Against All Odds: Women’s Ways to Mathematical Research Since 1800 [1st ed.] 9783030476090, 9783030476106

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Women in the History of Philosophy and Sciences 6

Eva Kaufholz-Soldat Nicola M. R. Oswald   Editors

Against All Odds Women’s Ways to Mathematical Research Since 1800

Women in the History of Philosophy and Sciences Volume 6

Series Editors Ruth Edith Hagengruber, Department of Humanities, Center for the History of Women Philosophers, Paderborn University, Paderborn, Germany Mary Ellen Waithe, Professor Emerita, Department of Philosophy and Comparative Religion, Cleveland State University, Cleveland, OH, USA Gianni Paganini, Department of Humanities, University of Piedmont, Vercelli, Italy

As the historical records prove, women have long been creating original contributions to philosophy. We have valuable writings from female philosophers from Antiquity and the Middle Ages, and a continuous tradition from the Renaissance to today. The history of women philosophers thus stretches back as far as the history of philosophy itself. The presence as well as the absence of women philosophers throughout the course of history parallels the history of philosophy as a whole. Edith Stein, Hannah Arendt and Simone de Beauvoir, the most famous representatives of this tradition in the twentieth century, did not appear form nowhere. They stand, so to speak, on the shoulders of the female titans who came before them. The series Women Philosophers and Scientists published by Springer will be of interest not only to the international philosophy community, but also for scholars in history of science and mathematics, the history of ideas, and in women’s studies.

More information about this series at http://www.springer.com/series/15896

Eva Kaufholz-Soldat Nicola M. R. Oswald •

Editors

Against All Odds Women’s Ways to Mathematical Research Since 1800

123

Editors Eva Kaufholz-Soldat Goethe University Frankfurt Frankfurt am Mainu, Hessen, Germany

Nicola M. R. Oswald Department of Mathematics and Informatics University of Wuppertal Wuppertal, Nordrhein-Westfalen, Germany

ISSN 2523-8760 ISSN 2523-8779 (electronic) Women in the History of Philosophy and Sciences ISBN 978-3-030-47609-0 ISBN 978-3-030-47610-6 (eBook) https://doi.org/10.1007/978-3-030-47610-6 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Series Foreword

Women in the History of Philosophy and Sciences The history of women’s contributions to philosophy and the sciences dates back to the very beginnings of these disciplines. Theano, Hypatia, Du Châtelet, Agnesi, Germain, Lovelace, Stebbing, Curie, Stein are only a small selection of prominent women philosophers and scientists throughout history. The Springer Series Women in the History of Philosophy and Sciences provides a platform for publishing cutting-edge scholarship on women’s contributions to the sciences, to philosophy, and to interdisciplinary academic areas. We therefore include in our scope women’s contributions to biology, physics, chemistry and related sciences. The Series also encompasses the entire discipline of the history of philosophy since antiquity (including metaphysics, aesthetics, philosophy of religion. We welcome also work about women’s contributions to mathematics and to interdisciplinary areas such as philosophy of biology, philosophy of medicine,sociology. The research presented in this series serves to recover women’s contributionsand to revise our knowledge of the development of philosophical and scientific disciplines, so as to present the full scope of their theoretical and methodological traditions. Supported by an advisory board of internationally esteemed scholars, the volumes offer a comprehensive, up-to-date source of reference for this field ofgrowing relevance. See the listing of planned volumes. The Springer Series Women in the History of Philosophy and Sciences will publish monographs, handbooks, collections, anthologies, and dissertations. Paderborn, Germany Cleveland, USA Vercelli, Italy

Ruth Hagengruber Mary Ellen Waithe Gianni Paganini

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Introduction

Virginia Woolf once famously claimed that “a woman must have money and a room of her own if she is to write fiction” (Woolf [1929], p. 4). Since most women depended economically on a male figure, whether a husband or father, they lacked such means; this, in Woolf’s eyes, provided an explanation for the small number of female authors up until her day. Clearly, just as was the case with careers in literature, so with mathematics; there were only a few female mathematicians before the middle of the twentieth century. Even today, far fewer women than men hold professorships in mathematics. Thus, this volume raises an analogous question: What did it take for a woman to become a mathematician? Woolf looked at historical examples and so do the essays in this volume, which describe the lives of a variety of mathematicians from the eighteenth through the twentieth centuries. But whereas Woolf came to her conclusion by looking at things women lacked, this book takes a different approach—by analyzing some of those women who actually did manage to receive academic training and go on to do creative work in mathematics. Since these women were able to find their way into a very patriarchal and hegemonic world, by looking at these cases certain possibilities, however limited, become evident. The plural already hints at a vital difference when compared with Woolf’s seemingly universal solution: there was and never will be a single way for women to become mathematicians, even if one can find plenty of common circumstances. As will become clear, each of the women portrayed here eventually found individual solutions that depended heavily on the specific situation in different countries during the period in question. Though we have grouped the essays in this volume into four thematic sections—three with introductory reflections by Nicola Oswald—I will here discuss these cases roughly in chronological order. My aim is to provide some historical background that will highlight and contextualize important milestones in the timeline of women’s education, especially in the field of mathematics. Let me remark to begin with that we, too, cannot ignore what was lacking. We need to familiarize ourselves with the obstacles in order to understand how some women were able to overcome them. Without higher education—or barely any education at all for most girls until well into the nineteenth century and in many vii

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countries long after that—mathematics could only be studied in private. Such women invariably came from progressive and, more often than not, wealthy families. Only in these special circumstances did young women have access to books and the time to read them. It is certainly no coincidence that many of the first female mathematicians—Celia Grillo Borromeo (1684–1777),1 Émilie du Châtelet (1706– 1749), Maria Gaetana Agnesi (1718–1799), Ada Lovelace (1815–1852)—came from a noble background. The same holds for Sophie Germain (1776–1831), whose career we will discuss shortly. First, however, let us set the stage by taking a closer look at the period she lived in by considering certain aspects that reflect the relationship between science and gender at that time. According to Londa Schiebinger, the eighteenth century was characterized by a rather rapid and finally peremptory exclusion of women from science, which from then on was predominantly practiced at academies and universities, which were exclusively reserved for men (Schiebinger [1989]). Criticism of this thesis was brought forward by scholars such as Lorraine Daston, who pointed out that Schiebinger was mainly talking about singularities when she pointed to artisan families and aristocratic salons as examples of a much friendlier attitude towards women in science during the Enlightenment (Daston [1989], p. 1502). Subsequent publications, however, such as (Mommerts [2002]), show that at least some women carried out scientific endeavors during this period, albeit well hidden from the public eye; they stayed behind the scenes, while the men took center stage. If we take into account the female mathematicians cited above, I believe the topic of exclusion of women from science during the eighteenth century stands in need of reassessment. Another line of argument in Schiebinger’s book, however, is indisputable. Aristotle (384–322 BC) argued that women were essentially inferior to men, in particular with respect to their bodies and their reproductive organs. This view became entrenched through Galen (129 AD–c. 200/c. 216), who shaped medical discourse for centuries to come. Schiebinger shows how this claim influenced the drawings of female skeletons, which appeared as mild modifications of male ones (Schiebinger [1989], chapter 7). Such drawings revealed few significant differences before the two sexes were invented/discovered and understood as incommensurable in the eighteenth century (Laqueur [1992], pp. 151–152). Another assumption made by Aristotle survived even longer. Understanding women to be inferior or incomplete males, the philosopher claimed they had characteristics complementary to those of men. Over time, those characteristics were linked more and more to what were considered the natural destinies of each sex: women were relegated to the private sphere of the home as wives and mothers; men were destined to be actors in the public sphere (Hausen [1976]). Within this tradition, one saw women as endowed with imagination and creativity, faculties that 1

Since Borromeo is rather unknown, let us just note in passing that her most important result was arguably the discovery of the Clélie, as in “the curve of Celia”. This spherical curve is given by the property that the angle of longitude is a multiple of the angle of colatitude. Thus, a spherical spiral, e.g. is a special case of the Clélie.

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were long associated with cunning and deception. During the eighteenth century, however, in the wake of a growing veneration for literati and the cult of (male) genius, these formerly female characteristics suddenly became desirable. Yet rather than now being ascribed to both sexes, they were simply declared male attributes (Daston [2008]). Imagination and creativity, being abilities demanded of poets, artists, and writers, were apparently talents that women simply lacked. This sweeping argument or prejudice was enough to discourage most women from pursuing these avocations, a topic Woolf touched upon in her essay (Woolf [1929], pp. 39–41). Yet, in fact, many of her contemporaries saw a clear connection with all forms of intellectual endeavor, including mathematics. For them, the female intellect was merely capable of imitating and, as in childbirth, reproducing. Hence, following this traditional view, most believed that while women could perform calculations following careful instructions, men alone could do original mathematical work, proving theorems and creating interesting hypotheses (Kaufholz-Soldat [2017], pp. 204–205). This overall background has clear relevance for the second part of Jenny Boucard’s essay on Sophie Germain. Born in 1776, Germain was excluded from academic circles and thus could only study mathematics in private. Her predicament led her to take on a false identity, and she presented herself in letters to Carl Friedrich Gauß (1777–1855) as Monsieur Antoine-Auguste Le Blanc. After presenting Germain’s biography and an overview of her mathematical works, Boucard takes a closer look at the reception of the French mathematician at the turn from the nineteenth to the twentieth century. This shows that Germain was not only presented as an inspirational role model for young girls. Indeed, critical voices were also raised, questioning her mathematical abilities and those of women in general, a trope that can also be found in the reception of Sofja Kovalevskaya (1850–1891) in the years after her sudden death (Kaufholz-Soldat [2019]). This view shifted later, as Boucard shows, when Germain’s mathematical works came to be appreciated more and more, especially after her correspondence with Gauß was rediscovered beginning at the end of the 1870s. The Belgian mathematician Paul Mansion (1844–1919) called her a “competent reader” of Gauß’s writings. Still, even this positive assessment was nonetheless consistent with the view of those who, while conceding women the possibility to comprehend at least some parts of mathematics, denied their ability to do creative work (cf. e.g. Loria [1903; 1904], Möbius [1905], Münsterberg [1897], p. 349). The other women portrayed in this volume were born quite a bit later, and thus experienced very different circumstances once institutions of higher education gradually began to open their doors to women. Throughout continental Europe— whether in France (Christen-Lécuyer [2000]), Germany, or Switzerland—foreigners were more often than not the first walk through those doors. This reflected a double standard that was particularly striking in the Swiss case, since already in 1867 the universities allowed international students to matriculate regardless of their sex. Swiss women, on the other hand, could not even attend a Gymnasium to gain the Abitur, which was required for Swiss citizens who wanted to study at a university (Creese & Creese [2004], p. 181).

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A similar situation prevailed in the German states during the nineteenth century up until 1893 when Karlsruhe in the state of Baden allowed women to attend a Gymnasium for the first time.2 Thus, it comes as no surprise that the first female students at German universities came from abroad as well. Most came from English-speaking countries after having studied beforehand at elite women’s colleges. The oldest of these was Girton College, Cambridge, founded in 1869; in the United States of America, Vassar College was the first to open in 1865, followed by Smith and Wellesley in 1871 and 1875, respectively (Parshall [2015], p. 72f.). However, Bryn Mawr, which opened in 1885, was the only women’s college in the United States to grant Ph.D.’s (Green & LaDuke [1987], p. 12). For this very reason, and the excellent reputation German mathematics enjoyed at the time, two Americans—Mary Frances Winston (1869–1959) and Magaret Maltby (1844–1960)—as well as Grace Chisholm (1868–1944) from England enrolled in the mid-1890s at the Prussian University of Göttingen. Not until 1908 would German women be allowed to matriculate at Prussian universities. It was partly due to Felix Klein’s (1849–1925) influential contacts in the Prussian Ministry of Culture that these foreign women gained permission to enroll as part of an experiment to ascertain women’s ability to undertake university studies. In her essay, Renate Tobies’ takes a closer look at how Klein and his colleague David Hilbert (1862–1943) made their university into a hub for this generation of women in mathematics, thereby serving as a forerunner for Prussia as well as the other German states. Drawing on archival sources, Tobies gives an extensive overview of the women attending Klein’s and Hilbert’s seminars between 1893 and 1912. This study thus further elaborates on the findings in (Tobies [1991/92, 1997, 1999]), which confirm and consolidate Göttingen’s reputation as a female-friendly university for mathematics and other related fields. Tobies’ essay offers an important supplement to the relatively large body of work on Göttingen, whereas the other essays in this chapter look at universities that have not received much attention from historians, insofar as female students of mathematics are concerned. Katharina Spiess examines the situation in Bavaria, comparing its university in Würzburg with its counterparts in Erlangen and Munich. While women were only allowed to study at Bavarian universities in 1903, already in 1886 the American Marcella O’Grady (1863–1950) was allowed to attend biology courses at Würzburg. Still, it was not until 1912 that Bavarian women began to enroll in mathematics in Würzburg. Spiess carefully highlights the most important factors for this delay, pointing clearly to the role of the Bavarian Ministry of Culture. Martina Bečvářová, on the other hand, considers the first women to earn their Ph.D.’s at the universities in Prague between 1900 and 1945. In a prosopographic study largely based on archival sources and interviews, she was able to detail their biographical background, more often than not for the first time. Moreover, the comparative overview of the situation at each university, in particular the careful distinction between German students and those of Czech nationality, will hopefully Baden was also the first German state allowing women to study in 1900.

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serve as a fruitful starting point for future research about female students in Prague and the changes that took place there during WWII. Another feature uniting these two essays is the fact that the names of the women presented in them will not sound familiar to the modern reader. The only likely exception would be Saly Ruth Ramler (1894–1993), the first woman to gain a mathematics Ph.D. from the German University in Prague who later married the noted mathematician and historian of mathematics Dirk Jan Struik (1894–2000). Moreover, the reader will notice that most of the women portrayed by Spiess and Bečvářová, became teachers after obtaining their degrees. This was, in fact, the most common profession for students of mathematics of both sexes, as research opportunities were slim at best for men, and practically nonexistent for women.3 This should be kept in mind in the next chapter, which deals with career opportunities for women in mathematics. At the beginning of the twentieth century other sciences, such as astronomy or chemistry, created a plethora of positions for women, albeit more often than not at the lowest ranks of the hierarchy, both with regard to responsibilities and salaries (cf. e.g. Rossiter [1984] or Lankford & Slavings [1990]). One of the reasons for this discrepancy in the job market stemmed from the fact that the experimental sciences require activities that involve data gathering before creative work can proceed. This type of work was tedious, involving lengthy calculations or repetitive work in laboratories, and therefore often relegated to women. These organized and collaborative efforts were typical for the sciences, whereas mathematics offered few such opportunities.4 Still, there was a special form of collaboration, which offered an opportunity for some women to pursue mathematical research without holding a formal position. Though by no means limited to mathematics, this arrangement has been studied in recent literature under the nomenclature of “Couples in Science”. This term refers to two researchers who work together and either publish together or “create a nurturing environment for the other’s scientific work, even if the scientific work itself did not involve joint research and publications” (Slack [2012], p. 272). This is certainly a highly specific form of scientific collaboration (Pycior et al. [1996], p. 3), since it requires that the respective partners either be married or at least live in a committed relationship through which the intersection of private and professional lives is acknowledged (Lykknes, Opitz & Van Tiggelen [2012]). This takes us back to the aforementioned Grace Chisholm and the women who studied in Göttingen. She was the first to complete her Ph.D. in 1895 and the first woman to take her degree by passing the oral exams normally required. Elisabeth Mühlhausen takes a closer look at the unusual relationship between Chisholm and her husband, William Henry Young (1863–1942), whom she married 1 year later. Afterward, they published much of their work together. Mühlhausen describes letters Grace sent in 1906 to Max Dehn (1878–1952), written in response to his criticism of an article published under her husband’s name alone. Chisholm 3

For Germany, e.g. compare Tobies (2012a). This changed during the twentieth century, however. Compare, for example, Tobies (2012b) or Tobies & Vogt (2014).

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Young’s detailed knowledge of the matter at hand, as well as her dominant role in the scientific relationship, surfaces clearly in this exchange. Mühlhausen substantiates that much of their joint work only bore her husband’s name as a way to further his chances of finding a permanent position, since she as a woman would have no such opportunity (cf. e.g. Jones [2009], chap. 4). This was a conscious decision made by both partners in the marriage, and Chisholm took it upon herself to promote her husband’s achievements in order to reach this common goal. One can detect certain parallels here with the chapter that follows, in which Annette Vogt discusses the eventful lives of Emma S. (1893–1968) and Wladimir Woytinsky (1885–1960). Banned into exile in Siberia, they left Russia for Berlin as political émigrés in 1922. Little more than a decade later, they had to flee again in 1933, both as Jews and Socialists, and after a longer journey through Europe they found a final home in Washington, DC. Strictly speaking, the Woytinskies have to be considered political economists, not mathematicians in the conventional sense, with publications on the “World Population and Production” (1953) or on “World Commerce and Government” (1955). While these books had both their names on the cover, Vogt shows the important role Emma played in the earlier publications as well, while providing insights on how the roles of women in scientific relationships mirror their economic and social situation in different societies. Finally, Danuta Ciesielska examines the relationship between Otto Nikodym (1887–1974) and his wife Stanislawa (1897–1988), a virtually forgotten figure. Nikodym was a member of the Polish School of Mathematics, known for his contributions to measure theory, especially the Radon–Nikodym Theorem. Although they worked in different mathematical areas and their careers were largely independent, they still were intertwined as a creative couple who took an active interest in each other’s research. Drawing from rich archival material, Ciesielska’s collective biography captures fascinating insights into the Polish mathematical community and the respective prospects for women under different political systems. The Nikodyms lived in Poland under Russian Rule and afterward through the First and Second World Wars. They immigrated to the U.S.A. in 1948 and taught at Kenyon College until their retirement in 1965. Around the same time, Dorothy Wrinch (1894–1976) was teaching at Smith College in Northampton, Massachusetts, her final position after stints at various institutions. In a sometimes very personal account, Marjorie Senechal details the major points in this very unusual woman’s life (see also Senechal [2013]). They became friends after Wrinch’s retirement, sharing a common interest in crystal-symmetry. Starting as a student of the noted philosopher Bertrand Russell (1872–1970), Wrinch began her career as a logician, but she later published on subjects as diverse as scientific method and motherhood as well. Her most famous works, however, stem from questions arising in biology, and she later became known not only as a mathematician but also as a distinguished theoretical biologist. Both Nikodym and Wrinch take us well into the twentieth century, a time by which women’s opportunities to become a mathematician had noticeably improved since the time of Sophie Germain, who had to disguise as a man in order to get in

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contact with academic circles. Looking at the modern situation, we see that, while institutional and formal conditions changed, the peoples’ minds did not always keep pace with those changes. In their text, Lisbeth Fajstrup, Anne Katrine Gjerløff, and Tinne Hoff Kjeldsen present results from interviews conducted with the first generation of female mathematics professors in Denmark, the youngest of whom was born in 1939. While all of them profited from a progressive spirit during their careers, only some expressed that they had experienced discrimination due to their gender. Still, drawing from the interviews, the authors find hints that indicate otherwise, and they introduce the notion of the “implicit girl” to explain this. All of the women interviewed noted that they had encountered a prevalent attitude during their education that suggested girls should have lower expectations due to their sex. These pervasive views with respect to the abilities of girls, in effect, set implicit standards for what they “can and cannot do, in school, in academia in mathematics, and in life as such.” That girls’ aptitude for mathematics is often questioned still today—and that these prejudices are accepted to some extent by society in general as well as by many girls in particular—can be seen from Andrea Blunck’s essay. These prejudices, Blunck argues, are one of the reasons why the number of women with a Ph.D. or a professorship in mathematics remains quite low even today, whereas the percentage of those who study the discipline is almost equal to that of men. Paola Govoni examines the long-term influence of the notion that mathematics is an intrinsically male discipline. Coming from a sociological point of view, she takes a closer look at discussions about women’s aptitude for mathematics, spanning an arch from antiquity to modern times. By looking at its origins, some already mentioned above, while discussing assumptions about creativity as a strictly male characteristic, she not only offers an important overview of the history of this specific prejudice. More importantly, she is able to show the impact historic views still have on modern culture, in particular the still prevalent conviction that women’s nature is different than that of men, a notion that more often than not implies inferiority in general. Both Blunck and Govoni agree that changes in structure cannot suffice if the underlying sociological factors continue to remain the same. They also believe that one measure to help encourage more women to choose a career in mathematics and approach the goal of a gender-free science is to introduce girls to role models, both historic and contemporary. This volume, presenting an overview of various ways that women were able to enter into the world of mathematics from the eighteenth century onward, indirectly contributes to this intention as well. Despite the singularities in their respective biographies, these women represent a common phenomenon. By overcoming the odds in different countries and societies with different educational systems, they were able to enter this “forbidden domain.” And while mathematics clearly

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continues to remain a predominantly male discipline, these women show that it was never exclusively so, and that members of the allegedly emotional and feeble sex managed to finds ways to contribute to a discipline long considered the most abstract and rigorous form of science.

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Rossiter, M. W. (1982). Women Scientists in America: Struggles and Strategies to 1940 (Vol. 1). JHU Press. Schiebinger, L. (1989). The Mind has no Sex? Women in the Origins of Modern Science. Harvard University Press. Senechal, M. (2013). I died for Beauty: Dorothy Wrinch and the Cultures of Science. New York: Oxford University Press. Slack, N. G. (2012). Epilogue: Collaborative Couples—Past, Present and Future. In A. Lykknes, D. L. Opitz & B. van Tiggelen (Eds.), For better or for worse? Collaborative couples in the sciences (pp. 271–304). Basel: Birkhäuser. Tobies, R. & Vogt, A. B., with the assistance of Pakis, Valentine A., (Eds.). (2014). Women in industrial research. Wissenschaftskultur um 1900. Stuttgart: Franz Steiner. Tobies, R. (1991/1992). Zum Beginn des mathematischen Frauenstudiums in Preußen. NTM– Schriftenreihe für Geschichte der Naturwissenschaften, Technik und Medizin, 28, 151–172. Tobies, R. (1999). Felix Klein und David Hilbert als Förderer von Frauen in der Mathematik. Acta Historiae Rerum Naturalium necnon Technicarum/Prague Studies in the History of Science and Technology, 3, 69–101. Tobies, R. (2012a). German Graduates in Mathematics in the first Half of the 20th Century: Biographies and Prosopography. In L. Rollet & P. Nabonnand (Eds.), Les uns et les autres… Biographies et prosopographies en histoire des sciences. Collection Histoire des institutions scientifiques (pp. 387–407). Nancy: Presses Universitaires. Tobies, R. (2012b). Iris Runge: A life at the Crossroads of Mathematics, Science, and Industry. Trans. V. A. Pakis. Science networks: Historical studies 43. Basel: Birkhäuser. Tobies, R. (Ed.). (1997). Aller Männerkultur zum Trotz: Frauen in Mathematik, Naturwissenschaften und Technik. Frankfurt am Main: Campus. (2nd ed.) (2008). Wobbe, T. (2003). Instabile Beziehungen: Die kulturelle Dynamik von Wissenschaft und Geschlecht. In T. Wobbe (Ed.), Zwischen Vorderbühne und Hinterbühne: Beiträge zum Wandel der Geschlechterbeziehungen in der Wissenschaft vom 17. Jahrhundert bis zur Gegenwart (pp. 13–38). Transcript Verlag. Woolf, V. (1935). [1929]. A Room of one’s own. Originally published with Hogarth Press in London.

Prologue

The composition of this anthology has been a long journey. First ideas arose during a small conference on women in mathematics in Würzburg in 2015, where some of the authors in this volume participated. In 2017, Tinne Hoff Kjeldsen (Copenhagen) and Renate Tobies (Jena), and Nicola Oswald (Wuppertal) organized the Workshop “Women in Mathematics: Historical and Modern Perspectives” (1702a), with funding from the Oberwolfach Research Institute for Mathematics, gaining momentum for this publication and further developing the ideas found herein. While overviews on women in science exist—most notably, Margaret Rossiter’s quantitative work about “Women Scientists in America” (1982, 1984, 2012), “Aller Männerkultur zum Trotz: Frauen in Mathematik, Naturwissenschaften und Technik” (2008) edited by Renate Tobies, and Claire Jones’s “Femininity, Mathematics and Science, 1880–1914” (2009)—there are few studies focused solely on the field of mathematics. Significantly, there is no such publication with a focus on European countries. This volume, which documents mathematicians in Italy, Poland, the Czech Republic, Germany, Denmark, France, and Great Britain, is thus another tile in the yet incomplete mosaic that is the contemporary history of women in scientific disciplines. It is well known that the success of marginalized groups in gate-kept institutions depends on a multitude of factors. In this book, we do our best to take stock of such a reality. We have grouped the essays into four subsections. In three of these––“Institutions,” “Couples,” and “Approaches”––biographies of women in mathematics or specific conditions at different places in time are (re-)evaluated. In the concluding chapter, “Perspectives,” sociological and sociographic factors are the main focus. The anthology presents a wide range of historiographical approaches to the topic, including classical biography, interdisciplinary questionnaire analysis, archival work, oral history, and analysis of quantitative statistics. In their rich studies, the authors investigate prestigious mathematical institutes as well as provincial universities, they reassess well-known mathematicians and introduce readers to those who have been forgotten, and they shed light on lesser

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acknowledged factors that historically influenced career paths and remain significant for women working in the sciences today. Thus, we believe this collection of diverse texts to be a valuable contribution to the series “Women Philosophers and Scientists” initiated by Ruth Hagengruber (Paderborn) and are thankful for the opportunity to publish it in this context. We would also like to thank Klaus Volkert and Jörn Steuding for nearly 2 years of consulting. Language support for this book was provided by Valentine Pakis, Miriam Hayland-Müller, and Carly Crane. Last but not least, a heartfelt thank-you is more than due to Pascal Stumpf (Würzburg), who significantly supported us in the formal processing of the book in the past few months before its completion without ever losing his patience with us. We wish you a pleasant read and inspiring insights! Wiesbaden, Germany Würzburg, Germany November 2019

Eva Kaufholz-Soldat Nicola M. R. Oswald

Contents

Part I 1

2

3

Institutions

Internationality: Women in Felix Klein’s Courses at the University of Göttingen (1893–1920) . . . . . . . . . . . . . . . . . . . Renate Tobies

9

Academic Education for Women at the University of Würzburg, Bavaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katharina Spieß

39

Women and Mathematics at the Universities in Prague . . . . . . . . . Martina Bečvářová

Part II

73

Couples

4

Grace Chisholm Young, William Henry Young, Their Results on the Theory of Sets of Points at the Beginning of the Twentieth Century, and a Controversy with Max Dehn . . . . . . . . . . . . . . . . . 121 Elisabeth Mühlhausen

5

Emma S. and Wladimir S. Woytinsky: An Unusual Couple in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Annette B. Vogt

6

Stanisława and Otton Nikodym . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Danuta Ciesielska

Part III 7

Approaches

Arithmetic and Memorial Practices by and Around Sophie Germain in the 19th Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Jenny Boucard

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Dorothy Wrinch, 1894–1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Marjorie Senechal

9

Living by Numbers: The Strategies and Life Stories of Mid-Twentieth Century Danish Women Mathematicians . . . . . . 249 Lisbeth Fajstrup, Anne Katrine Gjerløff, and Tinne Hoff Kjeldsen

Part IV

Perspectives

10 Hearsay, Not-So-Big Data and Choice: Understanding Science and Maths Through the Lives of Men Who Supported Women . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Paola Govoni Epilogue: Mathematics—Still a Male Domain?. . . . . . . . . . . . . . . . . . . . . 315 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Contributors

Martina Bečvářová Department of Applied Mathematics, Faculty of Transportation Sciences, Czech Technical University in Prague, Prague 1, Czech Republic Jenny Boucard Faculté des sciences et des techniques de Nantes, Centre François Viète d’histoire des sciences et des techniques, Nantes, France Andrea Blunck Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany Danuta Ciesielska Institute for the History of Science, Polish Academy of Sciences, Warsaw, Poland Lisbeth Fajstrup Department of Mathematical Sciences, Aalborg University, Aalborg Øst, Denmark Anne Katrine Gjerløff Natural History Museum of Denmark, University of Copenhagen, Copenhagen K, Denmark Paola Govoni Department of Philosophy and Communication, University of Bologna, Bologna, Italy Tinne Hoff Kjeldsen Department of Mathematical Sciences, University of Copenhagen, DK, Copenhagen Ø, Denmark Elisabeth Mühlhausen Krebeck, Germany Marjorie Senechal Clark Science Center, Smith College, Northampton, MA, USA Katharina Spieß Institute of Mathematics, Julius-Maximilians-Universität, Würzburg, Germany Renate Tobies Ernst-Haeckel-Haus, Universität Jena, Jena, Germany Annette B. Vogt Max Planck Institute for the History of Science, Berlin, Germany

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Introductory Reflections on Influencing Institutions Nicola M. R. Oswald Institutional Framework The institutional framework naturally played a decisive role concerning the participation of women in academic life. The possibility of regular enrolment at a university was a basic prerequisite to enable a higher educated woman to be considered as normal and also to allow her activity as a scientist.1 In the second half of the nineteenth century, certain institutions in Europe2 began to overcome this hurdle by offering women regular higher education or at least the chance to study under special conditions. In this process, usually, dedicated actors played a role, which, within its own framework, exerted influence on the current conditions. To illustrate this, we will consider two different examples:3 The first college in England to educate women, Girton College, was founded four miles from Cambridge in 1873.4 In particular, the co-founder Emily Davies (1830– 1921) had ambitions for the graduates of her college to be considered as equivalently qualified as educated men. Davies herself was a pioneer and campaigner for women’s

1 In her explanatory model for the access of women to academic careers in international comparison

Ilse Costas takes several political and social variables into account when measuring the hurdle of opening universities to women; see Costas (2003). Comparing conditions in the United States of America, France and Germany she underlines the important role of the socially influenced structure of educational systems together with the political organization of educational institutions. 2 In the United States, the institutional access for women to higher education also developed in this period, concerning mathematics see, e.g. Parshall (2015). 3 These brief introductory words may be considered as impulses und impressions. At this point, we do not aim to give a comprehensive overview of the complex and multifaceted details to the beginnings of women studying at European institutions. 4 There had already been a smaller forerunner in Hitchin, also near Cambridge, founded in 1869.

