the four other mathematicians who appear in “Diving into Math with Emmy Noether”: Bartel L. van der Waerden, Pavel Alexandrov, Helmut Hasse, and Olga Taussky. While none of these four took a doctoral degree under Noether, all were closely connected with her school in one way or another. Each, in fact, represents a strand of influence that ran through the Noether school, thereby contributing to its diverse and eclectic character. When 14 For

a detailed comparison, see [Corry 2017]. a sketch of the steps in her proof, see Jacobson’s introduction in [Noether 1983, 14]. 16 She exploited this new theory immediately afterward in [Noether 1927b] by proving a generalization of Dedekind’s discriminant theorem that applies to arbitrary orders in a number field. 15 For

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van der Waerden arrived in Göttingen from Amsterdam, his principal interests were closely related to Max Noether’s work in algebraic geometry. After studying under Noether’s daughter and then under Emil Artin in Hamburg, he published his classic two-volume textbook Moderne Algebra [van der Waerden 1930/31], which for decades afterward served as the standard introduction to the subject. The Russian topologist Pavel Alexandrov was a regular visitor in Göttingen during the summer months. As one of Emmy Noether’s closest friends, he spent countless hours “talking mathematics” with her, eventually joined by another topologist, Heinz Hopf. These conversations proved of vital importance for the emergence of modern topology, a field that began to take on clear form in their textbook [Alexandroff/Hopf 1935]. Both van der Waerden and Alexandrov very consciously adopted Noether’s conceptual approach in writing these two seminal works, which distilled and synthesized essential knowledge in two fundamentally new disciplines: abstract algebra and algebraic topology. During the final phase of Noether’s career, Helmut Hasse was her closest collaborator. As a student of Kurt Hensel in Marburg, Hasse developed a new local-global principle, based on Hensel’s p-adic numbers, that proved highly fertile for research in algebraic number theory. As he began to explore a new research agenda for class field theory, Emmy Noether pointed out the relevance of ongoing work on hypercomplex number systems (i.e., non-commutative algebras) for generalizing the number-theoretic investigations of Hasse and Artin. Thanks to the carefully edited publication of her letters to Hasse, published in [Lemmermeyer/Roquette 2006], one can easily recognize how Noether’s ideas had a catalytic effect on Hasse’s work after 1927. Her constant, unrelenting prodding, mixed with praise and encouragement, played a major part in their symbiotic relationship, underscoring the importance of purely human factors in mathematical research. Noether’s parallel collaboration with Richard Brauer soon led to a threesome, who together succeeded in proving the Brauer-Hasse-Noether theorem. Emmy Noether’s relationship with Olga Taussky was unlike any other, not least because Taussky, too, was a woman with a mind of her own. Their first lengthier interactions took place during the academic year 1931/32 when Taussky came to Göttingen as a young Viennese post-doctoral student, having been hired by Richard Courant to lend help in editing Hilbert’s early works on number theory. She was highly qualified to do so, having studied under Phillip Furtwängler, a leading expert on class field theory. After returning to Vienna for two years, Taussky rejoined Emmy Noether at Bryn Mawr College in 1934, a difficult time in the lives of both women, as Taussky would recall late in her life. Olga Taussky never became an enthusiast for Noether’s abstract style of mathematics, and yet her encounters with Emmy Noether, particularly during the last year of her life, proved to be of great importance for the young woman’s career. Indeed, none of these four mathematicians – van der Waerden, Alexandrov, Hasse, and Taussky – who went on to write hundreds of papers and produce dozens of Ph.D.s in the course of their careers, can really be called a disciple of Emmy Noether, even though all of them were inspired by her ideas and personality in significant ways.

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During the 1920s, Richard Courant and Emmy Noether actively promoted the trend toward internationalization that became a hallmark of Göttingen mathematics during this period. Their efforts received a major boost from American philanthropy and the vision of Wickliffe Rose, who founded the International Education Board (IEB) in 1923, backed by financial support from John D. Rockefeller, jun.17 Several of those who visited Göttingen during these years were IEB Fellows. Two who came from France were André Weil and Paul Dubreil; both attended Noether’s lectures, as did another co-founder of the Bourbaki group, Claude Chevalley. The Norwegian Øystein Ore visited Göttingen twice, the second time as an IEB fellow working under Noether. He was afterward recruited by James Pierpont, who invited him to join the faculty at Yale University, where he would remain throughout his career. He also joined Emmy Noether and Robert Fricke in editing the collected works of Richard Dedekind, [Dedekind 1930–32] (see Section 6.4). As Hermann Weyl emphasized in his memorial address, Noether stood at the very heart of mathematical life in Göttingen, just as its larger scientific community was a manifestation of Weimar Germany’s vibrant cultural life. 18 As one of Weimar culture’s leading representatives, Albert Einstein later wrote about Emmy Noether’s highly significant role in this ultimately tragic story. 19 Chapter 7 describes the traumatic events of 1933 that dramatically ended that life, as Noether had known it. She and Richard Courant, the director of the Mathematics Institute, were both forced to take refuge in the United States. Helmut Hasse would ultimately be appointed to Courant’s chair, but while still in Marburg he initiated a campaign to maintain Noether’s modest position in Göttingen. Predictably, this effort failed, though through the intercession of friends in the United States Emmy Noether gained a temporary appointment at Bryn Mawr College, a distinguished institution of higher learning for women. Chapter 8 briefly recounts Bryn Mawr’s importance for the history of mathematics before relating various events and circumstances connected with Noether’s association with the college. During her 18 months there, she also began to spread the gospel of modern algebra in weekly lectures at Princeton’s Institute for Advanced Study, where her seminar attracted a number of prominent, as well as up and coming mathematicians. Her collaborator from Germany, Richard Brauer, attended regularly, as did Nathan Jacobson. The latter filled in for Emmy Noether at Bryn Mawr the following year, and he would later edit her Collected Papers [Noether 1983]. Her sudden death on 14 April 1935, following an operation, came as a huge shock to everyone, perhaps most of all to her brother Fritz, who also had been forced to leave Germany with his two sons. Emmy had tried to find work for 17 For a detailed account of the IEB’s impact on mathematics, especially in Western Europe after World War I, see [Siegmund-Schultze 2001]. 18 This interpretation of Göttingen mathematics as a phenomenon within the larger context of Weimar culture is addressed in [Rowe 1986]. 19 Einstein’s obituary of Noether, which appeared in the New York Times, is discussed in Chapter 9; it was first analyzed in [Siegmund-Schultze 2007].

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him in the United States (see Section 8.3), but after her efforts failed he took a position at Tomsk Polytechnic University in Western Siberia. She was spared by her premature death from learning about the tragic events that afterward befell her brother and his family, described in Section 9.2. Emmy Noether’s last two years in the United States were filled with all kinds of worries, few of which she spoke about even with her closest friends. One of these was Anna Pell Wheeler, chair of the Mathematics Department at Bryn Mawr College, who in many ways helped her to adjust to life in the United States. Chapter 9 recounts some of the memories other friends of Emmy Noether shared with each other as well as with the Bryn Mawr community. Many sensed the grandeur of her intellectual legacy, but it would take some time to recognize clearly her importance for subsequent mathematical developments. This closing chapter cannot, of course, do justice to Noether’s legacy; nevertheless it seems appropriate to end with some reflections on her place in the mathematics of the last century. A contemporary mathematician once told David Hilbert, the man who first brought Noether to Göttingen, “You have made us all think only that which you would have us think.” 20 Those very same words could just as aptly have been said about Emmy Noether, whose ambitions for directing and shaping mathematical research spring to life in her correspondence, but also from later recollections written by contemporaries who knew her very well. By the end of her life, she had many admirers who recognized in her unique personality and boundless vitality the marks of a genial mathematician. As pointed out in Chapters 7 and 9, even some of those who knew Emmy Noether’s work very well considered it somehow “Hebraic,” and hence foreign to what they imagined to be good, sound “German” mathematics. Faced with seeing their teacher banned from the German universities, Noether’s faithful students tried to counter this by underscoring how her research was rooted in the tradition of Richard Dedekind, one of the great German mathematicians of the nineteenth century. Perhaps this was simply a matter of political expediency, but more likely it reflected a genuinely felt conviction that Emmy Noether’s mathematics was truly “Germanic” and was therefore not be be conflated with a “Jewish style.” A standard stereotype presumed that Jews had a distinctly different way of thinking about mathematics stemming from a Talmudic tradition that favored abstract theorizing, while neglecting fields with close ties to the physical sciences. Such stereotypes were particularly widespread in the German mathematical community during the period considered here, and yet for every Emmy Noether representing the first tendency, there was a Fritz Noether practicing the second. This larger point was brought out forcefully in the traveling exhibition “Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture,” which presented a wide array of books and articles written by GermanJewish scholars. These impressive works completely refute the claim that there 20 From Constantin Carathéodory’s funeral speech for Hilbert, Hilbert Nachlass, SUB Göttingen, 750.

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was a “typical form of ‘Jewish mathematics’, remote from geometrical intuition or from applications” [Bergmann/Epple/Ungar 2012, 134].21 “Transcending Tradition” grew out of an earlier effort that went on display at the Poppelsdorf Palace in Bonn in September 2006.22 This soon led to a broader undertaking, organized by Moritz Epple at Frankfurt University, that produced a traveling version of the original German-language exhibition. The latter was shown in a number of cities during 2008, the “Year of Mathematics” in Germany, and aroused considerable interest among the public at large. As a result, support was sought and obtained from governmental agencies for an English-language traveling exhibition that went on view in cities throughout Israel, the United States, and Australia. Naturally, Emmy Noether was accorded a prominent place in it, as was the Bonn mathematician Felix Hausdorff (see [Bergmann/Epple/Ungar 2012, 83–85; 94–104]), both of whom contributed in quite different ways to shaping the face of modern mathematics. In fact, the original impetus behind the 2006 exhibition arose from then ongoing work on the multi-volume Hausdorff edition, a highly ambitious project that was only completed quite recently. Although the names Emmy Noether and Felix Hausdorff are famous in the annals of mathematics, they are rarely mentioned together. And, in fact, it would be hard to imagine two mathematicians whose works, personalities, and influence differed so sharply. Nor does it appear that they had more than perhaps fleeting contacts with one another, since Hausdorff rarely attended the annual meetings of the German Mathematical Society, an event Noether rarely missed. Nevertheless, one merely needs to open [Hausdorff 2012], the correspondence volume in the Hausdorff edition, to recognize that Pavel Alexandrov, the great Russian topologist, acted as a kind of mediator between these two eminent figures.23 Indeed, his letters to Hausdorff clearly reflect the paths Alexandrov followed in an effort to link point set topology in the style of Felix Hausdorff with the then emergent algebraic topology promoted by Emmy Noether. This is but one of the many currents that ran through Noether’s life’s work. Here, as elsewhere in this book, effort has been made to illuminate her career through new findings based on recent research. Much more can be found in the many works listed in the bibliography, especially for those who read German. A book such as this one could obviously not have been written without the efforts of many others, including those whose names appear in the many works cited throughout. Rather than making this preface any longer than it is already, though, let me first extend thanks to all the unnamed individuals who have contributed 21 David Hilbert was one of the few who forthrightly claimed that “mathematics knows no races” (see [Siegmund-Schultze 2016]). His colleague, Felix Klein, thought that geometrical intuition (Anschauung) was deeply rooted in the Teutonic race. Yet as Klein and Max Noether well knew, after 1890 this impulse lost ground in Germany just as it was being taken up by leading Italian geometers: Corrado Segre, Guido Castelnuovo, and Federigo Enriques, all of whom were of Jewish descent. 22 On this prehistory, see Moritz Epple’s remarks in [Bergmann/Epple/Ungar 2012, 7–8]. 23 Alexandrov’s letters to Emmy Noether, cited hereinafter from [Tobies 2003], are part of the rich correspondence found in the Hochschularchiv der ETH Zürich, Hs 160.

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directly or indirectly to making this book possible. My inspiration to write about Emmy Noether came about through a most pleasant and fruitful collaboration with Mechthild Koreuber, whose passionate interest in this phenomenal figure quickly rubbed off on me. The same can be said for Sandra Schüddekopf and Anita Zieher of portraittheater Vienna, whose creative efforts provided another vital source of inspiration. As noted in our book Proving it Her Way: Emmy Noether, a Life in Mathematics, Mechthild and I are grateful for the cooperation we received from a number of institutions in the course of our work on both books. I would like once again to express appreciation for the help we received from archivists at the Austrian Academy of Sciences, Bryn Mawr College, Caltech, Göttingen State and University Library, Hebrew University, and Oberwolfach Research Institute for Mathematics. I am especially grateful to the grandchildren of Fritz Noether – Monica Noether, Margaret Noether Stevens, and Evelyn Noether Stokvis – for sharing records and documents in their family archives. Special thanks also go to Qinna Shen, Professor of German Studies at Bryn Mawr College, for her efforts in supporting this project, as well as to Ayse Gökmenoglu for the care she took in producing the photos included in this book. I also benefited from the helpful advice of Catriona Byrne and Rémi Lodh at Springer Nature, who both supported this venture from the outset. Among those who read and commented on the text at some stage, I wish to thank Leo Corry, Joe Dauben, Manfred Lehn, Jemma Lorenat, Monica Noether, Volker Remmert, Peter Roquette, Erhard Scholz, Reinhard Siegmund-Schultze, Margaret Noether Stevens, Evelyn Noether Stokvis, and Cordula Tollmien. Finally, special thanks must go to Walter Purkert, former coordinator of the Hausdorff editorial project and coauthor with Egbert Brieskorn of the monumental Hausdorff biography [Brieskorn/Purkert 2018]. Walter read the entire manuscript and offered a number of very helpful ideas with regard to Alexandrov’s role in the larger story told here. Without his knowledge and advice, this important dimension in modern mathematics – linking the work of Noether and Hausdorff – would have been far murkier. Much else, of course, remains to be told, as what I have attempted to do here might be likened with opening a few of the windows onto Emmy Noether’s world. May others feel inspired to go further. David E. Rowe

Chapter 1

Max and Emmy Noether: Mathematics in Erlangen Until 1933, most of Emmy Noether’s life was spent in two middle-sized cities: Erlangen, her birthplace, and Göttingen, where she began her mathematical career. Noether was already thirty-three when she left Erlangen for Göttingen in 1915. Although her brilliant career as an algebraist only began after her habilitation in 1919, one can trace many roots of her later mathematical activity and the work that would later make her famous back to Erlangen. The university’s mathematical faculty, one of the smallest in Germany, had only two members. Both also happened to be of Jewish descent: Emmy’s doctoral advisor, Paul Gordan, and her father, Max Noether, a leading algebraic geometer. These circumstances were highly unusual, making Erlangen an important locale for gauging the careers of Jewish mathematicians, as will be seen in this chapter. In Erlangen, but also during her first years in Göttingen, Emmy Noether was primarily known as the daughter of Max Noether. Today he is mainly known as the father of the famous “mother of modern algebra.” Aside from this not uninteresting observation, Max and Emmy Noether have seldom been compared, even though there are plenty of indications that she studied her father’s works in detail. Moreover, careful examination of her earlier work clearly reveals streams of thought from her Erlangen period that flowed into her later work in Göttingen. Like her father, Emmy was an impressive scholar, a mathematician whose work evinced broad and detailed knowledge of the mathematical literature. In this respect, both were outstanding representatives of Germany’s high mathematical culture, to which they made fundamental contributions. Yet Max Noether has rarely received serious attention in the by now quite extensive literature devoted to his daughter Emmy. Not only in this chapter but elsewhere in this book, Max Noether’s name comes up often, and this for good reason: he was most definitely a major formative influence on Emmy Noether’s life. This chapter thus aims, among © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_1

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1 Max and Emmy Noether in Erlangen

other things, to shed a small beam of light on the relationship between these two great mathematicians, who in several respects belong together.

1.1 Max Noether’s early Career To gain a sense of Emmy’s early development, one must go back to her years in Erlangen (Fig. 1.2), beginning with her home life there as the oldest of four children and the only daughter of Max and Ida Noether. Emmy Noether’s mother grew up in a large and very wealthy family from Cologne; she was one of eleven children of Markus Kaufmann and his wife Frederike Kaufmann née Scheuer. Two of Ida Kaufmann’s brothers assisted Emmy Noether financially after the death of her parents. These were her two uncles in Berlin: Paul, a wholesale merchant, and Wilhelm, a university professor who specialized in international economics [Dick 1970/1981, 1981: 8]. After her father’s death in 1866, Ida moved with her mother to Wiesbaden, a city known for its spas and aristocratic culture. Up until that year, Wiesbaden had been the capital of the Duchy of Nassau, but having sided with the Austrians in the Austro-Prussian War it fell into the hands of the Hohenzollern monarchy. During the era of the Kaiserreich, the emperors began making annual summer trips to Wiesbaden, which led to a construction boom that continued up until the First World War. It was in this glamorous city in 1880 that Max Noether married Ida Kaufmann, who would spend the remainder of her life looking after their household in Erlangen. Although little is known about Emmy Noether’s mother, Auguste Dick reported that she enjoyed playing the piano all her life, a talent she tried to pass on to her daughter, but without success [Dick 1970/1981, 1981: 9–10]. How she and her husband first met is also unknown; since marriages were still quite often arranged during this era, the couple may have barely known one another when they wed.1 Ida Noether’s family no doubt offered a substantial dowry at the time, which surely made life in Erlangen for the young family more comfortable. Max Noether’s salary as an associate professor was considerably less than that of a full professor (Ordinarius), and he would only gain that coveted title eight years later. Beginning with the nineteenth century, the city of Erlangen belonged to Bavaria. Its citizenry was fairly equally divided between Catholics and Protestants, whereas Jews were only allowed to settle in the city after 1861. Until then, fairly large Jewish communities existed in outlying villages, where life was hard and poverty widespread. A decade later, after the unification of Germany under the domination of Prussia, 65 Jews were living in Erlangen, a city of some 12,500 inhabitants. That number steadily rose to around 240 in 1890, which was roughly 1.5% of the total population. In the meantime, Jewish life in the outlying villages nearly disappeared as the flight to larger cities took place throughout large parts 1 Marion A. Kaplan describes the era of the Kaiserreich as a transitional period for Jewish families, as they began to allow young couples limited freedom in choosing a partner [Kaplan 1991].

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of Germany.2 Here real economic opportunities awaited them, and the German Jews contributed greatly to the modernization of urban centers in nearly all parts of the German Empire. When economic misfortune struck, on the other hand, as happened in the early 1870s, the blame often fell on Jewish financiers. This was a new form of antisemitism, a hatred tinged by envy rather than the loathing of Christian society.3 To what extent Max and Ida Noether’s children were exposed to milder forms of prejudice against Jews no one will likely know. They belonged to a special elite, as the offspring of a university professor, and their parents may well have avoided talking about antisemitism in their presence. Although Ida Noether was eight years younger than her husband, she died six years before him on 9 May 1915. As the oldest of the children, Emmy thereafter took on major responsibility for running household affairs. Up until her father’s death on 13 December 1921, she often left Göttingen to care for him in Erlangen. One year before his passing, she officially left the city’s Jewish community, though her family’s religious orientation had probably never been strong.4 She and her father were not only personally close; Emmy also developed a deep appreciation for Max Noether’s place in the mathematics of his time. Emmy grew up with three younger brothers: Alfred, Fritz, and Gustav Robert. By all reports, she enjoyed a happy childhood, but her mother’s life was hardly carefree, as two of her sons had serious health problems. Alfred, the eldest, had a weak constitution and died near the end of the war at age 35. Robert, the youngest, was mentally handicapped and spent his last years in a sanatorium; he died before reaching the age of 40. Fritz, on the other hand, was healthy and robust. He and his sister were very close all their lives, though temperamentally they differed quite strikingly. Fritz was more serious and sober-minded, whereas Emmy had a fun-loving spirit. As a professor’s daughter, she looked forward to dancing parties at the houses of Max Noether’s colleagues [Dick 1970/1981, 1981: 11]. Her easy-going manner no doubt led people to overlook that she was also ambitious and self-disciplined; already as a teenager she knew that she wanted to study mathematics, perhaps even follow in her father’s footsteps [Tollmien 2016a]. When she first had such a dream is impossible to say, but since her brother Fritz had similar thoughts, it seems more than likely that both talked about such plans for the future. Moreover, in one sense they were in a privileged situation. How many teenagers could even imagine the kind of life their father led, constantly steeped in thought about matters neither they nor their mother could comprehend? Yet this was a natural part of their home life, and so they grew up knowing 2 A very similar pattern can be seen in the case of Göttingen, where the Jewish population nearly tripled between 1867 and 1885; for a detailed study, see [Wilhelm 1979]. 3 The distinction between modern antisemitism and traditional religious forms was made by Hannah Arendt in the first essay in her study The Origins of Totalitarianism (1951). 4 According to an article written by Ilse Sponsel to commemorate the fiftieth anniversary of Emmy Noether’s death, she resigned from the Erlangen Jewish community on 29 December 1920 (Erlanger Tageblatt, 12 April 1985).

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instinctively how mathematicians think and talk, and also how they demand peace and quiet to concentrate on their work. Very few published sources contain information about Max Noether’s early life, and those that happen to report on his youth invariably obtained those facts from his daughter.5 Max Noether (Fig. 1.1) was born in Mannheim on September 24, 1844 as the third of five siblings. His father and an uncle ran a well-established wholesale iron business that provided their families with financial stability. According to Emmy Noether, her father was very close with his mother, although she knew this only through him; Emmy’s grandmother died long before she was born. Max was an ambitious child and went straight into the third grade of the Gymnasium after primary school. At the age of 14, however, he contracted polio, which resulted in paralysis in one of his legs. For the next few years he could barely walk at all. He took private lessons during this time, but as Emmy reported, he also spent many hours reading. It was thus through self-initiative that “he laid the foundation for a very extensive literary and historical education. At home he took it upon himself to work through the usual university curriculum in mathematics” [Brill 1923, 212–213]. Under very different circumstances, Emmy Noether would later do the same when preparing to take the examination required for admission to a university. Max Noether’s first love was astronomy, which he pursued at the local Mannheim observatory. His first publication was a short paper on the paths of comets; it appeared in Astronomische Nachrichten in 1867 when he was still a student at Heidelberg University. More than twenty years later, having long since made a name for himself in algebraic geometry, he published a lengthy review of Henri Poincaré’s famous prize-winning study of the 3-body problem [Barrow-Green 1997]. In Heidelberg, Noether mainly studied theoretical physics under Gustav Kirchhoff. Emmy Noether commented briefly on how Kirchhoff indirectly kindled her father’s early mathematical interests by way of mapping problems in theoretical physics. These led him to Riemann’s works and then to the geometric theory of algebraic functions, which he learned by reading Riemann as well as the monograph by Clebsch and Gordan [Brill 1923, 213]. Noether needed only three semesters to complete his doctorate in Heidelberg. At that time, a dissertation was not even required, but he nevertheless submitted his astronomical work as a doctoral thesis, only to have it returned to him. In the end, Noether merely had to endure an easy “oral examination in the dean’s apartment, for which the doctoral student was obligated to supply the wine” [Brill 1923, 213]. These details we owe to Emmy Noether’s recollections of her father’s early life. In Heidelberg Max Noether also befriended Jakob Lüroth, who habilitated there after studying under Alfred Clebsch in Giessen. On Lüroth’s advice, Noether left for Giessen in 1868, a decision that would decisively influence the course of his subsequent career. Five years earlier, Clebsch had published a paper that gave a new impulse to algebraic geometry, connecting it with Riemann’s theory of al5 This

applies to [Brill 1923] as well as for [Castelnuovo/Enriques/Severi 1925].

1.1. Max Noether’s early Career

5

gebraic functions, while exploiting the notion of the genus of an algebraic curve. This concept was closely connected with Abel’s Theorem as well as Riemann’s central result, which established that the connectivity of a Riemann surface was given by the number of everywhere bounded integrals the surface supported. Clebsch identified this number as a birational invariant of the corresponding algebraic curve, which opened the way to develop a purely algebraic approach to this theory, thereby evading some problematic aspects in Riemann’s geometric approach to function theory. Clebsch pursued that goal together with Paul Gordan, who had briefly studied with Riemann in Göttingen. Their joint work led to the treatise Theorie der Abelschen Funktionen, published in 1866. Clebsch had invited Gordan to habilitate in Giessen, where he taught as a Privatdozent until his promotion to associate professor in 1865. Three years later, Clebsch assumed Riemann’s chair in Göttingen, and in 1869 Gordan married Sophie Deurer, the daughter of a professor of law in Giessen. Noether was by now strongly drawn to Clebsch, so he left Giessen to continue working under him in Göttingen. In the meantime, Clebsch had found a way to extend the notion of genus for algebraic curves to surfaces. He published this new birational invariant – later dubbed the “geometric genus” of a surface – in the Comptes Rendus of the French Academy in 1868. Originally this invariant was only defined for surfaces whose singularities were double and cuspidal curves, but Noether showed that Clebsch’s theorem on the birational invariance of the geometric genus could be extended to surfaces with more general singularities. In a letter from Göttingen, written to his future collaborator Alexander Brill on July 7, 1869, Noether soberly noted: “The work I hereby send to you, as you will see, stems from the sphere of Clebsch’s findings, though I claim for myself the ideas developed and hinted at therein” [Brill 1923, 214]. Clebsch was much impressed by Noether’s new results; he later told Brill, he would have been even happier had he found them himself [Brill 1923, 215]. Around this same time, Felix Klein came to Göttingen from Bonn to study with Clebsch, who was by now the head of a prominent mathematical school [Tobies 2019, 37–48]. One year earlier, Clebsch and Carl Neumann founded the journal Mathematische Annalen, which later would become the main publishing organ for mathematicians with close ties to the Göttingen network. Noether and Klein soon became close friends – adopting the more intimate “du” form when they addressed each other – a friendship they maintained up until Noether’s death in 1921. Although their time together in Göttingen was brief, it was also very significant for both of them. Klein left for Berlin in the fall of 1869, and then in the spring of 1870 he went to Paris, where he joined his new-found Norwegian friend Sophus Lie. Their stay, however, ended abruptly, when in midJuly France declared war on Prussia. Klein returned home quickly, joined a crew of emergency volunteers, and returned to France, where he witnessed the battle sites around Metz and Sedan, before falling ill. After spending several weeks recovering from gastric fever at his family’s home in Düsseldorf, he habilitated in January 1871 in Göttingen, under the watchful eye of Clebsch. By now, how-

6

1 Max and Emmy Noether in Erlangen

ever, Noether was already back in Heidelberg, where he habilitated in the winter semester 1870/71. During all this time, Klein and Noether corresponded regularly, not least because their mutual mathematical interests were very close during these years. The friendship that developed between Klein and Noether clearly had much to do with the fact that both enjoyed close ties with Clebsch. Three years later, in November 1872, both were deeply shocked when they learned that their revered master, who was not yet 40 years old, had suddenly died from an attack of diphtheria. Only a short time before his death, Clebsch had paved the way for Klein – who was then only 23 years old – to be appointed as the new professor of mathematics in Erlangen. In so doing, Clebsch passed over two far older candidates from his school, namely, Gordan and Noether. Klein remained in Erlangen for only three years, yet his name remains prominently associated with this university owing to his famous Erlangen Program [Klein 1872], which he published in 1872 when he joined its philosophical faculty. He was then its only mathematician, but in 1874, the year before he succeeded Otto Hesse in Munich, Klein managed to gain a second position for Erlangen. He also arranged for Paul Gordan, Emmy Noether’s future doctoral supervisor, to fill this associate professorship. This enabled Gordan to assume Klein’s chair one year later, thereby opening the door for Max Noether to fill Gordan’s post as associate professor. It was a classical case of networking, but with long-term significance, since these arrangements helped to stabilize the precarious state of the Clebsch school and its journal, Mathematische Annalen. Klein and Gordan continued to collaborate during the years that followed, often meeting in the small town of Eichstätt, which was conveniently located halfway between Erlangen and Munich. Later, and up through the final phase of Klein’s highly successful career in Göttingen, he continued to cultivate close relations with his longtime allies in Erlangen.6

1.2 Academic Antisemitism These events from the early 1870s led to an unusual situation in Erlangen. During an era when very few German Jews could hope to attain a professorship in Germany, both mathematicians on the small faculty at Erlangen University were of Jewish background. This unusual circumstance certainly did not go unnoticed at the time, and the present section attempts to gauge the effects of academic antisemitism on their careers. Unlike Max Noether, who remained a non-practicing Jew all his life, Paul Gordan converted to Christianity at the age of 18. 7 Still, in the eyes of many, a baptized Jew was not to be confused with a “real German.” 6 Over the course of their friendship, Klein and Noether exchanged some 340 letters, from which 280 are still extant in Klein’s estate (SUB Göttingen). 7 A finding due to Cordula Tollmien, who kindly sent me a copy of Gordan’s baptismal certificate dated July 1857. Tollmien points out that Gordan’s baptism took place before he began his academic studies, though nothing is known about his motives at this time.

1.2. Academic Antisemitism

7

Figure 1.1: Max Noether (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna)

This pervasive attitude surely helps to explain why both Gordan and Noether never had a chance to leave Erlangen. In Gordan’s case, he may have been quite content to stay in Erlangen since he was already a full professor, but Noether, as an associate professor, could hardly feel the same way. Yet he was passed over on numerous occasions; in some cases Klein informed him in advance that certain localities were simply opposed to any and all Jewish candidates. Over time, Noether came to realize that his best chance for promotion would likely come if Gordan were to receive an outside offer; that was Klein’s frank opinion, too. When a mathematics professorship opened in Tübingen in 1885, Noether hoped this might indeed transpire.8 A short time before, Max Noether’s friend and collaborator, Alexander Brill, was appointed to a newly established second chair there, a situation that lifted Noether’s hopes Gordan might well be chosen. 8 This was the position formerly occupied by Paul Du Bois-Reymond, who one year earlier accepted an offer from the Technische Hochschule in Berlin. Hermann von Stahl from Aachen Institute of Technology was eventually appointed his successor in Tübingen.

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1 Max and Emmy Noether in Erlangen

Instead, however, Gordan’s candidacy received no serious consideration at all, as Brill informed him in a letter from July 1, 1885: You should know that it was not the faculty or the Senate that blocked Gordan’s candidacy, nor was it the chancellor nor the government: the entire country [meaning the state of Württemberg] is currently of such a mind that a professor of Jewish origin in Tübingen is impossible. This can and will change, but as a newcomer I am unable to make the first breach in this prejudice. [Seidl, et al. 2018, 23] Brill gave no clear indications as to what was behind this disaffection for Jewish candidates. He merely stated that this specific appointment had caused a great deal of controversy because of differences between Paul Du Bois-Reymond, the previous chair holder, and the faculty, a circumstance that obviously had no bearing on the issue of antisemitism. More than likely, Brill alluded to this merely in order to explain why he, a newcomer, had only limited influence on the faculty’s decision. As a matter of fact, before this time only one mathematician of Jewish origin, Sigmund Gundelfinger, had ever been a member of the Tübingen faculty. 9 It should be noted that Brill’s prediction, according to which future prospects for Jewish mathematicians would improve in Tübingen, never materialized. Although his friendship with Max Noether apparently remained firm over the years, his general view of German Jewry became increasingly hostile, reflecting opinions held by conventional antisemites. On January 5, 1914, not long before the outbreak of World War I, Brill wrote this entry in his diary: The effect of the Jews on Germanic peoples is like alcohol on the individual! In small doses they are stimulating and invigorating, but in large quantities devastating like poison. The organism of our people requires time to assimilate them. Therefore they should be warded off because otherwise the flood from the east threatens to destroy the body of the people, like aphids attacking a plant, which will then perish. Fend them off! They know nonetheless how to smuggle themselves in. [Seidl, et al. 2018, 23-24] This theme of Germania as the victim of merciless and conspiring Jews, who threatened to invade the young nation from the East, would become a standard trope in the period after the monarchy fell in November 1918. The fact that Brill had already adopted this viewpoint even before the outbreak of the Great War suggests how deep-rooted these types of fears must have been among Germany’s educated classes. Conditions in Erlangen during the Wilhelmian age may have been more liberal, at least in some academic circles, but German Jews who managed to attain 9 Coincidentally, Gundelfinger had studied under Clebsch and Gordan in Giessen and, like them, he worked mainly on invariant theory and its application to algebraic curves. After taking his doctorate in Giessen in 1867, he habilitated two years later in Tübingen, where he was appointed associate professor in 1873. Six years later he joined the faculty at the Technical University in Darmstadt as a full professor.

1.2. Academic Antisemitism

9

professorships were acutely aware that their presence on university faculties was rarely welcomed [Kaplan 1991, 137–150]. In some disciplines, classical philology being a noteworthy example, scholars of Jewish origin had virtually no chance of advancement. Mathematics, on the other hand, was long seen as a field in which high-quality research was recognized objectively and judged accordingly. If that was the ideal, then the reality was very different indeed. In today’s universities, mathematics is strongly allied with the natural sciences, in part due to the current importance of applied mathematics. Historically, however, these relationships were by no means self-evident. During the nineteenth century, the ties between mathematics and the human sciences were, in some ways, the stronger ones. First, it should be remembered that the research interests of most mathematicians at the German universities were devoted to some branch of pure mathematics. It was not until the advent of the twentieth century that applied fields began to receive strong attention. Second, throughout most of the nineteenth century, humanists and natural scientists were colleagues in a single philosophical faculty. Mathematicians could therefore interact just as easily with philologists and philosophers as with their colleagues in astronomy and physics. Third, and perhaps most important, it was mainly the humanists who set the tone at faculty meetings and in broader forums outside the university proper. The most prominent among them spoke as Kulturträger, an elite class of intellectuals often called “Mandarins” (Bonzen). This group reached its zenith during the last decades of the Wilhelminian era. Its demise began with the fall of the German Reich, accelerating as Germany descended into Nazism; this familiar story is described and documented in detail in [Ringer 1969]. Looking backward to the early decades of the nineteenth century, mathematicians often had a stronger affinity for idealistic philosophy than for the materialism many identified with the natural sciences. The latter fields had, in any case, a lower status than the human sciences, and since mathematicians saw themselves as purveyors of pure knowledge they naturally followed the lead of their colleagues in classical philology, who were the first to establish research-oriented seminars. One seminar that was particularly influential for physics and mathematics was founded in 1834 in Königsberg [Olesko 1991]. Initially, this seminar was under the direction of the physicist Franz Neumann and the Jewish mathematician Carl Gustav Jacob (“Jacques”) Jacobi, who later went to Berlin in 1844. Jacobi, the son of a Potsdam banker, became a model figure for numerous Jewish mathematicians who pursued careers in Germany after him, one of whom was Leo Koenigsberger, Jacobi’s biographer [Koenigsberger 1904]. Koenigsberger delivered a speech in honor of his hero at the Third International Congress of Mathematicians, which was held in Heidelberg in 1904, one hundred years after Jacobi’s birth. He ended this oration by proclaiming: “We are all students of Jacobi.” This, of course, was an exaggeration, and a well-known German mathematician10 wrote to Felix Klein, 10 He

was Klein’s former student Walther von Dyck; see [Hashagen 2003]

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1 Max and Emmy Noether in Erlangen

wondering how anyone could make such a claim, overlooking Gauss and Riemann [Rowe 2018b, 28-29]. Koenigsberger, who was himself a baptized Jew, knew very well that Jacobi had to undergo baptism in order to pursue an academic career. Yet, this circumstance went unmentioned in his biography, which even ignored Jacobi’s Jewish origins.11 This reticence to address the “Jewish question” stands in stark contrast to Koenigsberger’s autobiography [Koenigsberger 1919/2015], which he published in 1919. There he went into detail about the painful conflict that many young Jews had to face in deciding whether they should be baptized. This was less the case for Koenigsberger himself, but the question plagued his longtime friend Lazarus Fuchs, with whom he studied in Berlin in the 1860s: Fuchs and I had to ask ourselves whether we should sacrifice our entire scientific life and existence because of the prevailing, narrow-minded views of the government or, after we had stripped away all religious prejudices, instead convert to Christianity. Fuchs had already let three years go by in hesitation and indecision because he had to take diverse concerns of his family into consideration, whereas I was free from such bonds, coming from a household that was hardly religious, and so my firm urging on Fuchs, who had been fearful and timid throughout his life when making important decisions, resulted in his future, too, being saved. [Koenigsberger 1919/2015, 18] If one compares this passage with Koenigsberger’s Jacobi biography, in which he completely avoided the topic of the Jewish question, one can hardly escape the impression that this topic was completely taboo during the Wilhelmian era. It seems likely that the critics of this state-sanctioned form of academic antisemitism only felt free to speak about it in private circles. In the context of careers in mathematics, one is reminded of the long struggle faced by Gauss’ student, Moritz Abraham Stern [Schmitz 2006]. Stern refused to be baptized and consequently had to wait 30 years before he was appointed full professor in Göttingen in 1859. Stern’s earlier appointment to an associate professorship in September 1848 came at a time of symbolic significance for German Jews. Many had pinned their hopes for social progress and true emancipation on the success of liberalism, a movement that lost momentum after 1848 and then fell into decline. Yet with increasing trends toward secularization, compulsory baptism gradually declined as well. Nevertheless, subtler forms of academic antisemitism continued to prevail at German universities. One only rarely finds references to the religion or ethnic background of the candidates in official documents concerning appointment procedures, but these aspects surfaced quite often in private correspondence. Such considerations often came into play, but one should not overlook 11 The only place in which Judaism comes up at all is in a letter from F.W. Bessel to C.F. Gauss, written shortly after Jacobi came to Königsberg as a private lecturer. Bessel described him as “very talented” but also tactless. He also believed he had heard that “his father was a Jew and money changer in Potsdam” [Koenigsberger 1904, 27].

1.2. Academic Antisemitism

11

another factor that obviously discouraged many talented young men who might have dreamed of a life as a university professor, namely the competition for such positions. In mathematics, there were roughly only a hundred full professorships in all the German states combined! Not until the latter half of the twentieth century did the number of positions increase dramatically. Little wonder, then, that many Jews chose to study law or medicine, fields they could practice privately, rather than trying to pursue a university career [Richarz 2015]. By the second half of the nineteenth century, the natural and engineering sciences were also opening new doors, whereas the humanities continued to prepare most of the candidates for the teaching profession and other civil service positions. University careers in mathematics thus posed daunting challenges, though this field offered better prospects for Jews than did many others. Their chances, in fact, depended heavily on attitudes regarding the “Jewish question” within the respective faculties. The fact that Göttingen later attracted many young Jewish mathematicians, especially after 1900, had much to do with the comparatively liberal atmosphere that prevailed there ([Rowe 2004], [Rowe 2018a, 171 –232]). Young Jews, like Richard Courant and Max Born, were encouraged by witnessing the extremely close friendship between David Hilbert and his Jewish colleague Hermann Minkowski, who had studied together in Königsberg. By the end of the Weimar period, Göttingen was seen in the eyes of the antisemites as the stronghold for a Jewish conspiracy within German mathematics and physics. Albert Einstein’s numerous enemies also counted him as part of the “Jewish network” centered in Göttingen.12 As noted already, the careers of Gordan and Noether were closely intertwined, despite notable differences in their temperament and outlook. Both deferred to Felix Klein as the acknowledged new leader of the Clebsch school. One of its younger members, Ferdinand Lindemann, studied briefly with Clebsch before completing his education under Klein in Erlangen. Soon thereafter, he made a name for himself by publishing the first volume of Clebsch’s lectures on geometry [Clebsch/Lindemann 1876]. During the period from 1880 to 1883, Lindemann was professor of mathematics in Freiburg. This turned out to be a turning point in his illustrious, though singularly exotic career. Near the end of that period, Lindemann found a way to prove that π is a transcendental number, a finding that had eluded Charles Hermite, who had proved the transcendence of e ten years earlier. Lindemann’s proof was confirmed by Karl Weierstrass, who quickly found a way to improve the argument. Not long afterward, Lindemann succeeded Heinrich Weber as professor of mathematics in Königsberg, a major breakthrough for him as well as for members of the Clebsch school. Up until this time, none had managed to gain a professorship at a Prussian university, as these chairs invariably went to graduates from Berlin University or their allies. Lindemann’s departure from Freiburg momentarily raised Max Noether’s hopes that he might be chosen to succeed him. Klein, who was eager to help, made 12 See

[Rowe 1986] and [Rowe 2018a, 258–261].

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inquiries, and then sent Noether his personal assessment of his friend’s chances both in Freiburg and elsewhere: I don’t yet have any definitive news from Freiburg, but I’m afraid you have little to hope for there, despite the fact that I once again vigorously recommended you as did [Theodor] Reye.13 And Tübingen, where a new full professorship has in fact been approved, will also bring you nothing, because I heard from there in no uncertain terms that your confession would constitute an obstacle.14 These are two pieces of bad news that I take no pleasure in writing. It would seem that Erlangen remains the place that offers you the best prospects. If only there were some desire to bring Gordan somewhere else! But in the eyes of the world he looks like a permanent Erlangen fixture. That’s what happens when one remains isolated and stops seeking new working relationships. I lectured him about this over the Easter holidays, but he has little desire to change. 15 A similar situation arose four years later in Giessen, where Moritz Pasch (1843–1930) was a member of the search committee called upon to fill a vacancy there. Pasch, himself of Jewish origin, informed Klein about final candidates for this position, none of whom were of Jewish extraction.16 He explicitly pointed this out by noting that he would have otherwise recommended Max Noether and Klein’s student Adolf Hurwitz. When Klein learned in 1888 that Noether would be promoted to full professor in Erlangen, he wrote him to express his congratulations, but also his personal relief that this ordeal was finally about to end.17 Three months later, he read about Noether’s new appointment in the newspaper.18 Paul Gordan’s last best chance to attain a more prominent position came five years after this. In 1893, Klein declined a call from the Ludwig Maximilians Universität in Munich, at which time he was asked to recommend candidates for this professorship [Tobies 2019, 337–338]. Instead of Gordan, Klein named Lindemann, who would spend the remainder of his career in Munich. Gordan was most unhappy about this turn of events, which temporarily strained his relationship with Klein, who was mainly focused on trying to bring David Hilbert to Göttingen.19 His plans for doing so came to fruition one year later. And so it transpired that the University of Erlangen, with its two mathematicians of Jewish origin, remained a singularity in the mathematical landscape 13 The

chair went to Jacob Lüroth, a close friend of Noether. concerns the position that ultimately went to Alexander Brill, who remained in Tübingen until his retirement in 1918. 15 Klein to Max Noether, May 29, 1883, Klein Nachlass, SUB Göttingen; reproduced in [Bergmann/Epple/Ungar 2012, 189]. 16 Pasch an Klein, 1887, Nachlass Klein, SUB Göttingen. 17 Klein to Max Noether, January 31 1888, Nachlass Klein, SUB Göttingen. 18 Klein to Max Noether, April 24, 1888, Nachlass Klein, SUB Göttingen. 19 The tensions between Klein and his longtime Erlangen friend are apparent in Gordan to Klein, December 5, 1892, Nachlass Klein, SUB Göttingen. 14 This

1.2. Academic Antisemitism

13

of Germany. By the early 1890s, Max Noether turned his attention more and more to the broader mathematical literature, whereas his colleague Paul Gordan continued to pursue his special research interests. Aside from Emmy Noether, Gordan supervised only one other doctoral student, the American Harry W. Tyler. How that came to pass reflects once again the close connections between Klein and his two friends in Erlangen. After studying at MIT, Tyler arrived in Göttingen in 1887 during a period when several young American mathematicians were flocking to Klein’s courses [Parshall/Rowe 1994]. During his first year of study, Tyler thought about moving on to Berlin, but decided instead on Erlangen. He reached this decision following a conversation with Gordan, who had come to visit Klein in Göttingen. Presumably these two friends and former collaborators encouraged Tyler to test the waters in Erlangen, and so the latter took the plunge. One of the Americans who came to Göttingen at the same time as Tyler was William Fogg Osgood from Harvard, later to become a fixture of its mathematics faculty. Osgood remained in Göttingen after Tyler left, but he was also curious to learn about mathematical life in Erlangen, which led to an interesting correspondence with his American friend. Tyler not only offered advice about how Osgood should prepare in the event he might decide to write his dissertation in Erlangen, he also contrasted his new academic environment with the one he experienced one year earlier. “I’m very glad I went to Göttingen first,” he wrote, “a first semester here would have been a mournful experience for me. . . . I understand in a measure the superiority which Klein and Gordan each credited to himself or to the other when I met the latter in Göttingen, and although I have at present more admiration for Kl[ein] I feel better satisfied not to have worked with him alone” [Parshall/Rowe 1994, 229]. Tyler had not come to Erlangen empty-handed, as Klein had given him a thesis topic closely connected with the coursework Tyler did with him. This concerned Abelian integrals over a ground curve whose only singularities were double points. Meanwhile, however, Tyler took up an independent study on resultants under Gordan, never imagining that this topic might be a suitable dissertation topic. Yet little more than one month after his arrival in Erlangen, Gordan informed him that he could use it for his doctoral thesis. This came as a big surprise to Tyler, who knew very well that he was only grazing in pastures that were long familiar to experts in algebra. He reported to Osgood that the coursework in Erlangen was in no way comparable to the offerings in Göttingen, but as compensation he had nearly daily contact with Gordan and Noether. Since the latter was unable to walk to classes due to his physical handicap, Harry Tyler and another student took turns transporting him from his home to the university in his wheel chair. A few months later, after Osgood inquired again, Tyler offered a quite lengthy estimation of the pros and cons of study in Erlangen as opposed to Göttingen. In his view, Americans who had three years to pursue graduate studies in Germany would be well advised not to spend the entire time studying under Klein, who though a brilliant lecturer was less than ideal as a role model. Klein offered his auditors the chance to gain a sweeping overview of major fields of research, but he

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only occasionally discussed a complex theorem in any detail. As Tyler saw it, “this broad view of things is very attractive and something that any student may go to Göttingen for, but there seems to me danger that in attempting to follow in Klein’s direction [one] will produce only rubbish.” He also underscored another drawback for those who chose Klein as their thesis advisor, namely, “[so] busy a man cannot and will not give a student a very large share of his time and attention; so too he will not study out of interest himself especially in the painstaking elaboration of details, preferring to scatter all sorts of seed continually and let other people follow after to do the hoeing” [Parshall/Rowe 1994, 231–232]. In this respect, Tyler considered the conditions in Erlangen far better suited for dissertation work. He had now completed his own thesis and assured Osgood that he would easily be able to do the same during his final year of study. “Anyone coming here from Klein,” he wrote, “would be sure to look at mathematical things from a new standpoint and . . . would be practically certain of a degree of interest and attention almost out of the question in Göttingen, and especially valuable when one is beginning original work. I have been and am still embarrassed by the opportunities.” This did not mean, however, that Tyler always found it easy to stay on good terms with his two teachers, who were temperamentally extremely different. “Both men,” he pointed out to Osgood, “are so peculiar and so irreconciliable that . . . [personal relations] must be cultivated with some tact especially if one tries to divide his attentions equally. So far as I know N[oether] like G[ordan] confines himself to pure mathematics – though he studied physics with Kirchhoff – and both I think run to Tiefe [depth] rather than Breite [breadth], as compared with Klein. If they have that much in common, that’s about all. G[ordan] is outspoken, irrascible, exasperating, violent; N[oether] is taciturn, serious equable, patient” [Parshall/Rowe 1994, 232]. Gordan and Noether were, indeed, very different types of personalities; though they apparently never became close friends, their relationship seems to have remained fairly collegial. A contrary opinion was once expressed by Erhard Schmidt, who taught in Erlangen for just one year, arriving in 1910 as Gordan’s successor. During a celebration in his honor in Berlin, Schmidt recalled that time from four decades earlier. He imagined this would be a rich and rewarding experience during which he would enjoy a good deal of contact with Gordan and Noether. After he arrived, however, he noticed: . . . a small difficulty in that these two luminaries had not been able to stand one another for thirty years – and with a deep antipathy that typically develops in smaller cities, where one tends to encounter the object of one’s anger on nearly every street corner. [Schmidt 1951, 22] 20 Schmidt was a witty man, and on such an occasion he no doubt enjoyed embellishing on such anecdotes from his long career. 20 Thanks

to Reinhard Siegmund-Schultze for pointing out this source.

1.3. Emmy Noether’s Uphill Climb

15

Tyler ended his long letter to William Osgood by offering the following summary advice: . . . come here if you want . . . detailed work in pure mathematics. If you want to work especially with Gordan I wouldn’t suggest any preparation unless the first volume of his book [Gordan 1885/1887]. If you had anything underway very likely it wouldn’t interest him. For Noether, on the other hand, I think it would be worthwhile to have something yourself to propose – in Abelian functions if you like or any of his subjects that you know from the Annalen as well as I could tell you. I wouldn’t advise you to come unless you feel sure your tastes lie in these directions. [Parshall/Rowe 1994, 232–233] Osgood took his friend’s words to heart. The following semester, he arrived in Erlangen already prepared to write his dissertation on a topic in Abelian functions, which he completed the following academic year. His chosen mentor, not surprisingly, was Max Noether. Presumably Noether would have also supervised Tyler’s dissertation on the topic Klein had given him, had not his impulsive colleague jumped into the fray with his own proposal for the American. Not that Paul Gordan was keen to supervise students’ doctoral dissertations, a chore he usually left to Noether. The latter served as mentor to eighteen doctoral students over the course of his career, compared with two for Gordan. Nevertheless, Gordan’s voluble personality and eccentric mannerisms made him highly popular with students as well as colleagues. Osgood, too, learned a great deal from Gordan during the year he spent in Erlangen. When he returned to Harvard, he published [Osgood 1892], an expository paper on the German symbolic methods for calculating invariants in an effort to make these accessible to the English-speaking mathematical world. Tyler’s letters to Osgood were written a decade before Emmy Noether began auditing courses at Erlangen University; still, there is every reason to believe that mathematical life there had changed very little in the meantime. No doubt she and her brothers took turns wheeling their father from home to campus on a daily basis. Surely he guided her studies during these early years, and at some point along the way she began reading [Brill/Noether 1894], the monumental historical study on algebraic functions that he and Alexander Brill wrote for the newly founded German Mathematical Society. In the following section we sketch some of the most important early influences on Emmy Noether’s life and how these shaped her outlook as a mathematician.

1.3 Emmy Noether’s Uphill Climb Mathematical talent, if it is to be realized, requires ample amounts of nature and nurture. In the case of the Noether family, this leads to an obvious question. What kind of influence did Max Noether exert on his two mathematically gifted

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1 Max and Emmy Noether in Erlangen

children, Emmy and Fritz? Although barely any documentary evidence survives that provides information about the encouragement and/or pressure they might have received from their father, it is striking that both of his children went their own way in mathematics; and both chose very different paths. It may well be that Max was an ambitious and demanding father, fulfilling what was then the standard role in those days. Still, one finds nothing that suggests his parenting had an adverse effect on Emmy and Fritz. Both began their studies in Erlangen, as did their brother Alfred, who took his doctorate in chemistry in 1909 (Fig. 1.3).

Figure 1.2: Emmy Noether’s Place of Birth in Erlangen, Photo from March 1982 (Auguste Dick Papers, 12-14, Austrian Academy of Sciences, Vienna) As a young woman (Fig. 1.4), Emmy had to overcome many obstacles before she could even begin her studies at the university.21 Since her formal schooling ended with the tenth grade, she had to concoct a plan that would later enable her to attend classes as an auditor. Her French teacher, Mathilde Koenig, may well have been a source of inspiration in this respect, since she was one of three women who received permission to audit classes at Erlangen University in 1897, the year Emmy Noether graduated from the local high school for girls. In all likelihood, Emmy was well aware of this since her father, who was then dean of 21 The

following is based on information in [Tollmien 2016a].

1.3. Emmy Noether’s Uphill Climb

17

Figure 1.3: Emmy Noether and her three brothers, l. to r., Alfred, Fritz, and Gustav Robert (Auguste Dick Papers, 12-14, Austrian Academy of Sciences, Vienna)

the philosophical faculty, had written to the Ministry seeking approval for this request. Two years later, Emmy passed the Bavarian state examinations that qualified her to teach French and English language courses at schools for young women. Characteristically, she received the highest grade (1,0) in all parts of these exams except for “classroom teaching,” for which she only received a grade of 2,0. As her biographer Auguste Dick commented, the lower grade was no doubt justified, as “even later, as a university lecturer, she would not have done better” [Dick 1970/1981, 1981: 12]. It hardly mattered. Emmy Noether apparently never applied to teach at a girls’ school, and probably never intended to do so. She merely took this exam so that she could attend university courses as an auditor, following in the footsteps of her former French teacher. Beyond mathematics, during the next three years she also studied history, romance languages, and archaeology [Tollmien 1990, 160]

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1 Max and Emmy Noether in Erlangen

What clearly set Emmy Noether apart at this early age was her strong desire to study mathematics rather than pursue a more conventional career, such as to become a school teacher. Her family supported this dream by paying for private lessons, though they must have wondered how she would ever be able to take up studies at a university. Only boys were allowed to take the examination required for admission to Bavarian universities, though pressure was mounting to open this exam in special cases to girls, if they were sufficiently prepared. In the meantime, Emmy was allowed to attend courses in Erlangen as a guest auditor. At the same time, she studied alongside her brother Fritz, who was preparing to take the Abitur examination at Erlangen’s humanistic Gymnasium. After completing 13 years of education there, he passed that exam, whereas his sister had to apply for permission to take this test at the Realgymnasium in Nuremberg. Since she had never set foot in this school, she asked for permission to attend classes one week before the exam took place in July 1903. The school director would have approved her request, but the Ministry refused to go along. As it turned out, all went well, but even after she passed this exam, it was still not certain that Emmy Noether would be allowed to take up formal studies in Bavaria. Then, two months later, Prince Regent Luitpold signed a decree that gave women the right to matriculate at Bavarian universities if they were properly certified, a true stroke of good luck. Despite this happy turn of events, Emmy decided to begin her studies at the Prussian university in Göttingen, possibly on the advice of her father. As a close friend of Felix Klein, head of Göttingen’s stellar mathematics faculty, Max Noether knew that there was no better place on earth for an aspiring young mathematician to take wing. In any event, she enrolled at the Georgia Augusta in the winter of 1903, but only as an auditor since women were still not allowed to attend Prussian universities as regularly matriculated students. This bold venture did not go well, however, possibly due to Emmy’s excessive ambition. She not only took courses offered by Klein, Hilbert, and Minkowski, but also attended the lectures of Otto Blumenthal, and the astronomer Karl Schwarzschild. It was all too much for a first semester student. When she returned home at the end of these studies, she was seriously ill. Her family then sent her to a quiet place in the countryside where she could rest and recover (Max Noether to David Hilbert, April 27, 1904, cited in [Tollmien 1990, 160]). This period of convalescence proved to be just what Emmy Noether needed, and so she afterward took up her studies again, but this time in Erlangen, where she could live at home (Max Noether to Felix Klein, November 14, 1904, ibid.). From 1904 onward, Emmy and Fritz Noether spent five semesters together at the University of Erlangen before Fritz decided to continue his studies in Munich. Both were friends of Hans Falckenberg, whose father, Richard Falckenberg, was professor of philosophy in Erlangen [Dick 1970/1981, 1981: 16]. Hans and Fritz served together during the war, called up from their respective teaching posts in Brunswick and Karlsruhe. Fritz Noether’s academic career was more conventional than his friend’s, as the young Falckenberg began, like Karl Weierstrass, by study-

1.3. Emmy Noether’s Uphill Climb

19

Figure 1.4: Earliest known Picture of Emmy Noether, ca. 1900 (Courtesy of MFO, Oberwolfach Research Institute for Mathematics)

ing law, before switching to mathematics. Although he was no Weierstrass, he found the support he needed in Erlangen, not least from Emmy Noether. Hans Falckenberg’s doctoral dissertation dealt with a topic in analysis closely related to the research interests of Erhard Schmidt, who succeeded Gordan in 1910, but then left for Breslau after just one year. Ernst Fischer, who was well versed in analysis, thus served as Falckenberg’s official advisor. At first glance, one would never imagine that Emmy Noether would have had a role in this: what did she know about the “branching of solutions of nonlinear differential equations”? Yet reading her curriculum vitae at the time she finally habilitated in Göttingen in June 1919, we find at the very end: “Finally, I would like to mention that, beyond those mentioned above, another Erlangen dissertation by H. Falckenberg was prompted by me . . . . This concerns the investigation of reality relations in connection with work by Schmidt on integral equations” [Koreuber 2015, 21]. Clearly, Emmy Noether’s eyes were wide open after she completed her own doctorate, and in this case Hans Falckenberg was the beneficiary of her new-found expertise, for which he thanked her profusely. After his stint as a private lecturer, in 1922 Fal-

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1 Max and Emmy Noether in Erlangen

ckenberg was appointed associate professor in Giessen; he became a full professor there in 1931. Both Emmy and Fritz also surely knew Emil Hilb, who grew up in Württemberg as the son of a Jewish merchant. His family later moved to Bavaria, where Hilb attended the Realgymnasium in Augsburg. Afterward he studied in Berlin and Göttingen, before taking his doctorate in 1903 in Munich under Ferdinand Lindemann. Hilb then got a teaching position at his old secondary school in Augsburg, where he worked for three years until Max Noether discovered his mathematical talent. Noether brought him to Erlangen in 1906 as his assistant. Two years later, Hilb became a private lecturer at the University of Erlangen, before being appointed to an associate professorship in Würzburg in 1909. Hilb was highly respected during his lifetime, above all because of his work on linear differential equations. He was one of four chosen speakers for a special session on uniformization theorems held during the annual meeting of the German Mathematical Society in Karlsruhe in September 1911. The other three speakers at this memorable session, organized by Felix Klein, were Paul Koebe, Ludwig Bieberbach, and L.E.J. Brouwer.22 Emmy Noether attended this meeting, where she met Brouwer for the first time. As for Hilb, he did not become a full professor in Würzburg until 1923; he died just six years later following a long illness. 23 As a mathematician, Emmy was much closer to her father than was her brother, who turned to applied mathematics. Fritz Noether’s broad interest in technical applications was already noticed by his professor of physics, Arnold Sommerfeld, who gave him a free hand to complete the last part of the four-volume study [Klein/Sommerfeld 1910] on the theory of gyroscopes. Emmy, too, studied physics in Erlangen, where her professors were Rudolf Reiger, Arthur Wehnelt, and Eilhard Wiedemann. Her early interests, however, centered on pure mathematics, especially problems in classical invariant theory, which she learned while working under the supervision of Paul Gordan. Her Doktorvater was an impulsive man with a sarcastic sense of humor. Hermann Weyl thought of him as a typical representative of the 1848 generation, the eternal “Bursche” who went around in a nightshirt that smelled of beer and tobacco ([Weyl 1935]). Gordan’s mathematics somehow matched his moody personality, too. He was one of the last great algorists; his contemporaries knew him as the “King of Invariants,” the man who proved the finite basis theorem for binary algebraic forms by showing, in principle, how to calculate a basis for the invariants and covariants of any system of forms of arbitrary degree. His colleague, Max Noether, was familiar enough with invariant theory, but from a broader standpoint, not as a specialist like Gordan. Max Noether presumably held strong views when it came to the mathematical education of his children. The fact that neither Emmy nor Fritz pursued 22 For

an account of this session and its aftermath, see [Rowe/Felsch 2019, 60–69]. was buried in the Jewish part of the Prague cemetery in Stuttgart. His wife Marianne tried to leave Hitler’s Germany with her two daughters, but only the older daughter obtained a visa and was able to emigrate to England in 1939. Marianne and her younger daughter were abducted and killed in the Treblinka extermination camp. 23 Hilb

1.3. Emmy Noether’s Uphill Climb

21

algebraic geometry, their father’s main field of expertise, would seem telling. This could very well reflect a conscious decision on the part of Max Noether to let both of them strike out on their own. In 1909, Fritz completed his doctorate in Munich under Aurel Voss with a thesis in kinematics (rolling motions of a sphere on surfaces of revolution). He spent the next year assisting Arnold Sommerfeld in finally completing the final volume of [Klein/Sommerfeld 1910]. This paved the way for a similar one-year appointment in Göttingen, where Noether assisted Carl Runge, who also took him in as a boarder. During this year, Fritz Noether befriended Hermann Weyl, a talented analyst and the rising young star in Hilbert’s fast-growing school. Weyl probably only met Fritz’s sister much later, when he was a visiting professor in Göttingen during the winter semester of 1926/27. Following this twoyear stint as a post-doc, Fritz Noether gained a position as assistant to Karl Heun at the Karlsruhe Institute of Technology (KIT). Soon thereafter, he submitted his habilitation thesis, published the following year as [F. Noether 1912], 24 and in the summer of 1911 he became a private lecturer at KIT. Noether served on the Western front until he was wounded there, after which he was employed to do research on ballistics. Once the war ended, Fritz Noether returned to KIT, where he was promoted to an associate professorship. After three years, he took a leave of absence to work at the Siemens-Schuckert electrical company in Berlin, after which he obtained a full professorship in 1922 at the Breslau Institute of Technology. He there joined Emmy Noether’s protégé Werner Schmeidler, who had succeeded Max Dehn just one year before (see Section 5.7). On 20 December 1911, only months after his appointment as a private lecturer in Karlsruhe, Fritz Noether married Regina Maria Würth, the daughter of a customs official. She grew up in a large Catholic family, originally from the Black Forest region in southwestern Germany, and her husband converted to Catholicism around the time they wed. Not long thereafter, their two sons were born: Hermann and Gottfried Noether.25 Fritz Noether was still only a private lecturer at the time he married, so he presumably relied on subsidies, either from his mother or possibly from wealthy members on her side of the family. Emmy later received financial support from two wealthy uncles, Wilhelm and Paul Kaufmann, who lived in Berlin [Dick 1970/1981, 8]. Since Emmy had every opportunity to learn algebraic geometry from her father, he no doubt wanted her to explore other terrain, and in Erlangen this left only one alternative: she would learn to do invariant theory under Paul Gordan. Indeed, her doctoral dissertation was written entirely in his algorithmic style. In her younger years, she clearly had a gift for calculation, as the task Gordan gave her was a truly daunting one: to develop the complete invariant theory for ternary forms of degree four. Whether Gordan had ever attempted this is hard to say, but one can safely say that Emmy Noether has had the last word with 24 The topic concerned the range of validity of Stokes’ law, published by G.G. Stokes in 1851, which deals with fluid flow around a spherical body. 25 The fate of Fritz Noether and his family after the Nazis came to power is described in Section 9.2.

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1 Max and Emmy Noether in Erlangen

regard to this problem. She was able to construct 331 associated covariants; who is to say whether her list was really complete? After finishing this ambitious work and publishing it in Crelle [Noether 1908], she passed her oral exam “summa cum laude.” If she herself later dismissed her dissertation as a piece of juvenalia (she once called it “dung”), it should not be overlooked that this study demonstrated her ability to manipulate algebraic expressions of a most unwieldy variety. Indeed, in her youth the “mother of modern algebra” was a first-rate algorist, and though her later publications were models for abstract clarity, these works were always grounded in a thorough mastery of relevant special cases. Noether’s mature studies reflected her deep interest in substantive mathematics; she was never content with empty abstraction. During the following eight years, Emmy Noether continued to mature as a mathematician, particularly under the influence of Gordan’s successor, Ernst Fischer, who arrived in Erlangen in 1911. Fischer grew up in a musical family; his father was a composer and taught at the Vienna Academy and his maternal grandfather was also a musician. Ernst Fischer studied mathematics at the University of Vienna under Franz Mertens, who was a leading expert in invariant theory. Before he came to Erlangen, Fischer taught in Brünn. Few sources have survived that shed real light on Emmy’s intellectual development during the period 1908 to 1914, but these were clearly important years during which her mathematical capabilities became ever more apparent to those who knew her best. Although she undoubtedly spent a good deal of time at home studying the books in her father’s library, she also continued to take part in mathematical life at the university. The directors of the mathematics seminar in Erlangen – Max Noether as well as Gordan’s successors, Erhard Schmidt and Ernst Fischer – called on her regularly to assist them with seminar lectures and practice sessions. Gradually, she began to make a name for herself as an expert in modern invariant theory, not the old-fashioned version she had learned under Gordan and by reading [Gordan 1885/1887]. This new direction in her research actually dated back to Hilbert’s papers from the late 1880s, which reoriented research in invariant theory. It was through her interactions with Fischer that Noether discovered her real strengths and interests in abstract algebra. Noether’s first biographer, Auguste Dick, aptly described Ernst Fischer’s impact on her subsequent career in these words: With him she could “talk mathematics” to her heart’s desire. Although both lived in Erlangen and saw each other frequently at the University, a large number of postcards exist from E. Noether to E. Fischer, containing mathematical arguments. Looking over this correspondence, one gets the impression that immediately after a conversation with Fischer, Emmy Noether sat down and continued the ideas discussed in writing, whether so as not to forget them, or whether to stimulate another discussion. Ernst Fischer has succeeded in carefully preserving these communications through all the havoc of war. The correspondence ex-

1.4. Classical vs. Modern Invariant Theory

23

tends from 1911 to 1929 and is most frequent in 1915, just before Emmy Noether moved to Göttingen and Ernst Fischer was drafted by the military. There can be no doubt that it was under Fischer’s influence that Emmy Noether made the definite change from the purely computational distinctly algorithmical approach represented by Gordan to the mode of thinking characteristic of Hilbert. [Dick 1970/1981, 23] Unfortunately, almost nothing seems to have survived from this correspondence between Fischer and Noether, as these documents would surely have provided many insights into their mathematical discussions and growing relationship during the critical transitional years before Emmy left Erlangen for Göttingen. Nevertheless, her publications from this period contain numerous hints and references to ideas she discussed with Fischer, so that by scrutinizing these carefully it becomes possible to draw a general picture of their mutual interests. Looking backward, Emmy always emphasized that Fischer’s presence in Erlangen had opened her eyes to a very different world of mathematical ideas, a realm that had been closed to her as a student of Gordan and her father.

1.4 Classical vs. Modern Invariant Theory Emmy Noether’s expertise in invariant theory caught the attention of senior mathematicians in Göttingen, but it was particularly Hilbert who took notice of her work. He and his colleagues soon began to contemplate inviting her to habilitate there, a plan that met with considerable resistance, as will be seen in the chapter that follows. By 1915, few mathematicians had any familiarity with classical invariant theory in the tradition of Noether’s teacher, Paul Gordan. Hilbert’s modern approach found few followers as well, in large part because he had abruptly left the field after publishing his final paper on the subject [Hilbert 1893]. Noether, on the other hand, had deep knowledge of both of these directions of research, which explains why the Göttingen mathematicians were willing to promote her candidacy for Habilitation, knowing full well that this would be a difficult struggle. In later years, Emmy Noether occasionally gave lectures on research trends in the field of invariant theory, and on one such occasion she referred to Hilbert’s earlier periodization of the subject in [Hilbert 1896]. According to this scheme, there were three distinct periods of development – naive, formal, and critical – each largely associated with the work of British and German invariant theorists, followed by Hilbert’s own works, which constituted the critical phase. 26 The fact that Noether, some thirty years later, evidently still found this tripartite scheme illuminating only confirms that she saw her own work as strongly linked with past developments. Indeed, the standard view that she and Ernst Fischer both followed 26 In [Noether 1923b, 436] she highlighted Riemann’s works as marking the onset of the critical phase for differential invariants.

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1 Max and Emmy Noether in Erlangen

in Hilbert’s wake stands in need of a corrective. While this accounts for part of the story, it should not be overlooked that both developed and refined methods they took from the arsenal of the formalist tradition, techniques due to Clebsch, Gordan and others. The field of invariant theory was first launched by Arthur Cayley and J.J. Sylvester around mid-century. The latter introduced much of its standard terminology in a paper from 1853. For a single binary form f (x, y), an invariant J is a homogeneous polynomial in the coefficients of f that remains unchanged when the variables x, y undergo linear substitutions. The latter form a group – the projective linear group– and classical invariant theory was exclusively concerned with this special group of linear transformations. One also studied covariants, which are expressions in the variables and coefficients of f (x, y) that satisfy the same property (up to a fixed power of the determinant of the substitution). During the naive period, one studied specific invariants and covariants of a given form. Homogeneous forms in the variables x, y, z were closely linked with plane analytic projective geometry, which studies the properties of figures that remain invariant under linear transformations, such as the degree of an algebraic curve. Curves of degree two correspond to a quadratic expression (in non-homogeneous coordinates) satisfying the general equation: ax2 + 2bxy + cy 2 + 2dx + 2ey + f = 0. Geometrically, these curves are conic sections that possess two invariants: a + c and ac − b2 . In the general case, this curve will be either an ellipse or a hyperbola, whereas if the discriminant ∆ = ac − b2 = 0 it will be a parabola. In older textbooks on analytic geometry, one typically finds detailed explanations about how to rotate and translate the coordinate axes in order to rewrite the above equation in simplified form: x2 y2 ± = 1, p2 q2 where the sign distinguishes the type of conic. It is then easy to see that a+c=

1 1 1 ± 2 and ac − b2 = ± 2 2 . p2 q p q

So these invariants have a clear geometric interpretation, since 2p, 2q are the respective lengths of the axes. In 1858, Siegfried Aronhold introduced a symbolic method for calculating invariants and covariants for ternary cubic forms f (x, y, z). A well-known example of such a covariant was the Hessian determinant Hf , which is also of degree three. The curve Hf = 0 has the property that it intersects f = 0 in its nine inflection points. Three years later, Alfred Clebsch showed how this symbolic calculus could be extended to arbitrary forms of any degree and with any finite number of variables. This Clebsch-Aronhold symbolic calculus afterward became the standard method used by German algebraists, thereby inaugurating the “formal period” of

1.4. Classical vs. Modern Invariant Theory

25

research on algebraic variants, following Hilbert’s scheme.27 Paul Gordan worked closely with Clebsch in Giessen, and he later became the leading exponent of invariant theory by using and refining these symbolic methods. In the meantime, British investigators, lacking any comparable system, gradually fell behind in a situation reminiscent of the competition over the differential and integral calculus during the eighteenth century. Newton’s followers were certainly less talented and productive than the Bernoullis and Euler, but the latter group could also exploit the advantages of Leibnizian notation, which made calculations far easier and much more transparent. Sylvester championed invariant theory during the years 1876 to 1883, when he taught at Johns Hopkins University. Among his lasting contributions from that time was the founding of the American Journal of Mathematics, which served to showcase works by members of the Sylvester school [Parshall/Rowe 1994]. A decade after Sylvester’s return to England, W.F. Osgood published his expository paper on the Clebsch-Aronhold notation in this journal, beginning with the following comments: This notation has been adopted generally among German mathematicians. As yet it has not, however, met with the acceptance in English literature to which its merits entitle it. The writer believes that this is due, in part, to the lack of an easily accessible and systematic exposition of the notation and of those fundamental properties of the same that render it so well adapted to the expression of functional invariants. The object of the present paper is to supply such an exposition. [Osgood 1892, 251] Passing from the naive to the formal period essentially meant shifting attention from concrete invariants to a theory concerned with all possible cases. In 1856 Cayley published the first finiteness results for binary forms, but he blundered in asserting that for forms of degree five and higher the number of irreducible invariants was necessarily infinite [Parshall 1989, 167–179]. Gordan later showed that Cayley’s assertion was incorrect and that, in fact, for any finite system of binary forms of any degree whatsoever a finite set of fundamental invariants and covariants could always be found (theoretically) so that all others would be expressible algebraically in terms of these. His methods of proof, as presented in [Gordan 1868] and later in [Gordan 1885/1887], were purely algorithmic and constructive in nature. They were also exceedingly complicated, so that subsequent attempts, including Gordan’s own, to simplify the theory or extend it to general ternary forms produced only rather meager results. Still, this does not mean that the symbolic method came to a complete standstill. In fact, one of its leading exponents after 1907 was none other than Emmy Noether, who presented a modification of Gordan’s technique at the 1909 27 Hilbert gave an introductory presentation of the symbolic method in his lecture course from 1897 [Hilbert 1993, 105–113].

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1 Max and Emmy Noether in Erlangen

meeting of German Society of Natural Scientists held in 1909, [Noether 1910], her first public lecture outside Erlangen.28 Whether or not she counted this lecture successful, it was clearly fruitful and no doubt encouraged her to become a regular participant at future conferences. Already in Gordan’s “Erlangen Program,” 29 he had noted that the symbolic method cannot furnish a complete set of invariants and covariants for forms with several variables. Eduard Study showed, however, that two fundamental theorems that applied to the symbolic method for binary forms were also valid for three variables. In Salzburg, Emmy Noether noted that Study’s results did not apply for forms with four or more variables. She motivated this by recalling that the theory of quaternary forms corresponds to the geometry of lines in space, where one has coordinates not only for points and planes but also line coordinates, which are obtained by forming 2 × 2 determinants from these. In the case of n variables, one similarly forms determinants from subsets of the variables and symbols, but these cannot all be represented symbolically. In the course of her dissertation work on invariants of ternary forms, Noether had also studied a method introduced by Franz Mertens for representing invariants of quaternary forms by means of certain normal forms. By modifying Mertens argument, Noether showed how the usual method based on determinants could be replaced by certain types of matrix products that allow for a general representation. After she gave her talk in Salzburg, Noether was approached by Emil Müller, professor of descriptive geometry at the Vienna Institute of Technology. Müller also happened to be very well versed in the work of Hermann Grassmann, and so he was able to inform Noether that her matrix products seemed to be closely related to Grassmann’s combinatorial products in his Ausdehnungslehre. She afterward carefully studied the 1862 edition of the Ausdehnungslehre and confirmed that this was indeed the case. Most authors would probably only mention such a connection in a footnote, and Noether’s papers almost always were filled with references to relevant literature. In this case, though, when she wrote up her results for publication in Crelle’s Journal, she spelled out how her matrix products and Grassmann’s were related. She even credited Emil Müller with having pointed this out to her in Salzburg [Noether 1911, 107]. Noether’s standpoint in this paper fully reflected the spirit of classical invariant theory as practiced by her teacher. Gordan’s approach to invariant theory still dominated the field up until the early 1890s, when Hilbert began recasting the entire subject in terms of general theorems for algebraic systems. Between 1888 and 1890, the beginning of what he called the “critical period,” Hilbert succeeded in generalizing Gordan’s finiteness theorem from binary forms to families of forms in any number of variables. Initially, Gordan followed Hilbert’s work with enthusiasm, but by 1890 he was going 28 From its founding in 1890,the German Mathematical Society held its annual meetings together with this older scientific society. 29 Felix Klein’s much more famous Programmschrift from 1872 was only one of several such inaugural publications, which were then a requirement in Erlangen for all newly appointed full professors.

1.4. Classical vs. Modern Invariant Theory

27

around telling anyone who would listen that Hilbert’s approach to invariant theory was “theology not mathematics” [McLarty 2012]. Max Noether certainly heard this pronouncement from Gordan himself, since he recorded it in [M. Noether 1914]. Part of what Gordan disliked stemmed from the fact that Hilbert relied heavily on existence arguments. Rather than exhibiting an algorithm or general procedure for constructing a finite basis, he showed that such a basis had to exist out of sheer logical necessity. Hilbert’s first general result, Theorem I in [Hilbert 1888], is today known as Hilbert’s basis theorem for polynomial ideals. This states that for any infinite sequence of algebraic forms in n variables, φ1 , φ2 , . . . there exists an index m such that any form φ of the sequence can be written in terms of the first m forms, that is φ = α1 φ1 + α2 φ2 + · · · + αm φm , where the αi are appropriate n-ary forms. Thus, the forms φ1 , φ2 , . . . , φm form a basis for the entire system. Today Hilbert’s Theorem I is usually stated as a central result in ideal theory, namely, that every ideal of a polynomial ring over a field is finitely generated. This new interpretation reflects the transformation from classical to modern algebra that Emmy Noether inaugurated thirty years later. In fact, she incorporated Hilbert’s Theorem I (from [Hilbert 1890]) as well as his Nullstellensatz (from [Hilbert 1893]) into an abstract theory of ideals (see [Gilmer 1981]). Hilbert’s new ideas departed radically from the traditional ground rules for research in invariant theory, and his results bore virtually no resemblance to those of Gordan and other practitioners. Little wonder that the older man felt offended and scoffed at this pretentious “theology.” Felix Klein took a decidedly more openminded attitude. When he received the manuscript for [Hilbert 1890], Klein wrote back one day later: “I do not doubt that this is the most important work on general algebra that the Annalen has ever published.” 30 He then sent the manuscript to Gordan, asking him to report on it. Klein probably anticipated some criticism, but surely not the kind of sweeping negative reaction he received. Gordan claimed that Hilbert’s Theorem I failed to meet even the most modest standards for a mathematical proof.31 Once he learned about Gordan’s criticisms, Hilbert dashed off a fierce and lengthy rebuttal,32 reminding Klein that Theorem I was by no means new; he had already proved it some eighteen months earlier in [Hilbert 1888]. 33 He also took pains to refute the ad hominem side of Gordan’s attack, namely his insinuation that Hilbert’s new proof of Theorem I was not even meant to be understood, since its author thought it sufficed to claim that no one could contradict the argument. Hilbert ended his long defense with a none-too-veiled threat that he 30 Klein

to Hilbert, 18 Feb. 1890, [Frei 1985, 62]. to Klein, 24 Feb. 1890, [Frei 1985, 65]. 32 Hilbert to Klein, 3 March 1890, [Frei 1985, 64]. 33 In fact, Hilbert’s original proof was faulty; see [Rowe 2018a, 164]. 31 Gordan

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might withdraw his manuscript: “I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised.” Klein was surely impressed by the self-assurance and brassiness that Hilbert evinced in this letter. After all, he was still only a Privatdozent in Königsberg, where Klein’s former student, Adolf Hurwitz, had taken him under his wing. Clearly, Klein had not anticipated the explosive nature of this conflict. Being well aware of Gordan’s irascibility, Klein now realized he had to handle this matter with extreme delicacy. He thus decided not to reply to Hilbert until he had first conferred with Gordan personally. More than a month passed, before Hilbert heard anything more. In the meantime, Klein arranged a meeting in early April in Göttingen, and he invited Hilbert’s trusted friend, Hurwitz, to join him and Gordan during their negotiations. Afterward, he informed Hilbert of the results in a short letter, referring him to Hurwitz for details. This began with assurances that Gordan’s opinions were by no means uniformly negative. “His general opinion,” Klein noted, “is entirely respectful, and would exceed your every wish.” 34 He then attached a postscript that contained the message Hilbert had been waiting to hear: Gordan’s criticisms would have no bearing on the present submission; these remarks should merely be construed as guidelines for future work! Hilbert thus got what he had demanded, whereas Gordan had at least been given the opportunity to vent his views. In short, Klein’s diplomacy succeeded in averting a potential crisis within the network he was trying to build around the Mathematische Annalen. Klein consistently valued youthful vitality over age and experience, and Hilbert represented the wave of the future. So while this conflict, in and of itself, had no major ramifications, it clearly foreshadowed a highly significant methodological shift that played a central role in Emmy Noether’s education as a mathematician. For her mentor, Paul Gordan, mathematics was essentially a playground for manipulating complicated algebraic formulas. As Gordan described himself in a letter to Klein: “I can only learn something that is as clear to me as the rules of the multiplication table.” 35 Although Hilbert heaped scorn on Gordan’s pronouncements, he clearly saw the larger issue at stake. His general basis theorem proved that for an infinite system of algebraic forms in any number of variables there always exists a finite collection that generate all the others, but his methods of proof were of no help when it actually came to constructing such a finite basis. Theorem I asserts that for any infinite sequence of forms, there must be an m for which the first m forms suffice to generate all the others. So Gordan’s principal complaint had nothing to do with the logic behind Hilbert’s proof; he was annoyed that the argument said absolutely nothing about the number m beyond its mere existence. Two years later, however, Hilbert managed to find a new proof that was constructive in spirit. In an elated letter to Klein, he described this latest break34 Klein

to Hilbert, 14 April 1890, [Frei 1985, 66]. to Klein, 24 February 1890, [Frei 1985, 65].

35 Gordan

1.4. Classical vs. Modern Invariant Theory

29

through which allowed him to bypass the controversial Theorem I completely. He further noted that although this route to his finiteness theorems was more complicated, it carried a major new payoff, namely “the determination of an upper bound for the degree and weights of the invariants of a basis system.” 36 When Minkowski learned about Hilbert’s latest triumph, he fired off a witty letter congratulating his friend back in Königsberg: I had long ago thought that it could only be a matter of time before you finished off the old invariant theory to the point where there would hardly be an i left to dot. But it really gives me joy that it all went so quickly and that everything was so surprisingly simple, and I congratulate you on your success. Now that you’ve even discovered smokeless gunpowder with your last theorem, after Theorem I caused only Gordan’s eyes to sting anymore, it really is a good time to decimate the fortresses of the robber-knights [i.e., specialists in invariant theory] – [Georg Emil] Stroh, Gordan, [Kyparisos] Stephanos, and whoever they all are – who held up the individual traveling invariants and locked them in their dungeons, as there is a danger that new life will never sprout from these ruins again.37 It would be a mistake to imagine that the clash between Gordan and Hilbert was merely a momentary episode that pitted two headstrong personalities against one another. For Hilbert, much was at stake methodologically. In 1890, Gordan was in no position to contest Hilbert’s formalist views, but thirty years later the Dutch intuitionist L.E.J. Brouwer launched a serious attack on them. Hilbert later published an account of these new results in his classic paper [Hilbert 1893], his final contribution to invariant theory. Here he adopted an even more general standpoint by treating invariant theory as a special case of the general theory of algebraic function fields, while underscoring the close analogy with algebraic number fields. In his introduction, he set down five fundamental principles which could serve as the foundations of invariant theory. The first four concerned “elementary propositions of invariant theory,” whereas the fifth principle asserted the existence of a finite basis (or in Hilbert’s terminology a “full invariant system”). This highly abstract formulation would later become a hallmark for almost all of Hilbert’s work under the rubric of the modern axiomatic method. Some two decades later, Emmy Noether would take up this thread of ideas again, along with other strands of algebraic work developed by Richard Dedekind and Ernst Steinitz. Inspired by her many conversations with Ernst Fischer, she was already steeped in this world of mathematical thought when she left Erlangen for Göttingen in the spring of 1915. A brief, but telling glimpse of Noether’s research interests in the period before her departure can be seen from the text she prepared from a talk delivered 36 Hilbert

to Klein, 5 January 1892, [Frei 1985, 77]. to Hilbert, cited in [Rowe 2018a, 166].

37 Minkowski

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at the September 1913 meeting of the German Mathematical Society in Vienna [Noether 1913]. She began with two remarks, the first being that the questions presented originally arose through discussions with Fischer; the second, that certain special cases in a single variable had already been answered by Steinitz and held for more general fields. This was an allusion to his classical paper [Steinitz 1910], in which the abstract notion of a field was introduced axiomatically for the first time. Steinitz therein defined key concepts such as prime field, perfect field and the transcendence degree of a field extension, while proving the theorem that every field has an algebraic closure. Emmy Noether would later take full advantage of these ideas and results, but for her Vienna lecture she dealt with fields of rational functions defined over the more restricted and also quite familiar notion of number fields. Her preliminary results and long-term goals concerned finiteness theorems for these rational function fields. This involved finding conditions that lead to a finite set of basis functions for determining all others, of which she distinguished three types: a rational, minimal, or integral basis. For rational functions in n variables, there will always exist a rational basis with at least n functions. In cases where the number of functions is exactly n, the basis is minimal. Noether then cited work of Lüroth, Castelnuovo, and Enriques in asserting that one always has a minimal basis for function fields in a single variable. In the case of two variables, the same is true if one allows for extensions of the coefficient field. For three and more variables, a minimal basis does not exist in general, but Noether noted that investigations of special function fields with this property had not yet been undertaken. She then stated a theorem relating to finite groups G, such as Sn , the symmetric group on n symbols, whose function fields possess a minimal basis (for Sn , one has the n elementary symmetric functions). In such cases, one can then produce a rational parameterization for all algebraic equations associated with G, a result that pointed to the general problem: to determine the field of equations for a given Galois group G. Noether also gave a special result for a field that possesses an integral basis, thus a finite set of polynomials so that all others depend rationally on these. She also noted the related general question Hilbert raised with his fourteenth Paris problem. These types of finiteness issues – all quite closely connected with Hilbert’s style of algebraic research – would occupy Noether for the next several years. As part of the general shift from classical to modern invariant theory, several mathematicians had begun to investigate invariants of groups other than the general linear group. Felix Klein had already pointed in this direction in his Erlangen Program [Klein 1872], where he described how geometrical investigations could be pursued systematically by developing the invariant theory for the relevant transformations groups that play an important role in geometry. At the time, this was more akin to a vision than a research program, since group theory was then still very much in its infancy; but that was about to change. Klein’s intellectual partner, the Norwegian Sophus Lie, began to develop a general theory

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of continuous groups during the 1880s and 1890s, a large-scale research program that also included investigating associated differential invariants. As for algebraic invariant theory, it should not be overlooked that Hilbert’s interest in finiteness results continued well after he had ceased to publish in this field. A clear indication of this comes from his lecture course on algebraic invariant theory from the summer semester of 1897, originally transcribed by his doctoral student Sophus Marxsen and then published in English translation in [Hilbert 1993]. Shortly before this course began, Hermann Minkowski alerted Hilbert to the fact that their former mentor in Königsberg, Adolf Hurwitz, had “found really interesting new theorems about invariants and . . . [can] prove new finiteness theorems for which other methods did not suffice.” 38 Minkowski had recently arrived in Zurich as Hurwitz’s colleague, which meant this was fresh news. Two weeks later Hurwitz wrote to Hilbert directly explaining his method, which he used to prove finiteness for orthogonal groups, but which he believed would “suffice for the proof of the finiteness of invariants of an arbitrary algebraic group of linear transformations.” 39 Whether Hurwitz’s breakthrough prompted Hilbert to offer a course on algebraic invariant theory cannot be said with certainty, but in any event these circumstances indicate that he was well abreast of the latest developments. The first part of his course focused on formal methods and results that surfaced during what Hilbert called the naive and formal periods of invariant theory. As an example of the former, he presented certain results of J.J. Sylvester on the invariants of a fifth-degree binary form, characterizing Sylvester’s counting methods as very clever but nevertheless inadequate for a general theory [Hilbert 1993, 58– 61]. Next Hilbert turned to the theory of transvectants, which opened the way to finding in- and covariants systematically.40 It was by exploiting algorithms involving transvectants that Paul Gordan was able to prove the finite basis theorem for binary forms in [Gordan 1868]. Part II of Hilbert’s course was devoted to his own work on finiteness theorems, but he also made clear that analogous results for subgroups of the projective group remained a problem for the future. Three years later, he would have the opportunity to raise this question along with several others in his famous lecture “Mathematical Problems,” presented at the Second International Congress of Mathematicians held in Paris [Hilbert 1900]. A passage from his Göttingen lectures captures nicely how Hilbert saw the evolution of invariant theory in the light of his general views on mathematical theorizing: With each mathematical theorem, three things are to be distinguished. First, one needs to settle the basic question of whether the theorem is valid, one has to prove existence, so to speak. Second, one can ask 38 Minkowski

to Hilbert, 11 March 1897, cited from [Hawkins 2000, 387]. [Hawkins 2000, 387–388] for a discussion of this letter and Hurwitz’s publication from this same year. 40 [Hilbert 1993, 61–68]; for a modern approach to transvection processes, see [Olver 1999, 58–61]. 39 See

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In other words, if one can first prove that a certain mathematical entity exists and then develop a procedure for finding it in principle, the final step of actually producing the answer is a matter of little importance for the mathematician. Clearly, Paul Gordan was utterly opposed to this view, whereas Emmy Noether would gradually adopt a similar position by exalting theory over mere calculation. Hilbert’s general interest in finiteness theorems is also reflected in his fourteenth Paris problem, where he began by announcing that Ludwig Maurer had recently succeeded in proving finiteness for any subgroup of the general linear group. Maurer’s results drew heavily on Lie theory and his main theorem represented a broad generalization of Hilbert’s. He had only sketched a proof in 1899, however, although Hilbert indicated that Maurer would soon publish a complete proof in Mathematische Annalen. When that longer version finally appeared in 1903, it contained an impressive number of results on Lie groups and their invariants, ending with “continuation follows.” Apparently this published part of Maurer’s text had already been known to Hilbert at the time of his Paris address, since the manuscript is dated 30 January 1900. In the meantime, Maurer struggled to prove the theorem he had claimed, eventually giving up. Since Hilbert took for granted that Maurer had solved this important problem, he formulated his fourteenth Paris problem as yet another generalization of finiteness for systems of invariants. As it turned out, the problem of determining which continuous linear groups lead to invariant forms that can be expressed algebraically by a finite system is very difficult to answer (see [Humphreys 1978]). In any event, Emmy Noether was one of the first to solve the analogous problem in the case of finite linear groups.42 Moreover, her proof was constructive in nature, drawing on the Galois resolvents of the group to show that a finite basis can be constructed for which the order of the group served as an upper bound. In 1914 Noether corresponded with Hilbert about this and her other work, including [Noether 1915], which elaborated on results she announced in her Vienna lecture [Noether 1913]. In May, she sent him the manuscript for this article along with a letter, noting connections with [Hilbert 1893], his fourteenth Paris problem, and Steinitz’s paper on abstract fields [Steinitz 1910]. She then added: “I have tried to deal exhaustively with the question of the rational representation of functions of an abstractly defined system by means of a basis (rational basis), 41 Ironically, Hilbert illustrated these distinctions by considering a question that Brouwer later made famous: does there exist somewhere in the decimal expansion for π a sequence of ten consecutive ones 1111111111? [Brouwer 1921]. The question itself surely had no interest for Hilbert other than to illustrate his general views on methodology. 42 [Noether 1916a], in which her proof held for fields of characteristic zero. Ten years later, in [Noether 1926], she extended this result to fields of characterisitc p.