2

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rights to university access.5 Her strategy was to particularly include male-connoted contents into the education. Emily Davies, principal founder of Girton College, the Cambridge College of higher education for women, observed in 1868 that “the best girls” schools are precisely those in which the “masculine subjects have been introduced”. The subjects that she was referring to were mathematics, Latin and Greek. When it came to the contentious issue of higher education for women, Davies was convinced that only if women succeeded in subjects held to be prestigious for men would their educational achievements be recognized as equally valid. She rejected any idea of a special system or curriculum for women because, to opponents of women’s higher education, “different” would automatically mean “inferior”.6

The tripos, considered as an elite-graduation degree, should be a reasonable goal for her students. Mathematicss, in particular, was assigned as playing a key role: One of the reasons that Emily Davies encouraged her students at Girton to take the mathematics tripos was because this was the most prized degree for men and a subject generally held to be beyond the capabilities of women. When her students beat the men at mathematics it added ammunition to her argument for intellectual equality between the sexes.7

Girton College had some extraordinarily successful graduates and research fellows in mathematics (among them Mary Cartwright (1900–1998) and Olga Taussky-Todd (1906–1995) in the 1930s). Davies’ goal-oriented efforts in the early days certainly played a decisive role in paving the way. In the report, well worth reading “Eine Frauen-Universität in England” (Calm [1885]), published in 1885 in a German magazine for female teachers “in school and at home”, the writer and women’s rights activist Marie Calm (1832–1887) outlined in detail the foundation and financing of the college as well as a guided tour she had given through Girton College. Among other things, she described the classrooms where professors from Cambridge taught inquisitive female students. Regarding the room where, in addition to mathematics, scientific experiments were also carried out, she quoted the apologetic housekeeper with the humorous remark: “Mathematics makes such stains on the floor.”8 In her book Femininity, Mathematics and Science, 1880–1914 Claire Jones not only provides insights into the education of and opportunities for graduates from Girton College, particularly of Grace Chisholm Young (1868–1944), an unusual woman in mathematics, who is introduced in part II by Elisabeth Mühlhausen, and Hertha Ayrton (1854–1923), but furthermore sheds light on further universities in Britain. On the basis of “Statistics of women mathematicians at British Universities” from the Davis Archive of Female Mathematicians,9 she gives an overview of degrees achieved by women in mathematics at different institutions as well as the year of approval allowing women to study mathematics to degree level. Around 5 For more information on Davies and her influence on the development of Girton College, we refer

to Nightingale (1976). (2009), p. 8. 7 Jones (2009), p. 10. 8 Calm (1885), p. 168. 9 See http://www-history.mcs.st-andrews.ac.uk/history/Davis/info.html resp. Jones (2009), pp. 144– 145. 6 Jones

Part I: Institutions

3

1900 women could study at numerous universities. Concerning mathematics, the Scottish universities are certainly interesting (Aberdeen, Edinburgh, Glasgow and St. Andrews), where between 1896 and 1914 altogether 85 women completed their studies of mathematics. In total, the increase of professional female mathematicians was, however, accompanied by an often-described phenomenon:10 However, that women’s inclusion had coincided with the introduction of changes that radically reduced the prestige of a top place in the lists made this concession one that was less threatening to grant and, at the same time, turned it into a somewhat pyrrhic victory.11

Another famous example concerning the early fostering of female mathematicians is given by the University of Göttingen. The analysis in Tobies’ chapter in this book particularly highlights the important function of individual supporters within the group of influential professors. In 1897, the collection “Die Akademische Frau” with expert opinions from outstanding university professors, women teachers and writers on the qualification of women for scientific studies and professions12 was published. This was by Arthur Kirchhoff (1871–1921), the author of diverse essays on natural sciences, politics and cultural politics and in this the mathematicians emerged as quite favourable. Professor Albert Wangerin (1844–1944) of the University of Halle a.S. wrote, for example: This is supported above all by the fact that as well in olden times as also in recent times there have been women who have done outstanding work in mathematics. But that women are also capable of doing mathematics successfully, apart from such exceptions, is proven by the experiences of the universities in Göttingen and Zurich, and the English conditions also speak in favor of this.13

And also, Felix Klein (1849–1925), at this time already a professor at the University of Göttingen, supported the higher education of women.14 He underlined, in the same manner as Wangerin, that not only “extraordinary cases” should be taken into account and noted that in his seminars a certain number of women regularly take part (due to educational possibilities for foreigners). Such public statements by professors not only reflect the factual acceptance but also the positive personal attitude towards female students. Considering women in mathematics, respectively, at universities, as normal the general conditions are influenced step by step. Renate Tobies has presented in detail how affirmative the influence of Felix Klein on interested female mathematicians was. In a comprehensive table (pp. 37–45), she 10 See also comparable examples in Rossiter (1984), referred to in the introductory words of Part II of this book. 11 Jones (2009), p. 147. 12 “Gutachten hervorragender Universiätsprofessoren, Frauenlehrer und Schriftsteller über die Befähigung der Frau zum wissenschaftlichen Studium und Berufe”, see Kirchhoff (1897). 13 Dafür spricht vor allem, daß es sowohl in älterer als in neuer Zeit Frauen gegeben hat, die geradezu Hervorragendes in der Mathematik geleistet haben. Daß aber auch, abgesehen von derartigen Ausnahmen, Frauen imstande sind, mit Erfolg Mathematik zu treiben, beweisen die Erfahrungen der Universitäten in Göttingen und Zürich, dafür sprechen ferner die englischen Verhältnisse. 14 See Klein’s statement in Tobies’ chapter, p. 35.

4

Part I: Institutions

lists the names of women participating in his seminars and lectures. This precise archive work reflects a necessary part of the research on the history of women in mathematics. It is shown that already at the beginning of the era of officially permitted higher education for women there had been a great interest in mathematics. Before 1914 mathematics was under the top five of womens’ choices of a subject. The repeated emphasis of selected personalities, like Sofia Kowalewskaja (1850–1891), Sophie Germain (1776–1831) or Ada Lovelace15 (1815–1852) may sometimes be distorting the real picture. A careful analysis of archive material not only proves that there were more female students than often assumed, but also that it is worthwhile to focus on less famous examples of academic institutions. This is underlined by the research results in the chapters of Spieß and Beˇcváˇrová. First, the focus is on the provincial University of Würzburg (Bavaria). Here, the small-scale research of Katharina Spieß on the first female students in Würzburg in mathematics and at the university, in general, is used to gain an impression of the political background to women’s enrolment. On the basis of numerous file notes from the Main State Archive, discussions in the politics of the state of Bavaria are analysed. Furthermore, the necessary commitment of individuals as well as the influence of the respective ministers of cultural affairs in Bavaria are focussed. In this context, the contrast to the more intensely researched conditions in Prussia and the larger universities of Göttingen and Berlin is of particular interest. The author Martina Beˇcváˇrová, on the other hand, focuses on Prague’s big-city universities: the German University, the Czech University and the Charles University. Her work also yields an in-depth analysis of archival documents. She compiles numerous small-scale and detailed results from the individual university and state archives, thus providing a unique insight into the study of women in mathematics in Prague between 1900 and 1945. One of her concerns is to make visible the names and lives of the numerous female graduates who have been forgotten over the decades. In great detail, she puts together the mosaic of the individual biographies. In addition, she provides an empirical overview of the background of the women. Using the categories “social background”, “high-school education”, “duration of studies” and “dates of awarded doctoral degrees”, she analyses the situation of female students and thus obtains a comprehensive insight into the opportunities for female mathematicians in the Czech capital. The three examples of Göttingen, Würzburg and Prague not only show the different framework conditions for early women’s studies in mathematics at academic institutions. The chapters of this part of the book also raise one’s sights to different facets of historiographical analysis. The viewpoints of the individual contributions vary between influential supporters, political decisions and individual life paths. What unites them is the strong focus on a careful approach to archive sources.

15 Nevertheless

we refer to the especially entertaining comic Padua (2015).

Part I: Institutions

5

Other Kinds of Institutions As an example, we are turning to a country whose universities changed the institutional framework in such a way that women could be officially admitted to study very early on. Christine Riedtmann has been working on the historical reappraisal of female mathematicians in Switzerland, an area that has been little researched to date: The University of Zurich officially admitted women to study in 1867. Zurich was thus the second European university (after Paris) and the first in the German-speaking world to take this step. In the same year the Russian citizen Nadezda Suslova received her doctorate in medicine there. Women had already been accepted as students at the University of Zurich before this date.16

Also, the University of Bern had already accepted women since 1868, and here again, it was a Russian woman who pioneered. The first woman to receive a regular doctorate in mathematics in Europe was the Russian Yelisaveta Litvinova-Ivashkina. She wrote her dissertation in Bern under the supervision of Ludwig Schläfli and passed the doctoral examination in 1878.17 In fact, it is no coincidence that the first women to use the official permission to study were Russian. On the one hand, Swiss women (and women in Continental Europe in general) at that time did not usually have the possibility to obtain a high school graduation which they needed to enrol at universities; for foreign women different conditions applied. On the other hand, the Russian women’s movement had advanced rapidly by the middle of the nineteenth century. Between 1859 and 1864, women in Russia had already been briefly admitted to universities. After the right to study had been revoked, numerous Russian women went abroad to study, and Zurich was rather popular.18 At various points in this book, it will be emphasized that women from abroad have paved the way for women at universities in German-speaking countries. In addition to local institutions, it is also instructive to take a look at institutions in the broader sense, which have been and still are crucial for advancing in the mathematical community, for example, national societies. In chapter I.1, Renate Tobies discusses the German Mathematical Society (pp. 30f.). Another such institution is the Swiss Mathematical Society19 , founded on the initiative of the three mathematicians Rudolf Fueter (1880–1950) (University of Basel), Henri Fehr (1870–1954) (University of Geneva) and Marcel Grossmann (1878–1936) (Polytechnic School, since 1911 16 Die Universität Zürich liess Frauen 1867 offiziell zum Studium zu. Damit war Zürich die zweite europäische Hochschule (nach Paris) und die erste im deutschsprachigen Raum, die diesen Schritt wagte. Bereits im selben Jahr promovierte dort die russische Staatsbürgerin Nadezda Suslova in Medizin. Frauen waren schon vor diesem Datum als Hörerinnen an der Universität Zürich akzeptiert gewesen. 17 “Die erste Frau, die in Europa in Mathematik regulär promovierte, war die Russin Jelisaweta Litwinowa-Iwaschkina. Sie schrieb ihre Dissertation in Bern unter der Leitung von Ludwig Schläfli und legte 1878 das Doktorexamen ab.” See Riedtmann (2018), p. 4. 18 See Bankowski-Züllig (1988) as well as the introduction of this book, p. 9. 19 Of course, these are only selected examples. A comparison of different national societies would be interesting.

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Eidgenössische Technische Hochschule Zürich), who subsequently became the first presidents of the Society.20 Already in the founding year 1910, the mathematician Renée Rocques-Masson joined, followed by Grace Chisholm Young (1915) and her daughter Cecil Tanner Young (1923). In the first two decades, the proportion of women in the rapidly growing association was correspondingly low (2 out of a total of 135 members in 1920—almost a hundred years later, in 2009, the proportion is 45 out of a total of 543 members).21 The mathematical societies were places of networking and exchange, and were thus important for the appreciation of recent works. For example, the “Verhandlungen der Schweizerischen Naturforschenden Gesellschaft” regularly reported on current research results of members of affiliated societies (“Zweiggesellschaften”). Also, the works of Grace Chisholm Young are, for instance, mentioned here.22 The gradual desire to participate in the mathematical community is also reflected in the will of some women to attend the first International Congress of Mathematicians in Zurich in 1897, an institution of great interest to mathematicians of the time in Europe.23 The politics of the organization again reflect the open-minded spirit of the Swiss mathematicians: Only four female mathematicians attended the congress, which is not surprising given that the congress was held at the end of the 19th century. However, the congress organisers advanced a more modern view on female students than many of their international colleagues.24

In this context, archivist Evelyn Boesch (ETH Library) discovered an interesting letter exchange in the estate of Wilhelm Fiedler (1831–1912), Professor at the Polytechnic School in Zurich.25 On 4th July in 1897 the British pioneer of female mathematicians, Charlotte Angas Scott (1858–1931), wrote him a letter being interested in the conditions of participation at the ICM: Can you kindly tell me whether ladies will be welcome—as mathematicians, of course? And whether any special notification of my intention of being present should be made, and if so, to whom?

Fiedler replied positively, the permission for women was “without question”. Furthermore, he offered to send her all the required documents for the congress. That this direct and clear answer could be given by a professor who was not part of the organization committee itself might be due to the fact that already in the founding year 1855

20 See the review of the history of the Swiss Mathematical Society by Urs Stammbach, published online: https://math.ch/about-sms/HistorySMG.pdf (accessed April 25, 2019). 21 See Riedtmann (2018), pp. 16–17. 22 See among others: Volume 104 (1923), Teilband 2, p. 113. 23 For more details, we refer to the chapter “Der erste Internationale Mathematiker-Kongress 1897” in Frei & Stammbach (1994). 24 Eminger, 2015, p. 70. 25 See http://blogs.ethz.ch/digital-collections/2009/11/13/wissenschaftskongress-mit-frauenbeteil igung-oder-nur-mit-damenprogramm-ein-blick-zuruck-ins-jahr-1897/. The letter exchange is stored in (ETH Bibliothek, Archive und Nachlässe Hs 87:1182).

Part I: Institutions

7

women were explicitly allowed and since the 1870s some women actually studied at the Polytechnic School.26 This fundamental openness towards female participants plays a decisive role in the opportunities for a career in mathematics. Helena Mihaljevi´c and Marie-Françoise Roy,27 two mathematicians who are also committed to the European Women in Mathematics Association, describe the importance of the role of international conferences in their analysis of women among the International Congress of Mathematicians (ICM) speakers as follows: “Receiving an invitation to present a talk at an ICM signals the high international reputation of the recipient, and is akin to entering a ‘hall of fame for mathematics’.” (Mihaljevi´c & Roy [2019], abstract) They further analyse that out of 4.120 invited contributions between 1897 and 2018 a total of 202 was given by women, which amounts to only five percent.28 The participation of women in international congresses may be considered in a political and social context: Women’s participation over time, however, did not grow steadily but, instead, shows multiple trends. As presented [...], a comparatively large number of women presented their research at the congresses in 1928 at Bologna and in 1932 at Zurich. This reflects the overall progressive societal and political spirit of the 1920s, which had also enhanced the situation of women in science. In fact, the ICM in 1932 marks a pinnacle in the history of ICMs regarding the role of women. Emmy Noether gave the first plenary lecture by a woman; various women’s colleges and organizations of university women sent delegates, among them the Bedford College for Women (London), Hunter College (New York), the International Federation of University Women, and the American Association of University Women.29

We conclude that the institutional framework conditions for women in mathematics are quite diverse and go far beyond the mere admission to studies. Already the selected individual aspects, which we have outlined here, illustrate that acceptance in the mathematical community depended both on individual personalities and on the possibilities of a regular scientific and social participation. In order to gain an understanding of the achievement of a ‘normal state’ in the sense of equal study, a number of different institutional factors must be taken into account. Bibliography Bankowski-Züllig, Monika (1988). “Zürich–das russische Mekka.” In: Verein Feministische Wissenschaft Schweiz (ed.) Ebenso neu als kühn. 120 Jahre Frauenstudium an der Universität Zürich. Zürich: Efef. 127–128. 26 The first student was the Russian Nadezda Smeckaja, who studied mechanical engineering. A chronological list of events in the history of women at the ETH is given on https://ethz.ch/services/de/anstellung-und-arbeit/arbeitsumfeld/chancengleichheit/strate gie-und-zahlen/frauen-an-der-eth/geschichte-der-frauen-an-der-eth.html. 27 Marie-Fran¸ coise Roy was one of the founders of European Women in Mathematics (EWM) in 1986, she co-founded the French organization Femmes et Mathématiques in 1987. 28 In the first chapter of this book, Tobies takes a closer look at the first female participants at ICMs (p. 31). 29 Mihaljevi´ c & Roy (2019), p. 6.

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Calm, Marie (1885). “Eine Frauen-Universität in England.” In: Loeper-Housselle, M. (ed.) Die Lehrerin in Schule und Haus—Erster Jahrgang 1884/1885 (1). Verlag von Theodor Hofmann. 166–173. Online: https://goobiweb.bbf.dipf.de/viewer/object/1010997505_0001/1/ (accessed January 3, 2020). Costas, Ilse (2003): “Diskurse und gesellschaftliche Strukturen im Spannungsfeld von Geschlecht, Macht und Wissenschaft. Ein Erklärungsmodell für den Zugang von Frauen zu akademischen Karrieren im internationalen Vergleich.” In: Amodeo, I. (ed.) Frau Macht Wissenschaft. Wissenschaftlerinnen gestern und heute. Ulrike Helmer Verlag. Eminger, Stefanie Ursula (2015). Carl Friedrich Geiser and Ferdinand Rudio: The Men Behind the First International Congress of Mathematicians, Doctoral dissertation, University of St. Andrews. Online: http://www-history.mcs.st-and.ac.uk/Publications/Eminger.pdf (accessed May 15, 2019). Frei, Günther and Stammbach, Urs (1994). Die Mathematiker an den Zürcher Hochschulen. Basel: Springer. Jones, Claire (2009). Femininity, Mathematics and Science, 1880–1914. Palgrave Macmillan. Kirchhoff, Arthur (1897). Die akademische Frau. Gutachten herausragender Universitätsprofessoren, Fachlehrer und Schriftsteller über die Befähigung zum wissenschaftlichen Studium und Beruf . Berlin: Hugo Steinitz Verlag. Mihaljevi´c, Helena and Roy, Marie-Françoise (2019). A data analysis of women’s trails among ICM speakers. Online: https://arxiv.org/pdf/1903.02543.pdf. Nightingale Stephen, Barbara (1976). Emily Davies and Girton College. Hyperion (reprint of the original version from 1927). Padua, Sydney (2015). The Thrilling Adventures of Lovelace and Babbage: The (Mostly) True Story of the First Computer. Pantheon Graphic Library. Parshall, Karen Hunger (2015). “Training Women in Mathematical Research: The First Fifty Years of Bryn Mawr College (1885–1935).” Math Intelligencer 37: 71. https://doi.org/10.1007/s00 283-015-9540-2, 71–83. Riedtmann, Christine (2018). Wege von Frauen: Mathematikerinnen in der Schweiz. Online: https:// www.europeanwomeninmaths.org/wp-content/uploads/2018/08/frauenmathch.pdf. Rossiter, Margaret W. (1984). Women Scientists in America. Struggles and Strategies to 1940. The John Hopkins University Press (Paperbacks edition).

Chapter 1

Internationality: Women in Felix Klein’s Courses at the University of Göttingen (1893–1920) Renate Tobies

Abstract This contribution evaluates previous scholarship on the beginning of women’s study of mathematics at German universities and analyzes the special efforts of Felix Klein to advance this cause. It will also be shown when the first female mathematicians joined the German Mathematical Society, which was founded in 1890, and when female authors first published in the journal Mathematische Annalen, the chief editor of which was Klein himself. The study is based on materials from Klein’s archive in Göttingen, especially on the lists of students enrolled in his courses and on the protocols from his mathematics seminars. A special result of our research is that non-German women paved the way in Germany. Foreign women, like men, wanted to be qualified to study where the highest standards of scholarship could be expected, for some time they attempted to gain access to German universities even while official status as students could not yet be granted to them.

Renowned for his achievements in mathematics and its applications, Felix Klein (1849–1925) was also instrumental in spearheading the reform of mathematical education. From the early stages of his career, he was internationally oriented and supported mathematically gifted students regardless of their sex, religion, and nationality. The focus of this paper is Klein’s role as one of the foremost promoters of women studying mathematics. In these efforts, of course, he was not alone. Klein cooperated

1 See

Tina Richter (2015), pp. 18–20; and Tobies (2016). and Johan Ludvig Heiberg (1854–1928) supervised Thyra Eibe (1866–1955), the first women in Denmark to complete her doctorate in mathematics in 1895 (see below). 3 As early as 1878, Christine Ladd-Franklin (1847–1930) was admitted to study at Johns Hopkins University under the supervision of the British mathematician Sylvester (see Fenster and Parshall (1994), p. 234). Felix Klein was asked to succeed Sylvester in 1883, when the latter returned to Great Britain. 2 Zeuthen

R. Tobies (B) Ernst-Haeckel-Haus, Universität Jena, Kahlaische Str. 1, 07745 Jena, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_1

9

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R. Tobies

with a number of international colleagues who likewise supported women mathematicians, including Gaston Darboux (1842–1917) in France,1 Luigi Cremona (1830– 1903) in Italy, Arthur Cayley (1821–1895) in the United Kingdom, Hieronymus G. Zeuthen (1839–1920) in Denmark,2 and James Joseph Sylvester (1814–1897).3 Since the 1890s, when he founded the acclaimed international center of mathematics at the University of Göttingen, Klein admitted not only male mathematicians from abroad into his courses but non-German women as well. The goal of this contribution is to evaluate previous scholarship on the beginning of women’s study of mathematics at German universities and to analyze the special efforts of Felix Klein to advance this cause. It will also be shown when the first female mathematicians joined the German Mathematical Society, which was founded in 1890, and when female authors first published in the journal Mathematische Annalen,4 the chief editor of which was Klein himself. My study is based on materials from Klein’s archive in Göttingen, especially on the lists of students enrolled in his courses and on the protocols from his mathematics seminars.

1.1 Non-German Women Paving the Way in Germany Although women were not legally permitted to enroll in German universities during the nineteenth century, the first women to earn a doctoral degree in mathematics there nevertheless did so at the University of Göttingen in 1874: the Russian Sofia Kovalevskaya (1850–1891). The life and work of Kovalevskaya—and the circumstances of her doctorate in absentia—have received sufficient scholarly attention.5 It should be stressed that Kovalevskaya’s career, until she became a full professor in Stockholm, had been assisted by mathematicians from Sweden, Germany, France, and Italy.6 Felix Klein, too, then a young professor at the University of Erlangen, immediately recognized the significance of Kovalevskaya’s thesis and praised it to the Norwegian Sophus Lie (1842–1899).7 Kovalevskaya’s career, however, was an exception, and it was not until 1895 that the next women mathematicians completed their doctorates at German universities.8 4 See

Tobies and Rowe (1990). Ph.D. examination records are published in Tollmien (1997); see also Eva Kaufholz-Soldat’s dissertation “A Divergence of Lives. Zur Rezeption Sofja Kowalewskajas um die Wende vom 19. zum 20. Jahrhundert,” Johannes Gutenberg-Universität Mainz. 6 On the support that Kovalevskaya received from French and Italian mathematicians, see Coen (2012), pp. 477, 509–515. 7 Kovalevskaya’s thesis was published as “Zur Theorie der partiellen Differentialgleichungen,” Journal für die reine und angewandte Mathematik 80 (1875), pp. 1–32. Klein’s remarks to Lie about the thesis begin as follows: “What do you think about Sophia Kovalevskaya’s study in Borchardt’s journal? By means of direct series expansion, she proves the existence of integrals as well as their definiteness within certain limits.” Quoted from a letter from Klein to Lie dated July 8, 1875. 8 Regarding the laws governing the enrollment of women in several German states, see the details in Tobies (1997) and Birn (2015). 5 Kovalevskaya’s

1 Internationality: Women in Felix Klein’s Courses …

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Marie Gernet (1865–1924) was the first German to do so, namely at the University of Heidelberg and under the direction of Leo Königsberger (1837–1921),9 with whom Kovalevskaya had begun her studies in 1869. In 1895, too, the Englishwoman Grace E. Chisholm (1868–1944)10 and the American Mary F. Winston (1869–1959) submitted their doctoral dissertations, both of which were written under the direction of Felix Klein in Göttingen. It was in the same year, incidentally, that the aforementioned first female Danish mathematician received her doctoral degree in Copenhagen: Thyra Eibe.11 From an international perspective, between the year 1874 (the year of Kovalevskaya’s doctorate) and 1895, five other female mathematicians received a doctoral degree: the Russian Elizaveta Fedorovna Litvinova (1845–1919)—a friend of Kovalevskaya’s—at the University of Bern in 1878; the Englishwoman Charlotte Angas Scott (1858–1931) in London in 1885; and three North Americans: Winifred Harington Edgerton (1862–1951), who was the first to do so in the United States, with a thesis submitted at Columbia University in 1886; Ida Martha Metcalf (1857– 1952) from Cornell University in 1893; and one year later, likewise from Cornell, the Canadian Annie Louise MacKinnon (1868–1940).12 MacKinnon went on to conduct postdoctoral research in Göttingen (see Table 1.1), after which continued her career in the United States, where there were better job opportunities. At the University of Cambridge in the United Kingdom, a remarkable number of women graduated in mathematics, first from Girton College in 1873 and then from Newnham College in 1875.13 But they could neither earn a doctoral degree there nor receive a research position. On these grounds, some of them tried their luck abroad. Recommended by Arthur Cayley, Charlotte Angas Scott became chair of the mathematics department at the newly founded Bryn Mawr College for women in Pennsylvania (United States), where she maintained contact with Göttingen.14 Her association with Göttingen was based above all on her achievements in algebraic geometry, which included a proof of a theorem by Max Noether (1844–1921). The latter proof was published in the journal Mathematische Annalen in 1899.15 Scott was actively involved in the American Mathematical Society, for which she served as vice president in 1905 and 1906. Moreover, she became the first female member of the 9 Marie Gernet became a teacher at the first German secondary school for girls where it was possible

to take the Abitur, the examinations required for entrance to German universities. On the school, which was founded in Karlsruhe in 1896, see Tobies (2001a, b). 10 See Elisabeth Mühlhausen’s Chap. II.1. 11 See Footnote 2. Regarding the development of the Danish educational system, see Lisbeth Fajstrup, Anne Katrine Gjerløff and Tinne Hoff Kjeldsen’s Chap. III.3. 12 On MacKinnon (married name: Fitch, as of 1901) and other early North-American women in mathematics, see Fenster and Parshall (1994), p. 235; and Green and LaDuke (2009). 13 See Davis’s archive of female mathematicians, which includes a chronological list of graduates from the University of Cambridge (1873–1940): http://www-history.mcs.st-and.ac.uk/Davis/Ind exes/xCambridge.html. 14 See Grinstein and Campbell (1987); see also Parshall (2015). 15 Charlotte Angas Scott, “A Proof of Noether’s Fundamental Theorem,” Mathematische Annalen 52 (1899), pp. 593–597.

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Table 1.1 Women enrolled in Felix Klein’s courses Semester

Lectures and seminars (Hours per week)

WS 1893/1894 Hypergeometric functions (4)

SS 1894

11

Mary F. Winston, USA Grace E. Chisholm, UK

Math. seminar (2): Linear differential equations and p-functions

11

M. F. Winston* G. E. Chisholm*a

Linear differential equations of second order (4)

10

M. F. Winston G. E. Chisholm

Selected problems of elementary geometry (2)

14

M. F. Winston G. E. Chisholm

Math. seminar (2): Linear differential equations and spherical functions

12

M. F. Winston* G. E. Chisholm*

12

G. E. Chisholm Ada M. Johnson (b. 1870), UKb Ada Isabel Maddison, UK, later USA Annie Louise MacKinnon, USA

Math. Seminar (2), concluded by Heinrich Burkhardt (1861–1914), with the participation of Arnold Sommerfeld (1868–1951): Foundations of functions with one variable

19

M. F. Winston* A. I. Maddison* A. L. MacKinnon*

Differential calculus (4)

24

Lillien Jane Martin (1851–1943), USAc Frieda Hansmann Helene v. Bortkewitsch (1870–1939)d Alexandrine v. Stebnitzky (b. 1868 Tbilisi), Russiae

Exercises (1)

14

Lillien Jane Martin Frieda Hansmann

Math. seminar (2), conducted with Hilbert and Ernst Ritter (1867–1895): Differential calculus

17

M. F. Winston* H. v. Bortkewitsch* A. v. Stebnitzky* Ada M. Johnson* A. L. MacKinnon* A. I. Maddison*

WS 1894/1895 Number theory (4)

SS 1895

Number of participants Female students (Male and female)

(continued)

1 Internationality: Women in Felix Klein’s Courses …

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Table 1.1 (continued) Semester

Lectures and seminars (Hours per week)

WS 1895/1896 Number theory (2)

Theory of the top (2)

SS 1896

14

Ljubov N. Zapolskaya H. v. Bortkewitsch A. v. Stebnitzky A. L. MacKinnon Ada M. Johnson

19

L. N. Zapolskaya, H. v. Bortkewitsch A. v. Stebnitzky A. L. MacKinnon Mary F. Winston

Math. seminar (2), conducted with Hilbert and Arnold Sommerfeld: Number theory

9

Technical mechanics (2)

13

A. L. MacKinnon** A. v. Stebnitzky* Ada M. Johnson* H. v. Bortkewitsch* L. N. Zapolskaya* Elsa Neumann

Number theory (2)

9

L. N. Zapolskaya A. L. MacKinnon Ada M. Johnson

Math. seminar (2): Number theory

8

L. N. Zapolskaya* Ada M. Johnson* A. L. MacKinnon*

WS 1896/1897 Integral calculus (4)

SS 1897

Number of participants Female students (Male and female)

26

L. N. Zapolskaya H. v. Bortkewitsch Charlotte Wedell (1862–1953), Denmarkf

Math seminar (2), Klein and Hilbert: Theory of functions and conformal mapping

26

Charlotte Wedell* L. N. Zapolskaya, H. v. Bortkewitsch

Differential Eqs. (4)

27

Math. seminar (2), Klein and Hilbert: Theory of functions

19

L. N. Zapolskaya

49

Emilie Norton-Martin, USA Virginia Ragsdale, USA Fanny Cook Gates (1872–1931)g , USA Katharina Hogdon (b. 1871), USA L. N. Zapolskaya Grace E. Chisholm Young

WS 1897/1898 Mechanics I (4)

(continued)

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Table 1.1 (continued) Semester

SS 1898

Lectures and seminars (Hours per week) Exercises in mechanics (1)

31

Math. seminar (2), Klein and Hilbert: Mechanics

12

Grace E. Chisholm Young*

Mechanics II (4)

37

Emilie Norton-Martin Virginia Ragsdale Fanny Cook Gates Katharina Hogdon L. N. Zapolskaya Grace E. Chisholm Young A. L. Bosworth

Exercises in mechanics (1)

10

Emilie Norton-Martin V. Ragsdale Fanny Cook Gates Katharina Hogdon L. N. Zapolskaya

Math. seminar (2), Klein and Hilbert: Mechanics

18

WS 1898/1899 Theory of functions (4)

SS 1899

54

Math. seminar (2): Analysis of real functions

14

Theory of functions and theory of potential (4)

21

Math. seminar (2): Theory of functions (2)

23

WS 1899/1900 Mechanics, especially hydrodynamics (3)

SS 1900

Number of participants Female students (Male and female)

28

Analytic geometry (4)

42

Math. seminar (2) conducted with Max Abraham (1875–1922): Technical applications of elasticity theory

12

Exercises (1)

Nadeschda Nikolaevna von Gernet*

36

Math. seminar (2): Ship movement

WS 1900/1901 Projective geometry (4)

A. L. Bosworth* Anna Helene Palmié* (married name: Therriel) (1863–1946), USA

88

Mary Esther Trueblood (married name: Paine) (1872–1939), USA

29 (continued)

1 Internationality: Women in Felix Klein’s Courses …

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Table 1.1 (continued) Semester

SS 1901

Lectures and seminars (Hours per week) Math. seminar (2): Applications of projective geometry

16

Applications of calculus on geometry (4)

77

Math. seminar (2): Geodesy

29

WS 1901/1902 Mechanics I (4)

SS 1902

30

Mechanics II (4)

53

Math. seminar (2), conducted with Karl Schwarzschild (1873–1916): Astronomy

26

44

Missh Wassielieff Tatyana Afanasyeva (-Ehrenfest) (1876–1964)i , Russia Elizabeth Stephansen (1872–1961)j , Norway

Math. seminar (2), conducted with Karl Schwarzschild: Principles of mechanics

27

T. Afanasyeva*

Encyclopedia of mathematics II, geometry (4)

61

T. Afanasyeva, V. Lebedeva, Russia A. H. Palmié

Math. seminar (2): Graphical statics and the strength of materials

24

WS 1903/1904 Calculus II (4)

Math. seminar (2), conducted with Karl Schwarzschild: Selected chapters of hydrodynamics SS 1904

Mary Esther Trueblood

78

Math. seminar: Selected chapters of mechanics

WS 1902/1903 Encyclopedia of mathematics I (4)

SS 1903

Number of participants Female students (Male and female)

94

Miss Fleer; Miss Gamm Miss Paschen Miss Nobbe

22

T. Afanasyeva

Introduction to the theory 121 of differential Eqs. (4)

Miss Belteneff Miss Kucharskajak Miss Wassiljewa Miss Becker (continued)

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Table 1.1 (continued) Semester

Lectures and seminars (Hours per week)

Number of participants Female students (Male and female)

Math. seminar (2), conducted with Schwarzschild and Martin Brendel (1862–1939): Selected chapters of the theory of probability

35

WS 1904/1905 Mathematical pedagogy (4)

115

V. Lebedeva Miss Kucharskaja Miss Belteneff Mrs. Joukovsky Miss Paschen

Math. seminar (2), with Prandtl. Runge, and Woldemar Voigt (1850–1919): Theory of elasticity

23

Miss Kucharskaja

Elementary mathematics from a higher standpoint: arithmetic, algebra, analysis (4)

75

Math. seminar (2), with Prandtl, Runge, and H. T. Simon (1870–1918): Electrotechnology

20

SS 1905

WS 1905/1906 Projective geometry (4)

Math. Seminar (2), with Hilbert and Hermann Minkowski (1864–1909): Lectures of Klein on linear differential equation and automorphic functions SS 1906

Theory of functions (4)

144

51

125

V. Lebedeva* T. Afanasyeva

Vera Lebedeva Mrs. Joukovsky Miss Haccins Miss Schaeffer Gertrud Lange (b. 1879)l V. Lebedeva Mrs. Joukovsky Miss Schaeffer

Miss Beltenewa Mrs. Joukowsky Miss Potylizyn Miss Schestokow Miss Schirok Miss Stehogolewa Miss Hahn Miss Lehmann Klara Löbenstein (continued)

1 Internationality: Women in Felix Klein’s Courses …

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Table 1.1 (continued) Semester

Lectures and seminars (Hours per week) Math. Seminar (2), with Hilbert and Minkowski: Differential equations

WS 1906/1907 Theory of functions (4)

Math. seminar (2), (Klein, Hilbert, Minkowski, and Herglotz): Linear differential equations and automorphic functions SS 1907

Curves and planes (differential geometry) (4)

Number of participants Female students (Male and female) 32

100

25

117

Math. seminar (2) (Klein, Hilbert, and Minkowski): Lectures of Klein on linear differential equations and automorphic functions

22

WS 1907/1908 Elementary mathematics from a higher standpoint: arithmetic, algebra, analysis (4)

79

Math. seminar (2), Klein, Carl Runge, Ludwig Prandtl (1875–1953), Emil Wiechert (1861–1928): Hydrodynamics

24

Elementary mathematics from a higher standpoint: Geometry (4)

79

Math. seminar (2), (Klein, Runge, Prandtl, Emil Wiechert): Hydrodynamics

15

SS 1908

Miss Beltenewa Mrs. Joukowsky A. H. Palmié Anna Johnson (Pell Wheeler) (1883–1966),m USA Miss Beltenewa Miss Kucharsky Miss Lehmann A. H. Palmié T. Ehrenfest

V. Lebedeva Olga Polossuchina Miss Gray Miss White Gertrud Lange Margarete Kahn Klara Löbenstein

Miss Gray Miss E. Meyer Margarete Kahn Klara Löbenstein

Margarete Kahn Klara Löbenstein

(continued)

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Table 1.1 (continued) Semester

Lectures and seminars (Hours per week)

WS 1908/1909 Mechanics of point systems (4)

SS 1909

Number of participants Female students (Male and female) 116

Math. seminar (2), (Klein, Runge, and Prandtl): Structural design and mathematics

14

Mechanics of continua

72

Math. seminar (2), (Klein, Runge, and Prandtl): Elasticity theory and the strength of materials

Gertrud Lange Miss I. Lehmann Iris Runge

8

WS 1909/1910 Projective geometry (4)

109

Math. seminar (4), conducted with Felix Bernstein (1878–1965) and Leonard Nelson (1882–1927): Mathematics and psychology

9

SS 1910

Margarete Kahn Klara Löbenstein Gertrud Lange Iris Runge (1888–1966)n

Elisabeth Klein Johanna Droop (b. 1885)o Anna Hoffa (b. 1876)p Miss Kochler Miss Landsberg Miss I. Weinmeister

A semester of leave

WS 1910/1911 Mathematical pedagogy (2)

67

Clara Dittmar (b. 1880)q Käthe Heinemann (b. 1889)r Miss M. Kellner Miss B. Meese Miss G. Merker Miss E. Petersen Miss I. Pohlmann Miss Thiele

Math. seminar (2): Mathematical pedagogy, especially at elementary schools

18

Käthe Heinemann* Miss B. Meese* Miss E. Petersen* (continued)

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Table 1.1 (continued) Semester

Lectures and seminars (Hours per week)

Number of participants Female students (Male and female)

SS 1911

Calculus I (4)

145

Math. seminar (2), (Klein and Felix Bernstein): Insurance mathematics WS 191/1912

Miss F. Berger Miss E. Bodenburg Miss M. Borchers Hildegard Ehrenberg (b. 1885)s Käthe Heinemann Johanna Hilmer (b.1889)t Christel Jasmund (b. 1888)u Miss K. Klußmann Miss E. Kreuse Berta Kuck (b.1890)v Miss W. Lingelbach Miss G. Merker Miss E. Neussel Miss B. Paulssen Clara Pietzsch (b. 1887)w Miss I. Pohlmann Miss R. Scharlau Irma Schiersand (b. 1883)x Miss W. Schönklinz Miss M. Stennes Miss O. Lobanoff

8

Calculus II (4), from the end of November, held by Wilhelm Behrens (1885–1917) and Hermann Weyl (1885–1955)

145

Math. seminar (2), from the end of November, held by Rudolf Schimmack (1881–1912): History of differential and integral calculus

31

14 women, including Elisabeth Klein Iris Runge Alma Willers (b. 1881)y

Iris Runge** Elisabeth Klein* Erna Bockmann* Käthe Heinemann* Margaret Elisabeth Brusstar* (continued)

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Table 1.1 (continued) Semester

Lectures and seminars (Hours per week)

SS 1912

Klein on medical leave Math. seminar (2), announced by Klein (took leave) and Rudolf Schimmack: IMUK-Literature

a See

Number of participants Female students (Male and female) Miss Xenia Kucharsky*z

the Protokolle of Felix Klein’s seminars. The titles of the presentations given by women in Klein’s seminars are published in Appendix 7 of Tobies (1991/1992), pp. 166–169. Those who gave one or two presentations in his seminars are indicated here by one or two asterisks (*) (**) b Ada Maria Jane Elizabeth Johnson was educated in Newnham College at University of Cambridge; see http://www-history.mcs.st-and.ac.uk/Davis/Indexes/xCambridge.html c From 1894 to 1898, L. J. Martin studied a variety of subjects in Göttingen, especially psychology d See Vogt (2014), p. 24 e A. von Stebnitzky was the daughter of the Polish general I. I. von Stebnitzky (1832–1897), a geodesist and a corresponding member of the Academy of Sciences in St. Petersburg. She became an astronomer in Pulkovo near St. Petersburg and later at the Fort Skala Observatory in Krakow. The famous physicist Piotr L. Kapitza (1894–1984) was her nephew. I am indebted to Danuta Ciesielska for this information f Wedell, who had studied with Klein’s former doctoral student Adolf Hurwitz (1859–1919) at the ETH Zurich, completed her doctorate at the University of Lausanne in Switzerland. The title of her thesis is “Applications de la théorie des fonctions elliptiques à la solution du problème de Malfatti” (1897). Many thanks to Nicola Oswald for this reference. See also https://en.wikipedia.org/wiki/ Charlotte_Wedell g Gates became a pioneering woman in the field of radioactivity h All female students were listed as “Frl.” (Miss) with the exception of Joukowsky, who appears as “Frau” (Mrs.). Whenever first names could be identified, I have used those instead of “Miss” in the table i She married the Austrian physicist Paul Ehrenfest (1880–1933) on February 21, 1904. Ehrenfest studied in Göttingen as well. Together, they wrote the famous contribution on statistical mechanics for the fourth volume (mechanics) of the great Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, which was managed by Klein j Stephansen was the first women in Norway to receive a Ph.D. in mathematics. From 1892 to 1896, she studied at the ETH Zurich, where she earned a diploma. Later, in 1902, Heinrich Burkhardt (1861–1914), who did his postdoctoral degree under Klein in Göttingen, arranged for Stephansen to be granted her doctorate in absentia (the title of her thesis was “Über partielle Differentialgleichungen vierter Ordnung die ein intermediäres Integral besitzen”). It was not until 1971 that the next Norwegian woman would complete a doctorate in mathematics; see Hag & Lindquist (1997) k Kucharsky (= Kucharskaya), who first appears on Klein’s list in 1904, gave her first presentation in the summer semester of 1912. It is in that entry where her first name is mentioned (Xenia). I have not been able to discover any other information about her life l Gertrud Lange completed her doctorate in physics with the thesis “Beiträge zur Kenntnis der Lichtbogenhysteresis,” which was supervised by Hermann Theodor Simon in Göttingen and published in Annalen der Physik 337 (1910), pp. 589–647

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m Wheeler, a daughter of Swedish immigrants, is known for her work on linear algebra. She became

the chair of the mathematics department at Bryn Mawr in 1925 and was instrumental in bringing Emmy Noether there in 1933. On the close contact between Noether and Wheeler, see Tobies (2003), p. 107 n See Tobies (2012b) o [BBF] Personalblatt. Droop studied for one semester in Göttingen, passed her examination (math, physics, philosophy) in May of 1912 in Bonn, earned a doctoral degree with a thesis in philosophy there in 1920, and became secondary school teacher in Prussia p [BBF] Personalblatt. Hoffa completed her teaching examination on February 17, 1911 in Göttingen (philosophy, French, German, and mathematics for the middle grades). In 1930, she became the principal of a girls’ school in Frankfurt am Main q [BBF] Personalbogen. Dittmar passed her teaching examination (geography, mathematics, philosophy) in Göttingen in November of 1912 and became teacher in Wernigerode r [BBF] Personalbogen. After passing her teaching examination (mathematics, physics, chemistry/mineralogy, botany/zoology), Heinemann earned a doctorate in botany (1922) and became a school principal s [BBF] Personalbogen. Born in Strasburg, Ehrenberg studied for five semesters in Göttingen, passed her teaching examination (English, mathematics) in November of 1913, and became a teacher in Berlin t [BBF] Personalbogen. Hilmer became a teacher in Uelzen (mathematics, botany/zoology, physics) u [BBF] Personalbogen. Jasmund completed her teaching examination (mathematics, physics, geography) in Göttingen in July of 1916 and later (as of 1919) worked as a secondary school teacher in Barmen v [BBF] Personalbogen. After studying in Göttingen, Kuck passed her examination in Münster (physics, mathematics, chemistry/mineralogy) in 1914 and became a secondary school teacher w [BBF] Personalbogen. Pietzsch studied in Göttingen from 1911 to 1918; her examination took place in February of 1924 (chemistry, physics, mathematics). She taught as a secondary school teacher in Spremberg x [BBF] Personalkarte. Schiersand finished her examination (mathematics, geography) in November of 1915 and became secondary school teacher at the Cecilienschule in Breslau y [BBF] Personalbogen. Willers passed her examination (mathematics, botany/zoology) with distinction in Göttingen in June of 1913 and became a secondary school teacher in Hildenheim, her hometown z Xenia Kucharsky gave a presentation on July 24, 1912 titled “Über die Organisation der Schulen in Russland” (see Protokolle 29, pp. 400–404)

German Mathematical Society (Deutsche Mathematiker-Vereinigung) in 1898, when Felix Klein was its president.16 Scott, who supervised doctoral theses at Bryn Mawr, arranged for some of her students to pursue postdoctoral studies at the University of Göttingen. On March 19, 1897, for example, she wrote the following to Klein: “I expect to send two of my best students to Göttingen next year. Both have been awarded a College Fellowship, and both are eager to study under your direction for a year, if this is agreeable to you.”17 Thus, in the fall of 1897, Emilie Norton Martin (1869–1936) and Virginia Ragsdale (1870–1945)18 arrived in Göttingen along with 16 Toepell

(1991), p. 354. It was not until its one hundredth anniversary that the DMV would elect a female mathematician to its council, in 1991: the algebraist Ina Kersten, a scholarly descendent of Emmy Noether. She became president from 1995 to 1997. 17 [UBG] Cod. Ms. Klein XI, p. 947. 18 Ragsdale contributed to Hilbert’s “sixteenth problem” (Ragsdale conjecture), as did Hilbert’s German doctoral students Margarete Kahn and Klara Löbenstein.

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other American students. Having benefitted from their time with Klein and Hilbert, both went on to complete their doctorates under Scott’s supervision at Bryn Mawr. In 1910, as though in exchange, Klein’s youngest daughter Elisabeth (1888–1868) spent a semester abroad at Bryn Mawr. It was at that welcoming university, too, where Emmy Noether would find refuge in 1933, and Hilda Geiringer (1893–1973) would do the same from 1939 to 1944.19 It should be stressed that David Hilbert (1862–1943), who became a full professor at the University of Göttingen in 1895 with Klein’s endorsement, was also a staunch supporter of women’s right to study. Together, Klein and Hilbert conducted seminars in which women took part (see Table 1.1). Hilbert would supervise 69 doctoral students in all, 6 of whom were women: Anne Lucy Bosworth (1868–1907) from the United States; Nadeschda Nikolaevna von Gernet (1877–1943), Ljubov Nikolaevna Zapolskaya (Sapolsky, Sapolski) (1871–1943),20 and Vera Lebedeva (1880–1970) from Russia; and Margarete Kahn (1880–1942) and Klara Löbenstein (*1883) from Germany.21 I should add that it was not in Germany alone that foreigners paved the way for native women interested in science and mathematics. The situation was similar in France.22 A well-known case is that of Maria Skłodowska–Curie (1867–1934), who earned a Lizenziat in physics (1893) and mathematics (1894) at the Sorbonne. However, the aforementioned Sofia Kovalevskaya had already become a member of the French Mathematical Society (La Société Mathématique de France) as early as 1882. Kovalevskaya’s Finnish student Ebba Louise Nanny Lagerborg (later Cedercreutz; 1866–1950) would also complete her Lizenziat at the Sorbonne and become a member of the French Mathematical Society (in 1890).23

1.1.1 The First Female Members of the German Mathematical Society Whereas the French Mathematical Society had been founded in 1874, the German Mathematical Society (Deutsche Mathematiker-Vereinigung = DMV) was not formed until in 1890. The first women to become members of the DMV were 19 On

Elisabeth Klein (married name: Staiger), see Tobies (2008). Regarding women mathematicians in Germany who had to emigrate because of the Nazi dictatorship, see Tobies (2011b); and Siegmund-Schultze (2009). 20 Regarding Zapolskaya’s biography, see Makeev (2011). She received a teaching position at the University of Moscow, published her results in books, and became the first Russian woman with a postdoctoral degree and a professorship in 1905. The professional path that led her there was winding, however. She had been the headmaster of a secondary school and a lecturer at institutions in Moscow, Ryazan, Saratov, and Yaroslavl. 21 See Tobies (1999); and König et al. (2014). 22 This is based on a lecture given by Catherine Goldstein at the University of Würzburg in October of 2015. 23 See https://fi.wikipedia.org/wiki/Nanny_Cedercreutz (accessed October 30, 2016).