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and from there also to gain points of attack for the treatment of the finiteness problem. So new finiteness theorems have emerged; with conditions of a different kind than those that could be mastered up to now.” 43 However, she decided to omit the results announced in her Vienna lecture on the equations associated with a prescribed group because this would have required introducing new terminology. So she wished to publish this separately (it later appeared as [Noether 1918c]). Hilbert’s earlier works on invariant theory involved real or complex numbers, the standard number fields of classical mathematics. Noether, on the other hand, was trying to expand the terrain of finiteness theorems by adopting Steinitz’s abstract theory of number fields, which would henceforth play a major role in much of her work. Another noteworthy feature of this paper is the generous praise Emmy Noether bestowed on Kurt Hentzelt44 a student in Erlangen who wrote his dissertation on polynomial ideals and resultants. Hentzelt perished in Belgium during the early months of World War I, but Noether kept his memory alive while recasting his work in a more conceptual form.45 In late 1914, Hilbert wrote to Max Noether with a query about the invariants associated with a certain system of forms, for which he conjectured that one could construct a finite basis by an operation known as polarization. 46 One year earlier, Hilbert had posed the same question in a short paper [Hilbert 1914], which appeared in a volume honoring the Berlin mathematician Hermann Amandus Schwarz. He received a reply, but from Emmy Noether rather than her father.47 She informed him that his conjecture was indeed correct, as could be easily proven using Franz Mertens’ technique for expanding the Clebsch-Gordan series or by using methods introduced by Alfredo Capelli, after which she sketched the proof. Hilbert, not surprisingly, was most impressed. One year later, when he supported her application to habilitate in Göttingen (see Chapter 2), he paid tribute to this unsolicited and also unanticipated solution. He apparently invited her to write this up for publication in Mathematische Annalen, and one month later she submitted her paper [Noether 1916b]. This consisted of two parts, the first showing that Hilbert’s conjecture followed immediately from a reduction theorem that Franz Mertens had established by generalizing an older method due to Clebsch and Gordan. Yet Noether was not merely content to answer Hilbert’s question. In the second part of her paper, she gave a more conceptual proof of this reduction

43 E.

Noether to Hilbert, 4 May 1914, Nachlass Hilbert, SUB Göttingen. [Noether 1915, 163]; Hentzelt’s work on ideal theory was the principal source for [Hermann 1926], the dissertation by Grete Hermann, Noether’s first doctoral student in Göttingen. 45 In [Noether 1921a] and [Noether 1923a]; see [Koreuber 2015, 98–99]. 46 See [Hilbert 1993, 95–96] or [Olver 1999, 34–35]. At Hilbert’s request, Emmy Noether explained the operation to Einstein in May 1916; see Chapter 3. 47 E. Noether to Hilbert, 1 December 1914, Nachlass Hilbert, SUB Göttingen. 44 See

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theorem based on methods set forth in [Fischer 1911] as well as other results that Ernst Fischer communicated to her privately [Noether 1916b, 94]. 48 But there was more in her letter to Hilbert: she also informed him about another manuscript on an entirely different topic, namely, a generalization of Ernst Zermelo’s theory of entire transcendental numbers [Ebbinghaus 2007, 133]. She wrote that: “in it I show which of the basic properties of a Zermelo domain result from the special construction and which consequences of the abstract definition are required in order to obtain the most general construction.” In spirit, this new paper did have some similarity with her earlier work, except that now the basis for representations was no longer finite. Hilbert’s student Georg Hamel had shown in 1905 that it was possible to find an (uncountable) infinite basis for representing the real numbers as a vector space over the rational numbers Q. Such a basis of real numbers {αν } has the property that for any real number β, there will be a of these basis elements along with a unique set of uniquely given finite subset ανiP n numbers ri ∈ Q such that β = i=1 ri ανi . One can introduce such a Hamel basis in the more general setting of infinitedimensional vector spaces, but their existence depends on invoking Zermelo’s axiom of choice, long a controversial principle in set theory. Zermelo’s theory of entire transcendental numbers invoked the principle of well-ordered sets, which is equivalent to the choice axiom, and Noether’s work was based on the same types of arguments, only on an even higher level of abstraction.49 This was the lofty topic Noether chose to speak about for her trial lecture to the philosophical faculty on 19 November 1915 (see 2.3); her paper [Noether 1916c] was already submitted on March 30 and in the meantime she also completed a sequel, [Noether 1916d], which was prompted by a query from Edmund Landau. This concerned determining the properties of functions that are isomorphisms between two number fields, a concept she noted was already to be found in §161 of Dedekind’s Supplement XI [Dedekind 1894a], where they are called “permutations of a field.” Here she proved that, aside from the identity and complex conjugation, all other isomorphisms are “extremely discontinuous” in a sense made precise in this paper. Seen against this background, it becomes clear why Hilbert had such high regard for Noether’s abilities. She had already accomplished a great deal before her arrival in Göttingen, in particular in [Noether 1915], which combined the formal methods of Fischer and his teacher Franz Mertens with the modern ideas of Steinitz as well as those Hilbert had introduced in his seminal paper [Hilbert 1893]. Emmy Noether’s work on invariant theory soon caught the attention of others, as Hilbert began to spread the word. Little wonder that Hilbert was willing to go to great 48 [Noether 1916b] was reviewed by Franz Meyer in Jahrbuch über die Fortschritte der Mathematik, but with little attention to her results. Much of the review concerns two results in [Hilbert 1914], one of which Meyer considered obvious, the other an immediate consequence of a theorem Meyer had published years earlier. The question Noether answered was easily settled as well. In [Hilbert 1933, 392] the editors added a footnote stating that Hilbert’s conjecture had already been answered affirmatively by Giuseppe Peano in 1882. 49 For a discussion of her work in relation to Zermelo’s, see Ulrich Felgner’s introductory note to Zermelo’s paper in [Zermelo 2012, 274–277].

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lengths to bring her to Göttingen, despite strong opposition, as will be seen in the chapter that follows. Yet comparing [Noether 1915] from her Erlangen period with the lofty perspective Noether took in [Noether 1923b], her survey article on algebraic and differential invariants, the contrast is immediately apparent. It would take several more years before she made the full transition to the abstract style that became her calling card after 1921. In closing this section, we cite the opinion of Colin McLarty, who offered this synopsis of how Noether’s work stood in relation to that of her predecessors: . . . Emmy Noether . . . was in the most obvious sense a joint heir of Gordan and Hilbert. And she passionately sought to unify all mathematics in an algebraic axiomatic way. Corry has shown how Hilbert’s axiomatics are never purely formal, nor even aim to found new subjects but always aim “to better define and understand existing mathematical and scientific theories” [Corry 2004a, 161]. He aimed to organize classical subjects by paring each problem down to its stark essentials. For that very reason his axioms always have reference. They refer to the classical structures that motivate them. Gordan’s algebra, on the other hand, was in his own terms “purely symbolic” so that “no meaning can be assigned to it.” . . . Noether’s axiomatics combined the two. Her axioms create new subjects. They need not have classical referents. They are generally taken to have no specific referent, and sometimes understood to create new referents for themselves. But there is no use grappling with those conceptual ontological issues until we can make it as clear as one times one equals one how all of this is Mathematics. [McLarty 2012, 125]

1.5 Max Noether’s Career in Retrospect The mathematical world of Paul Gordan and Max Noether harkened back to the time of Alfred Clebsch, whose posthumous influence continued on through the journal he co-founded, Mathematische Annalen. Gordan and Noether both played important supporting roles with the Annalen [Rowe 2018b, 37–44], whereas Felix Klein assumed principal responsibility for the journal after Clebsch’s death. However, by the 1890s Klein was intent on consolidating his power as Göttingen’s senior mathematician. This placed him on a collision course with his old mathematical allies in Erlangen, both of whom strongly identified with the Clebschian tradition. Emmy Noether had surely heard about the famous conflict between Gordan and Hilbert, whether from Ernst Fischer or directly from her father, and she likely knew that Klein had intervened to mediate in this dispute. As editor-in-chief of Mathematische Annalen, Klein increasingly took on the role of diplomat and visionary, leaving the routine work of editing the journal to others. Diplomacy in

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mathematical circles, however, can be a difficult matter, as Klein came to realize when he tried to forge new alliances while maintaining older ones. At the beginning of the new century, Klein wanted to place the Annalen firmly in the hands of his younger Göttingen colleague Hilbert, who had been a member of the editorial team since 1898. Klein had long been the driving force behind the journal, but he now proposed to resign from its editorial board and give Hilbert full authority to manage its affairs. He evidently hoped that the once sharp tensions between Gordan and Hilbert had by now subsided enough that no one in Erlangen would raise objections to this plan. Instead Klein received a letter from Max Noether pleading that he should reconsider: H[ilbert] is for a rejuvenation suitable, maybe even indispensable. But in matters concerning the DMV [German Mathematical Society] he has shown himself to be quite stubborn and one-sided, almost personal. You can perhaps judge him more accurately in this direction. Scientific interests alone can also affect a person’s judgment, and some heads are never open to argument. I don’t particularly like everything that he has included in recent issues of the Annalen . . . . – Under these circumstances [Gordan and I] would like to request that you remain on the editorial board for some time, and . . . that you retain control of its management. G[ordan] even wishes for a kind of counterweight against an over-representation of work in Hilbert’s direction. (Max Noether to Klein, 21 February 1901, cited in [Rowe 2018b, 43]) This private vote of no confidence in Hilbert’s leadership qualities prompted Klein to change his plans. In fact, he would, retain his position as de facto head of Mathematische Annalen until shortly before his death in 1925. This anecdote clearly shows that Max Noether’s opinions carried significant weight for Felix Klein. Noether wrote his most important works in the 1870s and 1880s. He was considered to be the leading representative of algebraic geometry in Germany. Later he was referred to as the last major algebraic geometer in Germany, as after around 1890 Italian mathematicians dominated this field of research [Klein 1922, 7]. Together with Brill, Noether undertook the daunting task of writing a historical report on the development of the theory of algebraic functions for the German Mathematical Society (DMV) [Brill/Noether 1894]. Their joint work on this project took them three years to complete. Brill dealt with the older history up to Riemann, while Noether reported on subsequent developments. The DMV had originally enlisted Leopold Kronecker to write about the arithmetical direction of research, but this part of the report had to be dropped following Kronecker’s unexpected death. In his obituary article, Brill reported how Noether “opted for a comparative treatment of the well-known theories,” leaving aside the work of Kronecker and the arithmetical theory initiated by the famous paper of Dedekind-Weber from 1882 [Brill 1923, 228 -229]. Toward the end of her father’s life, Emmy Noether wrote a

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much shorter report under the title “The arithmetic theory of the algebraic functions of a single variable in relation to other theories and to the theory of number fields” [Noether 1919a]. At the outset, she noted that this should be understood as a supplement to [Brill/Noether 1894] as well as “a binding link between the individual theories of algebraic functions” [Noether 1919a, 182]. Anyone who reads this would easily recognize that Emmy Noether had studied the literature on this subject thoroughly and mastered it completely. It was precisely the arithmetical direction of research that attracted her most, so one must imagine that she and her elderly father had interesting discussions about the relative merits of the various approaches to algebraic functions. Max Noether celebrated the 50th jubilee of his doctorate in Erlangen on March 5, 1918. Emmy surely played a key role in making the arrangements for this event, a major milestone in her father’s long career. Klein, of course, was among those who sent congratulations, to which Noether replied one week later: 50 I was pleased that you remembered the day of my 50th doctoral jubilee with heartfelt words. How everything during that space of time compresses together for me! Above all the decline and receding of directions that we considered indispensable, but also their passing over to other hands, in France and Italy. What remains, though, is our shared view of things, and that traces back to our strong point of departure with Clebsch. Such a bond is stronger than the effects of later influences; the influence [of Clebsch] was much stronger than I realized at the time. Here in Erlangen some very hard years are now behind me, but now that I have officially resigned from teaching the difficulties will perhaps gradually diminish. That our mutual relations have been revived again through the activities of my daughter has been a great source of satisfaction for me; I see every day how her creative powers grow and hope that these will lead to many new finds.51 Max Noether’s most significant single achievement – often called his Fundamental Theorem or the AF + BG theorem – was an essential tool for the Brill-Noether theory [Gray 2018, 256–257]. His daughter’s account of that theory showed how its concepts can be rewritten in purely algebraic language by drawing on Dedekind’s theory of ideals [Noether 1919a, 197–201]. One must imagine that this was one of the first texts she advised young B.L. van der Waerden to read when they met in Göttingen in 1924 (see Section 4.4). Emmy surely had a clear picture of Max Noether’s importance and reputation as an algebraic geometer, at the latest by 1908. With her new doctorate then in hand, she accompanied her parents on a trip to Rome, where they took part in the Fourth International Congress of Mathematicians. On this occasion, the Guccia medal was awarded to Francesco Severi. Corrado Segre presented the 50 Max

Noether to Felix Klein, 13 March 1918, Nachlass Klein 12, SUB Göttingen. just one day before, she sent Klein a postcard (Fig. 3.1) outlining the key ideas behind her classic paper [Noether 1918b]; see Section 3.5. 51 Indeed,

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report on Severi’s work in the name of the scientific committee, consisting of him, Henri Poincaré, and Max Noether. This medal was donated by Giovanni Guccia, the founder of the Circolo Matematico di Palermo, a prominent Italian mathematical society with some thousand members worldwide [Bongiorno/Curbera 2018]. Emmy Noether also became a member on this same occasion. Just as Guccia cultivated international relations through the Circolo Matematico, so was Max Noether deeply interested in mathematics far beyond the borders of Germany. This is particularly evident from the scientific obituaries he published over many years in Mathematische Annalen. Except for the one written for his colleague Paul Gordan, all the others were dedicated to foreigners: the three British mathematicians Arthur Cayley, James Joseph Sylvester, and George Salmon; the two Italians Francesco Brioschi and Luigi Cremona; the Frenchman Charles Hermite; the Norwegian Sophus Lie; and the Dane Hieronymus Georg Zeuthen. 52 When Noether died in December 1921, Felix Klein, writing in the name of the editorial board, characterized this achievement in the following words: “[Noether] honored the memory of nine deceased mathematicians, masters with whom he was a kindred spirit, in extensive analytical-critical obituaries that form monuments, which together constitute an important contribution to the history of modern mathematics” [Klein 1922, 9]. Noether himself was so honored with an obituary in the Annalen, penned by Italy’s three leading Italian algebraic geometers: Guido Castelnuovo, Federigo Enriques, and Severi [Castelnuovo/Enriques/Severi 1925]. For details about Max Noether’s life, they relied on information from Emmy Noether. It was probably in connection with this obituary that these three Italians came up with the idea of presenting Emmy with a gift that surely meant a great deal to her. We only know about this, however, from a later source, a letter written by Emmy’s colleague at Bryn Mawr College, Marguerite Lehr [Kimberling 1981, 56]. She recalled that on a wall in her office Emmy had hung a beautiful scroll. It depicted a metaphorical tree of algebraic geometry with many branches identified by various famous names. Prominent among them was a branch labeled “Max Noether.” Emmy’s father would not live to see his daughter’s most creative work, which had only just begun when he died in 1921. But he did experience the satisfaction of witnessing her enormous first successes in Göttingen during the war years, crowned by her Habilitation in June 1919.

52 Information on these obituaries and other scientific works by Max Noether can be found in the list of publications in [Castelnuovo/Enriques/Severi 1925].

Chapter 2

Emmy Noether’s Long Struggle to Habilitate in Göttingen 2.1 Opportunities for Women in Göttingen, 1890–1914 Doctoral degrees have a long prehistory, but the modern Ph.D. first arose as part of an educational reform launched at the German universities. Over the course of the nineteenth century, this degree came to be awarded not merely to those who displayed a command of established knowledge in an academic field. Instead, faculties reserved this highest academic title for those with a demonstrated ability to pursue original research, an innovation promoted especially by the Prussian universities, where this so-called “research imperative” also led to a major reform in teaching practices. A candidate for the Ph.D. was thus required to submit a doctoral dissertation which the faculty deemed to be an original contribution to a scholarly discipline.1 In some instances – C.F. Gauss and Sofia Kovalevskaya were two such cases – dissertation research was undertaken elsewhere. The norm in mathematics, however, called for intensive study with a mentor, who often dictated the student’s topic of investigation. As this trend grew, the professor’s role as a teacher shifted, too. Alongside the traditional task of imparting knowledge came another, namely to train future researchers. Those especially adept at doing so exerted a strong influence on the direction of research they chose to practice and promote. One of the most influential mentors for mathematicians was the Berlin analyst, Karl Weierstrass, who became practically a cult figure in the 1870s and 1880s. Weierstrass published little, using his lectures as the primary vehicle for presenting his latest work. Notes from his courses circulated widely, but to understand them was very difficult without attending; even for many who did, this, 1 As noted in the last chapter, when Max Noether took his doctorate in 1868 in Heidelberg this famous old university in the state of Baden still did not require a written dissertation.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_2

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too, did not really help. Adding to Weierstrass’ mystique was the special relationship he cultivated during the early 1870s with the brilliant young Russian, Sofia Kovalevskaya, who studied with him privately (as a woman, she had no opportunity to take courses at the university). Her subsequent career, unique in the annals of mathematics, began when she was awarded a doctorate in absentia from Göttingen University in 1874 [Koblitz 1983]. Ten years later, following a tumultuous period in Russia that led to her husband’s suicide, she rebounded in Sweden, where she was appointed to a professorship at Stockholm University. Her colleague, Gösta Mittag-Leffler, also made her an associate editor of Acta Mathematica, a leading international journal. In 1888, the French Academy of Sciences awarded Kovalevskaya the Bordin Prize for her work on gyroscopic motion. 2 Then, three years later, it all ended tragically with her death at age 41. With it came an outpouring of various reflections about what many thought was an unhappy life, a cautionary tale for any young woman who might wish to pursue a career in mathematics [Kaufholz-Soldat 2019]. By the decade of the 1890s, this topic fit into a wide-ranging discourse about the role of women in modern societies.3 At this time, resistance to opening the Prussian universities to women was strong and widespread [Tobies 1991/92]. Felix Klein recalled how in 1891 Göttingen’s Kurator, the university’s head administrator, refused his request to allow Christine Ladd-Franklin to attend his lectures (she had studied at Johns Hopkins under J.J. Sylvester and Charles Sanders Peirce ). “That’s worse than social democracy,” Klein was told, “since it merely wishes to abolish differences in property. You want to eradicate the differences between the sexes” [Tobies 2019, 369]. Outside Germany, a small number of women began to study mathematics at some of the elite English and American colleges [Green/La Duke 2009, Green/La Duke 2016]. An important pioneering figure was Charlotte Angas Scott, who studied at Girton College, Cambridge from 1876–1880, after which she finished eighth in the Tripos examination. In 1885, Scott joined the founding faculty at Bryn Mawr College, where she taught until her retirement in 1924. One of her doctoral students, Isabel Maddison, had studied at Girton alongside Grace Chisholm, and both did graduate work under Felix Klein in Göttingen. For Chisholm this led to a Ph.D [Grattan-Guinness 1972] in 1895, the first awarded in any field to a woman enrolled as a regularly matriculated student at a Prussian university. Klein had convinced the Prussian Ministry to pursue this experiment with foreigners at a time when pressure was mounting to admit women into leading universities in Europe as well as in the United States [Fenster/Parshall 1994]. 2 Emmy’s brother Fritz may well have read Kovalevskaya’s famous paper on the motion of a special type of top. Some of his early work dealt with gyroscopes, though mainly in connection with real physical problems. For a detailed analysis of Sofia Kovalevskaya’s mathematical work, see [Cooke 1984]. 3 On general conditions for men and women at Göttingen University during the Wilhelmian era, see [Tollmien 1999].

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By this time, women had already gained the right to study at Paris University, though none had yet taken a higher degree in mathematics. Grace Chisholm arrived in Göttingen in the fall of 1893, along with two young American women, Mary Winston and Margaret Maltby; the latter would later take her Ph.D. in physics under Walther Nernst. Klein invited all three to attend his lectures and personally facilitated their applications for permission to attend his courses in Göttingen. This time the Ministry overruled the Kurator, who resigned not long afterward. Still, the situation was anything but simple. In requesting permission for Chisholm to take final oral exams (the Rigorosum), Klein emphasized that this petition carried no further implications. This case will not establish a precedent for the rights of women to be admitted to doctoral studies. This request for special permission for Chisholm is based on her exceptional qualities but also out of fear, should it be rejected, of an exodus of foreign female students. Chisholm’s doctorate should also serve as a warning signal to the male fellow students in order to shake them out of their self-righteous laziness.4 In 1895, toward the end of her stay in Göttingen, Isabel Maddison wrote to Bryn Mawr about Chisholm’s triumph, but she also offered a vivid account of the kind of difficulties she and the other female auditors encountered. She reported that in the academic year 1894–95 fifteen women were attending courses in the philosophical faculty – three from Great Britain, eleven Americans, and one German – and seven of them were studying mathematics, astronomy, and physics. None, however, had been allowed to matriculate as regular students, so they were keenly aware of their status as “guinea pigs” in an educational experiment. Indeed, Madisson emphasized that: . . . one should not conclude from this rapid advance in numbers that Göttingen is opening wide its doors and welcoming all comers. This is far from being the case. By no means are all the professors in favor of the admittance of women; some utterly refuse to lecture to women, others take up a neutral position, and those who are most willing to extend a University training to women are most anxious that it shall be done with discrimination, that the women students shall come well-grounded and prepared for hard work and that no element of dilettantism shall appear. (Newsletter, undated, Bryn Mawr College Special Collections) Clearly, much depended on the attitudes of individual professors. Moreover, even the more liberal-minded among them took a wait-and-see attitude. This applied, in particular, to the question of admission to final oral exams. This had never been an issue with regard to German women, since they could not attend the secondary schools that administered the Abitur, the certificate required for 4 Universitätsarchiv Göttingen Kur.Alt.4.I.147: Zulassung von Frauen zur Promotion und Habilitation.

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admission to a German university. Once foreign women began studying toward a doctorate, however, the Göttingen philosophical faculty was forced to confront this issue. Rather than adopting a general policy, it opted to treat each case individually, in effect treating these as exceptions to the rule. Since there were numerous opponents of female higher education in the philosophical faculty, those candidates who successfully completed their dissertation had to hope that they would be allowed to take the Rigorosum, an oral examination administered by three or more professors in three subjects (often mathematics was combined with astronomy and physics). All three of the women who entered in 1893 managed to get over that hurdle, and after passing their exams left Göttingen with doctoral degrees. Three Americans who studied under Klein came from Cornell University: Virgil Snyder, J. Henry Tanner, and Annie MacKinnon. Snyder and Tanner would return to Cornell, where they became fixtures of the faculty. MacKinnon, one of three women who took doctorates under James E. Oliver at Cornell, would marry and give up doing mathematics. Oliver had earlier taken courses under Klein, thereby forging this link between their two programs. During her stay in Göttingen, MacKinnon met Edward Fitch, who taught classics at his alma mater, Hamilton College, a liberal arts school for men in Clinton, New York. Fitch was studying with the renowned classical philologist Ulrich von WilamowitzMoellendorf, under whom he took his Ph.D. in 1896. He afterward returned to Hamilton, where he spent the remainder of his career, joined by MacKinnon, whom he married in 1901. Before coming to Cornell, MacKinnon had grown up in Kansas, where she studied mathematics at the university in Lawrence. She took her master’s degree there in 1891, while teaching at the local high school, but continued her studies under Henry Byron Newson, Mary Winston’s future husband. Only a few of the women who studied abroad later gained regular teaching positions at colleges in the United States, and even fewer were hired by American universities. David Hilbert’s first female doctoral student was the American Anne Lucy Bosworth, who defended her dissertation in 1899. Unlike Klein, who tried whenever possible to maintain cordial relations with more conservative colleagues, Hilbert preferred direct and open confrontation. Knowing that Bosworth’s candidacy might meet with resistance in the faculty, he wrote up notes under the heading “Über Frauenstudium” in preparation for this meeting [Tobies 1999]. A passage from these notes reads: Among you, my gentlemen, there are some who indeed hold an unfavorable opinion of female students. I ask you, however, to refrain from activating that view with respect to the field of mathematics. . . . If you had been in our seminar yesterday, you would certainly have been amazed at the enthusiasm and temperament with which a lady can speak about mathematics – it was a lecture delivered by a Russian woman (ibid.).

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The woman in question was Ljubova Sapolsky, who took her doctorate under Hilbert in 1900 on a topic in number theory. Evidently her candidacy was criticized due to her lack of competence in Latin; Hilbert dismissed this as a mere accident of her nationality, while praising her independence and her mastery of the only foreign language he considered relevant in such a case: German. [Tobies 1997, 41]. In the summer semester of 1895, Klein and Hilbert offered a joint seminar on the differential calculus that drew 17 participants, six of them women. These six female students came to Göttingen from England, the United States, and Russia. All of them performed exceedingly well and went on to take doctorates under either Klein or Hilbert. One year later, Klein commented briefly about their performances in a survey that inquired about the future prospects for women who wished to pursue academic studies: I am all the more pleased to answer this question because of the still prevailing view in Germany that for females mathematical studies must, in any case, be virtually inaccessible, an opinion that remains a major obstacle to all efforts aimed at improving higher education for women. Here I am not referring to extraordinary cases, which do not prove much as such, but rather to typical ones based on our experiences in Göttingen. Without going into details, I only want to point out that in this semester, for example, that no fewer than six women took part in our higher mathematical courses and exercises, in which they continued to show that they are in every respect comparable to their male counterparts . . . . [Kirchhoff 1897, 241] All four mathematicians who responded to this survey expressed similar positive views, though even the more liberal-minded had obvious reservations. One respondent answered by documenting various prominent female mathematicians from antiquity to the present, but he then ended with a lengthy account of Kovalevskaya’s final years filled with sadness. Much of the popular literature devoted to the brilliant Russian in the years following her death had this same ambivalent quality, as myths and controversies about her life spread. For many, she stood as a warning to those women who imagined they could find happiness while competing in an intellectual field that was quintessentially suited for a select group of men. Women might entertain the idea of studying mathematics, but the notion that a woman might go on to become a research mathematician was, for nearly all at this time, almost unthinkable [Kaufholz-Soldat 2019]. Compared with Great Britain and the United States, Germany lagged far behind in the movement to create educational opportunities for young women. After 1900, various German states allowed females to take the final examination required for graduation from Gymnasien, which remained an exclusively male domain. Since the curricula at the corresponding schools for females fell far below the standards at a Gymnasium, candidates had to undertake intense private studies to prepare for these exams. Emmy Noether did so by studying alongside her

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brother, Fritz, as he prepared for his final exams, and in 1903, she passed the Bavarian Abitur-exam (see 1.3). Some, on the other hand, recognized the need to raise standards of mathematics education for young women at the secondary schools. Klein considered this a major failing of the German educational system, which systematically hindered progress in this direction. Speaking to a group of Gymnasien teachers, he offered this opinion: First, one teaches women as little mathematics as possible, and then one wonders afterward that mathematical thinking seems alien to them, . . . Gentlemen, for a long time foreign countries can teach us that the woman is just as capable of mathematical thinking, as it is taught at the high schools for boys, as is the man. [Klein 1907, 45]

2.2 Habilitation as the Last Hurdle Seen from a long-term perspective, habilitation was a ritual of enormous significance for the German universities. As semi-autonomous institutions with close ties to church and state, these institutions maintained the right to confer academic titles but also to admit new members following age-old customs. In many ways, these traditional universities bore a striking likeness with craft guilds, and since several of them dated back to the late Middle Ages the resemblance was hardly fortuitous. From them emerged the research-oriented universities of the eighteenth and nineteenth centuries, one of the first being Göttingen’s Georgia Augusta, founded in Hanover in 1734. Yet even these enlightened institutions, steeped as they were in a neohumanist ethos, tended to reinforce older patterns of training reminiscent of medieval guilds. Neohumanism exalted the cultures of classical antiquity, a tradition reflected in the curriculum of the Gymnasien, which included heavy dosages of Latin and Greek. Since these elite schools for boys served for many decades as the exclusive pathway to entrance in the university system, those who taught at these institutions of higher learning took for granted that an educated person – such as those they encountered in their courses – possessed reasonable command, if not complete mastery, of these ancient languages. These capabilities in foreign languages were thus part of the standard repertoire of the educated class of citizens – the Bildungsbürgertum – which helps to account for the special status enjoyed by professors of classical philology, such as Göttingen’s Ulrich von Wilamowitz-Moellendorff. The young men who came to the university – most of whom were intent on pursuing careers outside of academia – entered a world utterly apart from the strictly disciplined atmosphere of the Gymnasium. Ideally at least, professors and students were expected to be motivated entirely by a spirit of higher learning enshrined in the twin principles of Lehr- und Lernfreiheit, the freedom to teach and to learn. There were no exams, no set requirements, not even a standard

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curriculum. Professors of mathematics, to be sure, taught introductory courses on subjects that every student was expected to know, though their content depended heavily on the tastes of the instructor. Teaching was understood as an expression of the personality of the Dozent, who performed more as a role model than as an instructor in the modern sense. The subject matter was mainly conveyed in courses called Vorlesungen, which in former times had been just that – a Dozent reading out loud from a text. This later evolved into a freer form of oral communication, which might be tied to a text, but usually was not. Students were expected not just to take notes, but to expand on them and, if possible, further develop the ideas set forth during lectures. They might also read textbook literature on the side, but the main focus was working on the lecture and producing an elaboration (Ausarbeitung) of its contents. Clearly, this system assumed a great deal of self-initiative and enthusiasm for learning by doing that had tremendous appeal for a certain type of intellectual elite. While free to explore new frontiers, students were expected to develop a deep sense of purpose and seriousness, an ethos that an older generation of scholars would then pass on to the next. This devotion to Wissenschaft (scholarship and science) emphasized the autonomy and self-sufficiency of the academic disciplines, ideals that had long made the inner sanctum of the universities rather similar to life in a religious order or a special type of guild. The “apprentice-student” often studied at a number of universities before being promoted to a doctor of philosophy. As capstone for a course of study, he presented a doctoral thesis (Inauguraldissertation), which had to be successfully defended in a Disputation. This originally sufficed for habilitation, the procedure adopted for admitting new faculty members in the seventeenth century. By the nineteenth century, however, a second doctoral thesis was generally required. A post-doctoral candidate thus became a “journeyman-doctor” during this period of preparation for habilitation. To reach this first rung of the academic ladder, it was necessary to present the faculty with a finished “masterwork,” the habilitation thesis (Habilitationsschrift). If this was found to be satisfactory. the candidate was awarded the venia legendi, which granted him the right to lecture as a Privatdozent, but without salary. He was merely entitled to collect the usual course fees from students, which were minimal. Since it was next to impossible for a Privatdozent to maintain himself without substantial private means, this system discouraged young scholars whose families were less wealthy from pursuing academic careers. During the last decades of the nineteenth century, a bottleneck developed that left many in this state of limbo for many years. Hilbert had to wait six long years working as a Privatdozent in Königsberg before he was appointed associate professor there. Klein never had to face such a struggle: he acquired his doctorate at age 20, habilitated two years later in Göttingen, and was only 23 when he became full professor in Erlangen. In 1894, when he was dean of the Göttingen faculty, he engineered Hilbert’s appointment within a matter of weeks. One of his colleagues criticized him for wanting to appoint an easy-going, amiable (bequem) younger man. To this Klein replied: “I want the most difficult of all” (Ich berufe

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mir den allerunbequemsten) [Blumenthal 1935, 399]; he knew how difficult Hilbert could be. Temperamentally, Klein and Hilbert stood poles apart, though they seldom came into serious conflict with one another. Klein was the public face of Göttingen mathematics, the man who built its stellar staff and first gave the Georgia Augusta its central position in the modern mathematical world. He was also a universalist and an outspoken advocate for upholding the common interests within a unified philosophical faculty [Tobies 2019, 418–419]. On one occasion, he gave a muchpublicized speech in hopes of garnering support for his position [Klein 1904], but relations between humanists and scientists in the faculty steadily worsened during the years afterward. Hilbert took little interest in the “tasks and future” of the faculty at large, particularly since he had more than enough to do in pursuing his own vast research agenda. This was anything but a lonely pursuit, and by 1900 his courses were beginning to attract many gifted students from all over the world. A decade later, Göttingen was literally swarming with talented doctoral and post-doctoral students in mathematics, supported by the largest teaching staff of any German university. Hilbert and his wife Käthe were on friendly terms with several Jewish families, and they were especially close with Hermann and Guste Minkowski (Fig. 2.2). Their liberal views and lifestyle stood out in this small university town, whose guardians of civic virtue expected professors to appreciate that their social station demanded observance of traditional norms of behavior. Hilbert was having none of it, and with time his opposition to provincial attitudes became ever more evident and outspoken. Klein and Hilbert were well aware that Emmy Noether’s candidacy for Habilitation would face stiff opposition. Not that anyone was likely to challenge her general qualifications or mathematical competence, on the contrary. The issue at stake was actually very simple and straightforward: could a woman habilitate at one of the Prussian universities? In fact, this question had already been raised back in 1907 when a similar case arose at Bonn University. This concerned the zoologist Maria von Linden, who in 1908 gained a position at the new Institute of Parasitology, but without the right to teach since she lacked the venia legendi. Her case came up one year before the Prussian Ministry of Education decreed on August 18, 1908 that its universities would henceforth be open to qualified females. Maria von Linden’s petition to habilitate was forwarded to the Ministry, which decided to canvas the university faculties throughout Prussia as to whether the clause prohibiting women from Habilitation should be dropped from the regulations. The results of that survey showed that nearly all were opposed to making any such change. As a result, the Ministry decided not to change the Habilitationsordnung, a decision that spelled the end of von Linden’s candidacy. Nevertheless, in 1910 she received a titular professorship as a researcher in Bonn [Tollmien 1990, 165]. Within the Göttingen philosophical faculty, however, several professors indicated that exemptions should be allowed in exceptional cases. In fact, nearly as

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many wanted to alter the regulation as those who supported upholding its prohibition of women. When a vote was taken, the latter group prevailed by only a single vote. Yet even more telling, particularly as a portent for the future, was the composition of these two opposing sides. For with the exception of the historian Max Lehmann, all those who supported the possibility of allowing women to be considered as candidates for habilitation were mathematicians or natural scientists, whereas those who were in the opposing camp were nearly all from the humanities. After this vote was taken, the losing side submitted a minority report, composed by Hilbert, Runge, Lehmann, and the physicist Woldemar Voigt, and supported by Klein and Minkowski, the chemist Gustav Tammann, and the geophysicist Emil Wiechert. Its contents are less noteworthy than the response it elicited, in particular from the historian Karl Brandi, a proponent of the traditional, allmale university culture: . . . I also believe that the minority report makes it necessary not only to emphasize that the previous scientific production of women in no way justifies making such a deep change in the character of the universities, but also to express once again that a great many of us in principle regard the entry of women into the organism of the universities as an impediment to the human and moral influence of the male university teacher on his hitherto largely homogeneous audience. I must confess that already in a mixed auditorium I feel a curb on the kind of complete lack of self-consciousness so necessary for our activity, so that I do not wish to dispense with the friendly tone of forthright expression and forthright trust. Our instruction should have a personal character and in my view, gender uniformity is required for this to be fully effective. These views were certainly shared, in one form or another, by a large portion of the German professoriate, very few of whom had ever encountered the opposite sex in their own educational experience. A particularly noteworthy feature of this debate, however, was the way these sharp differences of opinion fell along disciplinary lines within the philosophical faculty. Not that this was unusual. Göttingen’s humanists and scientists often found themselves on opposite sides of an argument, and in 1910 they agreed to go their separate ways by forming two separate departments: one for history and philology, the other for mathematics and natural sciences. Klein had long opposed such a division, but most felt that the divergence of interests between the two groups was too great to warrant maintaining a unified faculty. It soon transpired, however, that this arrangement left the underlying policy differences unresolved. The situation became even more intense after the outbreak of the Great War, which brought forth strong political antipathies as well. Under these circumstances, it became increasingly clear that many of the earlier disputes and conflicts between these factions were bound to continue.

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2.3 Noether’s Attempt to Habilitate These earlier events were still very fresh in mind when Emmy Noether (Fig. 2.1) accepted the challenge of trying to habilitate in Göttingen. Klein realized, of course, that the university would have to seek an exemption from the Ministry of Education to evade the formal prohibition of women in the regulations. The first order of business, however, would be to convince reluctant colleagues in the philosophical faculty that Noether posed no threat to the status quo. To what extent she was made aware of the circumstances remains unclear, but in any event on 20 July 1915 she submitted her application to the Mathematics and Natural Sciences Department of the Philosophical Faculty of the University of Göttingen, along with the required fee of 100 Marks. Exactly one week earlier, she had delivered a lecture in the Mathematical Society on finiteness questions in invariant theory, a performance that presumably convinced any remaining doubters among those who heard her speak. As her habilitation thesis, she submitted the paper “Fields and Systems of Rational Functions” [Noether 1915], which she had sent to Hilbert one year before. On the very day Noether submitted her application, Edmund Landau, who was then head of the department as well as dean of the faculty, called a preparatory meeting to discuss how to proceed. As usual, a commission was formed, some of whose members agreed to write reports evaluating Noether’s abilities as well as her suitability for membership in the faculty. This commission consisted of; the four mathematicians Landau, Klein, Hilbert and Constantin Carathéodory; the applied mathematician Carl Runge; the physicists Woldemar Voigt and Peter Debye; and the astronomer Johannes Hartmann. Because of the larger implications of Noether’s case, those present also decided to invite the Egyptologist Kurt Sethe, who was head of the historical-philological department at the time, to be a member of the commission as well. Sethe accepted, but he took no active part in any deliberations in accordance with a general understanding that matters concerning habilitation proceedings were to be handled exclusively by the responsible department. The four mathematicians in the commission agreed to submit reports, which served as the basis for discussion when the members reconvened in October. One week after this meeting, Klein wrote to the Ministry of Education with some personal observations about the prospective candidate. Recalling the one semester she had spent in Göttingen over a decade before, he then thought that she was attempting something she could not attain. What changed his mind was a lengthy visit two years before, when Emmy accompanied her father, who wanted to gather information from both her and Klein in preparation for writing his obituary of Paul Gordan [M. Noether 1914]. Klein and Gordan had been close collaborators in the 1870s and had remained in fairly close contact for many years afterward, whereas Emmy had been one of only two mathematicians who wrote dissertations under the cantankerous Gordan. The discussions were no doubt far-

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ranging, but also consequential for Emmy, who greatly impressed Klein on this occasion. “I saw to my surprise,” wrote Klein, that she not only had a full command of one of my earlier research fields, the theory of quintic equations, but that she could inform me about several details that were new and gave me great satisfaction. Since then I am convinced that Miss Noether fulfills by all means the conditions that we customarily require of those who apply to habilitate, indeed that she is superior in quality to the average candidates that we have admitted in recent years.5 Their conversation evidently concerned Klein’s book on the icosahedron and fifthdegree equations [Klein 1884], an enduring classic in the mathematical literature. 6 Among the reports offered by the four mathematicians, one finds significant divergences of opinion, although all reached the conclusion that the committee should recommend to the department the approval of Noether’s application. Only Hilbert, who was appointed principal reporter (erster Gutachter), treated Noether’s candidacy in a gender-free fashion, in accordance with his conviction that scientific abilities and accomplishments were the only relevant criteria. Under normal circumstances, the philosophical faculty would have routinely rubberstamped this decision, passed it on to the Kurator, who might have added a few brief remarks about the candidate before sending the application to the Prussian Ministry of Education for final approval. In short, the rest would have been a formality. All members of the commission were acutely aware, though, that Noether’s case was anything but normal. Indeed, the mathematicians’ initiative raised a general issue that had been debated earlier in the Göttingen philosophical faculty, which remained sharply divided over whether a woman could under any conditions be allowed to habilitate. Many of the commission members had mixed feelings about this as well. In contrast to Hilbert, Landau’s report defended the traditional conception of an academic faculty as a body that needed to scrutinize the personal qualities of applicants for membership with an eye toward ensuring harmonious cooperation. In this regard, he emphasized that a number of male applicants had been rejected as unsuitable for personal reasons. Landau thus conceded that the highly subjective quality of “suitability” had to be recognized as an important factor, though his report contained no direct indications why this was relevant in Noether’s case. Indeed, since his concrete remarks regarding her character were entirely positive, one can safely conclude that, for Landau, her sex was the only real issue at stake. “How easy this decision would be for us if this were a man with exactly these accomplishments, skills as a lecturer, and earnest ambitions. I would much prefer it, if this extension of our teaching program could be accomplished without the habilitation of a lady” [Tollmien 1990, 176]. 5 Klein 6 For

to Ministry, 27 July 1915, Nachlass Klein 2G, SUB Göttingen. a brief introduction, see [Gray 2018, 171–175].

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In line with the consensus view, Landau noted Noether’s highly exceptional mathematical abilities, though he did so by drawing on his uniformly negative experiences with female students, especially when it came to their productive capacities. “I think the female brain is unsuitable for mathematical production,” he wrote, “but I regard Miss N[oether] as one of the rare exceptions” (ibid.). Landau did not consider her to be a genius, not even a first-rate scholar, and then went so far as to predict that she would be accorded much unjustified fame, just as had been the case with Sonya Kovalevskaya. Still, he could not begrudge giving her what she was due, namely the chance to habilitate. Landau’s report was the least enthusiastic of the four, mainly because his general hostility to female mathematicians strongly colored the views he expressed in it. Yet even Klein took a rather reserved view of the matter, stating at the outset of his report that he had no wish to advocate women’s rights to pursue academic careers. He stressed that Noether’s case was entirely exceptional, distancing himself from “those who generally recommend the extension of academic studies to women,” including opening the doors to habilitation. Instead, Klein argued simply from the point of view of self-interest: Noether was an extraordinary talent, so acquiring her services would serve to strengthen mathematics in Göttingen. In support of this, he noted that she was better qualified than the average candidate accepted by the faculty in recent years [Tollmien 1990, 175]. Carathéodory’s report, which was likely written after conferring with Klein, took essentially the same position. He affirmed Emmy Noether’s special qualifications, while emphasizing her knowledge of invariant theory and her background as a student of Max Noether and Paul Gordan. He further pointed out that classical invariant theory had fallen out of favor after Hilbert managed to solve its central problem in one fell swoop (see Section 1.4). The result was that the generation to which he and his colleague Landau belonged had little idea of the subject, which for many years was unjustifiably devalued. Alluding to Hermann Minkowski’s mathematical formalization of Einstein’s special theory of relativity, Carathéodory wrote: “. . . now the time seems to have come [. . . ] when invariant theory is called upon to form the basis for the youngest and deepest physical theories.” As a former student of Minkowski, it should come as no surprise that Carathéodory emphasized this connection in describing Emmy Noether as “someone who has picked up the broken thread again after 20 years and not only understood the existing theory in the best sense of the word, but who has also added something new and valuable.” This, he added, was no mere coincidence, since she received her training in Erlangen, one of the last places where classical invariant theory was still cultivated.7 Largely echoing Klein’s argument, Carathéodory concluded: 7 As further indication of this trend, one can point to the career of Eduard Study who, unlike Hilbert, continued to work on invariant theory. Remaining copies of Study’s important book, Methoden zur Theorie der ternären Formen, first published in 1889 by Teubner, had to be discarded by the publisher for lack of buyers. Decades later, when classical invariant theory enjoyed a true Renaissance, Springer produced a new edition in 1982 [Brieskorn/Purkert 2018, 708].

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It is, in my opinion, not to be assumed that there is anyone else in the world today, who would be obtainable and who could replace Miss Noether. As the matter stands, Miss Noether is a singular phenomenon, who can be beneficial for the further scientific development of the university, and this is the main reason why, disregarding everything else, I am in favor of opening this habilitation procedure. [Tollmien 1990, 177]

Figure 2.1: Emmy Noether, ca. 1920 (SUB Göttingen, Sammlung Voit, no. 4) Unlike the other three reporters, who only had a general impression of Noether’s work, Hilbert had studied several of her papers carefully. So he was uniquely qualified to offer a competent judgment of her abilities, not least because her most recent publications represented a continuation of his own earlier researches. He thus characterized her habilitation thesis [Noether 1915] as “the successful execution of a part of the large program that I set forth with regard to finiteness questions” (ibid.). He also wrote about his delight that she had been able to prove a result he had recently conjectured concerning the finiteness of a system of infinitely many basis forms. Finally, Hilbert emphasized her versatility in applying formal theoretical methods, as demonstrated in her recently published paper [Noether 1916c]. As the summer vacation loomed, Noether’s Habilitation-Commission only met again on 29 October, when the winter semester 1915/16 was underway. Since all four reports praised the candidates qualifications and recommended that the

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department request an exemption from the exclusion clause in the habilitation regulations, there was presumably little to debate at this point in the procedure. Landau moved that the commission recommend approval of Noether’s application, and all members with the exception of the astronomer Johannes Hartmann voted in favor. After this, the matter was taken up by the mathematics-natural sciences department. During the course of its deliberations, it became apparent that others shared Hartmann’s reservations and opposed the commission’s findings. Whereas proponents stressed the exceptional nature of Noether’s case, opponents argued that it would establish an undesirable precedent. When the final vote was taken, the motion to accept the commission’s recommendations passed 10 to 7 with 2 abstentions.8 Afterward, some of those who voted with the losing side announced they would file a minority report.9 Arrangements were also made for Emmy Noether to deliver a public lecture before the Mathematical Society, which interested members of the philosophical faculty were invited to attend. Although largely a formality, this lecture provided an opportunity for the faculty to judge whether she possessed the requisite teaching skills to be appointed as a private lecturer. Three days later, on 9 November 1915, Noether spoke on transcendental integers, the topic of her paper [Noether 1916c]. Afterward, she wrote to Ernst Fischer: “Even our geographer came to hear it and found it rather too abstract; the faculty wants to make sure it’s not going to be duped at the meeting by the mathematicians” [Dick 1970/1981, 31–32]. The results from the earlier meeting on November 6 were also communicated to the historical-philological department, which met on 10 November to discuss their implications. Afterward, the head of the department announced that this was a matter of urgent concern to the entire faculty, “. . . in view of the fundamental importance of the present case, which would be a complete novelty of the greatest importance for German university life.” Joint deliberations of its two sections were particularly justified in view of the fact that, while “the general responsibility in habilitation matters lies with the individual departments,” some members of the mathematics-natural sciences department had themselves expressed the wish for a joint meeting [Tollmien 1990, 171]. 8 The

ten who voted in favor were the commission members Carathéodory, Debye, Hilbert, Klein, Landau, Runge and Voigt as well as the chemists Gustav Tammann and Adolf Windaus, and the geographer Hermann Wagner. The seven opponents were Johannes Hartmann, the zoologist Ernst Heinrich Ehlers, the agronomists Conrad von Seelhorst and Wilhelm Fleischmann, the minerologist Otto Mügge, the experimental psychologist Georg Elias Müller, and the geophysicist Emil Wiechert. The two who abstained were the chemist Otto Wallach and the botanist Gottfried Berthold. They later explained their position in a text submitted to the Kurator. Both stated that they did not know Noether personally and did not feel competent to judge whether she was indeed uniquely qualified and could only make her talents useful to science by means of a proper habilitation; at the same time, they recognized and shared the general concerns about allowing a woman to join the faculty. [Tollmien 1990, 171]. 9 This was later filed as a dissenting opinion (Sondervotum) of Profs. Ernst Ehlers, Wilhelm Fleischmann, Johannes Hartmann, Otto Mügge, Georg Elias Müller, Konrad von Seelhorst, Emil Wiechert.

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This meeting of the entire faculty then convened on 18 November in a tense atmosphere. Edmund Landau presided as dean of the faculty. Although minutes from this meeting have not survived, it clearly turned into an explosive session, during which Hilbert apparently lost his temper. At one point, he scolded the humanists on the other side, an incident remembered by his perhaps legendary remark: “gentlemen, we’re in a university and not a bathing establishment” (“meine Herren, wir befinden uns an einer Universität und nicht in einer Badeanstalt”). 10 Whether or not he actually uttered these words, the discussion must have become unusually heated, as Hilbert definitely did criticize two members of the other department for their inability to focus on the merits of Noether’s case. This was documented by Cordula Tollmien, who reproduced a draft of Hilbert’s letter of apology to these two colleagues, the classical philologists Richard Reitzenstein and Max Pohlenz, both of whom had filed a written complaint with Landau, in which they charged Hilbert with uncollegial behavior [Tollmien 1990, 178]. Evidently, Hilbert’s attack on them had included remarks, which they interpreted as an invitation to leave the meeting. He denied any intention “to personally insult any of my colleagues” or to expel any faculty member. What he meant to express was the view that university professors should concern themselves with scientific matters alone, leaving social and political pursuits to the world outside. Regarding collegiality, Hilbert drew a line when it came to “protecting real interests.” In his draft of this communication, however, Hilbert struck out a highly significant passage that read: In order to prevent misunderstandings, I must add that it was my full intention to say that in the past 20 years the historical-philological division, under the leadership of the classical and Germanic philologists, has at every opportunity ([matters concerning] matriculation, foreigners, women, doctoral candidates, habilitations) tried to thwart by all means possible my efforts, which are solely aimed at promoting science. Even after the division of the faculty and the changes in personnel, I still thought I had to assume that this attitude persisted, and the reason for it is to be found in the vast and, I believe, insurmountable gulf that separates my view of the professor’s responsibility to science and that of my opponents. To appreciate these remarks, which illuminate the whole context of events that surrounded Noether’s habilitation, we only need recall the clear signs of the polarized relations between scientists and humanists in the Göttingen philosophical faculty discussed above. Relations between these two bitterly opposed groups only worsened over the course of the war years, during which the Noether affair was anything but forgotten. This can be seen from a petition, filed by the 10 There appears to be no credible source asserting that Hilbert made this remark, although Hermann Weyl treated this as an established fact. In his memorial address for Emmy Noether, he speculated further that “probably he provoked the adversaries [the humanists in the faculty] even more by that remark” [Weyl 1935, 431].

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historical-philological department toward the end of the war, which sought to create two distinct faculties. The humanists’ stated reasons for doing so included these remarks: Ever new and serious conflicts are to be expected, especially since, beyond disagreements over appointments, there has been and still remains no lack of matters for such conflicts, which are rooted in the completely different views and principles of the two sides. We only mention the questions involving foreign students, the educational background of students, and the admission of women to habilitate, an issue already raised in 1915 by the same group led by Prof. Hilbert. [Dahms 1999, 401] These disputes thus reflect marked differences in the political orientation of scientists and humanists in Göttingen, a “two cultures” phenomenon that strongly colored the overall academic atmosphere. Emmy Noether surely realized that a large part of the philosophical faculty was unalterably opposed to her desire to become a fellow member. Beyond this particular conflict, there were a number of others, in particular over various candidates who were being considered for chairs in philosophy. In all these battles, Hilbert played a pivotal role; indeed, he was singled out by the humanists as their nemesis and the ringleader of the opposing side. As noted in [Rowe 1986, 447], this situation clearly indicates the limitations of Fritz Ringer’s assumption that “in their attitudes toward cultural and political problems, many German scientists followed the leads of their humanist colleagues” [Ringer 1969, 6]. Far from following in the footsteps of their colleagues in the humanities, the Göttingen scientific community actively opposed the policies and mentalities of their colleagues in the humanities, who represented Ringer’s academic mandarin caste. Returning to the faculty meeting on 18 November, this was called at the request of the historical-philological department, though obviously not to discuss Noether’s qualifications. Hilbert’s outburst stemmed from his strongly held conviction that these should have been the only relevant issue, a view most of his allies clearly did not share, as revealed by the three other reports that Landau, Klein, and Carathéodory prepared for the habilitation commission. As for the outcome of the meeting, this hinged on the results of two motions put before the faculty. First, a preliminary vote was taken on the question: “who opposes the admission of a woman to habilitate under all circumstances?” The outcome – 17 to 14 with one abstention – clearly revealed the depth of the opposition. Since the scientists outnumbered the humanists by 19 to 13, this vote demonstrated that the astronomer Hartmann was hardly a lone voice of dissent: in total, seven of the 19 scientists categorically opposed extending the right of habilitation to women. Among the humanists, only two out of 13 voted against this resolution, one of these being the liberal historian Max Lehmann, a long-time ally of the Hilbert faction [Tollmien 1990, 172]. The faculty then voted on a second motion, which recommended that the Minister of Education (Kultusminister) reject the initiative taken by the mathematics-natural sciences department. Since this motion clearly

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represented direct interference of one department in the affairs of another, three supporters of the first motion decided to abstain during this second round of voting. The result was thus 14 votes for and against, which placed the outcome in Landau’s hands as dean. As a result, this resolution was defeated by the slimmest of margins. Hilbert clearly felt vindicated by this turn of events. In a letter from his department to the Ministry from 4 December 1915, signed by Hilbert, he noted that the faculty as a whole had defeated a motion recommending that the Minister refuse to allow Noether’s habilitation and, furthermore, that the faculty’s philological- historical department also abandoned its original plan to submit a counter-presentation [Tollmien 1990, 179]. Having survived this confrontation, the mathematics-natural sciences department now had a free hand to pursue this matter and did so by convening on the very same day. The five mathematicians – Landau, Klein, Hilbert, Carathéodory, and Runge – formulated a petition that would go to the Ministry, and this draft was approved by a vote of 10 to 6 with one abstention. The following day, the six who lost that vote filed a minority report. Their main argument against Noether’s candidacy was simply the fear that it would mark the beginning of a domino effect: . . . it cannot in any way be denied that with the admission of this first woman the question of whether women are allowed to habilitate at all would be answered in the affirmative. This would open up a new life path for numerous female students and the scientific level at the German universities would undoubtedly fall as a result of this increasing feminization. All members of the faculty agree – and those in the majority expressly concede this – that only in the most exceptional cases can a female head produce creative scientific achievements. But a woman, in particular, is altogether unsuitable for regular instruction of our students because of the phenomena connected with the female organism. With the admission to habilitation would, in principle, also follow admission of women to the next stages of an academic career, to professorships, and consequently membership in the faculty and the senate. For it would be an obvious hardship if women were permitted to begin an academic career, but were then prevented from progressing afterward. [Tollmien 1990, 173] One hardly needs to read between the lines to recognize the entirely selfserving nature of this document. If nothing else, however, it fully confirms that those who supported it simply wished to preserve the university as an exclusively male sanctum. That being so, the logic behind its argumentation was unassailable: even if habilitation was by no means the final barrier women would encounter, it was the decisive one. As a kind of postscript, the astronomer Johannes Hartmann added quite another kind argument that brought the situation in Germany in November 1915

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into clear focus. His considerations touched on the role of women in social and family life as well as the future lives of the young men presently serving their country in the Great War. If Germany is now able to successfully face a world of enemies, we owe it in large part to our German women and mothers, who have brought up large numbers of sons. Every measure that furthers the equality of women and facilitates their independent attitudes and lifestyles brings with it certain dangers for family life, for fulfilling those tasks from which women cannot be relieved but which to some women, once they have turned their interest to scientific work, might seem uncomfortable. In particular, a regular academic teaching position can hardly be reconciled with the tasks of a married woman. In the interest of our offspring, it would surely be undesirable if mentally superior women were more and more removed from family life. – Quite apart from this general consideration, the current point in time would seem particularly unsuitable for taking a step with such far-reaching consequences. While Germany’s sons must carry out the bloody business of war far from their home place of work, we have in many instances necessarily utilized women to replace the missing male workforce. As welcome as this kind of help is, it would be reprehensible if a whole great profession, which till now was exclusively practiced by men, were given over to women without any need. Our private lecturers returning from the field, each of whom sacrificed a greater or lesser part of their health for the fatherland, would surely greet such competition, which had emerged during their absence, with very mixed feelings. Precisely because academic teaching is one of the few professions which a man with a war injury can practice almost without any difficulties, we should expect that some students returning from the field will want to pursue this career. It would be irresponsible and deeply troubling if in this very profession their prospects were made more difficult due to competition from women. [Tollmien 1990, 173–174] The authors of the department’s petition had ample time to read and react both to the minority report as well as Hartmann’s emotionally charged addendum to it. They took pains to explain that in requesting an exemption for Emmy Noether from the prohibition of women in the decree of 1908 they were in no sense contesting the legitimacy of that stipulation. They noted, however, that it was issued shortly before women had been allowed to matriculate. Nevertheless, the burden of their argument rested on the contention that Noether’s talents were so exceptional as to make her case unique and, hence, in no way would it constitute grounds for subsequent applications. Moreover, as further substantiation of the latter claim, they underscored that this matter had not come about through her own initiative but rather she had been encouraged to apply for habilitation by mathematicians on the faculty. Their encouragement was apparently formalized

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after she delivered her lecture on November 9, which was deemed “pedagogically successful.” With regard to the minority report, the majority merely noted that its signers recognized Noether’s scientific qualifications, which were examined by a special commission, so their dissent was based solely on general reluctance to admit a woman. The majority further affirmed that “Miss Noether’s achievements exceed the average level of private lecturers in mathematics who had previously been accepted in Göttingen.” They then turned to Hartmann’s arguments, particularly his contention that allowing a woman to habilitate in wartime amounted to de facto discrimination against war veterans. The majority asserted that, on the contrary, Noether’s candidacy “poses no threat to men returning from the field or to future private lecturers in mathematics.” First, the department had never instituted a numerus clausus for lecturers in mathematics. Second, an effort had been made even before the war to recruit new lecturers in mathematics, but this search had failed to turn up even one candidate who met the department’s high standards. Third, although Miss Noether would only be covering a part of the present gap in course offerings, there was little chance that another woman would be recruited any time soon. The reason for this – as noted in the minority report – was the shared opinion that “only in exceptional cases can a female head be creatively productive in mathematics, let alone display Miss Noether’s achievements” [Tollmien 1990, 164]. Hilbert (Fig. 2.2) probably anticipated that the Ministry would not go along with the mathematicians’ request. As a precaution, he therefore signaled his willingness to compromise in the letter from December 4 cited above. This was written just after he and Einstein had ended a flurry of correspondence in November. During this time, Einstein published four famous notes that culminated with his new gravitational field equations, whereas Hilbert submitted his first note on general relativity [Hilbert 1915] (see Section 3.2). In his letter, Hilbert briefly alluded to these fast-breaking developments, adding that “here I have Miss Emmy Noether as my most successful assistant.” He then went on to say that if the Minister should be disinclined to approve her habilitation, he would like to have the opportunity to discuss the situation personally [Tollmien 1990, 179]. This communication bypassed Göttingen’s Kurator, Ernst Osterrath, the official responsible for transmitting faculty decisions to the Ministry. In the past, Klein and Hilbert had often taken up direct negotiations with the Ministry, in keeping with Friedrich Althoff’s administrative style as head of the Prussian university system. Hilbert was also surely aware that the Kurator opposed allowing women to habilitate. He thus had reason to be concerned that his negative opinion might tip the scales against Noether. On 9 December, Osterrath submitted the department’s petition as well as the other related documents to the Ministry along with brief remarks indicating why he could not support this proposal. Rather surprisingly, he named Klein rather than Hilbert as its prime initiator. Doing so surely lent gravitas to the whole matter, since Klein was unquestionably the leading spokesman for all matters touching on

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Figure 2.2: Hilbert’s 60th Birthday, 23 January 1922: l. to r. Auguste Minkowski, David Hilbert, Ernst Hellinger, Käthe Hilbert, Lily Minkowski, Emma Schoenflies, Ruth Minkowski, Paul Bernays, Hanna Schoenflies, Peter Debye, Franz Hilbert, Hilbert, Franz (1893–1969) Ferdinand Springer, Otto Blumenthal (Archives of the Mathematisches Forschungsinstitut Oberwolfach)

mathematics education in Prussia. Since 1908 he represented Göttingen University in the Prussian House of Lords (Herrenhaus), where he occasionally gave speeches on educational matters. Osterrath was already appointed Kurator before the division of the philosophical faculty in 1908, so he knew very well that Hilbert was a dogged fighter whose various liberal causes rankled opponents in the faculty. He likely also knew that Emmy Noether’s case was dear to Klein’s heart, not least due to his long friendship with her father. As Kurator, he characterized Klein’s position as generally opposed to the habilitation of women. However, he added, “[Klein] believes that Miss Noether is so exceptionally competent that an exception can be justified,” a view Osterrath found unconvincing. He instead supported the views expressed in the minority report as well as the reasoning in the decree of 1908, which remained in his eyes just as valid as before. What happened afterward remains somewhat unclear owing to lack of documentary evidence. In any event, the Prussian Ministry of Education must have made clear that it would not approve Noether’s habilitation at this time. Nevertheless, the meeting Hilbert had requested did take place, even though few details

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about it are known. Nor did Hilbert go to Berlin alone; he was accompanied by two colleaugues, the theoretical physicists Woldemar Voigt and Peter Debye. At this meeting, they reached an understanding with the Ministry’s Director of University Affairs, Otto Naumann. Up until 1907, Naumann had worked closely with Friedrich Althoff, the official who had done so much to promote Klein’s ambitious plans for expanding the mathematical sciences in Göttingen. Hilbert thus knew Naumann quite well, and the latter was surely predisposed to reach an unofficial compromise, namely to allow Noether to teach special seminars that would be offered under Hilbert’s name. Thus, beginning in the winter semester of 1916/17, Hilbert began announcing courses taught by “Prof. Hilbert with the support of Frl. Dr. Noether.” These courses, as everyone knew, were taught exclusively by Emmy Noether; the first was a two-hour seminar on invariant theory for mathematicians and physicists, which, as Carathéodory had emphasized, was now a topic of burning interest in Göttingen. In her study of the surviving documents pertinent to Noether’s original application, Tollmien found nothing to indicate that the Ministry issued an official reply to her petition [Tollmien 1990, 179]. This being so, it seems likely that this informal arrangement was undertaken in lieu of a formal response, which would have certainly resulted in rejection of the application. A final decision would have rested with Kultusminister August von Trott zu Solz, a conservative who served in the cabinet of Chancellor Theobald von Bethmann Hollweg from 1909 to 1917. One can easily imagine why Naumann preferred a quiet, possibly temporary solution that did not require any official action. Two years later, however, the Göttingen mathematicians again called attention to Noether’s case, this time because they were concerned she might leave Göttingen in order to habilitate at the newly founded Frankfurt University, an institution with an unusual prehistory. Prior to its founding in 1914, Frankfurt’s mayor Franz Adickes and other civic leaders forged plans to create a foundation university, funded by wealthy and socially active citizens of Frankfurt. Unsurprisingly, this initiative met with considerable resistance from already established universities in Prussia. In 1911 their rectors petitioned Kultusminister von Trott, asking that he withdraw support for this plan, which they claimed would weaken the nearby universities in Marburg and Giessen. Nevertheless, the Prussian Kultusminister continued to lend his support to this project, and in June 1914 Kaiser Wilhelm II, acting as King of Prussia, conferred the official title of royal university on this new institution. Later that year, Frankfurt University opened its doors to 618 students, 100 of whom were women, taught by 50 professors. Several faculty members were of Jewish background, and in later years Frankfurt would serve as a kind of haven for Jewish scholars who had difficulty finding jobs at the older German universities. Emmy Noether was on friendly terms with many of the mathematicians there, and in 1930 she taught for one semester in Frankfurt, substituting for Carl Ludwig Siegel. One of Frankfurt’s first professors was the mathematician Arthur Schoenflies, who had earlier worked closely with Klein in Göttingen. It seems likely that Schoenflies contacted Noether, whether directly or through someone else, to in-

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quire whether she would be interested in habilitating in Frankfurt. In any event, the mathematicians in Göttingen somehow got wind of this, and they urged the physicist Voigt, as head of the department, to write the Ministry with the hope of reactivating their earlier petition. Voigt’s letter, dated 17 June 1917, was answered by Otto Naumann just six days later. His message was emphatic: there was no chance that Noether would be able to habilitate in Frankfurt or anywhere else in Prussia: The same regulations apply to the University of Frankfurt as to the other universities regarding the admission of women to the teaching profession; i.e., they are not allowed to become private lecturers. It is also quite impossible to make an exception for one university. Thus, your fear that Miss Noether could go to Frankfurt to obtain the venia legendi is unfounded; she will not be admitted there any more than in Göttingen or at another university. The Minister of Education has repeatedly stated that he firmly holds to his predecessor’s stipulation that women should not be allowed to teach at universities. In any case, you will not lose Miss Noether as a private lecturer to the University of Frankfurt. [Tollmien 1990, 180] Clearly, Naumann knew the minister’s views on this matter when the Noether case first arose, which helps to explain why he had earlier steered toward a quiet local solution. His immediate reply to Voigt also reflected the fact that his boss had other, far more pressing things on his agenda. Indeed, this brief exchange took place only weeks before August von Trott zu Solz resigned as Kultusminister, a decision that coincided with the collapse of Bethmann Hollweg’s government during the crisis of July 1917, a fateful moment in modern German history. Trott’s successor, Friedrich Schmidt-Ott, had also worked closely with Althoff, so he, too, had longstanding relations with the mathematicians in Göttingen. Possibly for that reason, but surely also to clarify his Ministry’s position on the issue of allowing women to habilitate, Schmidt-Ott decided that he should issue a response to the petition from November 1915 on behalf of Noether’s application. His communique to Göttingen’s Kurator, issued on 5 November 1917, was probably the first and only time the Ministry responded in writing to the Noether case. In effect, Schmidt-Ott merely signaled that nothing could change until such time as the faculties themselves voiced a desire to liberalize the habilitation regulations: The admission of women to habilitate as private lecturers continues to face considerable misgivings in academic circles. Since this question can only be decided as a matter of principle, I am unable to approve any exceptions, even if this means that certain hardships are unavoidable in individual cases. Should the general opinion of the faculties change from those then taken into account in the decree of May 29, 1908, I would be willing to consider the question again. [Tollmien 1990, 181]

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The professors’ opinions, by and large, did not change, but the world they lived in certainly did. Bethmann Hollweg’s fall ushered in what amounted to a military dictatorship, led by Chief of the General Staff Paul von Hindenburg and his deputy, General Erich Ludendorff. Meanwhile, the Kaiser retreated more and more from view, leaving all meaningful decision-making to the German High Command. As the military situation deteriorated, pressure mounted for Wilhelm II to abdicate, a step he took most reluctantly on November 9, 1918, fleeing to Holland the following day. His departure spelled the end of the constitutional monarchies in Germany, as one by one the kings and princes of the German states willingly abdicated. Meanwhile, mutiny and chaos reigned, as many thought Germany was ripe for the revolution Karl Marx had long before predicted. The long-suppressed Social Democrats quickly tried to seize power, but internal dissension and fears of Bolshevist influences slowed these efforts. The Social Democratic Party of Germany (SPD) had long identified with Marxism, but with the outbreak of the Great War the party subordinated international solidarity of the working class to the pressing nationalist interests of the moment. As the fighting dragged on, dissension within its ranks grew, and in 1917 the party split, forming the Majority Social Democrats (MSPD), who supported the war, and the rival Independent Social Democrats (USPD), many of whom had strong pacifist leanings. Two years later, Emmy Noether joined the small, but growing Göttingen USPD, which over the course of that year tripled its membership from 146 to 450. The following year in the June elections, the USPD won 2,500 votes in Göttingen, second only to the MSPD, which gained 4,500 votes [McLarty 2005a, 438]. Noether remained in the party until 1922, when many in the USPD rejoined the Majority Social Democrats. Two years later, she dropped her membership in the SPD, perhaps as a result of the short-lived stabilization of political life in Germany during the Weimar Republic. To what degree was she politically active? And, if so, when and with what aims? All that can be said with any assurance is that Emmy Noether was branded a “Marxist” by many in Göttingen, a label that would have been applicable to any member of the intelligentsia who voted for the SPD.11 In any event, very few German academics sympathized with the MSPD and even fewer identified with the USPD. Still, Noether’s politics placed her in unusually interesting company. Einstein was elated by the collapse of the monarchy, and he too sympathized with figures on the left, such as the USPD politician Kurt Eisner, who was assassinated in Munich in February 1919. Probably Einstein had no idea of Noether’s political leanings, but soon after the war he wrote an emphatic letter to Klein pleading that something be done about her case. He even intimated that if the Göttingen faculty failed to act, he would intercede himself: “After receiving the new paper [“Invariante Variationsprobleme” [Noether 1918b]] by Frl. Noether I again felt that it is a great injustice that she is denied the venia 11 For a discussion of the contrasting views of Pavel Alexandrov and Hermann Weyl, see [McLarty 2005a].

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legendi. I would very much favor that we take energetic steps with the Ministry. If you do not regard this as possible, then I will make the efforts myself” (Einstein to Klein, 27 December 1918, [Einstein 1998, 976]). Klein needed no prodding, and he later gave the following assessment of Emmy Noether’s most recent accomplishments: “In the past year she has completed a number of theoretical studies that are superior to the achievements made by all others in the same period (including the work of the full professors) . . . .” Still, Einstein’s support came at an opportune time. The Ministry of Education was now headed by two prominent socialists: Adolph Hoffmann (USPD) and Konrad Haenisch (SPD). By the end of January the mathematics-natural sciences department voted to approve Noether’s petition to habilitate, and by mid-February the university communicated the same to the Ministry, this time with the approval of the entire philosophical faculty: The changed political conditions, which have led to a comprehensive expansion of women’s rights, have given our mathematicians the hope that an application in this direction would now be successful. At a meeting held on 31 January 1919, Frl. Dr. Emmy Noether’s wish to renew her application for habilitation, which had been rejected at the time, was endorsed by an overwhelming majority in the Department of Mathematics and Natural Sciences, which therefore renews her application from November 26, 1915 to be allowed to habilitate as an exceptional case. The department does not wish to prompt a general decision with regard to the admissibility of women in the teaching profession, but rather bases its application, as before, on the applicant’s extraordinarily high level of mathematical talent and academic performance. . . . [Tollmien 1990, 183–184] Much had changed, indeed, as women in Germany had now gained the right to vote, which they exercised in the election of 19 January 1919 for the Weimar National Assembly, the constitutional convention for the new Republic. Still, the stricture in the decree of 1908 that prevented women from habilitating had yet to be lifted when Noether applied for an exemption. Minister Haenisch (SPD) surely saw no reason not to do so, and on 4 June 1919, after delivering her trial lecture on “Questions of Module Theory,” Noether was finally given the venia legendi entitling her to teach courses as a Privatdozent. One year later, the philosopher Edith Stein, who had been denied the opportunity to habilitate on more than one occasion, presented a petition to the Ministry arguing that discrimination by gender should be disallowed in future habilitation proceedings. Haenisch agreed, thereby overturning the prohibitive provision in the decree from 1908. There was much to celebrate, not least the paper on “Invariant Variational Problems” [Noether 1918b] that Noether submitted as her habilitation thesis. The story behind this work takes us back to the time when she first arrived in Göttingen in 1915 and will be told in the following chapter.

Chapter 3

Emmy Noether’s Role in the Relativity Revolution Emmy Noether’s paper “Invariante Variationsprobleme” [Noether 1918b] is regarded today as one of her most important works, especially in view of its relevance for mathematical physics [Uhlenbeck 1983, Byers 1996]. Probably every student who is exposed to the role of action principles in physics learns about their connection with Noether’s first theorem, which occupies a prominent place in modern physics textbooks. Noether’s principal theorem has numerous applications both to classical mechanics as well as to field theories, providing insight into conservation laws (Noether currents) in many different types of physical settings. These include the familiar conservation laws for mechanics – linear and angular momentum as well as total energy – but also energy-momentum conservation in special relativity. During her lifetime, however, and even long after her death, few remembered that she also played a noteworthy part in unraveling the mathematical mysteries of Einstein’s general theory of relativity, particularly those surrounding the status of its conservation laws. Instead, her name was exclusively associated with her seminal role in launching modern algebra during the 1920s. Yvette Kosmann- Schwarzbach has shown that physicists were very slow in coming to appreciate the significance of Noether’s results.1 Even to this day, most commentators fail to emphasize that Noether’s original paper actually contains two fundamental theorems. Moreover, both are essential for understanding the original motivation behind her work, namely the distinction between what she called proper and improper conservation laws in physics. This work arose, in fact,

1 [Kosmann-Schwarzbach 2006/2011, 98–101] summarizes the negative results for the period up to 1950; the reception of Noether’s first and second theorems appears in Chapters 5 and 6, respectively.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_3

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amid complicated debates over the status of energy conservation in Einstein’s theory of general relativity, a context we briefly describe in this chapter. 2

3.1 Einstein’s Road to General Relativity In the late spring of 1915, only shortly after Emmy Noether was invited to habilitate in Göttingen, Einstein came to deliver a series of six two-hour lectures on his new theory of gravitation, the general theory of relativity. 3 Einstein was pleased with the reception he was accorded, and expressed particular pleasure with Hilbert’s reaction. “I am very enthusiastic about Hilbert,” he wrote Arnold Sommerfeld, “an important man!” [Einstein 1998, 147]. Although little is known about what transpired during the week of his visit to Göttingen, Einstein was clearly delighted by the response he received, writing to a friend afterward “to my great joy, I succeeded in convincing Hilbert and Klein completely” [Einstein 1998, 162]. As for Hilbert’s reaction to Einstein’s visit, this encounter inspired him to consider whether general relativity might provide a fruitful framework for combining Einstein’s gravitational theory with Gustav Mie’s electromagnetic theory of matter. Much as he had done in his other physical research, Hilbert hoped that by exploiting axiomatic and variational methods he would be able to place relativistic field theory on a firm footing. Hilbert was especially attracted to Max Born’s presentation of Mie’s theory because of its mathematical elegance and reliance on variational methods. Variational principles had a longstanding place in classical mechanics, particularly due to the influential work of J.L. Lagrange, but their use in electrodynamics and field physics brought about numerous challenges. In the context of Mie’s theory, Born derived its fundamental equations from a variational principle by varying the field variables rather than varying the coordinates for space and time. Einstein’s theory of relativity was initially conceived as a new framework for combining electrodynamics and mechanics. He thought of relativity as as a principle theory as opposed to a constructive theory [Howard/Giovanelli 2019], which implies that it could serve as a framework for the latter type of theory if sufficiently justified by experimental tests. Special relativity modified the principles 2 For further details on these debates, see [Rowe 1999], parts of which have been adapted for this chapter; for a brief introduction, see [Rowe 2019a]. 3 Whether Noether attended any of these lectures appears uncertain, however, as she may well have remained in Erlangen following her mother’s death in May 1915. On the other hand, he visited Göttingen on at least one other occasion during the war years, and they corresponded directly in 1926 ([Kosmann-Schwarzbach 2006/2011, 161–165]. Assertions that Noether and Einstein never met, while difficult to refute, lack plausbility [Kosmann-Schwarzbach 2006/2011, 76–77]. She was a regular visitor at Princeton’s Institute for Advanced Study, then housed in the old Fine Hall, so their paths almost surely crossed there as well. His letter of appreciation for Emmy Noether, first published in the New York Times [Dick 1970/1981, 37] and discussed in Section 9.1, also suggests that he had some direct personal knowledge of the deceased mathematician.

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of Newtonian mechanics by completely abandoning all appeal to absolute space and time. Both notions had gradually been undermined with the development of electrodynamics, but in the electron theory of H.A. Lorentz the notion of an absolute rest frame still remained in the form of a purported ether that filled space and served as the medium for propagating electromagnetic radiation. Einstein’s original theory from 1905 dealt only with so-called inertial frames of motion, but this special theory of relativity was not concerned with gravitational effects. Already in 1907, however, he conceived of a way to link gravitational effects with non-inertial frames, an idea he later called the equivalence principle. This paved the way for a generalized theory of relativity that incorporated gravitation by means of curvature effects in a 4-dimensional spacetime. Spacetime geometry was a mathematical innovation introduced by Hermann Minkowski in 1908. Just as Maxwell’s theory of electromagnetism eventually came to be presented in the language of vector analysis, Einstein’s special theory of relativity was soon repackaged mathematically in terms of 4-vectors and tensors, entities associated with the group of Lorentz transformations. Minkowskian spacetime then served as the underlying geometry for the physical objects under study. A number of investigators, including Minkowski himself, attempted to develop a theory of gravitation within this framework, but Einstein was convinced that the linkage between gravitational and inertial effects required the study of arbitrary frames of motion along with the acceptance of what he called the relativity of inertia. Put simply, this said that the distribution of matter in a closed physical system determined the curvature structure of spacetime, which in turn guided the force-free motion of bodies. A small body would thus move along a geodesic curve, a path that depended on the distribution of matter in the system. These geodesics correspond to inertial paths in classical physics, which are straight lines in a Euclidean 3-space. In Newtonian physics, bodies not subjected to external forces will move along straight-line paths with uniform velocity, and the same is true in Minkowski space, which is a flat spacetime geometry. In a general Einsteinian spacetime, however, the presence of gravitational fields produces curvature effects, which alters the underlying geometry as reflected in its geodesic curves. Even so, in a local setting these curvature effects can be transformed away, one of the features of Einstein’s theory that caused considerable confusion. Initially, Einstein resisted Minkowski’s mathematical formalism for special relativity, but he eventually realized its advantages once other physicists, principally Arnold Sommerfeld and Max Laue, recast its basic operations and created modern 4-dimensional vector analysis [Walter 2007]. By 1912, when Einstein began collaborating with his friend and colleague, the mathematician Marcel Grossmann, the full framework for special relativity as a spacetime theory was firmly in place. Grossmann’s principal contribution over the next two years was to guide Einstein into the complexities of yet another mathematical formalism that was well adapted for the latter’s general theory of relativity: the absolute differential calculus of Gregorio Ricci and Tullio Levi-Civita [Goodstein 2019], soon to be dubbed tensor analysis. Unlike the 4D-vector analysis of special relativity, which is

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a mathematical tool for handling the invariants of Lorentz (or Poincaré) transformations, the Ricci-calculus was designed to deal with invariants under practically any transformations relevant for a geometrized physical theory. Emmy Noether’s contributions to mathematical physics combined tensor analysis with variational calculus, both of which she treated as tools for constructing differential invariants. The variational calculus already had a long historical background, parts of which were directly relevant for her work (see [Kastrup 1987]). In the second edition of his famous Mécanique Analytique (1811), Lagrange elegantly derived a classical form of energy conservation by utilizing variational methods. The calculus of variations was a fairly new branch of mathematics that developed from efforts to solve minimization problems over a space of possible paths. Traditionally one identifies its origins with the famous brachistochrone problem. In 1696, Johann Bernoulli posed the problem of finding the path of free fall between two given points that minimizes time. He solved this problem himself by drawing on an optical analogy – Fermat’s principle of least time – and then deriving the differential equation satisfied by the path of quickest descent, which turns out to be a cycloid curve. Euler and Lagrange later invented variational methods for solving such problems related to action principles in physics. In classical mechanics, the action is given by an integral over time, Z

t2

L dt,

A= t1

where the integrand L is the so-called Lagrangian. The evolution of a physical system can then be determined by minimizing this action integral, which is taken over the virtual paths of the system from its initial state to final state between times t1 and t2 . In the language of the calculus of variations, this condition is written δA = 0, which leads to the equations of motion (Euler-Lagrange equations). These methods were eventually applied to formalize the law of least action, which later became known as Hamilton’s principle. Lagrange exploited variational methods to derive the law of conservation of energy for purely conservative forces, i.e. forces that derive from a potential function U that depends on the spatial coordinates alone. Introducing T as the kinetic energy and U for the potential energy, he defined L = T − U as the Lagrangian for an action integral, and then showed that this satisfies a symmetry property corresponding to T + U = E, where E remains constant over time. This thus showed that the total energy of such systems remains conserved. Variational problems in which time is the only independent variable correspond to problems in classical mechanics, whereas field theories, such as electrodynamics and general relativity, give rise to problems with several independent variables. Both special and general relativity involve an energy-momentum tensor, which encapsulates the relevant physical data in a region of spacetime. It is in several respects akin to the stress tensor in continuum mechanics, though this is a 3 × 3-matrix, whereas the energy-momentum tensor T µν is represented by a

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symmetric 4 × 4-matrix, so it has 10 independent components. These describe the density and flux of energy and momentum in spacetime, which act as the sources of the gravitational field for the field equations of general relativity. In Newtonian gravitational theory, on the other hand, the mass density alone serves as the source for Poisson’s equation. With this construct in hand, energy-momentum conservation has a straightforward formulation in special relativity. In Cartesian coordinates, one takes ordinary derivatives to get the four equations T,νµν = 0, which correspond to the conservation of the 4-momentum, a relativistic vector that represents the total energy together with the 3-vector for linear momentum. This is the differential form for energy-momentum conservation, which can then be used to express the total energy in a region of spacetime simply by integrating over its boundary. In general relativity, on the other hand, the coordinate derivative above must be replaced by the covariant derivative, following the operations of the Riccicalculus. This leads to the equation ν T;νµν = T,νµν + Γµσν T σν + Γσν T µσ = 0,

where the Γs are so-called Christoffel symbols, which Einstein interpreted in terms of the gravitational force field. He would rewrite this equation in a somewhat different form ∂(Tµσ + tµσ ) = 0, 4 ∂xσ where Tσµ represents the energy of matter fields and tσµ is a pseudotensor that corresponds to the energy of the gravitational field in a given coordinate system. So here the total energy has been separated in a transparent manner. As will be discussed later, Einstein attached great significance to this interpretation, whereas many felt that the theory should only employ true tensors, which are independent of any given coordinate system. As we will soon see, Hilbert introduced a very different approach to gravitational energy based on a vector el that was generally covariant, i.e. independent of the choice of coordinates.

3.2 Hilbert’s Approach to Einstein’s Theory Hilbert’s theory in [Hilbert 1915] was based on two axioms that concern the properties of a “world function” H(gµν , gµν,l , gµν,lk , qs , qs,l ). This H is taken to be a scalar-valued function that does not depend explicitly on the spacetime coordinates ws but rather only on the ten components of the 4 Note that this equation actually stands for four component equations given by µ = 1, 2, 3, 4, obtained by summing over the index σ, a notational convention introduced by Einstein that we use throughout this chapter.

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symmetric metric tensor gµν and its first and second derivatives as well as four electromagnetic potentials qs and their first derivatives. Axiom I then asserts that under infinitesimal variations of the field functions g µν → g µν + δg µν and qs → qs + δqs , Z √ δ H gdω = 0, where g = |g µν | and dω = dw1 dw2 dw3 dw4 . This variational principle is understood to apply throughout a finite region of spacetime. Axiom II then requires that the world function H be invariant under general coordinate transformations. Hilbert then derived ten Lagrangian differential equations for the ten gravitational potentials g µν and four equations for the four electrodynamic potentials qs . He called the first set the fundamental equations of gravitation and the second the fundamental equations of electrodynamics, abbreviating these to read: √ √ [ gH]µν = 0, [ gH]h = 0. In the course of developing his theory, Hilbert focused on the special case where the Lagrangian H takes the form H = K + L. HereP K is the curvature scalar obtained by contracting the Ricci tensor Kµν , i.e., K = µν g µν Kµν . He placed no special conditions on the Lagrangian L, but noted that it contained no derivatives of the metric tensor, so that H = K + L(g µν , qs , qs,l ). At the outset, Hilbert presented his main theorem – “the Leitmotiv for the erection of my theory” – which he promised to prove on another occasion. He worded this key result as follows: Theorem I. Given a scalar expression J depending on n magnitudes and their derivatives that is invariant under arbitrary transformations of the four world parameters ws , s = 1, 2, 3, 4, if one forms the n Lagrangian variational equations from Z √ δ J gdω = 0 with respect to these n magnitudes, then within this invariant system of n differential equations four of the n equations will always be determined by the other n − 4, in the sense that four linearly independent combinations of the n differential equations and their total derivatives are always identically satisfied [Hilbert 1915, 397]. Applying this theorem to the 14 fundamental equations, Hilbert concluded √ √ that four of the equations [ gH]µν = 0, [ gH]h = 0 can be deduced directly from the other ten. He then seized on this result to make a strong physical claim: . . . on account of that theorem we can immediately make the assertion, that in the sense indicated the electrodynamic phenomena are the effects of gravitation. In recognizing this, I discern the simple and very surprising solution of the problem of Riemann, who was the first to search for

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a theoretical connection between gravitation and light. [Hilbert 1915, 397–398]5 Hilbert would later drop this passage in [Hilbert 1924], a new version of his two notes [Hilbert 1915, Hilbert 1917], even though the theorem he stated was correct and certainly important. Indeed, it follows directly from Emmy Noether’s second theorem, which she proved in [Noether 1918b]. The other key result in Hilbert’s paper – he called it “the most important goal” of his theory [Hilbert 1915, 400] – concerned the determination of an invariant energy vector along with the proof of conservation of energy. The latter property, he emphasized, was to hold for any world function H that merely satisfied axioms I and II, and the results he derived in this direction broke new ground. But they also involved technical calculations of differential invariants that hardly anyone (Emmy Noether excepted) could follow. With Noether’s help, Klein was later able to rederive Hilbert’s invariant energy vector in a far more transparent way. At this juncture, it should merely be stressed that the thrust of Hilbert’s arguments depended on a combination of techniques from invariant theory and variational methods, two fields in his standard mathematical repertoire. Initially, Emmy Noether worked closely with Hilbert, but she also assisted Felix Klein in preparing his lectures on the development of mathematics during the nineteenth century [Klein 1926]. Starting in the summer of 1916, Klein broke off these lectures in order to begin a 3-semester course on the mathematical foundations of relativity theory (published posthumously in [Klein 1927]). Compared with Hilbert’s physically motivated research program, Klein’s agenda was entirely mathematical and fundamentally conservative, aiming to clarify fundamental issues. This orientation accorded very well with Noether’s strengths and interests, so that during the last two years of the war, she worked more and more closely with Klein. Moreover, by 1918 Hilbert had largely abandoned field physics in order to pursue his longstanding interests in the foundations of mathematics. 6 Noether presumably had little knowledge of variational methods when she joined Hilbert’s research group in 1915. What she knew very well, however, were related methods for using formal differential operators to generate algebraic and differential invariants. In November 1915, she wrote to her friend and former Erlangen mentor, Ernst Fischer, to tell him about her work in Göttingen. From Noether he now learned that Hilbert had created a buzz of excitement about invariant theory, so that even the physicist Paul Hertz was studying the classical literature [Dick 1970/1981, 30–31]. Hertz was learning these methods from her Doktorvater’s old lectures [Gordan 1885/1887], edited by Georg Kerschensteiner. Noether had herself learned these older methods in Erlangen from Paul Gordan, who supervised her dissertation, a detailed study of the invariants and covariants 5 Hilbert was alluding here to Riemann’s posthumously published “Gravitation und Licht,” in [Riemann 1876, 496]. 6 For documentation of this important shift in Hilbert’s research orientation, see [Ewald/Sieg 2013].

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associated with a ternary biquadratic form. She knew that Hilbert was pushing his team on with hopes for a breakthrough in physics, but freely admitted that none of them had any real idea how their calculations might be used for that purpose [Dick 1970/1981, 30–31]. Over time, Noether gained some understanding of what Hilbert hoped to achieve for physics by drawing on invariant theory. Yet in doing so, she surely never thought of her own work as motivated by Hilbert’s physical program. In fact, she was still pursuing a program for algebraic invariant theory inspired by her collaboration with Ernst Fischer (see Section 1.4). In the meantime, Klein had been urging her to work on linking differential and algebraic invariants by exploiting the formal identities that arise from variational methods. On 22 August 1917, Noether wrote to Fischer, announcing that she had now solved a problem that had occupied her attention since spring, namely the extension of the “reduction theorem,” proved by E.B. Christoffel and G. Ricci for quadratic differential forms, to forms of arbitrary degree [Dick 1970/1981, 33]. On 15 January 1918, she presented a lecture on her “Reduction Theorem” at a meeting of the Göttingen Mathematical Society, and ten days later Felix Klein submitted her paper [Noether 1918a] for publication. Drawing on methods in the calculus of variations introduced by Lagrange, Riemann, and Lipschitz, she showed how the problem of finding the differential invariants associated with a differential form can be reduced to classical invariant theory, i.e. to finding certain algebraic invariants under the projective group.7 Her treatment of Lagrangian derivatives as formal invariants reveals that this paper was closely related to her far more famous one on invariant variational problems, [Noether 1918b]. 8 Whereas Hilbert hoped to use Einstein’s gravitational theory as a framework for a new unified field theory, Noether remained what she had always been: a pure mathematician. Her work thus aimed to clarify the mathematical underpinnings of general relativity, an effort strongly promoted by Felix Klein, who took up this challenge around the time that Hilbert’s interests were turning back to the foundations of mathematics [Sauer/Majer 2009, 22].

3.3 Einstein reads Hilbert Contemporary interest in Einstein’s theory of gravitation gained considerable momentum after May 1916 when he published his classic article [Einstein 1916a], which came out as a separate brochure just as Einstein began an interesting correspondence with Hilbert. These exchanges came about because Einstein wanted to understand Hilbert’s complicated approach to energy-momentum conservation 7 Here, one regards the coordinate functions as parameters in the linear equations for the transformed differentials; see [Klein 1927, 196–197]. 8 Noether’s expertise in the area of formal variational methods can be seen from [Noether 1922b], her survey for the Encyklopädie; in all likelihood Klein asked her to write up this short summary.

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in [Hilbert 1915]. The complications were in part due to the fact that Hilbert had changed his mind about how to define a vector representing gravitational energy in his theory. Whether Emmy Noether was aware of Hilbert’s original definition for his energy vector remains an open question, but in any case she was very familiar with the version that ultimately appeared in [Hilbert 1915]. 9 Einstein struggled to understand the arguments in Hilbert’s first note as he prepared to speak about it in Heinrich Rubens’ colloquium. Twice he turned to Hilbert for clarifications, writing: “I admire your method, as far as I have understood it. But at certain points I cannot progress and therefore ask that you assist me with brief instructions.” 10 He was particularly baffled by Hilbert’s energy theorem, admitting that he could not comprehend it at all – not even what it asserted.11 Hilbert wrote back just two days later. He easily explained how, via the operation of polarization, an invariant J will lead to a new invariant P (J), its first polar. He then went on to say: My energy law is probably related to yours; I have already assigned this question to Miss Noether. As concerns your objection, however, you must consider that in the boundary case g µν = 0, 1 the vectors al , bl by µν terms and is differentiated no means vanish, as K is linear in the gσκ with respect to these quantities. For brevity I give you the enclosed paper from Miss Noether.12 Einstein was well aware that Noether was working closely with Hilbert and that the latter hoped she would be appointed as a Privatdozent (see Section 1.3)). Despite strong support from members of the natural sciences division, however, these efforts proved futile. Only after the fall of the German Reich and the advent of the Weimar Republic did this plan succeed. Hilbert’s conjecture regarding the relationship between his and Einstein’s versions of energy conservation was surely no more than a first guess. Even on the purely formal level, he could hardly assert that his energy vector el stood in some obvious relation to Einstein’s pseudo-tensor tσµ . Nevertheless, Emmy Noether was able to show that Hilbert’s el and Einstein’s tµσ both possessed a common property which seemed to reflect that the energy laws in general relativity differ from those in classical mechanics or special relativity. We will probably never 9 At

first Hilbert used this vector as a coordinate restriction, but he later decided to alter the definition in the page proofs of his original submission from 20 November 1915. Thus the version in [Hilbert 1915] actually reflects an important shift in Hilbert’s understanding of this aspect of the theory (for details, see [Sauer 1999] and Sauer’s commentary in [Sauer/Majer 2009, 11–13]). 10 Einstein to Hilbert, 25 May 1916, [Einstein 1998, 289]. 11 Hilbert claimed not only that the energy vector el depended solely on the metric tensor and its derivatives, he also showed that by passing to a flat metric its electromagnetic part turned out to be closely related to a formulation for energy derived from Mie’s theory. Einstein was puzzled about this derivation, since the argument seemed to show that not only the divergence of the energy term but this term itself would have to vanish. 12 Hilbert to Einstein, 27 May 1916, [Einstein 1998, 291].

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know precisely when she established this connection, for although she informed Hilbert of it, he apparently made no effort to publicize Noether’s findings. These survived, however, thanks to a transcription made by Rudolf J. Humm, a student in Göttingen (see below; for further details see [Rowe 2019c]). Einstein responded to Hilbert’s letter shortly afterward: Your explanation . . . delighted me. Why do you make it so hard for poor mortals by withholding the technique behind your ideas? It surely does not suffice for the thoughtful reader if, although able to verify the correctness of the equations, he cannot have a clear view of the overall plan of the analysis.13 Einstein was far more blunt about this in a letter he wrote to Paul Ehrenfest on May 24: “Hilbert’s description doesn’t appeal to me. It is unnecessarily specialized regarding ‘matter,’ is unnecessarily complicated, and not straightforward (= Gauss-like) in set-up (feigning the super-human through camouflaging the methods)” [Einstein 1998, 288]. After receiving Hilbert’s explanations, he may have felt somewhat more conciliatory. Certainly he made every effort to understand Hilbert’s arguments, and could report: “In your paper everything is understandable to me now except for the energy theorem. Please do not be angry with me that I ask you about this again” [Einstein 1998, 293]. After explaining the difficulty he still had, Einstein ended by writing that it would suffice if Hilbert asked Emmy Noether to clarify the point that was troubling him. This turned out to be a quite trivial matter, so Hilbert answered Einstein directly. The latter then responded with thanks, adding that “now your entire fine analysis is clear to me, also with respect to the heuristics. Our results are in complete agreement” [Einstein 1998, 295]. What Einstein meant by this would seem quite obscure. Perhaps he only meant to assure Hilbert that he would no longer be pestering him about these matters. Probably Hilbert had just as little interest to enter these waters further, for how else to account for the fact that he failed to publicize Emmy Noether’s findings, which clearly stemmed from this correspondence with Einstein? Not until Felix Klein began to take an interest in the status of conservation theorems in general relativity more than a year later did Noether’s name receive any attention in this connection. Noether’s original manuscript no longer survives, but fortunately R.J. Humm made a partial transcription, probably in early 1918.14 What this shows is that 13 Einstein

to Hilbert, 30 May 1916, [Einstein 1998, 293]. Humm’s copy of Noether’s manuscript contains no date, we can only fix bounds on the period during which she must have written it. In his correspondence with Einstein from late May and early June of 1916, Hilbert alluded to Noether’s efforts to reconcile their approaches to energy laws in general relativity. Much later, in January 1918, Hilbert and Klein both made reference to the results she had obtained more than one year earlier, so probably by December 1916 at the latest. Her text, on the other hand, contains no mention of [Einstein 1916b], which surely circulated in Göttingen soon after its publication in early November 1916. Had Noether known of this text at the time she wrote her manuscript, she would very likely have referred to 14 Since

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Noether was able to prove that Hilbert’s energy vector and Einstein’s pseudotensor representing gravitational energy shared an important formal property. Both can be decomposed into two parts, one of which has vanishing divergence, whereas the other vanishes as a result of the gravitational field equations. Her analysis draws closely on Hilbert’s own techniques in [Hilbert 1915], which she then applies in order to analyze Einstein’s construct in [Einstein 1916a]. Rudolf Humm later became a well-known writer in Zurich and a personal friend of Hermann Hesse [Rowe 2019b]. His family was Swiss, but he grew up in Modena until his teenage yaers, when he left Italy to attend the Kantonsschule in Aarau. This was the same institution Einstein had attended for just one year before he entered the Zurich Polytechnicum, a circumstance that may have given Humm inspiration to study relativity in Germany. After a brief stay in Munich, he went on to Göttingen, and by the winter semester of 1916/17 he was steeped in theoretical physics. His diaries from these years provide a highly subjective, yet very illuminating picture of the atmosphere at the university during wartime. 15 Several of those whom Humm mentioned quite often were fellow Swiss: Paul Scherrer and his wife, Paul Finsler, and Richard Bär. He also befriended Vsevolod Frederiks, a Russian who was studying physics and mathematics, and the astronomer Walter Baade, who took his degree in 1919. Many in Göttingen at this time were, like Humm, foreigners working more or less closely with Hilbert, so they likely knew Emmy Noether. Humm’s interactions with her seem to have been rather fleeting, but moving in such a small world they probably saw one another regularly. Both also had significant interactions with Felix Klein.