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foreigners. The aforementioned Charlotte Angas Scott (1898) was followed by Hilbert’s doctoral student Nadeschda von Gernet in 1901. In 1904, the American Helen Abbot Merrill (1864–1949) became a member.24 She had studied at the University of Göttingen from 1901 to 1902 and earned a doctoral degree from Yale two years later.25 Elizabeth Buchanan Cowley (1874–1945) became the next American female member in 1908.26 In that year, she had received her Ph.D. from Columbia and embarked upon further studies at the Universities of Göttingen and Munich. In 1907, she and Ida Whiteside (*1883) published an article together in the journal Astronomische Nachrichten, for which they were awarded a prize from the German Astronomical Society.27 The first Italian joined the DMV in 1905: Laura Pisati (1869/70–1908). She had earned her doctoral degree in Rome in 1903, taught at the Technical School “Marianna Dionigi,” and was also—as of February 26, 1905—a member of the Circolo Matematico di Palermo.28 Her article “Sulla estensione del metodo di Laplace alle equazioni differenziali lineari di ordine qualunque con due variabili indipendenti” was long a fixture in scholarly bibliographies.29 On account of her tragic premature death, Pisati was unable to deliver her lecture “Essay on a Synthetic Theory for Complex Variable Functions” at the Fourth International Congress of Mathematicians in Rome, where she would have been the first woman to have spoken at this event.30 Although Emmy Noether accompanied her father to this conference, she did not give a talk there. Having just finished her doctorate under Paul Gordan (1837–1912) at the University of Erlangen, she would deliver her first lecture at the conference held by the DMV in Salzburg one year later.31 In that same year, 1909, she became the first German woman to be granted membership to the DMV. In subsequent years, a number of other female German mathematicians earned a doctoral degree and became members of the DMV.32 Additional non-German women joined as well. Here, I would like to single out the aforementioned Austrian Hilda Geiringer, who came to Berlin after completing her doctorate in 1917 at the University 24 Toepell

(1991), p. 254. (2003), p. 93. 26 Toepell (1991), p. 75. 27 See https://www.agnesscott.edu/lriddle/women/cowley.htm (accessed August 20, 2016). 28 See Toepell (1991), p. 291; and Jones (2009), p. 91. 29 Laura Pisati, “Sulla estensione del metodo di Laplace alle equazioni differenziali lineari di ordine qualunque con due variabili indipendenti,” Rendiconti del Circulo Matematico di Palermo 20 (1905), pp. 344–374. See further Ganzha et al. (2008). 30 A memorial for Pisati was held during Section I of the Congress in Rome; see Curbera (2009), p. 44. At the first International Congress of Mathematicians (ICM, Zurich, in 1897), four female mathematicians took part: Iginia Massarini (Rome), Vera von Schiff (St. Petersburg), Charlotte Angas Scott (Bryn Mawr), and Charlotte Wedell (Göttingen); see Eminger (2015), p. 70. On Massarini, see Carbone and Talamo (2010). The first woman to give a talk at an ICM was H. P. Hudson, who spoke at Cambridge in 1912 (see Eminger (2015), p. 70). 31 Emmy Noether, “Zur Invariantentheorie der Formen von n Variablen,” Jahresbericht der Deutschen Mathematiker-Vereinigung 19 (1910), pp. 147–154. 32 See Tobies (2006). 25 Singer

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of Vienna under the supervision of Wilhelm Wirtinger (1865–1945). She, too, was in contact with Felix Klein, and she was made a member of the DMV in 1921.33 Olga Taussky (1906–1995), another Austrian who completed her doctorate with Klein’s former student Philipp Furtwängler (1869–1940) in Vienna, became a member in 1930 when she was called upon in Göttingen to edit certain chapters (on number theory) of Hilbert’s collected works. When the Nazis came to power, she, too, was forced to emigrate. Finally, the British mathematician Dorothy Wrinch (1894–1976), a biochemist who applied mathematical principles, joined the DMV in 1933.34

1.1.2 Felix Klein and the First Women to Study at the University of Göttingen Beginning with the winter semester of 1893, 3 years after the foundation of the DMV, Felix Klein made it possible for the first women to enroll at the University of Göttingen. As mentioned above, these women were at first exclusively foreigners. In order to understand this situation, it is necessary to examine it in closer detail. The Humboldtian university reform had provided a decisive impetus for mathematical research in Germany, one that attracted an increasing number of foreign students during the last third of the nineteenth century. Both women and men wished to study at the center of scholarly activity. As early as Klein’s time in Munich (1875– 1880), young men came to study with him from Italy, Norway, and elsewhere. With his move to Leipzig in 1880, French and American students came to learn from him, and when, in 1886, he became a professor in Göttingen, students from Russia came to him as well. Between 1886 and 1895, approximately ten Americans earned a doctoral degree under Klein’s supervision. The significance of these numbers becomes clear when we learn that, throughout the 1880s and 1890s in Leipzig and Göttingen, more Americans studied mathematics under Klein and his successor at Leipzig (the Norwegian mathematician Sophus Lie) than under any professor of mathematics in the United States. It goes without saying that the subsequent development of mathematics in that country was emphatically influenced by this contact.35 Sandra L. Singer has written a detailed book about North American women at German-speaking universities during the period of 1868–1915.36 Before her, in a pioneering study, Margaret Rossiter underscored Felix Klein’s special role in

33 While writing a popular science book on mathematics, which makes use of Klein’s educational reform ideas and his conceptual coupling of precision and approximation mathematics, Geiringer wrote two letters to Klein, dated November 7, 1921 and December 3, 1921 (see [UBG] Cod. MS Felix Klein 9, pp. 307–308). Klein had read the manuscript and sent comments to her (see Geiringer (1922), pp. 93–95). 34 See Toepell (1991), pp. 120, 381, 424; and Senechal (2013). 35 See Parshall and Rowe (1994). 36 Singer (2003).

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promoting American women mathematicians.37 Singer made good use of Rositter’s findings and drew upon additional archival sources.38 An outline of the achievements of the first women in the American mathematical community can be found in the work of Della D. Fenster and Karen H. Parshall.39 David Rowe has underscored the role of Felix Klein’s wife Anna (a granddaughter of the famous philosopher Friedrich Wilhelm Hegel), who integrated the first foreign students into her family and helped them to find their bearings in Göttingen.40 In my own work, which is likewise based on archival material, I have been able to document the earliest stages of women studying mathematics in Göttingen.41 There I have shown, for instance, how Klein managed to overcome the conservative attitude of the legal scholar and university curator Ernst von Meier (1832–1911), who in 1891 had prevented Christine LaddFranklin and Ruth Gentry (1862–1917) from participating in Klein’s courses.42 The massive amount of resistance faced by Klein and the perseverance required of him to change anything are indicated in a letter to him from Meier, which reads: “This is worse than social democracy, which only wants to do away with differences in property. They want to abolish the difference between the sexes.”43 While women in other countries were already able to take qualifying examinations and even to study at university, the ministerial decrees allowing women to matriculate in German states were first issued between 1900 and 1909.44 However, because foreign women, like men, wanted to be qualified to study where the highest standards of scholarship could be expected, for some time they attempted to gain access to German universities even while official status as students could not yet be granted to them. Up until the beginning of the 1890s, Berlin was regarded as the center of mathematics in Germany, with professors there such as Karl Weierstraß (1815–1897), Ernst Eduard Kummer (1810–1893), and Leopold Kronecker (1823–1891). Around that time in Göttingen, Felix Klein developed an international center for mathematics, which was further enhanced by Hilbert’s arrival in 1895. In order to accomplish his goals, Klein sidestepped the prescribed order of command (evading the conservative university curator) and communicated directly with the influential official Friedrich Althoff (1839–1908) at the Prussian Ministry of Culture. On May 20, 1892, the farsighted Althoff issued a new brief with the title “The Request of Persons of the Female Gender to Matriculate and Attend Lectures at the Royal State Universities.” 37 Rossiter

(1982), pp. 40–41. (2003), pp. 86–97. 39 Fenster and Parshall (1994). 40 See Rowe (1992), Chap. 5.7; and Parshall and Rowe (1994), pp. 239–253. For an interpretation of the roles played by the wives of mathematicians, see also Jones (2009), p. 37. 41 Tobies (1991/1992). 42 [UBG] Cod. MS F. Klein 9, pp. 310–311 (Gentry’s letters to Klein, dated July 21 and 31, 1891). 43 The original German reads as follows: “Das ist schlimmer als die Sozialdemokratie, die nur den Unterschied des Besitzes abschaffen will. Sie wollen den Unterschied der Geschlechter abschaffen” ([UBG] Cod. MS F. Klein, 22L, p. 7; also quoted in Siegmund-Schultze (1997), p. 31; and in English in Rowe (1992)). 44 See Costas (2002); and Tobies (1997). 38 Singer

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This document begins with excerpts from newspapers about the ability of women to study in foreign countries.45 Developments abroad, in other words, influenced the decisions made at the Prussian Ministry of Culture. Heinrich Maschke (1853–1908), who had studied under Klein in Göttingen and was a professor of mathematics at the newly established University of Chicago, wrote the following to his former teacher on April 8, 1893: One of our students of mathematics, Miss Mary F. Winston, is applying for a scholarship, on the basis of which she intends to go to Germany next year. She has […] talent, thinks independently, and is certainly above average. […] Bolza46 and I have encouraged her […] to go to Göttingen and have just as forcefully discouraged her from going to Berlin in order to keep her away from the stiff atmosphere there. Now the question remains whether female doctoral or post-doctoral students may enroll at Göttingen or whether, if that is not the case, you think you might exert your influence to make an exception.

On July 6, 1893, before he would first travel to the United States (for the world’s fair and a mathematics conference), Klein received the following promising message from Berlin: With respect to women studying, I can confidentially say that, as I know from Mr. Althoff, the Ministry will not hinder the matter, although it will not especially encourage such questions. Regarding their [women’s] participation in lectures, this custom will also become more entrenched than limited. And if American women will come to study in Germany, they will not have difficulties here. Mr. Althoff is of the opinion that, without asking, you could arrange that your numerous female American admirers can arrive.47

Klein’s attitude was reinforced in the United States, where he not only confirmed Mary F. Winston’s outstanding talent as a mathematician but also witnessed women in positions that a Swiss delegate to the world’s fair described as follows: The Americans find nothing unusual in the fact that, for instance, a woman is the director of a national bank, as in Texas, or that women have found a place on the supervisory committees of universities or in the national department of education, and this is not to mention professorial positions, of which there are many for women […]. Not only have universities been made available to women but also preparatory secondary education, be it in connection with schools for boys or in parallel institutions, as in Boston […]. America knows no difference in the practice of scientific careers between men and women […]. At the University of Chicago, there are six female professors.48

Having returned from Chicago to Germany, Klein proposed to the Ministry of Culture in Berlin that Winston, Grace Chisholm,49 and the American Margaret Eliza 45 See

Tobies (1991/1992), p. 151. Bolza (1857–1942), who had earned his doctorate in Göttingen in 1886, also became a professor at the University of Chicago. On the developments in the United States, see especially Parshall and Rowe (1994). 47 Tobies (1991/1992), p. 154. 48 Boos-Jegher (1894), pp. 8, 16. 49 Grace Chisholm had studied at Girton College at the University of Cambridge, where women could not earn a doctoral degree. One of her professors of mathematics, Andrew Forsyth (1865– 1942), recommended that she go to Felix Klein, with whom he was well acquainted (see Mühlhausen (1995), p. 197). 46 Oskar

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Maltby (1860–1944) be allowed to enroll in university. Despite another negative vote by the university curator, the ministry approved the application of these women within 6 days. The curator resigned from his position in a huff, and his successor was welcoming to women. All three women participated officially in lectures and seminars and earned doctoral degrees with distinction, Maltby with a dissertation in the field physical chemistry directed by Walther Nernst (1864–1941). The female students were not officially matriculated but rather possessed the status of auditors (every professor had to be asked individually for permission, which ultimately had to be granted on an individual basis by the ministry). In the meantime, additional female students had arrived at Göttingen. Klein assisted them personally to receive permission to attend courses at the university. In autumn of 1894, he wrote the following to the Prussian Minister of Education Robert Bosse (1832–1901): Your Excellence! In addition to the two women, Miss Chrisholm and Miss Winstaon, who for the last year have been studying mathematical subjects at the local university and whose diligence and abilities I have repeatedly commended, there are now two new applicants, Miss MacKinnon and Miss Maddison,50 who are likewise requesting permission from the appropriate instructors to participate, as of the next semester, in lectures on mathematics, physics, and astronomy. I have examined the qualifications of both women and am thus able to support their applications in every respect.51

On November 1, 1895, the mathematician Arthur Schönflies, an associate professor of descriptive geometry at Göttingen, wrote the following remarks: “We now have nine women studying mathematics, and yesterday they formed a club; they will meet once a week for coffee.”52 These meetings can be interpreted as the formation of the first women’s network of mathematicians. In the same year, because of a growing demand for access, the Prussian ministry determined that universities only to provide it with a list of the female participants in the courses (as auditors). When, in 1896–97, Klein was asked about his position concerning women studying and pursuing scientific careers, he offered the following response: I am far more pleased to answer this question than that concerning the opinion still prevailing in Germany, which is that the study of mathematics must be as good as inaccessible to women, that there should be an essential blockade to any efforts directed toward the development of women’s higher education. In this regard, I am not referring to extraordinary cases, which as such would not prove very much, but rather to our average experiences in Göttingen. Though this is not the place to get into the matter, I would simply like to point out that in this semester, for instance, no fewer than six women have participated in our higher mathematics courses and practica and, having advanced through them, have proven themselves to be equal to 50 Ada Isabel Maddison (1869–1950) was a British woman who, like Scott and Chisholm, studied with Arthur Cayley in Cambridge. After graduating from Girton College in 1892, she attended Bryn Mawr College in the United States, where she won a fellowship for studying abroad. After returning to Bryn Mawr, Maddison completed her Ph.D. and translated Klein’s article “The Arithmetizing of Mathematics,” Bulletin of the American Mathematical Society 2 (1896), pp. 241–249. 51 [UBG] Cod. Ms. Klein I C2, pp. 95–96. 52 Quoted from Tobies (1991/1992), p. 157.

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R. Tobies their male classmates in every respect. The nature of the situation is that, for the time being, these women have been exclusively foreigners: two Americans, an Englishwoman, and three Russians. No one would wish to assert, however, that these foreign nations possess some inherent and specific talent that evades us, and thus that, with suitable preparation, our German women should not be able to accomplish the same thing.53

Klein’s conclusion was that the infrastructure for educating girls should be enhanced in Germany.

1.2 Female Participants in Klein’s Courses Klein kept an account of the students who attended his courses. This record extends from his time as a university lecturer (the summer semester of 1871) until the year 1920, even though he had already become a professor emeritus in 1913.54 The first women appear on his list for the winter semester of 1893/94: Grace E. Chisholm and Mary F. Winston. They attended Klein’s lecture on hypergeometric series and also, though only in their first semester, gave presentations in his seminar. His first women students from the German-speaking region were Frieda Hansmann (*1873 in Northeim, Prussia) in the summer semester of 1895—who would later earn a doctoral degree in chemistry from the University of Bern—and Elsa Neumann (1872–1902), who attended Klein’s lecture on technical mechanics in the summer semester of 1896 and would become, in 1899, the first woman ever to be awarded a doctoral degree from the University of Berlin (her field was physics).55 The presence of women in university classrooms became an increasingly normal occurrence in Göttingen, although it was not until 1908 that they could attend as more than mere auditors. Together with new laws enacted in Prussia in 1908, which allowed the enrollment of women, and the establishment of new kinds of girls’ secondary schools, which could culminate in the Abitur, this led to even more women entering higher education. It is also apparent that an increasing number of German women began to enroll in Klein’s courses (see Table 1.1). At that time, the preferred career goal of both female and male students of mathematics was to become a secondary school teacher (of mathematics and other subjects).56 Table 1.1 does not contain all of the women who studied mathematics at the University of Göttingen at the time. A few women attended only Hilbert’s lectures. For instance, we know from the biography of Clara Eliza Smith (1865–1943), who completed her doctorate at Yale in 1904 with the thesis “Representation of an Arbitrary Function by Means of Bessel’s Functions,” that she conducted postdoctoral research in Göttingen from 1910 to 1911 (while she was a faculty member at

53 Quoted

from Kirchhoff (1897), p. 241. Cod. MS F. Klein VII E. 55 See Vogt (1999). 56 See Abele et al. (2004); and Tobies (2011a). 54 [UBG]

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Wellesley College). Otherwise, all of Hilbert’s female doctoral students mentioned above attended Klein’s courses as well. Although Klein, in 1913, was compelled to retire early for the sake of his health, he continued to hold lectures and seminars until the summer semester of 1920. These were mainly devoted to the history of mathematics during the nineteenth century and to questions concerning the theory of relativity. Women known to have attended some of these lectures include Klein’s widowed daughter Elisabeth Staiger, her friend Iris Runge,57 the aforementioned students Erna Bockmann and Käthe Heinemann, M. Jona, Antonie Stern (*1892),58 Helene Stähelin (1891–1970) from Switzerland,59 and others already in possession of a doctoral degree, such as Emmy Noether and the Austrian physicist Gerda Laski (1893–1928).

1.3 Noteworthy Trends It is not often that such thorough records exist about those who attended mathematics courses. On the basis of these lists and other sources, it is possible to draw the following conclusions about the early stages of women studying mathematics in Germany. (1) The beginnings of women studying mathematics in Germany may be associated with Göttingen, but an international perspective is needed to evaluate the situation. Klein developed an international network of connections that, among other things, brought women to study under his direction. His network was based on his scientific desires to acquaint himself with as many mathematical schools as possible, to maintain the highest international standards for his journal Mathematische Annalen, and to commission the foremost international experts to contribute to his monumental encyclopedia of mathematics, the six-volume Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. He collaborated as a peer reviewer for several journals of mathematics and he organized international exchanges of scholarship and bibliographical material. Klein himself took numerous research trips abroad and was a member of the most illustrious scientific

57 Iris Runge took part during the winter semester 1914–15 while she was already a secondary school teacher in Göttingen. This is before she would move to Bremen and elsewhere, and before she would become an industrial mathematician in 1923 (see Tobies (2011b)). 58 A. Stern passed her teaching examination in 1918 (mathematics, physics, chemistry) and completed her doctorate under the supervision of Richard Courant in 1925. Here thesis is entitled “Bemerkungen über asymptodisches Verhalten von Eigenwerten und Eigenfunktionen” ([UAG] Math.-nat. Fak. Prom. S, Vol. I 1922–25, Nr. 35). 59 Helene Stähelin completed her doctoral thesis—“Die charakteristischen Zahlen analytischer Kurven auf dem Kegel zweiter Ordnung und ihrer Studyschen Bildkurven”—at the university of Basel in 1924.

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academies and societies in multiple countries.60 He served three times as the president of the German Mathematical Society (for three separate yearly terms). When L’Enseignement mathématique was founded in 1899 as the first international journal devoted to mathematical instruction, he was made a member of its editorial board. When, in 1908, the first international committee for mathematical instruction was created at the Fourth International Congress of Mathematicians in Rome, Klein was elected its president (1908–1920), even though he was unable to be present at the meeting. (2) It was because of his activity on many committees and his many contacts that Klein, for the winter semester of 1893/94, was able to succeed in allowing the first women to attend his courses. Just three years later, he placed the mathematical ability of women on the same level as that of “their male classmates.” Klein endorsed the membership of women mathematicians in the German Mathematical Society, and he accepted submissions by women for publication in Mathematische Annalen. The first female contributors to this journal were Winston (1895), Scott (1899), Lebedeva (1907, 1909, 1911), Emmy Noether (1915, 1916 [4x], 1917, 1920 [2x], 1921, 1922, 1923, etc.), Tatyana T. Ehrenfest-Afanasyeva (1916), followed by Margarete (Grete) Hermann (1901–1984) in 1926,61 one of Emmy Noether’s doctoral students. Klein supported the initial efforts of women to study in Göttingen with personal letters to the ministry and to the university curator. From 1915 onward, he did much to promote Emmy Noether’s career,62 and he brought aboard Ehrenfest-Afanasyeva as a contributor to his Encyklopädie. (3) Foreign women paved the way for German women. The first female students came from the United States, England, and Russia. They ensured that professors of mathematics could set aside any doubt regarding the intelligence of women, and thus they played a part in the decision of German legislators to allow women to study in an official capacity. In other countries, secondary education was so developed that women could acquire all of the prerequisites required for university study. They chose to work with Klein in Göttingen because they were encouraged to do so by mathematicians who had been his students, had published in his journal, or had participated in one of his other projects. Klein had raised his profile in the United States by delivering lectures there during two research trips (1893, 1896). Approximately 12 female North Americans studied under him in Göttingen,63 one of whom (Winston) earned her doctorate under his supervision, and another (Bosworth) under Hilbert’s. American women went on to have careers primarily at women’s colleges. A number of them continued to produce valuable research as professors and were themselves able to inspire young women to pursue mathematics as a field of study. 60 See [UBG] Cod. MS F. Klein 114 (a collection of Klein’s nominations of potential members of academies and scientific societies). 61 See Herrmann (2019). 62 See especially Tollmien (1990). 63 The numbers are inexact because not all of the names could be identified.

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In Russia, advanced courses for women—established, for example, in 1876 in Kazan and in 1878 in St. Petersburg (the so-called Bestuzhev Courses)—provided a solid education (see Borisovna 2003). These courses, too, were taught by mathematicians who had close contact to Klein, among whom D. F. Selivanov (1855–1932) deserves special mention. As of 1895, 16 well-educated women came to study at the University of Göttingen. Most of them had graduated with a diploma from the Petersburg College for Women, including Helene von Bortkiewicz and her friend Alexandrine von Stebnitzky, who were born to Polish families of officers. Helene von Bortkiewicz was the sister of the statistician and later professor of economics Ladislaus von Bortkiewicz (1868–1931), who had completed his doctoral degree in Göttingen under the direction of Wilhelm Lexis (1837–1914) and who, by 1895, was already working as a lecturer in Straßburg. Born in Simbirsk (now Ulyanovsk), Nadeschda von Gernet, who earned her doctorate under Hilbert in 1901 with a dissertation on the calculus of variations,64 went on to become a lecturer at the aforementioned women’s college in St. Petersburg, from which she in turn sent students of her own to Göttingen. An active researcher, she published a book in 1913 on variational calculus and was a member of the German Mathematical Society from 1901 to 1938. After earning her degree, she returned often to Göttingen before the outbreak of the First World War. Before coming to Göttingen in 1903, Vera Lebedeva had also studied at the women’s college in St. Petersburg. Working under Hilbert on the latest field theory of integral equations, she defended her thesis in 1906, and her future husband, the Romanian Alexandru Myller (1879–1965) completed his own in the same year. Both became professors at the University of Iasi in Romania—he in 1910, she in 1918— and they created an influential school of mathematical thought. With her appointment, in fact, she became the second female professor of mathematics in all of Europe.65 Tatyana Afanasyeva, too, had studied at the women’s college in St. Petersburg (mathematics and physics). She arrived in Göttingen in 1902 and also met her husband there. (4) Göttingen became a role model for managing mathematical collaborations, and women were not excluded from this. The couples Chisholm-Young, LebedevaMyller, and Afanasyeva-Ehrenfest are examples of scientific couples in which the wife was able to carry on with research after marriage. The topic of academic couples has been discussed at length by Annette Lykknes and her collaborators.66 One of the earliest successful examples is the cooperation between Marie and Pierre Curie (1859–1906) in Paris, where Marie Skłodowska (1867–1934) had passed her examinations in physics (1893) and mathematics (1894). Marie Curie became a full professor after the death of her husband and went on to create a famous circle of scientists and thinkers. Although Marie Curie had been awarded two Nobel Prizes, she was never made a member of the Academy of Sciences in Paris. 64 The

evaluations of her dissertation are printed in Tobies (1999). Tobies (2004c). 66 Lykknes et al. (2012). 65 See

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Regarding the successful couple of Lebedeva and Myller, who both received professorships in mathematics, it should be noted that each of them continued to publish steadily in German and French journals. It should also be pointed out, however, that Lebedeva’s Göttingen dissertation received a higher grade than that of her husband, yet he was nevertheless offered a professorship 8 years before she was. That said, it is still remarkable that they were able to hold professorships simultaneously at the same university. This was possible in Romania at the time, but not elsewhere. Grace Chisholm Young and William Henry Young admittedly worked together, but she had to work for him and then had to resign from her position to care for their family. Nevertheless, she was pleased to be integrated into the community of mathematicians in Göttingen. In contrast, Saly Ruth Ramler (1894–1993), who was the first Czech woman to hold a doctoral degree in mathematics (earned in Prague in 1919), was quite displeased to be reduced to a housewife and mother. This change in her life took place when she followed her husband, the Dutch mathematician Dirk Struik, to the United States.67 Tatyana Afanasyeva and Paul Ehrenfest, who had four children as a couple, continued to work and publish together in several places. She was an active teacher of mathematics in Russia and later on in the Netherlands (after the tragic suicide of her husband). The aforementioned Olga Taussky and her Irish husband John Todd (1911–2007) were able to continue their creative mathematical research in the United States. The couple had no children; Olga was 5 years older, and both ultimately received permanent positions at the Californian Institute of Technology in Pasadena (near Los Angeles).68 (5) It is important to stress that Klein was excellent at recognizing the talents of every person in his sphere—women included—and at guiding them toward their own creative achievements. This was the case not only with his female doctoral students but also with other women mathematicians. Annie L. MacKinnon, for example, who during her time in Göttingen (1894–1895) had given five presentations in Klein’s seminars, went on to teach mathematics at Wells College in the United States (her courses included spatial geometry, analytic geometry, and differential and integral equations). Encouraged by Klein, she continued to conduct further research; in a letter to him dated January 2, 1897, for instance, she remarked: “As promised, I am writing to you now during the Christmas vacation about the work on number theory that I told you about last summer. I find that I have both the time and desire to undertake such a study and, following your suggestion, I would like to work on it for a year in order to see what I can do with it.”69 Grace Chisholm Young—who moved to Göttingen as a married woman, became a member of the Göttingen Mathematical Society, and later gave a lecture in one of Klein’s seminars (see Table 1.1)—was inspired by Klein to complete a number of works. These include the first English-language book on set theory (written with 67 On

developments in the Czech Republic, see Martina Beˇcváˇrová’s Chap. I.3. Binder (1998). 69 [UBG] Cod. MS. F. Klein 10, p. 905. MacKinnon gave a presentation, entitled “Die Smith’-sche Curve,” in Klein’s seminar during the summer semester of 1896. 68 See

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her husband) and an elementary book on geometry that Klein discussed in his courses, arranged to have translated into German, and recommended in his lectures on elementary mathematics (Vorlesungen über Elementarmathematik vom höheren Standpunkte, vol. 2).70 During the First World War, Klein presented lectures on the history of mathematics in the nineteenth century. His lectures from the winter semester of 1914–15 and the summer semester of 1915 were recorded by his daughter Elisabeth Staiger, and those from the winter semester of 1915–16 were taken down by Käthe Heinemann and Käthe Stähelin for later publication.71 The editors of Klein’s posthumously published Vorlesungen über die Mathematik im 19. Jahrhundert (Berlin: Springer, 1926–1927), however, made no mention of the women’s contributions. That is a surprise, given that Klein himself always emphasized the work of his associates. Stimulated by his correspondence with Albert Einstein (1879–1955), Klein initiated a new lecture course on the theory of general relativity in October of 1918, and it was attended by Emmy Noether, Gerda Laski, and other women. Emmy Noether assisted both Klein and David Hilbert with their research on Einstein’s theory. Klein recommended Emmy Noether’s most significant paper in this field (on invariant variation problems) for publication in the Nachrichten of the Academy of Science in Göttingen, and he repeatedly stressed the value of her contribution and results in the edition of his Collected Papers.72 (6) Another important issue was that of allowing mathematically educated women to work in appropriate jobs. Felix Klein had high regard for Thekla Freytag (1877– 1932), who was the first woman in Prussia to fight for the right to pass the examination for secondary school teachers (for mathematics and scientific subjects), and he wrote about all of the obstacles that she had to overcome to do so in 1905.73 Within the framework of the educational reform movement, Klein voiced his opinion on numerous committees, in numerous publications and lectures, and in speeches held as a deputy in the first chamber of the Prussian House of Representatives.74 Allowed to study as an officially matriculated student, his eldest daughter Elisabeth, mentioned above, reaped the benefits of his efforts. Because she became a widow in 1914, she worked as a secondary school teacher of mathematics, physics, and English, and she ultimately became the principal of a school for girls (though she was demoted when the Nazis came to power in 1933). With the right of women to matriculate and with the new opportunity of becoming secondary school teachers (and even a principal at a secondary school for girls), the number of female German students of mathematics began to increase. The names of many women who had studied under Klein appear in the Prussian records of female teachers, mentioned above. Because it was long obligatory for female civil servants

70 See

http://www.tollmien.com/pdf/chisholm.pdf. also Rowe (1992), p. 492. 72 Klein (1921), pp. 559–560, 565, 584–585. See also Rowe (2019). 73 See Tobies (2017); and Lorey (1909). 74 See Tobies (1989). 71 See

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to remain “celibate,” these teachers as a rule remained unmarried or had to leave their positions if they did marry.75 With the reform of mathematical and scientific education, which even during his lifetime was known as “Klein’s Reform,” the number of female students increased in general, for many new career options were made available to women. His new teaching program in applied mathematics also enabled women to become, for instance, an actuary at an insurance company or an industrial mathematician.76 Klein personally supported Emmy Noether’s Habilitation, a prerequisite for a professorship at German universities. In a letter to the ministry of education dated January 5, 1919, he wrote that Noether was then the most productive mathematician at the University of Göttingen.77 In 1919, Noether became the first woman mathematician to achieve this rank. Thus it is clear that Klein, along with other mathematicians in Göttingen, created conditions that would allow women to attain faculty positions at universities. Klein also emerges as a role model for promoting women if we look at what some of his former students accomplished on this front. A number of his students became the first mathematicians at their respective institutions to supervise female doctoral students. Examples include Adolf Hurwitz and Heinrich Burkhardt in Zurich, Wilhelm Wirtinger and Philipp Furtwängler (1869–1940) in Vienna, Georg Pick (1859–1942) in Prague, Virgil Snyder (1869–1950) at Cornell University in the United States, and Max Winkelmann (1879–1946) at the University of Jena.78 (7) Over time, the number of foreign women who earned a doctoral degree in mathematics at German universities fell as the number of German women to do so rose. Up to 1906, seven foreign women (four Russians, two Americans, and an Englishwoman) had defended a mathematical dissertation in Germany (all in Göttingen). From 1907 to 1945, only three foreign women did the same, two from Great Britain (at the Universities of Marburg and Göttingen) and one from Denmark (at the University of Freiburg).79 The cause of this regression was above all the First World War. Afterward, many nations, most notably the United States and Russia, established new research centers for mathematics, so that women were then more likely to pursue doctoral research in their home country. Nevertheless, Göttingen remained an important international center for research up until 1933, to which point it continued to attract both male and female mathematicians from abroad. Translated by Valentine A. Pakis

75 This law applied to women in all official positions; see Deutscher Juristinnenbund (1984), pp. 76–

77. 76 See

Tobies and Vogt (2014), and Tobies (2012b). letter was published for the first time in Tobies (1991/1992), p. 172. 78 On George Pick, see Martina Beˇ cváˇrová’s Chap. I.3. Regarding Winkelmann, see Tobies and Vogt (2014), p. 195; and Bischof (2014). 79 See Tobies (2006). 77 The

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Bibliography Archival Resources [BBF] Bibliothek für bildungsgeschichtliche Forschung, Berlin, Personal Records of Prussian teachers. [UAG] Universitätsarchiv Göttingen, Philos. Fak., Promotionsakten. [UBG] Handschriftenabteilung der Niedersächsischen Staats- und Universitätsbibliothek Göttingen, Cod. MS. Felix Klein. Protokolle. Protocols of Felix Klein’s mathematics seminars. Library of the Mathematical Institute of the University of Göttingen. Online: http://www.uni-math.gwdg.de/aufzeichnungen/klein-scans/ klein/.

Secondary Literature Abele, A., Neunzert, H., & Tobies, R. (2004). Traumjob Mathematik: Berufswege von Frauen und Männern in der Mathematik. Basel: Birkhäuser. Binder, C. (1998). Von Olmütz nach Pasadena: Die Zahlentheoretikerin Olga Taussky-Todd. In: R. Tobies (Ed.), “Aller Männerkultur zum Trotz”: Frauen in Mathematik, Naturwissenschaften und Technik. Frankfurt am Main: Campus. Birn, M. (2015). Die Anfänge des Frauenstudiums in Deutschland: Das Streben nach Gleichberechtigung von 1869–1918, dargestellt anhand politischer, statistischer und biographischer Zeugnisse. Heidelberg: Universitätsverlag Winter. Bischof, T. (2014). Angewandte Mathematik und Frauenstudium in Thüringen: Eingebettet in die mathematisch-naturwissenschaftliche Unterrichtsreform seit 1900 am Beispiel Dorothea Starke. Jena: Geramond. Boos-Jegher, E. (Ed.). (1894). Die Thätigkeit der Frau in Amerika. Weltausstellung in Chicago 1893: Berichte der schweizerischen Delegierten. Bern: Michel & Büchler. Borisovna, V. O. (2003). Duhovnoe prostranstvo Universiteta: Vysxie enskie (Bestuevskie) kursy (1878–1918 gg.): Issledovani i materialy. (Monografi). St. Petersburg. Carbone, L., & Talamo, M. (2010). Gi albori della presanza femminile nello studio della matematica presso l’Università di Napoli nell’Italia unificata. Rendiconto dell’ Accademia delle Scienze Fisiche e Matematiche Napoli, 77, 15–43. Coen, S. (Ed.). (2012). Mathematicians in Bologna 1861–1960. Basel: Birkhäuser. Costas, I. (2002). Women in science in Germany. Science in Context, 15, 557–576. Curbera, G. P. (2009). Mathematicians of the World, Unite! The International Congress of Mathematicians—A Human Endeavor. Wellesley, MA: CRC Press. Deutscher Juristinnenbund. (1984). Juristinnen in Deutschland: Eine Dokumentation (1900–1984). Munich. Eminger, S. U. (2015). Carl Friedrich Geiser and Ferdinand Rudio: The Men Behind the First International Congress of Mathematicians. Doctoral thesis, University of St Andrews (Scotland). Fenster, D. D., & Hunger Parshall, K. (1994). Women in the American Mathematical Research Community. In: E. Knobloch & D. E. Rowe (Eds.). The history of modern mathematics (Vol. 3, pp. 229–261). Boston: Academic Press. Ganzha, E. I., Loginov, V. M., & Tsarev, S. P. (2008). Exact solutions of hyperbolic systems of kinetic equations: Application to Verhulst model with random perturbation. Mathematics in Computer Science, 1(3), 459–472. Geiringer, H. (1922). Die Gedankenwelt der Mathematik. Berlin: Verlag der Arbeitsgemeinschaft.

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Green, J., & LaDuke, J. (2009). Pioneering women in american mathematics: The pre-1940 PhD’s. History of Mathematics 34. Providence: American Mathematical Society and London Mathematical Society. Grinstein, L. S., & Campbell, P. J. (Eds.). (1987). Women of mathematics: A bio-bibliographical sourcebook. Westport: Greenwood Press. Härdle, W. K., & Vogt, A. B. (2015). Ladislaus von Bortkiewicz: Statistician, Economist and a European Intellectual. International Statistical Review, 83, 17–35. Hag, K., & Lindquist, P. (1997). Elizabeth Stephansen: A pioneer. Det Kongelige Norske Videnskabers Selskab: Skrifter, 2, 1–23. Herrmann, K. (Ed.). (2019). Grete Henry-Hermann: Philosophie - Mathematik - Quantenmechanik. Texte zur Naturphilosophie und Erkenntnistheorie, mathematisch-physikalische Beiträge sowie ausgewählte Korrespondenz aus den Jahren 1925 bis 1982. Wiesbaden: Springer VS. Jones, C. G. (2009). Femininity, mathematics and science, 1880–1914. Basingstoke: Palgrave Macmillan. Kirchhoff, A. (Ed.). (1897). Die akademische Frau: Gutachten hervorragender Universitätsprofessoren, Frauenlehrer und Schriftsteller über die Befähigung der Frau zum wissenschaftlichen Studium und Berufe. Berlin: Steinitz. König, Y.-E., Prauss, C., & Tobies, R. (2014). Margarete Kahn, Klara Löbenstein: Mathematicians – Assistant Headmasters – Friends. Trans. Selker, Jeanne M. L. Jewish Miniatures 108. Berlin: Hentrich & Hentrich. Lorey, W. (1909). Die mathematischen Wissenschaften und die Frauen. Bemerkungen zur Reform der höheren Mädchenschule. Frauenbildung. Zeitschrift für die gesamten Interessen des weiblichen Unterrichtswesens, 8, 161–178. Lykknes, A., Opitz, D. L., & van Tiggelen, B. (Eds.). (2012). For better or for worse? Collaborative couples in the sciences. Basel: Birkhäuser. Makeev, N. N. (2011). Lubobj Nikolaevna Zapolskaya (on the 140th Birthday Anniversary [Russian]. Vestnik permskovo universiteta 3, 7, 82–87. Mühlhausen, E. (1995). Grace Emily Chisholm Young (1868–1944). In: T. Weber-Reich (Ed.), Des “Kennenlernens werth”: Bedeutende Frauen Göttingens (pp. 195–211). Göttingen: Wallstein. Parshall, K. H. (2015). Training Women in mathematical research: The first fifty years of Bryn Mawr College (1885–1935). Mathematical Intelligencer, 37, 71–83. Parshall, K. H., & Rowe, D. E. (1994). The Emergence of the American Mathematical Research Community 1876 –1900: J. J. Sylvetser, Felix Klein, and E. H. Moore. History of Mathematics 8. American Mathematical Society/London Mathematical Society. Richter, T. (2015). Analyse der Briefe des französischen Mathematikers Gaston Darboux (1842– 1917) an den deutschen Mathematiker Felix Klein (1849–1925). Wissenschaftliche Hausarbeit zur Ersten Staatsprüfung für das Lehramt an Gymnasien im Fach Mathematik. Friedrich-SchillerUniversität Jena. Rossiter, M. W. (1982). Women scientists in America. Baltimore: The Johns Hopkins University Press. Rowe, D. E. (1992). Felix Klein, David Hilbert, and the Goettingen Mathematical Tradition (2 vols). Doctoral dissertation, City University of New York. Rowe, D. E. (2019). On Emmy Noether’s role in the relativity revolution. The Mathematical Intelligencer, 41(2), 65–72. Senechal, M. (2013). I died for beauty: Dorothy Wrinch and the cultures of science. New York: Oxford University Press. Siegmund-Schultze, R. (1993). Hilda Geiringer-von Mises, Charlier series, ideology, and the human side of the emancipation of applied mathematics at the University of Berlin During the 1920s. Historia Mathematica, 20, 364–381. Siegmund-Schultze, R. (1997). Felix Kleins Beziehungen zu den Vereinigten Staaten, die Anfänge deutscher Wisenschaftspolitik und die Reform um 1900. Sudhoffs Archiv, 81, 21–38. Siegmund-Schultze, R. (2009). Mathematicians Fleeing from Nazi Germany. Princeton: Princeton University-Press.

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Singer, S. L. (2003). Adventures abroad: North American women at german-speaking universities, 1868–1915. Contributions in Women’s Studies 201. Westport: Praeger. Tobies, R. (1989). Felix Klein als Mitglied des preußischen ‘Herrenhauses’: Wissenschaftlicher Mathematikunterricht für alle Schüler – auch für Mädchen und Frauen. Der Mathematikunterricht, 35, 4–12. Tobies, R. (1991/1992). Zum Beginn des mathematischen Frauenstudiums in Preußen. NTM– Schriftenreihe für Geschichte der Naturwissenschaften, Technik und Medizin, 28, 151–172. Tobies, R. (1997). Einflussfaktoren auf die Karrieren von Frauen in Mathematik und Naturwissenschaften. In: R. Tobies (Ed.), “Aller Männerkultur zum Trotz”: Frauen in Mathematik und Naturwissenschaften. Frankfurt am Main: Campus. 17–67. Second extended edition (2008): “Aller Männerkultur zum Trotz”: Frauen in Mathematik, Naturwissenschaften und Technik (pp. 21–80). Tobies, R. (1999). Felix Klein und David Hilbert als Förderer von Frauen in der Mathematik. Acta Historiae Rerum Naturalium necnon Technicarum/Prague Studies in the History of Science and Technology, 3, 69–101. Tobies, R. (2001a). Femmes et mathématiques dans le monde occidental, un panorama historiographique. Gazette des mathématiciens, 90, 26–35. Tobies, R. (2001b). Baden als Wegbereiter: Marie Gernet (1865–1924) erste Doktorandin in Mathematik und Lehrerin am ersten Mädchengymnasium. In: M. Toepell (Ed.), Mathematik im Wandel (pp. 228–241). Hildesheim: Franzbecker. Tobies, R. (2003). Briefe Emmy Noethers an P. S. Alexandroff. Internationale Zeitschrift für Geschichte und Ethik der Naturwissenschaften, Technik und Medizin N. S., 11, 100–115. Tobies, R. (2004a). Berufswege im Lehramt in der ersten Hälfte des 20. Jahrhunderts. In: A. Abele, H. Neunzert, & R. Tobies (Eds.), Traumjob Mathematik (pp. 18–37). Basel: Birkhäuser. Tobies, R. (2004b). Berufswege promovierter Mathematikerinnen und Mathematiker in der ersten Hälfte des 20. Jahrhunderts. In: A. Abele, H. Neunzert, & R. Tobies (Eds.). Traumjob Mathematik (pp. 89–120). Basel: Birkhäuser. Tobies, R. (2004c). Mathematikerinnen und Mathematiker um 1900 in Deutschland und international. In: A Abele, H. Neunzert, & R. Tobies (Eds.), Traumjob Mathematik (pp. 133–146). Basel: Birkhäuser. Tobies, R. (2006). Biographisches Lexikon in Mathematik promovierter Personen. Algorismus 58. Augsburg: Dr. Erwin Rauner. Tobies, R. (2008). Elisabeth Staiger: Oberstudiendirektorin in Hildesheim. Hildesheimer Jahrbuch für Stadt und Stift Hildesheim, 80, 51–68. Tobies, R. (2011a). Career paths in mathematics: A comparison between women and men. In: K. François, et al. (Eds.), Foundations of the formal sciences VII: Bringing together philosophy and sociology of science (pp. 229–242). Studies in Logic 32. Milton Keynes: Lightning Source. Tobies, R. (2011b). Vertrieben aus Positionen seit 1933: Habilitierte und promovierte Mathematikerinnen und Physikerinnen – Trends, Ursachen, Merkmale. In: I. Hansen-Schaberg & H. Häntzschel (Eds.), Alma Maters Töchter im Exil: Zur Vertreibung von Wissenschaftlerinnen und Akademikerinnen in der NS-Zeit (pp. 114–131). Frauen und Exil 4. Munich: Richard Boorberg. Tobies, R. (2012a). German graduates in mathematics in the first half of the 20th century: Biographies and prosopography. In: L. Rollet & P. Nabonnand (Eds.), Les uns et les autres… Biographies et prosopographies en histoire des sciences (pp. 387–407). Collection Histoire des institutions scientifiques. Nancy: Presses Universitaires. Tobies, R. (2012b). Iris Runge: A life at the crossroads of mathematics, science, and industry (Trans. Pakis, V. A.) Science Networks: Historical Studies 43. Basel: Birkhäuser. Tobies, R. (2016). Felix Klein und französische Mathematiker. In: T. Krohn & S. Schöneburg (Eds.) Mathematik von einst für jetzt (pp. 103–132). Hildesheim: Franzbecker. Tobies, R. (2017). Thekla Freytag: Die Mädchen werden beweisen, dass auch sie exakt und logisch denken können… In: G. Wolfschmidt (Ed.), Scriba Memorial Meeting: History of Mathematics (pp. 330–379). Nuncius Hamburgensis: Beiträge zur Geschichte der Naturwissenschaften 36. Hamburg: Tredition.

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Tobies, R. (2019). Felix Klein. Visionen für Mathematik, Anwendungen und Unterricht. Heidelberg: SpringerSpektrum. Tobies, R., & Rowe, D. E. (1990). Korrespondenz Felix Klein – Adolph Mayer: Auswahl aus den Jahren 1871 bis 1907. Teubner-Archiv zur Mathematik 14. Leipzig: B. G. Teubner. Tobies, R., & Vogt, A. B., with the assistance of Pakis, V. A. (Eds.). (2014). Women in industrial research. Wissenschaftskultur um 1900 8. Stuttgart: Franz Steiner. Toepell, M. (Ed.). (1991). Mitgliedergesamtverzeichnis der Deutschen Mathematiker-Vereinigung 1890–1990. Munich: Institut für Geschichte der Naturwissenschaften. Tollmien, C. (1990). ‘Sind wir doch der Meinung, daß ein weiblicher Kopf nur ganz ausnahmsweise in der Mathematik schöpferisch tätig sein kann …’: Eine Biographie der Mathematikerin Emmy Noether (1882–1935) und zugleich ein Beitrag zur Geschichte der Habilitation von Frauen an der Universität Göttingen. Göttinger Jahrbuch, 38, 153–219. Tollmien, C. (1997). Zwei erste Promotionen: Die Mathematikerin Sofja Kowalewskaja und die Chemikerin Julia Lermontowa. In: R. Tobies (Ed.), “Aller Männerkultur zum Trotz”: Frauen in Mathematik und Naturwissenschaften (pp. 83–129). Frankfurt am Main: Campus. Vogt, A. (1999). Elsa Neumann: Berlins erstes Fräulein Doktor. Berlin: Verlag für Wissenschaftsund Regionalgeschichte Dr. Michael Engel.

Renate Tobies historian of mathematics and sciences at the Friedrich-Schiller-University at Jena (Germany), after studying mathematics, chemistry, physics, and pedagogy, completed her doctoral degree and Habilitation at the University of Leipzig working at the Karl Sudhoff Institute with Hans Wußing. She was managing editor of the International Journal of History of Natural Sciences, Technology and Medicine (Birkhäuser, Basel) for twenty years. After Wußing’s retirement, she became Visiting Professor at the University of Kaiserslautern, and held a chair of history of science and technology at the University of Stuttgart, and further visiting professorships at the universities in Braunschweig, Jena, and Saarbrücken in Germany, plus Linz and Graz (Austria). She is Effective Member of the Académie Internationale d’Histoire des Sciences (Paris) und Foreign Member of the Agder Academy of Sciences and Letters (Kristiansand, Norway). Her main research fields are the history of mathematics and its applications, and women in mathematics, sciences, and technology. She is the author of more than hundred publications in the history of mathematics and related fields. Her recent book is “Felix Klein: Visionen für Mathematik, Anwendungen und Unterricht” (Heidelberg: SpringerSpektrum, 2019).

Chapter 2

Academic Education for Women at the University of Würzburg, Bavaria Katharina Spieß

Abstract Bavaria was the second state in the German Empire that allowed women to enroll at universities. This chapter not only focuses on one of the three Bavarian universities of that time, Würzburg, but also provides a picture of the events and circumstances in this state in general. After having looked at the processes that happened in order to let women have access to academic knowledge, mathematics in Bavaria comes to the fore. It appears that Würzburg had far different conditions than, for example, Munich. Several female Bavarian mathematics students and their life stories are presented and, finally, governmental decisions, negotiations, and the main characters involved in them are discussed. The aim of this contribution is to examine a rather small, rural university and the proceedings there in contrast to large universities like those in Berlin and Göttingen, where the discipline of mathematics was flourishing.

2.1 (Territorial) Background In the nineteenth century, higher education for women was already established in North America, and, at that time, the United States of America took the lead in the subject of women’s and girls’ education. Women’s colleges like the ones in Mount Holyoke (1837) and in Bryn Mawr (1885)1 were founded.2 Developments happened on the European continent as well, and in many countries, like France (Paris, 1863) or Switzerland (Zürich, 1864), women were already admitted to universities at the

1 See Costas (1997), p. 20; and Tobies (2004c), p. 135. For additional information on female mathematicians and the development of the Bryn Mawr College, see Parshall (2015). 2 For further information on the beginning of women’s education in North America, see Rossiter (1984).

K. Spieß (B) Institute of Mathematics, Julius-Maximilians-Universität, Würzburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_2

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end of the nineteenth century.3 However, it took Germany rather long to follow suit. The German Empire was a federation of 25 German States (Prussia, Bavaria, etc.), and the Reichsland Alsace-Lorraine. It was under the command of the Prussian King, who served as the German Emperor. The German Empire was not a unitary construct, given that there were some large differences between its states in terms of size, population, law, and religion.4 In 1900, Baden was ahead of the other states of the German Empire in the matter of women’s education and made its universities accessible not solely for men but also for women. Three years later, the Minister of Cultural Affairs in Bavaria admitted women at Bavarian universities under the authorization of the Prince Regent Luitpold.5 By this incident, Bavaria became the second state in the German Empire to allow women to enroll at universities. Already some years before, it was decided that women, in most cases teachers who were willing to study, should at least be allowed to attend as auditors. This chapter focuses mainly on the general proceedings and circumstances in Bavaria (see Fig. 2.1), with special attention paid to governmental decisions and processes.6 Only in the section on female mathematicians in Würzburg we shall take a closer look at certain biographies. Our main interest lies on the incidents that took place at the University of Würzburg, which is located in the northern part of Bavaria (see Fig. 2.2). In the case of Würzburg (as well as in Germany overall), it must be underscored that the path to an academic education was paved by foreign female academics.7 Afterward, we shall see how local women found their way to education at the universities and take a closer look at the first female mathematicians at the University of Würzburg.8 The last part of the chapter concerns the opinions and the influence of several ministers of Cultural Affairs in Bavaria at that time. The ministers of Bavaria have not yet received much scholarly attention in discussions of women’s education in the nineteenth and twentieth centuries.9 3 For literature focusing on aspects of women finding their way into academic careers internationally,

see Costas (1997). See also Kaller (1992), pp. 361–362; and Hessenauer (1998), pp. 19–20. the distribution of size, population, and religion, and for more general information about the German Empire, see the website Gemeindeverzeichnis Deutschland 1900; Schwind (2004), pp. 25–30; Ullrich (2013) and there especially pp. 31–38 as well as Ullmann (2005); Greschat (1984); Müller (2015); and Lerman (2015). 5 See Hessenauer (1998); Kaiser (1995); and Spieß (2017). 6 For further reading on the matter of academic education for women in Bavaria, see Häntzschel and Bußmann (1997); Wilke (2003); Hessenauer (1998); Kaiser (1995); and Birn (2015). Most of these works include biographies of some of the first women who studied or worked at universities. 7 Here one can see a connection to the developments that happened in Göttingen (see Renate Tobies’s Chap. I.1). It is very interesting to compare the famous and renowned university of Göttingen with the rather provincial university of Würzburg. 8 The main sources for this part of the chapter are the files of the Archiv des Rektorats und des Senats der Universität Würzburg (items 8 and 9 in the bibliography), the files of the Bayerisches Hauptstaatsarchiv (items 1–4 and 6 in the bibliography), and the Jahresbericht der Sophienschule (1900–1920, 1920–1931, 1932–1937). I also draw upon Hessenauer (1998); Kaiser (1997); and Kaiser (1995). Those books also focus on female students in Würzburg. 9 The main sources for this part of the chapter are the files of the Archiv des Rektorats und des Senats der Universität Würzburg (items 8 and 9 in the bibliography), the files of the Bayerisches 4 On

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Fig. 2.1 The Kingdom of Bavaria within the German Empire (1871). Source Shadowxfox (https:// commons.wikimedia.org/wiki/File:German_Empire_-_Bavaria_(1871).svg), “German Empire – Bavaria (1871)”, altered by the author, https://creativecommons.org/licenses/by-sa/3.0/legalcode

Why should one look at Würzburg and Bavaria in the matter of higher education for women? Bavaria was by far the second largest state in the German Empire relating to the population as well as to the area,10 while Prussia was undoubtedly the largest (and probably the most influential) state and therefore has already been subject to many special studies. Moreover, even in literature that focuses on the whole German Empire, Prussia is often highlighted and all other states tend to be in the background. However, the matter of education was (and still is nowadays) in Germany mostly the responsibility of the individual states,11 and therefore it is important to mention that things did not happen in all states and universities similarly and at the same time and that issues that are applicable to the large state of Prussia need not be applicable to Hauptstaatsarchiv (items 1–4 and 6 in the bibliography), and the Jahresbericht der Sophienschule (1900–1920, 1920–1931, 1932–1937). I also draw upon Hessenauer (1998); Kaiser (1997); and Kaiser (1995). Those books also focus on female students in Würzburg. 10 See the website Gemeindeverzeichnis Deutschland 1900 and Albisetti (1988), p. xvi. To put things in further perspective, Bavaria had three universities while Prussia had ten; see Tobies (2004a), p. 19. 11 See Bußmann (1993), pp. 26–28; and Gaab (1931), p. 61, 78–79.

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Fig. 2.2 The Kingdom of Bavaria. Source Tk (https://commons.wikimedia.org/wiki/File:-Bayern_ von_1800_bis_heute.png), “Bayern von 1800 bis heute”, altered (Würzburg and Erlangen added) by the author, https://creativecommons.org/licenses/by-sa/3.0/legalcode

the agricultural country of Bavaria.12 Besides, the interactions between all German states were very important for the outcome of the academic education of women in the whole German Empire. Certainly, several governmental decisions, like the one in Baden, influenced resolutions in other states, for example in Bavaria, and the competitiveness between the individual states also played a role. Since the progress of higher education for women in Prussia is relatively well known,13 my focus below will be on the developments in Bavaria. There were several similarities between these two states, such as neither of them experimenting with coeducation in their school systems14 nor offering a curriculum in mathematics and the natural sciences at public higher schools for girls for a long time,15 but there were also some serious differences 12 See

Schwind (2004), p. 26. for example, Renate Tobies’s Chap. I.1. 14 Albisetti (1988), p. 284. 15 See Tobies (2004a), pp. 19–20; and Abele et al. (2001), p. 9. 13 See,

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between them, including religious differences and therefore a different canon of values.16 Moreover, the situation for women in Prussia became much different from that in Bavaria when Prussia initiated the so-called “Vierten Weg.”17 Prussia thus made it the easier for women to study, and therefore the situations at Prussian and Bavarian universities were quite different during those early years.

2.2 First General Steps in Würzburg Taken by Foreign Female Academics, and Bavarian Women Following At the turn of the twentieth century, there were only three Bavarian universities: Munich, Erlangen, and Würzburg (see Fig. 2.2).18 In fact, the University of Würzburg played an important role in the decision to make universities accessible to women in Bavaria. In the following, the main steps in this development are presented. As early as 1869, an inquiry was sent to the University of Würzburg by the “Central-Vorstand des Allgemeinen Vereins für Volkserziehung und zur Verbesserung des Frauenloses”19 concerning academic education for women. In the same year, an application was also submitted by an American woman, Laura ReuschFormes, who wanted to study medicine in Würzburg.20 After those incidents, little more was heard about the matter of education for women in Würzburg for quite a long time. It was only on July 21, 1894 when a local newspaper wrote: “Miss Dr. Derscheidt, who gained her licence to practice medicine at the university in Brussels, has been here for a few days, she has attended surgery lectures and also attended some

16 In Germany, Catholicism and Protestantism were the two main religious affiliations. While states like Prussia were mainly Protestant, others like Bavaria were predominantly Catholic; see Greschat (1984); and Bjork (2015). 17 The “Vierte Weg” (engl.: fourth way) was another possibility to gain admission to university. By this means, women were allowed to enter university after having finished a special teaching exam and did not have to take the Absolutorialsprüfung (engl.: the high school graduation exam); see Tobies (2004a), pp. 20–21. 18 Other Bavarian universities, such as those in Augsburg, Bamberg, or Bayreuth, were not founded until after World War II; see Quint (1997), p. 127. The TH Munich was renamed in 1970 and is now known as the Technische Universität München (TUM); see Mertens (1991), p. 157; Fuchs (1994), pp. 25–26; and Fuchs (1997), pp. 214, 316. Whenever we refer to the University of Munich in this chapter, the Ludwig-Maximilians-Universität (LMU) is meant. 19 In English: “Central Committee of the Association for Educating the People and Improving the Situation of Women.” This is the first document that appears in the files of the rectorate concerning higher education for women (see item 8 in the bibliography). For more information about this association, see Hessenauer (1998), pp. 20–23; and Birn (2015), pp. 33–34. 20 See Hessenauer (1998), pp. 24–26; Kaiser (1995), p. 7; Wilke (2003), pp. 16–17; Birn (2015), p. 34; and Gleichstellungsstelle für Frauen der Stadt Würzburg (1996), pp. 61, 111–112. This application has not been preserved in the files of the rectorate of Würzburg, but it is said to be the first extant application for medical studies by a women to a Bavarian university; see Bußmann (1993), p. 24; and Kaiser (1997), p. 57.