3.4 Klein’s Interests in General Relativity Just after the war broke out in August 1914, Klein began a series of lectures on the history of 19th-century mathematics. Lacking suitable “manpower,” he asked his youngest daughter, Elisabeth, to write these up during the first two semesters. Elisabeth Klein was a war bride, whose husband was killed at the outset of the fighting. She was assisted by her friend Iris Runge, daughter of Klein’s colleague, Carl Runge. During the third cycle of Klein’s lectures, held during the winter semester 1915/16, Emmy Noether was one of six women who attended. Two were responsible for preparing the Ausarbeitung: Käthe Heinemann and Helene Stähelin. These three texts served as the foundations for the published version of Klein’s lectures [Klein 1926], which appeared shortly after his death. Originally, Klein had planned to complete these lectures with a final course in the summer of 1916. In the midst of the excitement ignited by Einstein’s new theory of gravitation and Hilbert’s attempt to wed it to Mie’s electromagnetic theory of matter, he decided to drop his original plan. Instead, Klein began offering Einstein’s most recent arguments. These circumstances suggest that she probably completed her manuscript between June and October of 1916. 15 For details on Humm’s interests in general relativity during these years, see [Rowe 2019b].

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courses on the mathematical underpinnings of special and general relativity, but with a strong accent on the theory of transformation groups and the ideas he had first sketched in his Erlangen Program [Klein 1872]. He also began corresponding with Einstein about various mathematical aspects of importance for the theory of relativity. Toward the end of 1917, Klein sent Einstein a copy of the Ausarbeitung of his lectures. The latter’s response from 15 December was not very flattering: “. . . it seems to me that you highly overrate the value of formal points of view. These may be valuable when an already found truth needs to be formulated in a final form, but they fail almost always as heuristic aids” [Pais 1982, 325]. Abraham Pais noted the irony of this remark in view of how enamored Einstein later became with mathematical formalisms during his decades-long quest for a unified field theory. Emmy Noether, who regularly attended Klein’s seminar lectures from 1917– 1918 on mathematical methods in relativity, made explicit references to these in her survey article [Noether 1922b]. These lectures circulated for several years in mimeographed form – Sommerfeld and Einstein, for example, read parts of them – but they were only published later in [Klein 1927].16 During the last two years of the war, she was working closely with Klein and his assistant Hermann Vermeil. Their principal goal was to develop a general approach to differential invariants based on methods first explored by Bernhard Riemann and Rudolf Lipschitz, a program Klein outlined in his lectures, see [Klein 1927, 180–199]. By utilizing Riemannian normal coordinates, Vermeil was able to rederive Christoffel’s results characterizing a quadratic differential form by means of differential invariants. As he noted in [Vermeil 1919], his findings relied heavily on parallel investigations by Emmy Noether. In her case, this program led to a far-reaching generalization of a theorem due to Christoffel involving the reduction of results on differential invariants to corresponding algebraic invariants. In January 1918, Klein submitted her paper [Noether 1918a] to the Göttingen Scientific Society, and in May she sent a copy to Einstein, who reacted with enthusiasm. In a letter to Hilbert, he wrote: “Yesterday I received a very interesting paper from Frl. Noether on building invariants. It impresses me that one can view these things from such a general standpoint. It wouldn’t have hurt the Göttingen troops in the field if they had been sent to Frl. Noether. She appears to know her business!” 17 Klein was obviously not the only one who recognized Noether’s special talent for generalizing beyond known results or established theories. Felix Klein’s initial interest in relativity theory was related to spacetime geometry. Several years before, in [Klein 1910], he gave a projective interpretation of Minkowski space, which placed it within the larger context of metric geometries of constant curvature. Like Euclidean geometry, Minkowski space has zero curva16 Noether also attended some of Klein’s much better-known lectures from two years before, later published as [Klein 1926]. Both sets of lectures originally had more ambitious goals that Klein did not live to complete. Courant nevertheless decided to publish lightly edited versions of both posthumously; whether or not Klein ever assented to this plan remains unclear. 17 Einstein to Hilbert, 24 May 1918, cited in [Kimberling 1981, 13].

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ture; it corresponds to a flat space, but one in which the metric is indefinite rather than positive definite. Klein had been the first to exploit the possibility of attaching different types of quadrics to a projective space in order to introduce a metric, an idea inspired by Arthur Cayley’s realization of Euclidean geometry by means of the so-called Cayley metric.18 In this special case, the quadric is a degenerate imaginary figure, whereas the Minkowski metric corresponds to a real degenerate quadric. These possibilities for deriving different metrical geometries served as a major inspiration for Klein’s “Erlanger Programm,” especially since the same general approach could be applied in many different settings. After Minkowski geometrized “special relativity,” Klein proposed the general idea of calling “invariant theory relative to a group of transformations the relativity theory of a group” [Klein 1910, 539]. Emmy Noether pointed to this viewpoint at the very end of [Noether 1918b, 257] in order to emphasize how her results were fully in accord with Klein’s position. Klein’s interest in special relativity largely focused on the invariant theory of the Lorentz group, a key example being the Maxwell equations, which he wrote in a manifestly covariant form. While preparing his lectures on relativity theory, he came across an article by the English physicist Henry Bateman, in which he proved the invariance of the Maxwell equations under transformations by reciprocal radii [Bateman 1910].19 Klein took a special interest in the connections between group theory and physical laws, which he saw as further confirmation of the fertility of his Erlangen Program [Klein 1872]. In his famous work on the threebody problem, Henri Poincaré had drawn attention to the connection between the 10 classical conservation laws (energy, momenta, angular momenta, and uniform motion of the center of mass) and the corresponding spacetime symmetries for Galilean transformations [Poincaré 1890, Poincaré 1892]. In 1905, Poincaré showed that the Maxwell equations were invariant under the 10-parameter group of extended Lorentz transformations (today known as the Poincaré group, the semi-direct product of the 6-parameter rotations of the Lorentz group with the 4-parameter group of spacetime translations) [Kastrup 1987]. Bateman’s result thus extended this finding to the 15-parameter group of conformal transformations. When Klein wrote to Einstein about this finding in April 1917, he received the following disappointing reply: 18 In a metric geometry, the distance between two points is an elementary invariant under congruence transformations (isometries); whereas in projective geometry four collinear points determine an invariant, the cross ratio. By attaching a quadric to the space as “absolute” figure, one can introduce a metric, due to the fact that the line joining any two points will meet the quadric in two more. If the transformation group is then restricted to automorphisms that leave the “absolute” figure fixed, the invariance of the cross ratio ensures that the metric thus defined will remain invariant under these transformations. 19 It had long been known that these mappings play a significant role in classical physics, as Kelvin had employed them in his work on potential theory. In the 1840s Liouville proved that by adjoining inversive mappings to the group of similitudes one obtains the group of conformal transformations.

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3 Noether’s Role in the Relativity Revolution . . . I can’t well imagine that substitutions by reciprocal radii can have a physical significance. The Lorentz transformations leave not only the speed of light invariant but also the coordinates in two related coordinate systems have a simple physical meaning as measurement results taken on rods and clocks. This latter, essential property is certainly lost under the transformations of Bateman.20

Klein was not so easily dissuaded, however, and he encouraged Noether to study recent work by Gustav Herglotz on group invariants in classical physics. Herglotz’s paper [Herglotz 1911] was an important inspiration for Noether’s approach in [Noether 1918b], though he dealt only with the special case of continuum mechanics treated via the invariance theory of the Poincaré group. Utilizing an invariant action integral, he derived the 10 conservation laws for this classical field theory. His derivation was based on the infinitesimal generators of the Poincaré group, an approach closely related to the methods Sophus Lie had used in his group-theoretic studies of the theory of differential equations. Realizing this, Klein contacted Friedrich Engel, who had for many years been Lie’s collaborator, and urged him to undertake a similar study for the non-homogeneous group of Galilean transformations. The results of his investigations, [Engel 1916, Engel 1917], were published in the form of two letters to Klein, who surely shared their contents with Noether. Beginning in March 1918, Klein’s correspondence with Einstein intensified markedly following the appearance of [Klein 1918a], which was published in the form of an epistolary exchange with Hilbert concerning certain key results from the latter’s earlier note [Hilbert 1915]. Klein’s contribution arose from a presentation he made on 22 January to the Göttingen Mathematical Society, a talk that elicited a reaction from Hilbert one week later. The conclusions drawn from these two sessions were summarized as follows in the journal of the German Mathematical Society: “The ‘conservation laws’ valid for continua in classical mechanics (the impulse-energy theorems) are already contained in the field equations in Einstein’s newly inaugurated theory; they thereby lose their independent significance.” 21 This contemporary interest in the status of various conservation laws in general relativity can be seen from another piece of anecdotal information relating to Einstein’s views on this matter. One of those who attended Klein’s lecture was the Swiss student, Rudolf J. Humm, who may well have learned about Noether’s earlier results on energy conservation at this time.22 Humm had recently returned 20 Klein later cited this opinion of Einstein’s in the commentary to his collected works, but even Einstein’s authority could not dissuade him from pursuing the idea further [Klein 1921–23, 1: 552]. He persuaded Erich Bessel-Hagen to study the physical implications of this conformal invariance further. The latter did so in [Bessel-Hagen 1921], drawing on Emmy Noether’s theorems, which in this case lead to fifteen conserved quantities, appearing as divergence relations, and which can be reduced to fourteen by virtue of a single identity connecting them. 21 Jahresbericht der Deutschen Mathematiker–Vereinigung 27(1918) (“Mitteilungen und Nachrichten”), p. 28. 22 Since Humm’s manuscript left out the first several pages and thus only contains Noether’s main results, it seems very unlikely that his text was based on a course he attended. What seems

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from a semester in Berlin, where he attended Einstein’s lectures on general relativity [Rowe 2019b]. At the end of the final lecture, Humm wanted to learn Einstein’s opinion about the status of his pseudo-tensor for representing gravitational energy and got these remarks in reply: I asked Einstein if it would be possible to generalize the conservation equation ∂(Tσµ + tσµ ) =0 ∂xσ so that it would contain only real tensors. He thought not: one does not shy from writing ∂(T + U ) =0 ∂t in classical mechanics, where U in an invariant under Galilean transformations, but T is not. So it not so terrible to have the general tensor Tσµ next to the special tσµ . If one considers an accelerative field, then there will be a tσµ , even though the field can be transformed away. Ultimately, one can operate with any arbitrary concept, and it cannot be said that these must be tensor quantities; the [Christoffel symbols] are also not tensors, but one operates with them. The tσµ are the quantities that deliver the most. (Nachlass Rudolf Jakob Humm, Zentralbibliothek Zürich) Klein, too, was keen to peel away the many obscure aspects that surrounded energy conservation in general relativity. One of his first technical achievements was to give a simplified and much clearer derivation of Hilbert’s invariant energy equation. Hilbert’s version of this result had been worked out by using invariant theory to derive a very complicated entity, his energy vector el . Klein became convinced that Hilbert’s energy equation should not be regarded as analogous to conservation of energy in classical mechanics; instead he understood it to be a general identity that resulted from the Lagrangian expressions in the variational formalism. When he arrived at this conclusion is difficult to say, but his exchange of letters with Hilbert makes clear that Emmy Noether was thickly involved from the beginning. After discussing his main mathematical points, Klein added the following “essential” remark: “You know that Frl. Noether continually advises me in my work and that actually it is only through her that I have delved into these matters” [Klein 1918a, 559]. Klein explained further that already a good year before he presented his findings to the Göttingen Mathematical Society on January 22, Emmy Noether had found related results pointing in the same direction. She told him about her prior work and showed him a manuscript in which she had derived her findings. much more likely, given that Noether had shown Klein her manuscript, is that the latter called attention to it during his lecture. If so, Humm would surely have been curious to see it and probably then asked Noether if he could make a copy of it.

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This came as no surprise to Hilbert, who had asked her to investigate the relationship between this aspect of his theory and Einstein’s back in May 1916. In his reply, he began by expressing total agreement not only with Klein’s derivation but also with his interpretation of it as well (“. . . mit Ihren Ausführungen über den Energiesatz stimme ich sachlich völlig überein . . . ”). This assent, however, probably did not result from a close study of Klein’s argument, because Hilbert immediately followed with these brief remarks indicating that Noether had already clarified the central issue: Emmy Noether, whose help I called upon more than a year ago to clarify these types of analytical questions pertaining to my energy theorem, found at that time that the energy components I had set forth – as well as those of Einstein – could be formally transposed by means of the √ Lagrangian differential equations (4) and (5) [the equations [ gH]µν = √ 0, [ gH]h = 0 discussed above] in my first paper into expressions whose divergence vanished identically, that is without using the Lagrangian equations (4) and (5). [from Hilbert’s reply in [Klein 1918a, 560]]23 These remarks clearly point to the substantial nature of Noether’s contribution, but without actually making evident exactly what she had proved and how. Humm’s manuscript reveals, in fact, that her methods were very close to those Hilbert had employed in [Hilbert 1915]. If Hilbert had found her results at that time significant, it seems hard to understand that they only received mention in this somewhat contrived epistolary exchange, published probably more than one year later. Certainly by modern-day standards of fairness, Noether should have been treated as a coauthor rather than as a mere background figure. In the meantime, the status of Theorem I and its role in the formulation of conservation laws remained, for the time being at least, in the background. Of far greater concern was the general interpretive issue Klein had raised regarding the distinction between the conservation laws of classical mechanics and those of general relativity. Klein, not surprisingly, attached special significance to this distinction in the light of his Erlangen Program [Klein 1872], which he now promoted as a general doctrine applicable to the new physics. Relativity theory, according to Klein, should not be thought of exclusively in terms of two groups – the Lorentz group of special relativity and the group of continuous point transformations of general relativity – but rather should be broadly understood as the invariant theory relative to some given group pertaining to a particular physical situation. What he meant by this is illustrated by his subsequent publications on energy-momentum conservation and relativistic cosmology [Klein 1918b] and [Klein 1919]. Emmy Noether clearly saw her work on conservation laws derived 23 This characterization of Noether’s reulsts is, at best, misleading. What Noether showed was that Hilbert’s energy vector el and Einstein’s pseudo-tensor tσ µ could be decomposed into a sum of two entities, one of which had vanishing divergence independent of the field equations, whereas the other part vanished as a consequence of the field equations; for details see [Rowe 2019c]).

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from variational principles as part of Klein’s general program (see [Noether 1918b, 256–57]). In his reply to Klein, cited above, Hilbert not only expressed agreement with Klein about the lack of analogy between classical energy conservation and his own energy equation, he went on to say that this was a characteristic feature of general relativity. In fact, he even claimed one could prove a theorem effectively ruling out conservation laws for theories bases on general transformations, i.e., laws analogous to those that hold under transformations of the orthogonal group. In his counterresponse, Klein closed by saying: “It would interest me very much to see the mathematical proof carried out that you alluded to in your answer” ([Klein 1918a, 565]). By this time, however, Hilbert’s period of intense involvement with general relativity had reached an end. He and Klein continued to correspond about relativity into March of 1918 [Frei 1985, 141–144], after which time Hilbert only showed sporadic interest in these matters.24 It was thus left to Emmy Noether not only to settle this issue but to prove Theorem I, the Leitmotiv for Hilbert’s first foray into general relativity, in her fundamental paper “Invariante Variationsprobleme.”

3.5 Noether on Invariant Variational Problems On March 5, 1918, Max Noether celebrated the fiftieth anniversary of his Heidelberg doctorate, a major event in the life of a German professor during this time. Probably very few in academic circles then realized that their country’s proud monarchy would soon be in its waning hours. In late February, Emmy Noether was back home visiting her father in Erlangen, no doubt helping prepare for the celebration. Around this time, Felix Klein put the last touches on [Klein 1918a], the tripartite article he composed with Hilbert. Klein was already making plans for a second note, which he described in a no longer extant letter to Noether, who wrote back on 29 February: “I thank you very much for sending me your note and today’s letter, and I’m very excited about your second note; the notes will certainly contribute much to the understanding of the Einstein–Hilbert theory.” 25 After this she proceeded to explain where matters stood with regard to the key question Klein was hoping to answer, namely, the relationship between the classical and relativistic energy equations. Clearly, she was already deeply immersed in this problem. In the meantime, Klein was conferring with his colleague Carl Runge, who thought he had found an elegant way to finesse the problems associated with Einstein’s pseudo-tensor tσµ for gravitational energy in the equation ∂(Tσµ + tσµ ) = 0. ∂xσ 24 Hilbert’s interests in physics were very broad (see [Corry 2004b]) and his interest in quantum physics continued unabated; see [Schirrmacher 2019]. 25 E. Noether to F. Klein, 29 February 1918, Klein Nachlass, SUB Göttingen.

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On March 8, Runge presented this trick at a meeting of the Göttingen Scientific Society. His idea was to particularize the coordinate system so that tσµ would vanish, leading to the equation ∂Tµσ = 0, ∂xσ which would allow for the recovery of energy-momentum conservation just as in special relativity. Klein wrote to Noether, who was still in Erlangen, to tell her about this supposed breakthrough and send her drafts of two notes, one by Runge and another he himself had composed on a related idea. She replied only a few days later (Fig. 3.1) by throwing cold water on Runge’s idea; she cited concrete examples, such as the principle of least action, for which it failed. In this case, she indicated that Runge’s transformation leads to well-known identities that cannot be interpreted as energy laws. In analogy with gravitational theory, she went on, one could develop a version of mechanics whose laws were exclusively derived from an invariant principle of least action, but this theory would lack an energy law; instead it would have an identity linking the Lagrangian expressions derivable from the action principle. Noether thus emphasized that even in the most familiar settings “one cannot, in any manner, obtain an energy law unless one postulates it in place of Runge’s condition” (Noether to Klein, 12 March 1918 [Kosmann-Schwarzbach 2006/2011, 155]). Klein apparently did not understand Noether’s arguments, or possibly found them less than convincing. In any event, when he wrote Einstein nine days later he made no mention at all of her reservations, calling Runge’s idea a “pure egg of Columbus” [Einstein 1998, 688]. Delighted by this imagined breakthrough, Klein went on to say that Runge’s communication as well as his own additional note were now being prepared for publication in the Göttinger Nachrichten. Klein then awaited Einstein’s reply with great anticipation. This came four days later, on March 24, and it took the wind right out of Klein’s sails. The Berlin physicist dismissed Runge’s idea of particularizing the coordinate system to eliminate Einstein’s pseudo-tensor for gravitational energy, though his reasons for doing so were very different from Emmy Noether’s. Whereas her reservations were based on mathematical considerations, Einstein merely pointed out that he had earlier considered a similar approach but gave it up for physical reasons, since “the theory predicts energy losses due to gravitational waves” which undermine the conservation of the total energy of a closed system [Einstein 1998, 698]. Such considerations were surely new in Göttingen, so this brusque rejection caused Runge and Klein to reconsider the whole matter again; they thereafter withdrew their proposal. A few months later, Runge reported on Einstein’s paper “On Gravitational Waves” [Einstein 1918a] at a meeting of the Göttingen Mathematical Society. In it, Einstein rebutted a similar proposal that had been set forth by Tullio Levi-Civita.

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Figure 3.1: Emmy Noether to Felix Klein, 12 March 1918 (Nachlass Klein 22B, SUB Göttingen)

Emmy Noether’s letter to Klein from 12 March also contains indications that she was well on her way to obtaining key results she would later publish in [Noether 1918b]. “As a result of my further research,” she wrote, “I have now seen that the energy law fails to hold for invariance under every extended group of transformations induced by z.” 26 Her remarks by way of explanation were surely insufficient for Klein to understand what she meant by this, but after Noether returned to Göttingen, she briefed him in full about the import of this important breakthrough. Though still groping her way forward, she had begun to realize that the conventional conservation laws derived from various spatial and temporal symmetries underlying action principles correspond to systems invariant under finitely generated Lie groups. These groups lead to divergence relations, which then vanish when the corresponding Lagrangian derivatives vanish, and in the case of a first-order variational problem, they lead to n linearly independent first integrals, where n is the number of parameters for the group in question. These were familiar from classical mechanics, where the 10-parameter group of Galilean transformations corresponds to ten first integrals, which can then be interpreted 26 Noether to Klein, 12 March 1918 [Kosmann-Schwarzbach 2006/2011, 155–156]. Her idea was evidently to introduce a function φi that reflected the same transformational properties as those given by the dependent variables zi . This is elaborated at the beginning of Section 6 in [Noether 1918b].

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as conservation laws. Since a first integral reduces the number of independent variables in a system of differential equations by one, ten independent first integrals suffice to completely determine the motion of a mechanical system. For the general case, Noether would later refer to these as “proper conservation laws,” which she distinguished from those that are “improper” because they can be derived from general identities by particularizing the functions in an infinite Lie group. Emmy Noether prepared a first draft of [Noether 1918b] for a presentation she gave to the Göttingen Mathematical Society on 23 July. Three days later, Klein submitted this text to the Göttingen Scientific Society, and after receiving proofs toward the end of September, she then put the paper in its final form. 27 The fundamental results she obtained not only resolved the status of Hilbert’s Theorem I, they effectively described the precise manner in which conservation laws arise for Lagrangian systems in both classical mechanics as well as modern field theories. From the mathematical standpoint, Noether’s analysis also provided a strikingly clear and altogether general answer to the question Klein had raised about the status of conservation laws in Lagrangian systems. Her study [Noether 1918b] thus remains today nearly the last word on this subject.28 In her introduction, Noether described her paper as combining the formal methods of the calculus of variations with those of Sophus Lie’s group theory. 29 In several places she cited terminology, concepts, and results from [Lie 1891], a two-part study that Friedrich Engel wrote up based on manuscripts prepared by Lie. Klein was familiar with this work shortly after it was published, as he referred to it in his lectures on higher geometry, the second half of which was primarily devoted to Lie’s work [Klein 1893/1907, 303]. As in nearly all her work, she conscientiously cited earlier related publications; in the present case, beyond Lie’s study, these were papers by Georg Hamel, Gustav Herglotz, H.A. Lorentz, Adriaan Fokker, and Hermann Weyl. She further noted that Klein’s nearly simultaneous paper, [Klein 1918b], concerning the relationship between various forms for differential conservation laws in general relativity, resulted from their discussions, so that her results and his “mutually influenced one another” [Noether 1918b, 235–236]. In October, Klein sent a copy of his paper to Einstein, who responded on 22 October with enthusiasm: “I have already studied your paper most thoroughly and with true amazement. You have clarified this difficult matter fully. Everything is wonderfully transparent” [Einstein 1998, 917].30 Klein had apparently informed Einstein that he was now at work on another paper, [Klein 1919], the last of his 27 The English translation in [Kosmann-Schwarzbach 2006/2011, 3–22] contains corrections of several minor errors, most of them probably due to mistakes by the typesetter. 28 For a discussion of various modern refinements of Noether’s fundamental results, see [Kosmann-Schwarzbach 2006/2011, 133–144]. 29 Lie usually dealt with families of infinitesimal transformations that generated the corresponding Lie group; for historical background on his approach, see [Hawkins 1991]. 30 Einstein queried Klein, however, about a certain entity that Klein had claimed was a 4vector. After a few more exchanges, Klein wrote on 10 November that Emmy Noether had in the meantime informed him that a proof based on higher principles could be put together

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three contributions to general relativity; this clarified a number of points concerning the integral form of the conservation laws as formulated in [Einstein 1918b]. Despite his close involvement with Noether’s work, Klein made no substantial contribution to the results in [Noether 1918b], which she conceived and proved on her own. What he contributed was, first and foremost, framing the general question that led to Hilbert’s conjecture regarding the relationship between classical conservation laws and those that arise in general relativity, and then urging Noether to investigate this question. Klein’s knowledge of Lie theory was probably of some help to Noether as well, although she may well have been familiar with Lie’s writings in the course of her studies dealing with differential invariants. In his theory of continuous groups, Sophus Lie distinguished between three types: those with finitely many parameters, those defined by finitely many functions and their derivatives (infinite continuous groups), and mixed groups, which contain both types of parameters. These were essential ingredients for Noether’s two main theorems, which she formulated as follows: Theorem I. Let Gρ be a finite continuous group with ρ parameters. If the integral I is invariant with respect to Gρ , then ρ linearly independent combinations of the Lagrangian expressions become divergences, and conversely. The theorem also holds in the limiting case of infinitely many parameters. Theorem II. Let G∞ρ be an infinite continuous group depending on ρ continuous functions. If the integral I is invariant with respect to G∞ρ , in which arbitrary functions and their derivatives up to the σth order appear, then ρ identical relations are satisfied between the Lagrangian expressions and their derivatives up to the order σ. The converse also holds here. [Noether 1918b, 238–239] In the course of proving these very general results, Noether proceeded constructively. Her arguments thus lead, at least in principle, to concrete formulas that can either be computed directly or else used to model solutions based on approximation methods. One should further note the strong parallelism in these two formulations, both of which assert that the number of parameters in the two types of symmetry groups lead to an equal number of relations between the Lagrangian expressions (Noether’s term for the left-hand side of the Euler-Lagrange equations derived from I, which she writes ψi ) and their derivatives. Both theorems thus address the types of formal identities one can deduce from an invariant variational problem depending on whether the Lie group is finite or infinite. In the first case, the Lagrangian expressions alone enter, and so by invoking the Euler-Lagrange equations ψi = 0, Theorem I asserts that there are ρ conserved quantities whose divergences vanish. This is the usual formulation of “Noether’s Theorem,” which using ideas in Hilbert’s first note. Klein sketched her argument in a letter to Einstein from 10 November [Einstein 1998, 942–943].

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in her paper follows as an immediate consequence of Theorem I. Her Theorem II characterizes the manner in which identities satisfied by a combination of the Lagrangian expressions and their derivatives come into play. Hilbert’s Theorem I may thus be seen as a special case of Noether’s Theorem II in which the transformation group G∞4 consists of all possible functions of the four coordinates of the world points. Hilbert had done little more than notice that a system of generally covariant equations will possess four superfluous degrees of freedom that lead to four dependence relations. Noether’s Theorem II, on the other hand, not only generalizes Hilbert’s Theorem I, it also provides a constructive proof, while sharpening the result so that it becomes an if and only if condition. Indeed, key parts of her overall argument depend on knowing that the two theorems and their converses happen to hold. Writing to Einstein on 7 January 1926, Noether noted that “what mattered in ‘Invariante Variationsprobleme’ was the precise formulation of the scope of the principle and, above all, its converse . . . ” [Kosmann-Schwarzbach 2006/2011, 2011: 164]. With regard to its significance for physics, she pointed out in her paper that the first theorem generalized the formalism underlying the standard results pertaining to first integrals in classical mechanics, whereas she characterized her second theorem as “the most general group-theoretic generalization of ‘general relativity” ’ [Noether 1918b, 240]. “Noether’s Theorem” – often formulated in far less general terms – is usually the only result from her paper that one finds in physics textbooks. Its importance for physical theories became obvious once physicists began to realize the wealth of possibilities it provided for deducing conserved quantities from symmetries in variational systems. Noether’s motivation, on the other hand, had very little to do with what today’s physicists think about when they discuss “Noether’s Theorem.” Her principal goal was to use group theory as a tool for distinguishing between proper and improper conservation laws. The upshot of her analysis would show, in fact, that this distinction depends heavily on the choice of the underlying transformation group. What remains to be seen is how Noether combined the results from her two theorems in order to distinguish between real and apparent conservation laws in physics. Her idea is essentially the following: Suppose that the integral I is invariant with respect to G∞ρ , then one can particularize the functions pλ , λ = 1, 2, . . . , ρ to obtain a finite continuous subgroup Gσ of G∞ρ . The divergence relations that arise from Gσ are thus fully determined by the group Gσ . Hence, as a subgroup of G∞ρ , these divergence relations must also be derivable from the identities connecting the Lagrangian expressions and their total derivatives by suitably particularizing the pλ . Since such expressions are derivable as invariants of the general group G∞ρ , she called these improper divergence relations; all others were proper. From these considerations, she concluded that: The divergence relations corresponding to a finite group Gσ are improper if and only if Gσ is a subgroup of an infinite group with respect to which I is invariant [Noether 1918b, 254].

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Noether then turned to illustrate how this finding relates to the case of conservation theorems in special and general relativity. In the former context, the infinitesimal coordinate transformations xi → xi + i , i = 1, 2, 3, 4 generate a finite continuous group, which Noether calls a translation group. In classical mechanics, with three coordinates, these symmetries correspond to the three components of linear momentum, whereas invariance under the fourth time coordinate yields conservation of energy. One then shows that this energy-momentum 4-tuple transforms as a 4-vector, which can likewise be obtained by taking the derivative of the energy-momentum tensor T,νµν = 0, as noted above. Within the context of special relativity these translational symmetries constitute four of the ten parameters of the Poincaré group, and by Noether’s Theorem I they correspondingly lead to these four conserved quantities. On the other hand, if we place these transformations in the context of general relativity, where the corresponding infinite Lie group is G∞4 , then one can particularize the arbitrary functions p1 , p2 , p3 , p4 so that pi = i , obtaining the same result. And since this argument clearly holds in general, Noether could draw the following significant conclusion: If I admits a translation group G, then the energy relations are improper if and only if I is invariant with respect to an infinite group that contains G as a subgroup [Noether 1918b, 255]. These were clearly the kinds of considerations Noether had in mind when she sent Klein her postcard on March 12. As Noether noted, the conservation laws of classical mechanics as well as those of special relativity theory are proper in Noether’s sense. One cannot deduce these as invariants of a suitably particularized subgroup of an infinite group within the context of these two theories. In general relativity, on the other hand, every Lagrangian variational problem will contain four identities as a consequence of the principle of general covariance. These, however, play no role in proper conservation laws, which also cannot be derived from special groups of point transformations. In summarizing these findings, Noether wrote: Hilbert expressed his assertion regarding the absence of actual energy theorems as a characteristic attribute of ‘general relativity theory.’ If this assertion is to be literally valid, then the term ‘general relativity’ must be taken more broadly than is usual and extended to groups that depend on n arbitrary functions. [Noether 1918b, 256–257] From the mathematical standpoint, Noether’s analysis provided a strikingly clear and altogether general answer to the question Klein had raised about the status of conservation laws in Lagrangian systems.

3.6 On the Slow Reception of Noether’s Theorems Strange as it may seem, Noether’s novel theoretical results made little impression on contemporary investigators [Kosmann-Schwarzbach 2006/2011]. With the

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exception of [Bessel-Hagen 1921], one finds only scattered references to Noether’s “Invariante Variationsprobleme” [Noether 1918b] in the standard literature on relativity theory, despite the fact that Klein clearly regarded these particular findings as not only important but definitive (see his remarks in [Klein 1921–23, 1: 584– 585]). In his influential report on relativity theory, [Pauli 1921], Wolfgang Pauli discusses the four identities connecting the ten field equations for the gµν , noting only that “this was first recognized by Hilbert” [Pauli 1921, 160]. Pauli’s report contains detailed accounts of work on energy conservation, but not a single reference to [Noether 1918b] (he only mentions Noether’s earlier paper [Noether 1918a] once in a footnote, [Pauli 1958, 48]). Just as significant, Weyl’s Raum—Zeit— Materie [Weyl 1918], which went through five editions between 1918 and 1923, carries but a single reference to her paper [Noether 1918b], and this is tucked away in a footnote [Weyl 1952, 322, note 5]. Finally Einstein, who clearly admired Noether’s work but who also was less than generous when it came to citing other authors, apparently never so much as mentioned Noether’s name in any of his own publications. The curious lack of references to Noether’s two theorems, even as late as the period 1950–1980, is documented in [Kosmann-Schwarzbach 2006/2011, 103–132].31 Noether herself briefly referred to [Noether 1918b] in the context of surveying the importance of formal variational methods for the theory of differential invariants (see [Noether 1922b, 408]). She also made some interesting remarks about her paper in a letter to Einstein from January 7, 1926. He had requested her opinion on a paper submitted to Mathematische Annalen that dealt with a topic directly related to her work on invariant variational problems. In her reply, she described its contents as neither original nor intelligible, which surely accounts for why it was never published. She remarked further that . . . there is a reference to Bessel-Hagen’s work (citing me there is an error); then the author carries out the obvious integration of the conservation laws, which is not in Bessel-Hagen. In the following paragraphs he uses the calculus of variations to derive the field equations and their identities in the case of general relativity: first in the case where the electrical field vanishes, then without that assumption, and finally in Weyl’s case or even more generally. Since there are only calculations and not a single word of explanation (except in the introduction), this is difficult to comprehend. The systematization in earlier works – in particular with respect to Klein – consists in calculating the formulas for an arbitrary action function W and only then substituting the value of W in the final formulas. For someone who does not know the theory it would be impossible to understand the calculations. 31 [Kastrup 1987] gave the first detailed study of the published literature related to [Noether 1918b].

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Nor does this matter represent any essential progress, since nearly everyone has worked with variational principles in the end. For me, what mattered in ‘Invariante Variationsprobleme’ was the precise formulation of the scope of the principle and, above all, its converse, which does not play a role here. I cannot judge to what extent the integration of the conservation laws is of interest physically. Should that be the case, perhaps a physics journal might publish this short part, while referencing Bessel-Hagen; this note could then explicate my theorems by citing Courant-Hilbert [Courant/Hilbert 1924, 216] . . . . But here I must leave it to physicists to judge its value.32 The final paragraph suggests that Emmy Noether considered Courant’s influential textbook (which only carried Hilbert’s name as spiritus rector, not as a coauthor) an excellent vehicle for promoting awareness of her results among physicists. Few, however, seem to have taken any notice, as the Noether theorems (note the plural above) met with stoney silence. In the light of this, it should come as no surprise that few mathematicians and even fewer physicists ever read [Noether 1918b]. It hardly helped matters that her paper was published in the Nachrichten of the Göttingen Scientific Society, where it was mixed together with work from a wide variety of scientific disciplines. Klein’s articles on relativity theory originally appeared there as well, but in 1921 he gathered these and earlier papers together in the first volume of his collected works, for which he provided supplementary commentary. By doing so, he placed them in a special section entitled “Zum Erlanger Programm” [Klein 1921–23, 1: 411–612], precisely the perspective he wanted his readers to appreciate. Hilbert’s work also became more visible after 1924 when he published a revised version of his two papers on foundations of physics in Mathematische Annalen [Hilbert 1924].33 Unlike Klein, however, he minimized the significance of Emmy Noether’s achievement while at the same time deflecting attention from major weaknesses in his own prior work on unified field theory. In republishing this work, he advertised it as “essentially a reprint of the earlier communications [Hilbert 1915, Hilbert 1917] . . . and my remarks on them that F. Klein published in his communication ‘Zu Hilberts erster Note über die Grundlagen der Physik’ [Klein 1918a] – with only minor editorial alterations and changes in order to ease understanding” [Hilbert 1924, 1]. This “reprint” contains, in fact, major alterations of the contents of the first note that no careful reader could possibly miss. His central Theorem I from [Hilbert 1915] is here only mentioned in passing, with a footnote citing Emmy Noether’s paper for a “general proof” ([Hilbert 1924, 6]). Moreover, his main physical claim, namely that his results proved that “electrody32 This letter is replicated, transcribed, and translated in [Kosmann-Schwarzbach 2006/2011, 161–165]; the translation given here deviates slightly from the one given there. 33 His second paper [Hilbert 1917] dealt with causality conditions and the Cauchy problem in general relativity.

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namic phenomena are the effects of gravitation” was simply dropped from the text. In its place, he wrote instead that the four independent identities that derive from √ the gravitational equations [ gH]µν = 0 signify the “connection between gravity and electrodynamics” [Hilbert 1924]. Like Hilbert, Klein clearly took deep satisfaction in the part he was able to play in elucidating the mathematical underpinnings of key results in the general theory of relativity. In a letter to Pauli, Klein related how Einstein had responded after reading [Klein 1919], which made him “happy as a child whose mother had presented him with a piece of chocolate.” To this, Klein added that “Einstein is always so gracious in his personal remarks, in complete contrast to the foolish promotional efforts (“törichten Reklametum”) undertaken to honor him.” 34 Klein also made clear to Pauli that, while he liked the pre-publication version of his report that he read, the article “could not pass over Hilbert’s efforts in silence.” 35 The twenty-one-year-old Pauli clearly took heed of this advice. He apparently never heard a similar concern expressed regarding Emmy Noether’s work on relativity theory, despite the fact that it was she who resolved the central question Klein had raised. When it came to bestowing honor, Klein, as well as Hilbert, Einstein, Weyl, and Sommerfeld’s young pupil, Wolfgang Pauli, seem to have given little thought to the woman who went on to make Göttingen the world’s leading outpost for the study of modern algebra. In the next two chapters, we describe how this triumphal period in Emmy Noether’s extraordinary life began to unfold.

34 Klein 35 Klein

to Pauli, 8 March 1921, [Pauli 1979, 27]. to Pauli, 8 May 1921, [Pauli 1979, 31].

Chapter 4

Noether’s Early Contributions to Modern Algebra 4.1 On the Rise of Abstract Algebra As described in preceding chapters, Noether’s work on invariant theory broke new ground that led the Göttingen mathematicians, but first and foremost Hilbert, to invite her to habilitate there. At the time, no one would have imagined that her expertise in this field would prove decisive for clarifying how so-called conservation laws are related to symmetries of group actions that leave a variational system invariant. This caused a real stir of interest in 1918, but soon thereafter the Noether theorems were largely forgotten by the main actors, Einstein, Weyl, et al., even though they continued to pursue schemes for uniting gravity and electromagnetism into a single field theory.1 In the meantime, however, Noether’s earlier work on invariant theory was by no means overlooked. One of those who took a deep interest in it was the Ukrainian mathematician Alexander Ostrowski, who met Emmy Noether soon after the war ended. Since he attended the Kiev College of Commerce, rather than a regular high school, Ostrowski was not formally qualified to be admitted to a university. His talent did not go undetected, however, and his mentor, Dmitry Grave, wrote to Edmund Landau as well as to Kurt Hensel seeking their help. Arrangements were afterward made for Ostrowski to begin studying mathematics in 1912 under Hensel’s supervision at Marburg University. Then the war broke out, leading to his internment, though thanks to Hensel’s intervention he still had access to the university library. Taking full advantage of this opportunity, Ostrowski began reading entire volumes of mathematics journals from cover to cover; before long he was fully versed on much recent research, including work on invariant theory. 1 Gustav Mie’s electromagnetic theory of matter soon fell out of favor, however, which probably helps account for why Hilbert withdrew from this field after 1918.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_4

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Ernst Fischer’s article in Crelle [Fischer 1911] and two related papers by Emmy Noether ([Noether 1913] and [Noether 1915]) particularly caught his attention. Noether wrote these during her last years in Erlangen, when she and Fischer were still engaged in an intense collaboration. Nevertheless, her work on invariant theory – parts of which still used the formal methods she learned under Gordan – by no means ended at this juncture; two of her later papers ([Noether 1919b] and [Noether 1920]) reflect her ongoing interests in this area of research. After Noether moved to Göttingen in the spring of 1915, Ostrowski struck up a correspondence with her concerning some of his closely related results, which he would later publish in [Ostrowski 1918b]. Their paths would afterward cross on a great many occasions. When the war ended, Ostrowski moved to Göttingen, where he completed his doctoral dissertation, working closely with Hilbert and Landau. His primary responsibility, however, was to serve as Felix Klein’s principal assistant in preparing the first volume of his Collected Works [Klein 1921–23]. He met with Klein virtually every day for this purpose, which surely led to occasional conversations with Noether, who had already been working closely with Klein during the preceding two years. In 1920, Ostrowski completed the editorial work for volume 1, which contains Klein’s commentaries on his papers dealing with Einstein’s general theory of relativity as well as on Emmy Noether’s contributions from the same period (see Sections 3.5 and 3.4). During the next two years, Ostrowski worked as Erich Hecke’s assistant in Hamburg, completing his Habilitation there before returning to Göttingen. Over the next six years he taught alongside Noether as a private lecturer before accepting a professorship in Basel, the final outpost of his long career. Although their lives and mathematical interests crisscrossed often, Ostrowski’s and Noether’s names are seldom mentioned together.2 Invariant theory was only one of the threads that tied these two rising stars together. In a paper on absolute irreducible polynomials [Noether 1922a],3 Noether noted the close connection between her main result and a theorem published by Ostrowski a few years before. Ostrowski and Helmut Hasse were both in different ways exponents of the Marburg tradition founded by Kurt Hensel, whose theory of p-adic numbers emerged as a new cornerstone for modern algebra in the 1920s. In 1916, Ostrowski helped launch valuation theory by proving that the only non-trivial absolute values on 2 Many references to him and his work can be found in [Noether 1983]; for example, in the concluding section of [Noether 1920, 30], she notes that Ostrowski had alerted her to an inexact formulation in her paper [Noether 1918c]. Another instance is a lecture Noether gave at the Göttingen Mathematical Society, delivered on 11 November 1924, in which she referred to an explicit formula derived by Ostrowski for calculating the number of linear independent residue classes for ideals in a polynomial ring [Noether 1925a]. 3 In Steinitz’s classic paper [Steinitz 1910], he proved that every field has an essentially unique extension to an algebraically closed field F , a fact Noether likened to a generalized form of the fundamental theorem of algebra [Noether 1922a, 26], since every polynomial in a single variable reduces to linear factors in F . A polynomial in two or more variables is called absolutely irreducible if it remains irreducible in F .

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the rational numbers are the usual real absolute value and the assorted p-adic absolute values for prime numbers p. Four years later, Hasse uncovered his local global-principle, which has ever since played a central role in number-theoretic researches. During the late 1920s, when Hasse became her single most important collaborator, Noether forged a new line of abstract research aimed at wedding hypercomplex number systems to algebraic number theory (see Sections 5.6, 6.1, 6.3). As for invariant theory, Ostrowski’s work attracted the interest of the distinguished Berlin algebraist Isaai Schur.4 One of Schur’s most brilliant students was Richard Brauer, the “one who most carried on the [Berlin] tradition of Frobenius and Schur by making fundamental contributions to the representation theory of finite groups.” 5 His collaboration with Noether would later become famous in the annals of algebra by way of the Brauer-Hasse-Noether theorem. In reflecting on the enormous range of relationships Emmy Noether cultivated during her years in Göttingen, it is easy to forget that jointly written papers and collaborative partnerships were quite rare in mathematics before 1920. They became more common, at least in part, because of the atmosphere that surrounded her in that special time and place. This chapter attempts to provide a glimpse of the many facets of Noether’s school in abstract algebra, one of the most famous success stories in the history of modern mathematics. The examples just given should suggest the broader scope of the phenomenon we have in mind. Although Ostrowski, Hasse, and Brauer were major mathematicians with independent research agendas, they were also in varying degrees affiliated with Emmy Noether’s school, which was itself part of a larger cultural movement in early twentieth-century mathematics.6 This applies as well to B.L. van der Waerden, Pavel Alexandrov, and Olga Taussky, all of whom, like Hasse, entered Noether’s world from mathematical cultures very different from the one they found in Göttingen. Noether also supervised the work of several doctoral students, of course, some of whom made important contributions to modern algebra. Her influence, however, clearly extended far beyond the smaller circle of those who took their doctorates under her. Today Emmy Noether holds a special place in the history of algebra, as her work represents not only the culmination of a long-standing development but also a fresh beginning for a new form of abstract mathematics.7 Reflecting on this, Israel Kleiner wrote: . . . Noether was not the only, nor even the only major contributor to the abstract, axiomatic approach to algebra. Among her predeces4 Ostrowski’s work and its influence on Schur are part of the complex story leading up to Hermann Weyl’s seminal work in the mid-1920s that launched modern Lie theory, as told by Thomas Hawkins in [Hawkins 2000, 409–414]. 5 [Hawkins 2013, 552]; see also [Curtis 1999] and [Curtis 2003]. 6 The role of the Noether school in this cultural movement is described in [Koreuber 2015, Kap. 4]. 7 For a recent survey of the subject, see [Gray 2018].

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4 Noether’s Early Contributions to Modern Algebra sors who contributed to the genre were Cayley and Frobenius in group theory, Dedekind in lattice theory, Weber and Steinitz in field theory, and Wedderburn and Dickson in the theory of hypercomplex systems. Among her contemporaries, Albert in the U.S. and Artin in Germany stand out. . . . 8 . . . [Noether] dealt with just about the whole range of subject matter of the algebraic tradition of the nineteenth and early twentieth centuries (with the possible exception of group theory proper). What is significant is that she transformed that subject matter, thereby originating a new algebraic tradition – what has come to be known as modern or abstract algebra. [Kleiner 2007, 91–92]

In short, Noether helped to spearhead a broad cultural movement, and her influence within this dynamic sphere of activity was truly pervasive. Van der Waerden summed up her mathematics as flowing from a single unified principle that served as her maxim: All relations between numbers, functions, and operations become perspicuous, capable of generalization, and truly fruitful, when they are detached from specific examples and traced back to conceptual connections. [van der Waerden 1935, 469]

4.2 Noether’s Contributions to Abstract Ideal Theory Over the last centuries, the natural sciences have employed various concepts which serve as building blocks for more complex structures. One thinks of the periodic system in chemistry, elementary particles in physics, or molecular structures in cell biology. In classical number theory, these building blocks are the prime numbers, which still hide many secrets even to the present day. In group theory, they are the simple groups.9 Number fields and Galois groups, two key concepts in modern algebra, eventually assumed center stage in Emmy Noether’s work. Although she came to algebraic number theory relatively late in her career, from 1927 onward she took up this new field of research with real passion in an intensive collaboration with Helmut Hasse. During the period immediately before, from 1920 to 1926, ideal theory was the main focus of her interests. She was primarily concerned with decomposition theorems in ideal theory, which utilize special types of ideals as building blocks, in particular prime ideals. The roots of classical ideal theory can be found in two central developments from the 19th century: algebraic number theory and the theory of algebraic functions. Around 1880, Leopold Kronecker recognized that these theories have anal8 On

the Chicago tradition of Dickson and Albert, see [Fenster 2007]. groups are those with no nontrivial normal subgroups; just like prime numbers, they cannot be “factored” into smaller groups. One of the major research projects of the twentieth century, completed in 2004, was the classification of all finite simple groups. 9 Simple

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ogous structures. This analogy was also a great inspiration for Hilbert in his work on algebraic number theory. For Emmy Noether, who came to the theory of algebraic functions from invariant theory, a major challenge was to explain the deeper reasons underlying this parallelism. In doing so, she took a great deal of inspiration from the work of Richard Dedekind, who created classical ideal theory by introducing new algebraic concepts in number theory. Over time, Dedekind’s works exerted an ever-stronger influence on Emmy Noether’s orientation as an algebraist, and she never tired in recommending his Supplement XI to Dirichlet’s lectures on number theory [Dedekind 1894a] to her students. Nevertheless, her own work was decidedly more abstract than Dedekind’s [Corry 2017]. Moreover, her axiomatic approach to algebraic structures opened up new vistas not only for algebra but for other fields as well. The contrast between Dedekind’s approach to ideal theory and Noether’s was emphasized by Pavel Alexandrov, who attended her lectures during the summer semester of 1923: Of all the lectures I heard in Göttingen that summer, the apex were Emmy Noether’s lectures on general ideal theory. As is well known, foundations of this theory had been laid by Dedekind in his famous paper that was published as the eleventh supplement to the edition of Dirichlet’s lectures on number theory under Dedekind’s editorship. . . . Emmy Noether always said that the whole theory of ideals could already be found in Dedekind and that all she had done was to develop Dedekind’s ideas. Of course, the basis of the theory was laid by Dedekind, but only the basis: ideal theory, with all the richness of its ideas and facts, the theory that has exerted such an enormous influence on modern mathematics, was the creation of Emmy Noether. I can judge this, because I know both Dedekind’s work, and the fundamental work of Emmy Noether on ideal theory. [Alexandrov 1979/1980, 299] Emmy Noether’s contributions to ideal theory were part of a general trend that shifted mathematicians’ understanding of their objects of research. Modernists like her sought to define these objects in abstract terms by setting down axioms for the properties they satisfied. This approach differed from classical mathematics, which took the objects under investigation to be somehow known in a concrete sense. One thus started from fixed objects and examined their properties. For example, in algebra one analyzed systems of equations in which the coefficients came from very specific number systems. In geometry one examined the properties of curves and surfaces, which were often described by means of equations. With the use of abstract axioms, however, mathematicians could argue backwards: starting with a series of conditions, they derived further properties that must hold for all possible objects that satisfy the initial conditions. This trend toward axiomatization is often mentioned in connection with Hilbert’s formalist approach to mathematics, which led to some of his most famous

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achievements (especially for investigations in the foundations of geometry). 10 But, in fact, this was only part of a much broader movement that culminated in the work of the Bourbaki group in the 1940s and 1950s. Since that time, most mathematicians have taken this standpoint for granted. Thus, a great deal of modern mathematics is concerned with abstract structures and their interrelationships, starting with axiom systems that define precisely which properties hold for certain objects, but which otherwise require no further claims as to existence of these objects beyond the fact that they satisfy these axioms.11 Naturally, such axiom systems have to be formulated carefully in order to capture the essential properties of the objects characterized by them. In cases of doubt – and there have long been leading figures in mathematics who were doubters – the viability of an axiom system itself might be subjected to proof. Indeed, the modern field of proof theory begun by Hilbert was undertaken to nullify the criticisms of doubters like Kronecker and Brouwer. No one seriously questioned that Giuseppe Peano’s axioms for arithmetic were consistent, meaning that one could never prove a statement like 1 = 2 from them. Still, Kronecker famously claimed that the natural numbers were a creation of God, all else in mathematics being the work of mankind. Modernists, on the other hand, saw no need for appeal to divine agencies even when dealing with infinite objects, the bedrock of Georg Cantor’s set theory. As the leading spokesman for Cantorian mathematics, Hilbert thought the time was ripe to expel all doubt as to the absolute certainty of mathematical knowledge, starting with the axioms for arithmetic. Assisted by Paul Bernays, he developed a sophisticated proof theory that aimed to firmly establish the consistency of the axioms for ordinary arithmetic once and for all. This effort, however, famously failed: in 1930 Kurt Gödel showed that such a proof was impossible within the scope of Hilbert’s original proof theory. 12 Ideal theory had classical roots and thus stood apart from Cantorian modernism. It arose instead from problems in number theory, in particular the problem of representing numbers as products of primes. The fundamental Theorem of Arithmetic states that such a representation always exists in the realm of classical number theory, but over the course of the nineteenth century it became clear that this was not the case in other number fields. Gauss, Kummer, and Dedekind all struggled with this problem, which led to a sophisticated theory in which the primary building blocks for higher arithmetics had to be reinvented: these were now the prime ideals.13 To appreciate the nature of Emmy Noether’s contributions to ideal theory, we briefly review how Richard Dedekind came to invent ideal theory 10 On

Hilbert’s axiomatic approach to physics, see [Corry 2004b] and [Schirrmacher 2019]. formalist stance was already implicit in Hilbert’s early work in invariant theory, but became explicit in [Hilbert 1899] and especially in [Hilbert 1900]; its broader implications form a central theme in [Mehrtens 1990]. 12 For a comprehensive account of Hilbert’s works on foundations of mathematics, see [Ewald/Sieg 2013]. 13 A prime number p has the property that if p divides ab, then it must either divide a or b. Likewise, a prime ideal P in a commutative ring R has the property that if ab ∈ P , then either a ∈ P or b ∈ P . 11 This

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for the ring of integers in a number field. By Noether’s day, Dedekind’s theory had also been extended to other rings, in particular rings of polynomials. Her abstract approach to commutative ring theory embraced all previous results in ideal theories as special cases, thereby launching a new field of research: commutative algebra. In the terminology of modern algebra, this forms part of the general study of algebraic structures, beginning with the general notions of groups, rings, and fields. These, in turn, can best be understood by means of examples from classical mathematics, where these structures appear in concrete form in familiar number systems. In what follows, we sketch some of the relevant background for contextualizing Noether’s work in ideal theory. Dedekind’s theory was inspired by E. E. Kummer’s earlier approach to factorization in number fields using so-called ideal numbers [Edwards 1977]. Dedekind’s theory of ideals dealt with the ring of integers in a number field K, usually denoted OK , the algebraic integers in K. The algebraic integers A ⊂ C are the roots of monic polynomials (polynomials whose leading coefficient is 1) with coefficients in Z. A is a commutative subring of the complex numbers, and OK = K ∩ A. In this setting, Dedekind was able to prove that every ideal in OK can be uniquely expressed as a product of prime ideals. Dedekind’s theorem is the analogue of the fundamental theorem of arithmetic for the representation of ideals in a number ring. It states that if I is an ideal in the ring of integers of a number field (a Dedekind ring), then there are prime ideals P1 , P2 , . . . , Pk and positive integers (e ) (e ) (e ) e1 , e2 , . . . , ek such that I = P1 1 · P2 2 · . . . · Pk k . Dedekind’s theory served as a key role model for Emmy Noether’s more abstract theory of ideals in general rings. His theory, however, was situated in the special context of classical number theory, i.e. algebraic extensions of Z. Gauss had already done pioneering work in this field shortly after 1800 when he showed √ that in the ring Z[ −1] = Z[i] = {a + bi | a, b ∈ Z} unique factorization is valid. However, the prime numbers in Z are not necessarily prime in Z[i]. The number 5, for example, decomposes as: 5 = (1 + 2i)(1 − 2i), its factorization into prime numbers in Z[i]. Gauss introduced the notion of associates after realizing that the units {1, −1} for Z had to be extended. In Z[i] there are four units: {1, −1, i, −i}, the four roots of the equation x4 − 1 = 0. For each number α ∈ Z[i] there are four numbers associated with it: α, −α, αi, −αi. Gauss was able to prove that, up to associated elements, every α can be uniquely represented as a product of irreducible elements in Z[i].14 √ actually set forth a general theory of arithmetic for quadratic extensions Q( d), and √ for d < 0 he conjectured that in the ring of algebraic integers Z( d) the only cases with unique factorization are d = −1, −2, −3, −7, −11, −19, −43, −67, −163. This famous conjecture, the Gauss class 1 problem, was finally proved in 1952 by Kurt Heegner, an amateur mathematician. His proof was long thought faulty, however, and Heegner was already dead when in 1967 experts confirmed that it was correct. Although Hans Heilbronn proved in 1934 that a given class number has only finitely many cases, the general class number problem has only been solved for a few small values. 14 Gauss

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√ For the ring Z[ −5] this statement is not correct, which is why Dedekind had to introduce ideals.15 By doing so, he could introduce what today is called the ideal class group of an algebraic number field k.16 The order of this group is the class number of k. Moreover, this theory extends to Dedekind domains, so the class group is then trivial precisely when the ring is a unique factorization domain. In [Noether 1927a], Emmy Noether was able to give a general proof of Dedekind’s fundamental theorem and its converse on the basis of five axioms for a Dedekind ring. In her earlier paper [Noether 1921b], “Theory of Ideals in Ring Domains,” she introduced a general concept for rings that merely had to satisfy one axiom: the ascending chain condition. This acc now became Axiom 1 in [Noether 1927a] and its counterpart, the descending chain condition (dcc), was formulated as Axiom 2. She had not, however, explicitly stated that the ring R must possess an identity element for multiplication. Pavel Urysohn brought this oversight to her attention in 1923, and so she introduced this as Axiom 3, while pointing out that Urysohn had alerted her to it [Noether 1927a, 494]. Axiom 4 further stipulates that the ring must have no zero divisors. Finally, Axiom 5 introduces the decisive condition that the ring R must be algebraically closed in its associated quotient field (i.e. the smallest field that contains R). These are the five axioms for a Dedekind ring found in textbooks today. Emmy Noether announced her central results on Dedekind rings in a lecture at the annual meeting of the German Mathematical Society held in September 1924 in Innsbruck [Noether 1983, 482]. The final proof appeared three years later in [Noether 1927a]. She was clearly inspired by Dedekind’s ideas, referring several times to his Supplement XI to Dirichlet’s lectures on number theory [Dedekind 1894a]. On the other hand, it is also quite clear that Noether’s famous claim that “everything is already there in Dedekind” was a great exaggeration.

4.3 Noether’s Ideal Theory and the Theorem of LaskerNoether Already by 1920, Emmy Noether’s research interests had begun to take a decidedly new turn. She herself considered the paper [Noether/Schmeidler 1920], coauthored with Werner Schmeidler, to be the first sign of this new orientation.17 In 1917 Schmeidler took his doctorate in Göttingen under Edmund Landau, around which time he began working closely with Noether. Under her guidance, he completed his habilitation in 1919. In their joint paper, they developed a theory of modules for polynomials with a non-commutative multiplication operation, thereby introducing left and right modules, later a central concept in modern al15 This

ring has, in fact, class number 2. is defined as a quotient group formed from two groups of ideals in the ring of integers of k: Jk is the group of all fractional ideals and Pk the subgroup of its principal ideals. Then the quotient group Jk /Pk measures, in effect, the extent to which unique factorization fails. 17 This according to [Alexandroff 1935, 2]. 16 This

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gebra [Bourbaki 1960]. Soon thereafter, Noether published [Noether 1921b], her first fundamental contribution to modern ideal theory in which she distinguished between four different types of factorization theorems. Her starting point was a commutative ring in which the ideals are finitely generated, a property that follows from the ascending chain condition. She then applied this theory to rings of polynomials, focusing primarily on Lasker’s decomposition theorem [Lasker 1905]. Emanuel Lasker (1868–1941) is mainly remembered today not as a mathematician, but as the world’s chess champion for a period of 27 years (from 1894 to 1921), longer than any other player. He grew up in a Jewish family in Berlinchen (West Pomerania), then left for Berlin at the age of eleven to join his older brother Bertold, who was studying medicine at the time. Bertold Lasker was himself a talented player, and he introduced his younger brother to the Berlin chess scene. He later opened a practice in Elberfeld, where he met his future wife, the poet Else Lasker-Schüler; they were married from 1894 to 1903. Emanuel Lasker studied mathematics in Berlin and Göttingen, but he interrupted his studies to pursue his chess career in London and New York. After winning the world championship, he retired from chess in the late 1890s to continue studying mathematics in Heidelberg and Berlin. He finally completed his doctoral thesis in 1900 under Max Noether in Erlangen. There seems to be no record showing that Emmy Noether ever met him. After this, Lasker resumed his chess career, though he still somehow found time for several other intellectual pursuits. Five years after taking his doctorate, he published his groundbreaking paper [Lasker 1905] on polynomial ideals in Mathematische Annalen. Lasker’s theorem concerns polynomial rings with coefficients in C, a result Noether was able to extend to arbitrary Noetherian rings. More important still, she discovered the deeper reason why this theorem holds. Lasker’s proof was not only much more complicated, it also used methods from elimination theory that she showed were unnecessary. Noether built her proof on a new interpretation of Hilbert’s finite basis theorem, which he had published some thirty years earlier in the context of his pioneering work on invariant theory. In Noether’s language this theorem states that if R is a Noetherian ring, then every polynomial ring R[x1 , x2 , . . . , xn ] with coefficients in R is also Noetherian; in other words, the ideals in every polynomial ring over a Noetherian ring are finitely generated. Lasker had recognized that the ideals in a polynomial ring cannot always be decomposed into prime ideals. He therefore introduced the notion of primary ideals as the appropriate building blocks for these types of rings [Gray 2018, 259– 262]. An ideal P in a ring R is called primary if for all a, b ∈ R with ab ∈ P it follows that a ∈ P or bn ∈ P for a certain positive integer n (if n = 1, then P is a prime ideal). Lasker was able to show that every ideal I in a polynomial ring C[x1 , x2 , . . . , xn ], i.e. with coefficients in C, can be uniquely represented by a finite intersection of primary ideals: I = P1 ∩ P2 ∩ . . . ∩ Pk .

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In [Noether 1921b] Lasker’s fundamental result easily follows as a special case of Noether’s theorem, in which an arbitrary Noetherian ring replaces the field C. Thus, by introducing ideal theory axiomatically, instead of through concrete number systems, she went far beyond the range of ideas that had formed the framework for the works of Lasker and Hilbert. In her subsequent research, Noether connected this new ideal theory in polynomial rings with elimination theory, which concerns the common zeroes of polynomials. These ideas proved highly fruitful for the foundations of algebraic geometry, including the fundamental theorem of Max Noether. His daughter would promote this line of research in part by collaborating with a young visitor from the Netherlands named Bartel Leendert van der Waerden.

4.4 Van der Waerden in Göttingen Van der Waerden’s father, Theo, studied civil engineering at the Delft Technical University. He and his wife, Dorothea, moved to Amsterdam in 1902, where their eldest son, Bartel, was born one year later. Theo van der Waerden taught mathematics, though he apparently made no effort to push the young boy in that direction; nor did he need to, since B.L. van der Waerden was a born mathematician who learned all facets of the subject almost effortlessly. As a pupil at the Hogere Burger School, he taught himself trigonometry, and in 1919 he entered Amsterdam University at the age of sixteen. His scientific interests were extremely broad, so he took courses offered by the philosopher-mathematician Gerrit Mannoury (a close friend of his father), the invariant-theorist Roland Weitzenböck, the geometer Hendrik de Vries, and the famous topologist, L.E.J. Brouwer. By this time, however, Brouwer was no longer working on topology but rather pursuing his first love, a new approach to the foundations of mathematics known as intuitionism. Brouwer and Emmy Noether had been acquainted for some time; they first met one another during the DMV conference held in Karlsruhe during the late summer of 1911. She recalled those days with pleasure in a postcard she sent to Brouwer eight years later.18 During five years of study in Amsterdam, van der Waerden also learned a good deal of classical algebra, beginning with a course he took with de Vries. 19 As a voracious reader, he picked up Galois theory by studying Heinrich Weber’s Lehrbuch der Algebra, and he learned how finite linear groups can be treated geometrically to solve quintic equations from Klein’s Ikosaeder book. For an aspiring young mathematician in the early 1920s, van der Waerden surely had acquired 18 Noether

to Brouwer, 7 October 1919, Brouwer Papers, Noord-Hollands Archief, Haarlem. later recalled that this course dealt with topics like: determinants and linear equations, symmetric functions, resultants and discriminants, Sturm’s theorem on real roots, Sylvester’s index of inertia for real quadratic forms, and the solution of cubic and biquadratic equations by radicals [van der Waerden 1975, 31]. 19 He

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a solid background in Amsterdam, and yet he soon learned that marvelous new wholly unsuspected things awaited him in Göttingen. He later recalled his “apprentice years” there, including his first encounters with Noether. Brouwer recommended that I particularly stick with Emmy Noether. At that time he was quite hostile toward Hilbert due to the argument over intuitionism versus formalism.20 But Brouwer thought highly of the younger mathematicians in Göttingen, such as Emmy Noether, Ostrowski and Hellmuth Kneser. So I attended Emmy Noether’s lectures and soon met her personally. She was a very peculiar personality, roughly built with a large nose and inelegant movements, and she trudged while lecturing, sometimes she crushed a piece of chalk that had broken off . . . the opposite of an elegant lady. As Hermann Weyl expressed it in his obituary: “The graces were not at her cradle.” But these were outward appearances. More importantly, she was an altogether good person, free of all selfishness, free of all vanity, never posing and always willing to help everyone whenever she could. Her lectures were not well polished. She presented what she had just been thinking about, and she tried to improve the presentation during the lecture. It went like this: even before she had finished formulating a theorem, she quickly brought a better formulation. That, of course, did not make understanding any easier, on the contrary. But if you listened carefully and tried to think along, you could learn more than from a perfectly polished lecture. [van der Waerden 1997] It also helped to be young, bright, and ambitious. Most students who showed up unexpectedly in one of Noether’s courses never came back a second time, and nearly everyone agreed that her lectures were very difficult to follow, not only because of the abstract character of the subject matter but also due to her improvised explanations. Those who preferred polished performances usually stayed away, leaving behind a core of devoted listeners who learned how to grasp what she was saying; several of them took up questions she raised and began pursuing independent research. One of her auditors was Grete Hermann (Fig. 4.2), the first official graduate of the “Noether school.” Although she took up philosophy immediately afterward, her dissertation [Hermann 1926] constitutes an important contribution to ideal theory.21 In it she gave the first algorithm for computing primary decompositions of polynomial rings, a method still used today in computer algebra. Brouwer’s relations with Hilbert were particularly tense at this time, but he stood on very good terms with Emmy Noether, who was nearly his age. He had also gotten to know the far younger Hellmuth Kneser, then an assistant of Richard 20 On 21 For

intuitionism in the 1920s, see [Hesseling 2003]. Noether’s report on Hermann’s dissertation, see [Koreuber 2015, 320].

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Courant. This particular connection would also prove to be of great value for van der Waerden. On 21 October 1924, Brouwer wrote Kneser to to inform him of the latter’s imminent arrival: In a few days, a student of mine (or actually rather of Weitzenböck’s) will come to Göttingen for the winter term. His name is van der Waerden; he is very bright and has already published things (especially about invariant theory). I do not know whether the formalities a foreigner has to go through in order to register at the University are difficult at the moment; at any rate it would be very valuable for van der Waerden if he could find help and guidance. May he then contact you?22 Soon afterward, van der Waerden began meeting Kneser regularly for lunch, after which they would often take strolls through the woods just outside the town. Many years later, van der Waerden recalled how Kneser would . . . start on a certain subject and make a few remarks which I couldn’t understand at all. Then I would say to him that I would like to learn about that subject. Where could I find out about it? So he would give me the names of some books which I could find in the Lesezimmer. A day or so later I would be able to answer his questions and also make some significant remarks of my own, and then I learned much more. [Reid 1970, 162] During the 1920s, Hellmuth Kneser was a leading figure in topology, particularly known for his contributions to the theory of 3-manifolds. He introduced the concept of normal surfaces and used it to prove his theorem on the prime decomposition of 3-manifolds [Gordon 1999]. In 1924, at age 26, Kneser delivered a lecture on manifold theory at the annual meeting of the German Mathematical Society in Innsbruck [Kneser 1926]. While summarizing the current state of knowledge, he proceeded to sketch a program for future research that would only reach its zenith in the 1960s.23 One of its major objectives was to prove what Kneser dubbed the Hauptvermutung (principal conjecture).24 Five years later, van der Waerden presented an overview of research on combinatorial topology at the 1929 DMV conference in Prague [van der Waerden 1930a], at which Kneser was present. Van der Waerden began by referring back to Kneser’s Innsbruck lecture before proceeding to describe more recent work. Emmy Noether reported briefly about this lecture in a letter to Pavel Alexandrov: Prague showed that there is great interest in topology. In no lecture were there as many auditors as those who heard v. d. Waerden’s report. It was probably the content essentially of Pontryagin’s dissertation, . . . he 22 Brouwer

to Kneser, 21 October 1924, Nachlass Kneser, SUB Göttingen. the significance of Kneser’s work in this field, see [Hofmann/Betsch 1998, 9–11]. 24 In combinatorial topology, one studies triangulated spaces. It had been conjectured that if a space has two different triangulations, then these have refinements which are equivalent, but this turned out not to be true in general for dimensions four and higher; see [Scholz 2008, 865–866]. 23 On

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will send you the manuscript or page proofs, so that Moscow will be quoted correctly. As for topologists belonging to the guild, only H. Kneser was there, who you don’t even count as a guild member!25 Ironically, Kneser had shortly before discovered a gap in Max Dehn’s proof from 1910 of Dehn’s Lemma, a cornerstone in the theory of 3-dimensional topological manifolds. Probably this was first pointed out publicly by van der Waerden, who mentioned it at the end of his lecture [van der Waerden 1930a, 133]. After realizing that many of his results hinged on Dehn’s Lemma (which was first proved by Christos Papakyriakopoulos in 1957), Hellmuth Kneser gave up research in topology and turned to other fields. In any event, looking backward from 1929, one can see that Brouwer’s letter to Kneser from 1924 helped put van der Waerden on the fast track in a field that would soon occupy a central place in modern mathematics. Clearly, by this time Emmy Noether had her finger on the pulse of current developments in the fast-breaking field of topology. The relationship Bartel van der Waerden (Fig. 4.1) forged with Emmy Noether during the 1920s was of decisive importance for his career. Due to the success of his two-volume textbook, Moderne Algebra [van der Waerden 1930/31], he later came to be regarded as one of Noether’s leading disciples, a role he played only reluctantly. This book project actually dated from the year 1926/27 when van der Waerden was a Rockefeller fellow in Hamburg and attended Emil Artin’s lecture courses. Artin had informed Courant that he and van der Waerden would write a textbook on abstract algebra based on these lectures, a plan that Artin soon dropped after realizing that his coauthor was far more enthusiastic about this project than was he [Schneider 2011, 100–102]. B.L. van der Waerden was most definitely fascinated by Emmy Noether’s mathematical vision, even if he never entirely shared her enthusiasm for abstract algebra as an end unto itself. His own tastes and style were grounded, at least in part, in a more geometrical tradition. In fact, his original interests were closer to those of Max Noether, whose work he studied in Amsterdam, and one may fairly doubt that he ever felt completely at home with the purely algebraic direction that Emmy Noether promoted.26 Van der Waerden may have spent part of his career in Noether’s shadow, but he was never her epigone; in fact, his mathematical interests were far broader than hers. In Göttingen, he also deepened his knowledge of physics by taking Courant’s lecture course on methods of mathematical physics, taught from the newly published volume [Courant/Hilbert 1924]. This interest bore fruit later when he published his monograph [van der Waerden 1932] on group-theoretic methods in quantum mechanics, a study that appeared in Courant’s “yellow series” immediately after Moderne Algebra.27 25 Noether

to Alexandrov, 13 October 1929, translated from [Tobies 2003, 103]. a probing look at van der Waerden’s views on algebraic geometry over the course of his career, see [Schappacher 2007]. 27 For a detailed account of van der Waerden’s physical interests and his mathematical contributions to quantum physics, see [Schneider 2011]. 26 For

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Still, it is undeniable that Emmy Noether played a formative part in van der Waerden’s mathematical education, and he was the first to concede that he learned modern ideal theory, among several other things, from her. During his first year in Göttingen, he took her course on group theory and hypercomplex numbers (the older term for what later were called algebras). Already by this time, she was lecturing on Joseph Wedderburn’s structure theory of algebras over arbitrary fields, thereby laying the groundwork for what would later become her principal field of research.28 In a conversation with Auguste Dick, van der Waerden recalled a memorable scene from one of Emmy Noether’s lectures when she wanted to present a new proof of Maschke’s theorem in group representation theory. This was a classic result from the late 1890s, thus from well before the time when abstract algebra came into vogue. Not surprisingly, Noether was keen to present it from a modern point of view. In all likelihood, she had come up with the main ideas for her new proof not long before arriving in the lecture hall. As she began setting out her argument on the blackboard, she was feeling, as usual, buoyant and upbeat in the excitement of the moment. Somewhere along the way, though, her mood changed, as it dawned on her that there was a serious hole in her proof that she was not going to be able to repair after a short moment’s thought. To the surprise of her audience, she suddenly broke into a rage, threw down her chalk, and stomping on it yelled out: “Now I’m forced to do it the way I don’t want to!” She then picked up a new piece of chalk and presented the traditional proof flawlessly [Dick 1970/1981, 1981: 39–40]. As this anecdote shows, Emmy Noether was not only passionate when it came to her abstract approach, she also had complete mastery of the methods and results of classical algebra. During his initial stay in Göttingen, van der Waerden had already begun to grapple with problems in the foundations of algebraic geometry that would dominate his attention for many years to come [Schappacher 2007]. Already in Amsterdam, he had studied invariant theory and its importance for geometry, a viewpoint made famous by Felix Klein in his “Erlangen Program” [Klein 1872]. But as he later recalled, “when I studied the fundamental papers of Max Noether . . . and the work of the great Italian geometers, . . . I soon discovered that the real difficulties of algebraic geometry cannot be overcome by calculating invariants and covariants” [van der Waerden 1975, 32]. A swarm of related questions soon filled van der Waerden’s mind: how to define the dimension of an algebraic variety? what do the Italian geometers mean by the “generic points” of a variety? how can one rigorously define intersection multiplicities? how can one prove the ndimensional version of Bézout’s Theorem? is there a way to rigorously justify the Schubert calculus in enumerative geometry? And then there was this question, 28 Wedderburn’s 1907 doctoral thesis, On hypercomplex numbers, gave a classification theory for semi-simple-algebras, which are essentially Cartesian products of simple algebras (those with no non-trivial two-sided ideals). He showed, in turn, that all finite-dimensional simple algebras are isomorphic to a matrix algebra over some division algebra; see [Parshall 1985].

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Figure 4.1: B.L. van der Waerden and Emmy Noether, Göttingen, Summer 1929 (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna) already addressed by Emmy Noether in her report for the German Mathematical Society [Noether 1919a, 197–201]: Another problem that worried me very much was the generalization to n dimensions of Max Noether’s “fundamental theorem on algebraic functions.” Noether’s Theorem specified the conditions under which a given polynomial F (x, y) can be written as a linear combination of two given polynomials f and φ with polynomial coefficients A and B : F = Af + Bφ . More generally, one can ask under what conditions a polynomial F (x1 , . . . , xn ) can be written as a linear combination of given polynomials f1 , . . . , fr with polynomial coefficients: F = A1 f1 +· · · +Ar fr or in modern terminology, under what conditions is F contained in the ideal generated by f1 , . . . , fr . From the papers of Max Noether I knew that this question is of considerable importance in algebraic geometry, and I succeeded in solving it in a few special cases. I did not know then that Lasker and Macaulay had obtained much more general results. [van der Waerden 1975, 32] Van der Waerden went on to relate how Noether taught him that the tools needed to handle such questions “had already been developed by Dedekind and

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Weber, by Hilbert, Lasker and Macaulay, by Steinitz, and by Emmy Noether herself” [van der Waerden 1975, 32–33]. The bookish Dutchman had no difficulty absorbing the literature she advised him to read: Steinitz’s classic paper on abstract fields [Steinitz 1910], Macaulay’s Cambridge Tract on modular systems [Macaulay 1916], the famous paper of Dedekind and Weber on algebraic functions [Dedekind/Weber 1882], and Noether’s own papers on ideal theory [Noether 1921b] and elimination theory [Noether 1923c]. These works opened up a whole new world for him: The mathematical library of Göttingen was unique. Everything one needed was there, and one could take the books from the shelves oneself! In Amsterdam and in most continental universities this was impossible. So I started learning abstract algebra and working at my main problem: the foundation of algebraic geometry. [van der Waerden 1975, 33]

Figure 4.2: Emmy Noether and Grete Hermann, June 1926 (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna) Noether’s paper [Noether 1923c] elaborated on a new approach to elimination theory developed by her student Kurt Hentzelt, who defended his dissertation summa cum laude in Erlangen shortly before the outbreak of the war. 29 His 29 In [Noether 1924a] she showed how elimination could be presented in the context of her ideal theory for the zeroes of polynomials by combining Hentzelt’s methods with Steinitz’s ideas.

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official adviser, Ernst Fischer, stipulated that the text had to be rewritten prior to publication [Koreuber 2015, 15], but Hentzelt had no opportunity to do so: he was killed during the early months of the fighting. Years later, realizing the importance of Hentzelt’s methods [Koreuber 2015, 98–99], Noether took it upon herself to rewrite his dissertation in [Noether 1923a], after publishing a note about this work for the German Mathematical Society [Noether 1921a]. The paper van der Waerden read, [Noether 1923c], was a later refinement of Hentzelt’s theory. He later recalled how these ideas enabled him to give a precise notion of the generic points of an algebraic variety: After her lecture we, Noether’s students, often discussed mathematical problems with her. My problems mainly were concerned with algebraic geometry, since I found what I had learned in Amsterdam all very nice, but knew that it lacked rigorous grounding and I was searching for such a foundation. I needed algebra for that. So I presented her with my basic problems that I had already struggled with in Amsterdam. For example: How do you define the dimension of . . . an algebraic variety? One has a system of equations that define an algebraic variety in n-dimensional space. What does it mean when one calls this variety a curve or a surface? What do the Italians mean when they speak of a punto generico, a general point, of a variety? Well, . . . a general point on a curve cannot be a double point or an inflection point, nor can it be the point of contact of a double tangent. In short, a general point is required to have no special properties that do not belong to all points. Is there such a thing? I found the answer to this question in a work by Emmy Noether on elimination.30 [van der Waerden 1997] Soon afterward, however, van der Waerden saw that one did not need elimination theory for the concept of general points.31 He wrote up a paper on zero-sets of polynomials and then showed it to Emmy Noether, who told him it was very good and that she would submit it right away to Mathematische Annalen. She also gave him some tips on how to conceptualize the work and improve the presentation by changing the order of some definitions and theorems, and it soon appeared as [van der Waerden 1927]. What she never told him was that she had already come up with the same idea and had even presented it in her lecture course about a half year before he returned to Göttingen. Van der Waerden learned about this 30 Noether’s idea was to replace the coordinates x , . . . , x by indeterminates ξ , . . . , ξ and to 1 1 d d determine the other ξi as algebraic functions of these indeterminates in an algebraic extension field k(ξ). This field she called the Nullstellenkörper of the prime ideal p belonging to the variety. Van der Waerden saw that the point ξ with co-ordinates ξ1 , . . . , ξm was the concept of generic point he was looking for. 31 As he later explained: “I also saw Emmy Noether’s Nullstellenkörper was isomorphic to the quotient field of the residue class ring o/p, where o is the polynomial ring k[X]. Hence it was not necessary to go through Hentzelt’s elimination procedure; one could start with any prime ideal p 6= 0, construct the residue class ring o/p and its quotient field k(ξ), and thus find a generic point ξ of the variety of p” [van der Waerden 1971, 173].

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later from Heinrich Grell, who had attended that course [van der Waerden 1971, 173]. In recalling this story, a favorite that he told many times, he noted his amazement that Emmy Noether had not said a word to him about this. “She did not want to spoil the young man’s joy over his discovery! Isn’t that fabulous? Gauss was completely different; he spoiled the young Bolyai’s joy in discovering non-Euclidean geometry by writing to him: ‘I’ve known all that for a long time” ’ [van der Waerden 1997]. After van der Waerden’s first semester in Göttingen, Noether persuaded Courant to support his application for a fellowship from the Rockefeller Foundation, which promoted a program initiated by the International Educational Board. The IEB supported young mathematicians and scientists who wished to travel abroad to pursue post-doctoral research [Siegmund-Schultze 2001]. In van der Waerden’s case, the fellowship only covered nine months, as he returned to Amsterdam to finish his doctoral degree under Hendrik de Vries. His topic was the foundations of enumerative geometry, thus the Schubert calculus, which Hilbert had singled out as the fifteenth of his 23 Paris Problems. Van der Waerden would return to this theme in the 1930s, beginning with [van der Waerden 1930b]. 32 Noether remained in close contact with van der Waerden after he returned to Holland, as we learn from a letter she wrote to Brouwer on 14 November 1925: You have correctly foreseen that van der Waerden would give us special pleasure! His work on algebraic manifolds (zero sets for polynomial ideals), submitted in August to the Annalen, is truly excellent, and he is now in the middle of productive work; we are corresponding the whole time with enthusiasm. Over Christmas we can continue orally; I long ago promised Alexandrov that I would visit him then and I’ll be very happy to see you again soon and speak with you. (Brouwer Papers, Noord-Hollands Archief, Haarlem) Noether’s monthlong stay in the Netherlands would turn out to be both memorable and fruitful, but before taking up this thread of events, we first recount how Emmy became friends with the young Russian she named in this letter.

4.5 Pavel Alexandrov and Pavel Urysohn Emmy Noether’s first encounter with the Russian topologist Pavel Alexandrov took place during the summer of 1923. Although their mathematical interests ran along quite different tracks, which only occasionally intersected, they nevertheless shared an openness for everything new in mathematics that led to a strong personal bond. In this broader sense, Alexandrov must be counted as one of the leading representatives of the Noether school. Both loved to “talk mathematics” in the open air, but especially at the outdoor pool in Göttingen run by the lifeguard 32 In Leipzig, he wrote several related papers with his student, Chow Wei-Liang, who introduced Chow coordinates in intersection theory.

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Fritz Klie (Fig. 5.1). An indefatigable swimmer, Alexandrov later sang the praises of this particular locale in his recollections of Heinz Hopf [Alexandroff 1976]. But he also recalled the times he and Pavel Urysohn spent there when they first visited Göttingen, where they were nicknamed “the inseparables” (a play on the property of separability in topological spaces).33 Nearly every day we could “meet Emmy Noether and Courant . . . at the university swimming pool (mainly for students) on the river Leine . . . . There we could also often meet Hilbert but not Landau (who, when I asked him whether he bathed replied, ‘Yes, every day, in my bath at home’)” [Alexandrov 1979/1980, 316]. Pavel Alexandrov grew up in a refined home in Smolensk, where his father, a gynecological surgeon, was director of the local hospital. Owing to his many and varied responsibilities, the family lived in a two-story house on the grounds of the hospital. Pavel’s mother was a highly educated woman, who oversaw the upbringing of her children, in particular their early training in foreign languages. Thus, Pavel and his siblings learned French and German at home. His mother spoke both fluently, but she engaged a governess from Riga, who helped instill her son’s lifelong love of the German language. This woman spoke with the distinctive Baltic accent that Alexandrov later recalled when he heard Hilbert lecture. Alexandrov studied at the renowned Moscow State University, where he came under the influence of the Soviet analysts Dmitri Egorov and Nikolai Luzin, the latter a pioneer in the field of descriptive set theory. Luzin’s interests in this modern theory were first awakened in Paris, where during the Revolution of 1905 he attended lectures given by Émile Borel. He later spent three years in Göttingen as a research fellow, during which time he worked closely with Edmund Landau. After the October Revolution of 1917, Moscow State University began to admit students from proletarian and peasant families by enabling them to do preparatory work before taking their entrance examinations. At this newly reformed institution, Luzin established a famous research seminar during the 1920s whose members came to be called “Luzitania.” Among them were Alexandrov and his intimate friend Pavel Urysohn, along with Aleksandr Khinchin, Andrey Kolmogorov, Mikhail Lavrentyev, Alexey Lyapunov, Lazar Lyusternik, Pyotr Novikov, Lev Schnirelmann, and many others. Alexandrov and Urysohn soon began an intense collaboration that led to several fundamental results in general topology. Urysohn, who grew up in a Jewish family from Odessa, was an adventurous spirit. Both he and Alexandrov loved to travel, and since they both spoke excellent German, Urysohn easily convinced his friend that they should try to visit Germany in the summer of 1923. The only question, then, was how to finance their trip. Realizing that Russian intellectuals were keenly aware of and curious about Einstein’s new theory of relativity, they came up with the brilliant idea of offering public lectures on this subject at various locations around Moscow. And this venture actually paid off, or at least provided 33 Felix Hausdorff invented this nickname; he addressed the two Russians as the “Herren Inséparables” in a letter from 11 August 1924 [Hausdorff 2012, 22]. They, in turn, expressed their delight with this title in a footnote to an earlier letter [Hausdorff 2012, 16].

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them with enough money to travel and stay abroad [Alexandrov 1979/1980, 297– 298]. As budding topologists, they were particularly eager to meet Felix Hausdorff, a pioneering figure in the field. In a letter dated 18 April 1923, they announced a whole series of their new results,34 and Hausdorff reciprocated by warmly welcoming the two young Russians to visit him in Bonn. His return letter, however, dated 22 May, only arrived after they had left Moscow. Thus, they only received his message in Göttingen, a mishap due to the fact that Alexandrov and Urysohn had sent their letter to his old address in Greifswald, not realizing that he had been living in Bonn since 1921. In any event, their planned visit could not have taken place due to the Ruhr crisis, during which Bonn fell under French occupation. Despite this disappointment, they were delighted by the friendly reception that awaited them in Göttingen, a visit arranged by Landau. There they Alexandrov and Urysohn attended Hilbert’s course on “Anschauliche Geometrie,” later published as [Hilbert/Cohn-Vossen 1932]. They also heard Landau’s lectures on analytic number theory and Courant’s on differential equations in mathematical physics. Yet, most of all, Alexandrov remembered Emmy Noether’s lectures as particularly inspiring; he also recalled her famous saying – “es steht schon alles bei Dedekind” – a view he firmly dismissed. Many decades later, he related how “[h]er lectures enthralled both Urysohn and me. In form they were not magnificent, but they conquered us by the wealth of their content. We constantly met Emmy Noether on a relaxed basis and very often talked to her about topics both in ideal theory, and in our work, which had caught her interest at once” [Alexandrov 1979/1980, 299]. Noether’s lively enthusiasm for mathematical discussions left a deep impression on Alexandrov and Urysohn, both of whom were highly sociable: We were constantly meeting Emmy Noether on her famous walks which were first called algebraic and after our arrival came to be called topological algebraic. There were always many young mathematicians taking part in these walks, which were a model for the topology walks of our Moscow topology seminar, but of quite a different character. [Alexandrov 1979/1980, 316] Given that Alexandrov wrote these autobiographical notes late in his life, it should come as no surprise that he no longer accurately remembered various details from his first visits to Göttingen. During their first stay, he and Urysohn were invited to give talks before the Mathematical Society. Afterward, Hilbert requested that they write up their main results for publication in Mathematishe Annalen, and so they submitted six short notes, which Hilbert then passed over to Courant, although Brouwer normally handled all submissions in topology. Alexandrov evidently only remembered that the publication of these notes had been delayed, and 34 [Hausdorff 2012, 5–7]; Alexandrov and Urysohn were the first to recognize the central importance of what they called bicompactness in topological spaces (today one speaks of compact spaces).

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he mistakenly thought that Ostrowski was responsible for this. Here is what he wrote about what transpired when he and Urysohn returned to Göttingen for the second time in May 1924: [These circumstances] displeased Emmy Noether, who insisted that our papers should be sent immediately (it was already June 1924) to Brouwer, as the member of the editorial board in charge of topology. Brouwer gave his favorable opinion of these papers both to Emmy Noether and to Hilbert. She brought to Hilbert’s attention the fact that the papers, already approved by him, had lain for a year without action. Hilbert summoned Ostrowski, and a conversation between them took place in the presence of Emmy Noether, from whom I learned this whole story. [Alexandrov 1979/1980, 300]

4.6 Brouwer and the Two Russians This account might seem plausible in view of the many details Alexandrov provided, but the documentary evidence flies in the face of what he wrote (see [Rowe/Felsch 2019, 178–184]). The responsible referee was Richard Courant, not Ostrowski, and if there was any delay at all, then this was probably due to Courant or possibly even Brouwer’s own intervention in this matter. Otto Blumenthal had received the six papers from Courant in mid-May 1924, at which time he expressed his annoyance to Hilbert that these were to be published all at once. Then, in mid-June, Blumenthal received a letter from Brouwer requesting permission to correspond with the authors in the name of the editorial board, which was duly granted. Emmy Noether could have heard something about this and then spoken with Hilbert about the matter. But if this happened, then in all likelihood she was urged to do so by Brouwer, who was determined to intervene. Moreover, during this time Alexandrov was certainly well aware that Brouwer was pressuring Urysohn to withdraw one of these six papers.35 Brouwer’s concerns dated back to his first encounter with Urysohn, which took place in Marburg in September 1923 at the annual DMV conference. 36 Over these four days, the two Russian visitors listened to and talked about mathematics 35 In the end, only five of the six papers appeared because Brouwer was able to persuade both Blumenthal and Urysohn not to publish the note in which Urysohn corrected a mistake in an earlier paper by Brouwer. A summary of these events, which belong to the early history of dimension theory, can be found in [Rowe/Felsch 2019, 35–38]. General topology had been in full swing since 1914, spurred on by the publication of Felix Hausdorff’s Grundzüge der Mengenlehre [Hausdorff 2002]. On this basis, Alexandrov and Urysohn further developed the theory of general topological spaces, for which Urysohn was able to develop a new concept of dimension. However, it later turned out that the Austrian Karl Menger had also found a similar recursive concept of dimension at about the same time. Today this is usually referred to as the small inductive dimension or Urysohn-Menger dimension; see [Hurewicz/Wallman 1948]. 36 For details and documentation relating to the Marburg meeting and its aftermath, see “Brouwer und die Dimensionstheorie (1923–1924),” Chapter 5 in [Rowe/Felsch 2019, 165–194].

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non-stop, as they reported afterward to Hausdorff in a letter from 21 May 1924 [Hausdorff 2012, 17]. Lectures from 9 until 1 o’clock and from 3 to 7, with two hours to eat and discuss in between, and then afterward discussions that lasted until midnight. Toward the end, Ludwig Bieberbach scurried around to collect short abstracts, which he then published soon thereafter in the DMV’s official journal.37 At this meeting, Urysohn lectured on “general Cantorian curves” based on his new theory of dimension. During the course of his presentation, he mentioned that Brouwer’s theory was based on an untenable definition for separation of sets in metric spaces. Since Brouwer did not attend this lecture, he only learned later about Urysohn’s comment from Bieberbach. He then spoke with Urysohn, who told him that his proof of the dimension theorem in [Brouwer 1913] was unsound. Brouwer then asked Urysohn to send him a written explanation. About a month later, he received a very polite letter from Moscow, in which Urysohn produced a counterexample showing that Brouwer’s proof was incorrect.38 However, he was able to salvage Brouwer’s theorem by introducing a new definition of separation. After reading this letter, Brouwer realized that the young Russian had indeed found an error in his proof, a mistake he himself vaguely remembered and would later characterize as a “slip of the pen.” 39 He published a “correction” in [Brouwer 1924], but this was so tersely worded that probably no one would have understood what was behind this mistake, which was surely just what Brouwer intended. Although he was fully preoccupied with intuitionism and had not worked on topology for nearly a decade, Brouwer was deeply interested in securing his reputation as one of the foremost figures in the field. In the aftermath of the Marburg meeting, he began to realize that recent developments in point set topology threatened to eclipse some of his fundamental contributions from a decade earlier. 40 A few months before the Marburg meeting, Urysohn had given Hilbert a short note, one of the six that he and Alexandrov wished to publish in the Annalen; in this note he explained the nature of Brouwer’s mistake. The latter only learned about this when Urysohn sent him the printer’s proof along with a letter asking Brouwer to approve an addendum to his short article. This pointed to the fact that the original mistake had in the meantime been corrected in [Brouwer 1924], but that the matter stood in need of clarification since Brouwer had not indicated why his earlier definition was incorrect.41 This letter was written only shortly after Urysohn and Alexandrov spent several days visiting Brouwer at his home in Blaricum, during which time all was 37 These circumstances were described to Hausdorff by way of explaining how an error had crept into the abstract of Alexandrov’s talk on the theory of point sets. 38 [van Dalen 2011, 256–260]; this counterexample was similar to the topologist’s sine curve, which Brouwer had already encountered in [Hausdorff 2002, 558], showing a continuum that is not arcwise connected. See Brouwer to Urysohn, 9 April 1924 [Rowe/Felsch 2019, 174–175]. 39 For details, see [van Dalen 2013, 406–416]. 40 These developments were launched with Hausdorff’s Grundzüge der Mengenlehre, which was published one year after [Brouwer 1913]. 41 Urysohn to Brouwer, 21 June 1924 [Rowe/Felsch 2019, 182].