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surgical operations performed by Professor Dr. Schönborn.21 […] She visited several facilities of the university closely.”22 Archival sources prove that this article caused a lot of turbulence at the Ministry of Cultural Affairs. Two days later, the ministry wrote the following in a letter to the senate of the university (subject: “Attendance of university lecture by a female person”): According to a newspaper, there are rumours that there is a female person (foreigner) due to studies in Würzburg and she is said to have already attended some lectures. […] If this has not already happened, then the senate shall investigate this and if needed it shall immediately stop this attendance and it shall inform about the result of this investigation and about the execution as soon as possible.23

This shows how illiberal the Ministry of Cultural Affairs was at that time. According to them, universities were no place for women and, should they happen to be there, the people responsible for that had to be reprimanded. It seemed unthinkable that women should ever be allowed to enroll. The rectorate of the University of Würzburg replied immediately and tried to trivialize the circumstances.24 Nevertheless, the conclusion was: Even though the majority of the signing faculty [i.e. medical faculty] is indeed generally against allowing women to study medicine, it can only see the behavior of privy councillor Dr. Schoenborn as a harmless act of courtesy and believes that probably also in such exceptional cases a similar goodwill towards graduated ladies will be allowed.25

However, the ministry made it clear that something like that should not happen again and that no director or lecturer had the authorization to make exceptions. This 21 Karl Schönborn (1840–1906) was a professor of surgery in Würzburg and the director of the surgical clinic; see Hessenauer (1998), p. 43. 22 In the German original: “Frl. Dr. Derscheidt, welche an der Universität Brüssel die ärztliche Approbation erhalten, weilte seit einigen Tagen dahier, besuchte die Vorlesungen für Chirurgie, wohnte auch einigen Operationen bei Professor Dr. Schönborn an […] Den verschiedenen Attributen der Universität stattete sie eingehende Besuche ab” (cf. Würzburger Generalanzeiger). This article and most of the following quotations are in a rather outdated German language. However, the translation is as close to the original text as possible. 23 Original: “Zeitungsnachrichten zufolge soll sich eine Frauenperson (Ausländerin) studiumshalber in Würzburg aufhalten und in letzter Zeit bereits verschiedene Vorlesungen besucht […] haben. Der Senat der Universität wird, wenn es noch nicht geschehen sein sollte, dieser Nachricht sofort auf den Grund sehen, gegebenen Falls die Einstellung des Kollegienbesuches verfügen und über das Ergebnis der Erhebung bzw. den Vollzug gegenwärtiger Anordnung alsbald berichten.” Quoted from item 8 in the bibliography: A letter from the Ministry of Cultural Affairs to the Senate of the University of Würzburg dated July 23, 1894. 24 The director of the university at the time, Wilhelm Conrad Röntgen (1845–1923), who had discovered X-rays in Würzburg in 1895 (see Hessenauer (1998), p. 43) wrote to the ministry that the aim of Derscheidt’s visit was never to study there and that she had only been three times to the surgical clinic under the full permission of professor Schönborn but never elsewhere or in lectures; see Kaiser (1995), p. 8. 25 Original: “Die Majorität der unterz. Fakult. ist zwar prinzipiell gegen die Zulassung der Frauen zum medic. Studium, kann aber in der Handlungsweise des Hofr. Dr. Schoenborn nur einen unbedenklichen Act der Höflichkeit erblicken, und ist der Meinung, dass wohl auch ferner in solchen Ausnahmefällen ein ähnliches Entgegenkommen gegen graduierte Damen zulässig sein dürfte.” See Kaiser (1995), p. 8; and Hessenauer (1998), pp. 43–44.

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incident took its toll. Two years later, Professor Schönborn opposed the idea of letting a female teacher attend art history lectures because of the reprimand he had received from the ministry.26 Nevertheless, there were still some professors who were supportive of women after all. In 1896, the American scientist Marcella O’Grady (1863–1950)27 filed a request to the university to work together with the zoology professor Theodor Boveri (1862– 1915).28 On June 3, 1896, the philosophical faculty29 wrote that O’Grady should be admitted because of the educational background that she had already obtained in her home country.30 Marcella O’Grady had studied in Boston and at Bryn Mawr and had been a professor of zoology at Vassar College (a women’s college in the United States) for 7 years.31 The faculty stood unanimously behind this statement, and so the senate of the University of Würzburg wrote a recommendation letter for O’Grady to the Ministry of Cultural Affairs. Their argumentation was also heavily based on the fact that O’Grady already had an extraordinarily good education. Thereupon, the ministry allowed her to attend the university, though only as an auditor (a formal criterion for working at the university) and not as a fully enrolled student.32 Yet that was not the end of the story. Without informing the administration, they planned that O’Grady would conduct her doctoral studies in Germany under the supervision of Boveri. Within a year, she managed to finish her thesis, and she would have become Würzburg’s first woman to earn a doctorate33 had she not married Boveri before taking her oral examination. It is not known why she failed to take this last step and did not finish her Ph.D., given that only working was forbidden for married women. More surprisingly, she even published her thesis.34 As a married woman, she was not allowed to pursue a scientific career any further.35 The common argument for this was 26 See

Kaiser (1995), p. 8. ibid., p. 10. 28 For more information on Boveri, see Hessenauer (1998), p. 44. 29 At that time, mathematics belonged to the Philosophical Faculty; see ibid., 125–127. One of the signatories was the mathematician Friedrich Prym (1841–1915); see item 8 in the bibliography and, for further information about Prym, see Vollrath (1995). 30 The subject line was “Authorization for working at the Zoological Institute for a graduated female foreigner” (“Zulassung einer graduierten Ausländerin zum Arbeiten im zoologischen Institut”). See item 8 in the bibliography: A letter from the Philosophical Faculty to the Ministry of Cultural Affairs dated June 3, 1896. 31 See Kaiser (1995), pp. 9–13. 32 See Hessenauer (1998), p. 44. 33 The first woman to do so was a British woman named Beatrice Edgell (1871–?), who was awarded the degree in 1902; see Kaiser (1995), p. 35. The second was Helene Nanu (1874–?) from Romania (see Nanu (1904), p. 57), and the third was Edna Carter (1875–?). An American woman, Carter had studied at Vassar College and had dedicated her Ph.D. thesis in physics to her “dear friend Mrs. Marcella Boveri”; see Kaiser (1995), p. 35. 34 See ibid., p. 9. 35 It needs to be emphasized that, even after World War II, married women still had to rely on the approval of their husbands if they wanted to work. Married women were obligated to remain housewives until the 1970 s. Furthermore, until 1962, a married woman was not able to have her own bank account, and only after 1969 could a married women be considered legally competent. 27 See

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that married women were already “cared for” by their husbands and should not take away jobs from men.36 The group most affected by this happened to be teachers, and so the term “Lehrerinnenzölibat” (engl.: “female teacher celibacy”) became used for it37 (teaching was the most common and the highest possible profession a woman could have at that time).38 According to her daughter, O’Grady deeply regretted the decision to give up her scientific career and suffered from depression. After her husband’s unexpected death, O’Grady left Germany in 1925 in order to work again on an academic level in America.39 It needs to be stressed that in Würzburg, as well as in other German universities, foreign female academics paved the way for German women who were willing to study.40 This seems rather contradictory when one looks at the fact that one major argument against the allowance of women at the universities was that this would lead to a rush of foreigners to Bavarian universities.41 The year 1898 became an important landmark for Würzburg’s women who were aiming to study. In this year, women from the wealthier part of the society founded an association by the name of Frauenheil in order to fight for a better education for women. The aim of this club was “to promote a higher education for females and to better the earning capacity of those women who rely on their own livelihood.”42 Similar associations were spreading all over Germany from the end of the nineteenth century on.43 Munich became a sort of center for the women’s movement in Bavaria. In 1894, the Gesellschaft zur Förderung der geistigen Interessen der Frau (later the Verein für Fraueninteressen) was founded there, but also a much more radical association was established in Munich, the Frauenbildungs-Reform, which tried to hand in petitions in order to let women enter universities, unfortunately without much success.44 The association in Würzburg was, compared to the one in Munich, moderate. While the Frauenbildungs-Reform fought actively with their petitions, the Frauenheil-association tried to achieve university education for women through For more information on this topic, see Helwig and Nickel (1993), pp. 273–278; the Focus online article “Google ehrt Internationalen Frauentag mit Doodle. Die erste Frau, die ohne Erlaubnis ihres Ehemannes arbeiten darf”; and the website “Geschichte der Gleichstellung”. 36 Cf. Kaiser (1995), p. 9: “[…] wer heiratete wurde sofort entlassen mit dem Argument, daß diese Frauen dann ja ‘versorgt’ wären und keinem Mann den Arbeitsplatz mehr wegnehmen müßten.”. 37 See Wilke (2003), p. 103. 38 See Knauer-Nothaft (1997b), p. 152; Gleichstellungsstelle für Frauen der Stadt Würzburg (1996), pp. 18–19, 111; and Hessenauer (1998), p. 126. 39 See Kaiser (1995), pp. 9–13; and Gleichstellungsstelle für Frauen der Stadt Würzburg (1996), pp. 34, 112. 40 For more details on this topic, see Renate Tobies’s Chap. I.1 as well as Tobies (2004c); Meister (1997); Kaiser (1997); and Hessenauer (1998), pp. 56, 129–130. 41 See ibid., p. 80. 42 Original: “Der Verein hatte die ‘Förderung höherer Bildung des weiblichen Geschlechts und der Erwerbsfähigkeit der auf eigenen Unterhalt angewiesenen Frauen’ zum Ziel.” The founding of this association was planned as early as 1896; see Kaiser (1995), p. 13. 43 See Costas (1997), pp. 18–20. 44 See Bußmann (1993), p. 27; Meister (1997), pp. 40–42; and Kaiser (1997), p. 60.

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the back door, as is outlined in the following. In some literature, it is said that the interaction of both parties, the radical and the moderate one, helped to achieve their goal in the end.45 A large step in the build-up to opening universities for women in Bavaria was the aforementioned “back door” that was slipped through in Würzburg in 1898. It should be noted that this step could only happen with the help of some open-minded professors at the university.46 In particular, Karl Bernhard Lehmann (1858–1940),47 a professor of medicine in Würzburg and the husband of Amalie Lehmann, one of the co-founders of the Frauenheil-association, played an important role. It was he, who filed a request to the Ministry of Cultural Affairs with the subject line “Tutorials for ladies” in May, 1898.48 It was accepted that he could give an extracurricular lecture to the association members on the premises of the university.49 Other professors followed his example and also gave special courses for this association, including the chemistry professor Arthur Hantzsch (1857–1935) and the professor of Romance studies Heinrich Schneegans (1863–1914),50 but it is highly unlikely that a mathematics professor in Würzburg gave special lectures for these women, as no file of such activity could be found. By delivering those lectures, the professors of Würzburg opened a small door for local women to enter the university.51 In October of 1899, another great step was taken when the Bavarian woman Jenny Danziger (1879–?) applied to study medicine. After being rejected in Erlangen and Munich, she was accepted in Würzburg, though just as an auditor and not as a fully enrolled student. Later, she was eventually accepted at Munich, and from the winter semester of 1903/04 on, she studied there as a completely enrolled student.52 Only weeks after this incident, 13 other women also applied to be auditors.53 After some back and forth, the ministry allowed a general admission for female teachers of the city of Würzburg to the lectures of the philosophical faculty under the

45 In the third University city of Bavaria, Erlangen, a similar association for women’s education was founded, but not until 1906. According to Kaiser, “Comparable actions relating to academic education for women did not happen there” (ibid., p. 60; translated from the original German). 46 See ibid., pp. 59–61. 47 For more information about Lehmann, Professor for hygienics in Würzburg, see Hessenauer (1998), p. 37; and Spieß (2017). 48 Original: “Unterrichtskurse für Damen.” See item 8 in the bibliography, which contains this letter from Lehmann to the Royal State Ministry of Cultural Affairs dated May 5, 1898. 49 On the entire process of this request, see Hessenauer (1998), pp. 59–60. 50 More information about Hantzsch can be found in “Gelehrter des Monats: Arthur Hantzsch.” For details about Schneegans, see Deutsche Biographie: Schneegans. 51 For more information on these extramural lectures and the Frauenheil-association, see Hessenauer (1998), pp. 57–62; Kaiser (1995), p. 19; and the archival material cited as item 8 in the bibliography. 52 On the case of Danziger’s acceptance and for more information about her, see Ebert (2003), pp. 32–33; Kaiser (1997), p. 61; Kaiser (1995), p. 16; and Spieß (2017). 53 See Hessenauer (1998), pp. 47–50. According to Kaiser (1995), pp. 16–18 and Kaiser (1997), pp. 62–63, all of these women were teachers and also members of the Frauenheil-association.

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condition that these teachers had already passed the qualifying test and could demonstrate that.54 Ever since then, the number of female auditors at Bavarian universities increased permanently, whereupon one has to mention that the number of female auditors in Würzburg was larger than that in Munich. Erlangen was far behind those two.55 Seeing women at universities became more and more common, the claim for equal rights grew louder, and debates about women’s education became more frequent.56 Finally, in 1903, the new minister for Cultural Affairs, Anton von Wehner (1850– 1915), requested the three Bavarian universities to give a comment on their view of letting women enter university. Except for the theological faculty, all other faculties of the University of Würzburg voted for the admission of women in principle.57 The University of Erlangen was divided on this decision, and in Munich the majority was against it. However, on September 21, 1903, Wehner granted permission with a warrant of “his royal highness, Prince Luitpold,”58 that women would be allowed to enroll if they were able to verify their Reifezeugnis (engl.: university entrance certificate). This, however, happened to be the “biggest objective obstacle” for women who wanted to study.59 In Bavaria, there was only the possibility that girls could take their Absolutorialsprüfung as day pupils at schools for boys, and prior to that often costly private lessons had to be given to girls.60 In Munich, there was already in the 1890s a group that fought for establishing a Mädchengymnasium (engl.: a higher secondary school for girls).61 After several rejections by the state of Bavaria, this initially led not to an actual Gymnasium for girls but to the installation of Private Gymnasialkurse für Damen under the direction of Adolf Sickenberger.62 In the beginning, these classes in Munich were the only ones for girls in Bavaria; in other states of the German Empire, for example in Berlin, where three of such seminars had already been established, these courses fared better. It must be mentioned that, in comparison to all the other courses, those in Munich were by far the most expensive.63 54 See

Kaiser (1995), p. 18. should also be noted that while in Würzburg most of the female auditors were teachers, in Munich the auditors were mainly foreign academics and members of the local women’s movement; see Kaiser (1997), p. 63; and Kaiser (1995), pp. 16–20. 56 See ibid., pp. 20–23; and Hessenauer (1998), p. 80. On the debate and the progress of women’s education in the Bavarian Landtag, see also the section below titled “The Ministers of Cultural Affairs in Bavaria and Their Role in the History of Women’s Education.”. 57 See Kaiser (1995), p. 20; and Hessenauer (1998), pp. 80–83. 58 Translated from the German original; see Kaiser (1995), p. 20; and Hessenauer (1998), p. 84. 59 Translated from the German original; see Hessenauer (1998), p. 63; and Mertens (1991), pp. 43– 46. 60 See Hessenauer (1998), pp. 64–66, 71–79. 61 Founded in 1894, this group was called the Verein zur Errichtung eines Mädchengymnasiums in München. See ibid., p. 66; Fuchs (1994), pp. 1–16; and Bölling (2010), p. 64. 62 See Fuchs (1994), p. 11; and Knauer-Nothaft (1997a), pp. 78–80. 63 The school’s fee for one year was 450 Mark; the second most expensive course was the one in Frankfurt, at 300 Mark per year. For some idea of how exclusive these courses were, it should be noted that the average income of a working person amounted to 849 Mark per year. See Hessenauer 55 It

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The school situation in Bavaria might be a good explanation for why university education for women developed so differently at the three Bavarian universities.64 While Munich expanded largely in the period from 1903 to 1913, the number of students overall at the universities Erlangen and Würzburg only grew slowly. The same circumstances were true for female students. In Würzburg, 3 female students enrolled in 1903, in Erlangen just 1, and in Munich there were already 26 women enrolled in that semester. In the winter semester of 1913/14, there were already 443 female students in Munich, in comparison to 36 in Würzburg and 32 in Erlangen. Some historians cite the high percentages of fraternities at these universities as another explanation for the slow development of women’s education in Würzburg and Erlangen.65 In the following section, we shall again see the outstanding role of Munich, now in the context of mathematics.

2.3 The First Female Mathematicians Now we turn to the topic of female mathematicians and mathematics in general in Bavaria and especially in Würzburg at the end of the nineteenth and beginning of the twentieth centuries. Here, we shall see that Munich was by far the most exceptional of the Bavarian universities and that the two small universities, Erlangen and Würzburg, faced rather similar developments and the female mathematics students there had quite typical curricula vitae for that time. In a volume about Bavarian history, it is said that “all four colleges [i.e. the three universities Munich, Erlangen and Würzburg plus the Technische Hochschule (TH; engl.: polytechnic university) in Munich] displayed outstanding personnel and accomplishments in mathematics.” The notable “Erlanger Programm” is mentioned as well as several famous mathematicians who worked at Bavarian universities, including Paul Gordan, Ernst Fischer, and Max Noether in Erlangen; Friedrich Prym, Friedrich Rost, Emil Hilb, and Aurel Voss in Würzburg (Voss also in Munich); and Alfred Pringsheim, Ferdinand Lindemann, Felix Klein (who was in Erlangen before), Constantin Carathéodory, Otto Hesse, Walther Dyck, and Oskar Perron (among others) in Munich.66 As one can already see by this list, Munich was way ahead of the other universities in Bavaria with regard to mathematics, though without any doubt the center of mathematics in Germany at that time was not in Bavaria, but

(1998), p. 66; and Bölling (2010), p. 60. Moreover, the school fee at public higher schools for boys was far lower than at the private higher schools for girls (see Fuchs (1994), p. 15), which is a reason why often just the male children could gain a good education. Regarding the school situation of girls in Bavaria, see also Panzer (1993), pp. 97–106. 64 See Kaiser (1997), pp. 63–65. 65 Those two universities were said to be “strongholds of student leagues.” See Spitznagel (1974), pp. 172–173; and Kaiser (1997), p. 65. 66 Translated from the German original. See Spindler (1979), p. 1064; as well as Vollrath (2017), pp. 67–75; and Tobies (2006), pp. 23, 32, 35.

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in Prussia, first in Berlin and later on in Göttingen.67 The University of Würzburg is said to be mainly a “medical university,” and medicine was the preferred subject among the first female students there.68 One of the first female mathematics students in Bavaria was Thekla Freytag (1877– 1932) from Berlin, who registered as an auditor at the TH Munich in 1899.69 Two years later, Hilde Mollier (1876–1967) followed as an auditor for courses in mathematics and physics.70 In Erlangen, probably the most famous Bavarian female mathematician, Emmy Noether, applied as an auditor for the winter semester of 1900/01 in order to attend lectures in mathematics, and she fully enrolled as a student in Erlangen as of the winter semester of 1904/05 (after having studied one semester at the prestigious university in Göttingen).71 In Würzburg, no female auditor who explicitly attended lectures on mathematics could be verified, but there were some who applied for the philosophical faculty in general, which included the mathematics department at that time.72 It was only in the winter semester of 1912/13, nearly ten years after women were allowed to study at Bavarian universities, that the first female student enrolled in Würzburg exclusively at the mathematical faculty.73 The next female mathematics student enrolled in 1914,74 and from then on the number of female students grew slowly so that in the summer semester of 1917, for example, there were five female mathematics students in Würzburg (see also Table 2.1). Let us now examine the situation in Würzburg in regard to mathematics more closely. The mathematician Friedrich Prym was appointed to Würzburg in 1869. The University of Würzburg was not able to pay a large salary but instead they was able to win him over because Prym, who was wealthy prior to the offer from Würzburg, put more emphasis on “other conditions and the sphere of activity” as well as on “a more independent position at a university instead of a job at a polytechnic university, which is more limiting.”75 When Prym arrived in Würzburg, he said the following 67 See

Tobies (2004c), p. 134. Kaiser (1997), p. 64; and Sticker (1932), esp. pp. 754–755. 69 See Tobies (2017). 70 See Fuchs (1994), p. 31. For more information on Freytag, see Tobies (2008), pp. 24–25; and Tobies (2004a), p. 20. Mollier (married name: Barkhausen) was able to study at the TH because one of her relatives, the mathematician van Dyck, stood up for her. For more about Mollier, see Fuchs (1997), pp. 219–226, 317. 71 For a historical account of Emmy Noether’s educational background and her path to university, see Tollmien (2016), pp. 1, 6–11. 72 See Kaiser (1995), p. 18. In Würzburg, the faculty of natural sciences did not split from the philosophical faculty until 1937; see Vollrath (2017), pp. 87–88; and Hessenauer (1998), p. 113. 73 The same circumstances as with the auditors before occurred here again. From 1905 on, there were already some female students who were enrolled at the philosophical faculty in general. See item 9 in the bibliography and Hessenauer (1998), pp. 166–171. Whether they also attended lectures in mathematics is unknown. 74 In order to appreciate these dates in the context of the German Empire: In the summer semester of 1914 there were already seventy-seven female mathematics students in Berlin, fifty in Bonn, forty-eight in Münster, and thirty-six enrolled at Göttingen; see Tobies (2004a), p. 21. 75 Quoted from Vollrath (1995), p. 165 (translated from the German original). This source focuses on the life of Friedrich Prym, who served as mathematics professor in Würzburg from 1869 to 1909. 68 See

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Table 2.1 Number of female students in mathematics and in general in Würzburg in the early years. All data from item 9 in the bibliography. The numbers in brackets are from Hessenauer (1998), pp. 158–171, when different Semester

Female students enrolled at the mathematical faculty

Female students enrolled overall

1903/04

0

3

…

…

…

1911/12

0

17

1912

0

15

1912/13

1

16

1913

1

18 (19)

1913/14

1

36 (31)

1914

1

40

1914/15

2

42

1915

2

43

1915/16

5

49

1916

5

56

1916/17

5

49

1917

5

69

1917/18

4

93

1918

4

115

1918/19

5

147

about the local situation in mathematics: “When […] I started my work here, there was no student in the field of mathematics. In the following school year […], there were three and in the ensuing school year […] there were four mathematics students and only gradually did this number grow in the years after that.”76 Moreover, he outlined that Munich was always a big competitor to Würzburg and even to Erlangen, because there one could study at the university as well as at the TH, and it was also in Munich where all the Examina took place. These circumstances were said to have changed only after the former mathematics professor Aloys Mayr (1807–1890) had passed away and Professor Aurel Voss (1845–1931)

76 German

Original: “Als ich […] meine hiesige Tätigkeit begann, war hier kein Studierender der Mathematik vorhanden. Im folgenden Schuljahre […] waren 3 und im darauffolgenden Schuljahre […] 4 Studierende der Mathematik da und nur allmählich stieg in den folgenden Jahren ihre Zahl.” For this and the following, see ibid., p. 165.

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and Professor Georg Rost (1870–1958) were appointed to Würzburg.77 After that, the entire apprenticeship could be provided to the mathematics students in Würzburg. Another comparison between the universities in Bavaria can show how much Munich was ahead of the other two78 : In the summer semester of 1932, there were more than three times the number of students in mathematics in Munich (309; 67 among them female) than in Würzburg (97 with 24 female mathematics students among them) or in Erlangen (67 mathematics students and 17 of them female).79 The fact that Würzburg was not that popular for mathematics students in general could explain why university education for women in mathematics there started so slowly (see Table 2.1). It should also be mentioned that there is practically no public statement from the mathematics professors in Würzburg known in regard to women’s education. Nevertheless, archival sources suggest that at least Prym was not averse to it. He signed several times for the admission of female individuals applying to the university.80 Moreover, Prym donated 300 Mark to the Frauenheil-association, of which his wife seemed to be a member as she appears in the list of members of this association.81 Due to the war, Georg Rost helped out and taught mathematics at the Sophienschule,82 one of the very few schools for girls in Würzburg. This school was supported by and closely linked to many professors of Würzburg, such as Karl Bernhard Lehmann and Arthur Hantzsch.83 However, Rost was not the first mathematician who worked at the University of Würzburg and assisted at the Sophienschule. There was already the Universitätsassistent Dr. Josef Zillig, who taught mathematics and physics for the higher classes (that is, for those classes that wanted to prepare for the Gymnasium) and at the Selekta (a kind of senior class that could also be attended by the mothers and sisters of the school girls in exchange for a fee).84 77 Aloys Mayr received an associate professorship in mathematics in 1837 and became a full professor in Würzburg from 1840 on. Aurel Voss was a professor of mathematics in Würzburg from 1891 to 1903. Georg Rost was first appointed to an Extraordinariat from 1903 to 1906 and then gained a tenured professorship, which he held until his retirement in 1935. See Vollrath (2017), pp. 58–59, 64, 69–71, 119. On the history of the mathematics professors in Würzburg in general, see ibid., passim. 78 It is important to mention that this circumstance was not just specifically for mathematics; in general, Munich was far ahead of the other two universities in the case of female students. See Hessenauer (1998), p. 96. 79 For these statistics (and those concerning other German universities), see Tobies (1997), p. 142. 80 For example, Prym approved the admission of Marcella O’Grady to the university (see item 8 in the bibliography). 81 The money was intended for the culinary school that was organized by the association Frauenheil; see Zweiter Jahresbericht des Vereins “Frauenheil” (1900), p. 12. His wife is listed as a member on page 16. However, whether or not she was an active member could not be verified. A special lecture given to the members of this association by Prym could also not be found. 82 See the Jahresbericht der Sophien-Schule (1917/18), p. 36. 83 On the history of the Sophienschule and its connection to the university, see Hessenauer (1998), pp. 67–79; and Direktorat des Mozart-Gymnasiums (1987), pp. 47–62. 84 See Hessenauer (1998), p. 68; and Jahresbericht der Sophien-Schule (1908/09), p. 53; and (1909/10), pp. 11, 50–51.

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Now we shall take a closer look at the first six women who were enrolled at the faculty of mathematics in Würzburg (see Table 2.2). All of them came from Bavaria, and three of them were even born in Würzburg.85 The first three were Catholic, the others were Protestants. The fathers of two of them worked as counselors at the court, while all the other fathers were teachers. Alma Wolffhardt’s father’s profession is declared as divisional chaplain in Table 2.2, but he also served as the headmaster at the aforementioned Sophienschule and gave religious education there.86 Hence, Alma Wolffhardt had a direct connection to the Sophienschule. Moreover, not only did she attend this school in her childhood; she also taught mathematics there during World War I because of a shortage of teachers while she was still studying the subject at the university.87 For a time, Wolffhardt also studied mathematics in Berlin.88 She stood up to the smear campaign against female students that arose in Germany shortly after World War I. However, it must be said that she certainly was active for equal (female) rights but never hesitated to state that the true profession of a woman is caring for her family, a perspective that she probably shared with her father. Because of the current circumstances (war, etc.), she argued, there were now more women than men and therefore certain prospects should be granted to this female surplus.89 In 1920, Wolffhardt married the second headmaster of the Sophienschule, Friedrich Kreiner, with whom she had ten children.90 In 1942, moreover, her rather propagandistic book Das Lerchennest: Ein Buch vom Glück des Kinderreichtums das alle angeht was published.91 But Wolffhardt was not the only one of these six mathematics students who had a link to the Sophienschule. Agnes Braun and Hilde Fuß had also attended this school,92 85 The parents of Elsa Trammer were the only ones who did not live directly in Würzburg during the student days of these women. 86 See the Jahresbericht der Sophien-Schule; Hessenauer (1998), pp. 68, 92–93; and Direktorat des Mozart-Gymnasiums (1987), p. 51. 87 See the Jahresbericht der Sophien-Schule (1903/04), p. 34; (1918/19), pp. 11, 28, 32, 36; as well as archival items 1 and 9 in the bibliography. 88 Changing universities was more common at that time than it is today; see Abele et al. (2001), p. 10. 89 See Hessenauer (1998), p. 93; Würzburger Universitäts-Zeitung (1920), pp. 112–113; and KreinerWolffhardt (1942). 90 See Kaiser (1995), pp. 32–33. There it is also said that Wolffhardt worked as a teacher of mathematics at the Sophienschule. What is meant here is probably the time during the war, because after her marriage Wolffhardt appears on the list of members of the Sophienschule only as a spouse of the headmaster and not in the teacher section; see the Jahresbericht der Sophien-Schule (1923–1933). 91 The title could be translated as “The Nest of the Lark: A Book about the Fortune of the Abundance of Children, which Concerns Everyone.” The title is probably an allusion to the street where the family in the book (and the Kreiner family itself) lived, the Lerchenweg (transl.: “lark road”) in Würzburg. See the Jahresbericht der Sophien-Schule (1936), p. 62; and Kreiner-Wolffhardt (1942), p. 143. Throughout this book, Wolffhardt often makes it clear that, according to her, a woman’s main job is caring for her family. 92 In the yearbooks from 1934 and 1935, there are lists of prior students of the Sophienschule. In the Jahresbericht der Sophien-Schule (1934), p. 58, Hilde Fuß is listed as “Studienrat (math.)” at a school for girls in Fürth, and in the Jahresbericht (1935), p. 68, Agnes Braun is marked as dead.

10/18/1915

10/16/1916

Alma Wolffhardt

10/15/1915

Elsa Trammer

10/18/1915

10/19/1914

Luise Bauer

Agnes Braun

10/17/1912

Olga Sauer

Hilde Fuß

Date of enrollment

Name

12/24/1897

04/04/1892

09/20/1895

05/09/1892

03/14/1899

07/30/1891

Birthday

Bavaria

Bavaria

Bavaria

Bavaria

Bavaria

Bavaria

Country of origin

Würzburg

Kirchheim-bolanden

Würzburg

Ober-viechtach

Würzburg

Lauf Protestant

Place of birth

Protestant

Protestant

Protestant

Catholic

Catholic

Catholic

Religion

Div. pfarrer (engl: divisional chaplain), Würzburg

Gym. Professor (engl: teacher),Würzburg

Hauptlehrer (engl: teacher), Würzburg

Landger. Rat (engl: counsellor at the court), Amberg

Reallehrer (engl: teacher), deceased, Würzburg

Oberlandgerichtsrat (engl: counsellor at the court), Würzburg

Profession of the Parents and Parents’ residence

Table 2.2 The first six female mathematicians at the University of Würzburg. All data (except for the birthdates) from item 9 in the bibliography. For the birthdates see items 11, 14, 13, 10, 12, 15, respectively

54 K. Spieß

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and the latter also taught there for a short time.93 Much could be found out about Hilde Fuß before and after her student days: She was born on September 20, 1895 in Würzburg as the only child of her parents. She first attended a primary school and then an upper private school (the Sophienschule). Initially, Fuß had wanted to study medicine, but she then followed the legacy of her great-grandfather and became a teacher of mathematics and physics. Apart from the Sophienschule in Würzburg, she also taught at a school for girls in Erlangen, in Reichenhall, and finally, as of 1928, at the Mädchenlyzeum in Fürth.94 In one source, it is said that her special interest was educating girls.95 Furthermore, she is said to have been “a very strict and a very correct teacher.”96 In 1952, Hilde Fuß ran for election to the city council for her women’s party, Fürther-Frauen-Liste, which received 1.24% of the vote.97 She was the eldest resident of Fürth in the recent history of the city, and she died on July 16, 2003 at the age of 107. Little could be found out about Elsa Trammer and Luise Bauer. Bauer worked as a teacher at a school for girls in Munich after finishing her studies.98 Trammer99 wrote in her letter for the admission of the Absolutorialsprüfung that she had private lessons in mathematics from her uncle Otto Trammer, a teacher in Landshut.100 Maybe he motivated her to study mathematics, but her actual reasons could not be found out. More can be said about the first female mathematician who studied at the University of Würzburg, Olga Sauer.101 Sauer was born on July, 30, 1891 in Lauf a. Pegnitz as the daughter of the k.[öniglichem] Amtsrichters (engl.: royal district judge) Karl

What happened to Agnes Braun between these years could not be discovered. For their application for admission to the Absolutorialsprüfung, see item 1 in the bibliography (Braun: April 23, 1915) and item 4 (Fuß: May 7, 1914). 93 See the Jahresbericht der Sophien-Schule (1923/24), p. 2. 94 See Trautwein (2007), p. 67; Trautwein (2003), pp. 22–23; Emmerig (1925), p. 80; Emmerig (1926), p. 98; Emmerig (1927), p. 52; Emmerig (1929), p. 58; Emmerig (1930), p. 58; and Emmerig (1933), p. 69. 95 See Trautwein (2007), p. 67. 96 See Trautwein (2003), p. 23 (translated from the German original). 97 For more about this party, see Trautwein (2007), pp. 66–68, 163–164; and Trautwein (2003), p. 23. 98 Bauer appears in lists of Bavarian teachers; see, for instance, Emmerig (1925), p. 79; Emmerig (1930), p. 59; and Emmerig (1933), p. 69. 99 In the yearbooks of the Sophienschule, there also appears an Else Trammer as a mathematics teacher, for example in Jahresbericht der Sophien-Schule (1916/17), p. 10. However, this is probably someone else because the mathematics student Trammer was still studying at the university at the same time. It is worth noting, however, that it is possible in several places in the sources to read the name as either Else or Elsa Trammer. 100 See archival item 2 in the bibliography. 101 The primary sources on Sauer are items 7 (which contains a curriculum vitae) and 1 in the bibliography; Wolf (2005); the Jahresbericht der Maria-Theresia Schule (1969/70); as well as reports about her from her former colleagues. Many thanks to Martin Aulbach, the assistant headmaster of the Maria-Theresia-Gymnasium in Augsburg, and Peter Wolf for their help in this research.

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Sauer.102 She attended the Volksschule in Lauf and the Volksschule in Karlstadt a. M., and after that she was at the Töchterschule der Englischen Fräulein in Würzburg from 1903 until 1909. After that, she prepared herself for the Absolutoriumspüfung with private tuition and with courses at the Privatrealgymnasium in Nuremberg (where she spent two years). On April 30, 1912 there was an application103 by the director of the Privat-Realgymnasialkurse für Mädchen in Nürnberg to the Ministry of Cultural Affairs concerning that year’s Absolutorialsprüfung. In the list of the pupils, Olga Sauer appears among nine others. One of the mathematics teachers of these private courses was a teacher from an Oberrealschule, M. Fronmüller.104 The Absolutorialsprüfung for these ten girls was suggested to take place at the school in Nuremberg, but this was turned down by the Ministry of Cultural Affairs, and the ten girls were sent to the Realgymnasium at Würzburg in order to take their exam. Sauer received her Absolutorium on July 13, 1912. In October of the same year, Olga Sauer enrolled at the mathematics faculty of the University of Würzburg.105 She was a student there until the summer semester of 1916. The title of her teacher’s thesis in mathematics is given as follows106 : Nach Königs (Comptes Rendus 114, p. 55, 1892; Darboux, Théorie générale des surfaces IV, p. 33) haben die Projektionen der Haupttangentenkurven einer Fläche die Eigenschaft, ein System gleicher Invarianten zu bilden; dieser Satz ist zu benutzen, um 1.) solche Systeme zu konsturieren, 2.) Flächen mit bekannten Haupttangentenkurven aufzustellen.

Sauer passed the 1. Abschnitt der Prüfung für den Unterricht (engl.: first section for becoming a teacher) in 1916 with a good final grade. From then on, Olga Sauer worked as a teacher for mathematics and physics at the municipal Maria-Theresia Schule für höhere Mädchenbildung, which is today the Maria-Theresia Gymnasium in Augsburg. She was described by a former pupil as “so ladylike that one did not dare to mess around in her presence.”107 Sauer was appointed commissarial headmaster in August 1945 and regular headmaster of the Maria-Theresia Schule from 1946 on until her retirement on August 1, 1956.108 A former teacher at this school remembered Olga Sauer very well. According to him, as a child he always had to visit the headmaster Sauer (who lived in a large apartment really close to 102 In

an obituary of Olga Sauer (Jahresbericht der Maria-Theresia Schule (1969/70), p. 48), it is mentioned that she was buried at the cemetery in Ursberg, where her father and her sister were also resting. Whether she had further siblings is not known. 103 See archival item 1 in the bibliography. 104 See the annual report of the Privat-Realgymnasialkurse für Mädchen (1909/10), pp. 4–6, in item 1 in the bibliography. 105 See items 9 and 7 in the bibliography. 106 Quoted from item 7 in the bibliography. No supervisor is mentioned for this thesis. 107 Quoted from Wolf (2005), p. 51 (translated from the original German). 108 See Emmerig (1925), p. 79; Emmerig (1926), p. 98; Emmerig (1927), p. 48; Emmerig (1929), p. 54; Emmerig (1930), p. 54; Emmerig (1933), p. 65; as well as “The School Year 1944/45 at the Maria-Theresia School”; and the Bayerisches Philologenjahrbuch (1950/51), p. 112; (1955/56), p. 90; (1960/61), p. 208.

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the school) at Christmas together with his father. He also remembered that Olga Sauer was unattached, that she was very fluent in French, and that she repeatedly used to say that she had only been able to study mathematics because of the explicit permission she received from the king of Bavaria.109 In her obituary,110 Sauer was described as highly virtuous and very committed to her working life as a teacher. It is also mentioned that her Catholic religion played an important role in her life, in particular during the time of National Socialism in Germany.111 Summarizing the biographies of these six women and comparing them to what was common at that time,112 one sees that their biographies and living conditions were typical. For instance, the majority of women studied near their parents’ homes,113 as did all six of the mathematics students in Würzburg. Moreover, the professions of each of the fathers of the six women can also be considered rather usual,114 since they all belonged to the wealthier and better-educated part of society. Even the distribution of their religious affiliations is rather typical.115 The later courses of the lives of the first six female mathematicians (at least the ones that could be verified by now) comply with the later lives of most of the first female mathematicians in Germany overall, given that all of those who are known either became teachers at schools for girls or dropped out of working life after they married, as Wolffhardt probably did.116 Having looked at the first female students in mathematics in Würzburg, we now turn to the first female mathematicians to earn a Ph.D. in Bavaria at large. Before 1945, there were ten women in Bavaria who did so: seven in Munich, two in Erlangen, and one in Würzburg. These were, at Würzburg, Maria Knoll (1938); at Erlangen, Emmy Noether (1908), and Ilse Sauter (1936); at TH Munich, Josephina Kapfer (1923); and, at the University of Munich, Else Schöll (1913), Auguste Gruber (1923), Margarete Haendel (1924), Josefa von Schwarz (1933), Erna Zurl (1934), and Süe-yung Kiang (1941).117 Here again, one sees the leading position that Munich held in Bavaria. A further explanation could be the poor school situation for girls in Würzburg in comparison

109 What Sauer probably meant by this was the permission of women to attend Bavarian universities

from 1903. See the section above: “First General Steps in Würzburg Taken by Foreign Female Academics, and Bavarian Women Following.” No explicit permission from the king exclusively for Sauer could be found in the files. 110 See the Jahresbericht der Maria-Theresia Schule (1969/70), pp. 47–49. 111 See ibid., p. 49. 112 On criteria such as the role of parents or the influence of schools in the lives of the first female mathematics students, see Tobies (2008). 113 See Tobies (1997), p. 140. 114 See Hessenauer (1998), pp. 88–101. 115 Catholics were rather under-represented in the group of female students in contrast to the high population of the mainly Catholic Bavarian population; see ibid., pp. 89, 98. 116 See Abele et al. (2001), pp. 12–13; and Tobies (2004a). 117 For more information on these women and their biographies, see Tobies (2006), pp. 133, 137, 178, 182, 190, 247, 285, 298, 308, 376.

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to the rather good one in Munich.118 In Erlangen, Emmy Noether became the second German woman to gain a Ph.D. in mathematics at a German university119 and the first to do so in Bavaria.120 After that, it took long for the next female to obtain a doctorate in mathematics in Erlangen: not until 1936 did Ilse Sauter follow.121 Both came from the surrounding area: Sauter from Nuremberg and Noether directly from Erlangen. Yet while Noether studied in Erlangen and briefly in Göttingen,122 Sauter studied there (in Erlangen) for six semesters, with one semester in Munich and Würzburg each. Moreover, the first woman to gain a Ph.D. in mathematics in Würzburg, Maria Knoll,123 also came from this area and had a life similar to Sauter’s. Knoll was born in Nuremberg-Eibach as the daughter of the upholsterer Michael Knoll. She attended the same school as Sauter and finished it with the Absolutorium just one year later.124 Unlike Sauter, Knoll studied exclusively in Würzburg, except for one semester in Vienna. Both of them took the state examination to become a teacher and both submitted their doctoral thesis in 1936 (even though Knoll would not gain her Ph.D. until 1938). Knoll did her thesis in the field of differential geometry (title: Flächen mit einer Schar geodätischer Krümmungslinien und Flächen mit einer Schar kongruenter Krümmungslinien) under the supervision of Otto Volk.125 Sauter’s dissertation was Zur Theorie der Bogen n-ter (Realitäts-) Ordnung im projektiven Rn 126 which was supervised by Otto Haupt (1887–1988), who himself was born, had studied, and had done his Ph.D. in Würzburg.127 After having finished their Ph.D., they both worked as teachers, even at the same school in Nuremberg.128 Maria Knoll must have died somewhere between 1955 and 1958, because in the list of Bavarian teachers from 118 In

Nuremberg (close to Erlangen), the school situation was clearly better than in Würzburg, but not as good as in Munich. For the school situation for girls in Bavaria and in those three cities in particular, see Fuchs (1994), pp. 3–16; Hessenauer (1998), pp. 63–79; as well as Gaab (1931) and Schwind (2004) for general information. 119 The first was Marie Gernet (1865–1924); see Tobies (2004c), p. 20; and Tobies (1997), pp. 137– 140. Even before Noether, in fact, Annie Reineck, a woman from Thuringia, earned her doctoral degree in mathematics in 1907 at the University of Bern, Switzerland; see ibid., p. 137. 120 Because her life and career are relatively well-known and have been the subject of many studies, we shall not take an in-depth look at Noether here. For more information about her, see for example Tobies (2004b), pp. 107–109; Tobies (2006), p. 247; and Koreuber (2015). 121 Unlike in many cases, Sauter’s mother held a job (as a teacher); see Tobies (2006), p. 285. 122 Noether studied in Göttingen (as an auditor) in the winter semester of 1903/04 and, after a semester off due to illness, she studied in Erlangen from the winter semester of 1904/05 on; see Tobies (2004b), pp. 107–109. 123 See Tobies (2006), p. 190. 124 Sauter in 1930 and Knoll in 1931 (see ibid.). For Knoll’s CV, see Knoll (1936), p. 35. 125 Otto Volk (1892–1989) was an associate professor of mathematics in Würzburg as of 1930 and a full professor as of 1935. For more information about Volk, see Vollrath (2017), pp. 76–80. See also Knoll (1936); and Spieß (2017). 126 See Tobies (2006), p. 285. 127 Otto Haupt was a professor of mathematics in Erlangen as of 1921. The supervisor of his Ph.D. was Georg Rost, and the recommendation for it came from Emil Hilb; see ibid., p. 146. 128 See the Bayerisches Philologenjahrbuch (1950/51), pp. 118, 120; (1955/56), pp. 98–99; and (1958/59/59), pp. 127, 175.

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Fig. 2.3 Olga Sauer (on the left, she is the woman in front pointing her finger; Source Spurensuche – Maria Theresia Schule Augsburg, Schulleitung; the photograph on the right is from the Jahresbericht der Maria-Theresia Schule (1969/70), p. 47)

those years, she is identified as deceased.129 Another interesting fact about Ilse Sauter is that she became a member of the Deutsche Mathematiker-Vereinigung (DMV) in 1953.130 Having closely examined the lives of the first women with a doctorate in mathematics in Erlangen and in Würzburg, we now look at what happened to the women in Munich after their graduation: One of them studied medicine afterward and became a medical doctor,131 one married,132 and another one worked at an insurance company in Berlin.133 None of the Bavarian women with a doctorate in mathematics seemed to have pursued an academic career, except for Emmy Noether. We have now seen the outsized status of Munich in Bavaria in the case of mathematics and women’s education, but if we look at the whole German Empire—and not just Bavaria—the University of Munich does not stand out as much. In Berlin, for example, nearly twice the number of women gained a Ph.D. in mathematics by 1945, and Bonn had almost three times the number (Fig. 2.3).134

129 See

ibid. (1958/59), p. 175. Tobies (2006), p. 285. 131 This was Josephina Kapfer; see ibid., p. 178. 132 Erna Zurl married the mathematician Othmar Baier (see ibid., p. 376). The Chinese woman Süe-yung Kiang was already married before her doctorate (see ibid., 182). 133 This was Josefa von Schwarz (see ibid., p. 308). 134 At the University of Munich, there were six women; at the University of Berlin, eleven; and at the University of Bonn, seventeen (see ibid.). 130 See

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2.4 The Ministers of Cultural Affairs in Bavaria and Their Role in the History of Women’s Education While the Ministry of Cultural Affairs in Prussia and one of its representatives, Friedrich Althoff (1839–1908) in particular, have been the object of research in various valuable studies,135 the Ministry of Cultural Affairs in Bavaria136 has not yet been the primary focus in studies that concern higher education for women as far as we know. In the early years (1869–1890) of the feminist movement in Bavaria, the incumbent ministers of Cultural Affairs, Johann von Lutz and Ludwig August von Müller, had no reason to discuss much about higher education for women.137 The inquiries of women willing to study at the university were too few and the women’s associations (with, for example, all the petitions that they would later file) were just in their infancy. The rejective stance of the Ministry of Cultural Affairs can be seen in the incident from 1894 concerning a well-educated woman, Marie Derscheidt, which was outlined already. Yet the call for education for girls and women became louder in the 1890s. In 1891, the Reichstag had its first debate about education for women, and it became a Germany-wide issue.138 With the new minister of Cultural Affairs, Robert von Landmann, a reversal to the classical negative attitude was starting in the Bavarian Ministry and State Parliament. This change can be seen in the case of Marcella O’Grady, who was accepted at the university, as already mentioned above. Moreover, individual women were accepted as guest auditors for single lectures at the Bavarian universities from then on.139 However, it did not come to the change that was hoped for by many activists fighting for women’s rights. Landmann pursued a policy based on stalling tactics. He himself stated that he was not against higher education for women in principle. In 1902, Landmann said that he faced the women’s movement “in a friendly manner” but that the “female Russian students need to be feared.”140 In this context, it is interesting to quote that “[…] Russian women […] opened up higher education to women in continental Europe” and they were also “committed as 135 For

discussions of Althoff as well as the ministries of Cultural Affairs and higher education for women, see Tobies (2004c); and Tobies (2008), pp. 29–32. 136 Dr. Johann von Lutz (1826–1890) was minister of Cultural Affairs from 12/20/1869–6/1/1890; Dr. Ludwig August von Müller (1846–1895) from 6/1/1890 to 3/24/1895; Robert von Landmann (1845–1926) from 3/31/1895 to 8/10/1902; Klemens von Podewils-Dürniz (1850–1922) from 8/10/1902 to 3/1/1903; and Dr. Anton von Wehner (1850–1915) from 3/1/1903 to 2/11/1912. See Bayerisches Staatsministerium für Unterricht (1997), pp. 314–15. An account on the history of the Ministry of Cultural Affairs in Bavaria in general can be found in Rumschöttel (1997). 137 For this and the following description of the situation in Bavaria, see Fuchs (1994), pp. 27–29. 138 See Bußmann (1993), pp. 27–28. 139 From 1896 on in Munich (Gertrude Skeat; on the process of her acceptance, see Birn (2015), p. 33; and Wilke (2003), pp. 20–21) and in Würzburg (Marcella O’Grady); and from 1897 on in Erlangen. See Meister (1997), p. 58; and Birn (2015), p. 38. On the connection between the first female guest auditors in Erlangen and the Noether family, see Tollmien (2016), pp. 4–5. 140 Meister (1997), p. 53.

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much to social reform as they were to their studies.”141 Landmann probably feared that Russian women would come to Bavaria and this would lead to politicization and radicalization there.142 Furthermore, in the Bavarian Chamber of Deputies in March of 1896, Landmann said “that he would think it would not be a misfortune for the state […] if individual ladies, who are characterized by special scientific efficiency, would be able to attend individual lectures provided that the ministry allows it on a caseby-case basis.”143 Nevertheless, he wanted the decisions in the matter of education for women to depend on the actual needs of female academics.144 In the Bavarian Chamber of Deputies in April 1900 there arose another insight on Landmann’s view on education for women: Personally, he has the opinion that a woman’s physical and mental predisposition is a contradiction to the admission to all professions; as a proof for this he refers to the statistical outcome that the average working time of a retired male teacher in Upper Bavaria is 34 years 10 months and that of a female retired teacher is 18 years 3 months, so it is nearly just half of it. This outcome is even more striking when one thinks about the things a woman could have done in her natural profession [i.e. marriage and family].145

His argumentation does not seem very convincing when one looks at the fact that married women had to give up their career. Thus it is expected that the average time of a female working teacher would be distorted and cannot be easily compared to the average time of a man’s teaching career. In August of 1901, the Bavarian press wrote the following about Landmann and his practices regarding education for women146 : Mister v. Landmann […] still seems to be a downright enemy of academic education for women. This declared opponency is once again shown in a decree, which the minister of cultural affairs gave to a request to study at the Munich University of a female foreigner who already had studied medicine at German universities for several semesters. […] Mister v. Landmann refused steadfastly to accept it without giving any explanation. The Frankfurter Zeitung writes about this case: The fact is that despite the known dislike of the Bavarian ministry of cultural affairs there is a number of females at Bavarian universities […]. In contrast to other German states, in Bavaria one cherishes the procedure down to the present 141 Koblitz

(1983), pp. xiv, xv.

142 For more information on these pioneering women, see Koblitz (2000), pp. 10–25. I am indebted

to Eva Kaufholz-Soldat for this useful reference. 143 Original: “[…] dass er es für kein Unglück für den Staat halte, wenn […] einzelne Damen, die sich

durch besondere wissenschaftliche Tüchtigkeit auszeichnen, durch besonderen Ministerialerlass von Fall zu Fall zum Besuch einzelner Vorlesungen zugelassen würden.” Quoted from Gemkow (1991), p. 144. 144 See Fuchs (1994), p. 27. 145 Original: “Persönlich neige er der Anschauung zu, dass die körperliche und geistige Veranlagung der Frau ihrer Zulassung zu allen Berufsarten widerspreche; als Beweis hierfür diene das statistische Ergebnis, dass die durchschnittliche Dienstzeit der pensionierten Lehrer in Oberbayern 34 Jahre 10 Monate, die der pensionierten Lehrerinnen dagegen 18 Jahre 3 Monate, also fast nur die Hälfte […] betrage. Dieses Ergebnis sei um so auffallender, wenn man erwägt, was die Frauen in ihrem natürlichen Berufe leisten können.” Quoted from Gemkow (1991), p. 145. 146 The following text is an excerpt from the Münchner Post on August 31, 1901. It is quoted here from Bußmann (1993), p. 28–29.

62

K. Spieß day that first the ministry has to allow it, then the rector and the docent have to give their allowance. There are no elementary laws in Bavaria and one handles each case at a whim, once granting, then denying, even when the circumstances are totally analogous.147

At the end of the article, moreover, a possible reason is given for why Landmann acted so strangely in the matter of academic education for women. He is said to have done all this because he wanted to influence the forthcoming Landtag (Bavarian state parliament), and he wished to keep on the right side of some parliamentarians and parties, the conservative Zentrumspartei in particular. Landmann was a “representative of the liberal Bavarian ministerial bureaucracy.”148 However, being Catholic, he was more liberal-conservative than national-liberal. Therefore, he endeavored to maintain a good relation to the catholic majority of the parliament. Nevertheless, none of his schemes worked out very well in the end. Because of intrigues and conspiracies, Landmann had to relinquish his office on August 10, 1902.149 Landmann’s actions and views in the matter of higher education for women are hard to summarize. With him began a change in the Bavarian Ministry of Cultural Affairs, as individual women, like O’Grady, were allowed as auditors at the universities and the first regulations for the education of women in Bavaria were made. In the early years, the decision fell to the Ministry of Cultural Affairs whether or not to allow individuals to be auditors (but never as fully enrolled students) without having strict regulations, even though regulations were demanded by the philosophical faculty of the university in Munich. At the end of the year 1901, there was a resolution to simplify the procedure, but it did not work out well in the end because the regulations (women should have the university entrance certificate of a German Gymnasium or Realgymnasium) were too strict and only few could meet the conditions.150 However, all these regulations are mainly attributable to the pressure of the many women’s associations and the change in the stance toward women that was happening in the whole German Empire—and not primarily to Landmann. 147 Original: “Herr v. Landmann […] scheint immer noch ein abgesagter Feind des Frauenstudiums

zu sein. Diese erklärte Gegnerschaft kommt wieder einmal deutlich zum Ausdruck in einem Bescheide, den der Kultusminister auf ein Gesuch einer schon seit mehreren Semestern an reichsdeutschen Universitäten Medizin studierenden Ausländerin um Zulassung zum Studium an der Münchner Universität gegeben hat. […] Ohne Angaben von Gründen lehnt Herr v. Landmann das Gesuch kategorisch ab. In diesem Falle wird von der Frankf. Zeitung geschrieben: Thatsache ist, daß trotz der bekannten Abneigung des bayerischen Kultusministeriums an bayerischen Universitäten eine Anzahl von Frauen […] zum Studium zugelassen sind. […] Im Gegensatz zu anderen deutschen Ländern hält man in Bayern bis auf den heutigen Tag an dem Verfahren fest, demzufolge zuerst der Minister zu erlauben und alsdann Rektor und Dozent zuzustimmen hat. Man stellt in Bayern keinerlei Grundsätze auf und verfügt von heute auf morgen nach Lust und Belieben, bald gewährend, bald ablehnend, auch bei völlig analogen Umständen.”. 148 Herde (1998), p. 249 (translated from the German original). 149 The quarrel between professors of the University of Würzburg was the straw that broke the camel’s back. However, this quarrel was not about allowing women to study. For more information about the overthrow, see Herde (1998); and Rall (1974), pp. 93–94. 150 See Hessenauer (1998), pp. 51–52; and Knauer-Nothaft (1997a), p. 70. Further ministerial regulations concerning the status of auditors at the universities were not made until 1907; see item 8 in the bibliography.