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harmony and light. Urysohn was clearly keen to maintain these good relations, so he sought Brouwer’s approval before proceeding further. His apologetic tone was surely music to Brouwer’s ears, as the Dutchman was prepared to go to extreme lengths to prevent this paper from appearing in print. Now he felt reassured that the young Russian could easily be persuaded to withdraw his note voluntarily; still, just to be sure, he contacted Otto Blumenthal in the event Urysohn might decide to act against Brouwer’s fatherly advice and return the corrected proofs. If this were to happen, then he asked Blumenthal to inform Urysohn that the editorial board of Mathematische Annalen had been informed by Brouwer that his note was unsuitable for the journal. Urysohn, however, simply acted according to Brouwer’s wishes, which spelled the end of this strange episode. One can only imagine that Urysohn lost no sleep over this matter, which Alexandrov had apparently completely forgotten when he tried to recall the circumstances that led to the delayed publication of their papers for the Annalen. For Brouwer, on the other hand, this was a traumatic episode that plagued him for many years afterward [van Dalen 2013, 404–416, 595–597].

Figure 4.3: Pavel Alexandrov and Pavel Urysohn, Blaricum 1924 (Hausdorff Papers, Bonn University)

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When he learned that Alexandrov and Urysohn were planning to return to Göttingen during the summer of 1924 as well as meet with Hausdorff in Bonn, Brouwer was more than willing to help make arrangements for them to visit him in the Netherlands. Brouwer’s earlier work stood closer to traditional combinatorial topology, which derived from geometrical investigations of polyhedra, whereas Hausdorff’s investigations grew out of Cantorian set theory and were far more general and systematic. The latter’s Grundzüge sparked a tremendous outburst of new researches in Poland and Russia, including those of Alexandrov and Urysohn, who helped launch the field of continuum theory.42 Their meeting with Hausdorff was, indeed, nothing less than an historic event in the history of modern topology, marking the beginning of Alexandrov’s friendship with Felix Hausdorff, as documented in their correspondence from 1924 up until 1935 [Hausdorff 2012, 20–133]. After spending several weeks in Göttingen, the two Russians arrived in Bonn on July 9. Despite warnings from Hausdorff’s wife, they followed their natural instincts and took to swimming in the Rhine River, evading the barges along the way. Their stay was prolonged by difficulties they encountered when applying for visas to enter the Netherlands, though this meant they had more time to discuss their work with Hausdorff, a sharp-minded listener who took an avid interest in their future plans. The two young Russians were also warmly welcomed in Blaricum by Brouwer, though their host had to leave only a few days later to give a lecture in Göttingen. During this visit, they discussed a future plan for both to return with fellowships from the International Education Board. Around this same time, Brouwer wrote to IEB President Wickliffe Rose in New York City: Two young Russian mathematicians full of promise, Dr. Alexandroff and Dr. Urysohn, of Moskau, desire to study topology under me. I already spoke of them to you this winter on our meeting in the Amstel Hotel at Amsterdam, and I can readily declare that the International Education Board would perform an action of decided scientific interest by enabling these very clever young scholars to come to Amsterdam and live there this winter. You find their request with biographic notice enclosed in this letter.43

4.7 Urysohn’s Tragic Death After their stay in the Netherlands, Alexandrov and Urysohn spent four days in Paris, but without seeing any mathematicians there. They then left to vacation in Batz-sur-Mer in Brittany, where they worked together during most of the day 42 On Hausdorff’s Grundzüge, see [Purkert 2002]; topological continua are compact connected metric spaces. 43 Brouwer to Rose, August 1924, Rockefeller Archive Center, IEB Collection, 1-1, Box 44, Folder f616, North Tarrytown, New York.

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and then went swimming in the Atlantic. They arrived around the first of August and sent a letter to Hausdorff, picking up on their last conversations in Bonn, but also reporting briefly on their days together with Brouwer. They were particularly happy to have seen that, despite his present focus on foundational studies, Brouwer remained keenly interested in other parts of mathematics, including topology in general spaces. Looking back on all they had experienced during the past months, the two highlights had been their visits in Bonn and Blaricum and their conversations there with Hausdorff and Brouwer, respectively.44 Indeed, in the course of two weeks they had won over two of the leading coryphaei in the field of topology. And then, just a fortnight later, tragedy struck. On a Sunday, August 17, 1924, they were caught in a storm that took Urysohn’s life, a horrible event for all who had known him. For a long time afterward, Alexandrov could not believe what he suddenly lost on that day. In the years that followed, his friendship with Brouwer was strengthened by their common interest in preserving Pavel Urysohn’s legacy [van Dalen 2013, 424–434]. Some forty years later, Alexandrov described that fateful day in detail: The main part of the day was spent on work, and in spite of our custom it was already five o’clock in the afternoon when we got ready to go swimming. When we got into the water, a kind of uneasiness rose up within us; I not only felt it myself, but I also saw it clearly in Pavel. If only I had said, “Maybe we shouldn’t swim today?” But I said nothing .... After a moment’s hesitation, we plunged into a not very large shore wave and swam some distance into the open sea. However, the very next sensation that reached my consciousness was one of something indescribably huge, which suddenly grabbed me . . . . A moment later I came to myself on the shore, which was covered with small stones it was the shore of a bay, separated from the open sea by two rocks between which we had had to swim as we made our way to open sea. I had been thrown over by a wave, right across these rocks and the bay. When I was on my feet, I looked out to sea and saw Pavel at those same rocks, already in the bay, passively rolling on the waves (which were comparatively small in the bay) in a half-sitting position. I immediately swam up to him. At that time I saw a large group of people on the shore. . . . After swimming to Pavel, I put my right arm around him above his waist, and with my left arm and my legs I began to paddle to shore with all my might. This was difficult, but no one came to my assistance. Finally, when I was already quite near the shore, someone threw me a rope, and within a few moments I reached land. Then eye-witnesses told me that the same great wave that had thrown me across the bay 44 Alexandrov

and Urysohn to Hausdorff, 3 August 1924, [Hausdorff 2012, 20–21].

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4 Noether’s Early Contributions to Modern Algebra had struck Urysohn’s head against one of the two rocks and after that he had begun to roll helplessly on the waves in the bay. When I pulled Pavel to the shore and felt the warmth of his body in my hand, I was in no doubt that he was alive. Some people then ran up to him, and began to do something to him, obviously artificial respiration. Among these people, there happened to be, as I was later told, a doctor, who apparently directed the attempts at life-saving. I do not know and did not know then how long they continued, it seemed like quite a long time. In any case, after some time I asked the doctor what the condition of the victim was and what further measures he proposed undertaking. To this the doctor replied “Que voulez vous que je fasse avec un cadavre?” . . . Some more time passed, and I went into my room and finally dressed. (Until then I had remained in my swimming clothes.) Pavel Urysohn lay on his bed, covered by a sheet; there were flowers at the head of the bed. It was here that I thought for the first time about what had happened. All my experiences, all my impressions of that summer, and indeed of the last two years, rose up in my consciousness, with such distinctness and clarity. All this merged into a single awareness of how good, how exceptionally good, things had been for each of us, only about an hour ago. And the sea raged. Its roaring, its crashing, its bubbling, seemed to fill everything. [Alexandrov 1979/1980, 318–319]

The next day, Alexandrov sent telegrams to Brouwer and to his brother, who informed the Urysohn family about what had happened. The funeral took place the day afterward, on 19 August; Alexandrov contacted a local rabbi, who performed the funeral rites. He then left Batz the next day, spent the following day in Paris, and arrived in Göttingen on the 22nd, where he was met by Brouwer, Courant and Emmy Noether. Hilbert and Klein also requested that Alexandrov visit them. On the evening before he departed for Moscow on the 3rd of September, Alexandrov wrote an emotional farewell letter to Hausdorff from Berlin. 45 He apologized for writing so directly and personally about how he felt, but he needed to describe his state of mind and how his euphoric life had suddenly turned to misery with Urysohn’s death. Beyond that, he wanted Hausdorff to know that his friend had left behind a great deal of important work; he assured him that one paper, in particular, once it appears, would truly astonish Felix Hausdorff by showing him “how deeply [Urysohn] could penetrate into the very most hidden secrets of topological space structure.” This would now become Alexandrov’s next great task, to prepare Urysohn’s posthumous papers for publication. 45 Alexandrov

to Hausdorff, 2 September 1924, [Hausdorff 2012, 27–28].

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4.8 Helping a Needy Friend Upon returning to Moscow, the distraught topologist received a consoling message from Emmy Noether, in which she addressed him as “my dear, poor Paul Alexandrov” and with the familiar “Du” form. She had wanted to suggest that they use “Du” already in Göttingen, but as she explained, she was not able to summon up the courage during his short last visit. Recalling her two Russian friends, “the inseparables,” she wrote: I have the image of both of you always in mind, with all that outpouring of life that you brought forth; and now you are all alone. But he lives on with you; and if his manuscripts now bring his thoughts back to life – and only you can bring them to full life – so he will return to you more and more. And the sharp pain will perhaps not hurt so much after all; and you can always then think back with more gratitude for what you had during these four years.46 During the next three years, Alexandrov attached himself closely to Brouwer, who also took a deep interest in preserving Pavel Urysohn’s legacy. Brouwer arranged an invitation for Alexandrov to Amsterdam with financing through the Rockefeller Foundation and the IEB. One month after Urysohn’s death, he wrote to Rose: I have to inform you that Doctor Urysohn, one of the two Russian mathematicians I proposed to you for a fellowship of the International Education Board, found a sudden death by a most tragical accident some weeks ago. He left a mass of posthumous scientific papers and notes which when elaborated and edited will prove, I am certain, to contain results of the highest scientific importance. I believe that this elaboration will come to the best end, if it is undertaken by Urysohn’s friend Doctor Alexandroff under my guidance. This is also the opinion of Doctor Alexandroff, whose desire to come to me to Amsterdam is in this way strengthened by a motive of scientific piety.47 Arriving in May 1925, the young Russian took up residence in the village of Blaricum, where Brouwer lived and spent most of his time. Blaricum long attracted an assortment of artists and elites, many of whom worked in nearby Amsterdam. Brouwer owned a good-sized house there, but he spent most of his time in a small cottage with a desk and piano, situated in a fairly large, but completely overgrown garden. This is where photos of Brouwer and his lively Russian friends were taken the year before (see Fig. 4.3), and where Alexandrov and Brouwer began their work editing Urysohn’s posthumous papers. 46 Noether

to Alexandrov, 1 September 1924, transcribed in [Tobies 2003, 102]. to Rose, 17 September 1924, Rockefeller Archive Center, IEB Collection, 1-1, Box 44, Folder f616, North Tarrytown, New York. 47 Brouwer

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Later that year, Alexandrov received an inquiry from Noether regarding plans for his forthcoming stay in Göttingen. She was eager to have him offer a course on topology during the summer term, and he was keen to reciprocate, as he was about to begin lecturing on the subject in Amsterdam. He replied to her from Blaricum as follows: . . . I hope not only to give a course on topology [in Göttingen] but also to regularly follow your lectures on the foundations of group theory. You know that I am very interested in your works in this area, especially because in terms of their content and methodology they are very close to the circle of ideas in general topology. I am also counting on stimulation from your side because, as you also know, the many mathematical discussions that Urysohn and I had with you were among the most lively and stimulating we have ever had, as I’ve told Brouwer on several occasions. Brouwer also knows how excited I was about your correspondence from last summer about group theory. So I ask you to count on me to be an attentive and eager auditor.48 Noether had discussed with Courant whether it might be possible to extend Alexandrov’s IEB Fellowship, a plan that turned out not to be feasible. In any event, Alexandrov cautioned her that she should take this up with Brouwer, whose prickly and oftentimes domineering personality made the Russian very wary: I consider it necessary to write you that I feel obliged to let this whole question rest with Brouwer’s decision: for moral reasons it would be quite impossible for me to accept something in this direction if Brouwer does not fully agree. . . . I don’t, in fact, want my purely moral dependence on Brouwer regarding these and some other questions, which I take on freely, and which consequently in no way violates my human and scientific freedom, to lead to any misunderstanding. . . . I don’t want to influence Brouwer’s decisions in any way, because I want to avoid at any price the possibility of internal friction with Brouwer. Three days later, Noether posted the letter already cited above, in which she sang the praises of van der Waerden. Along with it, she enclosed Alexandrov’s letter, and her own wishes for his visit: I would like to write you today about Alexandrov’s planned stay in Göttingen during the coming summer and the possibility of extending his Rockefeller scholarship. He wrote me about his plans and views in the attached letter, which he authorized me to show you. I would be so pleased if Alexandrov received an extension of his scholarship as much as I wish him anything at all that would make his life a little easier! I don’t think any formal reason stands in the 48 P.S.

160.

Alexandrov to E. Noether, 11 November 1925, Hochschularchiv der ETH Zürich, Hs

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way either, since he wishes both to learn as well as teach here just as in Amsterdam. Would it perhaps be possible to proceed as with van der Waerden: that you present the proposal and I countersign it, since Alexandrov wants to work with me? Of course, any other arrangement that you might propose would be fine with me. It would also be possible to submit an application from here; Trowbridge49 is said to have been generally very accommodating during a visit a few weeks ago, though I didn’t speak with him myself. In any case, I hope that either way the matter comes to a good end, and may I ask you to inform me of your intentions soon?50 As Noether soon learned, Brouwer was more than happy to cooperate with her in support of this plan. She was undoubtedly delighted to read his response: “You will surely experience a great joy in Göttingen from Alexandrov’s lectures; he recently began a course of lectures here that has completely captivated his auditors, not only due to the clarity and precision of the subject matter but also because of his enthusiasm as a lecturer.” 51 Brouwer went on to say that he would have been happy to promote Alexandrov’s career chances in the Netherlands were the latter not already so advanced that he could expect to gain a professorship in Russia. So these plans for Alexandrov’s stay in Göttingen had already been settled by mid-December when Emmy Noether came to spend nearly a month in Blaricum. She stayed there until January 10, residing in the village’s Villa Cornelia. 52 During her stay, Noether connected again with Alexandrov and van der Waerden, while meeting other mathematicians in Brouwer’s circle. Alexandrov offered a vivid recollection of a dinner party in her honor at Brouwer’s home, “during which she explained the definition of the Betti groups of complexes, which spread around quickly and completely transformed the whole of topology. When Emmy Noether arrived at Blaricum, her student van der Waerden, who was then 22 years old, also came. I remember extraordinarily lively mathematical conversations in which he took part” [Alexandrov 1979/1980, 324].53 This anecdote has often been taken 49 Augustus Trowbridge was an experimental physicist from Princeton, who was appointed as the IEB’s chief representative in Europe; his first visit to Göttingen in October 1925, mentioned by Noether here, is described in [Siegmund-Schultze 2001, 145–148]. 50 E. Noether to L.E.J. Brouwer, 14 November 1925, Brouwer Papers, Noord-Hollands Archief, Haarlem. 51 Brouwer to Noether, 21 November 1925, Brouwer Papers, Noord-Hollands Archief, Haarlem. 52 These details surface in a letter she wrote to Einstein on January 7, 1926, in which she advised him about a paper that had been submitted for publication in Mathematische Annalen; see Section 3.6. Einstein was one of the journal’s four principal editors, though he would resign during the tumultuous events of December 1928 when Hilbert summarily dismissed Brouwer from the board of associate editors [Rowe/Felsch 2019, 276–351]. 53 Here, again, one should treat this recollection with caution. Elsewhere, Alexandrov wrote that his “theory of continuous partitions of topological spaces arose to a large extent under the influence of conversations with her in December-January of 1925-1926, when we were both in Holland” [Alexandroff 1969]. As Colin McLarty has pointed out, the relevant conversations must have taken place earlier [McLarty 2005b, 228].

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to imply that Noether was the first to conceive of homology groups, a claim that Saunders Mac Lane challenged in [Mac Lane 1986]. This paper was written in reply to [Dieudonné 1984], which documents Noether’s influence on Heinz Hopf in this direction. Mac Lane notes that Vienna was also an important center for combinatorial topology, pointing to the work of Leopold Vietoris, whose homology theory preceded Hopf’s reference to homological groups. While Mac Lane was certainly right in pointing out that one should not simply assume that “mathematical ideas originated in Göttingen and then spread to lesser places” [Mac Lane 1981, 306], he himself wrote five years earlier: It was Noether who emphasized the fact that one should replace these numerical invariants [the Betti numbers and torsion coefficients] by the abelian homology groups of which they are the invariants. The clearest evidence of Noether’s early influence is a report [Noether 1925c] of her talk at the Mathematische Gesellschaft in Göttingen on January 27, 1925, where she discussed the introduction of homology groups . . . . This is the earliest example I know of the use of group theory in homology. According to the synopsis she gave of her talk,54 its main result was algebraic, namely to show how elementary divisor theory can be derived from the structure theory for finitely generated Abelian groups. She then pointed out, as an application of this result, that the Betti and torsion numbers in topology can be treated directly using group theory and do not require elementary divisor theory. This talk took place a full year before Emmy Noether’s visit in Blaricum, so there is every reason to treat Alexandrov’s anecdote as an accurate account. Whether or not her ideas might have influenced others, like Vietoris, who were in Brouwer’s circle of topologists, as suggested in [McLarty 2005b], it is undeniable that she exerted a strong influence on Alexandrov and Hopf, who would soon become two of the leading figures in the field. Already in a paper Alexandrov submitted to Mathematische Annalen in November 1925, he added the footnote: “The first, abstract part of the present work is closely related to the more recent studies by Miss E. Noether from the field of general group theory and is partly inspired by these studies” [Alexandroff 1927, 555]. During his stay in Blaricum, Alexandrov was able to finish most of the editorial work on Urysohn’s posthumous papers. Afterward, from the beginning of May until mid-July 1926, he was in Göttingen residing as Emmy Noether’s house guest at Friedländerweg 57. From there, Alexandrov wrote to Hausdorff on 13 May 1926, informing him that he had received the first set of page proofs for the new edition of Hausdorff’s book on set theory.55 Alexandrov promised to return the proofs with his comments in the next two or three days, adding that these 54 This

summary is reproduced with an English translation in [Bergmann/Epple/Ungar 2012,

71]. 55 Mengenlehre, published in 1927, reprinted in [Hausdorff 2008, 1–351]. This was actually an entirely new book, as described in [Hausdorff 2002, Hausdorff 2008].

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elements of set theory had by now crystallized to the point that there was hardly room for any substantial mathematical differences of opinion. I thank you again very much for giving me the opportunity to be one of your first readers. By the way, I notice during my present lecture course in Göttingen that I can already cite your first edition by heart (just like good conductors, for example, who conduct the Beethoven symphonies without the score!). I’m very happy to be teaching a systematic course on topology here . . . . The interest in the whole complex of questions in set-theoretic topology is very great, above all for Urysohn’s theory of dimension; Urysohn himself only experienced the initial stage of this interest in and recognition for his truly most significant mathematical achievement. [Hausdorff 2012, 42] Toward the end of the semester, Noether and Alexandrov organized a small meeting on group theory and topology, or as Alexandrov called it in a postcard sent to Hausdorff, a “Locarno conference” that had reached a peaceful conclusion at a favorite watering hole just outside Göttingen, the “Kehr” restaurant on Hainholzweg.56 Some weeks earlier, Alexandrov sent Hausdorff his reaction on reading the proofs of his new book Mengenlehre. Unlike the Grundzüge der Mengenlehre, Hausdorff’s new book dealt almost exclusively with metric spaces, a restriction Alexandrov apparently found disappointing. He expressed his general opinion in a letter from July 4, in which he also sketched his own research program for the future, namely to build bridges from point set topology à la Hausdorff to the more classical geometric topology dating back to the nineteenth century. 57 In August, he returned to Batz, where he completed the manuscript for [Alexandroff 1928], the paper in which he proved his fundamental theorem on the approximation of compact metric spaces by polytopes.58 He described his more general goals to Hausdorff in these words: I’m working on various, in general really difficult questions (with still limited success!), all of which concern filling the long-standing deep gorge separating general (set-theoretic) and classical topology, and hoping thereby to be able to clarify many topological features of our ancient, God-given physical space. I’m grateful for a great deal of stimulation in pursuing all these plans – for which some parts require entirely new methods – to the 56 [Hausdorff 2012, 45]; among the others who signed this postcard were Brouwer, Landau, Grell, and Erich Bessel-Hagen. 57 In a broad sense, this gulf was never completely bridged; see [Brieskorn/Scholz 2002]. 58 The bridge Alexandrov built in [Alexandroff 1928] was based on his notion of nerves of coverings, a novel way to pass from coverings of a compact subset of a metric space to polytopes, the traditional objects studied in combinatorial topology; see [Hurewicz/Wallman 1948, 67–72]. This theory was later exploited by Eduard Čech in creating Čech cohomology, a standard theory in algebraic topology.

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4 Noether’s Early Contributions to Modern Algebra young Berlin mathematician Hopf, a truly outstanding topologist who was here during this summer semester (he has carried some of Brouwer’s things further in remarkable ways and has completed them with essentially new methods); he is also personally a very nice human being, which is of course a necessary condition for purely scientific relations. [Hausdorff 2012, 44]

Alexandrov somehow neglected to mention one other (for him) important attribute of Heinz Hopf: he, too, was an avid swimmer.

Chapter 5

Noether’s International School in Modern Algebra 5.1 Mathematics at “The Klie” Pavel Alexandrov and Heinz Hopf met for the first time in Göttingen in the spring of 1926, soon after Alexandrov departed from Blaricum. Hopf had recently taken his doctorate in Berlin under Ludwig Bieberbach and Erhard Schmidt, and his research interests differed sharply from Alexandrov’s work in general topology. In his dissertation he proved a fundamental theorem in differential geometry, namely that a simply connected complete Riemannian 3-manifold of constant sectional curvature is globally isometric to one of three types of spaces: Euclidean, spherical, or hyperbolic. He also studied vector fields on manifolds and proved what came to be called the Poincaré-Hopf theorem. As Hopf and Alexandrov gradually discovered their common interests over the course of that summer, they soon became close friends as part of Emmy Noether’s circle. She knew, of course, that no one could ever replace Pavel Urysohn in Alexandrov’s life, but she surely felt that this new friendship with Hopf was a turning point for him. Noether’s disarming frankness and warmth spilled over very quickly whenever outsiders came to Göttingen, creating a memorable atmosphere for those who shared her addiction for mathematics and the simple pleasures of life. Such as swimming. Hilbert had long been an inveterate swimmer, going back to his years in Königsberg when he often spent vacations at nearby locales on the Baltic. His legendary quip in rebuking those who opposed Noether’s habilitation – “we’re a university, not a bathing establishment” – takes on more vivid meaning in the light of how important swimming was for many of the mathematicians in Göttingen.1 And when they thought about topology while swimming, then very likely Pavel Alexandrov and Heinz Hopf were in their midst. 1 This

was pointed out by Cordula Tollmien in her delightful essay [Tollmien 2016b].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_5

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Figure 5.1: “The Klie”: a breeding locale for mathematics and mosquitos (Städtisches Museum Göttingen)

B.L. van der Waerden, who had picked up some of the latest trends in topology two years earlier on afternoon walks with Hellmuth Kneser, now eagerly listened to the latest ideas that would soon take form as standard concepts in algebraic topology. Decades later he recalled one of these conversations, in which Hopf, Alexandrov, and Emmy Noether discussed Lefschetz’s fixed point formula. This formula, which had just been published by Lefschetz, made it possible to calculate the number of fixed points of a continuous map, or more precisely, the sum of the indices of the fixed points for a mapping of a manifold into itself. Emmy Noether said that one should not do this with matrices, not by calculation, but with concepts, with additive groups and homomorphisms of these groups. Then everything becomes much more transparent and beautiful. And so the old concepts, such as Betti numbers and torsion numbers, were to be retained, but based on group theory. The basic concept from which these older ones were derived was that of a homology group, a notion familiar to every topologist today. . . . When Lefschetz’s fixed point formula was later formulated

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and proven by group theory, Emmy Noether was thrilled. In such cases Emmy Noether . . . sometimes used to say: “The proof is now conceived abstractly and thereby made transparent.” For her, that was the point of modern, abstract algebra: to avoid all special calculations, matrices, etc. and through abstraction to do away with all insignificant features of a particular problem in order to make the essential concepts visible by placing these up front so that the entire proof becomes transparent. [van der Waerden 1997] Years later, in his personal memories of Heinz Hopf, Pavel Alexandrov recalled the lively atmosphere that both of them enjoyed that summer, including musical evenings with the Courants, boat rides on the Leine River, and afternoons spent swimming and chatting at “the Klie” (Fig. 5.1), this being the colloquial name for the university swimming pool run by its lifeguard, Fritz Klie, located next to the river and just south of the town. Many a mathematical, but not only mathematical conversation took place at the Klie, either in the moving waters of the Leine, which were not always particularly clean, even quite brown after it had rained, or in the sun or else the shade of the lovely trees, a favorite spot for the mosquitos. And many a mathematical idea was born there as well. . . . The Klie swimming pool was exclusively for men; females were only represented by Miss Emmy Noether and Mrs. Nina Courant, both of whom exercised their exclusive privileges on a daily basis, no matter what the weather conditions. [Alexandroff 1976, 114] Van der Waerden was also a regular guest at the Klie. In a letter to Felix Hausdorff, Alexandrov reported on an interesting new paper by David van Dantzig and van der Waerden, which also happened to betray the influence of Göttingen’s summer culture: “for the first time in the mathematical literature a certain surface (a sphere with three holes) has been ‘officially’ designated the swimsuit. The stimulus for introducing this terminology can no doubt be traced to activities connected with my two-semester topology seminar in Göttingen.” 2 Hilbert, Courant, Otto Neugebauer, and numerous others were also often to be seen at the Klie, and they were joined that summer by Brouwer as well. Two years earlier, in the midst of the Grundlagenstreit with Hilbert, Brouwer had been invited by the philosopher Moritz Geiger to give a lecture on intuitionism in Göttingen. He spoke at that time before a large crowd, few of whom sympathized with his position. According to Hans Lewy, who heard this lecture as a doctoral student, a heated discussion afterward ensued. Hilbert did not take part until the very end. Then he stood up and said: “With your methods most of the results of modern mathematics would have to be abandoned, and to me the important thing is not to get fewer results but to get more results” [Reid 1970, 184]. The audience’s response to this retort was predictable – thunderous applause – after all, this was 2 Alexandrov

to Hausdorff, 20 December 1928, [Hausdorff 2012, 78].

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Hilbert’s crowd. Still, this could hardly pass as a defense of formalism, which Brouwer had long been attacking. Hilbert was merely expressing a pragmatic view, one that appealed to many mathematicians, like Hans Lewy, who simply wanted to carry on with their research projects. A similar sympathy for Hilbert’s position can be seen from letters Alexandrov wrote to Hausdorff during this time.3 Four months later, Alexandrov assured Hausdorff (who detested intuitionism4 ) that he remained a formalist and fully accepted Georg Cantor’s credo that “the essence of mathematics lies in its freedom.” Even though he did not share Brouwer’s views, he still had high respect for his position, which he regarded as an heroic attempt to draw earthly knowledge from mathematics, whereas he saw the latter as primarily an art form. From a neighboring hotel room in Berlin, Alexandrov was listening to the funeral march from Beethoven’s “Eroica” symphony, which led him to express the thought that it contained more real knowledge than all of science and mathematics, formalism and intuitionism included.5 During Brouwer’s visit in 1926, Emmy Noether and Alexandrov were eager to restore harmonious relations between Hilbert and the Dutch topologist turned philosopher, who evidently felt warmly received by those closest to Courant and Noether. She decided to invite a group of mathematicians, including Hilbert and Brouwer, to her apartment one evening (Fig. 5.3). Those present included Courant, Landau, and Hopf, all seated around Emmy’s table in her cozy quarters. She and her Russian friend decided that he should think of a good way to break the ice between Hilbert and Brouwer, and Alexandrov decided to try an age-old expedient: what better way to bring two persons together than by innocently mentioning the name of a third person who was persona non grata for both? The list of potential candidates might well have been long, but none could compete with the widely disliked and famously vain Leipzig mathematician Paul Koebe, who became notorious in Göttingen for stealing other people’s ideas – a practice many there normally viewed as rather harmless. (In fact, appropriating ideas that found their way into the mathematical atmosphere in Göttingen became so commonplace that a special term was adopted for this: this was called “nostrifying” the thoughts of another.) Brouwer had tangled with Koebe already in 1911 at the Karlsruhe conference, where he first met Emmy Noether.6 Alexandrov could hardly believe his luck once he had merely dropped this fellow’s name. Before long, Brouwer and Hilbert were trying to outbid each other with nasty remarks 3 Alexandrov expressed dismay when he learned that his former mentor Luzin took the “ignorabimus” position regarding certain difficult mathematical questions, a view Hilbert had famously rejected when he spoke about mathematical problems at the 1900 ICM in Paris (Alexandrov to Hausdorff, 29 November 1925, [Hausdorff 2012, 38]). 4 In a letter to Abraham Fraenkel, Hausdorff wrote: “Both you and Hilbert treat intuitionism with too much respect; one must for once roll out heavier guns against the senseless destructive anger of these mathematical Bolshevists!” [Hausdorff 2012, 293]. 5 Alexandrov to Hausdorff, 4 April 1926 [Hausdorff 2012, 38–40]. 6 Brouwer’s ensuing difficulties with Koebe were legendary; for details see [van Dalen 2013, 175–192] and [Rowe 2018b, 266–290].

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about Koebe’s character, all the while nodding agreement and warming to their heartfelt consensus, which finally ended in a toast to everyone’s health for the betterment of mathematics [Alexandroff 1976, 115]. After their stay in Göttingen, Brouwer joined Alexandrov on his nearly annual pilgrimage to Batz, where they commemorated the genial and much-missed Pavel Urysohn.

5.2 The Takagi Connection In 1927 two young Japanese mathematicians arrived in Göttingen to study under Emmy Noether. These were Zyoiti Suetuna and Kenjiro Shoda. Both had studied number theory under Teiji Takagi in Tokyo. Suetuna’s principal interests centered on analytic number theory, so he worked closely with Landau, but also later with Artin in Hamburg as he was deeply attracted to the latter’s general reciprocity law. When he returned to Tokyo University, Suetuna introduced a research seminar modeled on those in Göttingen, and in 1936 he was appointed to Takagi’s chair after the latter’s retirement. Shoda had already become interested in algebra as a student, so he chose to spend the year 1926/27 working with Issai Schur in Berlin.7 He had never heard of Emmy Noether until the summer of 1927, so his plan to leave Berlin and travel to Göttingen had nothing to do with her. Shoda wanted to study at this famous university especially to hear Hilbert’s lectures, knowing that he had been Takagi’s teacher. Soon after his arrival, he went to look for Emmy Noether’s apartment in the Friedländerweg. Once he found it and rang, a badly dressed stocky woman came to the door and so he asked if Prof. Noether was at home. “I’m Noether and you must be Herr Shoda,” came the reply, quite to his amazement. Then, as soon as he sat down, she bombarded him with questions in a very friendly way. He began to realize that Schur must have written her, because she already seemed to know various things about him. She advised him to read works by Steinitz and Krull as background for her lectures; it was very good, she told him, that he had studied representation theory in the style of Frobenius and Schur. He would soon learn about her approach to the subject. Kenjiro Shoda found Noether’s lecture style very difficult to follow, but the study tips she had given him proved extremely useful, especially the literature on abelian groups. Learning this material provided him with concrete examples that enabled him gradually to grasp the general concepts, which Emmy Noether only rarely illustrated by means of simple cases. In 1975, Shoda recalled one of Noether’s famous sayings concerning methods of proof in mathematics. She maintained that one could, of course, prove the equality of two real numbers a and b by first showing that a ≥ b and then that a ≤ b, but one should not 7 The following information is taken from Shoda’s personal recollections, which he wrote for the Japanese edition of Auguste’s Dick’s biography of Emmy Noether [Dick 1970/1981], published in 1975. Shoda’s text was translated into German and can be found in Dick’s literary estate in the Austrian Academy of Sciences, item 10-26.

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be satisfied with such a proof. Instead, one should search for the true reason that actually reveals why a and b must be equal. One must imagine that she repeated this cryptic advice nearly as often as her most famous saying, namely “that’s already in Dedekind” (das steht schon bei Dedekind).8 Yet Shoda came to understand that these favorite remarks had a much deeper meaning, namely, that for Emmy Noether understanding a mathematical result only began with a provisional argument or the sketch of a proof. Her lectures were often confusing and difficult to follow – not only for foreigners trying to read her handwriting (she wrote on the blackboard using the old Gothic script), but for anyone, because she was often still struggling to clarify the arguments for herself. During his year in Göttingen, Shoda rented a room in a boarding house just down the street from Noether’s apartment. In fact, his landlady was on friendly terms with his teacher, who would occasionally drop by. Best of all, he often had the chance to accompany Emmy Noether on the fairly long walk from her residence to the Auditorienhaus where she taught. On such occasions, she normally did all the talking, and what she had on her mind, of course, was the lecture she was about to deliver. Listening to her ramble on without any visual aids at all was definitely a challenge, but it could also serve to sharpen a person’s ability to think about abstract mathematics. After he grew accustomed to her speaking style, Shoda found that such “preview” performances – which were almost like hearing her give the lecture twice – contributed greatly to his understanding. Repeat performances later became standard fare for Emmy Noether in the United States, where she would lecture on Monday at Bryn Mawr College as a warm-up for the teaching she would do the next day in Princeton. Shoda was one of only around ten auditors who attended Noether’s course. Others included his compatriot Suetuna, Jakob Levitzki, and van der Waerden, who often interrupted to ask questions, since he was serving as Noether’s official notetaker. The material she taught that semester on “Hypercomplex numbers and representation theory” was by no means new; van der Waerden had attended the same course three years earlier. This time, though, she was ready to put everything in better shape, and so she enlisted his help in writing the lectures up for publication. Shoda and many others would later have the opportunity to study this theory in print form in [Noether 1929a].9 The term “hypercomplex numbers” harkens back to an older tradition that began with the discovery of number systems that lie beyond the real and complex numbers. In 1843, William Rowan Hamilton introduced the earliest example, the quaternions. Thirty years later, the Harvard mathematician Benjamin Peirce began a project to classify hypercomplex number systems, highlighted when he published his Linear Associative Algebra (see [La Duke 1983]). These early studies have very little in common with modern algebra, however, as the latter deals with abstract structures defined via a system of axioms. For example, a group is any 8 One

finds the exact same admonition in [Weyl 1968, 3: 442]. of the novel features in this paper were described by Nathan Jacobson in [Noether 1983,

9 Some

17–18].

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set of elements with an operation that satisfies three axioms: the associative law, the existence of a neutral element, and the existence of inverse elements (if the commutative law also holds, then this is called an Abelian group). One can then define abstract operations on groups, analogous to the usual arithmetical operations applied to numbers, to form products or quotients of groups, etc. Likewise, one can introduce more complex structures by means of additional axioms. Thus, a ring is a group with two operations linked by the distributive law, whereas a field is a ring with the additional property that its elements have multiplicative inverses. In modern algebra, two objects are called isomorphic when they have the same structure. This can be established by means of an isomorphism, a mapping between the elements of the two sets that preserves their structures. For example, one can easily show that every infinite cyclic group is isomorphic to the group Z. In other words, there is essentially only one such group, since all infinite cyclic groups have exactly the same properties as Z. In A History of Algebra [van der Waerden 1985], B.L. van der Waerden describes the transition from the earlier period in algebra to the one inaugurated by Noether.10 In [Noether 1929a], she united earlier results on hypercomplex systems by means of the representation theory of groups and algebras. In other work from this period, she exploited abstract ideal theory to shed new light on the structure theory of associative algebras and the representation theory of groups. During this time, she worked closely with Hasse and Richard Brauer; their publications would prove to be of major importance for subsequent developments in modern algebra and algebraic number theory. Like most of those who fell under Noether’s wing, Kenjiro Shoda had to learn a new way to think about mathematics. He already knew how to read and study, and he quickly learned how to listen and “talk mathematics,” but he also gained some important insights from her about writing, advice he clearly took to heart. One such insight she imparted was actually a remark she had heard from Hilbert, and since the latter loved to package everything in threes, we can easily imagine that this bit of wisdom originated with him. According to Hilbert, mathematicians should realize that there are three types of papers: those that no one reads, others that people skim through quickly, and then there are the rare few that other mathematicians actually study carefully. This being the case, an author need only be concerned about papers in the second category, those that will be browsed rather than read. For such publications, authors should make sure to have a clearly written introduction that informs the reader about what the remainder of the paper contains (the part no one is likely to read). Noether also informed Shoda that Schur had criticized a manuscript he had written, a remark that suggests she thought her Japanese student would not be offended if she imparted constructive criticism. One piece of common-sense advice he remembered all his life. She told him, “an author should not try to include all of his or her results in a single paper. Mathematical research is a lifelong activity 10 For

more recent accounts, one can consult [Corry 2004a] or [Gray 2018].

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and the publications are only signposts along the way. One should leave some things out that one can continue to work on later.” Through this type of advice, he reflected, Noether gave him the foundation he needed to become a research mathematician. They remained in contact from a distance after 1929, when Shoda returned to Japan. Soon afterward, he began to write his textbook Abstract Algebra, which was first published in 1932 and then reprinted many times later. Its impact on mathematics in Japan has sometimes been compared with that of van der Waerden’s Moderne Algebra on research in Europe and the United States [Sasaki 2002, 246]. Kenjiro Shoda went on to a highly successful career, beginning in 1933 with his appointment as professor in the Faculty of Science at the newly founded Osaka University. After the Second World War, he was elected to preside over the Mathematical Society of Japan, and in 1949 he became Dean of the Faculty of Science. He served in that office for six years, after which he was appointed President of Osaka University, a position he held for another six years. He is remembered today by students and alumni there as the founder of the Shoda Cup, awarded each year to the winning team in an athletic competition.

5.3 Bologna ICM and Semester in Moscow Kenjiro Shoda’s encounters with Emmy Noether were anything but unique, as similar stories and recollections of her personality can be found over and again. Her passion for “talking mathematics” – long a favorite pastime in the intense mathematical atmosphere that emerged in Göttingen after 1900 – was simply boundless. One of her students, Heinrich Kapferer, recalled a walking tour with Noether, very likely to her favorite spot in the midst of the woods outside town, the Kerstlingeröder Feld: It was a beautiful summer’s day, but not a word was spoken about the gorgeous surroundings that we encountered, instead it was one ongoing mathematical conversation without interruption that lasted for at least two hours. In essence, it was a monologue delivered by E.N. with touching efforts to gain my interest in her problems, but without any kind of written material and with no opportunity for me to take notes that I could refer to later. As such, this was for both of us a strenuous undertaking, for E.N. in a productive sense and for me in a receptive sense, constantly nodding my head out of politeness, even though by no means everything had been fully understood. Finally we came to a clearing flooded with light in the woods , which offered us a well-earned resting spot, no bench, only a soft, rising grassy area, where we could lie down.11 11 Translated

from [Tollmien 2016b, 190–191].

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Yet not only students found it difficult trying to keep up with Emmy’s rapidfire thought processes. Even Emil Artin, one of the giant figures in twentiethcentury mathematics, often found it very difficult to keep up with her. His former wife, Natascha Artin Brunswick, once described how he devised a method for coping with this problem: They would go for walks, and he would ask her a question, and she would talk very, very fast. He knew he couldn’t keep up with her, so he would let her talk for about half an hour and then say “Emmy, but I didn’t understand a word; could you please tell me again.” But in the meantime they would walk very fast, and she would get a little slower and go through it a little more slowly again. The second time he would say, “Emmy I haven’t understood it yet.” On the third rendition he would understand what she was talking about. By that time, you see, she was so tired that her speed would slow down. She was so amazingly lively! [Kimberling 1981, 34] Hermann Weyl, who first got to know Noether during the winter semester of 1926/27 when he was a guest professor in Göttingen, later recalled lively conversations with her and John von Neumann following his lectures on the representation theory of continuous groups. Weyl and von Neumann were both then deeply immersed in developing aspects of this theory that were central for the mathematical foundations of quantum mechanics.12 At this time, Weyl had already submitted his paper “Quantum Mechanics and Group Theory” [Weyl 1927] when von Neumann informed him that he had independently introduced one of Weyl’s main ideas for it.13 In his memorial lecture for Noether, Weyl recalled those days: I have a vivid recollection of her when I was in Göttingen . . . 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 von Neumann, who was in Göttingen as a Rockefeller Fellow, through the cold, dirty, rain-wet streets of Göttingen. [Weyl 1968, 3: 432] It was during the academic year 1927/28 that Alexandrov’s friendship with Hopf really intensified. Both were then Rockefeller fellows in Princeton, by now the leading international center for topology, led by Lefschetz, Veblen, and J.W. Alexander. During this time, Alexandrov and Hopf began planning a multi-volume work on topology, though only the first volume was completed and did not appear until 1935. They spent the following summer in Göttingen, much of the time together with Otto Neugebauer, who also enjoyed swimming in scenic surroundings. Toward the end of their stay, all three traveled to Bologna to attend the International Congress, the first postwar congress open to mathematicians from 12 On

Weyl’s mathematical interests during the mid-1920s, see [Hawkins 2000, 465–511]. noted this in a footnote added in the proofs, [Weyl 1968, 3: 90–135].

13 Weyl

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countries who were on the losing side of the First World War. Emmy Noether, who had attended the 1908 ICM in Rome, was not about to miss out on this event, even though many in Germany refused to attend for political reasons. Over the summer, she and van der Waerden polished up the manuscript from her wintersemester lectures, which Noether then submitted to Mathematische Zeitschrift, where it was published as [Noether 1929a]. This new work opened the way to the third main period in her research career.14 Noether no doubt took great pleasure in presenting some of her first main results in Bologna [Noether 1928]. As noted, the Bologna ICM was a politically contentious event that exacerbated tensions between nationalists and internationalists within the German mathematical community. Hilbert clashed openly with Ludwig Bieberbach, one of several Berlin mathematicians who chose to boycott the congress, though Hilbert was convinced that behind the scenes Brouwer was the true ringleader behind this boycott effort, which he was determined to foil [Siegmund-Schultze 2016]. For some time, Hilbert had been fighting to stay alive after learning that he was suffering from pernicious anemia, which was then a lethal disease. His gloomy private thoughts fixated on Brouwer, whom he saw as a dangerous influence on the German mathematical community: In Germany there has arisen a form of political blackmailing of the worst kind: you are not a German, unworthy of German birth, if you do not talk and act as I prescribe. It is very easy to get rid of these blackmailers. You have only to ask them how long they were in German trenches. Unfortunately, German mathematicians have fallen victim to this blackmailing, Bieberbach for example. Brouwer . . . has cultivated the instigation of discord among the Germans, all the more in order to set himself up as the master of German mathematics. With complete success. He will not succeed a second time.15 Alexandrov’s relations with Brouwer had also grown strained, though for other reasons. This came about because of a long-brewing conflict over the origins of dimension theory, which in its modern form emerged during the early 1920s through the work of Pavel Urysohn and Karl Menger, a student of Hans Hahn in Vienna. This conflict erupted around the time of the Bologna Congress with the publication of [Menger 1928], a study that contained a number of notes relating to prior historical contributions, including those of Brouwer, who published a rebuttal in [Brouwer 1928]. Hahn, as principal editor of Monatshefte für Mathematik und Physik, made a heroic effort to calm passions on both sides. To Brouwer he wrote: 14 In the winter semester of 1929/30, she offered a course on “Algebra of hypercomplex numbers” that was written up by Max Deuring; these lectures, however, were only published posthumously in [Noether 1983, 711–762]. 15 Translated from [Rowe/Felsch 2019, 260–261]; a full account of the Brouwer-Hilbert conflict appears there in Chapters 7 and 8. Hilbert’s allusion to Brouwer’s earlier success refers to a propaganda campaign directed against French mathematicians, particularly Paul Painlevé.

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. . . you may be interested to know that I was with Mr. Alexandrov both in Bologna and while he was traveling through Vienna, and on both occasions he spontaneously expressed the wish that the priority dispute in dimension theory be forgotten and buried,16 since he is convinced that Urysohn and Menger had found the essentials independently of one another, and he asked me to signal to Menger a restoration of friendly and loyal relations, which I believe I did successfully.17 This letter marks only the beginning of a long and dreary correspondence that must have dearly tested Hahn’s nerves. Symptomatic for Brouwer’s attitude with regard to this matter is the following passage from a later letter: “As for your advice on the priority issue in dimension theory, I believe that the smallest moral matter is more important than all of science, and that the only way to preserve moral uprightness in the world is to oppose every immoral undertaking. This applied before to the Bologna Congress and it applies now to Menger’s false claims.” 18 Hahn nevertheless kept pleading with him: Under no circumstances should you devote so much time and attention to these matters as to affect your scientific work. The same is true in my view of priority questions in dimension theory. Your own contributions to this theory, on the one hand, and those of Menger and Urysohn, on the other hand, are so evident to any informed person that nothing is gained by sacrificing time and energy to this matter.19 After the ICM in Bologna, Alexandrov Hopf, and Neugebauer vacationed in Italy on the shores of the Mediterranean. When they departed, Alexandrov traveled to Venice to meet with Emmy Noether, in all likelihood to discuss plans for her forthcoming stay in Moscow. She arrived there on 13 October 1928 and took up quarters in a dormitory near the Krymskii Bridge within walking distance of Moscow University. There she taught a course on abstract algebra, a novelty in the Soviet Union, as well as a seminar on algebraic geometry at the Russian Academy. One of those whom she met during this time was 20-year-old Lev Pontryagin, the blind topologist who would soon emerge as one of the giants in the field. Alexandrov asserted that he was strongly influenced by Noether’s mathematics [Alexandroff 1935, 9], though most likely this influence was mediated by Alexandrov himself; no one made a stronger impression on Pontryagin than his teacher. The latter did, however, record these impressions from Noether’s Moscow lectures: Around the beginning of my fourth year P.S. Alexandrov returned from abroad and brought Professor Noether with him. So I returned to topology in my fourth year and also attended the lectures of Miss Noether 16 In [Hurewicz/Wallman 1948], long the standard work on dimension theory, there are virtually no traces of the contentious disputes that dominated its early history. 17 Hahn to Brouwer, 6 November 1928, Brouwer Papers, Noord-Hollands Archief, Haarlem. 18 Brouwer to Hahn, 24 February 1929, Brouwer Papers, Noord-Hollands Archief, Haarlem. 19 Hahn to Brouwer, 6 March 1929, Brouwer Papers, Noord-Hollands Archief, Haarlem.

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5 Noether’s International School in Modern Algebra on contemporary, modern algebra. These lectures had an impressive comprehensive quality quite different from Alexandrov’s lectures, but they were by no means dry and I found them very interesting. Miss Noether spoke in German, but her lectures were understandable due to the unusual clarity of the presentation. The opening lecture of this famous German female mathematician was attended by a huge crowd.20

Alexandrov’s friendship with Emmy Noether intensified during the winter she spent in Moscow. In an undated letter to Oswald Veblen from 1928, he wrote: This winter (as you know) we have Miss Noether here in Moscow as a guest professor, and, of course, her presence enlivens our mathematical life greatly, especially since algebra notably belongs to those mathematical fields that unfortunately have been cultivated little so far in Moscow. Partly under the influence of Miss Noether, partly also stimulated by my algebraic lectures here in Smolensk, I begin to become very interested in algebra, for the time being only “from afar” of course, without trying to work in it myself. The topological lecture which Hopf is holding in Berlin this winter (and which is very interesting [. . . ]) is also very algebraically influenced. [Merzbach 1983, 168] Veblen had corresponded with Noether a year earlier in connection with her work on differential invariants. His curiosity about this was surely piqued by reading in [Noether 1918a, 44] that she planned to publish a more detailed account in Mathematische Annalen. As so often happened, however, she never found time to complete such a paper and had to inform Veblen that her interests were now focused on arithmetical matters [Merzbach 1983, 165].21 During that winter, Alexandrov was commuting back and forth between Moscow and his native Smolensk, where he taught at the Pedagogical Institute. He arranged this so that he could attend Noether’s course in Moscow, making use of their conversations while he was teaching algebra in Smolensk. He described this as “a long algebra course in which apart from the obligatory material I presented the fundamentals of modern algebra (the theory of groups, rings and fields). I brought all these new ideas from Emmy Noether” [Alexandrov 1979/1980, 325]. This experiment apparently bore real fruit, since one of those who attended Alexandrov’s course was A.G. Kurosh, later to become one of the leading algebraists in the Soviet Union. The summer that followed was the first since 1923 during which Alexandrov decided not to visit Göttingen. Instead, he and Kolmogorov undertook a long journey (1300 km.) by rowboat on the Volga River. Returning on board a steamship, 20 Translated from the German in [Koreuber 2015, 172]; the original Russian was published in 1988. 21 Veblen referred to [Noether 1918a] in noting: “A generalization of the normal coordinates of Riemann to a still larger class of differential invariants was indicated by Emmy Noether” [Veblen 1927, 101].

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Alexandrov wrote a letter to Felix Hausdorff describing his trip, but also events of the recent past. The winter flew by quickly with much pedagogical (only a little scientific) work. You probably know that Frl. Noether adorned Moscow with her presence throughout the entire winter . . . . Her presence was a great joy for all of us and she performed a real service for Russian mathematics: she succeeded in creating great interest in a field that has long constituted a gaping hole in the Russian mathematical tradition, I mean algebra.22 During the summer semester after her return, Noether spoke about her experiences in Moscow in a “travel report” for the members of the Göttingen Mathematical Society.23 In past years, she corresponded only sporadically with Alexandrov, but after her return from Moscow she tried to keep him informed about mathematical life in Göttingen. On the first day of vacation after the semester, she wrote with various bits of news: . . . van der Waerden has made many things more transparent and we now know what it is about; but we still don’t know how it should work – at least in the algebraic-analytical part, a completely radical “transparency” will be necessary! And some of the “banalysis” – I don’t know who invented that word – will still have to be thrown out! You’ll see for yourself.24 How is the book progressing? At Landau pace, 40 pages per day?25 Incidentally, Kerekjarto26 also reappeared, in transit to Hamburg, where he performed. But Courant did not let him come up with any wishes. Cohn-Vossen is editing Hilbert’s “intuitive geometry” to make a yellow book out of it; but he has the right feeling, that he should only give notes and references to existing proofs, and otherwise add nothing. So the type of lectures you always talked about with so much joy will still remain the same.27 The final product, [Hilbert/Cohn-Vossen 1932], appeared with 230 figures that guide the reader through a vast range of geometrical knowledge, all of it presented informally. This was the furthest thing from Noether’s mathematics, and yet she wrote about it respectfully. 22 Alexandrov

to Hausdorff, 10 July 1929, [Hausdorff 2012, 85–86]. 4 June 1929, Jahresbericht der Deutschen Mathematiker-Vereinigung 38: 142; unfortunately no further record of what she reported seems to have survived. 24 This passage is almost surely a reference to the topic of Noether’s Prague lecture [Noether 1929c], discussed below. 25 Edmund Landau’s lectures were famously well organized and presented in rapid-fire fashion. He apparently composed his books at the same pace. 26 Béla Kerékjártó was a Hungarian topologist who often visited Göttingen. 27 Noether to Alexandrov, 1 August 1929, translated from [Tobies 2003, 103]. 23 Delivered

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5.4 Helmut Hasse and the Marburg Connection During her third mathematical period, beginning in 1927, Emmy Noether worked closely with Helmut Hasse, a mathematician who made fundamental contributions to algebraic number theory and especially class field theory. Their paths no doubt crossed in Göttingen during 1919 when Hasse began his studies there, drawn especially by Erich Hecke’s lectures on number theory. When Hecke soon thereafter accepted a chair at the newly founded University of Hamburg, however, Hasse decided to continue his studies under Kurt Hensel in Marburg. Not many serious young mathematicians would have considered leaving Göttingen to study at this much smaller university, but Hasse had picked up a copy of Hensel’s Zahlentheorie [Hensel 1913] at a local bookstore and became enthralled with the author’s treatment of p-adic numbers. Hensel had studied in Berlin under the distinguished algebraist Leopold Kronecker, whose collected works he later edited. At the time Hasse met him, he had long been chief editor of the Journal für die reine und angewandte Mathematik, the Berlin journal commonly called simply Crelle after its founder. In short, Kurt Hensel, whose grandparents were the painter Wilhelm Hensel and the composer Fanny Mendelssohn, was a leading representative of the Berlin algebraic tradition.28 Hasse thus entered a world quite distinct from the intellectual atmosphere familiar to Emmy Noether from Erlangen and Göttingen. Not that this proved to be a barrier; on the contrary, when they met again in the mid-1920s it took little time before their mutual interests set off fresh sparks. Hasse revered his mentor, Kurt Hensel, under whom he completed his doctoral dissertation in 1921 and his habilitation thesis one year later. In 1925 he was already a full professor in Halle, where he assumed the chair long held by Georg Cantor. While there he became co-editor of Crelle alongside Hensel, whom he succeeded in Marburg when the latter retired in 1930. Hasse remained editorin-chief of Crelle until the end of his life. His subsequent career, beginning with his appointment in Göttingen in 1934 as Hermann Weyl’s successor, will occupy us in the final two chapters. Kurt Hensel’s fame rests on his invention of the p-adic numbers, which he had already introduced in 1897. American mathematicians could get a crash course in this new theory by reading Leonard E. Dickson’s review of [Hensel 1908] in the Bulletin of the AMS [Dickson 1910]. In that same year, Ernst Steinitz published his fundamental paper on the the modern concept of fields, “Algebraische Theorie der Körper” [Steinitz 1910], noting how the invention of p-adic numbers influenced this work. “I was particularly stimulated,” he wrote, “to undertake these general investigations by Hensel’s theory of algebraic numbers (Leipzig, 1908), in which the field of p-adic numbers forms the starting point, a field that is neither a functionnor a number field in the usual sense of the word.” 28 For insights into the long-term influence of this Berlin tradition, see [Hawkins 2000, chaps. 4, 5, 10] and [Hawkins 2013].

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Another mathematician who took inspiration from Hensel’s p-adic numbers was the Hungarian József Kürschák, who is credited with inventing the theory of valuations. Kürschák introduced the p-adic number field Qp as the completion of the rational number field Q with respect to its p-adic valuation, thereby dispelling the mystery surrounding the status of Hensel’s contruct. In describing his fundamental paper [Kürschák 1913], Peter Roquette commented: Before Kürschák, Hensel had defined p-adic algebraic numbers through their power series expansions with respect to a prime element. This procedure was quite unusual since Hensel’s power series do not converge in the usual sense, and hence do not represent “numbers” in the sense understood at the time, i.e., they are not complex numbers. Accordingly there was some widespread uneasiness about Hensel’s p-adic number fields and there were doubts whether they really existed. Kürschák’s paper was written to clear up this point. [Roquette 2003, 6] During the First World War, when he was studying under Hensel in Marburg, Alexander Ostrowski studied Kürschák’s work and in 1916 he proved two fundamental theorems in valuation theory.29 As a consequence, he showed that the p-adic number fields Qp were the only possible completions of the rational numbers besides the usual completion leading to the real numbers R. Ostrowski sent a preprint of his paper to Emmy Noether, who reacted with typical enthusiasm: I have started to read your functional equations [paper] and I am very interested in it. Is it perhaps possible to characterize the most general field which is isomorphic to a subfield of the field of all real numbers? [Roquette 2003, 6] As Roquette noted, she was asking a question that was both natural and deep, namely which fields can be isomorphically embedded into R. Questions of this sort would later be answered after Emil Artin and Otto Schreier developed the algebraic theory of real closed fields in the mid-1920s [Artin/Schreier 1926, Artin/Schreier 1927]. This theory played a major role in the rise of abstract algebra, in part because it went hand-in-hand with the emergence of modern Galois theory, one of the cornerstones of Artin’s mathematical legacy (see Section 9.5). Galois theory concerns special types of number fields which extend the field of rational numbers Q.30 A real closed field shares many properties of the real numbers R; in fact, the latter is an example of the former. 29 His

paper [Ostrowski 1918a] was already submitted √ in 1916. simple example is Q[i] = {a + bi | a, b ∈ Q, i = −1}. This is a 2-dimensional extension of Q with basis 1, i, since all numbers in Q[i] are linear combinations of 1 and i, i.e. of the form a·1+b· i. All number fields are finite-dimensional extensions of Q and therefore have a basis consisting of the solutions of algebraic equations with coefficients in Q. Such extensions form an n-dimensional vector space over Q. The theory of algebraic number fields was developed by Dedekind, Hilbert, et al., and was presented in detail in Hilbert’s Zahlbericht [Hilbert 1897, Hilbert 1998]. 30 A

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In classical mathematics, one starts with the natural numbers N, from which one can then build a tower of number systems: N ⊂ Z ⊂ Q ⊂ R ⊂ C. From the standpoint of modern algebra, Q is small, in fact it is the smallest infinite field. The field of real numbers R, on the other hand, lies far beyond the realm of all algebraic number fields. In between them, though, lie the real algebraic numbers, A ∩ R, where A ⊂ C is the set of all algebraic numbers, which consists of all possible numbers that can arise as solutions of any algebraic equation with coefficients in Q. That may sound like a huge set, too, but Georg Cantor proved that A is countably infinite, whereas R is not.31 So, in the language of measure theory, almost all numbers in R are not algebraic, they are transcendental, like π, which can never be expressed as the solution to an algebraic equation with rational coefficients.32 The complex numbers C, on the other hand, lie only slightly beyond R. C is R[i], which is a two-dimensional extension of R, where equivalent to the the field √ the imaginary number i = −1 solves the second-degree equation x2 +1 = 0. Most importantly, the field C is the algebraic closure of R because every polynomial equation with coefficients in C also has its solutions in C. This, in fact, is the key property that holds in the theory of Artin and Schreier: their real closed fields F are, like R, not algebraically closed, but by adjoining i, the field F (i) will be algebraically closed. Furthermore, as they proved in [Artin/Schreier 1926], if F is any ordered field, then there exists a unique extension field K which is a real closed field and preserves the ordering on F .33 Whereas Noether clearly sensed the importance of all these new ideas, recognition of valuation theory and the importance of p-adic number theory emerged only gradually in the wake of Hasse’s work. An early turning point came in October 1920, when Hasse discovered his Local-Global Principle.34 This breakthrough awoke real interest in Hensel’s theory, transforming it from what many in Göt31 On

Cantor’s mathematical career, see [Dauben 1979]. the original Festschrift edition of his Grundlagen der Geometrie [Hilbert 1899], Hilbert avoided the thorny problem of axiomatizing the continuum of real numbers R. Since every construction problem is theoretically solvable in the geometry over R – which Hilbert referred to as Cartesian geometry – this case had little relevance for his axiomatic program. Soon afterward, however, he added an additional completeness axiom, which appeared in all subsequent editions. This extended system of 18 axioms was designed to characterize the real numbers as a complete ordered Archimedean field. Hilbert’s second Paris problem called for a proof of the consistency of this axiom system [Hilbert 1900]. 33 For a detailed discussion of the importance of these concepts for the foundations of modern mathematics, see [Sinaceur 2003]. 34 In number theory, one seeks to find rational solutions to equations with coefficients in Q. When such a solution exists, one calls this a global solution, which automatially yields local solutions in the the reals R and p-adic number fields Qp , which are extensions of Q. Hasse’s Local-Global Principle involves reversing this process by using information from local solutions to produce a global solution. Minkowski gave an early example of this in the theory of quadratic forms; see [Schwermer 2007]. 32 In

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tingen regarded as a mere curiosity into a central tool for research in number theory. Like Noether, Hasse rarely missed a chance to speak at the annual meetings of the German Mathematical Society (DMV), and both were on the program for the 1924 conference held in Innsbruck.35 Hasse spoke on his explicit reciprocity formula for higher exponents, which drew a critical comment from Hermann Weyl (see [Schwermer 2007, 171–173]). Noether presented the axiomatic basis for factorization of ideals into prime ideals [Noether 1924b], a prelude to her classic paper [Noether 1927a]. This was a generalization of a property Dedekind had proved for the ring of integers in an algebraic number field (today called “Dedekind rings”). Hasse was in attendance at that lecture, which left a deep impression on him. The following year, at the annual DMV meeting in Danzig, Noether spoke about her new approach to representation theory in the context of the theory of abstract algebras. In her abstract [Noether 1925b], she wrote: “Frobenius’ theory of group characters [for representations of finite groups] is seen as the ideal theory of a completely reducible ring, the group ring.” She then took up Wedderburn’s structure theorems for algebras and indicated how these can be used as a new framework for representation theory. Finally, she announced that this theory would be elaborated in Mathematische Annalen, though she never realized that plan. Instead she continued to develop these ideas in her courses, culminating with the major publication [Noether 1929a], written with the help of van der Waerden. In the same session of the Danzig meeting, Hasse gave a general survey talk about class field theory, in particular the broad new vistas opened by the recent work of Teiji Takagi. This marks the beginning of Hasse’s famous “class fields report.” 36

5.5 Takagi and Class Field Theory Takagi’s life story underscores how even a true mathematical genius may have a difficult time finding his or her own way forward.37 A stimulating intellectual atmosphere will usually be very helpful, but at times isolation can be even more beneficial, particularly if it frees a person from the influential views of leading authorities. Takagi had already studied Hilbert’s Zahlbericht in Tokyo when he arrived in Göttingen in 1900 [Sasaki 2002, 240], but his timing was unlucky: Hilbert’s research interests were no longer focused on number theory. They met occasionally, but Hilbert was more than a little skeptical when Takagi told him he 35 Alexandrov and Urysohn were on the initial program as well, and were looking forward to the meeting with great anticipation; see [Hausdorff 2012, 28]. 36 Part I appeared as the expository paper [Hasse 1926], followed by proofs in [Hasse 1927a]; for background see [Frei/Roquette 2008]. For an approach to class field theory based on historical developments, see [Cohn 1994]. 37 The story is briefly told in [Yandell 2002, 219–230].

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was working on Kronecker’s Jugendtraum, one of the early cornerstones of class field theory.38 After three semesters in Göttingen, Takagi returned to Tokyo and gradually came to feel he had been too quick to accept Hilbert’s approach to class fields. 39 Heinrich Weber had developed a more general theory, though it was also more complicated than Hilbert’s approach.40 After the outbreak of the Great War, Takagi lost all possibility of staying in touch with European mathematicians, a circumstance that may have emboldened him to drop Hilbert’s approach and take up Weber’s more general theory. As he later recalled: I was freed from that idea and suspected that every abelian extension might be a class field if the latter is not limited to the unramified case. I thought at first that this could not be true. Were it false, the idea should contain an error and I tried my best to find this error. At that time I almost suffered from a nervous breakdown. I dreamt often that I had resolved the question. I woke up and tried to remember my reasoning but in vain. I tried my utmost to find a counterexample to the conjecture which seemed all too perfect. Finally I made my theory confirming this conjecture, but I could not rid myself of the doubt that it might contain an error which would invalidate the whole theory. I badly lacked colleagues who could check my work. [Takagi 1935] After a long struggle, Takagi convinced himself that his theory was sound, but he still faced the difficult task of convincing leading experts in the mathematical world at large. He set out his case in two long papers, [Takagi 1920] and [Takagi 1922], both written in German. His first opportunity came in September 1920, when he spoke about his results at the International Congress in Strasbourg. Again, his timing was unlucky, since German mathematicians were not allowed to attend due to the post-war boycott implemented by the International Research Council. Takagi first spent a month in Paris before leaving for Strasbourg to take part in the congress, where he would speak in French. The session in which he spoke was chaired by the eminent American number-theorist Leonard E. Dickson from the University of Chicago.41 Unfortunately, Takagi’s presentation left little impression on him or the few other mathematicians in the audience who might have been able to follow him under other circumstances. 38 Hilbert referred directly to Kronecker’s Jugendtraum in formulating his twelfth Paris problem, but his interpretation of this conjecture turned out to have been both misleading and incorrect; see the lengthy discussion in [Schappacher 1998b]. 39 Hilbert’s theory was essentially a string of conjectures for what came to be called the Hilbert class field; see 6.5. 40 See [Edwards 1990] and Hilbert’s theory was restricted to unramified class fields, where factorizations of the prime ideals of the ground field contain no repeated factors in the extension fields. From the standpoint of the theory of algebraic functions, which are defined by Riemann surfaces, it was natural to limit consideration to this unramified case. 41 On Dickson as an international figure, see [Fenster 2002].

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Before returning to Japan, Takagi visited Hilbert in Göttingen, but apparently missed the chance to meet Carl Ludwig Siegel, who was then studying under Edmund Landau. Siegel later received a copy of [Takagi 1920], and in 1922 he loaned this to Emil Artin, who had come to Göttingen that year. Artin had no difficulty digesting its 133 pages, and came away duly impressed. The following year, as a private lecturer in Hamburg, Artin had ample opportunities to meet with Hasse, who held the same position in Kiel. He urged Hasse to read Takagi’s two papers, both of which he studied carefully. As reported in [Honda 1975], after reading the first paper, “Hasse was deeply fascinated by its generality, its clearness, its effective methods, and its wonderful results.” The second paper, on reciprocity laws, inspired him even more. Thus, by 1923 there were three young mathematicians in Germany who had read and appreciated Takagi’s papers on class field theory: Siegel, Artin and Hasse. Each of them immediately started to integrate Takagi’s results into his work with striking results, and in this way these results became quickly known among mathematicians. . . . [Frei/Roquette 2008, 14] Hasse soon began preparing a lecture course on Takagi’s class field theory, which he taught in Kiel during the summer semester of 1924. Reinhold Baer, who later became Hasse’s assistant in Halle, wrote up a text based on his notes from this course. This served as the basis for Hasse’s lecture the following year at the DMV meeting in Danzig as well as his “class fields report” [Hasse 1926]. In underscoring the impact it had, Frei and Roquette quote from a postcard Erich Bessel-Hagen sent to Hilbert, dated 17 August 1926: A few days ago there appeared . . . a report by Hasse on class field theory, which is written with excellent clarity. The design of the whole theory is wonderfully uncovered by presenting only the main ideas, while the proofs are reduced to their skeletons. Reading this article is a real pleasure; now all obstacles are eliminated which may have hampered access to the theory. . . . [Frei/Roquette 2008, 16]

5.6 Collaboration with Hasse and Brauer Not long after this, Emmy Noether began to take a serious interest in these developments, which she recognized were closely connected with the arithmetical properties of hypercomplex number systems. She and Hasse (Fig. 5.2) had already been corresponding since 1925, but in 1927 they began an intense and mutually fruitful collaboration. As Lemmermeyer and Roquette described this: . . . Hasse and Noether had somewhat different motivations and aims in their mathematical work. Whereas Hasse is remembered for his great concrete results in number theory, Emmy Noether’s main claim to fame is not so much the theorems she proved but her methods. . . . their

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5 Noether’s International School in Modern Algebra letters show [they shared] a mutual understanding on the basic intellectual foundations of mathematical work (if not to say of its “philosophy”). Both profited greatly from their contact. [Lemmermeyer/Roquette 2006, 8]

Figure 5.2: Emmy Noether, Helmut Hasse, and an unidentified woman, ca. 1930 (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna) Noether reacted to Hasse’s report [Hasse 1926] in a letter from 3 January 1927: I am very interested in your ideas on class field theory. It runs in the direction I’ve always thought about in connection with DedekindWeber (Crelle 92):42 I. formal part, II. Abstract Riemann surface. The formal part – which characterizes an integral domain – is essentially ideal-theoretical; the prerequisites are my five axioms, from which – by transition to the quotient ring – you can also deduce your 6th axiom of the principal ideal property. Here the requirements are wider 42 The goal in [Dedekind/Weber 1882] was to prove the theorem of Riemann-Roch without recourse to analysis, that is by appealing to concepts drawn from the theory of algebraic number fields. Noether analyzed the methods and results of this paper in her review article [Noether 1919a].

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than you assume . . . . On the other hand, I think that you are right with your conjecture in Part II – number fields or algebraic functions of a single variable; here, however, all integral domains must be taken into account. [Lemmermeyer/Roquette 2006, 62] Soon afterward, Hasse began to explore the theory of algebras and their arithmetics, whereas Emmy Noether began to turn her gaze toward class field theory. Around the time that the Noether-Hasse collaboration was warming up, B.L. van der Waerden returned to Göttingen as Courant’s assistant. During 1926/27 he was a Rockefeller fellow in Hamburg [Remmert/Schneider 2010, 193] – a leading center for modern algebra with Emil Artin, Otto Schreier, and Erich Hecke. Artin had agreed to write a book on modern algebra for Courant’s yellow series, beginning with a course on the subject that van der Waerden attended. They planned at first to write the book together, starting from the material van der Waerden extrapolated from his course notes. Before the course had ended, however, Artin dropped out of this project, which would occupy the young Dutchman’s attention for another three years, during which time he worked closely with Emmy Noether. In Göttingen, van der Waerden decided to solidify his knowledge of ideal theory by teaching a course on it. He also attended Noether’s lecture course on the theory of algebras, the same topic he had heard before, but this time presented in a new form based on representations of groups and algebras. Van der Waerden took careful notes from these lectures and together they turned these into a readable text. These were then elaborated in her groundbreaking article [Noether 1929a], published in Mathematische Zeitschrift.43 In the meantime, Noether had struck up a friendly correspondence with the algebraist Richard Brauer, then a private lecturer in Königsberg. Brauer’s research focused on the structure theory of simple algebras, which he studied by means of representation theory and his theory of factor sets. He used factor sets to introduce the structure of an Abelian group, since known as the Brauer group, on the classes of central simple algebras over a given field.44 By 1927, he and Noether, working independently, had arrived at the concept of splitting fields for simple algebras. This led to their joint paper [Noether/Brauer 1927], submitted to the Berlin Academy by Brauer’s former mentor, Isaai Schur. The background behind this publication sheds considerable light on Noether’s role in this story as well as subsequent developments that culminated with the Brauer-Hasse-Noether Theorem [Brauer/Hasse/Noether 1932].45 In an early letter to Brauer, dated March 28, 1927, Noether commented: 43 The contents of this paper were summarized by Nathan Jacobson, who noted its importance for representation theory in [Noether 1983, 17–18]. 44 For a given field k, the Brauer group is defined on equivalence classes of central simple kalgebras, which by Wedderburn’s theorem are isomorphic to matrix algebras over some division algebra. Thus, in effect, the Brauer group classifies division algebras over k; see [Curtis 2003]. 45 This alphabetical order of names, while today quite standard, has not always been adopted. Some authors also add Abraham Adrian Albert to the list (e.g., [Curtis 2007]), whereas the ordering Hasse-Noether-Brauer appears in [Koreuber 2015]. In support of the latter, we cite the final remark of Nathan Jacobson following his discussion of [Brauer/Hasse/Noether 1932]:

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5 Noether’s International School in Modern Algebra I am very glad that you have now also recognized the connection between representation theory and the theory of noncommutative rings or ‘algebras,’ and the connection between the Schur index and division algebras. In regard to these fundamentals our investigations are of course in agreement; but then it seems to me there is a divergence. [There follows a description of four results on splitting fields of central division algebras.] . . . I hope to elaborate on these matters in the course of the summer. The above theorems are not formulated very clearly, and I cannot judge how far you understand them. [Curtis 2003, 667–668].

Noether and Brauer picked up their discussion again in September at the annual meeting of the German Mathematical Society held in Bad Kissingen. On that occasion, he showed her a counterexample to one of her claims. They also discussed plans for their joint publication. Noether was eager to bring Hasse into the picture, which eventually led to their three-way collaboration.

Figure 5.3: Emmy Noether lived from 1922 to 1932 in the attic apartment of this house, Friedländerweg 57, Göttingen (Photo form 1966, Auguste Dick Papers, 12-12, Austrian Academy of Sciences, Vienna) A few weeks after the Bad Kissingen meeting, Noether sent a postcard to Hasse, posing a question she hoped he could answer. Two days later, on 6 October 1927, Hasse already wrote back with the answer: “Your conjecture is correct, though it’s not a direct consequence of my earlier existence theorems. But I prove it with entirely similar methods . . . ” [Lemmermeyer/Roquette 2006, 72]. Noether “One must conclude that the major share of this achievement should be attributed to Hasse” [Noether 1983, 21].

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then wrote to Brauer, sending him Hasse’s letter that solved the question under discussion; she now proposed writing a joint note, to be published together with a note of Hasse. This led to a series of letters within the triangle Brauer-HasseNoether, as they discussed various details about the planned notes and possible generalizations. As Peter Roquette commented: “in Hasse’s letter of October 1927 [we see] the nucleus of what in 1931 would become the Local-Global Principle for algebras in the Brauer- Hasse-Noether paper [Brauer/Hasse/Noether 1932]” [Roquette 2004, 53]. Noether’s evident delight shines through in her letter to Hasse: 46 Your proof brought me much joy; so the matter lies somewhat deeper! I was thinking of a publication in the Reports of the Berlin Academy, where up to now pretty much all short communications on representation theory can be found. I sent a 5-6 page note to R. Brauer – Königsberg so that he could add his part; it should appear under our names . . . . Your proof could then follow immediately afterward as a short note; perhaps with the subtitle “from a letter to E. Noether” so that no textual changes would be needed! What should the main title be and do you agree with my ideas at all? Should the notes be bound together or separately? . . . Hasse decided to rewrite his letter as [Hasse 1927b], which appeared immediately after [Noether/Brauer 1927]. The contents of these two papers were perhaps less important than the new dynamic that Emmy Noether had created by bringing Brauer and Hasse closer together. In the years that followed a close cooperation developed between these three talented mathematicians, whose special expertise blended together remarkably well. The year 1927 also saw the publication of the German edition of Leonard Eugene Dickson’s Algebras and their Arithmetics [Dickson 1927].47 As Günther Frei has described, this book exerted a strong influence on algebraists in Germany, in particular Artin, Brauer, Hasse, and Noether [Frei 2007, 124–129]. After learning about it from Artin, Hasse wrote a lengthy review of Dickson’s book [Hasse 1928]. The following year, he and his colleagues in Halle, Heinrich Wilhelm Jung and Reinhold Baer, organized a lecture seminar on the theory of hypercomplex numbers based on Dickson’s book. In an undated letter to Kurt Hensel, Hasse wrote: “we expect to gain a lot from a thorough study of this beautiful new theory, which will certainly be of crucial importance for the further development of arithmetic” [Lemmermeyer/Roquette 2006, 91]. Hasse soon enriched this theory by introducing algebras over local fields, which proved to be a powerful tool for research on the foundations of local class field theory. Emmy Noether’s connections with Hasse and other mathematicians in Halle were by no means the only ties she cultivated. She and her school were, in fact, 46 Noether 47 On

to Hasse, 19 October 1927, [Lemmermeyer/Roquette 2006, 76]. his work in this field, see [Fenster 1998].

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part of a large network of mathematicians who, in one form or another, were promoting modern algebra in the 1920s and early 1930s. An often overlooked figure was Alfred Loewy, who spent his entire career in Freiburg. He habilitated in 1897, then was promoted to an associate professorship in 1902, before gaining a full professorship in 1919. His teaching career ended abruptly, however, in 1933 when he was forced to take early retirement. Loewy’s position then went to Wilhelm Süss, who soon became one the most influential mathematicians in Germany.48 Two close associates of Emmy Noether, Wolfgang Krull and Friedrich Karl Schmidt, began their careers working under Loewy, and three others with close ties to Noether also studied under him in Freiburg: Ernst Witt, Richard Brauer, and Reinhold Baer. Witt later became one of Noether’s star students in Göttingen, Brauer was one of her most important collaborators, and Baer became part of the Noether-Hasse network in 1928 when he joined Helmut Hasse as his assistant in Halle. There they published a new edition of Steinitz’s classic study on field theory [Steinitz 1930] together with an appendix on modern Galois theory, which appeared in the same year as volume 1 of van der Waerden’s Moderne Algebra [van der Waerden 1930/31]. Noether and Hasse were again both on the program for the DMV conference in 1929, which convened in Prague. Noether’s lecture topic was briefly summarized in [Noether 1929c], where she noted that a detailed account would follow in Mathematische Annalen. She had prepared such a manuscript well before the Prague conference, but never found the opportunity to revise it for publication. Her former student, Heinrich Grell, obtained this manuscript or a copy thereof, which he submitted to Hasse after the Second World War had ended. Hasse, as editor of Crelle, published this sketch in [Noether 1950]. Nathan Jacobson underscored the originality and importance of this posthumously published paper in [Noether 1983, 15–16]. Hasse’s remarkable lecture in Prague, mentioned already in the Preface, surely warmed Noether’s heart. His message was the one she had long believed in and steadfastly promoted, and he duly noted the leading role she played in making applications of ideals a central feature of modern mathematics. “It is thanks to the rich and beautiful successes of E. Noether and to her tireless enthusiasm in writing and teaching that this concept, along with the concept of fields, today ties together through its methodological band the various areas of algebra into a unified whole” [Hasse 1930, 32–33]. He ended with these telling remarks: One must not forget that the algebraic method is only a method, and thus to apply it one needs substance. I mean by that: ideal theory, for example, would have never come about of its own accord through an interest in the definition of an ideal, but rather it required the concrete problems posed by algebraic number theory. And so it is with all of modern algebra. This is why abstract algebra in the long run cannot exist apart from concrete mathematical theories any more than the 48 On

Loewy’s career and mathematics in Freiburg, see [Remmert 1995].

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latter can assert themselves in a lasting way without the systematizing and energizing effects of abstract algebra. [Hasse 1930, 34] Following Hasse’s impassioned performance, Oskar Perron, van der Waerden, and Ernst Zermelo offered comments. Zermelo recommended dropping the adjective “abstract” in favor of “general” field or group theory, in particular because some representatives of the older tradition use the former term pejoratively. 49 Considering that Zermelo’s comment was made in the year 1929, it should come as no surprise that “abstract algebra” later became a principal target for proponents of “Deutsche Mathematik” during the Nazi years.50

5.7 Noether’s “Wish List” for Favorite Foreigners A year had now passed since Noether’s arrival in Moscow, where she and Alexandrov hoped to import her new brand of algebra. She wrote him in August 1929 with various local news, in particular about how Hilbert had engaged Stefan Cohn-Vossen to turn his lecture course on “Anschauliche Geometrie” into a book for Springer’s yellow series. She knew how enthusiastic Alexandrov was about this informal approach to topics in geometry, and so did Hilbert, who asked Alexandrov if he would write a short appendix on elementary problems in topology. He tried, but found it difficult to write something so compressed, so instead he wrote a short monograph with a preface by Hilbert, entitled Einfachste Grundbegriffe der Topologie, which was later published in [Alexandroff 1932a] alongside the volume [Hilbert/Cohn-Vossen 1932]. During the summer of 1929, Alexandrov had struck up a new friendship with Andrey Kolmogorov, another great mathematician with a deep love for nature. 51 Together they took a long boat trip along the Volga, then journeyed further in the Caucuses and Crimea as well as the south of France. Alexandrov reported on these adventures in a letter to Emmy Noether that she answered on 13 October 1929: Your travel descriptions are fantastic – I have never been so aware of the mixture of East and West in you as in this letter! And basically this summer gave you much more than the usual Göttingen summers! We all expect you next summer as a matter of course, and trust to your skills in overcoming all difficulties. I’m really looking forward to the “algebraic methods in general topology”. The final chapters of the book will probably be in this direction as well? I can easily imagine that the group-theoretic approach . . . allows for a much easier transfer to closed sets than using matrix calculations. Is that not so? 49 Cited

in Jahresbericht der Deutschen Mathematiker-Vereinigung 39: 17. “Deutsche Mathematik,” see [Mehrtens 1987] and [Segal 2003, 334–417]. 51 For Kolmogorov’s recollections of these times together with Alexandrov, [Shiryaev et al. 2000, 145–157]. 50 On

see

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After this, she went on to tell Alexandrov something about what had taken place in Prague at the annual conference of the German Mathematical Society, writing about “die Algebra” allegorically (for the missing portion of this letter, see Section 4.4): . . . Prague showed that there is great interest in topology. . . . Algebra was much more strongly represented; in the evenings she was sitting in thick piles, and there were still some missing, e.g. Krull because of his honeymoon. Artin, on the other hand, appeared with his young wife, a Russian math student whom Artin brought up on ideal theory!52 Talking math with Artin is always a pleasure. Algebra has altogether been busy getting married this year, starting with Reinhold Baer – the millionaire without millions,53 as Zermelo has called him . . . . Van der Waerden was the last; he married in late September and had to start lecturing in early October. In his thoughts, though, he is not yet focused, as he writes.54 The Galois theory is ready and will be typed next week;55 so you will get it in time. Tell me how your lectures are going and whether, unlike me, you have already won over the masses in Moscow!56 Noether also inquired about the short note she had submitted during her stay in Moscow (published as [Noether 1929b]), and mentioned that she had recently sent six offprints of her lengthy paper on hypercomplex systems ([Noether 1929a]). She also passed on the recent news that Hasse would succeed Hensel in Marburg, so she hoped this would lead to future guest lectures there. Emmy ended her rambling letter with a teasing remark made by Alexandrov’s former landlady: “Did I write to you how, at van der Waerden’s engagement party, Frau Bruns declared: ‘Now Alexandrov cannot be left alone.’ So take it to heart!” Noether had already written to Hasse one week earlier. She sent him her congratulations, but also let Hasse know that she hoped his appointment would open up new opportunities for two gifted foreigners. One of these was her student Jacob Levitzki, who was born in the Ukraine but grew up in Tel-Aviv after his family left for Palestine in 1913. In 1922, after graduating from the Gymnasium in Tel-Aviv, he took up studies in Göttingen, where he completed his doctorate under Noether and Landau in 1929.57 In her letter, she expressed hopes that Levitzki might be considered for a position as Hasse’s assistant. “He has presently returned home for the first time in seven years; but it is very possible he will come 52 Artin

married Natascha Jasny on August 29, 1929. wife, Marianne Kirstein, was the daughter of a (perhaps not so wealthy) bookseller in Leipzig. 54 In October 1928, van der Waerden was appointed professor of geometry in Groningen; he remained in that position until 1931 when he moved to Leipzig. 55 Probably a reference to notes from one of Noether’s lectures. 56 Translated from [Tobies 2003, 104]. 57 Noether’s report on Levitzki’s dissertation is transcribed in [Koreuber 2015, 323–324]. 53 Baer’s

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back in the winter, and he will surely come if he has prospects for employment.” She then went on to ask: The other question is whether it would be possible to put Alexandrov on the list in Halle. I know that Alexandrov very much wishes at some time to come to a German university. And since all his work is published in German journals – or even in America in the German language – perhaps having a foreigner there would not be too difficult, especially since his scientific importance is undisputed. In addition, he is now writing his “Topologie” for the yellow series; and finally, for the two summers when he was a guest lecturer in Göttingen he received a salary from the government, even with tax deductions. Besides his topology course, he is also teaching Galois theory this winter, of course modern; and he always works intensively with his people in seminars. Works by some of his students have already appeared in the Annalen, and more will come. You will know that he has complete and perfect command of the German language. That’s the end of my wishes for foreigners!58 She wrote Hasse again on 13 November apropos Levitzki, but also to ask whether Hasse planned to attend the ceremony for the opening of the new Mathematics Institute in Göttingen, which would take place on December 3. If Levitzki had no chances in Marburg, she planned to buttonhole whomever she might bump into at that gathering. Eventually, Noether wrote to Øystein Ore at Yale University, with whom she co-edited Dedekind’s Collected Works [Dedekind 1930–32], and Ore arranged for Levitzki to spend two years at Yale on a Sterling Fellowship. In 1931, he returned to Palestine as professor of mathematics at Hebrew University, where he lay the groundwork for algebraic research in Israel (see Section 9.6).59 Hasse’s eventual successor in Halle was Heinrich Brandt, who later became a sharp critic of Noether’s abstract style (see Section 9.4). This particular episode represents only one instance among many. Everyone knew that Emmy Noether took a deep interest in the professional development of all her associates, but especially her doctoral candidates. Although she never held a regular university professorship, her success in promoting the careers of others was truly remarkable. Even as a post-doc in Erlangen, where she had no official position whatsoever, she served as adviser to two students who did their thesis work under her.60 Nevertheless, her larger influence extended far beyond the immediate circle of her doctoral students. One of the first to benefit from her support was Landau’s student, Werner Schmeidler. Already in early 1920 she 58 Noether

to Hasse, 7 October 1929, translated from [Lemmermeyer/Roquette 2006, 92]. Landau gave a speech in Hebrew at the opening ceremony in 1925; it was rumored afterward that he learned his Hebrew from Levitzki; for background see [Corry/Schappacher 2010, 455–456]. 60 For detailed information on the work of Noether’s doctoral students, see [Koreuber 2015, 159–195, 310–336]. 59 Edmund

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took full advantage of her connections within the far-flung Göttingen network to promote his future career. The circumstances involved vacancies at two provincial outposts, Kiel University and the recently founded Breslau Institute of Technology. One should keep in mind that the number of permanent positions in mathematics at this time was very small, numbering no more than 200 at all institutions of higher education throughout the German states. It was a tiny world. In 1920 Otto Toeplitz, who had recently been elevated to a full professorship in Kiel, sought the advice of Klein and Hilbert regarding a vacancy there. Toeplitz also wrote to Eduard Study in Bonn, who advised him to consult the “Old Testament,” after which he named several qualified individuals, all of whom happened to be Jewish. The two he mentioned first were Emmy Noether and Arthur Rosenthal, neither of whom were named by Klein or Hilbert.61 Eight years later, Noether was also briefly considered for this very same position in Kiel, but no one there or elsewhere in Germany had a serious interest in promoting her candidacy for a professorship [Siegmund-Schultze 2018]. Back in 1920, the position in Kiel eventually went to Ernst Steinitz, who was a colleague of Max Dehn at the Breslau Institute of Technology. Dehn and Toeplitz were both protégés of Hilbert, though it remains unclear whether they were personally acquainted with Noether at this time. Dehn was briefly an associate professor in Kiel, where he was succeeded by Toeplitz in 1913. During his tenure in Kiel, Dehn took on a student named Jakob Nielsen, whom Noether apparently also knew, at least by reputation. This all becomes clear from a letter Emmy Noether wrote to Max Dehn on 8 January 1920, in which she answered a no longer extant letter from him. Dehn must have anticipated that Steinitz would soon be leaving Breslau for Kiel, so he wrote to Noether asking her opinion of Schmeidler’s abilities. After receiving her positive report, he informed her that it was very helpful, though he did not plan to include Schmeidler’s name on the short list of candidates. Noether then replied, in turn: I had already heard that Schmeidler will not be on the Breslau list, but I’m pleased that you have a favorable impression of him based on my report. Meanwhile, Toeplitz has offered to let him re-habilitate in Kiel and to take on a large teaching contract (with an inflation allowance of around 9,000 Marks). He will no doubt accept, if certain conditions (per diem expenses until he has found a place to live and similar things) are met. You surely know that Toeplitz had originally turned to Nielsen. 62 This kind of unusually strong personal engagement for young talent was altogether typical for Emmy Noether, who obviously took a strong interest in Schmei61 Study to Toeplitz, 1 July 1920, quoted in [Koreuber 2015, 40]. There was widespread awareness that many talented Jewish mathematicians had been passed over during the past decade, a circumstance that placed Toeplitz in an awkward position in conducting this search, since he, too, was of Jewish background; his correspondence with Klein is discussed in [Bergmann/Epple/Ungar 2012, 206–207] and in [Rowe 2018a, 349–350]. 62 Noether to Dehn, 8 January 1920, Dehn Papers, Dolph Briscoe Center for American History, University of Texas at Austin.

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dler’s personal situation. Still, it is quite astonishing in this context, considering that it was only a half-year earlier that she herself had been able to habilitate. What transpired not long afterward suggests that Noether was not just well connected; her opinions carried weight, too. Schmeidler and Steinitz both accepted the offers from Kiel, and Dehn then brought Nielsen to Breslau, though he would leave for Copenhagen after just one year. Dehn also decided to leave Breslau in 1921 when he was offered a prestigious chair in Frankfurt. In all likelihood, before departing he acted on Noether’s advice in helping to choose his successor, who was none other than Werner Schmeidler. When another position became vacant in Breslau just one year later, it went to Fritz Noether, whose sister afterward often visited him there during holidays. Pavel Alexandrov was probably never seriously considered as a candidate for the position in Halle or any other professorship in Germany. He kept coming back to Göttingen, though, joined in May 1930 by Kolmogorov. During that summer semester Alexandrov offered a lecture course on his new combinatorial theory of dimension, his latest effort in attempting to fill that “deep gorge separating general (set-theoretic) and classical topology” that he described to Hausdorff in his letter from July 4, 1926 (see Section 4.8). Two years later, during his stay in Princeton, Alexandrov felt he was narrowing in on this goal: I’m personally more and more interested in a combinatorial construction of dimension theory; it is truly surprising how many – by appearances set-theoretic properties of geometric figures – are ultimately of combinatorial origin and in a different sense than I thought just a short time ago; completely unexpected connections arise along this path, sometimes as new results, often as problems, with ties even to the theory of knots.63 A few weeks before he began his Göttingen lectures, Alexandrov provided Hausdorff with a preview in a lengthy letter from March 30, 1930 [Hausdorff 2012, 92–96]. After much searching, he had now found a way to absorb the usual dimension theory of Brouwer-Urysohn-Menger into his combinatorial structures in such a way that the former can be derived as a limiting case. Back in Moscow, he turned his lecture notes into one of his major papers, which he announced with much fanfare: It was only by applying the combinatorial homology-concept to settheoretic constructs that we were able to recognize that dimension theory is not itself a theory but rather merely the first paragraph in the incipient general investigations of boundary- and intersection- (in particular link-) constructions with closed sets . . . . Only from this standpoint do we attain a proper perspective that leads to a natural and rich field of problems. . . . this foundation . . . means however – as so 63 Alexandrov

to Hausdorff, 20 April 1928, [Hausdorff 2012, 56].

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5 Noether’s International School in Modern Algebra often – the end of an independent theory, which passes into a more comprehensive chapter of mathematics. [Alexandroff 1932b]64

We do not know what Emmy Noether actually thought of this work, but it is easy to imagine her smiling and nodding in agreement. What we do know is that Felix Hausdorff, who had long harbored a deeply skeptical view with regard to combinatorial topology, began to express serious interest in these newest developments. After reading the proofs for [Alexandroff 1932b], he wrote Alexandrov to express his admiration for the way in which he and others “move about so securely in the upper reaches of this skyscraper, whose foundations seem not really trustworthy,” at least not based on works he had read. Still, he found this whole field enticing and so he awaited the work of Alexandrov and Hopf with great anticipation. 65 Around this same time, Hausdorff began moving into this field himself, though quietly, while also maintaining contact with Alexandrov. In the summer semester of 1933, he offered a 2-hour lecture on algebraic topology, for which he prepared a manuscript of 100 pages [Hausdorff 2008, 893–976].66 Alexandrov saw Hausdorff for the last time in Locarno in 1932; their correspondence apparently came to an end in 1935. Years later, Alexandrov thought back fondly of the summer of 1930, when he and Kolmogorov enjoyed the frequent convivial gatherings and socializing with various Göttingen mathematicians. He remembered meeting . . . most of all with Courant (and I also met Emmy Noether very often), but also with Hilbert and Landau. We were sometimes invited to Landau’s house for a (usually very grand) supper, one feature (and the main attraction) of which was an enormous dish of lobster. The guests were asked to demonstrate their skill at eating these arthropods. Kolmogorov was awarded first prize, as he managed his portion of lobster without even once touching it with his hand, and using only knife and fork. [Alexandrov 1979/1980, 327]

5.8 Paul Dubreil and the French Connection Probably Alexandrov saw Noether less often that summer than he remembered, as she was then teaching in Frankfurt, having switched places with Carl Ludwig Siegel. Her student, Max Deuring, attended Siegel’s course on analytic number theory and was assigned the challenging task of writing up a readable manuscript from these lectures using the notes he took. On learning this, Emmy Noether sent him a postcard in which she remarked: “it is very nice that you are preparing the Siegel lectures; then I can read his breakneck proofs in peace in the 64 Alexandrov’s two main theorems are the principal results in the concluding chapter of [Hurewicz/Wallman 1948]. 65 Hausdorff to Alexandrov, 7 October 1931, [Hausdorff 2012, 112]. 66 On Hausdorff’s growing interest in algebraic methods in topology, see [Scholz 2008].

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winter, which I much prefer to hearing them” [Lemmermeyer/Roquette 2006, 92]. Siegel was a leading authority on analytic number theory and, in later years at least, a sharp critic of abstract algebra. Since Noether approached number theory entirely from the perspective of algebra, her efforts were aimed at finding algebraic constructs that could be substituted for the classical analytic tools. The latter had first been developed by Gustav Lejeune Dirichlet and Bernhard Riemann in their studies of the properties of prime numbers.67 Thus, Dirichlet had introduced L-series to prove that there are infinitely many primes in every arithmetic sequence,68 whereas Riemann extended Euler’s ζ-function to complex values in studying the distribution of primes. The Riemann conjecture regarding the zeroes of the ζ-function has long been one of the outstanding unsolved problems in mathematics. As mentioned below, Helmut Hasse may have ventured to solve an analogue of this conjecture (later proved by André Weil). Emmy Noether, on the other hand, rarely exerted her energies trying to solve specific problems: her forte was forging new tools for building abstract theories. She found Dedekind’s works particularly insightful insofar as they inspired her urge to algebraicize number theory by striping away everything that depended on analysis. During her summer in Frankfurt, Alexandrov did come over to visit Noether and to speak in the mathematics seminar. One of those who heard him lecture was the young French mathematician Paul Dubreil, who also recalled memorable lectures delivered by two other guests: Louis Mordell spoke on number theory and Wolfgang Krull on non-noetherian rings. Dubreil was an important link in the chain connecting Paris and Göttingen, but especially for the “French connection” running from Emmy Noether to the founding figures of the Bourbaki group. His recollections of his years abroad contain numerous interesting anecdotes that nicely capture the atmosphere from that time (see [Dubreil 1982] and [Dubreil 1983]). In 1929/30 Dubreil was a Rockefeller fellow studying with Artin in Hamburg; this was the first of five locales he would visit over the course of 15 months, the others being Groningen (van der Waerden), Göttingen, Frankfurt, and Rome. During parts of this journey he was accompanied by Marie-Louise Jacotin, a fellow normalien whom he married in June 1930. Before their marriage, she had a scholarship from the École Normale Supérieure that took her to Oslo, where she worked on fluid dynamics under Vilhelm Bjerknes. In Hamburg, Dubreil was swept into Artin’s world, which centered around informal group discussions that took place both before and after Artin’s lecture course, which met twice a week. In February 1930, Emmy Noether came to Hamburg, having received an invitation from its Philosophical Society. Dubreil got to meet her over lunch following Artin’s lecture, and that evening he went to hear her speak in the university’s 67 In a letter from 1846 to Alexander von Humboldt, C.G.J. Jacobi informed him that “. . . Dirichlet has created a new mathematical discipline by applying the infinite series that Fourier had introduced for heat conduction to study the properties of prime numbers” (quoted in [Scharlau/Opolka 1980, 135]. 68 More precisely, sequences {a + bn|n ∈ N}, where a and b are relatively prime.

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aula, where she addressed a large audience composed of philosophers and mathematicians. In a letter to Marie-Louise Jacotin, he afterward conveyed his “total disappointment”: I saw a short, corpulent woman with a ruddy complexion and dressed without any elegance. Obviously very intelligent, she spoke non-stop, very quickly, in a jerky manner. She fell headlong into the trap that this conference was for philosophers and mathematicians: incomprehensible to the first, she was breaking open doors for the others. And all in a big mess: definition of a field by the usual system of axioms, equivalence classes, isomorphisms, theory of ideals with Teilerkettensatz (axiom . . . now called the ascending chain condition), return to fields to say a few words about real fields, no doubt to pay homage to Artin. [Dubreil 1983, 64] He later came to realize, after he got to know Noether personally, that this talk was meant to illustrate two principles central to her work: the construction of concepts (Begriffsbildung) and the formulation of increasingly strong axiom systems within the same theory, thereby yielding more and more precise results. The day Dubreil left Hamburg, he went to Artin’s office to say good-bye and thank him for all his help. This prompted a quite unexpected and memorable exchange. “Oh no,” Artin responded, “I have helped you so little in your personal work!” Dubreil: “But, Herr Professor, I had to do my thesis on my own!” Artin then insisted that he tell him about his work dealing with a rigorous general treatment of the “multiplicity of (Max) Noether,” which Dubreil described as a method consisting of three-quarters part ideal theory and the rest analytical-geometrical techniques à la Puiseux. It took him almost an hour to explain this, and then to finally say good-bye [Dubreil 1983, 65]. After spending some time in Groningen with van der Waerden, Dubreil went on to Frankfurt, where Noether was teaching during the summer semester of 1930. After greeting him, he was in for his next surprise. “By the way,” she said, “you are speaking in the seminar next week.” Dubreil looked at her, taken aback. “Yes, about your results on the multiplicity of Noether” she added. He paused to ask: “And, in German?” “Yes, yes of course!” And so Dubreil got straight to work, feeling a little comforted that he had managed to give Artin an improvised explanation of his work before his departure from Hamburg. Moreover, the relaxed atmosphere in Frankfurt appealed to him right from the start. He got to know Wilhelm Magnus and Ruth Moufang and generally felt “supported by a kind of sympathetic prejudice that seemed to float in the air!” After giving his presentation, Dubreil remembered Emmy Noether’s comment that he had treated a problem of her father with the methods of his daughter, and Max Dehn joked about the terminology (Unterresultante) that was the key to his main result. The weather that day was splendid, and so the group made an Ausflug to the Taunus, where Dehn loved to hike. They surely made a stop at the Fuchstanz restaurant for cake and coffee; perhaps this was the day Dehn’s daughter Maria remembered,

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when Emmy ordered a second piece of Torte, laughing that it couldn’t hurt her figure anyway [Dubreil 1983, 66]. Like nearly everyone else, Paul Dubreil found Noether’s lectures difficult to follow; she spoke very fast and her presentations were untidy, so he had his troubles trying to take notes. Still, she frequently returned to important concepts, and so he eventually began to feel at home with her way of thinking. He recalled how one day, though, he was having difficulty with a certain assertion which seemed to him unsubstantiated by her methods, but which he could prove without difficulty using matrices. So after the lecture, he showed her his calculation – knowing full well her strong aversion to matrix methods. After a few moments reflection and with a bit of skillful juggling with her modules, she showed him how the matter was all very clear [Dubreil 1983, 67–68].

Chapter 6

Emmy Noether’s Triumphal Years 6.1 The Marburg “Schiefkongress” When Emmy Noether returned from the September 1929 conference in Prague – where she and Hasse surely spoke about their mutual mathematical interests – she belatedly answered a postcard he had sent here. He was interested to find out what she knew about the relationship between hypercomplex algebra and class field theory. She began by saying, “not much and mainly just formal,” and then proceeded to write a long letter describing eight points and then ending abruptly: “Now make what you want of these fantasies” [Lemmermeyer/Roquette 2006, 90]. By this time, Noether’s and Hasse’s research interests were beginning to converge. After receiving his manuscript for [Hasse 1931b] – on the arithmetic in ℘-adic skew fields1 – she wrote back 25 June 1930 to express the great pleasure it gave her, but also to point out connections with a number of other recent works. 2 Noether served as an unofficial editor for Mathematische Annalen, so her remarks were aimed at improving the text before it appeared in print. Since Otto Blumenthal, its managing editor, was attending a mathematical congress in Kharkov, she planned to forward Hasse’s manuscript to Blumenthal two weeks hence. In September, Hasse received a letter from Emil Artin that described his most recent work on L-series and his concept of conductors (Führer) in class field theory. A few weeks later, Hasse forwarded Artin’s letter to Noether, who responded on 10 October: “Many thanks for Artin! These things are really beautiful! I am particularly drawn to the formal foundations; in the meantime, quite independently, I have been considering some hypercomplex ideas. . . ,” which she proceeded to elab1 A skew field (Schiefkörper) satisfies all the axioms for a field except one: multiplication need not be commutative. A standard example is Hamilton’s quaternions. 2 As discussed in [Roquette 2004, 23–25], this paper contains a fundamental result that Hasse used for the principal theorem in [Brauer/Hasse/Noether 1932], namely the local-global principle. Here it was applied to a valuation ideal D℘ in a non-commutative division algebra, hence the peculiar notation using the Weierstrass ℘ symbol, as explained in [Roquette 2004, 37].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_6

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orate. These ideas, she conjectured, could be connected with Artin’s, though she called this a “fantasy of the future” [Lemmermeyer/Roquette 2006, 97]. Since she also saw connections with Deuring’s work, she showed him Artin’s letter before returning it to Hasse. Toward the end of 1930, Hasse formulated some far-reaching conjectures on simple algebras over number fields, including the assertion that they are all cyclic.3 When he sent these to Noether, her first reaction was disbelief; in fact, when she responded on 19 December, she thought that a recent paper by A.A. Albert provided counter-examples. “Yes, it is a shame,” she wrote, “that all your beautiful conjectures are merely floating in the air and not standing with firm feet on the ground: because some of them – how many I cannot oversee – crash beyond saving due to counterexamples in a completely new American work by Albert [Albert 1930] . . . ” [Lemmermeyer/Roquette 2006, 99]. Only five days later, however, on December 24, she wrote again with a pater peccavi (Father, I have sinned against heaven and before you). “Your castles in the air (Luftschlösser) have not fallen at all,” she wrote, because what she had originally read into [Albert 1930] was, in fact, pretty much the opposite of what he had proved. Hasse had sent her an example that made this clear to her. She then proceeded to summarize the results in three of Albert’s recent papers, and then concluded: “So you can see that your question about when direct products of skew fields are again such is not an easy one. This is always the case when the degrees are relatively prime, as Köthe and Brauer have proven; you may already know it.” She ended by sending “good wishes for Christmas and New Year, for people and skew fields” [Lemmermeyer/Roquette 2006, 102]. In January 1931, Heinrich Brandt, Hasse’s successor in Halle, invited Noether to give a talk there. She gladly accepted, proposing to speak about the relations between hypercomplex systems and commutative algebra. Since she realized that such a topic could not be easily digested in a single two-hour lecture, she proposed to hold a follow-up meeting the next day, “a kind of colloquium, where I could give explanations, or possibly elaborate on the foundations, or similar, more elemental things” [Jentsch 1986, 6]. It was probably during her visit in Halle that Noether met the talented young French mathematician Jacques Herbrand for the first time. He was then on a Rockefeller fellowship in Berlin, and she very likely informed him in advance about her lecture there. Hasse sent Noether another postcard with queries in early February 1931, and again she answered him with a long letter that began in a humorous vein. 4 She had lots of hypercomplex things to tell him – more perhaps than he really cared to know. She began with a brief synopsis of key results in the forthcoming dissertation of her student Hans Fitting. She noted that his general structure theorem was in the hypercomplex case simply the conceptual interpretation of a theorem found in the German edition of Dickson’s book [Dickson 1927, 130]. 3 A cyclic extension is one in which the Galois group is cyclic, thus generated by a single element. 4 Noether to Hasse, 8 February 1931, [Lemmermeyer/Roquette 2006, 103–104].

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Although she conceded that Fitting’s work was influenced by her approach, she emphasized that he had conceived both the question and the method for answering it himself. Her report on his dissertation [Koreuber 2015, 316–317] indicates that she had earlier found special results in the same abstract direction and published these in [Noether 1929a].5 Fitting habilitated in Königsberg in 1936, but died two years later from bone cancer. Noether’s letter also contained references to some new results that appeared in volume 2 of van der Waerden’s Moderne Algebra [van der Waerden 1930/31], which she was then reading in page proofs. This confirms, again, that she was thickly involved in this book project, just as she was in the case of Max Deuring’s monograph [Deuring 1935].6 Moderne Algebra was perhaps the most successful of all the works published in Courant’s “yellow series,” which he and Ferdinand Springer had launched immediately after the war. This series itself was a novelty at the time, as it began largely by transforming oral knowledge from the KleinHilbert era in Göttingen – which had in the meantime been accumulated in official lecture transcriptions – into internationally accessible knowledge in printed form (see [Rowe 2018a, 351–354] and [Remmert/Schneider 2010, 169–173]). A third matter of interest that Noether raised concerned plans for a forthcoming meeting in Marburg, the so-called “Schiefkongress” that would take place there from 26–28 February. She and Hasse had surely discussed this already on January 13, when Hasse came to deliver a lecture before the Göttingen Mathematical Society. Its title was simply “On Skew Fields (Schiefkörper),” which gave him the opportunity to present his conjectures on skew fields or algebras over number fields, in particular his claim that any central simple algebra over a number field is cyclic.7 Noether had at first been skeptical regarding these conjectures, but she quickly became convinced that Hasse’s hunches were probably correct; so they both set off in a quest to prove them. This would culminate near the end of the year when they and Richard Brauer proved the Brauer-Hasse-Noether theorem [Brauer/Hasse/Noether 1932] (see Section 6.3). Emmy was not quite a one-woman organizing committee, but she gave Hasse a clear idea of what she had in mind for the forthcoming “Schiefkongress” in Marburg, where she would speak about hypercomplex structure theorems with applications to number theory. She was hoping to have something new to say by then about a theory of conductors that generalized Artin’s version, but if 5 The later publication [Fitting 1932] goes beyond the original dissertation, which contained various partial results. His central idea was to analyze the structure of a general ring satisfying acc and dcc by its corresponding ring of endomorphisms [Lemmermeyer/Roquette 2006, 105]. 6 The latter appeared in Springer’s series Ergebnisse der Mathematik und ihrer Grenzgebiete, which began in 1932. 7 See [Fenster/Schwermer 2007]; a central simple algebra (CSA) over a field K is a finitedimensional simple associative algebra A, for which the center is exactly K. For example, the quaternions H form a 4-dimensional CSA over R. The claim that A is cyclic means that there exists a maximal field extension L with K ⊂ L ⊂ A, where the Galois group of L/K is a cyclic group. Hasse’s conjecture concerned the case where K is an algebraic number field, thus a finite extension of Q.

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Figure 6.1: Marburg “Schiefkongress,” 26–28 February 1931: Back, l. to r.: Ralph Archibald, Max Deuring, Hans Fitting, Gottfried Köthe, Heinrich Brandt; Front, l. to r.: Ernst Zermelo (?), Heinrich Grell, Emmy Noether, Jacques Herbrand, Richard Brauer (Auguste Dick Papers, 12-4, Austrian Academy of Sciences, Vienna)

that was not quite ripe for public display she would talk about the applications Hasse already knew. She also recommended the following order for the first four lectures: Brauer, Noether, Deuring, Hasse. Her reasoning for this was that such an arrangement would enable speakers to take full advantage of the preceding talks, whereas she imagined the other presentations would be independent contributions. Noether also asked Hasse to invite Jacques Herbrand, who was then in Berlin working with John von Neumann. She noted that he was on the first leg of his Rockefeller fellowship, which would later take him to Artin in Hamburg and finally to Noether in Göttingen. She had already met him in Halle and reported that he had understood her latest ideas better than anyone else there. Calling this gathering – really only a small workshop – a “congress” was, of course, meant as a joke. For insiders, skewed fields (“Schiefkörper”) stood at the center of the “Schiefkongress,” for the rest of the German-speaking world, this would have sounded like a big conference that went badly (schiefgehen). Despite its small scale, during this era mathematics meetings were still quite rare, which explains why a short report about this one was published in the journal of the German Mathematical Society. Hasse ultimately arranged the program differently

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than Noether had in mind. He himself gave the first and only talk on Thursday afternoon, February 26, when his topic was “Dickson skew fields of prime degree.” Peter Roquette, who studied Hasse’s manuscript, noted that he spoke of cyclic algebras of prime degree rather than Dickson algebras in this talk. The main substance was based on results in [Hasse 1931b] and the Hilbert-Furtwängler norm theorem from class field theory. Hasse ended his talk by stating a number of open problems in keeping with the spirit of the workshop [Roquette 2004, 59]. 8

6.2 Rockefeller and the IEB Program Jacques Herbrand was invited to attend the Marburg workshop, as can be seen from the group photo (Fig. 6.1). Later that summer he would present two talks in Emmy Noether’s seminar, which was attended at this time by three other mathematicians from France. We have already alluded to this “French connection” in Section 5.8, following the vivid recollections of Paul Dubreil [Dubreil 1982, Dubreil 1983]. More familiar still are the ties linking Noether to the Bourbaki group, founded in 1935, the year of her death. As mentioned earlier, two of Bourbaki’s founding members, André Weil and Claude Chevalley, had attended her lectures in Göttingen. Weil was an especially avid traveler in his youth [Weil 1992, 45–60]. He came to Göttingen on a Rockefeller fellowship in 1926/27, during which time he interacted with many people, including Courant, Noether, van der Waerden, and Alexandrov. Before coming to Göttingen, Weil had studied in Rome with Volterra, Severi, and Zariski. His Rockefeller stipend first brought him to Berlin, where he met Heinz Hopf and Erhard Schmidt, but he especially liked to visit Frankfurt, home to Carl Ludwig Siegel, Max Dehn, and Ernst Hellinger. These were well-known, in some cases even famous names, but the mathematical world of the 1920s was still very small. Many of the talented young post-docs on IEB fellowships spent at least part of their time in Göttingen, Rome, Princeton, or Paris. The main purpose of the IEB program, which was financed by the Rockefeller Foundation, was to stimulate high-level research by promoting international ties and exchanges. Although the outcomes varied a good deal, within the realm of mathematics one can point to several cases – like those of van der Waerden, Alexandrov, and Dirk Struik (see Section 8.3) – where the IEB helped to launch successful careers. It clearly helped Weil as an itinerant mathematician to connect with a host of interesting personalities, and he snatched up new ideas wherever 8 The next day there were five talks: R. Brauer (Königsberg) Galois theory of skew fields; M. Deuring (Göttingen) Application of non-commutative algebra to norms and norm residues; E. Noether (Göttingen) Hypercomplex structure theorems and number theoretic applications; R. Archibald (Chicago) The associativity conditions in Dickson’s division algebras; H. Fitting (Göttingen) Hypercomplex numbers as automorphism rings of abelian groups. Finally, on Saturday there were two last talks: H. Brandt (Halle) Ideal classes in the hypercomplex realm; G. Köthe (Münster) Skew fields of infinite degree over the center.

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he went. During his stay in Rome, he heard a lecture by a Rockefeller fellow from the United States, a no longer young woman named Mayme Logsdon. She was an algebraist who had studied under L.E. Dickson, and is remembered today as being the first woman to receive tenure from the mathematics department at the University of Chicago.9 It was from her that Weil learned of Louis Mordell’s article from 1922, which answered Poincaré’s query about the rank of the group of rational points on an elliptic curve.10 Weil would afterward generalize this result by proving what is today called the Mordell-Weil theorem, which opened the way to a new modern discipline: arithmetic algebraic geometry.11 One of Weil’s friends from his student days at the École normale in Paris was Paul Dubreil, who was working on his doctorate when Weil returned from Göttingen. Dubreil was then unknowingly following in van der Waerden’s tracks, trying to prove a general form of Max Noether’s F = Af + Bφ theorem, and he felt seriously stuck. He later recalled what happened some time in 1927: These were my tribulations when I had a conversation with my friend André Weil, who had just returned from a visit to Göttingen (. . . as a Rockefeller Fellow) where he had met Emmy Noether (daughter of Max Noether) and van der Waerden. He pointed out to me that I was immersed, without knowing it, in the theory of ideals of polynomials, and he advised me to read van der Waerden’s “Zur Nullstellentheorie der Polynomideale” [van der Waerden 1927] and introduced me to the fundamental work of Emmy Noether and of Wolfgang Krull. Reading these works, which were clear and rich in new ideas, gave me enthusiasm. In July 1928, my thesis was almost finished . . . . [Dubreil 1982, 79] For van der Waerden, if he ever heard the story, this would surely have struck him as déjà vu, except that this time it took place in Paris, rather than in Göttingen when he spoke with Noether. Mathematical ideas nearly always spread fastest when traveling by word of mouth. Having recounted Paul Dubreil’s encounters with Emmy Noether in Hamburg and Frankfurt in Section 5.8, we pick up the thread of that story before he and Marie-Louise Jacotin-Dubreil arrived in Göttingen. Dubreil defended his dissertation in October 1930 in Paris, after which he and his wife traveled to Rome. He had received an extension of his Rockefeller fellowship to work with Federigo Enriques, while Jacotin-Dubreil continued her work on hydrodynamics with Tullio Levi-Civita. His recollections of these months in Rome bring out very clearly that both Enriques and Severi were keen to exploit the algebraic results of 9 Among the 130 mathematicians on Reinhard Siegmund-Schultze’s list of recipients of IEB/RF fellowships, Logsdon was one of four women [Siegmund-Schultze 2001, 124–125; 288– 301]. 10 Weil mentions this incident in passing; he could no longer remember the name of the American woman who cited Mordell’s paper in a paper she distributed, [Weil 1992, 49]. 11 The Mordell-Weil theorem states that for an abelian variety A over a number field K the group of K-rational points of A is a finitely-generated abelian group.

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Noether and van der Waerden in the classical setting of complex projective 3-space [Dubreil 1983, 71]. Dubreil’s fellowship ended on March 31, but in February Jacotin-Dubreil received a letter from Bjerknes in Oslo, asking if she would be able to resume work on the French translation of his Physikalische Hydrodynamik mit Anwendung auf die dynamische Meteorologie, which was not yet in press (it was published by Springer in 1933). His proposal was particularly attractive because he suggested that they meet in Göttingen, where her husband could resume work under his Meisterin Frl. Noether. In fact, once there they worked on this translation together. 12

Figure 6.2: A Göttingen Ausflug with l. to r.: Hans Heilbronn, Emmy Noether, Marie-Louise Jacotin-Dubreil, Paul Dubreil, and Chiungtze Tsen, Summer Semester 1931 (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna) By this time, both of them had become seriously interested in abstract algebra, which in her case was quite a leap from hydrodynamics. In Göttingen, though, one could study both at the highest level and so she attended Ludwig 12 This

translation project was never completed, however; [Dubreil 1983, 71–72].

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Prandtl’s course alongside Noether’s offerings.13 That summer in Göttingen the young French couple got to meet an extraordinary array of talented, not to say famous, mathematicians, beginning with Hilbert, who had just retired. His successor, Hermann Weyl, and his wife Helene, were charming hosts, as of course were Richard and Nina Courant. Two of the speakers in the Mathematical Society were Princeton’s Solomon Lefschetz and Constantin Carathéodory from Munich, the latter long a fixture in Göttingen, where he had studied under Hermann Minkowski and then succeeded Felix Klein. After the weekly lectures, the crowd usually gathered for dinner at the Hotel zur Krone, regularly attended by Emmy Noether, the logician Paul Bernays, and Courant’s right-hand man, the historian of ancient mathematics Otto Neugebauer. Paul Dubreil remembered meeting Noether’s doctoral student, Hans Fitting, Edmund Landau’s assistant, Hans Heilbronn (see Fig. 6.2), the number-theorist Kurt Mahler, and the group-theorist Helmut Ulm. He also recalled that Courant’s assistant, Hans Lewy, was one of the few who seemed deeply troubled by the recurring displays of Nazis marching in the streets [Dubreil 1983, 73]. The visitors who came to Göttingen that summer were an especially impressive group. One of them, the Finnish analyst Lars Ahlfors, was also on a Rockefeller scholarship. Five years later, at the 1936 International Congress held in Oslo, Ahlfors and Jesse Douglas attained instantaneous fame as the first recipients of the coveted Fields Medal, occasionally called the “Nobel Prize” for mathematics. Emmy Noether’s seminar drew three distinguished young foreigner’s, one being Francesco Severi’s assistant Beniamino Segre, who had spent the year 1926/27 as an IEB fellow with Élie Cartan in Paris. The other two were former normaliens, both of whom had studied together with Dubreil and Jacotin: André Weil, then returning from India, and Noether’s favorite, Jacques Herbrand, whom she had invited to the Marburg workshop that past winter. Paul Dubreil recalled that Herbrand gave two brilliant presentations. On July 4, 1931, Emmy Noether submitted his latest work on an ideal-theoretic approach to the arithmetic in extensions of a number field for publication in Mathematischen Annalen [Herbrand 1932a]. Soon thereafter, he left Göttingen to go mountain climbing in France with two friends. On 29 July 1931 Le Temps reported that a man had fallen to his death in the French Alps, and one day later it was confirmed that this had been Jacques Herbrand. A clipping from the paper was sent to Dubreil, who immediately informed Emmy Noether. She was deeply disturbed by this news and kept repeating over and again: “such a talent, 13 Afterward her research remained focused on fluid mechanics, which led to her thesis in 1934 on systems of waves that extended earlier known types. Still, her knowledge of algebra was such that she offered this as a secondary subject for her oral examination. After she and her husband took various academic positions in France, in 1943 Marie-Louise Jacotin-Dubreil was appointed to a full professorship in Poitiers, where she combined her interests in both fields while developing applications of algebra to problems in turbulence and information theory. She and her husband spent many years commuting and arranging their lives until 1955, when both had positions in Paris. For testimonials of Dubreil-Jacotin’s impressive career, see her biography at the MacTutor website.

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that’s just unthinkable.” Emmy Noether then decided to publish excerpts from two letters Herbrand had sent her concerning a problem he formulated and soon thereafter solved; this appeared in [Herbrand 1932b], immediately after his other paper. Emmy Noether’s final paper [Noether 1934] was one of many published in Actualités scientifiques et industielles to commemorate his memory.

6.3 Birth of the Brauer-Hasse-Noether Theorem In the meantime, Noether had been in steady touch with Hasse and Brauer. The Marburg workshop had helped to propel Helmut Hasse forward, and already on March 6 he sent out a postcard to the participants informing them that he had proved an extension of the local-global principle to cyclic algebras of any index, not necessarily prime.14 This was one of the problems he had presented in his talk in Marburg. That eventful summer of 1931 Hasse began to concentrate on his principal conjecture, and wrote to Brauer, Artin, and Noether asking for their thoughts. On July 27 he sent this message to Brauer: . . . I would like to write to you about the only question which is still open, the question whether all central simple algebras [over number fields] are cyclic. For I believe that this question is now ripe and I would like to present to you the line of attack which I have in mind. [Roquette 2004, 21] The other three seemed to agree that this question was ready to be attacked, though they offered no very concrete ideas for doing so. In the meantime, Hasse was working on a paper in English that he later submitted for publication in the Transactions of the American Mathematical Society [Hasse 1932b]. When he sent Noether some of the results for this paper, she reacted in a postcard from April 12 with a mixture of excitement and advice for a still larger vision: I have read your theorems with great enthusiasm, like a thrilling novel; you have really gotten very far! Now . . . I wish to have also the reverse: direct hypercomplex foundation of the invariants . . . and thus hypercomplex foundation of the reciprocity law! But this may take still some time! Nevertheless, if I remember correctly, the first step is already done in your skew field paper [Hasse 1931b] with the exponents ep ? [Lemmermeyer/Roquette 2006, 109] As Roquette pointed out, Hasse became inspired by this vision to reverse his argument, and he would soon succeed in finding what Noether called a “hypercomplex proof of Artin’s Reciprocity Law.” In this way, “a close connection between the theory of algebras and class field theory became visible” [Roquette 2004, 44]. 14 The index of a central simple algebra A is the degree of the associated division algebra D, where by Wedderburn’s theorem A = Mn (D).

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Hasse had been corresponding with A.A. Albert, then at Columbia University, since January 1931. Quite conceivably, Albert suggested to Hasse that he submit a paper to the Transactions of the AMS, but even if not, his article [Hasse 1932b] was surely written in the main for Albert’s benefit. The middle section of it presents Noether’s theory of factor systems for the first time. She gave him permission to write this part of his paper, and in her letter from June 2 she expressed general satisfaction with his handling of her ideas. 15 There she wrote: “I think you have managed to make the thing bite size for the Americans, and also for the Germans, without sacrificing too many concepts.” Still, she made a number of critical remarks and probably felt that the overall presentation was not abstract enough to really please her. On the other hand, Hasse was writing “for the Americans,” who were less familiar with Noether’s abstract ideas. He submitted the paper in late May, and the editors of the Transactions then sent it on to Albert for a referee’s report [Roquette 2004, 62–67]. Hasse explicitly referred to communication difficulties as one of the motivations for his paper: The theory of linear algebras has been greatly extended through the work of American mathematicians. Of late, German mathematicians have become active in this field. In particular, they have succeeded in obtaining some apparently remarkable results by using algebraic numbers, ideals, and abstract algebra, theories that have been greatly extended in Germany in recent decades. These results do not seem to be as well known in America as they should be in view of their importance. This fact is due, perhaps, to language differences or to inaccessibility of the widely scattered sources. [Hasse 1932b, 171] The main results and methods in this paper were known to the protagonists long before it appeared in print, but the contact between Albert and Hasse broke off for several months and was only restored when Hasse wrote again in October. At that point Hasse had found a proof of his conjecture for the case of an abelian central simple algebra.16 Albert wrote back on November 6, remarking that he was “very glad to read of such an important result. I consider it as certainly the most important theorem yet obtained for the problem of determining all central division algebras over an algebraic number field” [Roquette 2004, 68]. Hasse, however, only received this important letter about one week after the joint efforts of the three German algebraists had cracked the general problem, which occurred before they had time to study what Albert had published in the meantime.17 15 The term “crossed products” (for Noether’s verschränkte Produkte) appeared here for the first time, even for her eyes. In a postcard, she asked Hasse whether this was his “English invention” and added: “das Wort ist gut” [Lemmermeyer/Roquette 2006, 109]. 16 A central simple algebra A/K is abelian if it admits a splitting field which is an abelian field extension of K. 17 For details, see [Roquette 2004, 67–72] and [Fenster/Schwermer 2005].

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The Brauer-Hasse-Noether theorem, as Hasse’s conjecture came to be known, has a precise birth date: it was first proved on November 9, 1931. One day after this, Noether sent a postcard to Hasse: “This is beautiful! And completely unexpected to me, notwithstanding that the last argument, due to Brauer, is quite trivial (Every prime number dividing the index is also a divisor of the exponent)” [Roquette 2004, 8]. Only two days before, Noether had sent Hasse a long letter containing her ideas for improving his paper dealing with the abelian case, which she was able to extend (to the case where the finite group of the required extension field only needed to be solvable). She called her new results “trivializations and generalizations” of those in his manuscript, and then summarized these in the form of a reduction theorem with a simple proof, followed by five easy consequences. Noether proposed that he append these refinements to his text [Lemmermeyer/Roquette 2006, 124–126]. When Hasse saw that Noether had now extended his original theorem to the case of solvable groups, he remembered that Brauer had written him earlier that the general case could be reduced to p-groups using Sylow theory, and since these are solvable, Hasse saw how to piece together a proof of the theorem in three steps.18 Up to this time, Hasse had been pursuing a different line of argument, which he communicated to Brauer in a 10-page letter. This ends with a description of the roadblock he had hit: “I have to admit that here I am at the end of my skills and I put all my hope in yours” [Roquette 2004, 9]. Only two days later, though, having received Noether’s letter, he sent off a postcard to Brauer with the good news: “Just now I received a letter from Emmy which takes care of the whole question . . . .” Brauer then wrote back to say: It is very nice that the problem of cyclicity is now solved! Just today I had meant to write you and to inform you in detail about Emmy’s method; but I have to admit that I feared making a silly mistake because I had the feeling that the thing was too simple. I just wanted to ask you about it, but now this is unnecessary. By the way, right from the beginning it was clear to me that with your reduction, the essential work had been done already. [Roquette 2004, 9]. On November 9, Hasse wrote up a first draft of the proof and sent this to Emmy Noether, who responded the next day via the postcard cited above. In it, she already pointed out how the proof could be simplified further still, a true work in progress! The final breakthrough that led to the Brauer-Hasse-Noether theorem thus came in early November 1931. This arose, however, after a long and complex intellectual struggle that also involved the birth of local class field theory. This larger context reflected Noether’s ongoing interest in non-commutative algebras as tools for both number theory and representation theory. Hasse took charge of 18 For a detailed account of this whole complicated story, see [Roquette 2004]; for a less technical account that emphasizes Noether’s crucial role in the discussions leading up to the proof, see [Koreuber 2015, 123–138].

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writing [Brauer/Hasse/Noether 1932], and he did so essentially by following the historical course of events rather than giving a more conventional and systematic presentation. The proof then involved three reductive steps, the first accomplished by Hasse, then came Brauer’s contribution, and finally Noether’s. This paper was also written in great haste, in just two or three days between receipt of a postcard from Emmy Noether, dated November 8, and submission of the manuscript on November 11. This quick rush into print had nothing to do with securing priority, though Hasse knew that Albert was working in the same direction. Indeed, Albert had in the meantime proven results that enabled him to derive the same theorem independently [Fenster/Schwermer 2005]. Hasse had another, quite special reason for pushing this paper into the pipreline. As co-editor of Crelle alongside his former mentor Kurt Hensel, he was preparing a special issue to honor Hensel on his 70th birthday, and this important new theorem made the perfect present for that occasion. Indeed, the first copy was already off the press in time for Hasse to present it to him on December 29, 1931, when Hensel turned 70 [Roquette 2004, 7]. As we shall see in Section 6.6, a similar gift that Courant and Springer had planned to give Hilbert on his 70th birthday failed to be ready on time. The Brauer-Hasse-Noether theorem was important not only for the theory of algebras; it also pointed the way to a broad new approach to class field theory much as Emmy Noether had long imagined. Her vision for such a theory was also shared by Emil Artin, who sent this reaction to Hasse when he learned of the breakthrough: “. . . You cannot imagine how ever so pleased I was about the proof, finally successful, for the cyclic systems. This is the greatest advance in number theory of the last years. My heartfelt congratulations for your proof.” 19 Perhaps even more important than the theorem itself, though, were the methods used to prove it, in particular Hasse’s local-global principle, to which he indirectly alluded in the introduction: At last through our joint endeavors we have finally succeeded in proving the following theorem, which is of fundamental importance for the structure theory of algebras over number fields as well as beyond: Main Theorem. Every central division algebra over a number field is cyclic (or, as is also said, of Dickson type). It gives us special pleasure to dedicate this result, essentially due to the p-adic method, to the founder of that method, Kurt Hensel, on the occasion of his 70th birthday. [Brauer/Hasse/Noether 1932, 399] This dedication prompted a brief, but telling remark from Noether, who in a letter to Hasse from November 12 wrote: “Of course I agree with the bow to 19 [Roquette 2004, 6] comments further: The Main Theorem opens new vistas into one of the most exciting areas of algebraic number theory at the time, namely the understanding of class field theory – its foundation, its structure and its generalization – by means of the structure of algebras. (This had been suggested for some time by Emmy Noether. It was also Artin’s viewpoint when he praised the Main Theorem . . . .).

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Figure 6.3: Emmy Noether on a ship crossing the Baltic to attend the DMV conference in Königsberg, 1930; photo by Helmut Hasse, courtesy of Peter Roquette

Hensel. My methods are really methods of working and of thinking; which is why they have crept in everywhere anonymously” [Lemmermeyer/Roquette 2006, 131]. As Roquette points out, this remark “puts into evidence that she was very sure about the power and success of ‘her methods’ which she describes quite to the point” [Roquette 2004, 11], though what she meant by this may be open to interpretation. In Roquette’s view, Noether promoted two major methodological shifts with her work on hypercomplex systems.The first involved a new approach to representation theory based on an abstract theory of algebras rather than matrix groups in the tradition of Isaai Schur. Secondly, she strongly advocated the idea that the theory of non-commutative algebras would lead to a deeper understanding of commutative algebraic number theory, including class field theory. Emmy Noether was quite unhappy with Hasse’s style of presentation, but she of course recognized the urgency of the situation. She, too, had written a paper ([Noether 1932a]) for the Hensel Festband, a major undertaking that involved

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papers from many eminent mathematicians. In a letter from 12 November, she urged Hasse to clarify that he was the actual author: This is now very nice and extremely convenient for us that we did not have to trouble with the text! But I think you should state in a footnote that you wrote the text – even if one can recognize your style; this, if only because we get a “bow” in the footnotes and you do not! ... Then I would like to have some more precise “historical” information. What she meant by that related to certain remarks in the text that clouded the picture of their respective contributions. These were easily repaired, of course, but what bothered Emmy most could not be. She complained about the unsystematic form of presentation, which Hasse defended by pointing to the authority of Otto Toeplitz, who strongly advocated a “genetic” approach as didactically superior. Noether remained unconvinced: “For me personally, the converse [form], which I did not understand, produced the opposite of Toeplitz’s joy; I only refrained from a ‘motion to systematize’ because of the time urgency . . . .” 20 She insisted, however, that Hasse add a footnote stating that the presentation follows the order of discovery rather than a systematic ordering of the results.

(a) Noether, flanked by Artin and Köthe

(b) In front of the Mathematics Institute

Figure 6.4: Emil Artin with Emmy Noether and Gottfried Köthe in Göttingen, 29 February to 2 March, 1932 (Photos by Natascha Artin, Courtesy of Tom Artin) Peter Roquette has suggested, however, that in the wake of this scurried activity, Hasse was eager to make amends: Three months later Hasse seized an opportunity to become reconciled with Emmy Noether by dedicating a new paper [Hasse 1933] to her, on the occasion of her 50th birthday on March 23, 1932. There 20 Noether

to Hasse, 14 November 1931, [Lemmermeyer/Roquette 2006, 133].

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he deals with the same subject but written more systematically. Those three months had seen a rapid development of the subject; in particular Hasse was now able to give a proof of Artin’s Reciprocity Law of class field theory which was based almost entirely on the theory of the Brauer group over a number field. Thereby he could fulfill a desideratum of Emmy Noether, who already one year earlier had asked him to give a hypercomplex foundation of the reciprocity law. [Roquette 2004, 11] A few weeks before this, Hasse visited Göttingen to attend three lectures on class field theory presented by Emil Artin, a special event organized by Emmy Noether (see Fig. 6.4). Others who attended included Edmund Landau, Gustav Herglotz, Gottfried Köthe, and Saunders Mac Lane, as well as several of Noether’s students. One of the latter, Ernst Witt, was most impressed and afterward spent his vacations in Hamburg to deepen his knowledge of the subject, which he later transferred to class theory of function fields [Roquette 2002, 45]. 21 Saunders Mac Lane was also highly impressed by Artin’s elegant presentations. Since the latter had taken his doctorate under Herglotz in Leipzig, Mac Lane attributed some of this flair to the influence of his former mentor, whose lectures were a major attraction in Göttingen (Herglotz was appointed to the chair formerly held by Carl Runge in 1925). During the years he studied in Göttingen, from 1931 to 1933, Mac Lane regularly attended whatever Herglotz happened to be teaching, whereas he only took a single course from Emmy Noether, whose lecture style he described in a letter to his mother: “Prof. Noether thinks fast and talks faster. As one listens, one must also think fast – and that is always excellent training. Furthermore, thinking fast is one of the joys of mathematics.” 22 During the break Mac Lane remembered chatting in the hallway with Paul Bernays, himself a regular attendee of Noether’s. Eventually Mac Lane wrote his dissertation under Bernays. Since Artin’s lectures were considered the latest word on class field theory, Emmy Noether was keen to have them written up for distribution as mimeographed copies, a task she delegated to Olga Taussky. By early 1933, when Taussky was still at work on the Ausarbeitung, she learned that Hasse’s Marburg lectures on class field theory were also being prepared for circulation. This made Taussky wonder whether her project to write up Artin’s lectures might be pointless, so she wrote to learn Noether’s opinion. Emmy agreed that it was now probably superfluous to pursue the idea of preparing an elaborated version of the Artin lectures – her way of gently needling Taussky for not having finished this work the previous summer. Noether nevertheless advised her to contact Hasse’s assistent, Wolfgang Franz, in order to find out when he expected the Marburg lectures would 21 His dissertation, “Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen,” was submitted in 1934; for an assessment of its contents and their significance, see Noether’s report in [Koreuber 2015, 334–335]. 22 S.Mac Lane to his mother, 8 December 1931, quoted from [Brewer/Smith 1981, 77].

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be ready.23 A few months later, Noether learned from Hasse that she and others could expect to receive the text of his lectures by May 1. “I didn’t mean to criticize you recently,” she now informed Taussky, “but rather only wanted to confirm this fact.” 24 Noether would later use Franz’s elaboration of Hasse’s lectures for her teaching in the United States, whereas Taussky’s version of Artin’s lectures apparently never circulated, although many years later she published an English translation in [Cohn/Taussky 1978]. Olga Taussky remembered how Emmy Noether one day told her that she had just turned 50, but that no one in Göttingen had taken notice. She then commented wistfully, “I suppose it is a sign that 50 does not mean old“ [Taussky 1981, 84]. Three days after her birthday, Noether wrote to thank Hasse for his unexpected present: “I was terribly delighted!” (Ich habe mich schrecklich gefreut!) . . . , she wrote, followed by her usual way of responding to a mathematical text: two pages of detailed comments on his paper. Her subsequent letters to Hasse stayed on message; she was as intent as ever on generalizing class field theory by purely algebraic methods. Yet the tone of her letters underwent a marked change that reveal her heartfelt affection for her younger colleague.

6.4 Olga Taussky and the Hilbert Edition Back in March 1918, Emmy Noether wrote to Felix Klein about her work on energy conservation in variational systems (Section 3.5). Her letter ended with a brief remark about how her father had taken pleasure in the festivities in Erlangen from the week before [Kosmann-Schwarzbach 2006/2011, 156]. She was referring to the 50th anniversary of Max Noether’s doctorate in Heidelberg, a standard occasion for celebration in the German academic world in those days. In Klein’s case, the corresponding event took place in Göttingen in December of that same year, though on a much grander scale, despite the surrounding political chaos. Leaving nothing to chance, Klein planned the whole ceremony in advance, including a most important announcement: Alexander Ostrowski had arrived from Marburg and would be assisting him in preparing his collected works for publication [Tobies 2019, 464]. As so often, Emmy Noether was not officially appointed to work on this project, though she supported it from the very beginning. Formally, this work was overseen by Robert Fricke, Klein’s former student and closest collaborator. Since 1894 Fricke was professor of mathematics at the Brunswick Institute of Technology, where he served as Rector from 1921 to 1923. The year he assumed that professorship he also married Leonora Flender, the daughter of Klein’s elder sister, after which Fricke addressed his letters to his former mentor “Dear Uncle Felix.” 23 Noether to Taussky, 4 February 1933, Papers of John Todd and Olga Taussky-Todd, Box 11, Folder 11, Caltech Archives. 24 Noether to Taussky, 24 April 1933, Papers of John Todd and Olga Taussky-Todd, Box 11, Folder 11, Caltech Archives.

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To what extent Noether knew Fricke personally is difficult to judge, but these circumstances alone suggest they must have been in fairly close contact during the three years when both worked on Klein’s collected works [Klein 1921–23]. Robert Fricke was a native of Brunswick, the birthplace of Carl Friedrich Gauss. The famous “Prince of Mathematicians” had first attended the city’s Collegium Carolinum, forerunner of the present-day Technical University, before taking up studies in Göttingen. The chair Fricke assumed in 1894 had been occupied by Gauss’ last student, Richard Dedekind, himself a native of Brunswick. As Dedekind’s successor, Fricke had ample motivation to assume responsibility for editing his collected works, though he clearly recognized that most of his predecessor’s creative work fell outside the range of his own mathematical expertise. Fricke may have long harbored a plan to pursue this project, but in any case, he surely realized that Emmy Noether was the ideal candidate to carry it through. When he approached her remains unclear, but circumstances suggest this may have occurred in 1926, one year after Klein’s death. At that time, Øystein Ore was working closely with her in Göttingen, and he, too, had strong interests in Dedekind’s works. An agreement was thus reached that all three would serve as editors, with Noether and Ore doing the lion’s share of the work; this included undertaking a careful study of Dedekind’s previously unpublished papers and assorted scientific correspondence. Around the time that Noether’s work on [Dedekind 1930–32] was winding down, Richard Courant was busy orchestrating a similar project aimed at publishing the works of his former mentor, David Hilbert. Courant and the Berlin publisher, Ferdinand Springer, had already hatched a plan for Volume 1, which would contain Hilbert’s works on number theory. Springer promised to visit Göttingen on the 23rd of January 1932 to attend the festivities in celebration of Hilbert’s 70th birthday, at which time he would present Hilbert with this volume hot off the press. In the meantime, Courant had engaged Wilhelm Magnus from Frankfurt and Helmut Ulm from Bonn to proofread Hilbert’s papers and correct any small mistakes. Neither could claim to be an expert in number theory, but both at least had solid backgrounds in algebra. Such was the situation in Göttingen when Courant set off in September 1931 to attend the annual meeting of the German Mathematical Society, held that year in Bad Elster. There he met with various colleagues, including Hans Hahn from Vienna, with whom he chatted about the status of the Hilbert project. Hahn told him about a young woman named Olga Taussky, who had done her doctoral work on class field theory under Philipp Furtwängler. Taussky also happened to be present in Bad Elster, where she presented a short lecture in hopes of attracting attention and landing a new job. Knowing this, Hahn introduced her to Courant, who was naturally well aware of Furtwängler’s stature as a number theorist. He may even have read the latter’s proof of the principal ideal theorem in [Furtwängler 1929], a by now famous result that answered the last remaining

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open conjecture in Hilbert’s paper [Hilbert 1902].25 In any event, Courant was eager to have Taussky join his editorial team in Göttingen, no doubt hoping that her expertise would help speed up this work. A few weeks later, she received a letter from him, inviting her to spend the winter semester in Göttingen. Courant therein mentioned a principal duty as well as possible second task: “firstly, to work very intensively in a collaboration to complete the number theory volume of Hilbert’s works, and secondly, to possibly also help a little with routines and operations of the Institute.” He might have been more forthcoming about what he meant by “routines and operations,” but at least he did not conceal that she would be working under time pressure. “The term starts on November 1,” he wrote, and “the Hilbert volume must be ready by mid-January.” 26 Never one to be intimidated by hard work, Taussky gladly accepted this offer. At the tender age of 25, she would be editing the papers of the greatest mathematician of the past era, a man she would soon meet personally.

6.5 From Vienna to Göttingen Olga Taussky left behind numerous vivid recollections of her difficult early years in Vienna and Göttingen, including various interesting anecdotes about her encounters with Emmy Noether. She first met her one year earlier at the 1930 DMV meeting in Königsberg (Fig. 6.3), where Taussky gave a short talk about her dissertation, in which she sharpened her mentor’s proof of the principal ideal theorem. After she had finished speaking, . . . Emmy jumped up and made a quite lengthy comment which, unfortunately, I was unable to understand because of insufficient training. However, Hasse understood it and replied to it at some length, and there developed between these two mathematicians some sort of duet which they enjoyed thoroughly. Clearly, Emmy was pleased and I even overheard some nice remarks she made about my lecture. She spoke to me frequently later, but not about the subject of my talk. She was very friendly; so was Hasse. All this was very helpful to me, for prior to my lecture I had been justifiably very nervous, for this was a meeting to which very famous mathematicians came, including even Hilbert. . . . 25 Hilbert had formulated four conjectures for so-called Hilbert class fields, beginning with the assertion that for any algebraic number field k there exists a unique finite Galois extension K for which the Galois group G(K/k) is identical with the ideal class group of k, and thus finite and Abelian. The latter group has roots in Gauss’s classification of the genera of quadratic forms, and its rank represents the degree of deviation from unique factorization in k. As Hasse emphasized in [Hasse 1932a], Hilbert based his claims on just one special case, namely a quadratic number field k whose ideal class group has rank 2. By 1911, Furtwängler proved three of Hilbert’s four conjectures, and in 1929 he proved the fourth, which stated that every ideal in k became a principal ideal in K. 26 Courant to Taussky, 7 October 1931, cited from the translation in [Goodstein 2020, 682].

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When lunch came, I sat down next to Emmy, to her left. . . . Emmy was very busy discussing mathematics with the man on her right and several people across the table. She was having a very good time. She ate her lunch, but gesticulated violently when eating. This kept her left hand busy too, for she spilled her food constantly and wiped it off with her dress, completely unperturbed. [Taussky 1981, 79–80]

Figure 6.5: On an Ausflug to Kerstlingeröder Feld, l. to r.: Otto Schilling, Emmy Noether, Olga Taussky, Hans Schwerdtfeger, Ernst Witt, Paul Bernays, unidentified, Erna Bannow, unidentified, summer 1932; Papers of John Todd and Olga Taussky-Todd, Caltech Archives. Little more than two decades earlier, Emmy Noether had undergone a similar initiation rite when she spoke for the first time at the 1909 meeting held in Salzburg. She understood instinctively how important such an experience can be for a young mathematician’s future, and Taussky surely left this conference with a new sense of reassurance. She also probably began to get over some of her ill feelings toward Furtwängler. At any rate, she realized now that she had been lucky to have the chance to work on class field theory, a prestigious area of number theory that stood at the very heart of current interest. Furtwängler had many other doctoral students, but nearly all of them worked on other topics. Olga Taussky went on to a long and illustrious career, recently described by Judith Goodstein in [Goodstein 2020]. She published some 300 research papers in

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algebraic number theory and matrix representations in algebra and analysis. During the period 1957 to 1977 she taught at the California Institute of Technology. Soon after her retirement, she recounted her life for the Oral History Project of the Caltech Archives, a text later reprinted in [Taussky-Todd 1985]. This provides many details about her early life in Vienna, including the hardships she faced as a student of Philipp Furtwängler. Her teacher came from Elze, an old town on the Leine River roughly 100 kilometers due north of Göttingen going downstream. Furtwängler’s grandfather founded an organ construction company there, a business that his father later took over. The family name was thus well-known in musical circles throughout the region, but attained international fame through another branch, to which the conductor Wilhelm Furtwängler belonged. Philipp Furtwängler was deeply inspired by Hilbert’s work on number theory, but as Olga Taussky reported, her mentor never met him personally. This may well have been the case, even though Furtwängler had studied in Göttingen, where he completed his doctorate under Felix Klein in 1896, one year after Hilbert joined the Göttingen faculty.27 In any event, his introduction to number theory came via Klein, who developed a geometric approach using lattices, not unlike Hermann Minkowski’s better-known theory. Beginning in 1902, however, Furtwängler began spinning out a series of articles in which he took up various ideas and conjectures that Hilbert had published in the late 1890s. Furtwängler’s first breakthrough came when he successfully answered a prize question that Hilbert posed as a member of the Göttingen Scientific Society. This question was related to the problem of establishing a reciprocity law analogous to quadratic reciprocity – which dates back to Euler and was first proved by Gauss – but for general algebraic number fields. As Helmut Hasse later emphasized in [Hasse 1932a, 530], Hilbert’s work launched a reorientation of number-theoretic research, which would henceforth move to ever-higher spheres of abstraction. Furtwängler, who like many others learned algebraic number theory by reading Hilbert’s Zahlbericht [Hilbert 1897], remained at a relatively low level in this process. As his student, Olga Taussky took a similar approach to number theory, which for her was ultimately about numbers, not abstract concepts. By the time she began her studies with him, Furtwängler was paralyzed from the neck down, so he had to lecture from a wheelchair, while one of his assistants wrote the equations on the blackboard. Olga was called upon to do this at times, a task she remembered as very challenging, although her mentor’s lectures were models of clarity; Kurt Gödel – whom Taussky met while attending Moritz Schlick’s famous philosophical seminar in Vienna – reputedly called Furtwängler’s lectures the best he had ever heard. In other respects, though, she remembered him as a poor adviser, whose lack of support only added to her doubts and suffering. His doctoral students – he had some 60 over the course of his career in Vienna – had to wait in a long line outside Furtwängler’s office on such occasions when he happened 27 According to his eventual successor in Vienna, Anton Huber, Furtwängler already passed his doctoral exam on 1 March 1895 [Huber 1940, 168]. Huber’s obituary was a blatantly obvious effort to represent Furtwängler as a hero of the Third Reich.

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to make himself available. Olga felt very disoriented, but kept studying the difficult literature on class field theory, hoping he would soon give her a dissertation problem. She later recalled how, around this time, Emil Artin found a way to reduce Hilbert’s conjecture regarding principal ideals to a problem concerning certain finite non-abelian groups. Furtwängler did actually tell me a little about this, but without explanations, and made me almost desperate. In the meantime, he proved Artin’s group-theoretic statement to be true and hence solved the principal ideal theorem. This was a tremendous achievement, but the world of mathematics was not very grateful and considered his proof as ugly. In fact, they had little appreciation for his earlier pioneering work either. In spite of my grievances against him as a teacher, I feel his work deserved better credit. [Taussky-Todd 1985, 316–317] Taussky found the atmosphere in the middle-sized town of Göttingen vastly different than in Vienna, but unfortunately she had barely any time available to immerse herself in its mathematical life. Not only did she, Magnus, and Ulm have to spend long hours each day working on the number theory volume, she was also burdened with the unpleasant task of correcting papers for the assignments Courant gave to students in his course on differential equations. Nevertheless, she still found time to attend Noether’s seminar, to which she had been specially invited (Fig. 6.5). In fact, when she arrived from Vienna, “[Emmy] immediately announced to me proudly that she and [Max] Deuring, her favorite student, had studied class field theory . . . and that she was going to run a seminar on this subject because of my visit” [Taussky 1981, 80–81].28 On one occasion in that seminar, Furtwängler’s proof of the principal ideal theorem came up. When Noether echoed the opinion Olga had heard all too often – it’s so “unattractive” – Taussky blew up at her, calling this an unfair criticism: it had taken decades to resolve Hilbert’s conjecture – and, after all, it could have been wrong! “I was amazed by my daring,” she later recalled, “and so were the others. However, Emmy was completely calm, and I felt certain, was not in the least angry with me. I recognized for the first time that she was a person who did not mind criticism” [Taussky 1981, 81]. 29 28 In her letter to Hasse from 8 November 1931, Noether wrote: “I’m offering a seminar on number theory: local class field theory, Artin’s conductors, principal ideal theorem (Frl. Taussky is here), etc.; the established proofs shall be presented, but my personal goal is to reach a hypercomplex understanding” [Lemmermeyer/Roquette 2006, 124–125]. 29 In [Lemmermeyer/Roquette 2006, 108] the editors point out that various attempts were made to find a simpler proof of the principal ideal theorem before Shokichi Iyanaga broke through. In his paper, published in 1934 in the Hamburger Abhandlungen, he credited Artin with the greater part of the work. Iyanaga’s proof appeared soon thereafter in the textbook by Hans Zassenhaus Lehrbuch der Gruppentheorie.

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6.6 Taussky on Hilbert’s 70th Birthday Noether was naturally interested to know about the state of progress with the Hilbert edition, while she and Ore continued their work on volume 3 of the Dedekind Werke. According to Taussky, Emmy was the only one in Göttingen whom she and her co-workers could consult on technical matters, and they probably did not turn to her often, given that she was far more an algebraist than a number theorist. In any event, if Noether did provide help, it was treated informally, since Courant did not wish for her to be directly involved in the Hilbert editorial project [Rowe 2020, 5]. Nor did he want his staff of proofreaders to append any editorial remarks to the texts; if alterations were required, these were to be entered without any indication that such a change had been made to the original text.30 This situation stands in sharp contrast with the Klein edition [Klein 1921–23], even though that project also began just as Felix Klein was approaching age 70. In that case, however, Emmy Noether had been one of several younger mathematicians who worked closely with him to prepare those three volumes. Unlike Klein, Hilbert never enjoyed looking backward. He had no patience for “scholarly studies,” even if this only involved supervising such work. So he put Courant in charge of this editorial project, just one of the many tasks the latter managed as Director of the Mathematics Institute. Part of Courant’s success stemmed from his ability to uphold and to capitalize on the traditional prestige of the Göttingen mathematical tradition, which Klein and Hilbert had so long embodied. Emmy Noether proved to be a tremendous asset for Courant’s Göttingen, but Olga Taussky realized that many there showed little respect for her achievements, despite the recognition she enjoyed elsewhere. Taussky once heard the distinguished Erlangen algebraist, Wolfgang Krull, emphatically say: “Miss Noether is not only a great mathematician, she is a great German woman!” [Taussky 1981, 84]. Richard Courant was a skillful, if unorthodox manager, but he was far too busy to cultivate close personal relations with Emmy Noether. He was also exceedingly protective of Hilbert’s image and knew full well that his former mentor, especially in his old age, demanded unconditional loyalty.31 As guardian of the Hilbert edition, Courant aimed to mobilize the support of several leading younger mathematicians, whom he enlisted to write commentaries on Hilbert’s contributions to various disciplines. In the end, five essays of varying length were published along with Otto Blumenthal’s biographical essay [Blumenthal 1935]:32 Helmut Hasse on number theory (Band 1); B.L. van der Waerden on algebra and Arnold Schmidt on foundations of geometry (Band 2); Ernst Hellinger on integral equations and Paul Bernays on foundations of arith30 As discussed in Section 3.6, Hilbert made major changes in [Hilbert 1924], while characterizing it as “essentially a reprint” of two earlier publications. 31 Hilbert’s first doctoral student, Otto Blumenthal, presents an even more striking example of selfless dedication and loyalty to Hilbert. see [Rowe 2018b] and [Rowe/Felsch 2019]. 32 Reprinted in [Rowe/Felsch 2019, 43–491].

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metic (Band 3). One might well wonder why Courant did not ask Emmy Noether to write an essay on Hilbert’s contributions to invariant theory. Perhaps he did and she deferred to van der Waerden, who merely summarized the significance of Hilbert’s work for subsequent developments, in particular Noether’s important studies, in two pages. Yet Taussky-Todd’s passing comment that Courant did not want Noether involved with the Hilbert edition was surely correct, and although there may have been a number of reasons for this, the most obvious problem stemmed from an evident conflict of interest. Emmy Noether’s outspoken enthusiasm for Dedekind’s work, which only grew with each volume she edited, was by this time known throughout the Göttingen institute. As Taussky recalled, “[s]he came to appreciate Dedekind’s work to the utmost, and found many sources of later achievements already in Dedekind. Occasionally she annoyed even her friends by this attitude. She managed to rename the Hilbert subgroups the Hilbert-Dedekind subgroups” [Taussky 1981, 81]. Courant very likely heard about, or may have even read, Noether’s commentary on [Dedekind 1894b] and [Dedekind 1895], where she sided entirely with Dedekind’s position in his rebuttal of Hurwitz’s approach to ideal theory.33 This alone would surely have disqualified her in Courant’s eyes, since he knew that Hilbert had been involved in this controversy and stood firmly on the side of Hurwitz. In any case, Courant had but one thought in mind: Taussky and co. needed to finish proofreading the papers for volume 1, which included Hilbert’s Zahlbericht, by early January; that gave them less than three months. The celebration of Hilbert’s 70th birthday would be a major event in Göttingen, but as that time neared it became increasingly clear that Courant’s deadline would be very difficult to meet. In later years, Olga Taussky told various versions of this story, one of which Constance Reid retold as follows: In the course of her work on Hilbert’s papers, Fräulein Taussky was astonished to discover many errors. These were not typographical errors. Perhaps the bound of a function would be wrongly computed, a theorem incorrectly stated, a step omitted in a proof, or an entire proof necessary to the argument dismissed as “easily seen” when it was not. Although she recognized that, because of Hilbert’s powerful mathematical intuition, the errors had not affected the ultimate results, she felt that they should be corrected in his collected works. She was encouraged in this by Emmy Noether, who was editing the work of Dedekind and frequently announced, loudly, that no one would be able to find a single error – “even with a magnifying glass!” [Reid 1970, 200]. 33 Her commentary began: “The latest developments have fully and completely affirmed the correctness of Dedekind’s views both with respect to the definition of ideal and divisibility as well as the foundation of the decomposition theorem” [Dedekind 1930–32, 2: 58].

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Although she was the source for this story, Olga Taussky-Todd was not at all pleased by Reid’s account of it, as the latter learned some years later. 34 In view of these misgivings, Reid revised the above passage in order to satisfy her concerns. Whether by design or mere accident (due to lack of space on the page), Emmy Noether was no longer mentioned at all in [Reid 1996, 200]. Yet she clearly was interested in the Hilbert project, at least from afar. Taussky-Todd, on the other hand, was eager to set the record straight about certain “false rumors” that spread around Göttingen at that time. People were saying that “our work concerned small defects only and delayed the publication unnecessarily, and this was even mentioned in C. Reid’s volume on Hilbert” [Taussky 1981, 82] (thanks to TausskyTodd’s input). To refute these grumblings, which she clearly attributed to Courant’s influence, Taussky-Todd tried to emphasize that the delay had been due to serious mistakes, not just a few typos. Although she could no longer remember precisely what these were, she did recall speaking to Emmy Noether about the difficulties she had encountered: “Emmy was truly amazed when I told her that Hilbert’s work contained many errors. She said that Dedekind never made any errors. Hilbert’s errors were on all levels” [Taussky 1981, 81]. This invidious comparison with Dedekind, who was without question an exceedingly exacting thinker and precise writer, rings rather hollow without any evidence to support the claim that Hilbert’s papers were full of substantive errors. If anyone really cared to find out, they could run a comparative analysis of the original papers with the printed edition. Short of that, one can easily imagine that back in 1931 Olga Taussky experienced a serious conflict of interest when she read and then edited [Hilbert 1902], with its various claims that her former teacher had struggled so mightily to prove. Although Furtwängler’s papers were lauded by cognoscenti – Edmund Landau and Hilbert’s student Rudolf Fueter reviewed them for the Jahrbuch über die Fortschritte der Mathematik – Olga clearly felt they were little appreciated by others. Moreover, since all his works postdated Hilbert’s publications on number theory, one finds no mention of them in [Hilbert 1932], except for Hasse’s closing essay [Hasse 1932a]. Taussky did, however, take the opportunity to add an editorial footnote on p. 506, in which she wrote somewhat vaguely that one of Hilbert’s conjectures in [Hilbert 1902] was incorrect as stated. She knew this, of course, because Furtwängler had corrected that mistake. Taussky-Todd also provided details about the celebration in the Hilberts’ home, which Reid described as follows: On the day of the birthday itself, Ferdinand Springer, who was the publisher of the collected works, came to Göttingen to present personally to Hilbert the special white and gold leather-bound copy of the first volume. The beautiful cover contained, however, not the printed pages, but only the proofs of the pages; for Fräulein Taussky was still 34 For

details about their ensuing correspondence, see [Rowe 2020].

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not satisfied. Hilbert made no comment on the unfinished nature of the volume. But later, in his presence, Fräulein Taussky declined a certain brand of cigarette as being too strong for her. Somebody said that one really couldn’t tell one brand from another. “Aber nein!” Hilbert said. “Fräulein Taussky can tell the difference. She is capable of making the finest, the very finest distinctions.” She was not sure, but she thought he was making fun of her for taking so seriously errors which he himself considered unimportant. [Reid 1970, 201]. Olga Taussky-Todd was particularly displeased about the final sentence, which put the whole episode in an ironic light. She later wrote Reid, . . . there is a real misunderstanding there. Hilbert made his remark by no means in a joking tone, it was quite an unfriendly one and he needled me quite a bit later at that party, too. I was afraid that he had seen that footnote! You see, I never discussed anything with him, nor was I expected to. He had moved for years through different streams of subjects, finally to logic and had no knowledge, nor even interest in that type of work any longer.35 Olga Taussky-Todd’s memories of her stay in Göttingen all point in the same direction: this was a most unpleasant time for her. One of the few positive aspects, though, came from the friendship she managed to strike up with Emmy Noether, despite their differences in temperament and mathematical orientation. As for her recollections of this early work on the Hilbert edition, she emphasized that a good deal of controversy concerned Hilbert’s Zahlbericht.36 Emmy Noether seems to have been one of its critics, if we can judge by a brief comment she made by way of praising Hasse’s first report on class field theory: “I’m happy to see that you’re bringing Takagi in order. I notice more and more how much your report helps make it easier to penetrate [the theory]; one only has to compare it with Hilbert’s; how much unnecessary effort that requires today!” 37 Taussky-Todd wrote that Noether was not among those who criticized the Zahlbericht when she was proofreading it, but then added that in Bryn Mawr Emmy once burst out against it, claiming that Artin said “it delayed the development of algebraic number theory by decades” [Taussky 1981, 82]. Presumably what Artin meant was Hilbert’s class field theory, not the Zahlbericht, since Artin’s work on general reciprocity laws – his solution of Hilbert’s ninth Paris problem – followed in the wake of Takagi’s general theory of class fields. What seems particularly baffling is that Olga Taussky apparently never mentioned [Hasse 1932a], the short essay at the end of the volume. This was the one 35 Taussky-Todd

to Reid, 5 December 1977, [Rowe 2020]. opinions about this led to a number of bizarre claims published in [Rota 2008]; these were exposed as “fake news” in [Lemmermeyer 2018], though on last viewing of the biographical article for Taussky-Todd in Wikipedia, Rota’s book is still cited there. 37 Noether to Hasse, 17 November 1926, [Lemmermeyer/Roquette 2006, 57]. 36 Taussky’s

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place in it where the reader was offered a larger overview of Hilbert’s work in the light of subsequent developments. Moreover, this brief survey touches on studies by Furtwängler, Takagi, Artin, and by Hasse himself. One year earlier, he used the local-global principle to prove his Norm Theorem for cyclic Galois extensions of an algebraic number field [Hasse 1931b]. This theorem not only generalized earlier results of Hilbert and Furtwängler, Hasse also showed how it could be applied to deduce new results on the structure theory of algebras over number fields. In this connection, he did not neglect to mention Emmy Noether’s vision for a general class field theory, about which he commented: As a result of these applications and by means of the general conceptual structures of E. Noether, this theory appears to open the way conversely to still unexplored important problems in algebraic number theory, namely to generalize class field theory to general relative-Galois number fields. However, this development is still too young to be a subject that could already be reported on here. [Hasse 1932a, 535] Hasse’s survey was written with a good deal of verve and subjective opinion, and Noether clearly read it with real interest when it was still in page proofs. Her reaction, transmitted by Taussky in a letter to Hasse, reflects not only her desire to set the record straight but also her sense of self-irony. This letter was written on 15 March 1932, thus roughly two months after Hilbert’s birthday party. Noether had apparently gone to speak with Taussky, Ulm, and Magnus after she noticed a glaring inconsistency in the text. As Olga Taussky put this: Professor Noether pointed out to us just now that there seems to be a contradiction in your afterword to Hilbert’s works on algebraic number theory in regard to the historical information about the HilbertDedekind theory of Galois number fields, since this speaks of the completely new discoveries of Hilbert, but then also about the earlier studies by Frobenius. . . . Since we will receive another revision, it might still be possible to make some minor changes without causing difficulties with the printing, such as the following . . . Taussky then suggested two minor emendations, which Hasse adopted for the published text.38 But Hasse also had the pleasure of reading Emmy Noether’s handwritten comments, which also appeared in this letter, quickly softening its tone: The Dedekind edition is to blame for my appearance as a historical complainer! But the letter is not intended to be as official – or as demanding – as it looks! What bothered me, though, was that you present the connection between Galois theory and ideal theory as the fundamentally new thing with Hilbert, whereas Dedekind had published at least 38 For

details, see [Lemmermeyer/Roquette 2006, 150–151].

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as much about it . . . And the density theorem from 1880 (although first published in 1896) is precisely this connection! Formally, this inaccuracy comes to haunt you in the contradiction pointed out by Miss Taussky! But by no means do I want to deny that it was only through Hilbert that these things came to life for number theorists – and that in all likelihood Hilbert found everything completely independently. Apart from that, the others are probably not such hair splitters as I am, in case you want to leave everything as it is! Kind regards, Your Emmy Noether.

6.7 Zurich ICM in 1932 In September 1932, Olga Taussky attended the International Congress of Mathematicians in Zurich (Fig. 6.6) with its several highlights, one being Emmy Noether’s plenary lecture, “Hypercomplex Systems in their Relationships to Commutative Algebra and Number Theory” [Noether 1932b]. According to her protégé, Pavel Alexandrov, this event marked a major triumph for Emmy Noether and the structural approach to mathematics. Speaking before a huge audience, she unveiled her visionary idea for a research program that aimed to generalize the classical theory of algebraic number fields by appealing to the properties of noncommutative algebraic systems. She called this the “principle of application of the noncommutative to the commutative”, by which she hoped to extend class field theory to Galois extensions of algebraic number fields for which the Galois groups were nonabelian [Curtis 2007]. A few months before the ICM opened, Noether wrote Hasse about her preparations for this major event: . . . while working on my Zurich lecture, I for once read Gauss. It has been claimed that a half-way educated mathematician knows the Gauss principal genus theorem, whereas only exceptional people know class field theory. Whether that’s true, I don’t know – in my case that knowledge went in the reverse order – but at least I learned a lot from Gauss in terms of comprehension . . . . [T]he transition from my version to the Gaussian is direct . . . . . . . What I’m doing is generalizing the definition of genera by means of characters. [Lemmermeyer/Roquette 2006, 165]. It was in Zurich that Noether first presented her version of the principal genus theorem for number fields, an idea she first developed in early 1932; she published a complete proof in [Noether 1933b]. Her letter to Hasse is particularly interesting, not to say surprising, in view of Dedekind’s great appreciation for Gauss’s theory, as Noether surely knew from reading his Festschrift contribution [Dedekind 1877].

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Figure 6.6: Group Picture of Participants at the 1932 ICM in Zurich (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna)

She, in fact, wrote the notes that appear in [Dedekind 1930–32, 1: 158], but with no mention of Gauss whatsoever. Her comments begin with reference to a letter Dedekind wrote to Frobenius in 1883, which reveals that [Dedekind 1877] was conceived in the context of a general theory of reciprocity laws. After noting that Dedekind’s theory resembles modern class field theory, as developed by Heinrich Weber, she pointed to Takagi’s work as the next stage in these developments. Her concluding remarks then read: “For ideal theory the significance of this study is that it treats for the first time relationships between ideals in different rings by means of mappings of intersection- and extension ideals. The concepts developed here can be extended to general rings . . . .” Noether then referred the reader to the dissertation written by her student Heinrich Grell.39 In summary, it would seem very clear that Emmy Noether read Dedekind à la Bourbaki, i.e. looking forward and not backward. The long, convoluted story of the principal genus theorem from Gauss to Noether can be read in [Lemmermeyer 2007]; apparently few picked up later where she had left off. Noether’s version was based on an important innovation, however, her theory of crossed products and factor systems, which Hasse had introduced 39 [Grell

1927]; Noether’s report on Grell’s dissertation appears in [Koreuber 2015, 318].

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Figure 6.7: Emmy Noether at the Podium (Courtesy of MFO, Oberwolfach Research Institute for Mathematics)

to American mathematicians in his paper [Hasse 1932b]. Noether had developed these ideas as part of her lecture course from the summer semester of 1929. Max Deuring wrote this up, and copies of his Ausarbeitung circulated within Noether’s network, which is how Hasse came to learn about it.40 In [Noether 1933a, 644], she noted that readers could refer to Hasse’s paper for the theory or another version closer to her lectures in Deuring’s forthcoming report, which appeared two years later in [Deuring 1935]. Nathan Jacobson, as editor of Noether’s collected papers, decided to include Noether’s original lectures, as prepared by Deuring, which were published in [Noether 1983, 711–763].41 40 The basic idea behind Noether’s crossed products is to construct an associative algebra A by means of a Galois field E and its associated Galois group G of order n. One can then define the elements of A to be linear combinations of the n elements of G subject to certain conditions. 41 Jacobson commented that Dickson had earlier defined Noether’s crossed products, but he neglected to give conditions on the factor set to ensure the associativity of the algebra. Richard Brauer was the first to construct non-cyclic division algebras using cross products, whereas

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Another highlight of the ICM was a private party thrown by Teiji Takagi at the Hotel Eden on the Zurichsee, one of those rare occasions when a whole group of leading experts on class field theory could meet face-to-face. A number of younger Japanese mathematicians attended, including Takagi’s star student, Shokichi Iyanaga. Among the Europeans were Emmy Noether and several close associates of her school: Taussky, Hasse, and van der Waerden. Others included the Ukrainian Nikolai Chebotaryev, whose density theorem gave Emil Artin the tool he needed to prove his reciprocity law, and Claude Chevalley, a friend of Iyanaga with whom he studied class field theory under Artin in Hamburg. Chebotaryev had already met Noether and Hasse at the 1925 DMV conference in Danzig, and they soon got into a spirited discussion. The young Japanese watched this display of exuberance by these raucous Europeans with a mixture of amusement and disbelief. Most of all they were struck by the loudest of them all: Emmy Noether. As they were getting ready to depart, she asked the Japanese to show her how to bow properly, a scene that left a lasting memory with some of them [Yandell 2002, 229]. Taussky greatly revered Takagi and even tried to pick up some Japanese so she could speak with him in his native language. He spent five months in Europe, much of it traveling with Iyanaga. During his stay in Vienna, Taussky introduced him to her teacher, Philipp Furtwängler, whose work represented the culmination of Hilbert’s vision for class field theory. Takagi’s publications, on the other hand, marked the beginning of a new vision that inspired the work of Hasse and Artin. In Hamburg, he had the opportunity to meet with Artin, an encounter very unlike his reunion with Hilbert in Göttingen. “Observing my old master grumbling as if speaking to himself,” he later wrote, “I wept in my heart” [Honda 1975, 165]. For Takagi, the Zurich Congress marked a highpoint in his life, much as it did for Emmy Noether. By this time, in fact, Noether was finally beginning to receive outward signs of long overdue recognition. In 1932, she and her Hamburg colleague Emil Artin were awarded the Ackermann-Teubner Memorial Prize for their achievements in modern algebra. During the eighteenth and nineteenth centuries, mathematical prizes were mainly conferred by scientific academies for special works submitted as answers to problems set by these institutions [Gray 2006]. Two of the most famous female mathematicians, Sophie Germain and Sofia Kovalevskaya, both gained fame by winning prizes offered by the Paris Academy. In the modern era, but before the establishment of the Fields Medals and other prizes for distinguished mathematical work, the Ackermann-Teubner Prize represented the highest award offered in Germany. Created in 1912 by Alfred Ackermann-Teubner, the Leipzig publisher and long-time benefactor of the German Mathematical Society, it was first conferred on Felix Klein in 1914 following the wishes of its founder. Every two years, one of eight areas of research in pure Noether’s main interest was to use them as a tool for studying the structure of Brauer groups [Noether 1983, 20–21].

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or applied mathematics became eligible, at which time a five-member jury would select the winner. In 1932 this area was arithmetic and algebra, two fields in which Emil Artin had recently made highly significant contributions. In 1927 he solved two of Hilbert’s 23 Paris problems – the 9th (to establish a general reciprocity law for algebraic number fields) and the 12th (to express a nonnegative rational function as a quotient of sums of squares) – making him an obvious candidate for the prize. Martina Schneider recently uncovered some of the actions behind the scenes that led to Noether’s selection along with Artin. Two of the five jurors were van der Waerden and Erich Hecke, who was Artin’s older colleague in Hamburg. In a letter addressed to Hecke, but sent to the other three jurors as well, van der Waerden argued forcefully in support of Emmy Noether’s case: . . . not only because of her mathematical achievements, but above all because of the extraordinarily stimulating and directive effect she has exerted on a whole generation of algebraists. It was she who made all of us, including Artin, aware of Steinitz’s work, and who in her work on elimination initiated applications of field-theoretical methods and concepts to algebraic geometry. It was she who created general ideal theory, with the chain conditions and the proof of the general decomposition theorem . . . . It was she who, against all odds, repeatedly emphasized Dedekind’s module methods and extended these to group theory and ring theory, until these methods led to her triumph with the theory of ideal and module classes of hypercomplex systems, namely the long-desired unification of hypercomplex numbers and representation theory. Finally, it was she who long ago foresaw and pointed the way to the number-theoretical applications of hypercomplex theory, which Hasse and others are now successfully pursuing. . . . You can already see from the above that I would favor awarding the prize to Noether. There is still another motive that influences me to tend more toward Noether in this difficult comparison of the diverse achievements of Artin and Noether: the consideration that she has not yet received as many honors as Artin, who has already received various offers [for professorships], whereas she is only a private lecturer with a teaching contract. One could try to use this opportunity as a form of compensation.42 This turned out to be the only instance when two individuals were chosen as recipients of the Ackermann-Teubner Prize. After Olga Taussky returned to Vienna, she remained in contact with Noether, who wrote her a friendly letter on 12 November 1932.43 Emmy began by informing her that she had just sent off a letter of recommendation in support of Taussky’s 42 Van

der Waerden to Hecke, 29 May 1932, translated from [Schneider 2021]. of John Todd and Olga Taussky-Todd, Box 11, Folder 11, Caltech Archives.

43 Papers

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application for an undisclosed academic position. At this time, Taussky had no income aside from the money she made by tutoring students. As usual, Emmy Noether’s letter brought various bits of news, such as the recent arrival of Wolfgang Gröbner, who had just taken his doctorate under Furtwängler in Vienna. He came to spend the year working with Noether and regretted that Taussky was no longer in Göttingen. Emmy included a few comments about her lectures, which her student Wolfgang Wichmann was writing up for her. Apparently Taussky had plans to visit Göttingen again, as Noether commented that she hoped Wichmann’s version would enable her to work through the material. In the meantime, Noether had been forced out of her small apartment in the house at Friedländerweg 57, where she had often entertained in the past. Pavel Alexandrov was familiar with the circumstances behind her expulsion, and he recalled these in his memorial address from 1935: She did not hide her sympathy toward our country and its social and governmental structure, despite the fact that such expressions of sympathy were considered shocking and improper by most representatives of Western European academic circles. It went so far that Emmy Noether was literally expelled [from her boarding house] at the insistence of the student boarders, who did not want to live under the same roof as a “pro-Marxist Jewess” – an excellent prologue to the drama that came at the end of her life in Germany.” [Alexandroff 1935, 8]. The building in which she formerly lived belonged to a fraternity, the Burschenschaft Thuringia, one of several fraternal organizations in Göttingen, many with a long tradition. in Göttingen, many with a long tradition. The Corps Hannovera Göttingen, a fraternity with compulsory academic fencing (mensur) was particularly proud to count Otto von Bismarck as a former member. Given the ultra-nationalist political orientation of these groups, one might imagine Noether was glad to live somewhere else. She found a new apartment in the large building located at Stegemühlenweg 51, not far from her former dwelling. In her letter to Taussky, she remarked, My apartment is also very nice and pleasant with the central heating. Alexandroff has been living with me for 14 days, but he has to return to Russia at the end of November. He would very much like to have the little picture of Veblen, where you cut me and Frau Veblen away; he thinks it’s very good, even said it should be enlarged. Olga Taussky had occupied an office next to Oswald Veblen’s at the Mathematical Institute; she could sometimes hear him practicing his lectures on relativity theory through the wall that separated them [Taussky-Todd 1985, 323]. Two years later, he would help arrange a fellowship for her to spend one year at Bryn Mawr College, where she would rejoin Emmy Noether.

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The fact that Pavel Alexandrov was her house guest at the new apartment on Stegemühlenweg explains why he would have known about the circumstances that led her to move out of her old quarters. Little did either of them know that this would be their last time together. In his autobiography, Alexandrov remembered this last stay with Emmy Noether nostalgically: I spent my last day in Göttingen making farewell visits. Emmy Noether accompanied me. We went first to the Landau’s; then to Hermann Weyl’s; then we were invited to the Hilbert’s for coffee at 5 o’clock. I never saw Hilbert or Weyl or Landau again. I was invited to a farewell supper at the Courant’s at 8 o’clock. Courant’s closest mathematical friends were there – Neugebauer, Friedrichs, Lewy, and, of course, Emmy Noether. My train was to leave at 5 a.m. and it was decided that all the people assembled at Courant’s house should spend the whole night with him and that we should then all set out for the station together. After a very long drawn-out supper we had a musical evening, which chiefly consisted of the Schubert trio in E-flat Major. It was played by Stefan Cohn-Vossen (piano), Hans Lewy (violin) and Frau Courant (cello). All three played superbly, with a great uplift. I had always loved this Schubert trio, but after this performance on my farewell night in Göttingen it came to occupy a special place in my appreciation of music and altogether in my consciousness and my life. Finally we went through the dark avenues of Göttingen by night to the station, and I left. I never saw Emmy Noether again, so that this parting was also forever. [Alexandrov 1979/1980, 328–329] Olga Taussky would, however, get to see Noether again, but not in Göttingen; after a two-year pause their world lines eventually converged during the fall of 1934 in the town of Bryn Mawr, Pennsylvania, a place neither had probably ever heard of before they took flight from Europe.

Chapter 7

Cast Out of Her Country 7.1 Dark Clouds over Göttingen When Pavel Alexandrov wrote about his last visit to Göttingen some four decades later, he could hardly look past the traumatic events that were to follow in the wake of his departure. His description of the atmosphere in the town little more than two months before Hitler would come to power set the stage: But in November 1932 clouds were already thickening over Germany. Often I was woken in the morning by the sounds of “Deutschland, erwache.” This was sung by the young people of the “Hitler-Jugend”, as they marched up and down the streets. It was clear that things were about to happen and that it was time for me to go home. The day of my departure finally arrived. As I have said, it was at the very end of November. Göttingen had long been a stronghold of the Nazi Party in northern Germany, which adds plausibility to these personal recollections. Yet like most such retrospective accounts, this one reflects a highly filtered view of the past, which is nothing unusual, of course, since this is how human memory typically operates. Alexandrov surely did hear the Schubert trio in E-flat Major performed in the Courants’ home some hours before his train departed that night. What he just as surely did not experience was a deep foreboding that all this was about to end. For, in fact, he was planning to come back to Göttingen in 1933 along with his friend Kolmogorov, and Emmy Noether was very much looking forward to their future visit. This is what she wrote to Alexandrov on 5 March 1933: My dear Alexandrov! The beginning of the holiday – we’re lazing around in March and April – gives me the opportunity to finally answer your lovely Christmas and New Year’s letter . . . . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_7

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7 Cast Out of Her Country I’ve lived pretty much from hand to mouth since then, partly for my lecture course, but even more for a lecture in Marburg, where I had promised Hasse the newest findings – result, everything else was left behind. It’s again hypercomplex number theory, in the sense of my Zurich lecture . . . . We’re, of course, still far from a general theory of non-Abelian fields, but the hypercomplex offers the possibility at least to formulate questions and conjectures and to create work for years, even for a whole lot of people! I’m very happy that everyone in your institute is working so well; you’ll soon be raising a younger topological generation, and since Pontryagin and my other acquaintances already have positions as parents, you will be able to enjoy your grandchildren while you are young. (“I hear you feel like a grandmother,” Fischer once said to me in regard to F.K. Schmidt-Erlangen).1 It is nice that Kolmogorov will come in the summer and that you will keep him company here in winter at least. Frau Bruns will be interested to know whether Kolmogorov will stay with her in the summer. Lately I’ve had lots of visitors under my Noether roof, but only for short stays. You can follow that next year by reading the guest book, where all sorts of mathematical poets have emerged, above all Neugebauer with his dedication. These last days van der Waerden was here for the habilitation of his brother-in-law [Franz] Rellich.2 V. d. Waerden is now much fresher than he was not long ago. . . . [Tobies 2003, 104–105]

Van der Waerden had left Groningen in 1931 to become professor of geometry in Leipzig; this was the prestigious chair previously occupied by Felix Klein, Sophus Lie, and Otto Hölder. Noether probably knew that van der Waerden had applied for funding from the Rockefeller Foundation in order to spend a semester in Rome working with Francesco Severi, whom he had met six months earlier at the ICM in Zurich.3 She alluded to this when she wrote: Courant thinks that with time the planned stay in Rome can take place, just not so quickly. Severi continues meanwhile to make propaganda for Italian algebraic geometry, now even in notes for the Annalen!4 1 Friedrich Karl Schmidt took his doctorate in Freiburg in 1922, formally under Alfred Loewy, though the topic for his doctoral thesis was suggested by Wolfgang Krull, who also took his degree under Loewy. Schmidt was also strongly influenced by Helmut Hasse before he habilitated in Erlangen in 1927. Ernst Fischer was presumably alluding to Noether’s connection with Krull, implying that she could think of herself as Krull’s academic mother. 2 Franz Rellich was a close protégé of Richard Courant; after the war ended, he was appointed director of the Göttingen Mathematics Institute. 3 Three days earlier, Courant wrote a letter of support for van der Waerden’s application, transcribed and translated in [Schappacher 2007, 265]. This plan was never realized. 4 Here Noether seems to have been referring to [Severi 1933], a paper reviewed by Hellmuth Kneser. The latter commented about it, “general and personal remarks scattered throughout

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Courants will drive to Arosa tomorrow with the whole family, including Miss Maier and Rellich; at the end of the month I’ll again go to the North Sea for 2-3 weeks. Now that Weyl has finally rejected the offer from America,5 Courant will be protected for a number of years from the burning question of having to bring a successor to Göttingen. You said it right: there was a mood of disaster over the possibility of needing to find a successor. By the way, this matter caused Weyl to become ill; he has been on leave since Christmas to cure his nerves, even gave up the trip to America he had planned for February. But he plans to be here again in May, which will probably be pleasant for Kolmogorov. . . . [Tobies 2003, 105] Emmy Noether’s letter went on with all kinds of other mathematical news, plus wishes for pleasant skiing weather. But there was not a word about politics or any sense that she and her colleagues might be affected by recent and certainly imminent Nazi policies and threats. Perhaps that was only prudent, but the tone in all her correspondence from this time onward was consistently apolitical. On February 4, she sent a short note to Olga Taussky in which she mentioned how Hasse and Brauer came to speak in her seminar on the same day. This was shortly after Christmas in the dead of winter, but with the obligatory walk to Kerstlingeröder Feld (Fig. 6.5), where Brauer held his lecture under petroleum lights. Or a few months later, after she learned that the government had suspended her from teaching: “I just spent three weeks on the North Sea with wonderful weather; one can then follow contemporary events as an observer.” 6 Hermann Weyl would not have been surprised, since that was how he knew her: “There was nothing rebellious in her nature; she was willing to accept conditions as they were” [Weyl 1935, 435]. In her world, as reflected in these letters, the general mood during the past few months had only been darkened by a seemingly trivial matter, namely, the uncertainty over who might take Weyl’s place were he to decide to leave Göttingen. But now that this dark cloud had passed, life could go on as usual. Clearly, Weyl had little more inkling than Noether about what the future held in store; otherwise, he could have saved his nerves, accepted Flexner’s offer, and boarded a ship bound for New York in February. If there was anyone who might have sensed trouble, it should have been Courant, who had many real enemies both inside the university and beyond it. He knew this well enough, but he, too, underestimated the danger. So the moment the winter semester ended, he drove off with his family to go skiing in Switzerland. the article impart even to the non-initiated reader a lively impression of the peculiarity and achievements of the author and the Italian school” [Schappacher 2007, 266]. 5 Abraham Flexner, director of the newly founded Institute for Advanced Study in Princeton, sought to gain both Einstein and Weyl for the new faculty. Weyl initially rejected the offer, but took up negotiations again after the dismissal of Courant and co.; see [Schappacher 1998a, 524–525]. 6 Noether to Taussky, 24 April 1933, Papers of John Todd and Olga Taussky-Todd, Box 11, Folder 11, Caltech Archives.

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One week before Noether wrote this letter, the Reichstag had gone down in flames as a result of arson. The day afterward, Hitler induced President Hindenburg to sign the Reichstag Fire Decree, which suspended most civil liberties in Germany; they would never again be restored under Nazi rule. New elections were called for March 5, the very day Noether wrote her letter. Having successfully suppressed the Communists (who formerly held 17% of the seats), the Nazis won an impressive victory, gaining 44% of the seats, enough to have a majority in alliance with the German Nationalist Party. On March 23, the newly assembled legislature convened in the Kroll Opera House, where it passed the Enabling Act that gave Hitler quasi-dictatorial powers. That event marked the end of the ill-fated Weimar Republic. On March 30, still in Arosa, Courant wrote a remarkable letter to his colleague, the physicist James Franck. During this vacation, he, Rellich, and Kurt Friedrichs were hard at work on the long-awaited second volume of Courant-Hilbert Methoden der mathematischen Physik. Yet Courant had recently learned through his house servants that rumors were spreading about how he and his family would not be returning to Göttingen, so he sought Franck’s advice. He had read that a nationwide boycott of Jewish businesses had been announced for April 1, and he hoped it could be deterred. He also expressed his anger over reports in the newspapers, citing public comments Einstein had issued about the political situation in Germany. Courant was appalled by this, since it made “Germany’s internal situation a butt for general political agitation abroad” [Reid 1976, 139]. He also resented that Einstein had taken steps to renounce his German citizenship (he was still a Swiss citizen). Shortly before his death, when Constance Reid showed him this letter and others, he could only comment that he really could not believe what he had written back then. Surely that applied to this passage from his letter to Franck: What hurts me particularly is that the renewed wave of antisemitism is . . . directed indiscriminately against every person of Jewish ancestry, no matter how truly German he may feel within himself, no matter how he and his family have bled during the war and how much he has contributed to the general community. I can’t believe that such injustice can prevail much longer – in particular, since it depends so much on the leaders, especially Hitler, whose last speech made a quite positive impression on me. [Reid 1976, 140] On Franck’s advice, Courant left his family behind and returned to Göttingen. He then learned that the SA had not only carried out the planned boycott on April 1st, they had five days earlier given the local Jewish community a foretaste of what was later to come by staging a march through the town, during which they demolished store fronts, vandalized property, and brutalized Jewish citizens. Several of those who committed acts of violence on that day were brought to court, though in all but one case the charges were dropped [Bruns-Wüstefeld 1997, 59– 63]. No doubt many who participated – a group of well over 100 SA men – were

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among the perpetrators who burned down the city’s synagogue during the infamous nationwide pogrom on November 9/10, 1938, the Night of Broken Glass. On April 7, 1933, the government passed a new Law for the Restoration of the Professional Civil Service (BBG) [Schappacher 1998a, 527–532]. The word “restoration” (Wiederherstellung) was, of course, a euphemism for “purging” undesirable elements from government employment. It contained an array of criteria, racial as well as political, and was directed not only against Jews but also against anyone whose political views might be regarded as suspect, which included those who were merely critical of the then fervent German nationalist ideology. Others, like Courant and Franck, who had fought in the war were to be exempted. What they and most others did not know was that this would only be the first of several waves of measures directed at individuals whom the Nazis portrayed as enemies of the German people. Courant conferred with the physicists James Franck and Max Born as well as his trusted senior assistant, Otto Neugebauer.7 The latter was Austrian and, unlike the other three, unaffected by §3 of the BBG, since he was an “Aryan.” 8 They considered filing a symbolic protest, but Franck was considering a far more drastic step, namely to resign his professorship. The Prussian Ministry had extended the vacation period in order to implement the new law. Before the summer semester began on May 1, faculty members were to fill out questionnaires as part of a bureaucratic procedure to determine whether their personal biography met the conditions for dismissal.9 Yet already by mid-April various individuals learned that they had been placed on leave of absence while their cases were under review. James Franck, a Nobel laureate and highly esteemed member of the faculty, deliberated for a week before reaching his decision. On Easter Sunday, April 16, he wrote the Minister of Education announcing his resignation. He also wrote to the then serving Rektor, Siegmund Schermer,10 and contacted the Göttinger Tageblatt by telephone [Rosenow 1998, 555–556]. The newspaper ran a major article that appeared on the day Franck’s announcement reached the Ministry. It contained a statement from Franck that created a furor within the faculty: We Germans of Jewish descent are being treated as aliens and enemies of our homeland. It is required that our children will grow up with the 7 On

Neugebauer’s early career and his relationship with Courant, see [Rowe 2016]. term Aryan as used in §3 essentially meant non-Jewish, which would be defined more precisely in 1935 in the Nuremberg Laws. 9 Statistics on the implementation of the BBG and other measures at 15 German universities were published in [Grüttner/Kinas 2007]; this study included adjunct lecturers, like Emmy Noether, who did not hold civil service positions. 10 The Rektor (or Präsident) is the elected highest official at German universities. In Göttingen, a new Rektor, the Germanist Friedrich Neumann, was elected on May 1. Neumann joined the NSDAP that same day, and pursued policies fully in accord with the Nazi regime. On May 10, 1933, he opened the book-burning ceremony in Göttingen, part of a nationwide academic demonstration of extreme intolerance for literature deemed anti-German. After the war, Neumann was eventually rehabilitated and in 1971 Marburg University awarded him its Brother Grimm Prize. 8 The

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7 Cast Out of Her Country knowledge that they will never be able to prove themselves as Germans . . . . Those who fought in the war are supposed to have permission to continue to serve the State. I refuse to avail myself of this privilege, even though I understand the position of those who consider it their duty to remain at their posts.11

On April 24, the Göttinger Tageblatt published an article – signed by 42 professors (who claimed to be writing on behalf of all their colleagues) – that vehemently protested against the manner in which Franck had tendered his resignation, but especially the opening sentence cited above. They considered this not only an impediment to the government’s domestic and foreign policies of national renewal but as “an act of sabotage,” and expressed the hope that “the government would carry out the necessary purges expeditiously” [Hentschel 1996, 33]. Rektor Schermer had initially counseled delay with regard to Franck’s case, but in view of the local headlines it was making he reversed course and requested that the Ministry respond quickly, since “the impression has grown that Franck was acting in concert with a specific group of faculty members” [Rosenow 1998, 557].

7.2 First Wave of Dismissals These were the immediate background events that that now prompted the Ministry of Education to take action.12 On April 25, 1933, officials there sent a telegram to the University of Göttingen stating that six of its faculty members were to be placed on leave of absence, effective immediately. Three were mathematicians: Felix Bernstein, Richard Courant, and Emmy Noether, none of whom should have been dismissed according to the conditions stipulated in §3 of the BBG. Bernstein was already appointed associate professor in Göttingen in 1911, which meant he qualified for the exemption granted those who were already civil servants before the Great War. Courant served on the Western front during the war until September 1915, when he narrowly escaped death, and then returned as part of a unit engaged in underground telegraphy. Emmy Noether only held the honorary title of extraordinary professor, so she enjoyed none of the privileges of a civil servant. The fact that the government simply ignored this fact makes evident that the BBG was merely a pretext for removing those it deemed undesirable. 13 She had been forced to wait four long years before gaining the right to teach, the venia legendi, a title traditionally conferred by a university faculty and only routinely confirmed by the state. Now that the much-maligned Weimar Republic was dead and the “national renewal” had begun, the lion’s share of the Göttingen faculty clearly thought it high time for the state to take bold action; indeed, what better time to conduct a purge? 11 Translated

from [Rosenow 1998, 556]. an overview of the situation in Göttingen, see [Dahms 2008]. 13 In a third version of the original law, instated on 6 May 1933, it was extended to include employees in state positions who were not civil servants. 12 For

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As it happened, the framers of the BBG had not neglected to include an elastic formulation for §4 that could serve as a catchall condition for removing those suspected of harboring sympathy for liberal causes. This read: “Officials, who by virtue of their previous political activities do not offer a guarantee that they will stand firmly for the national state at all times, can be dismissed from their positions.” In this connection, the German League of Human Rights (DLfM) was just one of the organizations that the Nazis considered subversive. 14 Courant was totally shocked when he learned about the Ministry’s action, which was reported in the papers almost immediately. Having received no official information at all, he began to wonder whether this was merely an oversight, but he also worried that his brief activity as a local Social Democratic politician after the war might have been used to defame him. In a letter to his former assistant, Hellmuth Kneser, he mulled over all the possible reasons why he had been targeted, one being that unfriendly colleagues called his institute a “fortress of Marxism” [Reid 1976, 144]. Emmy Noether’s case presented a similar problem, and she made no attempt to hide the fact that she, too, had taken an active interest in leftist politics during the early years of the Weimar Republic. In filling out the required questionnaire, she listed her membership in the pacifist-oriented Independent Social Democratic Party (USPD) from 1919 to 1922, after which she (like most USPD members) joined the mainstream SPD. After 1924, she dropped her membership altogether, and in a letter to Hasse she remarked, no doubt with a twinkle in her eye: “I never voted further left than that!” 15 She surely must have known that in the eyes of her conservative colleagues (and presumably Hasse’s too), she was if not a Marxist, then at least a sympathizer. Many would have known that she spent a semester teaching in the Soviet Union, and that she and Courant spent ample time hosting Russian visitors. Although the axe had not yet fallen, the situation looked ominous for both of them. Yet Emmy just kept doing mathematics as before; she mainly seemed upset over the fact that her course on hypercomplex methods in number theory could no longer take place (though she later decided to convene with a small group of students in her new apartment). Courant, on the other hand, saw his whole life’s work crumbling before his eyes. He was deeply troubled, but also determined to salvage whatever he could in this bizarre situation. To be sure, the difficulties both of them faced arose locally, not from the Ministry in Berlin. In his first letter to Kneser, Courant speculated “that it was Franck’s resignation which actually provoked the ministry’s action,” though he would later realize that this dramatic episode had little to do with the fundamental problem, which went far deeper. Long before the Nazis came to power, the student body in Göttingen had succumbed to their hyper-nationalist, antisemitic ideology. 14 The DLfM was explicitly named on the Ministry’s questionnaire; Otto Blumenthal’s membership in this organization led automatically to his dismissal from his professorship in Aachen [Rowe/Felsch 2019, 360–374]. 15 Noether to Hasse, 21 July 1933, [Lemmermeyer/Roquette 2006, 197]; Noether’s politics are discussed in [McLarty 2005a].

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This brand of far-right thinking prevailed at several German universities, but particularly those where Jewish scholars were prominently represented, as was the case in Berlin, Frankfurt, and Hamburg, as well as in Göttingen. A tell-tale event came in 1927 when students at the Prussian universities cast votes on whether to form a larger student organization by merging with those in other states, including Austria, where Jews had been excluded. The Prussian government demanded that such exclusionary clauses be dropped from the proposed constitution, which was then rejected by a nearly 80% majority of those who voted (in Göttingen the vote was well over 80%) [Dahms 1983, 84]. By the early 1930s right-wing radicalism was running amok. In a second letter to Kneser from 29 April, Courant described the chaotic atmosphere at the Mathematics Institute: Not only are the students apparently determined to try to prevent Landau and Bernays from lecturing, but they are also attacking Neugebauer as ‘politically unreliable’ – that is, communistically oriented. The dean has stood up for Neugebauer, but it does not look as if Neugebauer can endure the pressure. He now holds – as my representative – the position of director of the institute; but I am afraid he will give it up if the students don’t withdraw their threat to boycott him. Since Weyl is not a strong and stable personality and since Herglotz cannot be considered for the directorship either, I see the future of our institute as very dark. . . . [Reid 1976, 144–145] On 28 April, Neugebauer was instructed by the Rektor to assume the directorship, a position he held for exactly one day. In the meantime, the Ministry of Education had informed the deans of all faculties in Göttingen that controversial instructors should be suspended from teaching, a move obviously designed to placate those who demanded their removal. When Neugebauer learned that he was on the list of those who did not enjoy the trust of the students, he informed the Rektor that he would follow the dean’s advice, which obviously meant he could no longer assume responsibility for directing the institute.16 Hermann Weyl was then appointed acting director following Neugebauer’s resignation; he thenceforth did his utmost to have Courant reinstated as director and Noether reappointed to the faculty. On May 5, over a week after the matter had been reported in the Göttinger Tageblatt, the Ministry finally sent official notice to the six affected members of the faculty informing them of their status; they were placed on temporary leave pending review of their respective cases in view of the conditions stipulated by the BBG. A few weeks later, Max Born tendered his resignation to the Ministry, expressing his essential agreement with Franck’s views. In the meantime, Fritz Haber had also stepped down from his post as director of the Kaiser Wilhelm 16 Others who appeared on the dean’s list were Edmund Landau, Paul Bernays, Hans Lewy, and the physicists Paul Hertz and Kurt Hohenemser. All but Neugebauer were of Jewish descent [Schappacher 1998a, 528,542]

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Institute for Physical Chemistry in Berlin. In view of the impending dismissals of their Jewish associates, Haber’s colleague, Otto Hahn, enlisted the support of several others for a plan to announce a general protest. He then asked Max Planck to join them. Years later, Hahn remembered Planck’s response: “If today thirty professors stand up and protest . . . then tomorrow 150 will come to declare their solidarity with Hitler because they want the [vacated] positions” [Stern 1999, 53]. The same kind of struggle between political opportunists and old-guard conservatives would soon play out within the German mathematical community (see [Mehrtens 1987], [Schappacher 2000], and [Remmert 2012a]). Courant had little interest in symbolic protests: he was determined to press his case as a loyal German who had served his country with distinction. Knowing this, Neugebauer and Kurt Friedrichs composed a long and glowing testimonial that was sent to 65 prominent individuals who had known Courant in one capacity or another. In the end, only 28 agreed to sign, although many more expressed their support in various way.17 Erich Hecke, a friend of Courant’s since their days together as students in Breslau, doubted the wisdom of writing directly to the newly appointed Minister, Bernhard Rust. Kneser presumably shared this view and recommended instead that a small group of colleagues and students – best of all, any who might have been members of the Nazi Party before January 30, 1933! – could offer personal testimonials to Minister Rust. Kneser had spoken with the prominent Nazi mathematician Theodor Vahlen about Courant’s case and had received reassurances that such testimonials would be relevant for the Ministry’s scrupulous handling of these matters [Reid 1976, 148–149]. Courant was obviously a high-profile figure, and so his case was dealt with differently than Emmy Noether’s. On legal grounds, he could only be dismissed in accord with §4, which meant determining that he was politically suspect. Weyl obtained a detailed report from Courant about his past military and political activities, which he forwarded on 23 May to the Kurator,18 Theodor Valentiner, who sympathized with Courant. In doing so, Weyl emphasized that Courant’s report was “of the greatest importance for judging the case . . . and must not be overlooked” [Schappacher 2000, 21]. Valentiner, a jurist who had previously served as Kurator from 1921 to 1932, attached a lengthy personal and legal analysis to these documents. Concerning Courant’s “political sins” during the early Weimar years, he wrote: In any case, I have clearly sensed, as I have told trusted people as early as 1925 or 1926, that he evidently had found his way back to the middle class. . . . my confidential inquiries with suitable, thoroughly right-wing professors led to the result that no one remembers a single 17 The list contained many prominent names, including Artin, Blaschke, Carathéodory, Hasse, Heisenberg, Herglotz, Hilbert, von Laue, Mie, Planck, Prandtl, Schrödinger, Sommerfeld, van der Waerden, and Weyl; [Reid 1976, 151–152]. 18 The Kurator was the official representative of the Minister at German universities.

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7 Cast Out of Her Country utterance from his mouth since 1919 that could justify a different view. ... There is no doubt in my mind that he will stand up for the national state at all times, and he made this declaration directly to me. [Schappacher 2000, 22]

While Courant’s case dragged on, Nazi students were already pushing forward their preferred candidate for his chair: the Darmstadt mathematician Udo Wegner. Toward the end of the semester, Weyl submitted a lengthy memorandum on the sad state of mathematics in Göttingen, underscoring the need to hire suitably qualified personnel, and pointing in particular to the potential vacancy that might arise with Courant’s professorship. Realizing that Courant’s position was untenable, he took the opportunity to attack the politically motivated candidacy of Wegner, which he predicted would be “catastrophic for mathematics in Göttingen” [Schappacher 2000, 23]. Soon thereafter, Weyl left to vacation in Switzerland. From there, he renewed negotiations with Flexner, who was fully informed about the situation at the German universities. In October 1933, Weyl informed the Ministry that he had accepted an offer from the Institute for Advanced Study. 19 In the meantime, Courant’s case remained undecided, but he arranged to take an official leave of absence (without pay) in order to spend the coming academic year in Cambridge. Leaving his family behind, he began to contemplate new plans for a future life in the United States.20 Before turning to Noether’s case, it might here be noted that she and Courant were two among the 52 members of the Göttingen faculty who lost their positions directly as a result of Nazi policies. To put that number in perspective, this represents slightly more than one-fifth of the entire teaching corps; Noether was the only woman in this group (her dismissal then left one other woman on the faculty).21 Forty of these cases were due to racial discrimination, and in 32 instances the dismissals led to emigration. Three individuals voluntarily resigned (Franck and Born being two such cases), raising the total losses to 55. The numbers (both absolute and in percentage) were even higher at four other German universities: Frankfurt, Berlin, Heidelberg, and Hamburg [Grüttner/Kinas 2007, 140,166]. In view of this sudden and sweeping calamity, it should come as no surprise that very few of these dismissals drew efforts even remotely comparable to those undertaken on behalf of Courant. Still, the testimonials and support for Emmy Noether were every bit as strong, perhaps even more so, thanks to the efforts of Helmut Hasse, who intervened just as the meltdown in Göttingen had begun. Very 19 He did not neglect to add that, since his wife came from a Jewish family, he presumed the Ministry would have no objection to releasing him from his position in Göttingen [Schappacher 2000, 24]. 20 His case was eventually dropped after Courant negotiated reasonable conditions for his family to emigrate; in 1934 he began a new career at New York University [Reid 1976, 155–168]. 21 Women held slightly more than 1% of the teaching positions at the German universities; more than one-third of them lost their positions due to NS-policies [Grüttner/Kinas 2007, 142].

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few of his letters to Noether have survived, but she answered one of those many lost letters on the 10th of May: Thank you very much for your most friendly letter! Still, the matter is much less unfortunate for me than for many others: first, I have a small savings (I was never entitled to a pension); at the moment, I’m still collecting a salary, so I can wait until the definitive decision or a little longer. In the meantime the faculty will try what it can to prevent termination; the success of that, though, is quite doubtful at the moment. Finally, Weyl (see Fig. 7.1) told me that a few weeks ago, when everything was still up in the air,22 he had written to Princeton, where he still has contacts. . . . Weyl thinks that something could open up over time, especially since last year Veblen was keen to let Flexner know about me.23 . . . This “do not teach until further notice” is quite catastrophic here at the institute . . . I’m reading your lectures with great pleasure; I think that once in a while I’ll invite the “Noether community” (Noethergemeinschaft)24 to my apartment to talk about them. [Lemmermeyer/Roquette 2006, 187–188].

7.3 Hasse’s Campaign for Noether Hasse contacted Neugebauer before proceeding with his plan, namely to approach “a number of well-known mathematical scholars from Germany and abroad for assessments of the scientific significance of Miss Noether’s work and her entire scientific personality.” 25 Beyond his personal relationship with Emmy Noether, Hasse had a second motive for supporting her case. In his letter to Valentiner, the Kurator, he expressed this in these words: I also intend to draft a detailed report of this kind on my own, since I am convinced that Miss Noether is one of the leading German mathematicians and that German science, and especially the younger generation, will suffer a very serious loss should Miss Noether be forced indirectly to move abroad. As I hear, Miss Noether’s special students have also prepared a report that should be sent to you in a short time. 22 Presumably

this means before the Ministry’s telegram from 25 April had arrived. appears likely that Veblen introduced Emmy Noether to Flexner during the International Congress held in Zurich the previous September. [Lemmermeyer/Roquette 2006, 189]. 24 This is a play on the word Notgemeinschaft, reflecting her own situation and the state of science in general, since the Notgemeinschaft der deutschen Wissenschaft, founded in 1920 and the forerunner of the present-day German Research Foundation, was established expressly as an Emergency Foundation. 25 Hasse to Kurator of Göttingen University, 3 June 1933, translated from [Roquette 2008]. 23 It

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Figure 7.1: Emil Artin in Göttingen, July 1933: l. to r. Ernst Witt, Paul Bernays, Helene, Hermann, and Joachim Weyl, Artin, Emmy Noether, Ernst Knauf, unidentified person, Chiungtze Tsen, Erna Bannow. Photo by Natascha Artin; for dating and details, see [Eckes/Schappacher 2016] (Auguste Dick Papers, 12-14, Austrian Academy of Sciences, Vienna)

Indeed, Wolfgang Wichmann soon submitted this testimonial, undersigned by friends of Emmy Noether as well as her students in the narrower sense.26 She was certainly well aware of this text, which to a remarkable degree sought to paint her as a kind of nativist German mathematician (eine urdeutsche Mathematikerin). Carl Ludwig Siegel’s report pointed in a similar direction, though in a much subtler manner: Through investigations by Dedekind, Frobenius, and Kronecker, Germany attained a leading position in algebra toward the end of the last century. That the country has maintained and even strengthened its position until this day is largely due to Emmy Noether. In particular, Miss Noether’s publications and lectures have greatly promoted the so26 The signatories were E. Bannow, E. Knauf, Tsen, W. Vorbeck, G. Dechamps, W. Wichmann, H. Davenport (Cambridge, Engl.), H. Ulm, L. Schwarz, Walter Brandt (?), D. Derry, and WeiLiang Chow; see [Roquette 2008].

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called theory of hypercomplex systems so that the problems arising from it are now the focus of interest among algebraists around the world. In the past decade, E. Noether has stimulated productive work among young mathematicians probably more than any other Göttingen docent. In the interest of our young academics, it is therefore urgently wished that Miss Noether be able to continue teaching in Germany.27 The latter group went a good deal further in setting forth their case that Noether belonged in Germany: We doctoral students of Prof E. Noether, students of mathematics at the local university, request that the following considerations be taken into account: As much as we welcome the national revolution and all its effects, we also regret that Prof. Noether was placed on leave of absence, thereby preventing her from carrying out her work effectively, and this for the following reason. Ms. Noether has founded a mathematical school that has brought forth the most capable of the younger mathematicians, some of whom are now lecturers or professors at German universities. Her work has always consisted of special lectures with small groups of auditors, most of whom wish to pursue an academic career. The fact that her courses span over several semesters has also meant that students are given a deeper insight into the interconnections. It is no coincidence that all her students are Aryans; this is due to her essential conception of mathematics, which corresponds entirely to an Aryan way of thinking. This does not concern detached individual results, but rather a way of recognizing and understanding the whole, and E. Noether succeeds in doing this based on a conceptual method that she has developed in recent years. Through the research field she explores and the lively questions she poses, she has filled all of her students with enthusiasm and passion for mathematics. Despite differing political views, our personal relations with her are in no way disturbed, as she has never had any political influence on her students. The close connections she has managed to establish between herself and her students, as well as among the students themselves, comes from her great personal stimulation. These connections can hardly be maintained for long without further contact. Some students have already left this semester for other universities. This is the reason why we would be pleased if Professor Noether were given the opportunity to exercise again her profession as a teacher, one who is unique in all of Germany. Emmy Noether was not only well aware of this ongoing effort on her behalf, she took an active part in planning it. In particular, she surely read and pre27 C.L.

Siegel, Frankfurt (Main), 14 June 1933, [Roquette 2008].

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sumably approved the above text with its assertion that Noether’s mathematics “corresponds entirely to an Aryan way of thinking.” Could this have been intended as a ploy to counter §3 in the BBG, which clearly applied to her as a non-Aryan? 28 One also has to wonder why this text brought up the political differences between Noether and her “good Aryan students”? In any case, she did take an avid interest in mobilizing support, both from within Göttingen as well as from the mathematical world at large, as can be seen from her letter to Hasse, dated 21 June 1933: You really are making quite a job for yourself with the reports! As if you didn’t otherwise have enough work to do! Wichmann had just given the Kurator the report with student signatures – mainly from the algebraists – before he [Valentiner] left for Berlin, which the latter said was very correct, though of course at the moment it is difficult to get past §3. Then he got your letter as well! It would, however, seem to me good, in case a sufficient number of reports have arrived by the end of the semester, to send these already to the Kurator and forward the others (Takagi etc.) later; it has been said that the matter will not be decided beforehand – i.e., during the semester. And it also seems to me a good idea to make copies of the reports beforehand (but at my expense!), so that it will be easier to refer back to them later should they not be successful this time. [Lemmermeyer/Roquette 2006, 189–190] Emmy spent early August in the small beach resort town of Dierhagen on the Baltic, where she met her brother Fritz and his family (Figs. 7.2, 7.3). They were joined by Herbert Heisig and his wife Lotte, together with Hans and Eva Baerwald. Heisig was a native of Breslau and, like Baerwald, studied under Noether at the Institute of Technology; in 1931 he took his doctorate in engineering mathematics there. Only a few months before they vacationed together, Herbert Heisig was appointed head of the mathematics and natural sciences division at the Teubner publishing firm in Leipzig.29 Fritz Noether was in nearly the same situation as his sister, waiting to receive definitive news about whether he would lose his professorship at the Breslau Institute of Technology. He was hired there in 1922, one year after Emmy’s protégé Werner Schmeidler assumed Max Dehn’s chair in Breslau. On 21 July, before she left for vacation, Noether wrote to Hasse. She was excited about his new results on a function-theoretic version of the Riemann hypothesis and asked if he would lecture about that at the forthcoming DMV meeting in Würzburg. Obviously, she wanted very badly to attend that meeting, but felt she needed encouragement to do so under the circumstances. She had in the 28 At the time Noether was placed on leave, the BBG applied only to those who were civil servants, but soon afterward it was extended to anyone certified to teach at an institution of higher education. 29 Heisig remained with Teubner-Leipzig until 1952 when he assumed the same position for the West German branch in Stuttgart, which he headed until 1969.

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Figure 7.2: Emmy with Fritz and Regina Noether and Herbert and Lotte Heisig, Dierhagen, August 1933 (Auguste Dick Papers, 12-14, Austrian Academy of Sciences, Vienna)

meantime spoken with Weyl, who told her he would find out from the Kurator about the submission date for the reports Hasse had gathered. Weyl assumed that would be soon, but he also agreed that copies ought to be made in advance, whether in Marburg or in Göttingen was a matter of indifference to her. Hasse had probably inquired as to whether a de facto case could be made that she would have qualified as a private lecturer before the outbreak of the war – a potential argument for gaining an exemption to §3. In the meantime, she had filled out the questionnaire for the Ministry, and so informed Hasse: “I stated that Klein and Hilbert brought me to Göttingen in the spring of 1915 to fill in for the private lecturers. In order to conclude from this that I already met all the preconditions in August 1914 would require quite a lot of imaginary benevolence!” [Lemmermeyer/Roquette 2006, 196]. On July 31, the day Noether left for the Baltic, Hasse submitted all 13 reports to Valentiner, the Göttingen Kurator. Hermann Weyl may have received them first or else read them shortly afterward. In his memorial lecture for Emmy Noether at Bryn Mawr on 26 April 1935, he remarked: “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” [Weyl 1935, 435]. In his own testimonial, Weyl compared her with the two most famous women in the recent history of mathematics, Sophie Germain and Sofia Kovalevskaya, claiming that Noether surpassed them both in her originality and depth. He also took up the defense of “abstract algebra” as practiced by Noether,

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Figure 7.3: From left to right: Hermann Noether, his girlfriend Nora, Emmy Noether, Eva Baerwald, Fritz Noether, Lotte Heisig, Regina Noether, , Herbert Heisig, with Hans Baerwald in the foreground, Dierhagen, August 1933 (Archives of the Mathematisches Forschungsinstitut Oberwolfach)

who grasps problems with “seeing thoughts and through the formation of concepts as appropriate as possible for the object, rather than through blind calculation. In this respect Ms. Noether is the legitimate successor of the great German number theorist R. Dedekind.” Moreover, thanks to quantum theory, algebra is the area of mathematics that stands in the most intimate relationship with physics, and “in this field, in which mathematical research is currently developing most rapidly, Emmy Noether is recognized at home and abroad as the true leader” (Hermann Weyl, Göttingen, 12 July 1933, [Roquette 2008]).30 Hasse clearly took some care in selecting the foreigners whom he asked to write on behalf of Noether, drawing in part on his own personal connections in the world of algebraic number theory. He evidently also informed her about whom he had contacted, since she referred directly to Takagi as one of these persons. Had 30 In connection with quantum physics, Weyl was alluding to the role of group representation theory, which had recently been elaborated in monographs by Eugene Wigner, van der Waerden, and Weyl himself. Noether’s more abstract approach to this part of modern algebra was largely independent of these currents, although Martina Schneider has noted how [van der Waerden 1932] drew on the theory of groups with operators, a topic developed by Wolfgang Krull and Emmy Noether [Schneider 2011, 191–205].

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he asked for her advice directly, she surely would have suggested Kenjiro Shoda, but Takagi reached this conclusion himself, so there were two letters from Japanese mathematicians. Shoda expressed his personal appreciation for his former teacher with these words: Frl. Noether, who has brought forth so many new and fundamental ideas in the theory of hypercomplex numbers, representation theory, ideal theory, etc., is generally regarded by us foreigners as the most outstanding representative of German algebra. Like so many other Japanese algebraists, I remember with special thanks the time in Göttingen when I studied with Ms. Noether and gained so much invaluable scientific and personal encouragement from her. All of us wish very much that your efforts shall succeed in maintaining Ms. Noether for German mathematics. (Kenjiro Shoda, Osaka, 16 July 1933, [Roquette 2008]) Noether may have suggested the name Beniamino Segre, who was Severi’s assistant in Rome in 1928, the year she and Segre likely met at the Bologna ICM. Three years later, Segre gained a professorship in Bologna, but as a Jew he fell victim to the racial laws implemented in 1938, which forced his emigration to England. His testimonial for Emmy reflects the reverence Italian geometers held for her father. The tremendous scientific value of Max Noether’s geometric work is recognized by everyone who has been profoundly influenced by Riemann’s immortal work on the theory of algebraic functions and their integrals. This formed the starting point for an astonishing flowering of studies in France and especially in Italy since 1890, so that Max Noether can rightly be regarded as the founder of the great structure of today’s algebraic geometry. Ms. Emmy Noether is the worthy successor of the paternal name, although her works have a somewhat different direction and, above all, a purely algebraic emphasis. Miss Noether’s work is both remarkable and important. Together with Artin, Hasse, and van der Waerden, she is the recognized head of a school in modern studies of abstract algebra and general number theory that continues and crowns the fundamental ideas of Grassmann, Dedekind, Kronecker, and Weierstrass. (Beniamino Segre, Bologna, [Roquette 2008]) Two of the testimonials came from Viennese number theorists, Philipp Furtwängler and Anton Rella, neither of whom knew Noether well. Furtwängler offered this opinion: “Ms. E. Noether holds a leading position in the development of modern algebra. Not only has she furthered this theory partly through her own work and partly through the work of others, she has also through her selfless teaching, led by ideal goals alone, established a large circle of students, who have already today made a name for themselves in the mathematical world” (Philipp Furtwängler,

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Vienna, 29 June 1933, [Roquette 2008]). Rella was Gottfried Köthe’s doctoral adviser, which suggests perhaps a somewhat closer relationship to Noether’s sphere of influence. Rella apparently did meet her at the congresses in Bologna (1928) and Zurich (1932), but otherwise refers to secondhand knowledge of her enormous talents as a teacher: It actually seems inappropriate for me to offer a statement about a scientific personality of such eminence as Emmy Noether. I regard Ms. Noether as simply the leading figure in her special field of abstract algebra. From reports of my own former students who continued their mathematical training in Göttingen, I know that Ms. Noether, in her impulsive manner, is able to exert the greatest scientific influence through personal communication with young mathematicians, as is shown by the large number of leading young algebraists in Germany, almost all of whom regard themselves as her personal pupil or at least have been inspired by her to carry out their own research. (Anton Rella, Vienna, 9 July 1933, [Roquette 2008]) As one of Richard Courant’s closest collaborators, Harald Bohr was a frequent visitor in Göttingen. His interests in analytic number theory also drew him to Göttingen’s Edmund Landau, whom many considered the leading authority in this field. Another leading number theorist was the Cambridge mathematician G.H. Hardy, who happened to be visiting Bohr in Copenhagen when the latter received Hasse’s request. The testimonial Bohr wrote was signed by both men: Miss Noether is of paramount importance for the development of modern algebra and she is rightly regarded as the head of a school of young algebraists in and outside of Germany. The fact that algebra has experienced a new flowering and now stands at the forefront in the entire mathematical world, expanding into geometry and other areas of research, this has mainly been due to Miss Noether and her school. The influence this has had reaches far beyond the borders of Germany, and throughout the world hers is one of the best known names. (Harald Bohr and G.H. Hardy, Copenhagen, August 1933, [Roquette 2008]) The Swiss mathematician Andreas Speiser was another number theorist who took his doctorate in 1911 under Hilbert in Göttingen, although he was actually a student of Hermann Minkowski. Speiser also had strong interests in modern algebra. He arranged that Johann Jakob Burckhardt, his assistant in Zurich, translate Dickson’s seminal algebra book (Algebren und ihre Zahlentheorie, [Dickson 1927]), for which Speiser added an appendix on ideal theory. He answered Hasse’s request by writing in the name of all mathematicians in Zurich: [Emmy Noether] is undoubtedly the most important living female mathematician and possesses a highly extensive knowledge. We invited her to deliver a plenary lecture at the international congress in Zurich in

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1932 because she is one of the leading experts on modern algebra and especially since she works at the center of an excellent school in Germany. She brought the important ideas of Wedderburn and others to fruition, and as an extremely stimulating personality and creative mathematician she has exerted the greatest influence in her country. Since she lives for her science like few others, she also exercises a moral influence that should not be overlooked. To us she appears to be indispensable for Göttingen’s reputation and influence in the mathematical world. All of the younger mathematicians who studied there went through her school. (Andreas Speiser, Zurich, 1 July 1933, [Roquette 2008]) Two testimonials were written by Dutch mathematicians, one by B.L. van der Waerden. As one of her closest associates, he was especially well qualified to judge Noether’s work and its significance. Dr. Emmy Noether is a personality of unique importance in the mathematical world. Some 13 years ago, she began indicating the direction in which algebra and arithmetic should develop in her opinion, and now, in fact, they are developing under her recognized leadership in this very direction. She has held on to her own methods and problems with firm energy, even in times when the problems were considered too abstract and the methods too sterile, and now these methods have been successfully applied everywhere, especially in Germany, but also in France, Holland, Russia, America, and Japan, and they have delivered the most beautiful results. Before her leave of absence, algebraists from all over the world came to Göttingen to learn her methods, get her advice, and to work under her leadership. Her merits were recognized by the Faculty of Mathematics and Natural Sciences in Leipzig and by the German Mathematical Society, which in 1932 awarded her and E. Artin the Ackermann-Teubner Memorial Prize. (B.L. van der Waerden, Leipzig, 8 June 1 933, [Roquette 2008]) One might have anticipated that the second Dutch mathematician to write on Noether’s behalf would have been L.E.J. Brouwer, since he had known her even longer than had van der Waerden. Little had been heard from Brouwer, however, after he was dramatically purged from the editorial board of Mathematische Annalen in December 1928, and even Pavel Alexandrov was no longer in touch with him as before. Instead, Brouwer’s old nemesis, Jan Arnoldus Schouten, wrote an impassioned report, in which he expressed his outrage over the Nazi racial policies. Given the delicacy of the situation, one wonders whether Hasse might not have had second thoughts about including such a forthright political statement among these testimonials. In my opinion, Miss Noether is a first-class mathematician, indeed, she is the greatest living female mathematician in the world! Through

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7 Cast Out of Her Country hard work and sacrifice, she has done an incredible amount for mathematics and for her numerous students. Many of her students live here in Holland and gratefully acknowledge how much they have learned from her. . . . It would be a great scandal if such a powerful figure were lost due to racial prejudice. Apparently people in Germany have no conception of the outrage over this in those foreign countries friendly toward Germany. This is not a matter of Jews being beaten up here or there by misguided youth. What is happening here through the official authorities themselves causes the utmost anger among us, the friends of Germany. Only France and the Little Entente can feel happy! ... For orientation, I am of purely Aryan descent and 50% German, my mother comes from a family of Prussian officers; thus, in the eyes of the current rulers, of absolutely unobjectionable heritage. Before, during, and after the war, I was always extremely pro-German, which brought me a good deal of unpleasantness. I have never been ashamed of Germany and of my parentage, though I would be now if I were not of the conviction that what is happening is completely contrary to the essence of the German people’s soul. (Jan Arnoldus Schouten, Delft, 27 June 1933, [Roquette 2008])

Emmy Noether and the Munich mathematician Oskar Perron would seem an unlikely pairing. Perron’s tastes ran strongly toward classical mathematics, including number theory, with no signs of interest in abstract mathematics. Yet his report reflects real understanding of Noether’s work and the nature of her influence, which makes it highly likely that he knew her personally. Emmy Noether belongs among the leading personalities in modern mathematics. Already her older work on invariant theory testifies to a high level of skill; her studies on the general theory of fields and ideals are groundbreaking. Yet even more than through her printed publications, Emmy Noether has always understood how to inspire interest in young people for great scientific ideas through oral communication and personal interchange, and so a whole younger generation of algebraists has been influenced by her to a high degree and follows in her footsteps. One, however, who took inspiration from her spirit can no longer walk in her footsteps because he died in the war. For him, she erected a scientific monument with touching humility and piety; without Emmy, the name Kurt Hentzelt would be forgotten today. (Oskar Perron, Munich, [Roquette 2008]). All of the testimonials cited above took the form of short statements rather than actual reports (Gutachten). Helmut Hasse wrote a somewhat lengthier account that in some respects resembles the petition submitted by Noether’s students. He, too, emphasized that her mathematics was firmly rooted in a Germanic

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Figure 7.4: Emmy Noether departing from Göttingen for the USA, October 1933 (Courtesy of MFO, Oberwolfach Research Institute for Mathematics)

tradition. Like Weyl, but with a far stronger emphasis on her ties to Germany, Hasse sought to deflect the idea that Noether’s abstract mathematics was somehow “alien” (artfremd) or without substance. Miss Noether is the founder and leader of the modern algebraic school that developed in Germany after the war. Through a series of profound works that tie in with the life’s work of the German mathematician Richard Dedekind, and through her personal influence on numerous young German mathematicians, she laid the foundation for the transformation of traditional algebra by way of completely new general methods that have proven their strength not only in algebra itself, but which have also penetrated into other areas of mathematics, such

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7 Cast Out of Her Country as number theory, function theory, and geometry, while continuing to do so through her own collaborations as well as the impulses she gives to her numerous students. A flourishing school, comprised largely of young German mathematicians, would be robbed of its universally recognized leader if Miss Noether were forced to go abroad as a result of the current political situation. Most of these younger German mathematicians, those who have directly or indirectly gone through her school, engage with her in lively exchanges of ideas during frequent visits to Göttingen or at the conferences of German mathematicians, meetings that have a particularly stimulating and fruitful effect on their own work. All these mathematicians, and thus German mathematics in general, would suffer significant nonmaterial damage should the opportunity for such personal exchanges of ideas be taken away. Miss Noether has always felt like a German, and she still has a strong desire to remain in Germany in her position at the Göttingen Mathematical Institute, where her entire personality belongs. There, like nowhere else in the world, she has the opportunity to meet the kinds of people needed to pursue her far-reaching ideas, which she can only work out to a small extent herself. She can then, over and again, point them toward the research paths she has in mind. Because there alone, following an old tradition, one finds a select stream of mathematically talented youth capable of taking up these ideas, at least in higher semesters. In no sense can one call her mathematics “alien”. On the contrary, it has a quality much like the typical German mindset, which in its nature favors the intellectual, the theoretical, and the ideal rather than such qualities as purpose, material success, or the real. That this is so can be seen from the fact that the vast majority of German mathematicians who have found their way to her school over the past two decades are of Aryan descent. As proof of her truly outstanding significance one may further point to the fact that she is recognized as a leading contemporary mathematician even in Anglo-American countries, which favor the completely opposite orientation with more reality-based mathematical conceptions. Finally, it should be noted that Miss Noether’s father, Max Noether, was an outstanding German mathematician, who in his day received an honorary award from the Academy in Rome for his achievements in algebraic-geometric fields. Still today, for the important new Italian school of mathematics, he enjoys the greatest esteem among all German mathematicians. (H. Hasse, Marburg, 31 July 1933, [Roquette 2008])

Since Emmy Noether had already left on her vacation when Hasse sent these reports to Göttingen, she could not have read the original texts. She must have

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read them later, however, since Hasse (or possibly Weyl) ordered that copies of them be made, as she had requested. Emmy clearly felt a great deal of gratitude for the support she received during such a difficult time, even knowing that a positive outcome was improbable. In his cover letter to Valentiner, Hasse stressed that this action had a limited purpose, namely “to maintain her existence in any form at the Göttingen Mathematical Institute,” to which he added: Not only for Göttingen, but for German mathematics in general, it would be an immeasurable loss if Miss Noether found no further opportunity in Germany to continue teaching mathematics. Since her teaching stands outside the framework of the training plan for teaching candidates, but involves instead the stimulation of a relatively small group of advanced students, who mostly have academic careers in mind, I dare to hope that such an activity might not completely cross with the basic considerations and principles that led to her temporary leave of absence.31 Unlike in Courant’s case, Valentiner took a very critical view of Emmy Noether’s presumed political views, about which he wrote: Although I am aware of Miss Noether’s scientific importance, I refrain in this regard from commenting on the expert opinions herewith enclosed from a number of competent authorities. From a political point of view, to my knowledge Miss N. has stood on Marxist ground from the time of the Revolution of 1918 to the present day. And although I hold it for possible that her political views are more theoretical than conscious and practical, I do believe with certainty that her sympathies so strongly favor a Marxist political worldview that she cannot be expected to stand up wholeheartedly for the national state.32 So with all due respect for the scientific importance of Miss Noether, I am unable to support her. 33 Whether this remark played any role at all is impossible to know; it seems most likely that her case was handled in a routine manner based on the fact that she failed to satisfy §3, the Aryan paragraph. Having returned from her vacation, Noether wrote to Hasse, thanking him again for all his efforts. She then remarked: “If not for now, then the reports may help later! And it seems only right to me that they are now available!” 34 This seems to confirm that copies of the reports had indeed been made. In the meantime, Emmy began making plans for the coming academic year: 31 Translated

from [Tollmien 1990, 206]. German reads “ein rückhaltloses Eintreten für den nationalen Staat”, in conformance with §4 of the BBG. 33 Göttingen Kurator to Prussian Ministry, 9 August 1933, translated from [Tollmien 1990, 206]. 34 Noether to Hasse, 6/7 September 1933, [Lemmermeyer/Roquette 2006, 199]. The only known copies, however,are those in the Geheimes Staatsarchiv Preußischer Kulturbesitz, Berlin, available online at [Roquette 2008]. 32 The

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7 Cast Out of Her Country You have heard from Davenport that I want to go to Oxford for a term after Christmas. In the meantime, I received another offer for a research professorship in Bryn Mawr for one year (1933/34), but which I accepted for the following 34/35. I don’t have an answer yet, but I doubt that the postponement – I can’t very well be in England and America at the same time – should cause any problems. It’s a joint fellowship from Rockefeller and the Committee in Aid of Displaced German Scholars. 35 Bryn Mawr is, by the way, another woman’s college, but as Veblen later wrote me, it’s the best of these; it’s also so close to Princeton that I can come over often.. . . [Lemmermeyer/Roquette 2006, 199]

Noether further reported meeting Hamburg’s Wilhelm Blaschke and Hans Rademacher from Breslau during her vacation on the Baltic, where they discussed the forthcoming DMV conference in Würzburg. Rademacher suggested that those who had been placed on leave should feel free to attend, since this was a gathering for mathematicians and not merely professors.36 Blaschke counseled, on the other hand, that the DMV should “maintain its purely scientific, neutral character” and not take up the politically sensitive matter of the government’s recent actions. Emmy agreed with this, and thus wrote Hasse that she would probably not be coming to the Würzburg conference, one of the very few annual meetings of the DMV she did not attend. One week later, on September 13, Noether wrote Hasse again to report that she had now received definitive news: her teaching certification had been withdrawn on the basis of §3 of the BBG, thus owing to her racial status. Payment of her salary would then terminate at the end of the month, leaving her no choice but to go abroad (Fig. 7.4). She thanked Hasse again for his efforts, and reassured him that the reports could well be valuable later. Max Deuring, her star student, would be coming to the DMV meeting in Würzburg, so Hasse would be able to learn about his and her latest ideas from Deuring. As will be described in the next chapter, Emmy’s conjecture that she could postpone accepting the offer from Bryn Mawr was incorrect. Her plans then quickly shifted and she readied herself for the transatlantic voyage that brought her to the small college outside Philadelphia that would become her new academic home. The following section offers a brief account of mathematics at Bryn Mawr College in the years preceding Noether’s arrival.

35 The Emergency Committee in Aid of Displaced Foreign Scholars was created in 1933 to assist those who were barred from teaching by the Nazis. The program was later expanded to include Austria, Czechoslovakia, Norway, Belgium, the Netherlands, France, and Italy. 36 In February 1934 Hans Rademacher lost his professorship in Breslau under §4 of the BBG (see Section 8.3).

Chapter 8

Emmy Noether in Bryn Mawr 8.1 Bryn Mawr College and Algebra in the United States In the annals of higher education for women, two elite colleges were particularly important for mathematics: Girton College, in Cambridge, England and Bryn Mawr College, near Philadelphia, Pennsylvania. Both had significant historical ties with Göttingen.1 A central figure in this story was Charlotte Angas Scott, who studied at Girton from 1876–1880 (see Section 2.1). After completing her studies, she finished eighth in the Tripos examination, though her achievement went officially unacknowledged. Later, in 1885, she was awarded a doctorate from the University of London on the basis of an external examination. That same year, Scott joined the founding faculty at Bryn Mawr (Fig. 8.1), where she taught until her retirement in 1924. Two of her seven doctoral students, Isabel Maddison and Marguerite Lehr, also became fixtures of the Bryn Mawr faculty [Green/La Duke 2009, Green/La Duke 2016].2 Maddison had studied alongside Grace Chisholm at Girton College. Then, in the mid-1890s, both went on to do graduate work under Klein in Göttingen, the first German university to spearhead opportunities for (foreign) women to earn doctoral degrees. By this time, females had already gained the right to study at Paris University, though none had yet taken an advanced degree in mathematics. Scott’s first doctoral student, Ruth Gentry, spent one semester studying at the University of Paris. Chisholm took her Ph.D. magna cum laude in 1895 after passing the usual qualifying examinations. Sofia Kovalevskaya’s case twenty years earlier was altogether different; she never set foot in Göttingen and earned her doctorate in absentia, a quite common practice that the Prussian Ministry prohibited soon thereafter. In their study of American women mathematicians during the period 1891 to 1906, Fenster and Parshall found that of the 18 most active 1 On

Girton, see [McMurran/Tattersall 2017]; on Bryn Mawr, see [Parshall 2015]. of her students were recently portrayed in [Lorenat 2020].

2 Three

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fully half had gone to Göttingen for part of their studies [Fenster/Parshall 1994, 241].

Figure 8.1: Faculty and Students at Bryn Mawr College, Spring 1886: standing just left of the entrance is Charlotte Angas Scott, seated next to her is M. Cary Thomas, first dean and second president of the college, seated in the middle is President James E. Rhoads, and standing at the right of the entrance is Woodrow Wilson, who taught history and political science (Bryn Mawr College Special Collections) As the only college for women in the United States with a doctoral program in mathematics, Bryn Mawr naturally drew on talented graduates from various undergraduate institutions around the country, but especially the Seven Sisters colleges, to which Bryn Mawr belonged. The others were Mount Holyoke, Smith, Wellesley, and Vassar, all four independent liberal arts colleges for women, as well as Radcliffe and Barnard, which were associated with Harvard and Columbia, respectively. Hunter College in New York City, which specialized in training teachers, was another important outpost for aspiring female mathematicians. In their

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study of women who took doctoral degrees in the United States during the first four decades of the twentieth century, Judy Green and Jeanne LaDuke found that females constituted over 14 percent of the degrees granted, a proportion that would not be reached again until the 1980s [Green/La Duke 2009]. In 1896, Bryn Mawr established a Mathematical Journal Club, which met every other week to hear special reports and lectures presented by faculty members as well as graduate students. Two distinguished professors from nearby Haverford College, Frank Morley and E.W. Brown, also gave talks at these meetings. Bryn Mawr also offered a European Fellowship Program to outstanding students, several of whom took up studies in Göttingen. Little wonder that Felix Klein took a special interest in Bryn Mawr College, where his youngest daughter, Elisabeth, spent a year abroad before completing her studies in 1911. Her father briefly visited the college in 1896, when he met with Charlotte Angas Scott. She wrote him the following year to report: “I am expecting to send two of my best students to Göttingen next year; to both of them have been awarded College Fellowships . . . . One of them you met when you were here that Sunday afternoon.” 3 Scott was referring to Virginia Ragsdale and Emilie Norton Martin, who spent the academic year 1897/98 studying with Klein and Hilbert. They were joined by a third Bryn Mawr student, Fanny Gates, along with Anne Bosworth, a graduate from Wellesley College who went on to take her Ph.D. under Hilbert. Martin afterward returned to Bryn Mawr and took her doctorate there in 1901. She later joined the faculty at Mount Holyoke College, where she went through the ranks from instructor to full professor, serving as department chair from 1927 through 1935. By the time Emmy Noether habilitated in Göttingen in 1919, far fewer Americans were studying there. Still, she clearly made an effort to stay abreast of mathematical research in other countries, including the United States. This is confirmed by numerous reviews she wrote for the Jahrbuch über die Fortschritte der Mathematik, long the premier abstracting journal for mathematics. Among the American authors whose works she abstracted were Leonard Eugene Dickson, Eric Temple Bell, Joseph Ritt, Joseph Wedderburn, C.C. MacDuffee, Constance R. Ballantine, and Olive C. Hazlett. The latter three were all doctoral students of Dickson, the leading algebraist in the United States, a role he assumed from his mentor at the University of Chicago, E.H. Moore.4 Noether reviewed Dickson’s preliminary article [Dickson 1923a], which preceded the publication of his book Algebras and their Arithmetics [Dickson 1923b]. Dickson’s book was also reviewed by Hazlett in [Hazlett 1924], where she pointed out that its approach drew on the earlier work of Wedderburn on division algebras. Four years later, Dickson’s book appeared in a revised German edition, [Dickson 1927], which received a lengthy review in [Hasse 1928]. Both reviews sang the praises of these two books, but from quite different perspectives. Hasse noted, for example, that the German edition might have profited from a presentation 3 Scott 4 On

to Klein, 19 March 1897, cited in [Green/La Duke 2016, Entry: Martin]. Dickson see [Fenster 2007]; on Moore, see [Parshall/Rowe 1994].

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that made use of the by now standard terminology of ideal theory introduced by Emmy Noether and her school [Hasse 1928, 93]. Olive Hazlett reviewed her mentor’s book [Dickson 1923b] along with a much lengthier Italian monograph [Scorza 1921], which Hasse also mentioned in his review. Hazlett found Scorza’s book hopelessly pedantic and almost unreadable, adding this additional criticism: One notes with dismay that there is no index. This lack is an inconvenience in any book of this size and is only slightly ameliorated by putting in bold-face type the caption of every section and of every definition. In fact, one might almost say that the presence or lack of an index is a characteristic invariant which distinguishes Anglo-American texts from Continental ones. [Hazlett 1924, 270]5 Hasse, by contrast, wrote that Dickson might have given his readers some guidance as to the prehistory of this theory, referring to Scorza’s book for more details [Hasse 1928, 97]. These reviews in some ways reflect the oft-mentioned tension between American and German algebraists during this era.6 Olive Hazlett was Dickson’s second female doctoral student, following Mildred Sanderson, who died of tuberculosis at age 25. In a tribute to the latter, Dickson wrote these comments about her doctoral dissertation [Sanderson 1913]: “This paper is a highly important contribution to this new field of work; its importance lies partly in the fact that it establishes a correspondence between modular and formal invariants. Her main theorem has already been frequently quoted on account of its fundamental character. Her proof is a remarkable piece of mathematics.” 7 Dickson also considered Olive Hazlett to be a particularly talented mathematician. In a letter recommending her to Edgar J. Townsend, a former student of Hilbert who became chairman of the department at the University of Illinois, Dickson wrote: . . . She has shown more independence in research than any of our Doctors for [the] past ten years and has published perhaps a dozen excellent papers in several branches of algebra showing real originality and the ability to attack successfully quite fundamental problems. Her tested ability and her continued eagerness for research make it certain she will have a very successful career in research . . . .8 Hazlett gained an appointment at Illinois that year, and she spent the remainder of her career there. One of her students from the mid-1930s was the Hungarian-born Paul Halmos, who remembered her course on algebra. He and his fellow students thought of her as a “famous and important mathematician: 5 One

might wonder about the size of her sample space for making such a sweeping claim. topic lingers in the background in several of the papers in [Gray/Parshall 2007]. 7 Cited in [Green/La Duke 2016, Entry: Sanderson]. Hazlett drew heavily on results from [Sanderson 1913] in her study [Hazlett 1921], a paper Emmy Noether reviewed for the Jahrbuch. 8 Dickson to Townsend, 30 March 1925, cited in [Green/La Duke 2016, Entry: Hazlett]. 6 This

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she published papers and she taught advanced courses” [Halmos 1985, 45]; they learned algebra from her mainly from volume 1 of van der Waerden’s Moderne Algebra, which within only a few years had found an international readership. Olive Hazlett had previously taught for two years at Bryn Mawr College before she was hired by Mount Holyoke in 1918. That same year, Anna Johnson Pell (Wheeler) moved from Mount Holyoke to Bryn Mawr as associate professor. In their comprehensive study [Green/La Duke 2016], the authors noted that three women were singled out as leading research mathematicians during the period 1900 to 1940: Charlotte Scott, Olive Hazlett, and Anna Pell Wheeler, all three of whom spent part of their careers at Bryn Mawr. Emmy Noether met Hazlett in 1929, when the latter took a leave of absence to study in Göttingen. By this time, Noether was regarded as one of the leading algebraists in the world. A small, but noteworthy confirmation of the esteem she enjoyed in the United States occurred in 1931 when mathematicians at the University of Chicago requested that she send them a photo they could hang on one of the halls in Eckhardt Hall, their new headquarters built in 1930. She was eager to oblige, but turned to Hasse for help;9 his photo of her on the ship (see Fig. 6.3) that took them to Königsberg was, in her opinion, the only decent picture of her she could send. Her problem: she had only one enlargement of it and this was already very battered. Hasse apparently came to the rescue, and Noether’s smiling face adorned the walls of Echardt Hall for many years. Chicago was long the preeminent center for algebraic research in the US, which surely made this photo a fitting acquisition. When Noether began teaching a seminar at Princeton’s Institute for Advanced Study in 1934, one of those who attended was A.A. Albert, Chicago’s leading algebraist. Her strongest new American connection, though, was with the head of the mathematics program at Bryn Mawr College, Anna Pell Wheeler, who became one of Emmy Noether’s good friends.10 Born Anna Johnson, the daughter of Swedish immigrants, she grew up in Iowa and studied as an undergraduate at the University of South Dakota in Vermillion. This prairie institution was founded in 1862, thus 27 years before South Dakota became a state. Anna and her older sister Esther boarded with Alexander and Emma Pell, both of whom were immigrants from Russia. Alexander Pell was Anna’s mathematics instructor, and he soon discovered her unusual talent. After graduating in 1903, she went back to Iowa for her master’s degree, then went on to Radcliffe, where she took a second master’s, and finally arrived in Göttingen in 1906 on a one-year fellowship from Wellesley College. There she attended courses offered by Hilbert, Klein, Hermann Minkowski, Gustav Herglotz and Karl Schwarzschild. Since leaving Vermillion, Anna Johnson had remained in touch with Pell, whose wife died in 1904. He traveled to Göttingen, where they married, returning to the USA in August 1907. What she knew about her husband’s earlier life 9 Noether 10 On

to Hasse, 2 December 1931, [Lemmermeyer/Roquette 2006, 139]. her career at Bryn Mawr, see [Parshall 2015, 77–80].

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remains unclear, but the gist of this became known after she died [Pipes 2003]. Alexander Pell was born in 1857 in Moscow as Sergei Petrovich Degaev. He attended military schools, but then left this career behind him in 1879 to study engineering science in St. Petersburg. During this time, he became involved with the revolutionary cell known as the People’s Will, a group that became famous for the assassination of Czar Alexander II in March 1881. Degaev was arrested in Odessa the following year, but escaped punishment after agreeing to collaborate with the secret police, headed by the much-feared Georgy Sudeykin. Degaev informed Sudeykin of the whereabouts of several leading members of the People’s Will, including Vera Figner, and subsequent arrests nearly destroyed the military wing of the organization. Sudeykin later approached Degaev’s brother, Vladimir, who informed Sergei and others. They then hatched a plan to assassinate Sudeykin and carried it out successfully in December 1883. Afterward, Degaev fled to Paris, where members of the People’s Will financed his escape to America on condition that he should never set foot in Russia again. Once in the United States, he and his wife began new lives in 1891 as Alexander and Emma Pell. Six years later, Pell was awarded a Ph.D. in mathematics from Johns Hopkins University. Pell’s second wife, Anna, was 26 years younger than he, and when she returned with him to the University of South Dakota in the fall of 1907 she remained determined to finish her doctorate in Göttingen. She had begun work on a dissertation under Hilbert, and in the spring of 1908 she returned to Germany in order to complete it. By the end of that year, however, shortly before she planned to take her final examinations, her relationship with Hilbert had worsened to the point that she decided to return to the US without her degree.11 In the meantime, her husband had taken a new position as assistant professor at the Armour Institute of Technology (now Illinois Institute of Technology) in Chicago. That proved convenient for Anna Pell, who in January 1909 enrolled at the University of Chicago, where she took her doctorate the following year under E.H. Moore, graduating magna cum laude. Moore accepted her thesis work in functional analysis under Hilbert, which she only had to rewrite in English. The following year, Alexander Pell suffered a stroke while teaching, so Anna taught his class for the remainder of that semester. Though he recovered, the stroke effectively ended Pell’s career; his young wife cared for him until he died in 1921. In the fall of 1911 Anna Pell became an instructor at Mount Holyoke College, where three years later she was promoted to associate professor. In 1918, she was appointed at the same rank to Bryn Mawr College, filling the vacancy left by Olive Hazlett. Five years later, Anna Pell became the first woman to deliver an invited address at a meeting of the American Mathematical Society; the second was Emmy Noether, who a decade later spoke to a large audience at Columbia University. In 1924, following Charlotte Scott’s retirement, Pell became head of the mathematics department. In July 1925 she married Arthur Leslie Wheeler, 11 Nothing more seems to be known about the nature of this conflict, but Anna Pell’s experience was by no means unique; Hilbert could be both demanding and fickle as a mentor.

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a classicist who taught at nearby Princeton University. They resided together in Princeton until his death in 1932, one year before Emmy Noether’s arrival in Bryn Mawr.

8.2 Emmy Noether’s New Home Well before 2 September 1933, when Emmy Noether was officially removed from the Göttingen faculty, efforts had begun to find a teaching position for her abroad. The initial idea of bringing her to Bryn Mawr came from Princeton’s Solomon Lefschetz, who had met Noether in the summer of 1931 during a visit to Göttingen. Lefschetz made this suggestion to Anna Pell Wheeler some time during the spring of 1933. This was noted in a letter from July 11, written by President Marion Edwards Park to the Rockefeller Foundation, in which she inquired about supplemental funding for a foreign scholar who would be chosen by Bryn Mawr College. Park referred to a letter she had received from Edward R. Murrow, then assistant secretary of the Emergency Committee in Aid of Displaced Foreign Scholars, an organization which during the 1930s helped place some 300 Germans who had been dismissed from academic positions. Murrow’s organization offered Bryn Mawr $2,000 for this purpose, assuming that the college would be able to pay another $2,000 in salary [Kimberling 1981, 30]. Park’s letter implied that Emmy Noether was the prime candidate her institution had in mind for this position. Bryn Mawr was not the only institution, however, that had taken an interest in her case. In fact, the Principal of Somerville College in Oxford, Helen Darbyshire, had already entered into preliminary negotiations with Noether as well as with representatives of the Rockefeller Foundation in Paris. Somerville College was founded in 1879 as the the sister school of Girton College in Cambridge University. It was the first non-denominational college in Oxford. The other women’s college, Lady Margaret Hall, which opened in the same year, was strictly Anglican. Initially, Emmy Noether hoped to spend at least one semester in England. President Park had in the meantime received a positive response from the Rockefeller Foundation, and on August 4, 1933, she wrote Noether to offer her an appointment at Bryn Mawr for the coming academic year as a research professor with a salary of $4,000 (the equivalent of over $76,000 today). These funds would be made available to support her research as well as consultation with advanced students. Noether’s temporary position at Bryn Mawr was to be part of the Rockefeller Foundation’s $1.5 million aid package for displaced scholars. It seems likely that Emmy Noether had not anticipated receiving such a generous offer from the United States. On the other hand, her negotiations with Somerville College had progressed to the point that she felt certain to receive a firm offer from Darbyshire. In view of this attractive possibility, Noether nonchalantly suggested postponing her stay at Bryn Mawr until the academic year 1934/35. President Park did not reply immediately, and since Noether was unaware of the conditions set by the Emergency Committee and the Rockefeller Foundation, she

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made her plans accordingly. She described these in a letter to Richard Brauer from September 13: . . . As for myself, I have been invited to lecture in Oxford for one term, I have chosen the time between Christmas and Easter. Subsequently, I was also offered a research professorship in Bryn-Mawr for 1933/34; I have asked to have it postponed for 1934/ 35 as I have already accepted the Oxford offer. I have no answer yet, but I think it should be all right – Bryn Mawr is a women’s college, but Mitchell12 and others are there as professors . . . [Shen 2019, 57] Richard Courant had been reluctant to leave Europe when he was placed on leave, and Emmy, too, felt disinclined to move suddenly overseas. Confusion must have reigned for a few weeks, as it was not until October 2 that President Park received definite word that Noether had accepted the invitation. The following day, the college opened its fall term, and Park thus had the opportunity to announce this important news in her convocation address. After a rapid fire of cables, I heard yesterday that we are to have a most distinguished foreign visitor . . . in the faculty for the year, Dr. Emmy Noether, a member of the mathematical faculty of the University of Göttingen. Dr. Noether is the most eminent woman in mathematics in Europe and has had more students at Göttingen than anyone else in the department. With other members of the faculty, Dr. Noether was asked to resign from the University in the spring. To our great satisfaction the Institute of International Education and the Rockefeller Foundation have united in giving to the college a generous grant which makes it possible for the Department of Mathematics to invite her here for two years. Her general field is Algebra and the Theory of Numbers. Dr. Noether does not, I understand, speak English well enough to conduct a seminar at once but she will be available for consultation by the graduate students and later I trust can herself give a course. I need not say that I am delighted Bryn Mawr College is one of many American institutions to welcome the scholars whose own country has rejected them. For the time only we must believe, Germany has set aside a great tradition of reverence for the scholar and for learning. I am glad also that the college can entertain so distinguished a woman and that the students in mathematics can profit by her brilliant teaching. [Shen 2019, 58–59] These remarks reveal that people at Bryn Mawr College, including Anna Pell Wheeler, clearly had little idea what to expect. No one even knew whether this distinguished foreign visitor could speak English. Upon her arrival, Park 12 Howard Hawks Mitchell was an algebraist on the faculty at the University of Pennsylvania. He was then teaching a course at Bryn Mawr on number theory to six students there [Parshall 2015, 79].

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was pleasantly surprised to realize that Noether’s “English proves to be entirely usable.” Originally, Prof. Noether was to arrive on November 3 in New York, but she sent a cable on October 27 stating that her trip had been delayed due to some problems obtaining a visa. By November 7, President Park was able to inform a reporter for the Philadelphia Record that “Dr. Emmy Noether has just arrived from Germany on the Bremen after a voyage which she greatly enjoyed” [Shen 2019, 60]. The college was eager to publicize this momentous event, but at the same time took due care not to allow the press to badger their guest with questions about events in her native country. Park noted that [she] cannot speak of German conditions during her American residence. She has a brother and many friends in Germany and she wishes herself to return for a summer. It is clear that discreet silence on her part is necessary if she is to feel at ease about her family and insure her own return. [Shen 2019, 60] Later that month, President Park invited mathematicians from the region (including Princeton, the University of Pennsylvania, and Swarthmore) to a special lecture that Noether would give on December 15. Park also wrote to Warren Weaver in New York, inviting him to come down to see Noether “in action” on that occasion [Parshall 2015, 80]. Emmy stayed at a boarding house, run by a Mrs. Hicks, that was located only a short distance from the campus.13 Since Anna Pell Wheeler often had students over for tea in her apartment on campus, Emmy wanted to reciprocate at her new lodgings. Many years later, Ruth Stauffer McKee remembered the scene this way: . . . Mrs. Hicks planned a lovely tea party and Miss Noether asked Mrs. Wheeler to preside at the tea table. The setting was complete, the guest arrived, and Miss Noether beamed happily; but soon she was noticeably upset and went out to the kitchen for Mrs. Hicks. It was obvious that pouring tea, rather than being an honor, was an onerous job; and she had asked Mrs. Hicks to pour so that her good friend could enjoy herself at the party. Once again all was sunshine and light. In other words, correct an apparent problem in the simplest way. [Quinn et al. 1983, 143] Some months later, Emmy wrote a long letter to Hasse, part of which touched on her life at Bryn Mawr: The people here are all very accommodating and have a natural warmth that’s truly winning, even if it doesn’t go very deep. You are constantly 13 In a postcard to Dirk Struik, she gave her address as The Clifton, 14 Elliott Avenue, Bryn Mawr.

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8 Emmy Noether in Bryn Mawr getting invitations; I’ve also gotten to know all kinds of interesting people who don’t belong to the college. Incidentally, I am doing a seminar with three “girls” – they are only rarely called students – and a lecturer; right now they are reading van der Waerden Bd. I with enthusiasm, an enthusiasm that has them working through all the exercises – certainly not demanded by me.14 In between I give them a little Hecke, opening chapter [Erich Hecke’s Vorlesungen über die Theorie der algebraischen Zahlen]. For next year, however, there is – really American – an Emmy Noether Fellowship, which will probably be divided between a MacDuffee student15 and one from Manning-Blichfeldt-Dickson;16 the former seems to have some level. Frl. Taussky is also likely to come with a Bryn Mawr stipend; she had applied for it last year, but it wasn’t granted due to lack of funds – in the past, they issued five such scholarships per year, the depression is everywhere! Finally, one of Ore’s students has applied for a National Research Fellowship to come here.17

That first year at Bryn Mawr, Ruth Stauffer was one of four “Noether girls” who struggled with van der Waerden’s Moderne Algebra. Two years earlier she had earned her undergraduate degree at Swarthmore, where her adviser was Arnold Dresden. He encouraged Stauffer to do graduate work and helped her gain a graduate scholarship at Bryn Mawr for the 1931/32 academic year; she already had her master’s degree when Noether arrived. Bryn Mawr then had four mathematicians on its faculty – Wheeler, Marguerite Lehr, Noether, and Gustav A. Hedlund – all of whom (except for Noether) taught undergraduate as well as graduate courses. Hedlund, an expert on topological dynamics who took his doctorate under Marston Morse at Harvard, had only recently come on board. None of the graduate students at Bryn Mawr had ever been exposed to abstract algebra, and Ruth Stauffer remembered the shock of trying to figure out how to translate all the strange concepts in German. Miss Noether gave some simple advice: don’t bother, just read the German. “That is the way our strange method of communication began. Although we students were far from conversant with the German language, it was very easy for us to simply accept the German technical terms and to think about the concepts behind the terminology. Thus from the beginning we discussed our ideas and our difficulties in a strange language composed of some German and some English” [Quinn et al. 1983, 142]. After more than four months in the United States, Noether wrote to Pavel Alexandrov with various news and some personal impressions. She had last written 14 When volume 1 of van der Waerden’s Moderne Algebra first came out it received an enthusiastic review from Olga Taussky and Hans Hahn (see [Koreuber 2015, 242]). One point of praise were the exercises, which they claimed could be solved by anyone who really understood the book! Taussky reviewed volume 2 on her own [Koreuber 2015, 243]. 15 Grace Shover Quinn took her doctorate at Ohio State University in 1931; her dissertation dealt with arithmetic in linear associative algebras. 16 Marie Weiss took her Ph.D. from Stanford University in 1928. 17 Noether to Hasse, 6 March 1934, [Lemmermeyer/Roquette 2006, 204].

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him in May 1933 from Göttingen and wanted to break the long silence. Apparently she had received a preliminary inquiry from Moscow about a potential professorship in algebra there, which she assumed was initiated by Alexandrov. 18 In the meantime, she had responded with a letter and postcard, but had received no reply in return. Clearly, she needed to keep all her irons in the fire, as the long-term situation remained very unclear. She now thought, though, that there might be chances of an extension after two years, to which she added: I hope we will meet in Princeton in the meantime! I play there once a week; it’s actually a Göttingen rendez-vous! The level of the faculty there is, of course, higher than here: but insofar as the students are concerned, the difference between male and female does not seem to be as great as I originally thought. My three girls, one of whom is a lecturer, are reading v.d. Waerden enthusiastically; an enthusiasm that goes so far that they’re doing all the exercises in it, an amazing feminine thing that to some extent frightens me. Next year will see an increase in the female contingent; they’ll have – truly American – an Emmy Noether Fellowship for a MacDuffee student19 who apparently knows something about algebra and number theory. In addition, Miss Taussky will probably get a foreign fellowship and an Ore student a local fellowship. By the way, my English is absolutely smooth, with the result that the last miserable remains of my Russian have disappeared.20 Since she taught mainly by conducting dialogues, one can easily believe that by this time her English had become very fluent. Noether was obviously looking forward to the new crop of post-docs who would arrive in the coming year: Grace Shover (Quinn), Marie Weiss, and Olga Taussky (Taussky-Todd). Ruth Stauffer would stay on and eventually become Noether’s only doctoral student in the United States. Emmy had rented out her furnished apartment, and hoped she could continue doing that in the future. When she left in the fall of 1933, she imagined the trip as a long vacation, but to her own surprise, it took little time for her to feel acclimatized. As she wrote Alexandrov, “the original impossible idea of staying [in the USA] no longer seems impossible at all.” She was still getting plenty of algebraic news, both from inside and outside of Göttingen, and it seemed everyone was continuing to work intensely. Her sources suggested that Hasse would probably be called to Göttingen, but this was still up in the air. F.K. Schmidt was teaching in her place, and people wrote her that he was contributing a lot to maintaining her tradition. Of course, these were fragmentary impressions, “what else will happen there remains in the dark!”, she added. 18 In his memorial lecture from 5 September 1935, Alexandrov mentioned various efforts on his part to offer her such a position [Alexandroff 1935, 10]. 19 Cyrus Colton MacDuffee taught at Ohio State University. 20 Noether to Alexandrov, 19 March 1934, [Tobies 2003, 105]

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Emmy asked again about the present state of [Alexandroff/Hopf 1935], the now long-awaited book project. She had noticed their research announcements in the Comptes Rendus of the French Academy, a clear indication that their collaboration was still going strong. Speaking of algebra, she was still very excited about the prospects for work she had done with her doctoral student Chiungtze Tsen during the previous summer. Together they proved a function-theoretic analogue of Frobenius’ theorem for division algebras over the real numbers [Tsen 1933]. 21 She gave this amusing account of the background to the “Tsen” theorem: . . . prodded by his constantly repeated questions and conjectures I got through, so that his part in this was psychologically not insubstantial, apart from the fact that he also made a few calculations. But it was surprising for everyone that there are no skew fields of finite degree over algebraic functions of a complex variable with coefficients in an algebraically closed field. Witt and others are following up intensively on that, and Artin, too, has used it for a dissertation topic. Emmy also wrote that she had passed on Alexandrov’s greetings to Anna Pell Wheeler, whom he had met during his year in Princeton with Hopf. At that time she lived in Princeton with her husband, but following his death she had moved back to Bryn Mawr. In general, Noether found the Americans she met very friendly, but also somewhat superficial. She made an exception for Wheeler, though, whom she found very impressive. Emmy was happy to report that Mrs. Wheeler had taken her on sightseeing trips in her car, so she had visited the Jersey coast, Manasquan, Atlantic City, and other places over the Easter holiday. She had also attended a mathematics conference held in Cambridge, and in the meantime had received invitations to visit other places in the fall, such as New Haven, Providence, and also a major conference in New York, where she was invited to deliver a lecture. She was even thinking of a trip to Canada and the Great Lakes. A woman from Toronto who had once attended lectures in Göttingen sent her an invitation, but Emmy suspected this was only a private undertaking, despite the fact that the letter came on official departmental stationery. Noether wrote this letter shortly before she made her first trip to New York to attend the spring meeting of the American Mathematical Society, which was held on March 30–31 at Columbia University [Kline 1934]. She was not on the program, but evidently she had already received an invitation to speak at the next meeting to be held on October 27. On that occasion she spoke to a large audience about “Modern hypercomplex theories” [Kline 1935]. At the March meeting, she made some remarks after one of the talks that caused Solomon Lefschetz and A.A. Albert to sit up and take notice. What she had in mind seemed to her 21 In 1877 Georg Frobenius showed that up to isomorphism there were only three finitedimensional associative division algebras over the real numbers: R, the complex numbers C, and the quaternions H. Tsen’s theorem states that over a function field Ω(x) where the coefficient field Ω is algebraically closed the only finite-dimensional division algebras are isomorphic to Ω.

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substantiated by ideas in a no longer extant letter from Hasse, which Noether interpreted as hinting at a solution to (or perhaps progress on) Hilbert’s twelfth Paris problem.22 She mentioned this in her letter to him: I also saw . . . that the geometers had long had the p-dimensional manifold corresponding to their field of Abelian functions; of course without any idea of number theory. Lefschetz and Albert were very astonished when I said, after a lecture (at the meeting in New York), that it seemed to me that by extending complex multiplication one has the material for solving Hilbert’s problem using Abelian functions. The next day your letter came saying you had really done it! However, with much deeper methods!23 Emmy Noether remained in touch with her Göttingen students as well, including Ernst Witt, Erna Bannow, and Chiungtze Tsen. She learned from them that Hasse might begin teaching there already in the coming summer semester. Since she was planning to spend a few weeks in Göttingen during early June, chances seemed very good that they could meet again. At any rate, she was keen to gain a teaching contract for Max Deuring, who had not yet habilitated, and hoped that Hasse would be able to bring that about.24 In late April, roughly three weeks before her departure, Noether learned from F.K. Schmidt that Hasse had indeed been offered Weyl’s chair in Göttingen. She then wrote to tell him how happy she was to get this news: “Now Göttingen will remain in the center! Congratulations! But actually I wish you even more luck with your latest mathematics: Hilbert’s problem of class field construction and Riemann’s conjecture for function fields at the same time, that’s something!” 25 Emmy’s euphoria over Hasse’s latest results leads one to wonder what was in his most recent letters. As Lemmermeyer and Roquette pointed out, there is no evidence Hasse thought he had found a way to solve Hilbert’s twelfth problem, so what he had in mind for the first topic remains unclear. His results on the second appeared in [Hasse 1934], which represents a first step toward “Riemann’s conjecture for function fields,” a problem André Weil famously solved in 1941. She wrote in a similar vein to Alexandrov: . . . [Hasse] has been doing splendid things lately; he has once again solved one of the “Hilbert problems” so that not many more are left. He can, namely, in generalization of complex multiplication of elliptic functions, now construct all “class fields”, i.e. all rel[ative] abelian number fields by “divisor values of abelian functions”. Up until now, we had nothing here except for Hecke’s results for quadratic base fields. 22 Noether’s interpretation of what Hasse claimed to have proved however; see [Lemmermeyer/Roquette 2006, 208]. 23 Noether to Hasse, 26 April 1934, [Lemmermeyer/Roquette 2006, 24 Noether to Hasse, 6 March 1934, [Lemmermeyer/Roquette 2006, 25 Noether to Hasse, 26 April 1934, [Lemmermeyer/Roquette 2006,

was apparently incorrect, 206]. 203]. 206].

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8 Emmy Noether in Bryn Mawr And Hasse actually arrived at this as a by-product to the analogue of Riemann’s conjecture for function fields, which he also proved.26

Emmy held out high hopes for the future of algebra in Göttingen As she wrote to Alexandrov, during the past winter semester F.K. Schmidt had carried this burden all alone. Gustav Herglotz was nowhere to be seen, no doubt due to the total disruption of normal relations at the Mathematics Institute. By this time, Noether’s travel plans had firmed up somewhat, and she asked Hasse again about when he would be in Göttingen. The courses at Bryn Mawr would end in mid-May, followed by exam week (which did not involve her), and then “a solemn ceremony at the end, with triple graded gowns, B.A., M.A., or Ph.D.” She would visit with the Artins in Hamburg for a few days and from there travel to Göttingen, arriving in early June. “I’m especially happy for Deuring about your coming,” she wrote; “I hope that the habilitation in Göttingen will go quickly; he stupidly missed that chance before going to America, and in Leipzig it seems to be dragging on because of the new regulations.” Noether regarded Deuring not only as her most talented student but also as the ideal candidate to fill her shoes. She had been very actively involved in supporting his report on algebras [Deuring 1935], a work that would replace the earlier standard study [Dickson 1927]. In fact, she had originally recommended him for this project, which took far longer than planned, primarily because of the fast-breaking developments in this field. Under normal circumstances, she would have been listed as coauthor of this work, which she went through carefully during her last visit to Göttingen in the summer of 1934.27 Hasse also held Deuring in high esteem and was therefore eager to appoint him. He soon came to realize, however, that the politicized student body, largely under the influence of his new colleague and co-director, Erhard Tornier, would make this very difficult. When, in April 1935, Otto Toeplitz wrote to Hasse inquiring about suitable candidates for Bonn, the latter replied: “Deuring has not yet even habilitated, though plans are now in place to do that here. I’m going to have a tough fight in this case because his type of quiet scholarly manner is not what is wanted here. Mathematically he is completely first-class” [Koreuber 2015, 249].

8.3 Emmy’s Efforts on Behalf of Fritz Noether Over the Christmas holiday, Emmy Noether traveled to Cambridge to attend the joint annual conference of the two mathematical societies.28 Although we know nothing about which lectures she heard or whom she spoke with on this occasion, 26 Noether

to Alexandrov, 3 May 1934, [Tobies 2003, 107] an account of Noether’s role behind the scenes, see [Koreuber 2015, 245–255]. 28 At some point it was decided that these joint conferences of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA) should take place in early January, and today they are often simply called the January meeting. 27 For

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the experience must have reminded her of the ICMs in Bologna and Zurich, if only for its sheer size with over 250 members in attendance. During her visit, Noether stayed with two of her European friends, Dirk Jan and Ruth Ramler Struik. She first met them in 1925 when Dirk Struik was a Rockefeller fellow in Göttingen. Norbert Wiener from the Massachusetts Institute of Technology (MIT) also happened to be visiting that year, and they, too, became friends. Through Wiener’s intervention, Struik received an invitation to join the MIT faculty the following year. They had thus arrived on the North American shore well before the huge wave of immigration that began in 1933. Dirk became a US citizen soon after Emmy Noether’s visit, his wife only in 1939. In the meantime, they had three young daughters, so their house in Cambridge was a lively place to stay.29 Dirk Struik was one of the organizers for this conference, which featured various venues, alternating between MIT and Radcliffe College. He chaired a special symposium on probability theory and presumably had other duties as well, as this was a lavish affair [Richardson 1934]. Over 300 attended the evening dinner held in the Walker Memorial Building at MIT, during which the geometer Julian Lowell Coolidge presided. An entertaining after-dinner speaker, Coolidge also spoke the next day on “The rise and fall of projective geometry.” Before the dinner, MIT offered an exhibition of various technical artifacts of importance for applied mathematics. For the less engineering-minded, one could instead visit the Isabella Stewart Gardner Museum, and there were afternoon tea parties at Harvard’s Lowell House and Radcliffe’s Agassiz House. The Boston Symphony Orchestra even gave a complimentary concert. How much of this, if any, Emmy Noether would have taken in, no one will ever likely know. Probably she spent her free time visiting with Struik’s wife and her three little girls. In fact, they had much in common, as Ruth Struik was herself an accomplished mathematician, having taken her doctorate in Prague in 1919 under Georg Pick30 at the German Charles University, the first woman to achieve this distinction [Bečvářová 2018]. Her expertise was in foundations of geometry, whereas Dirk Struik was mainly known for his work on differential geometry and tensor analysis, much of the latter done in collaboration with J.A. Schouten in Delft. Neither thus shared strong mathematical interests with Emmy Noether, but they were nevertheless strongly attracted to her personality and human warmth. Little is known about the circumstances of her stay in Cambridge, aside from what she briefly mentioned in a letter to Dirk Struik, written in Bryn Mawr on 25 January. There she expressed her thanks to Ruth Struik for her gracious hospitality and hoped that she had not been offended by her guest’s occasional refusals to accept various kind offerings – perhaps Emmy was trying to lose weight? 29 Information on their personal lives is drawn from [Freistadt 2010], [Rowe 2018a, 379–393]; see also [Rowe 1994]. 30 Pick befriended Einstein during the academic year 1911/1912 when the latter taught in Prague. In July 1942 the Nazis sent Pick to Theresienstadt, where he died two weeks later.

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Scientific couples have never had an easy time balancing their private and professional lives [Abir-Am/Outram 1987], which throughout most of the twentieth century meant that married women had no chance of pursuing a career. Saly Ruth Ramler came from a fairly well-to-do Jewish family in Prague. As the only woman studying mathematics at the Charles University – the oldest of all German universities – Ramler encountered more than her share of incredulous stares and slighting remarks. When she presented her thesis work on affine reflections and their role in the foundations of affine geometry in Pick’s seminar, the Austrian mathematician remarked afterward in perplexed disbelief: “Did you know, Miss Ramler, that you’re doing axiomatics?” She turned this into her dissertation, a manuscript of some 100 pages, which her husband described as “a product of tenacity and original thinking” (he did not believe Pick gave her much help). Dirk Struik first met Ruth Ramler in 1921 when both were attending the annual meeting of the German Mathematical Society held in Jena. Beyond their fondness for mathematics, they also shared an enthusiasm for leftist causes and Marxist ideas. After the conference, they headed off for Dresden, where they managed to see the Sistine Madonna in the Zwinger even though it was past closing hours. At this time, Ruth was teaching mathematics and physical education at the Deutsches Mädchengymnasium in Pilsen. Beyond her talents as a mathematician, she also loved creative dance. They were married on Bastille Day, July 14, 1923, in the old Town Hall of Prague with its famous astronomical clock. They spent their honeymoon in Germany during the period of inflation, which Dirk remembered as a most affordable experience: “with a fistful of Dutch guilders and Ruth’s krones we could afford transportation, pensions, and hotels.” Afterward, they lived in Delft, where Struik worked as Schouten’s assistant. The newlyweds soon learned that their life was not quite as simple as it had at first seemed. As Dirk later recalled, “to our astonishment and mild amusement, we discovered after a while that we were living in sin. We were informed that the Dutch authorities did not recognize a marriage issued by the yet still young state of Czechoslovakia.” Nor could they be married in Delft either, since according to Dutch law Dirk’s father had to be present to sign his approval since the groom was under 30 years of age! So they made a trip to Rotterdam, the city in which Dirk Struik was born and raised, and were married a second time. In the city office, they handed over their Czech marriage certificate and tried to explain to the clerk that Ruth’s name appeared there in the genitive form as Salca Ruth Ramlerova. He gave them a puzzled look, and then insisted he would copy it exactly as her name appeared in that document. Ruth Struik was keen to move on with their lives; she disliked Schouten’s Prussian manner and urged Dirk to get out from under him. This would not be easy, as academic opportunities in the Netherlands were few and far between. Their chance came, though, in April 1924 during an International Congress on Theoretical and Applied Mechanics, which took place in Delft. This event was co-organized by Jan Burgers, a friend of Dirk’s since their student days in Leyden. In the meantime, Burgers had been appointed to a professorship at the Institute of

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Technology in Delft. Both Struiks attended the Congress, and Ruth struck on the idea of inviting some of the guests – Richard Courant, Constantin Carathéodory, Theodor von Kármán, Tullio Levi-Civita, and others – to their home. There, over an evening meal, Courant and Levi-Civita told the two of them about the Rockefeller fellowship program of the International Educational Board (IEB) and encouraged them to apply. One month later, Struik sent a handwritten letter of application to Wickliffe Rose of the IEB in Paris with reference to this conversation in Delft. He enclosed letters of recommendation from both Levi-Civita and Courant, while briefly describing his own and his wife’s academic background and accomplishments.31 Rose’s response came in the form of a perfunctory rejection in which he merely noted that the IEB could not accept applications from individuals. Twelve days later, Rose wrote to Levi-Civita, informing him that his letter recommending Struik had been read with interest by the IEB, but that he would need a letter directly from Levi-Civita in order to activate the application. He added in conclusion: “The Board would not be interested in providing a fellowship for both Dr. Struik and his wife, but if the fellowship should be provided, it would be for Professor Struik on the basis of a fellowship for a married man.” 32 The Rockefeller funds were generous, but young women were almost never the beneficiaries. They left for Rome in September and spent the next nine months there, an unforgettable time for both of them. Struik greatly admired Levi-Civita, whom he regarded as a true internationalist in the spirit of the Risorgimento. His colleague and neighbor, Federigo Enriques, was not only a prominent geometer but also one of Italy’s leading historians of science. At the time, he was editor of a new Italian edition of Euclid’s Elements, and with his customary charm he managed to convince Ruth to prepare the longest and most difficult of its 13 books, the tenth. This turned into a project that occupied her attention for many years, but with the help of Enriques’ student, Maria Teresa Zapelloni, they succeeded. 33 After this period ended, they left in June 1925 for Göttingen, since Struik’s fellowship was extended to a second year, thanks to Courant’s intervention. After experiencing a leisurely life in Rome’s sunny and courteous surroundings, the far less easygoing atmosphere in Göttingen came as a real shock. Especially the younger lecturers loved to trade cynical remarks, packaged as sarcastic humor, a favorite target being Emmy Noether and her “boys,” otherwise known as the “Unterdeterminanten,” (the “minors”, an algebra joke). Dirk Struik later wrote that his wife especially admired Emmy, “not only because of her way in expressing her mathematical ideas . . . but also for her courage in facing the many handicaps she had to meet as a woman – and Jewish to boot – in a masculine society in which (contrary to what we met in Rome) courtesy was not always a form of life” 31 Struik to Rose, 13 May 1924, Rockefeller Archive Center, IEB Collection, 1-1, Box 60, Folder f1002, North Tarrytown, New York. 32 Rose to Levi-Civita, 17 June 1924. 33 The book was published in 1932 under the title Gli Elementi d’Euclide e la Critica Antica e Moderna. Libra X., Bologna: Zanicelli.

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[Freistadt 2010, 14–15].34 Pavel Alexandrov delighted the Struiks with his own brand of humor, telling them how he was glad to be away from Moscow just to escape starving. It wasn’t easy to be a topologist in the Soviet Union, he told them, because one had to reassure the authorities that the subject was useful for the economic recovery. So he liked to emphasize how his research field would prove useful in the textile industry (no doubt he was prepared to explain the topology of weaving patterns to anyone willing to listen). When Alexandrov learned that Dirk had purchased his winter coat with his fellowship money, he dubbed it the “paletot Rockefeller” [Rowe 2018a, 384]. Following these adventures, they returned to Delft in August 1926. At this point, Dirk was unemployed and Ruth Struik’s health had begun to fail, partly due to the couple’s financial insecurity. So he began searching again for other opportunities to go abroad. His brother Anton was already working as an engineer in the Soviet Union, and soon Dirk Struik received an invitation from Otto Schmidt, a mathematician and academician in Moscow, who was then planning a series of Arctic expeditions. In the meantime, Norbert Wiener had managed to arrange a visiting appointment at MIT. Both offers seemed tempting to Struik, but the first was clearly riskier, especially in view of his wife’s delicate health. So after deliberating, they decided to accept the one from MIT, and in late November 1926 they boarded a ship bound for New York. Some eight years later, Fritz Noether faced a similar situation, except that he was already 50 years old and had two grown sons. Emmy thought that there might be a chance he would be able to keep his position in Breslau, but when she wrote Dirk Struik on 25 January 1934, she knew that these efforts had now definitely come to an end. As a decorated war veteran, the Nazi’s BBG stipulated that he was exempt from §3, the Aryan paragraph, which had been applied in Emmy’s case. This left §4 open, which meant that Fritz had to face the far more nebulous charge of being politically unreliable. After successfully defending himself against this accusation, he lost his professorship anyway on the basis of §5, which could be applied in such cases where a person might qualify to be relieved of the stigma of being declared an enemy of the state. As Sanford Segal pointed out, Fritz Noether’s case illustrates the extent to which Hitler’s government succeeded in maintaining the illusion that Germany was still a country ruled by laws [Segal 2003, 60]. The old Prussian Civil Code, enacted in 1794, was long seen as a model for modern European states, and Prussian state officials took pride in upholding its rules. After the fall of the Hohenzollern monarchy in 1918, Prussia quite astonishingly emerged as a bastion of stability during the Weimar Republic. Unlike the federal government, which went through 34 In a letter to Helmut Hasse from 24 December 1930, she enclosed a mocking photomontage made by the students in Mapha, who had cut out her head from the photo Hasse had taken a few months earlier on their ship bound for Königsberg (see Fig. 6.3). Mapha was a student association of math and physics students, so this “present” was likely shown at a recent gathering, perhaps a Christmas party. The photomontage showed an African market with women selling their goods; one of them had the head of Emmy Noether [Lemmermeyer/Roquette 2006, 102].

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13 chancellors in 14 years, the government of Prussia remained in the hands of a solid coalition of democratic parties led by the Social Democrat Otto Braun. Not until July 1932, when the reactionary Chancellor Franz von Papen drove them from power – Papen threatened to unleash the German army if Braun and his ministers did not resign peacefully – could the Nazis begin to dream of taking control of the highly efficient Prussian state bureaucracy.35 Joesph Goebbels, who came to Berlin as Gauleiter in 1926, found it very difficult terrain for his party, though the Nazis gradually gained strength there, just as did the Communists. In the Reichstag elections on 20 May 1928 the NSDAP garnered only 1.4% of the vote in Berlin. Their road to power has been described many times as an appalling example of the failure of democratic institutions, and yet many Germans continued to believe they were still living in a constitutional state, a Rechtsstaat. Those who openly questioned that very assumption – Einstein, Franck, and Schouten – were few, and hence easily shouted down. As with many other cases, Fritz Noether’s problems began locally when he had to contend with radical students who demanded his dismissal. Two weeks after the promulgation of the BBG on April 7, 1933, a group of disgruntled youth protested to the Rektor in Breslau that Noether’s presence contradicted the Aryan principle as well as the spirit of the new national movement. Noether voluntarily suspended his teaching for a brief time, but then took up his courses again and completed the semester. In August, students brought a new set of complaints against him, but by this time the Ministry had already linked him with various left-oriented causes that led to his dismissal on the basis of §4. Fritz Noether then appealed that decision, arguing that he had been politically inactive throughout his career. Realizing that a reversal of the decision was hopeless, he filed to be released according to the conditions stipulated in §5. One of its option called for transfer to another (presumably lower) position; another possibility allowed for the affected official to enter early retirement, which Noether requested and obtained. This also should have allowed him to qualify for regular pension benefits (had he been released on the basis of §4, his pension would have been reduced by 25%). 36 Emmy had already spoken to Dirk Struik about her brother’s situation when she wrote him on January 25. As an applied mathematician, Fritz might have under other circumstances quite easily found work somewhere in industry, but now that the United States was deeply mired in the Great Depression the outlook looked very bleak indeed. She knew that whatever pension he might draw would be far too small to feed his family, but hoped that Struik’s connections might help him obtain a guest professorship in applied mathematics or mathematical physics or perhaps a scientific post in industry. Struik had apparently mentioned that John C. Slater, who chaired the physics department at MIT, might be interested in hiring her brother. Emmy asked him to speak with Slater and sent over Fritz’s CV 35 Prussian militarism clearly also suited Hitler’s Nazi dictatorship, which benefited immensely from the mentality of slavish obedience (Kadavergehorsam). 36 After he accepted the professorship in Tomsk, his pension from Germany was canceled [Segal 2003, 60–61]; see also [Schlote 1991].

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with a list of his publications. She hoped to receive offprints of his work, which she would send later if so requested. She noted, however, that he published technical work as an employee of Siemens–Schuckert from 1920 to 1922. For more detailed information about his research, she also mentioned the possibility of writing to Richard von Mises in Istanbul, since she knew that he held Fritz’s work in high esteem. Emmy’s brother had also named the Ukrainian-born applied mathematician Stephen Timoshenko, who taught at the University of Michigan, as a potentially useful contact. Perhaps Fritz knew that Timoshenko had worked for five years with Westinghouse Electric Company before he was hired in Ann Arbor and therefore assumed he still had connections with industry? Possibly they had met each other at the 1930 Stockholm Congress, but Emmy was unsure how well they knew one another. She imagined that either Struik or Slater knew Timoshenko, but otherwise she would contact him herself. Emmy was hoping for a prompt provisional reply, but when none arrived she sent Struik a postcard ten days later, on Sunday February 4, 1934. She wondered if he had in the meantime spoken with Slater or sent her brother’s documents to Timoshenko. She would be going to Princeton on Wednesday and would speak with someone there about contacting Timoshenko in case that would be more appropriate. She also asked Struik to speak with the Hungarian mathematician Otto Szász, whom Norbert Wiener had brought to MIT after he lost his position in Frankfurt. This concerned Ruth Moufang, who had taken her doctorate there in 1930 under Max Dehn. Moufang hoped to pursue an academic career in Germany, and she eventually succeeded, though it was not until 1951 that she became the first woman to be appointed to a regular professorship in mathematics. After taking her doctorate, Moufang spent a year in Rome on a post-doctoral fellowship and then taught in Königsberg during the academic year 1932/33, before returning to Frankfurt. By this time, the largely Jewish Frankfurt faculty was fast disintegrating [Bergmann/Epple/Ungar 2012, 114-132], so Dehn wrote to Emmy Noether on behalf of Moufang. Since nothing was available for her at Bryn Mawr, Emmy forwarded Dehn’s letter of inquiry to Szász, now Struik’s colleague at MIT. In the meantime, she wanted to know if Szász had contacted anyone at Radcliffe to see if Ruth Moufang might get a position there. None of Emmy Noether’s efforts on behalf of her brother or Ruth Moufang have ever been mentioned before, no doubt because they proved entirely futile. Moufang stuck to her plan to habilitate in Frankfurt, even though Max Dehn and Ernst Hellinger were forced into retirement in 1935 and Paul Epstein chose to resign. She completed the requirements for Habilitation in 1936, but the Ministry then refused to issue the venia legendi, so she had to work in industry for the remainder of the Nazi era. In another case, however, Emmy Noether’s intervention proved both successful and also influential. Hans Rademacher, a left-oriented non-Jewish mathematician at the University of Breslau, was dismissed from his position already in February 1934. Originally, Hasse had recommended Rademacher for a position at the newly founded British Salem School in Scotland, led by Kurt Hahn (who had

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fled from the Nazis). In all likelihood, Fritz Noether (who was also in Breslau) informed his sister about Rademacher’s situation, and so she spoke with mathematicians at the nearby University of Pennsylvania. On June 2, 1934, Rademacher wrote to Hasse: “Now I just found out from Miss Noether that Phil[adelphia] is completely secure for me just as soon as it becomes known there that the protest I lodged with the Ministry has been finally rejected, and that has now occurred . . . ” [Lemmermeyer/Roquette 2006, 214]. On 31 October 1934, Noether reported to Hasse that Rademacher was expected to arrive in Philadelphia that day. He would spend the remainder of his career at the University of Pennsylvania, where he established a major school in number theory [Siegmund-Schultze 2009, 284– 286]. In that same year, Emmy Noether and Hermann Weyl established the German Mathematicians’ Relief Fund, which aimed to support poorer immigrants by means of voluntary contributions from those who were better off (Weyl received a salary of $15,000 from the IAS). Initially, he and Noether asked for contributions of anywhere from 1 to 4% of the incomes from those who had been able to obtain positions in foreign countries. Among those who received some support from this fund were Ernst Hellinger, Fritz John, Hans Schwerdtfeger, and Wolfgang Sternberg [Siegmund-Schultze 2009, 197, 209].37 In all likelihood, Emmy received no potentially positive news from Dirk Struik regarding employment opportunities in the USA for her brother. Roughly one month later, she raised the same issue with her good friend Pavel Alexandrov. Writing him from Bryn Mawr on March 19, 1934, she mentioned that Fritz had been in touch with someone in Zurich.38 What mainly concerned him was finding a locality that would be suitable for his two sons; the elder, Hermann, was studying chemistry in Breslau, whereas Gottfried, two years younger at age 19, was considering perhaps taking up actuarial mathematics or possibly a career in a commercial field. Emmy noted that although Gottfried was the best student in his class, he seemed not to have inherited the family’s pure mathematical talent, as had once been imagined.39 She hoped to receive news from Alexandrov right away, as she would be leaving in mid-May to meet with her brother, either in Göttingen or Breslau.40 37 Weyl later also set up a special fund in Emmy Noether’s name that helped to finance her nephew Gottfried Noether’s education in the United States. 38 At this time, the economist Fritz Demuth headed the Notgemeinschaft deutscher Wissenschaftler im Ausland (Emergency Association of German Scientists in Foreign Countries), which had been founded one year earlier in Zurich by the physician Philipp Schwartz. This organization soon spawned into an international network aimed at finding employment for those who had lost their positions after the Nazi government came to power. Among many other prominent émigrés, Hermann Weyl served on its advisory board. It was through the Zurich office of this organization that Fritz Noether gained a professorship in Tomsk. 39 As will be seen in Section 9.2, Emmy Noether’s nephew had real mathematical talent, though not at all in her favored direction. 40 [Tobies 2003, 105]; it is unclear whether they met in either of these localities, but they did get together for another vacation on the Baltic.

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Emmy was looking forward to this trip with great anticipation, hoping to see Hasse and others in Göttingen as well as to visit with her relatives. In the meantime, she had received news both from her brother as well as from Alexandrov, to whom she wrote back on May 3. The news this time sounded good: Fritz had received an offer from the Soviet Union, so it was “almost certain that they will go to Tomsk, a university research institute for applied mathematics and mechanics. My brother has also received favorable information for his boys” [Tobies 2003, 106].41 Emmy was already contemplating thoughts of a trip to that remote part of the world, which would also offer her the chance for a shorter or longer stay in Moscow. Alexandrov was still trying to negotiate an algebra professorship for her there, but she made it clear that “for the time being, I don’t want to commit myself to anything at all, despite the temptation.” She had already made firm commitments up until the autumn of 1935, and Veblen had signaled that Princeton hoped she would stay, though it was unclear whether she would continue to commute or be given a regular position at the IAS. Memories of Moscow were now receding fast, as Emmy imagined a future life in a country she found quite fascinating. “Staying here,” she wrote, “has the great advantage that – despite the dollar valuation – one can travel almost anywhere; maybe my expectations have risen in America in this regard! A ‘trip abroad’ doesn’t mean going to the North Sea any longer! It’s also a fact that English seems to have devoured all my memories of the Russian language. But at least I would still have German and English available to communicate with your students!” What she honestly hoped for the future was a wish she and Weyl both shared, namely that Princeton would eventually take the place of Göttingen for Alexandrov and other leading foreign mathematicians. If that were to “become a reality, then we could still talk about everything every day.” At the same time, Emmy’s immediate thoughts focused on her brother and his family. Before her departure, she wrote to Fritz’s former student Hans Baerwald and his wife Eva, reporting that her brother would be leaving with his family for Tomsk at the end of the summer, and she was planning to see them off.42 The decision to move thousands of miles from Germany did not come easy, she noted, but Fritz Noether had received positive reports in response to his inquiries about educational opportunities for his sons. Emmy also knew, surely from Pavel Alexandrov, that the region offered excellent conditions for skiing, “no wind, strong sunshine, so that the cold weather should be easily bearable.” She also had heard that a sanatorium for tuberculosis patients had been built nearby. The family “appear now to be preparing themselves learning the language, which is not easy!” 43 Since Emmy Noether had a standing offer to visit Moscow, she 41 Arrangements for this position in Tomsk had been negotiated by the Zurich office of the Notgemeinschaft deutscher Wissenschaftler im Ausland. 42 Emmy Noether to the Baerwalds, 15 May 1934; copy in the possession of Monica Noether. 43 Herman Noether later recalled that learning Russian was a major part of the challenge they faced at Tomsk University; nevertheless, he and his brother Gottfried did very well during their

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could already imagine visiting her brother there, even though the journey from Tomsk would take three days.

8.4 Last Visit in Göttingen Emmy Noether arrived in Hamburg in early June and stayed for a few days with Emil and Natascha Artin. The Artins lived north of the city in Langenhorn, a region Emmy described to Hasse as like a real summer resort. Natascha was heavily pregnant with their second child, Michael, who was born on June 28. He would follow in his father’s footsteps to become one of the century’s premier mathematicians. Emil Artin, unlike Hasse, was a poor correspondent, so Emmy passed on his apologies to Helmut Hasse for not having written to congratulate him on his new appointment in Göttingen. She also hinted that news of the difficulties Hasse would face there had traveled to Hamburg, but without indicating any specifics. As usual, her main focus was on mathematics. Thus, she related with excitement some news about her presentation in Artin’s seminar – in which she spoke as “Noether, America” – concerning her ideas for a theory of class fields for general Galois fields. Artin had earlier tried to push something similar through without success, so both he and Emmy were now skeptical about proving a general existence theorem. Claude Chevalley happened to be in Hamburg at this time, and he offered a new hypercomplex proposal, but they quickly determined that this, too, would not work.44 Emmy Noether was certainly very excited to be back in Germany. Natascha recalled an exotic scene when her husband and Emmy were trying to carry on a mathematical conversation during a ride on the subway, probably before or after her talk in his seminar. As it happens, the technical vocabulary in German for concepts in class field theory could easily be mistaken for the typical political jargon of that era. In her excitement while trying to make herself heard over the surrounding noise, Emmy was drawing more and more attention to herself as the passengers heard words like Ideale, Führer, Gruppe, Untergruppe, and Natascha became more and more worried that they might get arrested. She was half-Jewish, which led to her husband losing his position in 1937; their flight to the United States turned out to be a precarious venture.45 Since Emmy Noether was very reticent in writing about the political situation in Germany, it is difficult to draw a clear picture of what she knew about the specific circumstances that had radically upended mathematical life in Göttingen since her departure. Still, she almost surely knew that her former student Werner three years of study there. After one year, Fritz Noether was able to teach his classes in Russian. (Notes from Herman Noether for his family, undated, courtesy of Evelyn Noether Stokvis.) 44 Noether to Hasse, 21 June 1934, [Lemmermeyer/Roquette 2006, 209]. 45 Natascha Artin Brunswick was presumably the original source for this story, and it seems likely that she told it to Olga Taussky-Todd, one of the people interviewed by Sharon Bertsch McGrayne for her book Nobel Prize Women in Science: Their Lives, Struggles, and Momentous Discoveries, 2nd ed., Washington, DC: Joseph Henry Press, 1998, pp. 77, 413.

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Weber, who had been Edmund Landau’s assistant since 1928, was one of the ringleaders behind a protest movement that ended Landau’s career. Weber may well have been radicalized by a 20-year-old student named Oswald Teichmüller, whom the faculty came to know as both a brilliant mathematician as well as a totally fanatical Nazi.46 Teichmüller had only just turned 18 when he joined the NSDAP in July 1931 and became a member of the Sturmabteilung (SA), whereas Weber did not join until May 1, 1933, the exact same day as Friedrich Neumann, the new Nazi Rektor of Göttingen University and henceforth one of Weber’s principal allies. During that summer semester, Landau had followed the Dean’s advice, allowing Weber to teach in his place. When the winter semester 1933/34 opened, only Landau and Gustav Herglotz remained from the original faculty. Having heard nothing to the contrary, Landau assumed he could continue teaching as usual. But when he walked over to the lecture hall a large group of some 80 students stood outside in the foyer. They parted way for him to pass, but on entering he found only one single student inside. SA members stood next to the doors, blocking entry to anyone who might have wanted to get inside. Landau withdrew to his office, soon followed by Teichmüller, who came to explain the reason for the boycott. This incident took place on November 2, at which time Landau requested that the students’ case be given to him in writing [Schappacher 2000, 25]. The following day, Teichmüller delivered a lengthy statement, which included these remarks: . . . this is not a matter of making difficulties for you as a Jew, but rather only of protecting German students in their second semester from being taught differential and integral calculus by a teacher of an entirely foreign race. I, like everyone else, do not doubt your ability to instruct suitable students of whatever origin in the purely international scientific aspects of mathematics. But I also know that many academic courses, in particular the differential and integral calculus, have at the same time a broader educational value, introducing the pupil not only to a conceptual world but also to a different intellectual sphere. Since that orientation depends very substantially on the spirit that shall be adopted, a spirit that depends, however, very essentially, following long known principles, on the racial composition of the individual, it follows that, for example, a German student should not be trained by a Jewish teacher. . . . So we were and remain all the more united with regard to the purpose of this action, which is mainly to restore the situation of the previous semester. Dr. Weber is prepared to substitute for you in lectures and exercises. Since the uncertainty of the previous semester no 46 Teichmüller was drafted in July 1939 and took part in the invasion of Norway in 1940. He was afterward stationed in Berlin doing cryptographic work until 1942 when he was released to take up a teaching post in Berlin. After the German army suffered a crushing defeat at Stalingrad in February 1943, he volunteered for combat duty on the Eastern Front, where he was killed in action in September 1943.

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longer exists, it would not be necessary for you to speak with him before each lecture; he would rather teach the course, whether completely or in parts, on his own. We, too, would prefer that. Considering that Dr. Weber is the only one who really has to make a sacrifice, since he would double his workload in the interests of younger fellow students, while all you need do is absent yourself from the lecture course without any pecuniary or other disadvantage, I think I am offering a really easy proposal for you to accept. [Schappacher/Scholz 1992] Two days after receiving this letter, Landau wrote to the Kurator, informing him that in view of the present situation he had no choice but to apply for early retirement. This dramatic confrontation soon set forth shock waves throughout the international community. Richard Courant, who had recently arrived in Cambridge, informed Abraham Flexner at the Institute of Advanced Study. Shortly before this happened, he had already written him about the need to help young scientists leave Germany: The Nazis have remained consistent only in regard to the so-called Jewish question. . . . They indoctrinate a ridiculous racial theory (the basis of which is anti-Semitism) through propaganda of all kinds . . . and it may easily happen that once this poisonous seed has germinated, an atmosphere much worse than that now existing will have been created. [Reid 1976, 155] Since Emmy Noether began making regular trips to Princeton in February 1934, she presumably knew by then, at the latest, that Werner Weber had staged a successful coup by organizing a boycott of Landau’s lecture course and thereby forcing him into retirement. She perhaps also knew that the Dean had named Weber acting director of the mathematical institute. In that capacity, Weber was also asked to recommend candidates for the professorship vacated by Hermann Weyl. Weber would have preferred the Nazi Udo Wegner from Darmstadt, but given the prestige attached to this chair, he named Helmut Hasse as the most suitable person to succeed Weyl as director. When Hasse was appointed in April, however, Weber reverted to his true self. Sensing he had made a terrible mistake, he began a quiet campaign to undermine Hasse’s position before the latter took control of the Göttingen Mathematics Institute. The crux of this conflict stemmed from two opposing views of the university, one that favored promoting mathematical excellence in the name of German culture, the other demanding mathematical engagement for the greater cause of the national revolution. Weber came to fear that Hasse merely stood for traditional conservatism; even though he was willing to serve the Nazi state by promoting German science, Hasse failed to meet the political standards of the day because he lacked a deep commitment to Hitler’s messianic mission. In normal times, a person holding a lowly assistantship position would never have posed a threat to someone of Hasse’s stature. But these, of course, were

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not ordinary times. Friedrich Neumann’s election as the new Rektor signaled that Gleichschaltung, Nazi jargon for the process of institutional accommodation, now applied at Göttingen University. Indeed, Neumann, working together with the politicized Dozentenschaft, would ensure that all future university appointments were compatible with the principles of National Socialism. What transpired in the months that followed is a complicated story, told in considerable detail in [Segal 2003, 124–166]. Weber went to great lengths to ensure that Hasse would not have a free hand in running the institute. Landau’s chair now stood vacant, and Weber’s Nazi friend, Udo Wegner, was seen as a strong candidate for it. But Theodor Vahlen, who was named head of the division on higher education in the Prussian Kultusministerium, instead chose the probabilist Erhard Tornier, another ardent Nazi. Tornier was also appointed as co-director of the Göttingen Mathematics Institute, a move designed to curb Hasse’s influence. These developments were preceded by a bizarre situation that took place on May 29, when Hasse arrived in Göttingen only to discover that Weber was scheming to prevent him from becoming director of the institute. This culminated in a scene during which Weber, as acting director, refused to hand over the keys to the building. Recognizing that he was powerless to do anything in Göttingen, Hasse returned to Marburg, informed Vahlen of the present impasse, and awaited further orders from Berlin. He would only return to Göttingen on July 2, by which time Tornier had already been installed as co-director. As a backdrop to the general atmosphere, just two days earlier Hitler had unleashed a sweeping purge known as the “Night of the Long Knives,” which only ended on July 2. By murdering Ernst Röhm, head of the SA, and numerous others, including several prominent conservatives, Hitler consolidated his dictatorship through lawless violence and terror. Nazism thereby revealed its true nature for all to see on the very day of Hasse’s arrival, when he posted a brief notice: “Today I have taken over the leadership of the institute together with Dr. Tornier” [Segal 2003, 151]. Noether had been in Göttingen since June 7, and she soon realized that nothing was as before. Two weeks later, she wrote to Hasse in Marburg about how disappointed she was not to have found him in the institute, though by now she had apparently heard that he would soon be coming.47 Tornier had grudgingly offered to let her use the library as a “foreign scholar,” a designation that surely must have felt like adding insult to injury. She no longer had any illusions about keeping her apartment, and so had begun making plans for shipping her furniture, books, and other belongings to the United States, even though it still remained unclear whether she would be offered an extension beyond the coming academic year. She also expressed her sadness over not being able to invite Hasse to her apartment. Realizing that if she came to Marburg this could also easily lead to problems, she suggested meeting in Bad Soden near Frankfurt, in the event Hasse was planning to visit his parents, who lived in that town. It seems unlikely, though, that they managed to meet before Hasse’s arrival in Göttingen in early July. 47 Noether

to Hasse, 21 June 1934, [Lemmermeyer/Roquette 2006, 209].

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After Hasse took up his professorship, he commuted between Göttingen and his home in Marburg. During the week, he lodged at Gebhardt’s Hotel near the train station. He and Emmy took great care not to be seen together at the institute, knowing full well that Tornier and his Nazi allies would have relished the opportunity to use such evidence to attack Hasse as a friend of the old “Jewish clique” in Göttingen. Noether’s major concern at this time was to ensure that Deuring could habilitate. Originally, she assumed he could do so in Leipzig, where Deuring was van der Waerden’s assistant. However, as she explained in a letter to Hasse, the state of Saxony had in the meantime introduced additional requirements that would lead to unnecessary delay. Deuring had already written his postdoctoral thesis and mailed a copy to Noether. Since Hasse was spending the weekend in Marburg, Emmy wrote him on Sunday, July 15 to ask if she should bring this copy to Hasse’s hotel or whether he could come to her apartment one evening. Hasse was highly impressed with Deuring’s post-doctoral thesis, as can be seen from remarks in his report.48 Referring to its principal theorem, he wrote: “this can be seen as the first real result approaching the Riemannian conjecture and it is also of the highest significance due to the depth of the proof” [Koreuber 2015, 249]. This was written in January 1935; one month later he added a brief notice in which he underscored the importance he attached to Deuring’s case: The initiative behind Mr. Deuring’s application came from me. I have the greatest interest in bringing this excellent mathematician to Göttingen, not only because of his outstanding scientific capabilities but also because his mathematical activity continues old Göttingen traditions in an exceptional way. (ibid.) It need hardly be said what traditions Hasse had in mind, but of course Emmy Noether’s name could no longer be mentioned in the year 1935. Even when it came time for Deuring to deliver his qualifying lecture, Hasse advised him not to choose a topic in abstract algebra. It surely made no difference to Tornier, who in his own report made his position very clear: “I vote for Deuring’s habilitation, but remark prophylactically that I would with all means oppose offering a lectureship” (ibid.). In light of Tornier’s opposition, the Ministry refused to confer the venia legendi until 1938, when Deuring became a private lecturer in Jena. Thus, as Lemmermeyer and Roquette aptly noted, Emmy Noether’s hopes that under Hasse “Göttingen would remain in the center” (im Mittelpunkt) could not be realized, especially not for her own principal field of research. Already on April 18, 1935, just four days after her death, Hasse wrote to Otto Toeplitz: What I find even more depressing is the fact that, on the one hand, I carry the responsibility to the mathematical world for restoring Göttingen to a place of rank, whereas, on the other hand, I am deprived 48 Excerpts from it appear in [Koreuber 2015, 248–249]; Noether’s report on his dissertation from 1930 is reproduced in [Koreuber 2015, 313–314].

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8 Emmy Noether in Bryn Mawr of almost all influence on the personal organization due to the existing university policy regulations. This concerns not only filling vacant professorships, but also applies to lectureships, assistantships, and tutoring positions. [Lemmermeyer/Roquette 2006, 207]

Although Emmy Noether traveled to several different places during her final trip to Germany, it remains unclear whether she visited her brother and his family in Breslau. They did, however, spend some time vacationing together in a small village on the Baltic, possibly in Dierhagen, where they stayed the previous summer.49 They were again joined by the Heisigs and the Baerwalds. Herbert Heisig and Hans Baerwald had both studied under Fritz Noether in Breslau. As a Jew, Baerwald no longer had any future in Germany, and he soon left for New York City.50 Since Fritz Noether had acquired a car, Emmy may well have decided to travel with him and his family on their vacation to the Baltic. Their much longer trip soon afterward, from Breslau to Tomsk, nearly 3,000 miles away, was a true adventure. Since the family had been promised a large apartment, they sent over all their furniture and books, a grand piano, and even some of the older furniture from Max Noether’s days in Erlangen. Throughout the summer, Emmy Noether corresponded regularly with Olga Taussky, though often with hurried little messages, as Emmy was constantly on the move. She lectured in Marburg and Kiel, but she also wrote about plans to visit Basel and Berlin as well. Toward the end of August, Noether wrote from Magdeburg, where she was visiting friends.51 In the meantime, Taussky learned that she would be able to postpone her fellowship at Girton College, so she could spend the coming year at Bryn Mawr with no worries about the immediate future. Noether was pleased to learn this; she had for some time been doing her best to support Taussky’s future career with advice and letters of recommendation. In September, she would board a ship in Hamburg bound for a country Noether still barely knew, though she now realized that this was likely to become her new home. Still, Emmy Noether was not one to ponder over her fate when, in fact, there was still much to do in making plans for the trip back. The week following, she would return to Göttingen so that she could organize the shipment of her furniture to the United States. In the meantime, her mind was firmly focused on what she and Frl. Taussky would experience on the other side of the ocean. I plan to present Chevalley’s thesis in the seminar since it provides such easy access to class field theory. But I’ll probably also have to use other books for the basics of Hilbert-Dedekind theory. During the previous semester, I worked through van der Waerden I and Mitchell’s number 49 This and the following information comes from a conversation between Herman Noether and his daughter Evelyn. 50 He and his family later moved to Cleveland, where Gottfried Noether visited them in 1939, just before he began his studies at Ohio State University. 51 Noether to Taussky, 30 August 1933, Papers of John Todd and Olga Taussky-Todd, Box 11, Folder 11, Caltech Archives.

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theory52 with the four seminar students. This was something between a seminar and tutorial and it worked quite well, so with that you will be able to help! In Princeton, where I teach once a week, I’ll do the same thing only from a slightly more advanced point of view.53

8.5 Lecturer at Princeton’s Institute for Advanced Study Emmy Noether’s association with Princeton’s Institute for Advanced Study (IAS) first began in February 1934 when she started making weekly visits, usually on Tuesdays, to deliver lectures there. She suggested a bit of the flavor of what this was like in a letter to Hasse: . . . I’ve been lecturing once a week in Princeton – at the Institute and not at the “men’s”-university, which does not admit anything female, whereas Bryn Mawr has more male lecturers than females, so is only exclusive regarding students. At the beginning, I started with representation modules and groups with operators; this winter Princeton will for the first time get treated algebraically, and quite thoroughly. Weyl also lectures on representation theory, although he will soon switch to continuous groups. Albert . . . lectured before Christmas on something hypercomplex in the style of Dickson, together with his “Riemannian matrices.” Vandiver, who is also on “leave of absence,” is lecturing on number theory for the first time in an eternity at Princeton.54 And after I had given my survey on class field theory in the Mathematics Club, von Neumann ordered twelve copies of Chevalley as a textbook (Bryn Mawr shall also get some!). I was also told that your Lecture Notes will be translated into English, hopefully now in sufficiently many copies – I had pestered the people about this already in the fall. My audience consists essentially of research fellows, along with Albert and Vandiver, but I noticed that I have to be careful; these people are used to explicit computations, and I have already driven some of them away! The University and Flexner Institute together have more than sixty professors and those who want to be; even if Princeton tries to draw many of them, all these research fellows are a sign of academic unemployment.55 52 Howard H. Mitchell, who was Oswald Veblen’s first doctoral student at Princeton, taught a number theory course at Bryn Mawr before Noether’s arrival. Since he never wrote a textbook on number theory, this reference suggests that he probably gave Noether some notes for a continuation of his earlier course. 53 Noether to Taussky, 30 August 1933, Papers of John Todd and Olga Taussky-Todd, Box 11, Folder 11, Caltech Archives. 54 Harry Vandiver was a professor at the University of Texas; in 1931 he won the Cole Prize of the AMS for his work on the Fermat conjecture. 55 Noether to Hasse, 6 March 1934, [Lemmermeyer/Roquette 2006, 204].

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Noether’s assessment of the general employment picture in the US was certainly accurate, even though Princeton served as a springboard for many, including her collaborator Richard Brauer. As concerns the physical surroundings for mathematicians, Emmy’s claim not to be at the “men’s”-university stands in need of clarification. During its early years of operation, the IAS was housed in Fine Hall along with the mathematics department. Or, to be more precise, they were together in the old Fine Hall, today called Jones Hall, which was built in 1930 and originally named in memory of the department’s founding chairman, Dean Henry B. Fine.56 Henry Fine had been one of Felix Klein’s first American doctoral students, taking his degree in Leipzig in 1885. He was by no means a leading researcher, but he played a pivotal role in making Princeton one of the three leading universities for mathematics in the United States (the other two being Chicago and Harvard) [Parshall/Rowe 1994, 438–451]. Shortly after Dean Fine’s death, in 1929, Jones and his niece provided for the erection of a mathematics building in his memory, and, what was rarer, an endowment for its upkeep. Feeling that “nothing is too good for Harry Fine,” Jones said that the building to bear his name should be a place which “any mathematician would be loath to leave.” The finished building featured a spacious wood-paneled library, common rooms, and faculty studies. It also contained a locker room with shower bath for faculty wishing to use the then-nearby tennis courts; this amenity inspired these lines about the department chairman in the Faculty Song: “He’s built a country-club for Math Where you can even take a bath.” Loath as the mathematicians were to leave, in 1969 the increased size of the department compelled them to move into the new Fine Hall, leaving their marks on the old one – mathematical formulas and figures in the leaded design of the windows, and Einstein’s famous remark over the fireplace in what is now the lounge: Raffiniert ist der Herr Gott, aber Boshaft ist Er nicht (God is subtle, but He is not malicious). Oswald Veblen found that saying so delectable, he decided to have it engraved there for posterity.57 Hasse and Noether remained in steady contact after she returned to the United States. Emmy reported to him that some 200 people from New York and surrounding colleges came to hear her speak at the recent AMS meeting held at Columbia. Probably she visited the Courants during this trip, as she wrote further that Richard “feels very well in New York – he lives 40 minutes away by car in a rural area not far from the beach – and is becoming more human.” 58 Despite his frustrations over the political turbulence in Göttingen, Hasse remained eager to 56 When the new Fine Hall was completed in 1969, the old one was renamed in honor of its donors, Thomas D. Jones and his niece Gwethalyn Jones. 57 This saying alludes to Einstein’s skepticism regarding the new orthodoxy in quantum physics, as reflected in his remark that “God doesn’t throw dice.” 58 Noether to Hasse, 31 October 1934, [Lemmermeyer/Roquette 2006, 213]. Courant apparently did not attend the meeting at Columbia.

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pursue his ideas for proving a function-theoretic analogue of Riemann’s hypothesis for cases of higher genus. Evidently with that program in mind, he asked Noether’s advice regarding relevant literature. She responded by naming three works, two of them by Lefschetz. She noted that Oskar Zariski, who was attending her lectures in Princeton, was the only one in the United States besides Lefschetz who worked on algebraic geometry in several variables. Emmy Noether also had the pleasure of reconnecting with Richard Brauer on her weekly trips to Princeton. During the previous year, Brauer taught at the University of Kentucky in Lexington, after which Hermann Weyl invited him to spend the academic year 1934/35 at Princeton’s Institute for Advanced Study. His principal duty that year was to prepare the text for Weyl’s lectures on Structure and Representation of Continuous Groups [Weyl 1934/35]. They also jointly wrote [Brauer/Weyl 1935], in which they constructed an n-dimensional representation for spinors. During this year, Brauer and his wife Ilse (who was also a mathematician) became very close friends with Emmy Noether. About her own lectures at the IAS, Noether gave Hasse this glimpse of her teaching activity: There are a number of interested people in Princeton this year; I am doing a seminar on class field theory, although it’s mainly a lecture course with occasional [active] participation by other people. For the time being, however, we are still stuck in Galois theory; but next time Ms. Taussky, who occasionally comes along [from Bryn Mawr], will present some simple number-theoretic examples. She held a rehearsal here, and I’m doing the same, but tailored for women, i.e. the girls replace what they lack in self-reliance with an uncanny diligence – this year there are two others besides Miss Taussky here on a scholarship. [Lemmermeyer/Roquette 2006, 212–213] Ruth Stauffer (Fig. 8.2) was now in her second year of studies under Emmy Noether, who liked to tease Stauffer in a friendly way about her name. It reminded her of Werner Stauffacher, a character in Schiller’s William Tell, so she kept assuring her student that she was surely of Swiss origin. Many years later, Ruth Stauffer McKee told an anecdote about a typical local excursion with her fellow students, led by their irrepressible teacher. In those days, one did not need to wander far from the Bryn Mawr campus to reach the countryside, and so they began walking across an open field when Stauffer noticed that they were headed straight for a rail fence: Miss Noether was immersed in a mathematical discussion and went merrily along, all of us walking at a good clip. We got closer and closer to the fence. I was apparently the weak sister, concerned mostly in how we would handle the fence. For those of us in our twenties it would be no problem but, from my point of view, however would this “old lady,” fiftyish, handle the fence? On we went right up to the fence and without missing a word in her argument she climbed between the rails and on we went [Quinn et al. 1983, 144].

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Figure 8.2: Bryn Mawr College, June 1935: (l. to r.) Marie Weiss, Ruth Stauffer, and Grace Shover, May 1935 (Auguste Dick Papers, 13-1, Austrian Academy of Sciences, Vienna)

Grace Shover Quinn also remembered trips to Philadelphia and Princeton in Mrs. Wheeler’s car, and another trip to Swarthmore College, a little more than a half-hour’s drive away [Quinn et al. 1983, 140]. Like Bryn Mawr and Haverford College, all nearby, Swarthmore was a traditional Quaker school going back to the nineteenth century.59 On this trip the car must have been quite full with the driver, Emmy, Ilse and Richard Brauer, and Grace Shover. They went to Swarthmore to visit Isaac and Charlotte Schoenberg, another émigré couple. Isaac Jacob Schoenberg was a Rumanian Jew who took his doctorate under Issai Schur in Berlin. He then worked under Edmund Landau in Göttingen on analytic number theory. Through Landau he got a position at Hebrew University in Jerusalem, before coming to the US on a Rockefeller fellowship in 1930, the year he married Landau’s daughter, Charlotte. From 1933 to 1935 he had a fellowship at the IAS. His sister Irma Wolpe was a pianist married to Hans Rademacher; they met in India when he was on leave at the Tata Institute in Bombay. Coming from Vienna, interrupted by her year in Göttingen, Olga Taussky already knew a number of the mathematicians she would meet again in Princeton. Wilhelm Magnus, her co-worker on the Hilbert edition was there, as of course were Veblen and Weyl. She had met Richard Brauer at the 1930 conference in Königsberg, and she had done coursework in Vienna with Walter Mayer, who came to the IAS as Einstein’s assistant. Taussky even knew John von Neumann from 59 Haverford admitted men only until the 1970s, whereas Swarthmore was from its founding co-educational; Bryn Mawr remains today a women’s college.

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talks he had given in Vienna and Göttingen. Little wonder that her trips with Emmy were the highlight of her stay in the United States. Their “rehearsals” for the girls took place on a Monday, the day before they took the train to Princeton Junction, a 3-hour trip. Olga Taussky could not afford to go every week, partly because her European fellowship did not include a waiver of tuition fees. Still, she went whenever she could, even just for the chance to “talk mathematics” with Noether, who in such a situation could be a very good listener. During the two-year interim since she left Göttingen, Taussky became interested in topological algebra after reading a paper from 1932 by Lev Pontryagin. When she told Noether about this on the way to Princeton, Emmy afterward introduced her to the topologist James Alexander, who then told her that Nathan Jacobson was very familiar with Pontryagin’s paper. This soon led to a joint publication by Jacobson and Taussky on locally compact rings. Olga Taussky-Todd later recalled that Emmy Noether often felt irritated during their year together at Bryn Mawr, and since those feelings were very often expressed in German, she became a kind of sounding board for Emmy. At the same time, Taussky sometimes became annoyed with Noether for directing quite a lot of petty criticism at her, beginning with her Austrian accent and green felt hat with a feather. Emmy said it reminded her of someone they both knew – no doubt Gustav Herglotz – when he wore Lederhosen. So Taussky gave the hat to Ruth Stauffer, since she knew how much Ruth loved it [Taussky 1981, 87]. Taussky was now 28 and had taken her doctorate four years earlier under Furtwängler, so she came to resent Noether’s motherly ways, all the while not knowing about Emmy Noether’s many worries, which “der Noether” kept very much to herself. Taussky called her a “tough guy,” and felt that Noether expected the same of her; Emmy surely recognized that Olga Taussky had the talent and ambition to go far in her career. For Emmy Noether, oddly enough, the key to success for women in mathematics was marriage. She even colluded with Ilse Brauer in trying to line up husbands for her four students, but with no success. As Taussky saw her in retrospect, Noether was very naive and knew very little about life. She saw women as being protected by their families and even admitted to me that she gave young men preference in her recommendations for jobs so they could start a family. She asked me to understand this, but, of course, I did not. [Taussky 1981, 91] There was also a clash in their mathematical styles. Noether was trying to teach these young women cutting-edge research based on Hasse’s mimeographed lecture notes on class field theory (Noether had reviewed this work in Zentralblatt in 1933). Clearly one needed a solid background in algebraic number theory to understand such advanced material, so she tasked Taussky with presenting lectures designed to fill in some of those gaps. Not surprisingly, given her familiarity with Hilbert’s work, Taussky chose to base her presentations on his Zahlbericht, which

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did not please Noether at all. For her, this was just computing, not the kind of conceptual algebraic approach she wanted to see, and Emmy Noether was not one to hide her opinions on mathematical matters. This put real pressure on Olga Taussky, who later made light of the situation in a little poem. Her inspiration for it came from a witty piece by Wilhelm Busch about a bird stuck in a tree, a poor creature that realized it was about to be devoured by a cat. Emmy Noether would have surely enjoyed it, but Taussky was too bashful to show her this creation: Es steht die Olga vor der Klasse, sie zittert sehr und denkt an Hasse. Die Emmy kommt von fern herzu, mit lauter Stimm’, die Augen gluh. Die Trepp hinauf und immer höher kommt sie dem armen Mädchen näher. Die Olga denkt: weil das so ist und weil mich doch die Emmy frisst, so werd’ ich keine Zeit verlieren, werd’ keine Algebra studieren und lustig rechnen wie zuvor. Die Olga, dünkt mir, hat Humor.60 Olga stands outside the class room with wrinkled brow and in deep gloom. Emmy, from far away comes along with a firm step and feeling strong. She climbs upstairs with a great swirl and gets quite close to the poor girl. Now Olga thinks: Of hope there is no ray and Emmy scolds me anyway. Merrily I will compute some more and algebra I will ignore. It makes me think that Olga had humor.61 Emmy thought a lot about Hasse, too, and in one of her letters to him she described her latest ideas for pursuing class field theory by combining transcendental methods with Chevalley’s new algebraic ideas. Hasse discussed this with Ernst Witt and both agreed that if one needed analytical methods to found class field theory – which seemed at that time to be the case – then one should point that powerful cannon straight at the fortress of the theory. More precisely, “one should use Witt’s remounting of Käte Hey’s cannon and then fire a return shot 60 Nach

Wilhelm Busch, “Es sitzt ein Vogel auf dem Leim.” German original in [Dick 1970/1981, 1970: 34], the English version in [Dick 1970/1981, 1981: 84–85]. 61 The

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on the classical class field theory as was done on your 50th birthday.” 62 Soon afterward, Hasse sent Noether a paper for Mathematische Annalen by Chevalley and Nehrkorn, which he considered the furthest one could go in the arithmetical direction.63 Noether was very enthused about her Princeton lectures during this second year, as can be seen from a letter to Hasse from 28 November. She reported that Morgan Ward was diligently studying Hasse’s lecture notes from her seminar. 64 Zariski was also attending and had begun to immerse himself in the arithmetic theory of algebraic functions. Before Christmas time, Noether sent a witty message to Hasse, who answered in turn: Thank you for your content-rich letter! I can well imagine that with the stormy weather and the hopeless classes you spend your time dancing on tiptoe, that you are exhausted, cannot enjoy the primitive meals, but are still happy to be there anyway. I am very honored that you need me and are willing to sacrifice money, love, beer, sleeping, eating, and vacationing, whereas I feel ashamed since you really have nothing for which to thank me.65 Emmy sorely missed Hasse and the life she once had in Göttingen. All things considered, though, she was adjusting remarkably well to her new situation; still, she was worried about her brother and his family, not to mention the state of her own deteriorating health. She had undergone an operation the previous summer, during which the surgeon discovered a large uterine fibroid. Her original plan was to have it removed when she returned to Göttingen in the summer of 1935. Presumably, Emmy Noether had informed Anna Pell Wheeler of this, but in any event, on the advice of another physician, Noether decided to undergo this second operation in Bryn Mawr. On top of these worries, her employment situation was still unsettled, and she still had no news about this as the year 1934 came to an end. Since her initial two-year appointment would expire the following summer, Anna Pell Wheeler contacted Oswald Veblen to inform him that Bryn Mawr had no funding available to keep her on the faculty. Veblen then wrote to Abraham Flexner on 13 December 13 1934, alerting him to the impasse: 62 Noether to Hasse, 31 October 1934, and Hasse to Noether, 19 November 1934, [Lemmermeyer/Roquette 2006, 212–215], referring to his paper for her 50th birthday [Hasse 1933]; Hey’s cannon is a reference to Käte Hey’s doctoral dissertation from 1927, written under Emil Artin, see [Dumbaugh/Schwermer 2018]. 63 Hasse to Noether, 17 December 1934, referring to C. Chevalley and H. Nehrkorn. Sur les démonstrations arithmétiques dans la theorie du corps de classes, Mathematische Annalen 111(1935): 364–371; later that same year Chevalley announced a purely arithmetical theory. 64 Ward took his doctorate at Caltech under Eric Temple Bell in 1928 with a dissertation entitled “The Foundations of General Arithmetic”; he joined the faculty there the following year. Olga Taussky met him again at a number theory conference held at Caltech, and he was later instrumental in arranging her appointment as the first woman on its faculty. 65 Hasse to Noether, 17 December 1934, [Lemmermeyer/Roquette 2006, 219].

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8 Emmy Noether in Bryn Mawr The professors of the Institute would be quite willing to recommend a small grant-in-aid for a year or two, especially in view of the fact that Miss Noether has been lecturing here during the last two years. And this might help to bridge the gap in case it is necessary to make temporary arrangements for a couple of years longer. In view of Miss Noether’s unique position in the world – the only woman mathematician of the first order – it ought to be possible to find some persons or group of people who would make it possible for Bryn Mawr to keep her permanently. [Shen 2019, 61]

Flexner had doubts about such a short-term commitment and also felt that the IAS had already done a great deal already for German scholars. He was concerned not to create the impression that his institution was overlooking Americans in order to help unfortunate foreigners. These years were, indeed, particularly difficult ones for young American mathematicians, one of whom was Nathan Jacobson, a Polish Jew whose family immigrated to the USA in 1918. Jacobson took his doctorate at Princeton under Wedderburn in 1934, spent the year 1934/35 at the IAS, where he attended Noether’s lectures on class field theory, and then, following her death, was hired by Anna Pell Wheeler to teach at Bryn Mawr for one year. After that he was awarded a one-year fellowship from the National Research Council to do post-doctoral work with Abraham Adrian Albert and Leonard Dickson at the University of Chicago. By the spring of 1937, he was again looking for a regular position. As he later recalled: “. . . this was the depths of the Great Depression. Salaries declined in some instances and there were very few new positions. Moreover, for the new Jewish Ph.D.’s the situation was further aggravated by anti-Semitism that was prevalent, especially in the top universities – the only ones that had any interest in fostering research” [Niven 1988, 220].66 Sidestepping these issues, Veblen was eventually able to secure a $1,500 grant, while he continued soliciting larger donations for a “permanent commitment on the part of the Institute.” As he put it, Noether was not merely unique as a “woman mathematician,” she offered the Institute an opportunity to capitalize on the brain-drain from Göttingen by supporting “one of the most important scientists” displaced by the events in Germany. A parallel effort was undertaken by the Dutch-American mathematician Arnold Dresden at nearby Swarthmore College. 67 After meeting in Philadelphia with Jacob Billikopf, executive director of the Federation of Jewish Philanthropies, Dresden contacted several leading mathematicians who wrote letters of support [Kimberling 1981, 34–36]. 66 Jacobson was hired by the University of North Carolina at Chapel Hill; he then taught at Johns Hopkins from 1943 to 1947 before joining the faculty at Yale University. 67 Dresden came to Swarthmore from the University of Wisconsin in 1927. He was recruited by President Frank Aydelotte in order to initiate an honors program in mathematics that would later serve as a model for other colleges and universities throughout the U.S. Aydelotte was a member of the board of the IAS and later served as its director, which may have played a role in Dresden’s initiative.

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On 31 December, Lefschetz wrote: As the leader of the modern algebra school, she developed in recent Germany the only school worthy of note in the sense, not only of isolated work, but of very distinguished group scientific work. In fact, it is no exaggeration to say that without exception all the better young German mathematicians are her pupils. Were it not for her race, she would have held a first rate professorship in Germany and we would have no occasion to concern ourselves with her. She is the outstanding refugee German mathematician brought to these shores and if nothing is done for her, it will be a true scandal. Norbert Wiener from MIT, writing on 2 January, was just as emphatic: Miss Noether is a great personality; the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie. Leaving all questions of sex aside, she is one of the ten or twelve leading mathematicians of the present generation in the entire world and has founded what is certain to be the most important close-knit group of mathematicians in Germany – the Modern School of Algebraists. Even after she was deprived of her position in Germany on account of her sex, race and liberal attitude, numbers of students (men as well as women) continued to meet at her rooms for mathematical instruction. Of all the cases of German refugees, whether in this country or elsewhere, that of Miss Noether is without doubt the first to be considered. G.D. Birkhoff from Harvard also encouraged this undertaking, which he implied should not be as costly as in other cases of similarly prominent mathematicians. Miss Noether . . . is generally regarded as one of the leaders in modern Algebraic Theory. Within the last ten or fifteen years she and her students in Germany have led the way much of the time. It is not too much to say that, since Sonia Kovalevski, she is the only woman mathematician of high absolute rank. Thus it is an opportunity for us to have her in this country and I hope very much that you will succeed in your efforts. Her continued presence at Bryn Mawr is sure to be a stimulus to everyone interested in modern Algebra in this country. As far as the desirable arrangements to be made in her case are concerned I find myself in general agreement with my colleague Professor Wiener. I might mention here the fact that at least when I was last in Germany her salary there was not large, and also that as far as undergraduate work is concerned, she will be probably of no use at Bryn Mawr. A brief note from Einstein, dated January 8, was also sent to Billikopf, who then informed the Emergency Committee that he could offer between $750 and

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$1,000 toward Noether’s support. An additional $1,000 was offered by the Lucius N. Littauer Foundation in New York soon thereafter. Somewhat complicated negotiations then began between Bryn Mawr and the two main agencies that had financed Noether’s position until this time, the Emergency Committee and the Rockefeller Foundation. Warren Weaver, from the Rockefeller Foundation, was eventually persuaded to support Noether for the coming academic year. He saw no prospect for a long-term appointment at Bryn Mawr, due primarily to her exclusive interest in research-level mathematics. That being the case, her supporters focused on the possibility of creating a permanent position for her at the IAS. On 28 February, Veblen wrote to Flexner about . . . the possibility that this might become a permanent commitment on the part of the Institute. There is no doubt that, apart from the uniqueness of her position as a woman mathematician, she is quite obviously one of the most important scientists who have been displaced by the events in Germany. Therefore even a permanent commitment could be nothing but creditable to the Institute. [Shen 2019, 62] Weaver decided that the Rockefeller Foundation would make an exception for Noether with regard to its standard policy on academic refugees. He was thus preparing to authorize her appointment at Bryn Mawr for another two years. As matters turned out, however, she would never finish her original two-year term. No one realized at the time that Emmy Noether was seriously ill when she underwent an operation to remove a large ovarian cyst. For three days afterward, she seemed to be recovering without problems. But on the fourth day she fell into a coma and her temperature shot up to 108 degrees following the rupture of a cerebral blood vessel. Her death on that Sunday, April 14, 1935, came as a complete shock to everyone who knew her as a primordial spirit bursting with energy. Grace Shover Quinn later recalled some of the events from the last two weeks of Emmy Noether’s life. In late March, the college was on spring break, so the dorms were closed and everyone went their separate ways. Olga Taussky went to Atlantic City, and Emmy took Grace Shover along to visit her there on Sunday, March 31. Exactly one week later, Shover visited Noether again, this time with a friend from Germany, who enjoyed chatting with the famous mathematician in the apartment she rented south of the Lancaster Pike. She had all her furniture shipped from Germany, which made her feel quite at home, especially with her massive desk. The next day, Monday April 8, Mrs. Wheeler summoned the four girls to let them know that Miss Noether would be entering the hospital that day for removal of a uterine tumor. The operation was performed that Wednesday. Her four students had visited their teacher the day before the operation and planned to see her again on Saturday, but on that day they were told she was not feeling well enough to see visitors. In fact, she had been doing fine, but her condition suddenly worsened that very day. The next afternoon, when they were in their dormitory rooms, Ruth Stauffer was called to the telephone and learned from Anna Pell Wheeler that Emmy Noether had passed away.

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Various stories about the cause of death apparently went around shortly thereafter, but her physician, Dr. Brooke M. Anspach, wrote to President Park the day after Noether’s death informing her that the patient was suffering from a cerebral lesion which led to a ruptured blood vessel. After she became unconscious on early Sunday morning, her temperature rose to 108 degrees. The operation no doubt aggravated this condition, but in the opinion of Dr. Anspach, if the tumor had not been removed it alone would have caused her death [Shen 2019, 62]. Emmy Noether knew that her condition was very serious; she may have even realized that her life was at risk when she entered the hospital on April 8, 1935. Before leaving home, she made a list of items she owned that should be distributed to those dear to her in the event she did not survive the operation. Grace Shover received a necklace and Olga Taussky a brooch and one of Dedekind’s books from his own personal library.68 That day Emmy also wrote a short statement about Ruth Stauffer’s doctoral dissertation: This thesis gives some very interesting results in the field of Modern Algebra, and it has shown that Miss Stauffer has a thorough knowledge of the modern theories and that she has a feeling for abstract methods. I consider this thesis is satisfactory as a partial fulfillment of the requirements for the Ph.D. degree. [Shen 2019, 63] One day earlier, in her last letter to Helmut Hasse, Noether described Stauffer’s work, which gave an explicit method for constructing a normal basis for certain finite extension fields.69 She wrote nothing about going in for an operation and only mentioned that she was unsure whether she would visit Göttingen again in the summer, but if so then not until the end of June. She had to be present at the commencement ceremony to present her student to the president, who would then place the doctoral hood on Ruth Stauffer’s head. It was not to be, and so Richard Brauer had to step in for Emmy.70 Fritz Noether received the sad news in Berlin on Monday, April 15. He immediately sent a telegram to Hasse as a member of the executive committee of the German Mathematical Society. On the same day, Hasse also received a 68 During the next year, Shover taught at the Shipley School in Bryn Mawr, a position she had taken the previous winter with the intention of continuing her studies with Noether. In the summer of 1936, she attended the International Mathematical Congress in Oslo, where she met Fritz Noether. Before the Congress, she traveled to Göttingen and attended lectures by Helmut Hasse. 69 Noether to Hasse, 7 April 1935, [Lemmermeyer/Roquette 2006, 221]; Noether was the first to prove the existence of a normal basis in [Noether 1932a] under the assumption that the base field was infinite; Deuring then proved the same result without this assumption in [Deuring 1932]. Stauffer’s results concern separable normal extensions K/k, where the characteristic of k does not divide the degree of K/k [Stauffer 1936]. 70 After receiving her degree in June 1935, Ruth Stauffer taught mathematics for one year at the Bryn Mawr School in Baltimore while studying with Oscar Zariski at the Johns Hopkins University. In 1937 she married George McKee, who like her grew up in Harrisburg. She worked for some 30 years as a statistician for a research agency attached to the Pennsylvania State Legislature.

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telegram from Hermann Weyl. He answered by requesting that Weyl purchase a wreath in the name of the Göttingen mathematicans for the funeral ceremony on Wednesday. This was a traditional Quaker service that took place in President Park’s living room. During the Bryn Mawr Symposium, Grace Shover Quinn recalled the scene on that somber occasion [Quinn et al. 1983, 141]: We heard music played softly by a string ensemble in a nearby room. A closed black box along the side of the room reminded us of the loss of our beloved professor. Four eulogies were scheduled: Mrs. Wheeler represented Miss Noether’s American colleagues; Richard Brauer, speaking in German, her German colleagues; Ruth Stauffer, her American students; and Olga Taussky,71 her foreign students. During the morning chapel service the day afterward, Emmy Noether’s colleague Marguerite Lehr recalled the excitement and confusion that preceded her arrival in November 1933: . . . there was much discussion and rearrangement of schedule, so that graduate students might be free to read and consult with Miss Noether until she was ready to offer definitely scheduled courses. For many reasons it seemed that a slow beginning might have to be made; the graduate students were not trained in Miss Noether’s special field – the language might prove a barrier – after the academic upheaval in Göttingen the matter of settling into a new and puzzling environment might have to be taken into account. When she came, all of these barriers were suddenly non-existent, swept away by the amazing vitality of the woman whose fame as the inspiration of countless young workers had reached America long before she did. . . . Professor Brauer in speaking of Miss Noether’s powerful influence professionally and personally among the young scholars who surrounded her in Göttingen said that they were called the Noether family, and that when she had to leave Göttingen, she dreamed of building again somewhere what was destroyed there. We realize now with pride and thankfulness that we saw the beginning of a new “Noether family” here. To Miss Noether her work was as inevitable and natural as breathing . . . . She loved to walk, and many a Saturday with five or six students she tramped the roads with a fine disregard for bad weather. Mathematical meetings at the University of Pennsylvania, at Princeton, at New York began to watch for the little group, slowly growing, which always brought something of the freshness and buoyancy of its leader. [Quinn et al. 1983, 144–145] 71 Soon after Noether’s death, Taussky moved to England, where she became a Fellow at Girton College, Cambridge University. Three years later, she married the Irish mathematician Jack Todd, with whom she immigrated to the United States after the war. Although she published over 300 research papers, Olga Taussky-Todd taught for many years at the California Institute of Technology before she was finally appointed to a professorship [Goodstein 2020].

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Abraham Flexner wrote to President Park on April 25: Her death has shed a deep gloom over us all, but it ought to make you and Mrs. Wheeler happy to know that a few weeks ago she remarked to Professor Veblen that the last year and a half had been the very happiest in her whole life, for she was appreciated in Bryn Mawr and Princeton as she had never been appreciated in her own country. [Shen 2019, 62–63] Fritz Noether and his cousin in Mannheim, Otto Noether, both expressed their gratitude to Bryn Mawr for providing Emmy with a second home. Fritz also wrote to Professor Wheeler on 23 May 1935: I know also from other reports that she felt at home at Bryn Mawr, and Bryn Mawr had become an absolute substitute for what she had to give up in her homeland. I also see from your report, how well you all know her, her idiosyncrasies, and her main traits – the unbreakable optimism which she evidently held till the last hours. Painful as the thought is to us all that she is no longer here with us, the greatest satisfaction remains that she herself kept living and working in her ideas until the moment that her thinking stopped, without her becoming aware of it. [Shen 2019, 64]

Chapter 9

Memories and Legacies of Emmy Noether 9.1 Obituaries and Memorials Those who knew Emmy Noether best were her fellow Germans in exile, in particular her former colleague in Göttingen, Hermann Weyl. On April 26, Weyl delivered [Weyl 1935], his well-known address for a memorial service held at Goodhart Hall on which occasion an urn containing Noether’s ashes was interred in the Cloisters at Bryn Mawr (Fig. 9.1). In his speech, Weyl made these comments about their time together before both left for the United States. 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 a mathematician in many respects. I did not succeed, nor did an attempt to push through her election as a member of the Göttingen 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. [Weyl 1968, 3: 432] At the very end of this address, Weyl’s words took on an emotional edge, but otherwise he spoke about Emmy Noether respectfully, but somewhat impersonally. Perhaps he felt his main task was to give an American audience at least a glimpse of the remote world she came from, the small city of Erlangen during the quiet era before the Great War. Hermann Weyl did not pretend to know much about © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8_9

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her early life, about which very little is known. Nevertheless, he fully recognized the importance of her family background, so he tried to paint a picture of life in Erlangen as he imagined it to have been. In describing mathematical life in Erlangen, he drew on on Max Noether’s portrait of his colleague Paul Gordan in [M. Noether 1914], written soon after Max and Emmy visited Felix Klein in the winter of 1912/13.1 He also tried to convey a sense of how their generation’s sense of solidity surely “prevailed in the Noether home [leading to] a particularly warm and companionable family life.”. This type of atmosphere, with “its comfort and bourgeois peacefulness” was now gone forever [Weyl 1935, 429]. Weyl cited Friedrich Nietzsche as the great voice for cultural renewal in Germany. Nietzsche, who once saw Richard Wagner in that role before becoming an outspoken anti-Wagnerian as an outcast philosopher and cultural critic, was regarded by Weyl and many others of his generation as a prophet of the future; it was he who first tore away the facade of moral hypocrisy long before the Great War brought down that whole decadent world symbolized by Kaiser Wilhelm II. What followed during the tumultuous years of the Weimar Republic was presumably the sort of chaos Nietzscheans would have welcomed, but certainly not staid middle-class Germans. And Hermann Weyl clearly identified the Noether family, but Emmy in particular, as “good Germans” in a generic sense. Insofar as her mathematics was concerned, Weyl spoke very directly about their differences with regard to the fertility of abstraction. He did so by translating his concluding remarks from a general expository lecture he had delivered four years earlier in Bern. On that occasion, he had warned against the tendency to “generalize, formalize, and axiomatize” without due attention to mathematical substance, a trend he saw as dangerous for the next generation of mathematicians. Only four years separated these two talks, which for Weyl must have felt like an eternity given all that had happened in that time. Still, he repeated several points from the earlier presentation when he spoke at Bryn Mawr. What he especially remembered, though, was how Emmy Noether had protested against his deeply pessimistic views. She thought Weyl was wrong, not on general grounds but rather because “she could point to the fact that just during the last years the axiomatic method had disclosed in her hands new, concrete, profound problems . . . and had shown the way to their solution” [Weyl 1935, 439]. In writing this, Weyl surely wanted to say that Emmy Noether had been right about this, even though he would long continue to harbor doubts about the future course of mathematical research. A few of Weyl’s less flattering remarks have also been repeatedly cited, which might easily lead to the impression that he had limited sympathy for her. This was hardly the case, as should be evident from Chapter 8. Until quite recently, in fact, it was completely overlooked that Hermann Weyl had also spoken at the more intimate funeral service held on April 17. On that private occasion, speaking 1 See Section 2.3; Klein described this as a lengthy visit, which makes it not unlikely that Weyl also had the opportunity to meet with them at this time (he only left Göttingen to accept Carl Geiser’s chair on the ETH Zurich in the fall of 1913).

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in German, Weyl expressed far more emotionally how he felt just a few days after her death.2 The hour has struck, Emmy Noether, for us to say goodbye to you forever. Your death will move many deeply; none more so than your beloved brother Fritz, who living almost half a world away cannot be here and can only speak his last farewell through my mouth. These flowers that I place for you on the coffin are from him. We bow in acknowledgment of his pain, which we are not entitled to put into words. But I feel it is an obligation at this hour to express the feelings of your German colleagues, those who are here and those in your homeland who have been loyal to our goals and to you personally. And at your grave I would like to do so in our mother tongue – the language you felt in your heart and in which you conceived your thoughts – which remains sacred to us no matter what power reigns on German soil. You will rest in foreign soil, in the earth of this great hospitable country that offered you a place of work after your own country closed itself off from you. At this moment we feel the urge to thank America for what it has done for German science in the past two pressing years, and especially to thank Bryn Mawr College, which was both happy and proud to include you among its teachers, and rightly so. Because you were a great female mathematician, I have no hesitation in calling you the greatest that history can record. Through your work algebra has acquired a new face. With many Gothic letters, you have inscribed your name indelibly on its pages. Perhaps no one has contributed as much as you in transforming axiomatic thinking, which before was only used to elucidate the logical foundations, into a powerful tool for concrete, forward-moving research. Amongst your predecessors in algebra and number theory probably Dedekind came closest to you. When I think of your being in this hour, two traits, above all, appear as most important. The first is the primal, productive power of your mathematical thinking. Like an over-ripe fruit, it seemed to burst through the shell of your humanity. You were the instrument and vessel of the intellect that broke forth from within you. There were no tender considerations, it was always the matter at hand that demanded attention. There was nothing gentle and well balanced in your being; you were not clay, formed into a harmonious shape by the hands of God, but rather a chunk of primordial human rock into which he breathed creative spirit. The power of your genius seemed to transcend the bounds of your sex, which is why we in Göttingen, in awed mockery, often spoke of you in the masculine form as “der Noether.” And yet you were a motherly woman with the warm heart of a child. You gave your pupils full and abundant intellectual support, and they gathered around 2 Translated

here from [Roquette 2007, 19–20].

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9 Memories and Legacies of Emmy Noether you like chicks under the wings of a large mother hen. You loved them, cared for them, and lived in close communion with them. And that is the second feature of your nature that I think is most significant: your heart knew no malice; you did not believe in evil, indeed, it never occurred to you that evil played a role in human life. This never impressed me more than in the last stormy summer of 1933 that we spent together in Göttingen. In the midst of the terrible struggle, collapse and rupture that raged around us in all factions, in a sea of hatred and violence, of fear and despair and burdensome worry – you went your usual way, pondering mathematical problems with the same zeal as before. When you were denied access to a lecture hall in the institute, you gathered your students in your own apartment; you remained friends with those who wore the brown shirt, you never doubted their honesty for a moment. Unconcerned about your own fate, fearless and open and conciliatory as always, you went your own way. Many of us believed that an enmity had been unleashed that could not be pardoned; none of this touched your soul. You were happy to return to Göttingen last summer, lived and worked in the circle of like-minded German mathematicians, as if everything had remained as before; you planned to do the same this summer. You truly deserve the wreath that the Göttingen mathematicians have asked me to lay on your grave. We do not know what death is. But isn’t it a comforting thought to imagine that after this earthly life our souls might recognize each other again, and how your father’s soul would then greet you? Has any father ever found a daughter who was such a great and independent successor? – In the midst of your full creative power you were suddenly torn from us; your sudden departure still stands like a flash of lightning written on our faces. But your memory will long remain alive in science and among your students, friends and colleagues; you have ensured this through your work and your personality. Farewell, Emmy Noether, you great mathematician and great woman. Your perishable remains shall pass away, that which is imperishable we want to preserve.

Both personally and professionally, Helmut Hasse stood even closer to Emmy Noether than did Hermann Weyl. No one knew this better than Richard Brauer, who wrote Hasse just one day after the funeral service at which he and Weyl both spoke. Brauer saw a good deal of Emmy during the past academic year, and her death clearly came as a huge shock to him. You will want to know more about Emmy Noether’s death, and I want to tell you what I know. She was here for the last time on April 2nd, as fresh as ever. She spoke with gaiety about a little trip she made with some of her students, and how she was able to hold out longer and was stronger than the young girls. When she postponed a proof in her afternoon seminar until the next meeting, no one could have imagined

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(a) Locality of Emmy Noether’s Ashes

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(b) Gravestone Marker

Figure 9.1: Final Resting Place at Bryn Mawr College (Courtesy of Qinna Shen)

that this meeting would never take place. The following Saturday, she wrote that she had to undergo surgery and would unfortunately have to cancel her seminar a few times, but she thought she would be able to continue in May and hoped all her auditors would still be there. The operation involved removing a fibroid. The tone of her letter seemed to indicate that she was taking it all very lightly. She also wrote about some very minor things that she couldn’t do now. If she was worried herself, no one can say. You know how Emmy didn’t like to show when she was troubled. Of course her friends here were pretty worried. The operation took place on April 10th and seemed to have gone smoothly. The next day she naturally had severe pain, but then recovered well. A Bryn Mawr colleague who visited her early on Saturday found she was particularly cheerful and optimistic. In the afternoon, no more visitors were admitted because the state of her health had suddenly deteriorated. We do not know exactly what it was, the stories and news from Bryn Mawr don’t provide a clear picture. On Sunday it was said that she had an embolism, but this was inconsistent with other stories. It’s now no longer important. On Sunday her condition was

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almost completely hopeless and Emmy had to suffer a lot, but she was at most half-conscious. She died then in the early afternoon. Whether she was aware of her approaching death isn’t possible to know. One can only hope that this was not the case. About the last year here it can be said that Emmy felt very well. She had most of her things brought over from Göttingen and had set herself up extremely nicely and comfortably. Bryn Mawr was proud of her and tried to arrange everything according to her taste. The plan was to establish a permanent research professorship for her. She had a circle of students who worked diligently with her on class field theory, and here in Princeton she also held a seminar in class field theory. A young girl, Miss Stauffer, had just finished a dissertation with her in Bryn Mawr. Emmy held her also very close to heart. Emmy has made many friends in Bryn Mawr and Princeton. She was not unmoved by the recognition that she generally received, and she was particularly pleased that she was celebrated not only as the unique mathematician but also as a full person in every respect. She was interested in many things here besides mathematics. The only downside for Emmy was the strongly felt homesickness for Göttingen and her friends. She wanted to go to Göttingen in the summer and was busy thinking about this trip. She spoke a lot about you. Her interest focused very much on your work on algebraic functions, and she liked to imagine how you would design the theory. In addition, she was naturally also interested in the work of her other friends and students, especially Deuring’s. Every letter from Germany was a great pleasure for her. Right up to her last few days, she was concerned about what would happen with her individual students and with their progress and future. She thought of everyone in the same way as before. Personally I am happy that I was able to live near her this winter. My wife and I were lucky enough to be able to approach her very humanly. In her we have lost a very dear friend.3 In Göttingen, Hasse’s local nemesis, Erhard Tornier, filed a formal report about him with the Nazi teachers’ organization, in which he tried to argue that Hasse should be disqualified from the operational direction of the Mathematics Institute. Since the Prussian Ministry appointed Tornier as Landau’s successor in order to placate the young radicals in Göttingen, this report was obviously only preaching to the choir. Tornier described Hasse’s research field (algebra and number theory) as “essentially Jewish” and his accomplishments as “one-sided” [Segal 2003, 158]. As an example of Hasse’s disregard for appropriate behavior, he mentioned how Hasse had sent a telegram ordering a wreath – with a ribbon inscribed “The Göttingen Mathematicians” – for Noether’s funeral, the one Weyl brought to the ceremony. 3 Brauer

to Hasse, 18 April 1935, [Lemmermeyer/Roquette 2006, 229–230].

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One year later, in April 1936, Hasse wrote to Berlin threatening to take a leave of absence if Tornier were not disciplined, and the Ministry quickly responded by moving his chair to Berlin University. Isaai Schur had been forced to take early retirement one year before, after which his chair remained vacant. In effect, the positions of Landau and Schur – two of Germany’s greatest mathematicians – had been fused into a single chair occupied by Tornier, a second-rate researcher whose career was based on political opportunism. Two years later, Werner Weber joined Tornier and Ludwig Bieberbach in Berlin, thereby consolidating its position as the citadel for advocates of a racially inspired nationalist mathematical tradition [Mehrtens 1987]. Previous chapters have illustrated the radically different atmosphere in Göttingen up until 1933. During the final years of the Weimar Republic, many of the talented foreigners studying there began gravitating into Emmy Noether’s growing circle of admirers. One particularly striking example was the French philosopher Jean Cavaillès, whose intellectual accomplishments are remembered in France, especially since he was a martyr for the French Resistance [Sinaceur 2013]. Cavaillès studied philosophy under Léon Brunschvicg, but he struck out on his own and developed a special interest in the philosophy of mathematics. In 1937 he would defend his doctoral theses at the University of Paris on the axiomatic method and formalism as well as on the foundations of abstract set theory. As one of the handful of normaliens who studied in Germany on a Rockefeller fellowship, Cavaillès crossed paths in Berlin with Jacques Herbrand, who had similar interests in foundational matters. Perhaps this was how Jean Cavaillès learned that Emmy Noether was preparing the final volume of Dedekind’s Collected Works [Dedekind 1930–32]. Apparently Cavaillès also spoke with Abraham Fraenkel about Dedekind’s correspondence with Georg Cantor, parts of which Ernst Zermelo was then preparing for publication in [Cantor 1932]. Initially, Noether tried to persuade Zermelo to publish all of their letters in the Cantor edition. On 12 May, 1930, she wrote him: “I would be very pleased if the Cantor-Dedekind correspondence would be published in your Cantor edition” [Ebbinghaus 2007, 161]. She felt this was far better than to publish these letters in the Dedekind edition, since Cantor was the one who made new mathematical claims. Zermelo decided otherwise, however. Once Jean Cavaillès arrived in Göttingen, he and Emmy Noether soon reached agreement on a joint project to publish the entire Cantor–Dedekind correspondence, which only appeared two years after her death in [Noether/Cavaillès 1937]. 4 Since the early 1920s, Noether was in fairly steady contact with Otto Blumenthal, the managing editor of Mathematische Annalen. Already in 1933, Blumenthal lost his professorship in Aachen, but Hilbert insisted that he nevertheless remain on as managing editor. He agreed to do so, but he also wanted to strengthen its editorial board by appointing B.L. van der Waerden (Hilbert and Erich Hecke were the other editors). When Blumenthal learned of Emmy Noether’s passing, he wrote van der Waerden to ask whether he would agree to write an obituary of her. 4 For

a recent discussion of this correspondence, see [Ferreirós 2007, 171–213].

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After gaining his assent, Blumenthal then wrote to gain approval from the other two board members [Rowe/Felsch 2019, 401]. In doing so, he noted that in recent years the Annalen had not so honored others, even those who had contributed greatly to the journal. In fact, even Felix Klein’s death in 1925 had only received a brief notice. Such an obituary was thus highly unusual. Indeed, were it not for the several scientific obituaries in the Annalen written by Max Noether, Blumenthal might well have characterized this belated gesture honoring Noether’s now famous daughter as totally unprecedented. Since B.L. van der Waerden was uniquely qualified to write this “Nachruf auf Emmy Noether” [van der Waerden 1935], his essay has long been regarded as one of the most important assessments of her mathematical work.5 He gave this description of her personality and unique intellectual approach: We have found these to be her outstanding characteristics: the ability to tirelessly and consistently pursue conceptual penetration of her subject matter in order to achieve utmost methodological clarity; a tenacious insistence on methods and concepts she found valuable, no matter how abstract and unproductive they might appear to her contemporaries; and the aspiration to arrange all specific connections within particular, general conceptual schemata. In a number of respects, her thinking does indeed differ from that of most other mathematicians. We all gladly lean on figures and formulas. For her, these aids were worthless or even disruptive. She only worked with concepts, not intuition or calculation. [van der Waerden 1935, 476] One can easily imagine the outrage this obituary must have provoked among those who were determined to bury all memory of Courant’s Göttingen. Yet their anger surely would have had little to do with personal antipathy toward Noether, an exotic figure who never made trouble for anyone. What rankled them was rather the way in which the editorial board of Mathematische Annalen, long seen as a clique composed of philo-Semites and Jews, continued to insult their sense of German honor. How else to explain this obituary, paying tribute to one of those who was forced to leave, someone who after 1933 clearly had no place in German society? Van der Waerden wrote barely anything about her dismissal; but across the ocean, another did: Albert Einstein. Chapter 3 describes some of Noether’s distant encounters with Einstein, but occasionally the question has been asked: did they ever really meet? One can find stories about them chatting together during tea breaks in Fine Hall, but since no one ever caught them on camera together, this would seem to fall in the category of hearsay evidence. On the other hand, given the size of the building and the fact that both were often in it together at the same time, one might just as well wonder: how likely is it that they never spoke with one another? Is it really so 5 For a careful assessment of van der Waerden’s obituary in the context of his delicate situation in Leipzig, see [Siegmund-Schultze 2011b]. A different reading that emphasizes van der Waerden’s resistance to Noether’s enthusiasm for abstraction is presented in [Schappacher 2007].

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hard to imagine Emmy Noether knocking on Einstein’s office door? Others might have felt too shy, but she? There is another reason to entertain these speculations, and this concerns the obituary article that appeared in the New York Times on May 5, 1935, signed by Einstein. An oft-repeated story about this piece is that the author was actually Hermann Weyl, but since the journalists who handled these matters had never heard of him, they asked for something from his super-famous IAS colleague. Such stories are usually difficult to refute, and something like this may well have happened. But in this particular instance, surviving textual evidence makes clear that Einstein wrote the obituary in question, or to be more precise, he drafted a German text (Fig. 9.2), which served as the basis for the published obituary. Since it appeared rather late, some three weeks after Noether’s death, it seems entirely plausible that Weyl might have submitted an obituary to the New York Times, only to have it rejected. Einstein rarely wrote in any language other than German, so this situation was in no way unusual. As pointed out in [Siegmund-Schultze 2007], this particular text was almost certainly translated by Abraham Flexner, who took some liberties in doing so. Einstein certainly did not know Noether well, so in certain respects what he wrote tells us more about him than about her. In fact, it tells us something about what he believed they shared as two prominent German Jewish intellectuals who met – perhaps for the very first time – in Princeton’s old Fine Hall. Einstein’s language is simple, eloquent and direct, not at all like Weyl’s often more elaborate prose; what is more, the central chords he strikes resonated deeply with his core belief in a higher spiritual life. Here is the opening paragraph: The efforts of most human-beings are consumed in the struggle for their daily bread, but most of those who are, either through fortune or some special gift, relieved of this struggle are largely absorbed in further improving their worldly lot. Beneath the effort directed toward the accumulation of worldly goods lies all too frequently the illusion that this is the most substantial and desirable end to be achieved; but there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual’s own feeling, thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors. If one were to ask a group of readers, who only got this far in the text, to guess – in the spirit of the old television game show “What’s my Line?” – the profession of the person so described, it is doubtful that even one percent would have guessed correctly. And if they were given the additional hint that this person was a woman, not even one person in a thousand would have likely imagined that

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Einstein was writing about a deceased female mathematician, whom he identified by saying: In the judgment 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. . . . Born in a Jewish family distinguished for the love of learning, Emmy Noether, who, in spite of the efforts of the great Göttingen mathematician, Hilbert, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators. Her unselfish, significant work over a period of many years was rewarded by the new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies. Farsighted friends of science in this country were fortunately able to make such arrangements at Bryn Mawr College and at Princeton that she found in America up to the day of her death not only colleagues who esteemed her friendship but grateful pupils . . . . The final sentence ends: “whose enthusiasm made her last years the happiest and perhaps the most fruitful of her entire career.” This, however, is not what Einstein wrote, as was pointed out in [Siegmund-Schultze 2007]. A more accurate translation would read: “Farsighted friends of science in this country made arrangements for her to work in a circle of friendly colleagues and grateful pupils up until her death, which swept her away in the midst of joyful and fruitful work.” So there was no mention in the original text of Bryn Mawr College or Princeton, and of course neither institution contributed financially to supporting Emmy Noether. It is difficult to say whether Einstein had a clear idea of who her “farsighted friends” were, but if he did, then he would surely have named Anna Pell Wheeler as perhaps the key figure who supported Noether during her last years. Abraham Flexner’s assessment that this period was the “happiest and perhaps the most fruitful of her entire career” clearly stems from what Noether told Veblen shortly before her unexpected death (and which Flexner then conveyed to President Park; see the conclusion of Chapter 8). In his letter to Helmut Hasse, Richard Brauer described Emmy’s state of mind during the last year of her life, including how the friendly recognition she received had brought her real happiness. Yet her life in the United States had been full of worries, in part because her prospects for a secure position remained quite unclear. All she really knew was that Bryn Mawr College could offer her nothing except a temporary home on condition that outside funding be obtained. Neither she nor the college had any interest in a plan that would have led to Noether teaching undergraduate courses. Thus Flexner’s claim, now embellished by the faulty assertion that her last two years had been particularly fruitful ones, is quite misleading. Still, especially attached to Einstein’s name, it was certainly a stroke of good public relations. As

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Figure 9.2: Original Manuscript for Einstein’s obituary of Emmy Noether, Einstein c Archive 05-138, Hebrew University, The Hebrew University of Jerusalem

we have seen, Oswald Veblen was the key figure who tried to promote the idea of creating a permanent position at the IAS for Noether. Flexner, on the other hand, had been reluctant to make any kind of commitment to her. So there seems to be no documentary evidence that would qualify him to be counted among her “farsighted friends.” Had she lived, Noether would surely have been very happy at the IAS, or possibly even with some kind of joint appointment that would have enabled her to continue training post-docs at Bryn Mawr. Yet, as Marguerite Lehr suggested, Emmy Noether’s hopes for growing a new “Noether family” in the United States never had enough time to be realized. As for Einstein’s own words, these too were misleading in some ways, though through errors of omission rather than commission. He first came to know about Emmy Noether’s talents in 1915/16, when she was working as David Hilbert’s assistant (see Section 3.4), and he may even have met her back then in Göttingen. In what manner Hilbert supported her has been described in Chapter 2, but there

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is no evidence at all of any later efforts on his part to secure a chair for her. Nor does it seem that she held out hopes for such a position, which is not to say that she was satisfied with the titular professorship she was awarded in 1922. Her name did come up twice for a professorship in Kiel, once in 1920 and then in 1928 [Siegmund-Schultze 2018], but her candidacy never reached the stage that she appeared on a list sent to a state ministry. When Einstein mentions that she came from a “Jewish family distinguished for the love of learning,” one imagines perhaps a rabbinical or other scholarly influence. Aside from her father, though, the males on both sides of Emmy Noether’s family were business people. It would have been more accurate to say that her family cultivated mathematical learning, especially since Max Noether was a very scholarly mathematician. And while Einstein underscores the social injustices Emmy Noether had to endure in Germany, he never points out that these mainly resulted from general discrimination against women. In the year 1935 he did not need to say why the new rulers of Germany dismissed her, since she was but one of many prominent Jews who had lost their positions.6 How Einstein truly felt about this – and why he rightly saw himself and Noether as the victims of an evil social psychosis – can be readily understood from the preface he wrote for the first edition of The World as I See It, which was published one year earlier. The following pages are dedicated to an appreciation of the achievements of the German Jews. It must be remembered that we are concerned here with a body of people amounting, in numbers, to no more than the population of a moderate-sized town, who have held their own against a hundred times as many Germans, in spite of handicaps and prejudices, through the superiority of their ancient cultural traditions. Whatever attitude people may take up towards this little people, nobody who retains a shred of sound judgment in these times of confusion can deny them respect. In these days of the persecution of the German Jews especially it is time to remind the western world that it owes to the Jewish people (a) its religion and therewith its most valuable moral ideals, and (b), to a large extent, the resurrection of the world of Greek thought. Nor should it be forgotten that it was a translation of the Bible, that is to say, a translation from Hebrew, which brought about the refinement and perfection of the German language. Today the Jews of Germany find their fairest consolation in the thought of all they have produced and achieved for humanity by their efforts in modern times as well; and no oppression however brutal, no campaign of calumny however subtle will blind those who have eyes to see the intellectual and moral qualities inherent in this people.7 6 Many, like Richard Courant, first went to Britain before settling in the United States, as described in [Medawar/Pyke 2012]. 7 This preface appeared only in the 1934 edition of The World as I See It; it was reprinted in [Rowe/Schulmann 2007, 291–292].

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9.2 Fate of Fritz Noether and his Family Owing to her sudden death, Emmy Noether was spared the sad news of learning what became of her brother and his family, including her two nephews Hermann and Gottfried Noether. Emmy tried to find some kind of position for Fritz Noether in the United States, but as described in Section 8.3, these efforts failed. Fritz began making plans in the late summer of 1934 to depart for Tomsk with his wife Regina and two sons. Before long, Regina found life in Tomsk unbearable. After she experienced a total breakdown and attempted to kill herself, Fritz brought her back to Germany so that she could be under the care of two of her sisters in Gegenbach, the town where her parents had lived in the Black Forest. Later that summer, Fritz Noether returned to Germany, followed somewhat later by his two sons, but they were all too late; shortly before, on 21 July 1935, Regina Noether committed suicide.8 Two months later, in early September 1935, Fritz Noether attended a meeting of the Moscow Mathematical Society at which Pavel Alexandrov delivered his famous address honoring the memory of Emmy Noether [Alexandroff 1935]. As the guest of honor on this occasion, Fritz Noether was surely very moved by this special event held five month’s after his sister’s death. Alexandrov spoke (in Russian) as President of the Moscow Mathematical Society to an international audience that included Heinz Hopf, Solomon Lefschetz, and numerous others who were attending a week-long topology conference (see [Whitney 1988]). Lefschetz also said a few words, highlighting the great significance Noether’s ideas had for modern topology. Alexandrov began by mentioning Weyl’s address as well as B.L. van der Waerden’s obituary article [van der Waerden 1935]. In noting these testimonials, he emphasized that his own remarks would have a rather different character. “I would like to evoke for you as accurate an image as possible of the deceased, as a mathematician, as the head of a large scientific school, as a brilliant, original, fascinating personality” [Alexandroff 1935, 2]. Citing Weyl’s remark that “no one could contend that the Graces had stood by her cradle,” he assented, but then immediately went on to express what really mattered. Fräulein Noether, as all who knew her well could attest, “loved people, science, life with all the warmth, all the joy, all the selflessness and all the tenderness of which a deeply feeling heart – and a woman’s heart – was capable” [Alexandroff 1935, 11]. Alexandrov shared none of Hermann Weyl’s ambivalent feelings when it came to Emmy Noether’s personality, and some of his remarks were a direct reply to what Weyl had said in Bryn Mawr. One of the most striking passages picks up on what Weyl had written about the danger of losing “mathematical substance,” 8 These details are taken from a letter Herman Noether wrote to a relative on October 10, 1989, not long after he and his brother first learned about the circumstances surrounding his father’s death. Three years later, this sad news regarding their mother’s suicide was conveyed to Hermann Weyl by Hermann and Gottfried, who contacted him from their refuge in Sweden; see below.

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and how Emmy Noether had objected to the tenor of those remarks. Alexandrov wanted to make very clear the nature of her objections, which had nothing at all to do with dismissing the importance of what Weyl called substance. “What she protested against was the pessimism . . . from Weyl’s speech of 1931” because for her “the substance of human knowledge, including mathematical knowledge, was inexhaustible” [Alexandroff 1935, 4]. He recalled the many times he had spoken with her about the nature of mathematical knowledge and creativity – on which occasions he sometimes cited a famous saying of Laplace about the limitations of empiricism – in order to emphasize how they both shared a view of mathematics grounded in realism. As he put it, “a profound feeling for reality lay at the foundation of Emmy Noether’s creativity; her entire scientific personality opposed the tendency (which is widespread in many mathematical circles) to transform mathematics into a game, into some sort of peculiar mental sport” [Alexandroff 1935, 5]. Alexandrov wrote movingly about the woman he and his friend Pavel Urysohn first met in Göttingen in 1923. Her school then had only just begun and consisted of a mere handful of German students, but its international character would soon thereafter unfold. A major breakthrough came the very next year with the arrival of B.L. van der Waerden from Amsterdam. Alexandrov called him “one of the brightest young mathematical talents of Europe” and credited him with mastering Noether’s theories, while adding significant results of his own, and “more than anyone else, [helping] to make her ideas widely known” [Alexandroff 1935, 5]. He recalled how, a few years later in Göttingen, van der Waerden taught a highly successful course on ideal theory that did much to spread awareness of Noether’s work. Some of Alexandrov’s most vivid memories of Emmy came from the winter semester of 1928–1929, when they were together in Moscow. He spoke of her keen interest in the Soviet experiment and her firm intention to visit again, though this was not to be. As an intimate friend with longstanding ties to the Göttingen community during the 1920s, Alexandrov left a truly striking portrait of Emmy Noether as he knew her. The following summer, Fritz Noether visited Oslo to attend the International Congress of Mathematicians. He presented a paper on the transmission of electrical waves, the only presentation made by a representative of the Soviet Union, as no one else had been allowed to attend, not even those who were invited to deliver plenary lectures: A.O. Gelfond and A.Y. Khinchin.9 Noether evidently reached Oslo traveling with his German passport. Grace Shover also attended the ICM and introduced herself to her mentor’s brother on this occasion [Quinn et al. 1983, 141]. The boycott of this major international congress by the Soviet government was surely an ominous sign, and indeed the Moscow show trials were just now getting under way. One year later, Fritz Noether was swept up in the frenzy. 9 On the political tensions in [Hollings/Siegmund-Schultze 2020].

the

background

of

the

1936

ICM

in

Oslo,

see

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In November 1937, Noether was arrested by the NKVD in his home; he was charged with spying on the Russian armament industry and even committing acts of sabotage. The verdict, however, was only pronounced on October 13, 1938: he was found guilty and sentenced to twenty-five years along with the confiscation of all his belongings. During the interim period, his sons were told nothing of his whereabouts, despite many inquiries. Instead they were forced to leave the country; in March 1938, both were given 10 days notice to depart from the Soviet Union. They then contacted some distant relatives in Göteborg, Sweden, who agreed to take them in for a short while.10 Their visas expired in July of that year, so they applied for extensions and at the same time for fellowships offered by MIT. In this limbo state, during which time the Nazi government revoked Fritz Noether’s citizenship and that of his sons, they wrote to Hermann Weyl in Princeton, informing him of all that had happened. Weyl recapitulated this news in a letter to Jacob Billikopf, the Jewish philanthropist located in Philadelphia who had been involved in a fund-raising effort for Emmy Noether four years earlier.11 Recognizing that time was of the essence (Weyl even noted the expiration dates on their German passports), he urged Billikopf to obtain affidavits for both young men, since they could not wait until receiving official word about their applications from MIT. Weyl later sent a copy of this fairly lengthy letter to Dr. Florence R. Sabin at the Rockefeller Institute for Medical Research.12 She was a graduate of Smith College, but was clearly well known to President Park at Bryn Mawr, to whom she wrote about this matter. Already in April 1938, soon after arriving in Göteborg, the Noether brothers had applied for entry visas at the local consulate of the US State Department. 13 Since both were non-practicing Catholics, they soon thereafter also turned to a Catholic refugee relief program for assistance. They were informed that the immigration quota for Germans had already been met, but they hoped to be able to enter the US on student visas, a plan that depended on receiving fellowships from MIT. In the fall of 1938, they received letters of admission to the graduate 10 Herman Noether later recalled that they asked not to be deported to Germany, knowing that they had no future there. The Swedish relative was a sister of their paternal grandmother, Ida Kaufmann Noether. (Notes from Herman Noether for his family, undated, courtesy of Evelyn Noether Stokvis.) 11 Weyl to Billikopf, 17 October 1938, Emmy Noether Papers, Bryn Mawr College Special Collections. 12 Weyl to Sabin, 23 November 1938, Emmy Noether Papers, Bryn Mawr College Special Collections. How he came to know her remains unclear, but since Sabin was one of the most influential women in American science, one need not wonder that Weyl had been introduced to her on some occasion in New York. Sabin was the first woman to hold a full professorship at Johns Hopkins School of Medicine, the first woman elected to the National Academy of Sciences, and the first woman to head a department at the Rockefeller Institute for Medical Research. 13 The information described below, unless otherwise stated, can be found in 298 letters and documents in the case file for Hermann and Gottfried Noether compiled by the American Friends Service Committee. This file, documenting events from 1938 to 1940 in considerable detail, was obtained by the Noether family through the Holocaust Memorial Museum in Washington, D.C. I am very grateful to Margaret Noether Stevens for making this invaluable resource available to me for this project.

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school of MIT. Gottfried was informed that the decision on his application for a fellowship in mathematics would be made some months later, whereas Hermann learned in a letter from October 31 that the competition in chemistry was very intense and that he had few chances of gaining financial support. Hermann Noether presumably conveyed this disappointing news to Weyl, who then sought help from Dr. Sabin at the Rockefeller Institute. Sabin advised him to write to the Harvard astronomer Harlow Shapley, a central figure in American efforts to support academic refugees. Weyl did so, and in a letter to Shapley from December 9, 1938, he spelled out the plight of the Noether boys quite vividly. For the better part of the past year, they found themselves stranded and penniless in Sweden, having been expatriated by the German government. “Their permits to stay in Sweden have once been extended, but have expired again and the police are after them (though I do not see what they will do with them. Dump them into the Baltic Sea?).” Weyl explained that the Committee for Catholic Refugees from Germany had facilitated the applications to MIT, and he fully expected that Gottfried would be offered a stipend, whereas his brother had no chance. This being the case, he wanted to recommend Hermann Noether to Harvard, which had recently established a program offering graduate fellowships to twenty qualified refugees. This fund was initiated by a small number of Harvard students in response to the atrocities perpetrated by Nazis during the infamous Kristallnacht of November 1938. Harvard’s president, James Bryant Conant, was himself a distinguished chemist, which led Weyl to speculate that “[President Conant] himself might be interested in this extraordinary, and extraordinarily tragic, case.” Weyl added that, while he did not know these young men personally, he knew their father and his sister very well, “simple, warmhearted [people], of a deep rather than quick intelligence, Fritz without brilliance, Emmy a genius.” As background information, he enclosed a copy of his memorial address at Bryn Mawr from April 1935. Hermann Noether had a very impressive academic record, indeed, beginning with his scientific studies in Breslau, followed by the three years he spent at his father’s institution in Tomsk, where he and his brother managed to become fluent in Russian. There he studied physical chemistry with a specialization in spectroscopic research that led to a thesis dealing with the photochemical decomposition of benzene. After learning Swedish, he continued his studies at the Chalmers Institute of Technology in Göteborg. Probably soon after Weyl contacted him, Shapley passed this information on to the selection committee. In any event, Weyl’s intervention was successful, as Hermann Noether received a fellowship to pursue graduate studies in chemistry at Harvard, beginning in the fall of 1939. 14 This, however, hardly solved all the problems he and his brother still faced back in Sweden. 14 He was one of sixteen recipients, whose names were listed in an article from May 15, 2019 in the Harvard Gazette.

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In the meantime, Jacob Billikopf had contacted Walter H. Bieringer, a Jewish business executive who chaired the Boston Committee for Refugees. Bieringer was able to locate a sponsor who agreed to sign an affidavit for the Noether brothers, thereby solving what Weyl saw as the principal hurdle. He assumed they could enter the US on students visas, which did not fall under the quota system. This possibility, however, appeared to be foreclosed in view of the fact that they were now officially stateless, which meant that they had to wait until a new contingent of visas for Germans became available.15 At this stage, no one anticipated any financial difficulties since Emmy Noether’s estate included some $2,500, which a lawyer had deposited in her brother’s name in a bank in Philadelphia. By tapping these funds, his sons would have a solid basis for financing their education in the United States. After receiving the affidavit, Hermann and Gottfried took it to the US Consulate in Göteborg, where it was immediately approved by the consular official. The Noether brothers imparted this good news to Walter H. Bieringer in a letter of thanks from December 6, 1938. A copy of their letter was then forwarded to Professor Arnold Dresden at Swarthmore, no doubt because it also contained information concerning certain unanticipated financial difficulties. In their most recent conversation with the Consul, the Noethers were informed for the first time that to obtain visas they would need to show proof that each had resources of $1,000 in order to enter the country. They apparently had already contacted the bank in Philadelphia, but they were informed that their father’s signature would be needed in order to withdraw funds from the account in his name. Dresden had longstanding connections with the Philadelphia office of the American Friends Service Committee (AFSC), which played a major role in placing refugees. He therefore forwarded a copy of this letter to Dr. Hertha Kraus, who headed the AFSC’s refugee program. Her subsequent role in this affair turned out to be crucial, hence the importance of the following brief biographical information. Hertha Kraus came from a Jewish family in Prague. She later studied social welfare statistics in Frankfurt, dropped her Jewish religion to become a Quaker, and soon became a prominent expert on social issues in Germany. At age 25, she headed the city of Cologne’s Welfare Department under Mayor Konrad Adenauer, who was deeply impressed by her energy and accomplishments (despite the fact that Kraus was a convinced Social Democrat!). She later became an important figure in postwar reconstruction efforts when Adenauer became Chancellor of the new Federal Republic of Germany, but from 1933 onward her home was in the United States. Due to her Jewish background, Kraus fled to the US in 1933 and quickly emerged as a leading Quaker activist. Three years later, she was appointed Associate Professor of Social Economy and Social Research at Bryn Mawr College, where she taught until 1962. Her involvement with the Noether refugee case appears to have begun in December 1938, shortly before Arnold Dresden forwarded 15 This reflects that the US quota system was at least partially based on ethnicity, not citizenship alone.

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the brothers’ letter describing the Göteborg Consul’s latest condition for their visa application: written proof that each of them had $2,000 (roughly $40,000 today) in resources. By this time, the AFSC office in Philadelphia was buried in work, as waves of German Jews were hoping to immigrate to the United States in the wake of Kristallnacht, the “Night of Broken Glass.” On November 9–10, 1938, SA thugs and their supporters launched a pogrom that destroyed over 250 synagogues across Germany, Austria and the Sudetenland. Mobs terrorized the Jewish population, trashing their businesses and looting their stores, while police stood by passively. Some 30,000 Jewish men were arrested and incarcerated in concentration camps. Afterward, the Nazi government levied a collective fine amounting to 20% of the wealth of the Jewish community, while confiscating six million Reichsmarks worth of insurance payments for property damage. Over the course of the next ten months, more than 115,000 Jews emigrated, many to western European countries and Palestine, while some 14,000 left for Shanghai.16 The annual quota for Germans and Austrians wishing to immigrate to the United States was set at 27,370; it was in the year 1939 that this level was filled for the first time during the Nazi era. In view of these circumstances, not to mention the vacillating information they received from the US Consul in Göteborg, Hermann and Gottfried Noether found themselves in a truly desperate situation. Reading through the nearly 300 documents in their file from the American Friends Service Committee, one can only conclude that it was a great stroke of luck when Hertha Kraus became aware of their case. By this time, she had become very familiar with US immigration policies; moreover, her staff at the AFSC worked in close cooperation with a wide network of contacts throughout the country. Kraus realized that this was a highly unusual situation, one she had probably never encountered before. On the one hand, the Consul approved the affidavit of support, but, on the other hand, he claimed it did not suffice for obtaining a visa to enter the country. Still, her initial inclination was to approach the Ninth Bank & Trust Co. in Philadelphia, seeking their cooperation in the matter. Unfortunately, this only led to a great deal of futile discussion and correspondence. After some months had passed, Kraus obtained a statement from the bank that described both the amount of money in Fritz Noether’s savings account (ca. $2,500) as well as his sons’ rightful claim to it. She then contacted the State Department in Washington, asking whether these circumstances might not satisfy the Göteborg Consulate. Soon afterward, a State Department official informed Hertha Kraus that the case would be handled expeditiously, meaning that the Noether brothers could expect visas to be issued soon after July 1 (when a new contingent for the German quota would be available). The letter also implied that the Consul in Göteborg had never required any proof of financial resources, surely a typical instance of bureaucratic coverup. 16 Probably one of those who made it to Shanghai at this time was Fritz Noether’s cousin Otto Noether. Another Hermann Noether (Otto’s nephew) also moved to Shanghai and spent the rest of his career there.

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When Hermann and Gottfried next spoke with the Consul, he told them that no further financial proof was required in view of the many people in the US who supported their immigration case. From the time she first became aware of their situation, Hertha Kraus and her staff essentially took responsibility for all facets of the case, which had originally been in the hands of the Committee for Catholic Refugees from Germany. She made sure, however, that this organization and all other interested parties were updated periodically. As it turned out, Hermann Weyl’s optimism regarding Gottfried Noether’s chances for a fellowship from MIT was misplaced, but the AFSC made arrangements for him to study at Ohio State University, where he completed his B.A. degree in 1940 [Fritsch 1999]. One year later, he earned an M.A. from the University of Illinois, after which he was drafted into the US Army. During the period when Gottfried Noether was studying at Ohio State and Illinois, his brother Hermann made rapid progress at Harvard, completing his M.A. degree after one year. He then began work on a Ph.D. under E. Bright Wilson, completing his dissertation research in 1943. His work dealt with the synthesis and structural analysis of certain organic compounds by means of infrared spectroscopy. Both brothers married in 1942 and received congratulations from Hertha Kraus, who also inquired as to whether they had any more news about their father. After the war, they asked for help from the AFSC in trying to obtain information about him, but these efforts proved futile. Hermann Noether met his future wife, Dorit Low, at a Christmas dance party for students at Harvard and Radcliffe, where she, too, was studying chemistry. They clicked on the dance floor to their own delight, winning the waltz prize that very night: a baby doll that became a family heirloom. Dorit was born in Vienna as Dorit Löw to Joseph Löw and Marianne Dorit Weinberger Löw, who originally came from Jewish families in two smaller cities in the present-day Czech Republic. One year after their marriage in September 1942, Hermann and Dorit Noether moved to Kew Gardens, Queens, New York City, where they lived close to her parents. From there he commuted to his research lab at the Celanese Corporation of America in Newark, New Jersey, which specialized in manufacturing synthetic fibers. His wife shared a similar passion for chemistry, and so after graduating from Radcliffe she went on to take a masters degree in chemistry at Columbia University; she later completed her graduate work at Rutgers University while raising their three children. Dorit Noether eventually earned her Ph.D. from Rutgers and taught there for several years. She later became an associate editor of ChemTech, a magazine of the American Chemical Society, and together with Herman they wrote the Encyclopedic Dictionary of Chemical Technology. In 1945, they became naturalized US citizens, Herman D. and Dorit Low Noether. Two years later, his commute became even longer when Celanese transferred him to its research laboratory in Summit, New Jersey. His career there prospered and he soon became a leading expert on applications of polymer chemistry. Finally, in 1959 they bought a house in nearby Milburn, NJ, and eventually moved to Summit, where they spent the rest of their long lives. At the Summit

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lab, Herman Noether was able to pursue industrial research based on spectroscopic techniques, which fit his educational background perfectly. Then came a major turning point in his career when he was asked to take over the company’s x-ray crystallography operation, an open-ended field of research about which he knew very little. Some spectroscopists were then convinced that synthetic polymers lacked any crystalline structure, but new techniques were just then emerging that used x-ray diffraction to determine their degree of crystallinity. Soon Herman Noether developed a world-wide network of contacts with other researchers who were breaking new ground in this field. This basic research led to investigations of new types of polymers with unusual properties of importance for fiber production at high speeds. Over the course of his career at Celanese, Herman Noether took out dozens of patents and became an internationally recognized authority in this new area of industrial research. After a whirlwind romance, Gottfried Noether married Emiliana Pasca, who was born in Naples and then emigrated to the United States with her family. She grew up in New York City in a family of Italian musicians and teachers, then went on to study history at Columbia University. Gottfried and Emiliana first met in New York in early 1942, when he was visiting during a short Easter break from the army. They were introduced to each other by a mutual friend, a German Jewish refugee who was eager to play matchmaker. Emiliana, who was then pursuing her Ph.D. in Italian intellectual history at Columbia University, was none to eager to meet this “wonderful mathematician,” but reluctantly agreed to spend part of a Saturday showing him around New York. When she first met him the night before, though, her matchmaker friend began by telling her that Gottfried had an extra ticket for a Toscanini concert at Carnegie Hall on Saturday night. “Wouldn’t you like to go?” she asked. It turned out that her escort never said a word about mathematics, and she quickly discovered they had much in common, a passion for music and art, as well as ambition to pursue an academic career. After a few months of intense correspondence, Gottfried returned for another visit in late July. Knowing that he would likely be shipped off to Europe later that year, they spontaneously decided to marry. With virtually no time left before Gottfried had to return to his army base, they managed to obtain a marriage license and an appointment with the “marrying judge,” who enjoyed nothing more than performing this service for soldiers and their sweethearts. Just before his departure, Gottfried married Emiliana on Saturday morning, August 1, 1942. They would not see each other again until late 1945. Some six months after the United States entered the Second World War, the Army established a top-secret military intelligence training center at Camp Ritchie in Maryland [Spracher/Kramar 2013]. Gottfried Noether was among the 19,000 soldiers who took the eight-week training course there before taking up duties in Europe. Presumably he was recruited for this special training, since the military was keenly interested in utilizing those who spoke German for intelligence gathering and other purposes. Noether was stationed in London, where he translated intercepted messages for the army; after the war, he worked as a translator at the

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Nuremberg trials. He resigned from the army with the rank of first lieutenant. Later in life, Gottfried rarely spoke about his experiences as one of the “Ritchie boys” 17 Emiliana began teaching in 1946 – first at New York University, while still completing her dissertation, and subsequently at Rutgers University – during which time Gottfried began his doctoral studies, also at Columbia, under Jacob Wolfowitz. A native of Warsaw, Wolfowitz came to the United States in 1920 when he was ten years old. He worked very closely with the statistician Abraham Wald, a convenient arrangement as both were attached to the Statistical Research Group at Columbia. Their first joint paper dealt with calculations of confidence intervals which are not necessarily of fixed width on the basis of empirical data without assumptions about an underlying distribution, such as normality. In 1942 Wolfowitz coined the term nonparametric statistical inference for this type of study, and it was in this new field of research that Gottfried Noether made his name. After taking his doctorate in 1949, he taught for two years at the fast-growing Courant Institute at New York University. Noether then joined the mathematics department at Boston University, passing through the ranks from assistant to full professor from 1951 to 1958. A decade later, he left Boston to become chair of the Department of Statistics at the University of Connecticut at Storrs. By this time, he had established a reputation as a leading authority in his field, having one year earlier published a well-known textbook Elements of Nonparametric Statistics [G.E. Noether 1967]. Not entirely unlike his aunt, Gottfried Noether was eager to overturn conventional approaches to teaching, as reflected in another textbook from this time, Introduction to Statistics: A Fresh Approach [G.E. Noether 1971]. He also played an active role in efforts to reform high school mathematics by introducing statistics as a part of the standard curriculum. Emiliana Noether graduated in 1948 with a doctorate in history from Columbia University, one year before her husband took his Ph.D. there. She later became a well-known authority on Italian history and cultural and intellectual ties between Italy and the English-speaking world. At the end of their teaching careers, both she and her husband were working at the University of Connecticut. 18 Together they prepared a short paper on “Emmy Noether in Erlangen and Göttingen” [E. & G. Noether 1983], as part of the panel discussion that took place on March 18, 1982 during the Bryn Mawr Symposium commemorating the 100th anniversary of Emmy Noether’s birth. On that occasion, Gottfried Noether apparently said nothing about his father’s fate, although he and his brother Herman had by no means given up searching for answers. Probably the most concrete information they had received after the war came from a fellow prisoner, Fritz Houtermans, a leading nuclear physicist who had studied in Göttingen under James Franck. Houtermans had been a member of 17 Roughly 2,200 of those who were trained for U.S. Army Intelligence at Camp Ritchie were Jewish refugees born in Germany and Austria. 18 At the time of her death in 2018, Emiliana Noether was survived by her daughter, Monica Noether, and two grandchildren.

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the German Communist Party since the 1920s and left the country for England when Hitler came to power. He and his wife then emigrated to the Soviet Union in 1935. When the Great Purge swept the country, Houtermans was working in Kharkov. Arrested by the NKVD in December 1937, he was tortured and confessed to being a German spy. After the signing of the Hitler-Stalin Pact, thousands of political prisoners were transferred to Butyrka Prison in Moscow. From there, many Germans were repatriated, which only meant that they fell into the hands of the Nazis. Jewish ex-prisoners were usually sent to a work camp in Lublin, where very few survived the war. In Houtermans’ case, he was released to the Gestapo in May 1940 and then spent the next three months in a prison in Berlin. Owing to his scientific expertise, he regained his freedom after Max von Laue intervened on his behalf.19 Houtermans was contacted after the war by a friend of Otto Noether, one of Fritz Noether’s cousins. In a letter from February 20, 1946, Houtermans described the situation in Butyrka Prison, where he and Noether spent two months together in a cell with some 30 German prisoners from December 1, 1939 to the end of January 1940. Butyrka had a long tradition, dating back to the time of Catherine the Great, connected with its brutal treatment of political prisoners, though apparently conditions at this time for the thousands of German civilians were reasonably civil. According to Houtermans, torture techniques were only used to extract confessions.20 He had no information about Fritz Noether’s fate after he was transferred from their common cell, but he remembered him as a man of great integrity. “His constitution is very strong,” Houtermans wrote, and “in spite of all the suffering he was always in good humor and had inexhaustible courage to live and was a real comfort to his comrades.” At the time they were together in Butyrka, Fritz Noether spoke with Houtermans, hoping he could contact his sons, should Houtermans be released. Noether assumed they were living in Göteborg, and he wanted to let them know he was alive and well. More than a year later, Houtermans imparted this information to the Jewish physicist Fritz Reiche, who very likely knew Fritz Noether, since he, too, had held a professorship in Breslau up until 1933. Afterward, Reiche worked in Berlin, but he managed to leave for the USA in early 1941. Just before he departed, Houtermans asked Reiche to pass on Fritz Noether’s message to Richard Courant and Hermann Weyl, as he assumed they would be in contact with Noether’s sons.21 19 Toward the end of the war, he lost his research position in Berlin, but then soon garnered a new appointment in Göttingen, where he remained until 1952. In that year he left for Bern, where he founded an important school for studying radioactive phenomena in astrophysics and geology. 20 Under pseudonyms, he and another cellmate later wrote a book describing these technqiues (Beck, F. and Godin, W.: Russian Purge and the Extraction of Confession, (Hurst and Blackett, 1951). 21 This anecdote is also noteworthy because of another message that Houtermans asked Reiche to convey, namely, his fear that Werner Heisenberg’s team would soon be on track to build a

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In his letter from 1946, Houtermans outlined four possible scenarios, the first of which assumed Fritz Noether had been released from Soviet detention and placed in the hands of Nazi authorities. If that had occurred, as seemed at the time likely, then he almost surely perished in a work camp. After several futile attempts to learn what had happened to their father, Hermann and Gottfried Noether reached the tentative conclusion that he had, indeed, been deported to Nazi Germany and suffered this fate. Having received no news after the war, they requested in 1950 that an orphans’ court open proceedings that would lead to an official declaration of Fritz Noether’s death. More than a decade had passed since the brothers first entered the United States; they would now finally be able to receive their inheritance from Emmy Noether’s estate. In 1980, Herman and Dorit Noether were in London and had occasion to visit with the biochemist Zhores A. Medvedev, a well-known dissident who had been expatriated by the Soviet Union in 1973.22 Medvedev related that their best chance of gaining information about the fate of Fritz Noether would be to file an application for his rehabilitation with the Supreme Court of the USSR. Soon afterward, Herman and Gottfried Noether submitted such an application with the Soviet Ambassador Anatoly Dobrynin, but they received no reply. In 1985, Gottfried Noether published a short article about his father’s career in the journal Integral Equations and Operator Theory, which ended as follows: Information about Fritz Noether after his arrest is fragmentary. At some time, most likely before the end of 1939, he was in Orel. In December 1939 and January 1940, he was seen in Butyrka Prison in Moscow. There is a report that he was seen in the center of Moscow toward the end of 1941 or the beginning of 1942. But numerous attempts, official and unofficial, to find out more about his fate remain unsuccessful. At the time of his arrest, Fritz Noether had completed the manuscript of a book on Bessel functions. The manuscript had been translated into Russian and was scheduled for publication in the Soviet Union. It is not known whether the book has ever been published. [G.E. Noether 1985, 576] In 1986, Mikhail Gorbachev adopted “glasnost” as one of his political slogans, an encouraging sign that the Soviet Union might finally enter a period of peace and prosperity. One year later, Herman Noether wrote to Gorbachev, describing all previous efforts to obtain information, and this time he received a brief reply from the Soviet Embassy stating that Fritz Noether had died on September 11, 1941. After learning this, Noether wrote to Gorbachev a second time, asking for details nuclear bomb. Soon thereafter, Reiche imparted this news to a group of physicists in Princeton, see [Frenkel 2011, 85–86]. 22 These details relating to Herman Noether’s efforts to obtain information from the Soviet Union were described in a letter from Herman Noether to Auguste Dick, 21 June, 1989 (Auguste Dick Papers, 9-26, Austrian Academy of Sciences, Vienna).

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Figure 9.3: Plaque for Fritz Noether in Gegenbach (Auguste Dick Papers, 12-14a, Austrian Academy of Sciences, Vienna)

about what had transpired. In May 1989, Herman and Gottfried Noether finally received a definitive message from Soviet authorities.23 This came by way of a letter from Andrei Parastaev, First Secretary at the USSR Embassy in Washington, D.C. It related that their father’s case was reviewed the previous December by the Plenum of the USSR Supreme Court, which found that the charges that had been filed against Fritz Noether were groundless, that he had been wrongfully convicted, and that the court had declared him fully rehabilitated. Noether had originally been sentenced to 25 years imprisonment for espionage. Then, on September 8, 1941, he was summarily sentenced to death for supposedly engaging in anti-Soviet agitation; just two days later, he was executed. Though not stated, the obvious reason for Fritz Noether’s swift execution was purely political. Some two months earlier the German army had launched a surprise attack on Soviet Russia, following Hitler’s decision to break with the Molotov-Ribbentrop non-aggression pact. In response, Stalin and his henchmen decided to liquidate many of the Germans held in Russian prisons, especially those in localities close to encroaching German forces. Herman Noether, who had been particularly vigilant in his efforts to learn what had happened to his father after his arrest in November 1937, designed a 23 Auguste Dick learned about this through her correspondence with Fritz Noether’s sons, and thereafter she published a brief report in [Dick 1990]. She also informed the Leipzig historian of mathematics Hans Wußing, which led to Karl-Heinz Schlote’s article [Schlote 1991]; see also [Segal 2003, 60–62].

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plaque in his memory. It stands at the base of his mother’s gravestone and shows a replica of his father’s signature (Fig. 9.3). The text reads: In Memoriam Prof. Dr. Fritz Alexander Noether 7. Oct. 1884 Erlangen – 10 Sept. 1941 Orel Iron Cross 1914–1918 Victim of Two Dictatorships 1934 driven from Germany due to race 1938 charged and convicted by the Soviets 1941 executed · 1988 declared innocent

9.3 Hasse’s Sympathies for Hitlerism By the mid-1930s, Helmut Hasse’s career became increasingly entangled with the fate of the Third Reich. After Erhard Tornier’s sudden ouster in April 1936, Hasse gained full control of Courant’s former institute in Göttingen. As a hardcore German nationalist, Hasse had no difficulty accepting the Nazi dictatorship since he was convinced that Hitler was uniquely qualified to restore his country’s honor and position in the world. In September 1938, he was pleased to confer with Francesco Severi, an outspoken supporter of Mussolini’s regime, when the latter arrived in Baden-Baden to attend the annual meeting of the German Mathematical Society. This conference took place just when Hitler was threatening to unleash a European war over his demand that Czechoslovakia cede the Sudetenland, its buffer zone containing large numbers of ethnic Germans. Mussolini played a significant role in defusing this conflict, which was resolved on September 30 when Prime Minister Neville Chamberlain signed the Munich accord that gave Hitler what he wanted (for the time being). On October 3, a few weeks after they had met in Baden-Baden, Hasse wrote an elated letter to Severi in which he lauded il Duce:24 All Germans are moved these days by the resolute faithfulness with which your incomparable Duce has stood beside our Führer, and by the united solidarity which the Italian people have acknowledged in the interest of our people. Down to the last one among us we have realized these days that the intended goal: the liberation of the SudetenGermans, would never have been attained, if the unfaltering will of our Führer and our people had not enjoyed this strong and resolute support by the other pole of our axis. You have heard it from the mouth of our Führer, how he acknowledges this and how he is prepared also to stand by the side of his friend, the Duce, if ever this should prove necessary. 24 On Hasse’s activities to support a German-Italian partnership in mathematics, see [Remmert 2017].

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It might be remarked that outside of Czechoslovakia very few voiced criticism of the Munich accords, though Winston Churchill was of course adamantly opposed to appeasing Hitler. Hasse made no secret of his political views, and at times he seems to have been quite oblivious to the impression his opinions left on colleagues abroad. Probably his most glaring faux pas came in a letter from 15 March 1939 to the American mathematician Marshall Stone, in which Hasse remarked: “Looking at the situation from a practical point of view, one must admit that there is a state of war between the Germans and the Jews.” 26 Two years earlier, Hasse hoped to become a member of the Nazi Party, albeit somewhat belatedly, but his application was turned down because he was not of “purely Aryan” ancestry. Questioned by British authorities after the war as part of the denazification proceedings, he was classified as an “ardent Nazi” who should not be allowed to teach.27 In an ironic twist on the events of 1933, when he gathered letters of support for Emmy Noether (see Section 7.3), Hasse was now the recipient of various testimonials. Most were written by German colleagues, but a few foreign mathematicians also vouched for him, contending that he had never curried favor with the Nazi regime or exploited his position at the expense of others [Reich 2018, 49–54]. These statements went beyond the standard “Persilscheine” (white-washing certificates),28 but the fact remained that Hasse was a Nazi party member in-waiting and an outspoken nationalist who refused to acknowledge that hideous crimes had been committed in the name of the German people.29

9.4 Noether and “Hebraic Algebra” Shifts in mathematical style can be subtle in nature and difficult to comprehend or to describe. For many of her contemporaries what stood out about Emmy Noether was not only her ability to think in abstract categories but also her strong preference for framing mathematical arguments in the most general possible terms. In this respect, her work differed from that of other leading German algebraists – 25 [Schappacher 2007, 277]; this message of political solidarity was coupled with an offer to join hands with Severi to form a mathematical partnership between their countries. Hasse’s initiative may have paved the way for the Rome congress from 8–15 November, 1942, at which Hasse headed the German delegation [Segal 2002, 370–371]. 26 For an analysis of the context surrounding this letter, see [Siegmund-Schultze 1993, 164–166]. 27 Academics not accused of committing war crimes could continue to do research, whatever their political views. Those deemed “ardent Nazis” were to be removed from teaching positions on the grounds that their political views might exert a deleterious influence on students. This applied automatically to members of the Nazi party, although this restriction was later lifted. 28 Persil is the brand name for a well-known laundry detergent. 29 On Hasse’s activities within the German Mathematical Society during the NS-era, see [Remmert 2012a].

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Figure 9.4: Helmut Hasse and Emil Artin, early 1930s (Courtesy of MFO, Oberwolfach Research Institute for Mathematics)

Schur, Hasse, Artin, and Brauer – and this at least partly explains why her name became synonymous with the modern structuralist approach that later came into vogue with the Bourbaki movement. Reinhard Siegmund-Schultze has described furthermore how the general trend toward abstraction helped create the myth of a homogeneous “German approach” to algebra symbolized by Noether’s work and influence [Siegmund-Schultze 2009, 289–290].30 Yet if Noether’s name was identified by some with “German algebra,” for others her penchant for abstractions was typical of a “Hebraic style.” In assessing academic antisemitism among mathematicians in Germany during the 1930s, Sanford Segal rightly noted that at that time the situation in the United States was little better [Segal 2002, 372]. Moreover, only a relatively small group of these Germans were outright Nazi ideologues, though a great many were opportunists and just as many shared at least some of the standard prejudices against Jews, even someone as innocuous as Emmy Noether. She had long followed and presumably appreciated the work of the algebraist Heinrich Brandt, who twice invited her to speak in Halle, where he succeeded Hasse in 1930 (see Section 6.1). On at least one of these occasions, Noether was a house guest during her visit, and her letters to him from this time suggest that they were on quite 30 Roquette also noted a gradual shift in Albert’s style, which was originally rooted in the Dickson tradition, but became more modern under the influence of Noether and van der Waerden [Roquette 2004, 79].

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friendly terms [Jentsch 1986, 7].31 Based on these circumstances, one would hardly suspect Brandt of being a closet antisemite, but some of his other correspondence suggests otherwise. Only a few months before leaving Aachen for Halle, Brandt wrote two letters to his former colleague Erich Trefftz, informing him about recent events in Aachen. Both letters are filled with nasty barbs directed at his Jewish colleagues at the Institute of Technology in Aachen, especially Otto Blumenthal, 32 who was then a candidate to become Rektor. As for mathematicians who might succeed him, Brandt noted that the name Friedrich Wilhelm Levi had come up, someone he considered very deserving, but who had no realistic chances. “It’s really an unfortunate thing,” he wrote Trefftz, “I would very much wish that . . . he could come further. Why must he be named Levi? Why don’t they name their children either after the mother or father, as they do in Russia. Then unpopular and ugly names would quickly disappear.” 33 Brandt predicted that the election for Aachen’s next Rektor would be close, since Blumenthal had many supporters but there were also several faculty members who opposed his candidacy. Whatever the outcome, though, Brandt also predicted that Blumenthal would be unhappy with the result, “three days if he is not elected and at least three years if he is.” Remarkably, what seemed to make Otto Blumenthal so unsuitable in Heinrich Brandt’s eyes was the way he dressed: If only he would go around a bit more decently at least during this time! But he no sense for that at all. His ancestors must also have lived in the ghetto. There is no other way to explain it. Who would walk around like that with rough peasant shoes made of sole leather and sewn together several times, a worn-out suit, worn-thin shirt with frayed sleeves and a dirty collar; this is his typical attire in rainy weather, maybe a shade better when the sun shines. [Rowe/Felsch 2019, 355] In Brandt’s second letter, written four days later in the same haughty, mocking tone, he describes how Blumenthal lost the election. These letters seem particularly distasteful in view of the fact that Trefftz was a close friend of Blumenthal and his family. That Brandt could write in such a way, apparently presuming not to give offense, strongly suggests that these types of antisemitic expressions were so commonplace that no one found them at all remarkable, though of course they would not have been uttered in public, at least not in good society. 31 In 1985 Brandt’s widow, Eva-Maria, remembered Emmy Noether’s friendly manner and willingness to talk about the problems of running a household (the Brandts had seven children). 32 Blumenthal had converted to Protestant Christianity in his youth, but as noted in Section 1.2, for academic appointments ethnic background rather than religious affiliation was decisive in most instances. 33 Brandt to Trefftz, 12 May 1930, translated from [Rowe/Felsch 2019, 354]. Friedrich Wilhelm Levi was a Privatdozent in Leipzig, but lost the right to teach after the Nazis came to power. In 1935 he accepted an offer to head the Mathematics Department at the University of Calcutta, though he returned to Germany and taught in Berlin and Freiburg after the war; see [Kegel/Remmert 2003] and [Remmert 2015].

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Hasse had reworked Brandt’s theory of maximal orders for algebras in his study [Hasse 1931b], and Emmy Noether followed his lead, while emphasizing at the outset of [Noether 1933b] how Brandt’s theory could be simplified by making use of automorphisms. In all likelihood, she had already pointed this out in her lecture in Halle on 7 May, 1932, and as noted by Werner Jentsch, her work on the principal genus theorem bypassed Brandt’s theory entirely. Brandt was almost surely unhappy about this, and Jentsch speculated that these events mark the beginning of his emotionally charged aversion to abstract algebra in the style of Emmy Noether [Jentsch 1986, 11: fn 19]. Yet, all the same, Brandt looked to Noether’s school to fill an assistant’s position in Halle, and on September 10, 1933 she wrote him a letter characterizing the qualifications of two of her students, Hans Fitting and Ludwig Schwarz.34 Not long afterward, Brandt chose Ludwig Schwarz as his assistant; he remained in that position from 1933–1938. One week after Emmy Noether wrote to Heinrich Brandt, he sent a letter to Hermann Weyl, who only received this communication much later, since he was on his way to Princeton. Weyl answered on 15 December: As little as “abstract” algebra appeals to me personally, I evidently value its achievements and their significance much more highly than you do. What particularly impresses me about Emmy Noether is that her problems have steadily become more concrete and deeper. Why shouldn’t the Hebrew woman be entitled to that which in the hands of the “Aryan” Dedekind led to great results? I leave it to [Oswald] Spengler and [Ludwig] Bieberbach35 to categorize mathematical ways of thinking by peoples and races. I agree with you that Göttingen has lost its claim as a mathematical center – what even remains of Göttingen at all? I hope and wish that some new men may be able to achieve a worthy continuation of its old tradition; but I am glad I no longer have to support that tradition against the stream of nonsense and fanaticism! (Weyl to Brandt, 15 December 1933, translated from [Jentsch 1986, 9]) Although Brandt’s letter apparently no longer exists, Weyl’s answer leaves little to the imagination; nor did Brandt ever change his mind. On the contrary, his rhetoric in 1953, the year before he died, was both loud and clear. In a letter to his former student Martin Eichler, he condemned “this infernal, so-called abstract method” that mires itself in concepts: “I cannot accept this as mathematics at 34 In her opinion, they were of fairly equal mathematical talent, but she regarded Schwarz as the more flexible of the two and thus probably more useful as an assistant. The latter, she noted further, hoped to apply his current methods to make further progress on Max Dehn’s word problem in combinatorial group theory [Jentsch 1986, 8–9]. 35 This coupling of Spengler and Bieberbach is particularly noteworthy in view of the the date of this letter, which was written four months before Bieberbach delivered a notorious lecture in praise of the boycott in Göttingen that ended Edmund Landau’s teaching career. Evidently Bieberbach’s views regarding race and mathematical creativity were known well before the stormy events of 1934 [Mehrtens 1987].

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all; for me it is the worst decadence that has befallen mathematics in the thousands of years of its history” (Brandt to Eichler, 7 April 1953, translated from [Jentsch 1986, 11]).

9.5 “German Algebra” in the United States The internationalization of modern algebra clearly gained tremendous momentum from Moderne Algebra [van der Waerden 1930/31], a work that synthesized mathematics that was mostly made in Germany.36 Given the political atmosphere during the ensuing era, it seems hardly surprising that some in the United States were less enthused.37 One of these was Harvard’s Garrett Birkhoff, son of that university’s most esteemed mathematician, George David Birkhoff. There was no love lost between the elder Birkhoff and Princeton’s Solomon Lefschetz, an outspoken Russian Jew, and the old tensions were still very much in the air in 1946 when Princeton hosted its Bicentennial Conference [Rowe 2018a, 434–443]. In the algebra session sparks flew after Garrett Birkhoff attempted to define the boundaries of the discipline, prompting Artin’s response that he was overlooking valuation theory. The most substantive presentation was Richard Brauer’s new approach to class field theory, which also seemed not to have impressed Artin, who had just joined the Princeton faculty. Nothing very concrete emerged from this conference, certainly no clear consensus about where algebraic research stood or where it was going, perhaps not even what it meant to do algebra in the year 1946. On the other hand, the group photo is highly revealing, as it shows 93 mathematicians from America and Europe, many of them distinguished figures in the field and all very smartly dressed in suits – and not a single woman among them [Rowe 2018a, 437]. Artin’s legendary career at Princeton played a pivotal role in bringing highbrow algebra to the United States at the very time when the Bourbaki movement was beginning to make its impact felt. One of Artin’s most influential works dates from the early 1940s, when he lectured on Galois theory as a member of the faculty at Notre Dame. These lectures were published by the University of Notre Dame Press and appeared in a second revised edition in 1944; the book thereafter circulated widely and was eventually reprinted by Dover in 1998 [Artin 1944/1998]. This short monograph was surely not intended for beginners, but its abstract style, which also involved representation theory, did much to promote research in modern algebra. Graduate level textbooks afterward adopted Artin’s approach 36 On

its reception during the Nazi era, see [Siegmund-Schultze 2011b, 217–225]. Weyl had surely heard criticism of such propaganda for Noether and “German Algebra,” a topic he touched on in his memorial speech for Emmy Noether. There he commented: “Her methods need not, however, be considered the only means of salvation. In addition to Artin and Hasse, who in some respects are akin to her, there are other algebraists of a still more different stamp, such as I. Schur in Germany, Dickson and Wedderburn in America, whose achievements are certainly not behind hers in depth and significance. Perhaps her followers, in pardonable enthusiasm, have not always recognized this fact” [3: 442]Weyl-6. 37 Hermann

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to the subject, starting with the notion of arbitrary groups of automorphisms of arbitrary fields and building to a structure theory that displays the parallelism between the subgroups of a Galois group and the corresponding field extensions. Classical Galois theory was rooted in questions concerning when an algebraic equation will be solvable,38 an issue that arose after N.H. Abel showed that the general fifth-degree equation cannot be solved by means of expressions using only radicals. This means one can never find a formula (however complicated) in the coefficients of the equation that involves taking roots of various expressions (as, for example, the quadratic formula, which involves a square root). Some fifthdegree equations can be solved this way, but not all, and what Évariste Galois invented was a method for deciding such questions. His main result came to be summarized by saying that any given algebraic equation is solvable exactly when its Galois group is a solvable group. This approach to Galois theory was based on the older idea of studying equations by means of permutations of their roots, which leads to so-called substitution groups. In the modern approach, this classical background simply appears as an application of the theory, which no longer has anything to do with solving equations. In its new form, Galois theory deals with the interplay between abstract groups and fields or similar analogies between algebraic structures. Yet if modern Galois theory came to be associated with Artin’s name in the United States, this surely had much to do with his near cult-like status in Princeton. The conceptual transformation in algebra clearly predates Artin’s arrival in the US and it was part of a far broader development, as Emmy Noether’s letter to Hasse from 7 October 1929 reminds us: “[Alexandrov] is also teaching Galois theory this winter, of course modern.” Artin was hardly a provocateur, like Lefschetz, but his personal magnetism was bound to cause resentment in rival circles. Birkhoff’s feelings about the “Noether tradition” or “Germanic algebra” surfaced in the 1970s when he planned to publish a retrospective account of the sources behind [van der Waerden 1930/31], by then a highly touted textbook. 39 In a letter, Birkhoff informed its author about this plan, including his criticisms of Noether’s light treatment of British and American algebraists, especially Wedderburn. Göttingen mathematicians, starting with Hilbert himself, had long been accused of “nostrifying” the work of others, but never Emmy Noether, who was famously scrupulous when it came to acknowledging the work of others. Birkhoff seems to have arrived at his singular opinion on the basis of one text alone, Noether’s lecture at the 1932 ICM in Zurich, about which he wrote: “Reading it today, one sees clearly, how completely German algebraists had taken over: no reference to Wedderburn, passing references to Dickson and Chevalley, and the rest is Germanic” (quoted from [Siegmund-Schultze 2009, 316]). 38 Jeremy Gray presents the transition from classical to modern Galois theory in [Gray 2018] with due attention to other readings of this history. 39 The discussion that follows is based on [Siegmund-Schultze 2009, 315–318].

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Better than nearly anyone, van der Waerden knew that Noether never engaged in nostrification, though his reply to Birkhoff’s earlier letter stuck to specifics. “As far as Wedderburn is concerned,” he wrote, this impression is definitely not true. I heard two courses of lectures by her, both on Group Theory and Algebras. The contents of the second course was published in Math. Zeitschrift [Noether 1929a] . . . ; it included the Theory of Representations. In the first course (Winter 1924/25) the culmination point was the structure theorem of Wedderburn . . . . Wedderburn was for her “la crème de la crème.” As for Dickson I cannot be so positive. . . . Wedderburn was the man who created the structure theory, and Molien and Peirce were his main precursors, not Dickson.40 As Siegmund-Schultze noted, this sniping between leading representatives of German and American algebra went on for some time. Gian-Carlo Rota bemoaned Artin’s strong influence in the United States, arguing that American textbooks gave Galois theory far too much weight. As for the dispute between Birkhoff and van der Waerden, this led to an important positive outcome as the latter, prodded by his antagonist, decided to give a detailed account of the sources he had drawn on in writing Moderne Algebra in [van der Waerden 1975].41

9.6 On Noether’s Influence and Legacies Emmy Noether’s long-term impact on modern mathematics manifested itself in various ways and in many different places. Her abstract approach to algebra merged well with the already strong research tradition that had long been in place at the University of Chicago, the institution that spearheaded modern mathematics in the United States [Parshall/Rowe 1994]. A somewhat similar situation emerged in Japan, where the leading research centers in Tokyo, Sendai, and Kyoto were strongly influenced by German models. A key figure behind these developments was Rikitaro Fujisawa, who took his doctorate in Strassburg in 1888 under E.B. Christoffel and then taught for many years in Tokyo. Politically well connected, he became a major institution builder in Japan, placing two of his students – Tsuruichi Hayashi and Matsusaburo Fujiwara – at the newly founded Tohoku University in Sendai [Sasaki 2002, 243]. A leading figure in Kyoto was the algebraist Masazo Sono, whose work on abstract ring theory preceded that of Emmy Noether. As Harald Kümmerle has shown, Emmy Noether later became aware of his work in ideal theory and she cited some of his results on isomorphism theorems 40 Van

der Waerden to Birkhoff, 7 November 1973, quoted from [Siegmund-Schultze 2009, 316]. had a few years earlier in [van der Waerden 1971] written a similar firsthand account of how his own work related to developments in modern algebraic geometry, and in [van der Waerden 1985] he traced the broader history of algebra culminating with Noether’s work. 41 He

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in [Noether 1927a], her seminal paper on the subject. However, in the opinion of Sono’s student and later colleague, the algebraist Yasuo Akizuki, Noether did not fully appreciate the importance of Sono’s works [Kümmerle 2019, 228]. 42 Insofar as Noether’s legacy in ideal theory is concerned, her most important immediate heir was Wolfgang Krull, author of the standard survey [Krull 1935]. 43 Although he was not a Noether student in the narrower sense, his research interests fell entirely within the scope of those she had surveyed during her second period, from 1920 to 1926. Over the course of his long career in Erlangen and Bonn, Krull supervised 42 doctoral dissertations, some written by mathematicians who had numerous students themselves. As for Noether’s third period, one notes a remarkable divergence of interests in the subsequent research and teaching activity of her former collaborators, Helmut Hasse (Fig. 9.4) and Richard Brauer. In Hasse’s case, we see him making a stunning volte-face in [Hasse 1952], turning away from the general approaches of Dedekind and Hilbert to return to concrete number theory in the tradition of Gauss and Kummer. His fascination with Emmy Noether’s ideas had apparently dissipated completely, and his views now sounded almost like he had found his way into Heinrich Brandt’s camp. Writing in August 1945, Hasse presented his latest contributions to class field theory as vindicating the true interests of number theorists, whose “aims and methods are based on the reality of the natural numbers . . . and which do not need to search for new nourishment in the terrain of abstract algebra or from topology, set theory, and axiomatics” [Hasse 1952, ix]. 44 Comparing this preface with Hasse’s paean of praise for abstract algebra, delivered at the Prague conference in 1929 (Preface and 5.6), it becomes apparent that a decade after Emmy Noether’s death he had distanced himself entirely from her legacy. In Brauer’s case, as we have seen, his personal friendship with Noether had deepened during the final year of her life. She also apparently played a major part in promoting his career by recommending him for the professorship in Toronto that he would assume in the fall of 1935.45 During the thirteen highly productive 42 Research interests in Kyoto gradually shifted from algebra to algebraic geometry, and two of Akizuki’s students went on to become Fields medalists for their work in this field: Heisuke Hironka and Shigefumi Mori. 43 For a discussion of this work in context, see [Koreuber 2015, 258–270]. 44 This preface from 1945 apparently only appeared in print seven years later, by which time Hasse was in Hamburg. Beyond its anti-modern orientation, Hasse also criticized Hilbert’s Zahlbericht for obliterating Kummer’s work on concrete cases, but also American studies that aimed to produce a “complete list of cases” without paying sufficient attention to general theory. He made this appeal by claiming that the return to classicism in number theory was in the same spirit as what had taken place in music, which had witnessed a return to the original source of pure and simple musicality following the wild fantasies of the romantic and post-romantic epoch [Hasse 1952, v–ix]. 45 Only a week before her death, Noether informed Hasse that Brauer was almost sure to be appointed in Toronto [Lemmermeyer/Roquette 2006, 220]. Some sources claim that she recommended him during a visit she made to Toronto, but documentary evidence for such a trip seems to be lacking.

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years he spent there, Brauer extended Frobenius’ theory of group characters and obtained major new results in representation theory. In 1949 he received the Cole Prize in algebra from the American Mathematical Society for his paper “On Artin’s L-series with general group characters,” and in 1952 he joined the faculty at Harvard University, where his research focused on classifying finite simple groups. His work in this field paved the way for the breakthrough achieved by his former student, Walter Feit, and John Thompson in the early 1960s, when they proved that every finite non-cyclic simple group has even order. By this time Brauer had moved quite a long way from the realm of Noether’s algebraic interests. Appreciation for their earlier work, however, was kept alive by Nathan Jacobson (see [Jacobson 1983]). Noether’s pioneering work in Galois theory was also recalled by the Chicago algebraist Richard Swan in [Brewer/Smith 1981, 115–124]. She was among the first to tackle a problem first posed by Dedekind, namely to decide for a given number field F and group G whether one can find a Galois extension K of F with Galois group isomorphic to the given group G. Noether already announced preliminary results on this problem in her Vienna lecture [Noether 1913], but she only published these in [Noether 1918c]. Albrecht Fröhlich, writing about the importance of Noether’s work for algebraic number theory, underscored her insights regarding Galois module structure: In this Noether was well ahead of her time. She seems to have been almost alone in realizing the significance and general interest of the Galois module structure of rings of algebraic integers or of their local counterparts, a topic which others, both at the time and for many years to come, apparently considered devoid of any deeper content. Noether was one of the first . . . to see the connection between module properties and ramification, which is now the subject of an extensive theory, and she was the only one to guess the interpretation of various arithmetic character invariants – in her case of the Artin conductors – in terms of Galois module structure. It is this tentative anticipation of developments to come 40 years later, rather than just the theorem46 given by her at the time, which secures her an important place in the history of the subject [Brewer/Smith 1981, 157–163]. Commenting on Fröhlich’s assessment of Noether’s contributions to algebraic number theory, Olga Taussky wrote: “It seems certain she could have done more there. It is futile to wonder what it might have been” [Taussky 1981, 85]. Another line of influence ran through the career of one of Noether’s favorite students, Jacob Levitzki (see Section 5.7), who returned to Palestine in 1931 as a member of the recently established faculty at Hebrew University in Jerusalem [Katz 2004]. Over the course of his career there, Levitzki supervised only one doctoral dissertation, a work written by Shimshon Amitsur in 1950. The latter, 46 This refers to Noether’s principal genus theorem, announced in [Noether 1932b] and proved in [Noether 1933b]. Fröhlich’s paper briefly discusses the background to this result as well as the theory set forth in [Noether 1932a].

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however, had 13 doctoral students and his work launched a noteworthy school for algebraic research in Israel. Moreover, Amitsur has been credited with reviving the relatively moribund field of division algebras in 1972, when he proved an important new result that harkens back to Noether’s work on crossed products (6.7). According to the Brauer-Hasse-Noether theorem, every central simple algebra A over a number field k is cyclic, i.e. there exists a maximal subfield L of A for which the Galois group for L/k is cyclic. A natural, but long open question was whether a maximal Galois extension could be found when k is an arbitrary field. In such cases, Noether’s crossed product construction applies and one calls such simple algebras crossed products (by Wedderburn’s theorem every simple algebra is uniquely given as a matrix ring over a division algebra). Amitsur was the first to prove the existence of a noncrossed product algebra, which thereby opened a whole new line of investigation for the theory of division algebras [Saltman 2001]. Saunders Mac Lane followed the course of modernization in mathematics for many years, including those he spent studying under Bernays and Noether in Göttingen from 1931 to 1933.47 Looking backward from on high, he sketched three periods in the history of abstract algebra, which he identified with its inception, development, and firm establishment. He describes these phases as follows: The first wave of abstraction, 1921–1941, was dominated by Emmy Noether, Emil Artin [Fig. 9.4], and van der Waerden’s book Moderne Algebra (1930–31) and was centered on the concept of ring and ideal. The second wave, 1942–1955, was led by N. Bourbaki under the slogan “What are the morphisms?”, and the third period, 1957–1974, was under the influence of Grothendieck, algebraic geometry, and category theory.48 The transition from van der Waerden’s Moderne Algebra to Bourbaki’s Éléments de mathématique [Bourbaki 1939–98], the multi-volume series that began to appear in 1939, was noted already by Jean Dieudonné in his widely read article [Dieudonné 1970], written at the height of Bourbaki’s influence. Grothendieck’s work represents a much more radical break, just as the fundamental conceptions of category theory essentially sidestep foundational theories by moving to a higher level of abstraction.49 Modernists from Hilbert to Bourbaki grounded mathematical structures in the soil of Cantorian set theory, which famously led to difficult foundational problems. These difficulties gradually receded into the background as structuralism progressed and abstract theories gained legitimacy. To what extent these trends have been fruitful or healthy for mathematics as a whole has been 47 When he first arrived, he took Noether’s course on representations of non-commutative rings, but found her lectures obscure and never took another course with her [Brewer/Smith 1981, 70]. Mac Lane and Samuel Eilenberg co-founded category theory, which has long served as a framework for abstract mathematical structures. 48 Quoted from [Koreuber 2015, 282]. 49 Several connections between Noether’s ideas, category theory, and more recent research are described in [Müller-Stach 2021].

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debated all along, and no one should expect experts ever to reach a consensus about what constitutes “good mathematics.” Rather than trying to draw a line from Noether to Grothendieck, it would seem more plausible to focus on her impact on algebraic topology, in particular homology groups. One usually traces the roots of algebraic topology to Henri Poincaré’s pioneering works from the 1890s, while conceding that with them Poincaré was, once again, well ahead of the times. Just prior to the First World War, Max Dehn had begun to study knot complements starting with their fundamental groups, and by the 1920s his topological investigations had spawned renewed interest in combinatorial group theory. Noether’s insight in the mid-1920s that one could also approach homological invariants via group theory was a parallel development. Around this time, it must have been noticed that by abelianizing the fundamental group one obtained the first homology group, but this was probably part of the folklore knowledge before the work of Dehn.50 In the mid-1930s, Witold Hurewicz introduced the higher homotopy groups and connected these via the Hurewicz theorem to their corresponding homology groups, thereby paving the way for a new field of research: homological algebra. Heinz Hopf developed these ideas further before they were taken up in [Eilenberg/MacLane 1945]. That same year, Samuel Eilenberg and Norman Steenrod set down their axioms for homology theories, thereby formalizing the subject. Much of this took place during the decade following Emmy Noether’s death. These developments led to a great deal of new research during the postwar era that closely reflected her strong interests in the algebraization of topology. She had followed the collaboration of Alexandrov and Hopf from the beginning, but unfortunately she did not live to see their Topologie [Alexandroff/Hopf 1935] in print as it only came out shortly after her death.51 Alexandrov did, however, send a first set of page proofs to Felix Hausdorff, and he received a good deal of critique in response. In his last preserved letter to Hausdorff, from 9 March 1935, Alexandrov described some of the alterations that he and Hopf had made in response to his criticisms.52 He also wrote to Hopf that he would send the next round of proofs to Andrey Andreyevich Markov Jr. rather than to Hausdorff, since the latter’s suggestions read like recommendations for a different book than the one they had written.53 In their preface, the authors remembered the tremendous encouragement they received from Noether dating back to the 1920s: 50 Erhard Scholz pointed out that Poincaré had already noted this connection in his paper “Analysis situs” from 1895, though without the explicit concept of the first homology group [Scholz 1980, 316]. 51 For a comparative analysis of this classic text and A Textbook of Topology by Seifert and Threlfall, see [Herreman 2005]. 52 Alexandrov to Hausdorff, 9 March 1935, [Hausdorff 2012, 131–133]. This letter alludes to Hausdorff’s lectures on algebraic topology from 1933 [Hausdorff 2008, 893–976], but in such a way that suggests he knew nothing at this time about their contents. 53 [Scholz 2008, 882]; Alexandrov nevertheless rewrote large sections of the text, thereby creating frictions with both Hopf and Ferdinand Springer; see [Scholz 2008, 879–883].

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Back then, the interest in topology in Göttingen was mainly concentrated in the lively mathematical circle around Emmy Noether. We think of her in gratitude today. Emmy Noether’s general mathematical insights were not limited to her special field of activity, algebra, but rather she exerted a lively influence on everyone who entered into mathematical relations with her. For us, this influence was of the greatest importance, and it is also reflected in this book. The strong tendency toward algebraization of topology on a group-theoretical basis, which we follow in our presentation, goes back entirely to Emmy Noether. This tendency seems self-evident today; it was not so eight years ago. It took the energy and temperament of Emmy Noether to make these questions and methods the common property of topologists and to enable them to play the role they play in topology today. [Alexandroff/Hopf 1935, IX] Hassler Whitney was still in his twenties when he met Alexandrov and Hopf at the 1935 Moscow topology conference. Their book with its 636 pages would thereafter become his bible for a long time. But Whitney also wrote that the conference proved topology was moving far too quickly for them to attempt to write the other two volumes, as they had originally planned [Whitney 1988, 97]. After 1945, things would really explode.

9.7 Courant, Alexandrov, and Grete Hermann Soon after the Artins arrived in Princeton in 1946, Natascha began spending a good deal of time in New York, where she became a key member of Courant’s group at NYU. In 1948 she became the technical editor of the new journal Communications on Pure and Applied Mathematics, and beginning in 1956 she worked as the primary translation editor for Theory of Probability and Its Applications, a position she held until 1989. Working for Courant was always an adventure, and she quickly became a key player within his expanding circle. In 1947, Richard Courant received funding from the Office of Naval Research to tour German institutions in order to assess his former country’s progress during wartime in the development of computing technology. The Rockefeller Foundation’s Warren Weaver also requested information about young German scientists who might benefit from a stay in the United States [Reid 1976, 259–260]. Arrangements for this trip were facilitated by a State Department official who would soon be making headlines; his name was Alger Hiss, a friend of Courant’s younger colleague, Donald Flanders. Courant was assisted on this trip by Natascha Artin. Together they visited facilities in the British and American zones, from Hamburg in the north to Munich in the south, meeting with and interviewing innumerable people along the way. Most of the time they were escorted around in a US Army jeep. From the moment they landed in Frankfurt on June 20, both were shocked by what they saw: rubble

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and devastation everywhere, surrounded by mingling crowds of half-starved people and beggars dressed in rags. Most of those they spoke with at the universities were either depressed or openly hostile. They were somewhat buoyed after visits with Kurt Reidemeister in Marburg and Ferdinand Springer in Heidelberg, but they soon realized that Weaver’s hopes for recruiting younger scientists from Germany were illusory. At the Darmstadt Institute of Technology, where Courant’s former assistant Alwin Walther headed a major computing center, practically all of the buildings had been destroyed during an air raid. After they reached Munich, Artin went on to Vienna and Courant set off for Göttingen; he would for many years afterward visit there nearly every summer. Under British occupation, the city played a key role in German science during the post-war years, in part due to the largely unlivable conditions in Berlin. Arriving on the evening of July 3, he found the buildings in the town barely damaged, but the atmosphere at the university was chaotic and his conversations with former colleagues were most depressing. In his journal, he recorded: “Absolutely bitter, negative, accusing, discouraged, aggressive” [Reid 1976, 261]. Initially, Courant had a different impression of Werner Heisenberg’s attitude, but then the latter, too, began to complain of mistreatment by the Allies. On returning, Courant and Artin composed a “Summary Report on Conditions of Science in Germany,” which painted a none-too-flattering picture based on many interviews, including with some twenty scientists in Göttingen. Courant, who was predisposed to the idea of offering financial support for rebuilding German scientific institutions, expressed at the same time extreme caution when it came to cooperating with disreputable parties who had no respect for Americans and democratic institutions. In their report, he and Natascha Artin wrote: Many German scientists who should have known better have made unnecessary concessions to Hitlerism; some are still in their hearts aggressive nationalists. Along with the majority of Germans, they do not really understand their position in the world. They have no clear conception of the misery inflicted by Nazi Germany on her victims. Self-centered, they indulge criticism of the Allies and are unwilling to see the present plight of Germany as a consequence of Hitlerism rather than as of Allied mistakes. . . . Of course German scientists differ greatly in their attitudes. There have been only relatively few genuine Nazis among them. A very great number, however, have willingly cooperated and gladly benefited. This majority of German opportunists is in the long run dangerous – ready to help build a new German nationalism which some day may become aggressive. We were assured by reliable sources that 90 percent of the academic faculties belong in this category. . . . Unfortunately, among the German scientists who shocked us with their attitude and hostility towards the Allies were not only men of medium caliber but also well known people of authority. Statements

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could be heard from them such as, “We disliked Hitler, but the British and Americans are worse. They have replaced Hitler with Stalin.” Time and again we were confronted with the accusation that the Allies are deliberately starving the German population. Often the food shipments from America were lightly dismissed as exaggerated by propaganda, or in another case, by the flippant remark from the mouth of one of the most renowned scientists that even the Nazis fed the inmates of concentration camps. [Reich 2018, 204–205] Not long afterward, Göttingen became the center for relaunching the Kaiser Wilhelm Society, headed by the chemist Otto Hahn. In 1948 it was re-founded as the Max Planck Society in honor of the famous physicist who died in Göttingen one year before. During these interim years, Helmut Hasse had few defenders among the faculty in Göttingen, whereas he enjoyed strong support from mathematicians in East Berlin, particularly from his former student Hermann Ludwig Schmid. This led to him becoming a founding member of the Research Institute for Mathematics of the Berlin Academy, which was established in October 1946. 54 Two years later, he was appointed to the last vacant professorship at Berlin University, which was now located in the Soviet sector [Reich 2018, 61–66]. 55 In 1951 Hasse was appointed to the professorship in Hamburg previously held by Max Deuring, who finally arrived back in Göttingen. Seven years later, Artin left Princeton to join Hasse in Hamburg.56 During the 1950s they had renewed their earlier correspondence, which grew into a warm friendship.57 Natascha and Emil Artin divorced in 1958, the year he left for Hamburg. European émigrés to the United States continued to view Helmut Hasse as persona non grata. Before he spoke in Boulder Colorado at a 1963 memorial conference held in honor of Emil Artin, several in the audience left their seats [Reich 2018, 113–114]. In 1948 Richard Courant turned 60, and his birthday was celebrated in the traditional mathematical manner. Two of his friends at NYU, K.O. Friedrichs and J.J. Stoker, joined with Otto Neugebauer (then at Brown University) in inviting other friends of Courant to contribute to a Festschrift in his honor. The volume that resulted, [Friedrichs/Neugebauer/Stoker 1948], is a publication that tells us a good deal about Richard Courant’s world at this time. Nearly all of the contributors were Europeans and a great many were, like Courant, émigrés 54 The Berlin Academy of Sciences was in the eastern sector of the city. On the attitude of Soviet authorities in connection with Hasse’s appointment, see [Siegmund-Schultze 1999, 68–71]. 55 Erhard Schmidt was the only other mathematics professor who remained on the faculty when the university reopened in early 1946. The more than ardent Nazis, Ludwig Bieberbach and his semi-competent ally, the applied mathematician Alfred Klose, were gone; the first dismissed, the second conscripted by the Soviet Union to do technical research. Their colleague Harald Geppert committed suicide when Berlin fell to the Red Army [Segal 2003]. 56 It was also in 1951 that Carl Ludwig Siegel, who had held a professorship at the IAS in Princeton for over a decade, returned to Göttingen. Among the many Jewish mathematicians who had been forced to leave Germany, only three chose to return: Friedrich Wilhelm Levi, Hans Hamburger, and Reinhold Baer, for details see [Remmert 2012b] and [Remmert 2015]. 57 Their relationship is documented in detail in [Reich 2018].

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to the US. Among them were Emil Artin, James Franck, Fritz John, Hans Lewy, Georg Pólya, Hans Rademacher, I.J. Schoenberg, Carl Ludwig Siegel, and Olga Taussky. Some of the other Europeans were Paul Bernays, Harald Bohr, Constantin Carathéodory, Jacques Hadamard, Heinz Hopf, Franz Rellich, and B.L. van der Waerden. Nearly all these names have been mentioned elsewhere in this volume, though two were conspicuously absent: Hermann Weyl and Helmut Hasse. Most of these studies and essays were written in English, though several were in German, and Hadamard wrote in French. Franck’s “Remarks about the Role of Pure Science in General Education” echo what many European emigrants found when they came to the United States, namely, a culture that barely distinguished between science and its technological applications. Reading this today, one is immediately struck by James Franck’s sense of idealism. Emil Artin, true to his muse, offered Courant a beautiful little addendum to his Notre Dame lectures on Galois theory. Van der Waerden’s essay, which was more personal than most of the others, was a kind of homage to Emmy Noether. From its title alone – “The Foundation of Algebraic Geometry. A very incomplete historical Survey” – one recognizes that this was a sketch for what he would later publish in [van der Waerden 1971]. In this early account, he looked back on his first encounter with “der Noether”: As I came to Göttingen in 1924, I brought with me a manuscript containing generalizations of M. Noether’s famous theorem concerning forms F = Af + Bφ. However, Emmy Noether taught me that Lasker and Macaulay had developed . . . much deeper generalizations and refined methods. She taught me that algebraic geometry ought to be based on Steinitz’ algebraic theory of fields, and on Dedekind’s arithmetical theory of algebraic functions and ideals. I saw at once that she was right, for I had learnt to see the importance of fields and ideals by my previous work. I was enthused by her foundation of the general theory of ideals . . . . Less satisfactory was her theory of zero points (Nullstellen) of polynomial ideals, based upon Hentzelt’s elimination theory. There was, however, an excellent idea at the bottom of it: the idea that every prime ideal possesses a generic zero point (allgemeine Nullstelle) . . . from which all other points of the manifold may be deduced by specialization. I worked out this idea and gave direct proofs of the main theorems, without using Hentzelt’s elimination theory. As I showed Emmy Noether the manuscript, it turned out that she had done quite the same thing in a lecture the year before. She did not claim, however, any priority right, but presented my manuscript, wholly inspired by her ideas, to the Math. Annalen (Vol. 96). [van der Waerden 1948, 437–438] A decade later, in 1958, Emil Artin returned to Hamburg as Helmut Hasse’s colleague. It was here more than thirty years earlier that they first discussed

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Teiji Takagi’s groundbreaking papers on class field theory. When they learned that Takagi had died in Tokyo in early 1960, Hasse sent his condolences to Zyoiti Suetuna along with these words: You know the high standing the deceased enjoyed in Germany, especially in our number-theoretic school, and you can be assured that his name will live on, not only in our generation but also in those to come. It was he who first gave a complete and conceptually simple account of class field theory and the higher reciprocity laws that was methodologically and systematically satisfactory.58 It was also in the year 1958 that Pavel Alexandrov made his last trip to Göttingen. He had maintained his longtime friendships with Heinz Hopf, Otto Neugebauer, and Richard Courant, whom he saw for the last time in Moscow in 1970. Their encounter in Göttingen brought back some pleasant memories: In 1958 once again as before I spent a whole summer semester in Göttingen: I was offered the very honorable Gauss professorship for that semester, and there I met those of my old Göttingen friends who were still alive. In the summer of 1958 Courant took me, as before, in his car to the River Weser, and we both swam across it. This was not easy. The river was completely swollen after several heavy rains, and its current was swift. Courant had reached the age of 70 a few months earlier and I must confess that during this swim I was rather worried about him (swimming was, as a matter of fact, not at all easy.) But all ended well. Courant then treated me to a very tasty meal at a country inn and we went back to Göttingen. [Alexandrov 1979/1980, 329] Three years after this, Anna Pell Wheeler sent Alexandrov a letter and enclosed some photos of Emmy Noether. She asked him whether he might have any letters from Emmy. He replied to her in German, remembering that she spoke the language, on 7 June 1961. . . . It was with great joy and great inner excitement that I read the lines you sent me. I remember very well the hours that I spent in your home, together with Heinz Hopf, in the winter of 1927/28, and the other times we came together in Princeton more than thirty years ago. The memories of Emmy Noether that you sent me in your letter are very precious to me; the photo of her from April 1935 is surely one of the very last, since she died on 14 April 1935. I thank you from the bottom of my heart for sending me these photographs. I once owned quite a few letters from Emmy Noether, and some of them were anything but short. Unfortunately, during the war years 1941–45 all my letters, except for a few exceptions, were lost. In particular, I still only possess a very few letters from Emmy Noether and it 58 Helmut

Hasse to Zyoiti Suetuna, March 8, 1960, translated from [Kümmerle 2019, 396].

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would be very painful for me to part with them. I already have the obituary of Emmy Noether by H. Weyl; he presented me with an offprint at that time. I thank you again very warmly for your letter; it brought back a swarm of precious memories!59 On 23 March 1982, the University of Erlangen celebrated the 100th anniversary of Emmy Noether’s birth. Three years before, B.L. van der Waerden spoke in Heidelberg about his years of apprenticeship in Göttingen [van der Waerden 1997], and he now offered a repeat performance in the city of Noether’s birth. Grete Hermann, Emmy Noether’s first doctoral student in Göttingen, received an invitation to this Erlangen colloquium at her home in Bremen. When she saw that van der Waerden would speak about his “Göttingen apprenticeship,” she sent him a letter about those nostalgically remembered times. “The title of your lecture,” she wrote, . . . awakens so many memories: I see lecture hall 16 on the second floor of the Göttingen auditorium in front of me; Emmy Noether stands at the blackboard, head bent, thinking. Sitting in front of her, intensely involved, is a small crowd of listeners, to which you and I also belong. I will not come to the colloquium – not only because my ears, which have now grown old, are no longer fully functional, but above all because I have lost contact with ideal theory in the decades that have passed since then. Even back then, when I became the assistant to my other Göttingen teacher, the philosopher Leonard Nelson, Emmy Noether said to me grudgingly after my exams: “So after studying mathematics for four years, she suddenly discovers her philosophical heart!” But this year, during which take place the hundredth birthdays of both my teachers, who were so important to me in my apprenticeship years, I think back with joy and gratitude on this mathematically so original yet so humanly warm woman, from whom I received much help not only in the field of mathematics but also with some rather annoying formal examination difficulties. On the blackboard in lecture hall 16, she wrote those many ideals, of which it was said in one of the obituaries dedicated to her after her death: “You wrote your name in the history of mathematics in many small Gothic letters.” 60

59 Alexandrov

to Wheeler, 7 June 1961, Bryn Mawr College Archives. 2007, 21]; Grete Hermann was here alluding to Hermann Weyl’s words, cited above: “Through your work algebra has acquired a new face. With many gothic letters, you have inscribed your name indelibly on its pages.” 60 [Roquette

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Name Index Abel, Niels Henrik (1802–1829), 5, 285 Ackermann-Teubner, Alfred (1857–1941), 184 Adickes, Franz (1846–1915), 59 Ahlfors, Lars (1907–1996), 162 Akizuki, Yasuo (1902–1984), 287 Albert, A. Adrian (1905–1972), 92, 141, 156, 164, 166, 217, 224, 225, 241, 248, 281 Alexander II, Czar (1818–1881), 218 Alexander, James Waddell (1888–1971), 129, 245 Alexandrov, Pavel (1896–1982), xiii, xvii, xv, 61, 91, 93, 100, 101, 106– 118, 121–125, 129–133, 145– 147, 149, 150, 159, 181, 186, 187, 189, 190, 207, 222–226, 230, 233, 234, 267, 268, 285, 290, 291, 295, 296 Althoff, Friedrich (1839–1908), 57, 59, 60 Amitsur, Shimshon (1921–1994), 288 Anspach, Brooke M. (1876–1951), 251 Archibald, Ralph G., 158, 159 Aronhold, Siegfried (1819–1884), 24, 25 Artin, Emil (1898–1962), x , xvii, 92, 125, 129, 135, 136, 139, 141, 143, 146, 151, 152, 155, 157, 163, 166, 168, 169, 175, 179, 180, 184, 185, 197, 200, 205, 207, 235, 281, 284–286, 289, 291, 293, 294 Artin, Michael (1934–), 235

Aydelotte, Frank (1880–1956), 248 Baade, Walter (1893–1960), 73 Baer, Marianne Kirstein (1907–1986), 146 Baer, Reinhold (1902–1979), 139, 143, 144, 146 Baer, Richard (1892–1940), 73 Baerwald, Eva, 234, 240 Baerwald, Hans (1904–?), 234, 240 Ballantine, Constance R. (1897–1974), 215 Bateman, Henry (1882–1946), 75, 76 Beethoven, Ludwig van (1770–1827), 124 Bell, Eric Temple (1883–1960), 215, 247 Bernays, Paul (1888–1977), 58, 94, 162, 169, 173, 176, 196, 200, 294 Bernoulli, Johann (1667–1748), 66 Bernstein, Felix (1878–1956), 194 Berthold, Gottfried (1854–1937), 52 Bessel, Friedrich Wilhelm (1784–1846), 10 Bessel-Hagen, Erich (1898–1946), 76, 86, 139 Bethmann Hollweg, Theobald von (1856– 1921), 59–61 Bieberbach, Ludwig (1886–1982), 20, 110, 121, 130, 261, 293 Bieringer, Walter H. (1899–1900), 271 Billikopf, Jacob (1882–1950), 248, 249, 269, 271

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. E. Rowe, Emmy Noether – Mathematician Extraordinaire, https://doi.org/10.1007/978-3-030-63810-8

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330 Birkhoff, Garrett (1911–1996), 284– 286 Birkhoff, George David (1884–1944), 249, 284 Bjerknes, Vilhelm (1862–1951), 151, 161 Blaschke, Wilhelm (1885–1962), 197, 212 Blichfeldt, Hans Frederik (1873–1945), 222 Blumenthal, Otto (1876–1944), 18, 58, 109, 155, 176, 195, 261, 262, 282 Bohr, Harald (1887–1951), 206, 294 Bolyai, János (1802–1860, 106 Borel, Émile (1871–1956), 107 Born, Max (1882–1970), 11, 64, 193, 196, 198 Bosworth, Anne Lucy (1868–1907), 42, 215 Brandi, Karl (1868–1946), 47 Brandt, Heinrich (1886–1954), 147, 156, 158, 159, 281, 283, 284, 287 Brauer, Ilse Karger (1901–1980), 243– 245, 260 Brauer, Richard (1901–1977), xvii, xviii, 91, 127, 141–144, 156–159, 163, 165, 166, 169, 183, 191, 220, 243, 244, 251, 252, 258, 260, 264, 281, 284, 287, 288 Braun, Otto (1872–1955), 231 Brill, Alexander (1842–1935), 5, 7, 8, 12, 15, 36 Brioschi, Francesco (1824–1897), 38 Brouwer, L.E.J. (1881–1966), 20, 29, 32, 94, 98, 100, 101, 106, 108–118, 123, 124, 130, 131, 207 Brown, Ernest William (1866–1938), 215 Brunschvicg, Léon (1869–1944), 261

Name Index Brunswick, Natascha Artin (1909–2003), 129, 146, 200, 235, 247, 291– 293 Burckhardt, Johann Jakob (1903–2006), 206 Burgers, Johannes Martinus (1895– 1981), 228 Busch, Wilhelm (1832–1908), 246 Cantor, Georg (1845–1918), 94, 124, 134, 136, 261, 289 Capelli, Alfredo (1855–1910), 33 Carathéodory, Constantin (1873–1950), xvii, 48, 50–52, 54, 55, 59, 162, 197, 229, 294 Cartan, Élie (1869–1951), 162 Castelnuovo, Guido (1865–1952), 30, 38 Catherine the Great, Empress (1729– 1796), 276 Cavaillès, Jean (1903–1944), 261 Cayley, Arthur (1821–1895), 24, 25, 38, 92 Chamberlain, Neville (1869–1940), 279 Chebotaryev, Nikolai (1894–1947), 184 Chevalley, Claude (1909–1984), xvi, 159, 184, 235, 241, 246, 247, 285 Chow, Wei-Liang (1911–1995), 106, 200 Christoffel, Elwin Bruno (1829–1900), 67, 70, 74, 77, 286 Churchill, Winston (1874–1965), 280 Clebsch, Alfred (1833–1872), 4–6, 11, 24, 33, 35 Cohn-Vossen, Stefan (1902–1936), 133, 145, 187 Conant, James Bryant (1893–1978), 270 Coolidge, Julian Lowell (1873–1954), 227 Courant, Nina (1891–1991), 123, 162, 187, 189

Name Index Courant, Richard (1888–1972), viii, xvii, xvi, 11, 100, 101, 106–109, 114, 116, 123, 124, 133, 141, 150, 157, 159, 162, 171, 175– 177, 187, 189–192, 194–198, 211, 220, 229, 237, 242, 262, 276, 279, 291–295 Cremona, Luigi (1830–1903), 38 Curie, Marie (1867–1934), 249 Darbyshire, Helen (1881–1961), 219, 220 Davenport, Harold (1907–1969), 200, 212 de Vries, Hendrik (1867–1954), 98, 106 Debye, Peter (1884–1966), 48, 52, 58, 59 Dechamps, G., 200 Dedekind, Richard (1831–1916), xiv, xiv, xvi, 29, 36, 37, 92–96, 103, 104, 108, 126, 137, 140, 147, 171, 177, 178, 180–182, 185, 200, 204, 205, 209, 257, 261, 294 Dehn, Maria (1915–2013), 152 Dehn, Max (1878–1952), 101, 148, 149, 152, 159, 202, 232, 283, 290 Demuth, Fritz (1876–1965), 233 Derry, Douglas (1907–2001), 200 Deuring, Max (1907–1984), 130, 150, 156–159, 175, 183, 212, 225, 226, 239, 251, 260, 293 Dick, Auguste (1910–1993), xiv, 22 Dickson, Leonard Eugene (1874–1954), 92, 143, 156, 159, 160, 166, 183, 206, 215, 216, 222, 226, 241, 248, 281, 285, 286 Dieudonné, Jean (1906–1992), 289 Dirichlet, Peter Gustav Lejeune (1805– 1859), 93, 96, 151 Dobrynin, Anatoly (1919–2010), 277 Douglas, Jesse (1897–1965), 162 Dresden, Arnold (1882–1954), 248, 271

331 Du Bois-Reymond, Paul (1831-1889), 7, 8 Dubreil, Paul (1904–1994), xviii, 151– 153, 159–162 Dubreil-Jacotin, Marie-Louise (1905– 1972), 151, 152, 160–162 Egorov, Dmitri (1869–1931), 107 Ehlers, Ernst Heinrich (1835–1925), 52 Ehrenfest, Paul (1880–1933), 72 Eichler, Martin (1912–1992), 283 Eilenberg, Samuel (1913–1998), 289, 290 Einstein, Albert (1879–1955), xv, xviii, 11, 33, 50, 57, 61–65, 67, 70–72, 74–76, 78–80, 82, 84, 86, 88–90, 117, 191, 192, 227, 231, 242, 244, 249, 262–264, 266 Eisner, Kurt (1867–1919), 61 Engel, Friedrich (1861–1941), 76 Enriques, Federigo (1871–1946), 30, 38, 160, 229 Epstein, Paul (1871–1939), 232 Euler, Leonhard (1707–1783), 174 Falckenberg, Hans (1884–1941), 18 Falckenberg, Richard (1851–1920), 18 Feit, Walter (1930–2004), 288 Figner, Vera (1852–1942), 218 Fine, Henry B. (1858–1928), 242 Finsler, Paul (1894–1970), 73 Fischer, Ernst (1875–1954), 22, 23, 29, 34, 35, 52, 69, 70, 90, 105, 190 Fitch, Edward (1864–1946), 42 Fitting, Hans (1906–1938), 156–159, 162, 283 Flanders, Donald (1900–1958), 291 Fleischmann, Wilhelm (1837–1920), 52

332 Flexner, Abraham (1866–1959), 191, 198, 199, 237, 241, 247, 248, 250, 253, 263–265 Fokker, Adriaan (1887–1972), 82 Fourier, Joseph (1768–1830), 151 Fraenkel, Abraham (1891–1965), 124, 261 Franck, James (1882–1964), 192–196, 198, 231, 275, 294 Franz, Wolfgang (1905–1996), 169 Frederiks, Vsevolod (1885–1944), 73 Fricke, Leonora Flender (1873–?), 170 Fricke, Robert (1861–1930), xviii, 170, 171 Friedrichs, Kurt (1901–1982), 197, 293 Frobenius, Ferdinand Georg (1849– 1917), 91, 92, 125, 137, 180, 182, 200, 224 Fröhlich, Albrecht (1916–2001), 288 Fuchs, Lazarus (1833–1902), 10 Fueter, Rudolf (1880–1950), 178 Fujisawa, Rikitaro (1861–1933), 286 Fujiwara, Matsusaburo (1881–1946), 286 Fumi, Shigefumi (1951–), 287 Furtwängler, Philipp (1869–1940), xvii, 171, 173–175, 178, 180, 184, 186, 205, 206, 245 Furtwängler, Wilhelm (1886–1954), 174 Galois, Évariste (1811–1832), 285 Gates, Fanny (1872–1931), 215 Gauss, Carl Friedrich (1777–1855), ix, 10, 39, 72, 94, 95, 106, 171, 174, 181, 182, 295 Geiger, Moritz (1880–1923), 123 Gelfond, Alexander O. (1906–1968), 268 Gentry, Ruth (1862–1917), 213 Geppert, Harald (1902–1945), 293 Germain, Sophie (1776–1831), 184, 203 Goebbels, Joesph (1897–1945), 231 Gödel, Kurt (1906–1978), 94, 174

Name Index Gorbachev, Mikhail (b. 1931), 277 Gordan, Paul (1837–1912), 1, 4–8, 11– 15, 20–29, 31–33, 35, 36, 38, 50, 69 Gordan, Sophie Deurer (1848–1921), 5 Grassmann, Hermann (1809–1877), 26, 205 Grave, Dmitry (1863–1939), 89 Grell, Heinrich (1903–1974), 106, 144, 158, 182 Gröbner, Wolfgang (1899–1980), 186 Grossmann, Marcel (1878–1936), 65 Grothendieck, Alexander (1928–2014), 289, 290 Guccia, Giovanni (1855–1914), 38 Gundelfinger, Sigmund (1846–1910), 8 Haber, Fritz (1868–1934), 196 Hadamard, Jacques (1865–1963), 294 Haenisch, Konrad (1876–1925), 62 Hahn, Hans (1879–1934), 130, 131, 171, 222 Hahn, Otto (1879–1968), 197, 293 Halmos, Paul (1916–2006), 216 Hamburger, Hans (1889–1956), 293 Hamel, Georg (1877–1954), 34, 82 Hamilton, William Rowan (1805–1865), 126 Hardy, Godfrey Harold (1877–1954), 206 Hartmann, Johannes (1865–1936), 48, 52, 54–57 Hasse, Helmut (1898–1979), ix, xi, xiv–xvi, 90–92, 127, 134, 136, 137, 139–144, 146, 147, 151, 155–157, 159, 163–167, 169, 170, 172, 174, 175, 178–181, 184, 185, 190, 191, 195, 197– 203, 205–212, 215–217, 221– 223, 225, 226, 230, 232, 234, 235, 237–239, 241, 242, 245–

Name Index 247, 251, 258, 260, 261, 264, 279, 280, 285, 287, 293, 294 Hausdorff, Felix (1868–1942), 108, 109, 112, 118, 119, 123, 124, 149, 150, 290 Hayashi, Tsuruichi (1873–1935), 286 Hazlett, Olive Clio (1890–1974), 215– 218 Hecke, Erich (1887–1947), 90, 134, 141, 185, 197, 222, 225, 261 Hedlund, Gustav A. (1904–1993), 222 Heegner, Kurt (1893–1965), 95 Heilbronn, Hans (1908–1975), 95, 161, 162 Heinemann, Käthe (1889–?), 73 Heisenberg, Werner (1901–1976), 197, 276, 292 Heisig, Charlotte, 202–204, 240 Heisig, Herbert (1904–1989), 202–204, 240 Hellinger, Ernst (1883–1950), 58, 159, 176, 232, 233 Hensel, Fanny Mendelssohn Bartholdy (1805–1847), 134 Hensel, Kurt (1861–1941), xvii, 89, 90, 134–136, 143, 146, 166, 167 Hensel, Wilhelm (1794–1861), 134 Hentzelt, Kurt (1889–1914), 33, 104, 105, 208, 294 Herbrand, Jacques (1908–1931), 156, 158, 159, 162, 163, 261 Herglotz, Gustav (1881–1953), 76, 82, 169, 196, 197, 217, 226, 236, 245 Hermann, Grete (1901–1984), 33, 99, 104, 296 Hermite, Charles (1822–1901), 38 Hertz, Paul (1881–1940), 69, 196 Hesse, Otto (1811–1874), 6 Heun, Karl (1859–1929), 21 Hey, Käte (1904–1990), 246 Hilb, Emil (1882–1929), 20 Hilbert, David (1862–1943), xiii, xv– xvii, xix, 11, 12, 18, 23–36,

333 42, 43, 45–55, 57–59, 64, 67– 72, 76, 78, 79, 82–90, 93, 94, 98, 101, 104, 106–109, 114, 117, 121, 123–125, 127, 130, 133, 137–139, 145, 148, 150, 162, 171, 172, 174–180, 184, 185, 187, 197, 203, 206, 215–218, 225, 244, 245, 261, 264, 265, 285, 287, 289 Hilbert, Käthe (1864–1945), 46, 58 Hindenburg, Paul von (1847–1934), 61 Hironka, Heisuke (1931–), 287 Hiss, Alger (1904–1996), 291 Hitler, Adolf (1889–1945), 20, 189, 192, 197, 230, 231, 237, 238, 278–280, 293 Hölder, Otto (1859–1937), 190 Hoffmann, Adolph (1858–1930), 62 Hohenemser, Kurt (1906–2001), 196 Hopf, Heinz (1894–1971), xvii, 107, 118, 121, 122, 124, 129, 131, 132, 159, 267, 290, 291, 294, 295 Houtermans, Fritz (1903–1966), 275– 277 Huber, Anton (1897–1975), 174 Humboldt, Alexander von (1769–1859), 151 Humm, Rudolf Jakob (1895–1977), 72, 76, 77 Hurewicz, Witold (1904–1956), 290 Hurwitz, Adolf (1859-1919), 12, 28, 31, 177 Iyanaga, Shokichi (1906–2006), 175, 184 Jacobi, C.G.J. (1804–1851), 9, 10, 151 Jacobson, Nathan (1910–1999), xviii, 141, 144, 183, 245, 248, 288 John, Fritz (1910–1994), 233, 294 Jung, Heinrich Wilhelm (1876–1953), 143 Kapferer, Heinrich (1888–1984), 128

334 Karman, Theodor von (1881–1963), 229 Kaufmann, Frederike Scheuer, 2 Kaufmann, Markus (1813–1866), 2 Kaufmann, Paul, 2 Kaufmann, Wilhelm (1856–1928), 2 Kerékjártó, Béla (1898–1946), 133 Kerschensteiner, Georg (1854–1932), 15, 22, 69 Khinchin, Aleksandr (1894–1959), 107, 268 Kirchhoff, Gustav (1824–1887), 4 Klein, Felix (1849–1925), xv, 5–7, 9, 11–15, 18, 26–30, 35, 36, 38, 40–48, 50, 57, 59, 61, 62, 64, 69, 70, 72, 74–82, 85–88, 90, 98, 102, 114, 148, 162, 170, 171, 174, 176, 184, 190, 203, 213, 215, 217, 242, 262 Klie, Fritz, 107, 123 Klose, Alfred (1895–1953), 293 Knauf, Ernst, 200 Kneser, Hellmuth (1898–1973), 99– 101, 122, 190, 195–197 Koebe, Paul (1882–1945), 20, 124, 125 Koenig, Mathilde, 16 Koenigsberger, Leo (1837–1921), 9, 10 Köthe, Gottfried (1905–1989), 156, 169, 206 Kolmogorov, Andrey (1903–1987), 107, 145, 149, 150, 189–191 Kovalevskaya, Sofia (1850–1891), xii, 39, 40, 43, 50, 184, 203, 213, 249 Kraus, Hertha (1897–1968), 271 Kronecker, Leopold (1823–1891), xi, 32, 36, 92, 94, 134, 138, 200, 205 Krull, Wolfgang (1899–1971), 125, 144, 146, 151, 160, 176, 190, 204, 287 Kürschák, József (1864–1933), 135

Name Index Kummer, Ernst Eduard (1810–1893), 94, 95 Kurosh, A.G. (1908–1971), 132 Ladd-Franklin, Christine (1847–1930), 40 Lagrange, Joseph-Louis de (1736–1813), 64, 66, 70 Landau, Edmund (1877–1938), ix, x , 48–50, 52–55, 89, 90, 96, 107, 108, 124, 125, 133, 139, 146, 147, 150, 162, 169, 178, 187, 196, 206, 236–238, 244, 260, 261 Lasker, Bertold (1860–1928), 97 Lasker, Emanuel (1868–1941), xvi, 97, 103, 104 Lasker-Schüler, Else (1869–1945), 97 Laue, Max von (1879–1960), 65, 197 Lavrentyev, Nikolai (1900–1980), 107 Lefschetz, Solomon (1884–1972), 122, 129, 162, 219, 224, 225, 243, 249, 267, 284 Lehmann, Max (1845–1929), 47, 54 Lehr, Marguerite (1898–1987), 38, 213, 222, 252, 265 Levi, Friedrich Wilhelm (1888–1966), 282, 293 Levi-Civita, Tullio (1873–1941), 65, 80, 160, 229 Levitzki, Jakob (1904–1956), 126, 146, 147, 288 Lewy, Hans (1904–1988), 123, 162, 187, 294 Lie, Sophus (1842–1899), 5, 30, 38, 76, 82, 83, 190 Lindemann, Ferdinand (1852–1939), 11, 12, 20 Linden, Maria von (1869–1936), 46 Liouville, Joseph (1809–1882), 75 Lipschitz, Rudolf (1832–1903), 70, 74 Littauer, Lucius N. (1859–1944), 250 Löw, Joseph (1876–1964), 273

Name Index Löw, Marianne Weinberger (1892–1950), 273 Loewy, Alfred (1873–1935), 144, 190 Logsdon, Mayme (1881–1967), 160 Lorentz, Hendrik Antoon (1853–1928), 65, 66, 75, 78, 82 Ludendorff, Erich (1865–1937), 61 Lüroth, Jakob (1844–1910), 4, 12, 30 Luzin, Nikolai (1883–1950), 107, 124 Lyapunov, Alexey (1911–1973), 107 Lyusternik, Lazar (1899–1981), 107 Mac Lane, Saunders (1909–2005), 118, 169, 289 Macaulay, Francis (1862–1937), 104 MacDuffee, Cyrus Colton (1895–1961), 215, 222, 223 MacKinnon, Annie (1868–1940), 42 Maddison, Isabel (1869–1950), 40, 41, 213 Magnus, Wilhelm (1907–1990), 152, 171, 175, 180, 244 Mahler, Kurt (1903–1988), 162 Maltby, Margaret (1860–1944), 41 Manning, W.A., 222 Mannoury, Gerrit (1867–1956), 98 Markov Jr., Andrey Andreyevich (1903– 1979), 290 Martin, Emilie Norton (1869–1936), 215 Marxsen, Sophus (1877–?), 31 Maurer, Ludwig (1859–1927), 32 Maxwell, James Clerk (1831–1879), 65, 75 Mayer, Walter (1887–1948), 244 McKee, Ruth Stauffer (1910–1993), 221–223, 243–245, 250, 251, 260 Medvedev, Zhores A. (1925–2018), 277 Menger, Karl (1902–1985), 109, 130, 131 Mertens, Franz (1840–1927), 22, 26, 33, 34 Meyer, Franz (1856–1934), 34

335 Mie, Gustav (1868–1957), 64, 71, 197 Minkowski, Auguste (1875–1944), 46, 58 Minkowski, Hermann (1864–1909), 11, 18, 29, 46, 47, 50, 65, 162, 174, 206, 217 Minkowski, Lily (1898–1983), 58 Minkowski, Ruth (1902–1983), 58 Mises, Richard von (1883–1953), 232 Mitchell, Howard Hawks (1885–1943), 220, 241 Mittag-Leffler, Gösta (1846–1927), 40 Moore, Eliakim Hastings (1862–1932), 215, 218 Mordell, Louis (1888–1972), 151, 160 Morley, Frank (1860–1937), 215 Moufang, Ruth (1905–1977), 152, 232 Mügge, Otto (1858–1932), 52 Müller, Emil (1861–1927), 26 Müller, Georg Elias (1850–1934), 52 Murrow, Edward R. (1908–1965), 219 Mussolini, Benito (1883–1945), 279 Naumann, Otto (1852–1925), 59, 60 Nehrkorn, Harald (1910–2006), 247 Nelson, Leonard (1882–1927), 296 Nernst, Walther (1864–1941), 41 Neugebauer, Otto (1899–1990), 123, 129, 131, 162, 187, 190, 193, 196, 197, 199, 293, 295 Neumann, Carl (1832–1925), 5 Neumann, Franz (1798–1895), 9 Neumann, Friedrich (1889–1978), 193, 238 Neumann, John von (1903–1957), 129, 244 Newson, Henry Byron (1860–1910), 42 Nielsen, Jakob (1890–1959), 148, 149 Nietzsche, Friedrich (1844–1900), 256 Noether, Alfred (1883–1918), 3, 16, 17 Noether, Dorit Low (1922–2017), 273, 277

336 Noether, Emiliana Pasca (1917–2018), 274, 275 Noether, Fritz (1884–1941), x, xviii, 3, 16–18, 20, 21, 40, 44, 149, 202–204, 226, 230–234, 251, 253, 257, 267–270, 272, 276– 278 Noether, Gottfried (1915–1991), x , 21, 233, 267, 269, 270, 273, 275, 277, 278 Noether, Gustav Robert (1889–1928), 3, 17 Noether, Herman (1912–2007), 21, 233, 267, 269–271, 273, 275, 277, 278 Noether, Ida Kaufmann (1851–1915), 2, 3, 269 Noether, Max (1844–1921), x, xii, xiv, xvii, 1–4, 6–8, 11–16, 18, 20–23, 27, 33, 35–38, 50, 97, 98, 101–103, 152, 160, 210, 262, 266 Noether, Monica, 275 Noether, Otto (1882–1979), 253, 272, 276 Noether, Regina Maria Würth (1882– 1935), 21, 203, 204, 267 Novikov, Pyotr (1901–1975), 107 Oliver, James E. (1829–1895), 42 Ore, Øystein (1899-1968), xviii, 147, 171, 176, 222, 223 Osgood, William F. (1864–1943), 13– 15, 25 Osterrath, Ernst (1851–1931), 57, 58 Ostrowski, Alexander (1893–1986), 89– 91, 99, 109, 135, 170 Painlevé, Paul (1863–1933), 130 Papen, Franz von (1879–1969), 231 Parastaev, Andrei, 278 Park, Marion Edwards (1875–1960), 219–221, 251–253, 264, 269 Pasch, Moritz (1843–1930), 12

Name Index Pauli, Wolfgang (1900–1958), 86, 88 Peano, Giuseppe (1858–1932), 34, 94 Peirce, Benjamin (1809–1880), 126 Peirce, Charles Sanders (1839–1914), 40 Pell, Alexander (Sergei Petrovich Degaev) (1857–1921), 217, 218 Pell, Emma (?–1903), 217, 218 Perron, Oskar (1880–1975), 145, 208 Pick, Georg (1859–1942), 227, 228 Pierpont, James (1866–1938), xviii Planck, Max (1858–1947), 197, 293 Pohlenz, Max (1872–1962), 53 Poincaré, Henri (1854–1912), 38, 66, 75, 76, 85, 160, 290 Pólya, Georg (1887–1985), 294 Pontryagin, Lev (1908–1988), 100, 131, 132, 190, 245 Prandtl, Ludwig (1875–1953), 161, 197 Quinn, Grace Shover (1906–1998), 222, 223, 244, 250–252, 268 Rademacher, Hans (1892–1969), 212, 232, 233, 244, 294 Rademacher, Irma Wolpe (1902–1984), 244 Ragsdale, Virginia (1870–1945), 215 Reiche, Fritz (1883–1969), 276 Reid, Constance (1918–2010), 177, 179, 192 Reid, Constance (1918-=2010), 178 Reidemeister, Kurt (1893–1971), 292 Reiger, Rudolf (1877–1943), 20 Reitzenstein, Richard (1861–1931), 53 Rella, Anton (1888–1945), 205, 206 Rellich, Franz (1906–1955), 190–192, 294 Reye, Theodor (1838–1919), 12 Rhoads, James E. (1828–1895), 214 Ricci, Gregorio (1853–1925), 65, 67, 68, 70 Riemann, Bernhard (1826–1866), 4, 5, 10, 23, 36, 68, 70, 74, 151

Name Index Ritt, Joseph (1893–1951), 215 Rockefeller, John D., jun. (1874–1960), xviii Röhm, Ernst (1887–1934), 238 Rose, Wickliffe (1862–1931), xviii, 112, 115, 229 Rosenthal, Arthur (1887–1959), 148 Rota, Gian-Carlo (1932–1999), 286 Rubens, Heinrich (1865–1922), 71 Runge, Carl (1856–1927), 21, 47, 48, 52, 55, 73, 79, 80 Runge, Iris (1888–1966), 73 Rust, Bernhard (1883–1945), 197 Sabin, Florence R. (1871–1953), 269 Salmon, George (1819–1904), 38 Sanderson, Mildred (1889–1914), 216 Sapolsky, Ljubova, 43 Schermer, Siegmund (1886–1974), 193, 194 Scherrer, Paul (1890–1969), 73 Schilling, Otto (1911–1973), 173 Schlick, Moritz (1882–1936), 174 Schmeidler, Werner (1890–1969), 21, 96, 147–149, 202 Schmid, Hermann Ludwig (1908–1956), 293 Schmidt, Arnold (1902–1967), 176 Schmidt, Erhard (1876-1959), 14, 22, 121, 159, 293 Schmidt, Friedrich Karl (1901–1977), 144, 190, 223, 225, 226 Schmidt, Otto (1891–1956), 230 Schmidt-Ott, Friedrich (1860–1956), 60 Schnirelmann, Lev (1905–1938), 107 Schoenberg, Charlotte Landau (1907– 1949), 244 Schoenberg, Isaac Jacob (1903–1990), 244, 294 Schoenflies, Arthur (1853–1928), 59 Schoenflies, Emma (1868–1939), 58 Schoenflies, Hanna (1897–1985), 58

337 Schouten, Jan Arnoldus (1883–1971), 207, 208, 227, 228, 231 Schreier, Otto (1901–1929), 135, 136, 141 Schrödinger, Erwin (1887–1961), 197 Schur, Isaai (1875–1941), 91, 125, 127, 141, 167, 244, 261, 281 Schwartz, Philipp (1894–1977), 233 Schwarz, Hermann Amandus (1843– 1921), 33 Schwarz, Ludwig (1908–?), 200, 283 Schwarzschild, Karl (1873–1916), 18, 217 Schwerdtfeger, Hans (1902–1990), 173, 233 Scorza, Gaetano (1876–1939), 216 Scott, Charlotte Angas (1858–1931), 40, 213–215, 217, 218 Seelhorst, Conrad von (1853–1930), 52 Segre, Beniamino (1903–1977), 162, 205 Segre, Corrado (1863–1924), 37 Sethe, Kurt (1869–1934), 48 Severi, Francesco (1879–1961), 37, 38, 159, 160, 162, 190, 205, 279, 280 Shoda, Kenjiro (1902–1977), 125–128, 205 Siegel, Carl Ludwig (1896–1981), ix , 139, 150, 159, 200, 201, 294 Slater, John C. (1900–1976), 231, 232 Snyder, Virgil (1869–1950), 42 Sommerfeld, Arnold (1868–1951), 20, 21, 64, 65, 88, 197 Sono, Masazo (1886–1969), 286 Speiser, Andreas (1885–1970), 206, 207 Springer, Ferdinand (1881–1965), 58, 157, 171, 178, 292 Stähelin, Helene (1891–1970), 73 Stahl, Hermann von (1843–1908), 7 Staiger, Elisabeth Klein (1888–1968), 73, 215

338 Stalin, Josef (1878–1953), 278, 293 Steenrod, Norman (1910–1971), 290 Stein, Edith (1891–1942), 62 Steinitz, Ernst (1871–1928), 29, 30, 32–34, 92, 104, 125, 134, 148, 149, 185, 294 Stephanos, Kyparisos (1857-1911), 29 Stern, Moritz Abraham (1807–1894), 10 Sternberg, Wolfgang (1887–1953), 233 Stoker, James J. (1905–1992), 293 Stokes, George Gabriel (1819–1903), 21 Stone, Marshall (1903–1989), 280 Stroh, Georg Emil (1859-1919), 29 Struik, Anton (1897–1945), 230 Struik, Dirk Jan (1894–2000), 159, 221, 227–231, 233 Struik, Saly Ruth Ramler (1894–1993), 227–230 Study, Eduard (1862–1930), 26, 148 Sturm, Jacques Charles François (1803– 1855), 98 Sudeykin, Georgy (1850–1883), 218 Süss, Wilhelm (1895–1958), 144 Suetuna, Zyoiti (1898–1970), 125, 126, 295 Swan, Richard (1933–), 288 Sylvester, James Joseph (1814–1897), 24, 25, 31, 38, 40, 98 Szász, Otto (1884–1952), 232 Takagi, Teiji (1875–1960), 125, 137– 139, 179, 182, 184, 202, 204, 295 Tammann, Gustav (1861–1938), 47, 52 Tanner, J. Henry (1861–1940), 42 Taussky-Todd, Olga (1906–1995), xvi, xv, 91, 170–181, 184–187, 222, 223, 235, 243–247, 250–252, 294 Teichmüller, Oswald (1913–1943), 236, 237

Name Index Thomas, M. Cary (1857–1935), 214 Thompson, John (1932–), 288 Timoshenko, Stephen (1878–1972), 232 Todd, Jack (1911–2007), 252 Toeplitz, Otto (1881–1940), 148, 168, 226, 239 Tornier, Erhard (1894–1942), 226, 238, 239, 260, 261, 279 Townsend, Edgar J. (1864–1955), 216 Trefftz, Erich (1888–1937), 282 Trott zu Solz, August von (1855–1938), 59, 60 Trowbridge, Augustus (1870–1934), 117 Tsen, Chiungtze (1898–1940), 161, 200, 224, 225 Tyler, Harry W. (1863–1938), 13–15 Ulm, Helmut (1908–1975), 162, 171, 175, 180, 200 Urysohn, Pavel (1898–1924), 96, 107– 116, 118, 119, 121, 125, 130, 131, 268 Vahlen, Theodor (1869–1945), 197, 238 Valentiner, Theodor (1869–1952), 197, 199, 202, 203, 211 van Dantzig, David (1900–1959), 123 van der Waerden, B.L. (1903–1996), xiii, xvi, xvii, 37, 91, 92, 98, 100–103, 105, 106, 116, 117, 122, 123, 126–128, 130, 133, 137, 141, 146, 151, 152, 159, 160, 176, 184, 185, 190, 197, 204, 205, 207, 217, 222, 239, 261, 262, 267, 268, 281, 286, 289, 294, 296 van der Waerden, Dorothea (?–1941), 98 van der Waerden, Theo (1876–1940), 98 Vandiver, Harry (1882–1973), 241

Name Index Veblen, Oswald (1880–1960), 129, 132, 186, 199, 212, 234, 241, 242, 244, 247, 248, 250, 253, 264, 265 Vermeil, Hermann (1889–1959), 74 Voigt, Woldemar (1850–1919), 47, 48, 52, 59, 60 Volterra, Vito (1860–1940), 159 Vorbeck, Werner (1911–), 200 Voss, Aurel (1845–1931), 21 Wagner, Hermann (1840–1929), 52 Wagner, Richard (1813–1883), xii , 256 Wald, Abraham (1902–1950), 275 Wallach, Otto (1847–1931), 52 Walther, Alwin (1898–1967), 292 Ward, Morgan (1901–1963), 247 Weaver, Warren (1894–1978), 250, 291 Weber, Heinrich (1842–1913), 11, 36, 92, 98, 104 Weber, Werner (1906–1975), 235–238, 261 Wedderburn, Joseph Henry Maclagan (1882–1948), 92, 102, 137, 207, 215, 248, 285, 286 Wegner, Udo (1902–1989), 198, 237, 238 Wehnelt , Arthur (1871–1944), 20 Weierstrass, Karl (1815–1897), 11, 18, 39, 40, 155, 205 Weil, André (1906–1998), viii, xviii, 151, 159, 160, 162, 225 Weiss, Marie (1903–1952), 222, 223, 244 Weitzenböck, Roland (1885–1955), 98, 100 Weyl, Helene (1893–1948), 162, 200 Weyl, Hermann (1885–1955), xi, xiii, xiv, xviii, 20, 21, 53, 61, 82, 86, 88, 89, 91, 99, 129, 134, 137, 162, 187, 191, 196–200, 203, 204, 209, 211, 225, 233, 234, 237, 241, 243, 244, 252,

339 255–258, 260, 263, 267, 269, 270, 276, 283, 294, 296 Weyl, Joachim (1915–1977), 200 Wheeler, Anna Johnson Pell (1883– 1966), xix, 217, 219–222, 224, 244, 247, 248, 250, 252, 253, 264, 295, 296 Wheeler, Arthur Leslie (1861–1932), 218, 224 Whitney, Hassler (1907–1989), 291 Wichmann, Wolfgang (1912–1944), 186, 200, 202 Wiechert, Emil (1861–1928), 47, 52 Wiedemann, Eilhard (1852–1928), 20 Wiener, Norbert (1894–1964), 227, 230, 232, 249 Wigner, Eugene (1902–1995), 204 Wilamowitz-Moellendorf, Ulrich von (1848–1931), 42, 44 Wilhelm II, Kaiser (1859–1941), 59, 61, 256 Wilson, Woodrow (1856–1924), 214 Windaus, Adolf (1876–1959), 52 Winston, Mary (1869–1959), 41 Witt, Erna Bannow (1911–2006), 173, 200, 225 Witt, Ernst (1911–1991), 144, 169, 173, 200, 225, 246 Wolfowitz, Jacob (1910–1981), 275 Young, Grace Chisholm (1868–1944), 40, 41, 213 Zapelloni, Maria Teresa, 229 Zariski, Oscar (1899–1986), 159, 243, 247, 251 Zassenhaus, Hans (1912–1991), 175 Zermelo, Ernst (1871–1953), 34, 145, 146, 158, 261 Zeuthen, Hieronymus Georg (1839– 1920), 38