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The competitiveness among the states in the German Empire also played an important role in the change that happened in Bavaria. For example, in a debate in the Bavarian Landtag in April of 1900, the social democrat Georg von Vollmar (1850–1922)151 demanded that Bavaria should not be “among the last [to open its universities]” and especially that Bavaria should not “lag behind Prussia.”152 It must be said about Vollmar that he was an enthusiastic advocate for higher education for women in the Bavarian Landtag and that he was the first to bring up the matter there in 1894.153 He also had connections to Hope Adams, a pioneer in the feminist movement.154 In addition, his wife, Julia Kjellberg (1849–1923), who had close contact to protagonists in the European women’s movement, also had a great effect on him.155 His wife was introduced to Vollmar by the mathematician Kovalevskaya when he visited her in Sweden. Besides, Vollmar and Kovalevskaya shared a deep and close relationship,156 and her case and circumstances were also brought up in the debate of the Bavarian Landtag, which is outlined below. As already mentioned, even female mathematicians were brought up in the debates of the Bavarian Landtag. On the one hand, a member of the conservative Bavarian Zentrumspartei, in a debate from 1900, cited Sofia Kovalevskaya157 as an example of a woman who perished and suffered because she worked in a male domain and could not act according to her real nature. The speaker in the Landtag even said: “One can rape nature but it will come back and take vengeance for that.”158 In addition, he declared that Kovalevskaya was a deterrent example for allowing women to academic education. The aforementioned social democrat Georg von Vollmar had already brought up the University of Heidelberg, where they had allowed Kovalevskaya159 as auditor, in a debate that took place in 1894. Opposed to the Zentrumspartei, Vollmar used this as a positive example and he appealed that such progress should also happen at Bavarian universities.160 The other example of a female mathematician mentioned at the Bavarian Landtag was Maria Gaetana Agnesi.161 She was 151 For

more information about Vollmar and his political attitudes and actions, see Spindler (2003), p. 347–350. 152 Translated from the German original. See Birn (2015), pp. 40–41; and Meister (1997), p. 51. For more information about the matter of higher education for women in the debates of the Bavarian Landtag, see item 5 in the bibliography. 153 See Meister (1997), pp. 51–53; Wilke (2003), p. 25; and Bußmann (1993), p. 28. 154 See Kraus (2000), pp. 143, 146, 149–150, 153; and item 5 in the bibliography, pp. 52–53. 155 See ibid., pp. 48, 52. 156 This was kindly brought to my attention by Eva Kaufholz-Soldat. 157 For more information on the mathematician Kovalevskaya, see Tollmien (1997). 158 Original: “Man kann die Natur vergewaltigen, aber sie kommt und rächt sich dafür.” See item 5 in the bibliography, p. 71; and Verhandlungen der Kammer der Abgeordneten des Bayerischen Landtags im Jahre 1899/1900, p. 102. 159 In this context, Vollmar referred to Kovalevskaya as “one of the most significant women of his time.” See Verhandlungen der Kammer der Abgeordneten des Bayerischen Landtags (1893/94), pp. 132–133. 160 See item 5 in the bibliography, p. 53. 161 For more information about the life and work of the Italian astronomer Maria Gaetana Agnesi (1718–1799), see Bernardi (2016), pp. 115–120.

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contrasted with Kovalevskaya because Agnesi later advocated what was at that time commonly thought as a woman’s real destiny and turned to traditional female roles such as caring for the sick. For this reason, she was singled out by a member of the conservative Zentrumspartei in 1900.162 However, around 1902, there seemed to be a shift in the attitude toward academic education for women, even among the rows of the conservatives.163 Even more important than Landmann was the new minister of Cultural Affairs, Anton von Wehner.164 Under his guidance, there was another twist in the matter when universities were opened for women to enroll in 1903. Wehner was born on November 16, 1850 in Schillingsfürst as son of the cloth-maker Nikolaus Wehner. He studied law at the University of Munich and earned his Ph.D. in 1876.165 As of August 1878, he worked at the Ministry of Cultural Affairs in Bavaria, and since 1903 as the minister of Cultural Affairs until his resignation in 1912.166 During his time in office, the political fronts in Bavaria were at a deadlock, which led to the dissolution of the Landtag and re-elections in 1912 in which the social democrats and liberals lost. In the aftermath of this election, the administration around Podewils-Dürniz resigned, and so did Wehner. He died on March 10, 1915. Regarding higher education for women and for girls, Wehner introduced some innovations, though it still took a very long time until women had similar opportunities as their male counterparts. What is seldom (if at all) found in literature about women’s education in Bavaria is a potential motivation behind Wehner’s positions.167 Wehner had a highly gifted daughter, Klara von Wehner (1881–?), together with his first wife.168 Klara von Wehner attended a higher secondary school for girls and an educational institute in Zangberg. After finishing that, she prepared herself autonomously for the Absolutorialprüfung. For this purpose, she had private lessons; for example, she was taught mathematics by Ernst Piechler, a professor of mathematics at the royal 162 See item 5 in the bibliography, p. 70; and Verhandlungen der Kammer der Abgeordneten des Bayerischen Landtags (1899/1900), pp. 101–102. 163 The finance committee discussed the matter again in June of 1902. The committee representative as well as his substitute favored the idea of treating women as equal to the male students in regard to preconditions and allowing them to enroll. See Meister (1997), p. 53. 164 After the overthrow of Landmann, Podewils-Dürniz shortly took over the Ministry of Cultural Affairs, but after being involved in the overthrow of Minister Crailsheim, Podewils-Dürniz himself replaced Crailsheim as Minister des Kgl. Hauses und des Äußeren und Vorsitzender im Ministerrat, and Wehner was appointed minister of Cultural Affairs. See Spindler (1979), p. 405. 165 The title of his dissertation was Die Gerichtsverfassung der Stadt München von der Entstehung bis zum Untergang der Ratsverfassung. He was considered highly talented and a top legal expert. For this and the following, see Hetzer and Stephan (2008), pp. 90–91, 100–101. 166 In addition to his work at the Ministry for Women’s Education, he also espoused cultural heritage preservation. On his activities in this regard, see ibid., pp. 89–101. 167 This motivation was also suspected in an article published in the Frankfurter Kurier on June 24, 1907; see item 6 in the bibliography. 168 This is mentioned neither in Häntzschel and Bußmann (1997) nor in Hessenauer (1998), but rather in a book on Wehner’s work in preserving the cultural heritage of Bavaria; see Hetzer and Stephan (2008), p. 102.

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Theresiengymnasium169 in Munich. Unfortunately, it is not known what exactly she learned in these lessons. She was accepted to take the exam at the royal Wilhelmsgymnasium170 in Munich171 in 1907. Furthermore, Klara von Wehner was among the first female medical students at the University of Munich.172 Giving his daughter an opportunity might very well have been part of Wehner’s motivation for opening the universities in Bavaria for women. A similar motivation is mentioned for the Prussian Minister of Cultural Affairs, Robert Bosse.173 Indeed, tackling this situation was one of his first actions when Wehner began as the Minister of Cultural Affairs.174 Nevertheless, “from his argumentation in favor of the admission of women to universities from 1903, it is clear that the whole enactment was more likely a compromise than a real concession and without real practical use.”175 In fact, the enactment from 1903 was even harmful for some women. Foreigners without the German Reifezeugnis were not allowed to enroll, and neither were women who had gained their Absolutorium at a German Mädchengymnasium, which, in particular, meant the Karlsruher Mädchengymnasium.176 Even without having regulations for schools for girls, meeting the requirements for enrollment was hard for women, and those school regulations were fixed only in 1911 by the Ministry of Cultural Affairs.177 In contrast to Bavaria, Prussia had a whole program when they made their universities accessible for women. When this was done in 1908, the Ministry of Cultural Affairs in Prussia also enacted regulations for schools for girls at the same time.178 Moreover, Prussia became the first state in the German Empire to introduce mathematics and the natural sciences as subjects at higher schools for girls, while Bavaria again lagged behind.179

169 At

that time the Theresiengymnasium was a school for boys; see “The Theresiengymnasium in Munich.”. 170 The Wilhelmsgymnasium is the oldest Gymnasium in Upper Bavaria. Girls have been allowed at this school since the 1970 s; see “The Wilhelmgymnasium in Munich.”. 171 In contrast to many other girls who applied to take their Absolutorialsprüfung, Klara von Wehner was allowed to take it in her home town. Often, girls had to take their exam in cities away from home. Olga Sauer, for instance, had to travel to Würzburg; see items 3 and 1 in the bibliography. 172 Hetzer and Stephan (2008), p. 102. 173 See Albisetti (1988), p. 230. 174 See Hessenauer (1998), p. 80. 175 Knauer-Nothaft (1997a), p. 72 (translated from the German original). 176 The Karlsruher Mädchengymnasium was the first Gymnasium for girls in Germany, and there they could take the exam for the Absolutorium directly at the school (as of 1899) and not, as was usual, at a school for boys. For more information about this school, see Kaller (1992). It was only in October 1905, after some back and forth, when students from this school were allowed at Bavarian universities, see Knauer-Nothaft (1997a), pp. 72, 299; Boehm (1958), p. 321; and item 8 in the bibliography, letter to the senates of the three Bavarian universities from the Ministry of Cultural Affairs from March 28, 1905. 177 See Hessenauer (1998), pp. 63–72. 178 See Hessenauer (1998), p. 85. 179 See Abele et al. (2001), p. 9.

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2.5 Conclusion All in all, one can see that, in the matter of women’s (mathematical) education, it can be interesting to look beyond Prussia and examine the procedures that took place in the smaller and more rural state of Bavaria.180 Even though the government allowed women to study there as early as 1903, we have seen that this was not as progressive as it seemed, given all the restrictions for enrollment and the lack of proper school education for girls. Moreover, we have shown that even though Würzburg initially seemed to play a special role among the Bavarian universities (with its extramural lectures and the efforts that some professors had put into them), in the end, Munich was the superior university by a large margin. Finally, we have examined the role and the stance of the Ministry of Cultural Affairs. Things could only develop when a new minister came into office with new ideas and a different motivation from the previous one. However, other outstanding male (who obviously had more influence and power at that time) and female personalities were still needed—such as Georg von Vollmar, who persistently brought up the matter in the Landtag and fought for the right for women to gain equal academic knowledge, or Karl Lehmann, who wanted to give his relatives opportunities, or Marcella O’Grady and Jenny Danziger, who disobeyed the rules and norms in Germany at the time and tried to gain access to university. In the end, they succeeded, sent a positive signal, and spread a spark of hope for other women.

Bibliography Archival Resources Bayerisches Hauptstaatsarchiv 1. [BayHstA MK 20729]: Bayerisches Hauptstaatsarchiv: MK 20729, Realgymnasien. Frauenstudium (Aufnahmegesuche, Zulassung zur Absolutorialprüfung). (Laufzeit: 1901–1917). 2. [BayHstA MK 20734]: Bayerisches Hauptstaatsarchiv: MK 20734, Oberrealschulen. Frauenstudium. Zulassung zur Absolutorialprüfung. (Laufzeit: 1909–1935). 3. [BayHstA MK 15032]: Bayerisches Hauptstaatsarchiv: MK 15032, Frauenstudium i.g. Gesuche um Zulassung zu Absolutorial- und Lehramtsprüfungen und Zuweisung an Studienanstalten. (Laufzeit: 1901–1908). 4. [BayHstA MK 15033]: Bayerisches Hauptstaatsarchiv: MK 15033, Frauenstudium i.g. Gesuche um Zulassung zu Absolutorial- und Lehramtsprüfungen und Zuweisung an Studienanstalten. (Laufzeit: 1909–1916). 180 An

interesting fact is that the first woman to habilitate in mathematics in Germany did so in Prussia (Emmy Noether, herself born in Bavaria, habilitated at the University of Göttingen in 1919). However, the first habilitation by a woman overall in Germany took place in Bavaria (Adele Hartmann at the University of Munich in 1918). See Wilke (2003), p. 33; and Tobies (1997), pp. 141, 181.

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5. [BayHstA MK ZA 6770]: Bayerisches Hauptstaatsarchiv: MK ZA 6770, Die Anfänge des akademischen Frauenstudiums in der politischen Debatte (1894–1904), Zulassungsarbeit von Sylvia Lederer. 6. [BayHstA MK 11117]: Bayerisches Hauptstaatsarchiv: MK 11117, Universitäten. Frauenstudium im allgemeinen. (Laufzeit: 1905–1916). 7. [BayHstA MK 47297]: Bayerisches Hauptstaatsarchiv: MK 47297, Sauer, Olga – Studienprofessor – 30.7.1891. (Laufzeit: 1916–1951).

Universitätsarchiv Würzburg 8. [ARS 1651]: Archiv des Rektorats und des Senats der Universität Würzburg: 1651 Zulassung von Frauen zum Universitätsstudium sowie Vorträge für den Verein Frauenheil. 9. [ARS 2012–2048]: Archiv des Rektorats und des Senats der Universitat Würzburg: 2012–2048 Immatrikulationsprotokolle und Semestralregister. 10. UA WÜ Inskriptionslisten Olga Sauer Wintersemester 1912/1913. 11. UA WÜ Inskriptionslisten Luise Bauer Wintersemester 1914/1915. 12. UA WÜ Inskriptionslisten Elsa Trammer Wintersemester 1915/1916. 13. UA WÜ Inskriptionslisten Hilde Fuss Wintersemester 1915/1916. 14. UA WÜ Inskriptionslisten Agnes Braun Wintersemester 1915/1916. 15. UA WÜ Inskriptionslisten Alma Wolffhardt Wintersemester 1916/1917.

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Jahresbericht der Maria-Theresia Schule in Augsburg (1969/70). Kaiser, G. (1995). Spurensuche: Studentinnen und Wissenschaftlerinnen an der Julius-MaximiliansUniversität Würzburg von den Anfängen bis Heute. Würzburg: Max-Schimmel-Verlag. Kaiser, G. (1997). Studentinnen in Würzburg, München und Erlangen. In: Häntzschel, H. & Bußmann, H. (Eds.), Bedrohlich gescheit: Ein Jahrhundert Frauen und Wissenschaft in Bayern (pp. 57–68). Nördlingen: C. H. Beck. Kaller, G. (1992). Mädchenbildung und Frauenstudium: Die Gründung des ersten deutschen Mädchengymnasiums in Karlsruhe und die Anfänge des Frauenstudiums an den badischen Universitäten (1890–1910). Zeitschrift für die Geschichte des Oberrheins 140 (pp. 361–375). Band. Koblitz, A. H. (1983). A convergence of lives: Sofia Kovalevskaia—Scientist, writer, revolutionary. Boston: Birkhäuser. Koblitz, A. H. (2000). Science, women, and revolution in Russia. Amsterdam: Harwood Academic Publishers. Koreuber, M. (2015). Emmy Noether, die Noether-Schule und die moderne Algebra: Zur Geschichte einer kulturellen Bewegung. Berlin: Springer Spektrum. Knauer-Nothaft, C. (1997a). Bayerns Töchter auf dem Weg zur Alma Mater: Das höhere Mädchenschulwesen. In H. Häntzschel & H. Bußmann (Eds.), Bedrohlich gescheit: Ein Jahrhundert Frauen und Wissenschaft in Bayern (pp. 69–83). C. H. Beck: Nördlingen. Knauer-Nothaft, C. (1997b). ‘Wichtige Pionierposten der einen oder anderen Weltanschauenen’: Die Gymnasiallehrerin. In: Häntzschel, H. & Bußmann, H. (Eds.), Bedrohlich gescheit: Ein Jahrhundert Frauen und Wissenschaft in Bayern (pp. 152–163). Nördlingen: C. H. Beck. Knoll, M. (1936). Flächen mit einer Schar geodätischer Krümmungslinien und Flächen mit einer Schar kongruenter Krümmungslinien. Doctoral dissertation, University of Würzburg. Kraus, M. (2000). Die Lebensentwürfe und Reformvorschläge der Ärztin Hope Bridges Adams Lehmann (1855–1916). In E. Dickmann & E. Schöck-Quinteros (Eds.), Barrieren und Karrieren: Die Anfänge des Frauenstudiums in Deutschland (pp. 143–157). Berlin: Trafo Verlag Dr. Wolfgang Weist. Kreiner-Wolffhardt, A. (1942). Das Lerchennest: Ein Buch vom Glück des Kinderreichtums das alle angeht. Munich: J. F. Lehmanns Verlag. Lerman, K. A. (2015). Imperial Governance. In M. Jefferies (Ed.), The Ashgate research companion to imperial Germany (pp. 13–32). Farnham: Ashgate Publishing. Meister, M. (1997). Über die Anfänge des Frauenstudiums in Bayern. In H. Häntzschel & H. Bußmann (Eds.), Bedrohlich gescheit: Ein Jahrhundert Frauen und Wissenschaft in Bayern (pp. 35–56). C. H. Beck: Nördlingen. Mertens, L. (1991). Vernachlässigte Töchter der Alma Mater: Ein sozialhistorischer und bildungssoziologischer Beitrag zur strukturellen Entwicklung des Frauenstudiums in Deutschland seit der Jahrhundertwende. Berlin: Duncker & Humblot. Müller, F. L. (2015). The German monarchies. In M. Jefferie (Ed.), The Ashgate research companion to imperial Germany (pp. 55–73). Farnham: Ashgate Publishing. Nanu, H. A. (1904). Zur Psychologie der Zahlauffassung. Doctoral dissertation, University of Würzburg, submitted May 17, 1904. Panzer, M. A. (1993). ‘Zwischen Küche und Katheder’: Bürgerliche Frauen um die Jahrhundertwende, 1890–1915. In S. Krafft (Ed.), Frauenleben in Bayern von der Jahrhundertwende bis zur Trümmerzeit (pp. 86–118). Bayerische Landeszentrale für Politische Bildungsarbeit: Munich. Parshall, K. H. (2015). Training women in mathematical research: The first fifty years of Bryn Mawr College (1885–1935). The Mathematical Intelligencer, 37(2), 71–83. Quint, W. (1997). Wissenschaft und Kunst in Bayern von 1945 bis heute. In W. Oberholzner (Ed.), Tradition und Perspektive: 150 Jahre Bayerisches Kultusministerium (pp. 126–155). Bamberg: St. Otto-Verlag. Rall, H. (1974). Zeittafeln zur Geschichte Bayerns und der mit Bayern verknüpften oder darin aufgegangenen Territorien. Munich: Süddeutscher Verlag.

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Rossiter, M. W. (1984). Women scientists in America: Struggles and strategies to 1940. Baltimore: Johns Hopkins University Press. Rumschöttel, H. (1997). Geschichte des bayerischen Kultusministeriums von der Errichtung bis zum Ende des Zweiten Weltkriegs. In W. Oberholzner (Ed.), Tradition und Perspektive: 150 Jahre Bayerisches Kultusministerium (pp. 45–101). Bamberg: St. Otto-Verlag. Schwind, H. (2004). Bildung für Mädchen und Frauen im Bayern der Kaiserzeit: Die institutionellen Bildungsmöglichkeiten, 1871–1918. Osnabrück: Der Andere Verlag. Spieß, K. (2017). The University of Würzburg as a case study for university education of women in mathematics in Germany in the first half of the 20th century. Oberwolfach Reports, 14(1), 120–122. Spindler, M. (Ed.). (1979). Handbuch der bayerischen Geschichte. Vierter Band: Das neue Bayern, 1800–1970. Zweiter Teilband. Munich: C. H. Beck. Spindler, M. (Ed.). (2003). Handbuch der bayerischen Geschichte. Vierter Band: Das Neue Bayern, 1800 bis zur Gegenwart. Erster Teilband: Staat und Politik (2nd ed.). Munich: C. H. Beck. Spitznagel, P. (1974). Studentenschaft und Nationalsozialismus in Würzburg, 1927–1933. Doctoral diss: University of Würzburg. Spurensuche – Maria Theresia Schule Augsburg, Schulleitung. Retrieved January 26, 2018, from http://www.datenmatrix.de/projekte/hdbg/spurensuche/content/medienschulchronik/zoom/ pop-up_chronic_einleitung_020.htm. Sticker, G. (1932). Entwicklungsgeschichte der Medizinischen Fakultät an der Alma Mater Julia. In M. Buchner (Ed.), Aus der Vergangenheit der Universität Würzburg: Festschrift zum 350jährigen Bestehen der Universität (pp. 383–799). Berlin: Julius Springer. The School Year 1944/45 at the Maria-Theresia School. Retrieved January 28, 2018, from http:// www.datenmatrix.de/projekte/hdbg/spurensuche/content/pop-up-schulchronikl-1944-45-03. htm. The Theresiengymnasium in Munich. Retrieved January 28, 2018, from https://de.wikipedia.org/ wiki/Theresien-Gymnasium_München. Tobies, R. (1997). Mathematikerinnen und ihre Doktorväter. In R. Tobies (Ed.), “Aller Männerkultur zum Trotz”: Frauen in Mathematik und Naturwissenschaften (pp. 131–158). Frankfurt am Main: Campus Verlag. Tobies, R. (2004a). Berufswege im Lehramt in der ersten Hälfte des 20. Jahrhunderts. In: A. E. Abele-Brehm, et al. (Eds.), Traumjob Mathematik! Berufswege von Frauen und Männern in der Mathematik (pp. 18–37). Basel: Birkhäuser. Tobies, R. (2004b). Berufswege promovierter Mathematikerinnen und Mathematiker: Wege in der ersten Hälfte des 20. Jahrhunderts. In: Abele-Brehm, A. E., et al. (Eds.). Traumjob Mathematik! Berufswege von Frauen und Männern in der Mathematik (pp. 89–120). Basel: Birkhäuser. Tobies, R. (2004c). Mathematikerinnen und Mathematiker um 1900 in Deutschland und international. In: A. E. Abele-Brehm, et al. (Eds.). Traumjob Mathematik! Berufswege von Frauen und Männern in der Mathematik (pp. 133–147). Basel: Birkhäuser. Tobies, R. (2006). Biographisches Lexikon in Mathematik promovierter Personen: WS 1907/08 bis WS 1944/45. Augsburg: Rauner Verlag. Tobies, R. (2008). Einführung: Einflussfaktoren auf die Karriere von Frauen in Mathematik, Naturwissenschaften und Technik. In: R. Tobies (Ed.), “Aller Männerkultur zum Trotz”: Frauen in Mathematik, Naturwissenschaften und Technik (pp. 21–80). Frankfurt am Main: Campus Verlag. Tobies, R. (2017) Thekla Freytag: Die Mädchen werden beweisen, dass auch sie exakt und logisch denken können… In: G. Wolfschmidt (Ed.), Festschrift – Proceedings of the Scriba Memorial Meeting – History of Mathematics (Nuncius Hamburgensis – Beiträge zur Geschichte der Naturwissenschaften, Bd. 36) (pp. 344–393). Hamburg: tredition. Tollmien, C. (2016). ‘Das mathematische Pensum hat sie sich durch Privatunterricht angeeignet’: Emmy Noethers zielstrebiger Weg an die Universität. In A. Blunck, et al. (Eds.), Mathematik und Gender (pp. 1–12). Hildesheim: Franzbecker Verlag. Tollmien, C. (1997). Zwei erste Promotionen: Die Mathematikerin Sofja Kowalewskaja und die Chemikerin Julia Lermontowa. Mit Anhang: Dokumentation der Promotionsunterlagen. In: R.

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Tobies (Ed.), “Aller Männerkultur zum Trotz”: Frauen in Mathematik und Naturwissenschaften (pp. 83–129). Frankfurt am Main: Campus Verlag. Trautwein, R. (2003). Frauenleben in Fürth: Spurensammlung und Wegweiser. Nuremberg: emweVerlag. Trautwein, R. (2007). 1000 Fürther FrauenLeben: Eine ergänzende Spurensammlung. Nuremberg: emwe-Verlag. Ullmann, H.-P. (2005). Politik im deutschen Kaiserreich 1871–1918. 2 (durchgesehene ed.). München: R. Oldenbourg Verlag. Ullrich, V. (2013). Die nervöse Grossmacht 1871–1918. Aufstieg und Untergang des deutschen Kaiserreichs. Erweiterte Neuausgabe. Frankfurt am Main: Fischer Taschenbuch. Verhandlungen der Kammer der Abgeordneten des Bayerischen Landtags im Jahre 1893/94. Stenographische Berichte. Vol. III, no. 92. Verhandlungen der Kammer der Abgeordneten des Bayerischen Landtags im Jahre 1899/1900. Stenographische Berichte. Vol. IV, no. 121. Vollrath, H.-J. (1995). Friedrich Prym (1841–1915), Mathematiker. In P. Baumgart (Ed.), Lebensbilder bedeutender Würzburger Professoren (pp. 158–177). Neustadt an der Aisch: Degener Verlag. Vollrath, H.-J. (2017). Würzburger Mathematiker: Aus der Geschichte der Julius-MaximiliansUniversität (2nd ed.). Würzburg: Verlag Königshausen & Neumann. The Wilhelmsgymnasium in Munich. Retrieved January 28, 2018, from https://de.wikipedia.org/ wiki/Wilhelmsgymnasium_München. Wilke, C. (2003). Forschen, Lehren, Aufbegehren. 100 Jahre akademische Bildung von Frauen in Bayern. Munich: Herbert Utz Verlag. Wolf, P. (Ed.). (2005). Spuren: Die jüdischen Schülerinnen und die Zeit des Nationalsozialismus an der Maria-Theresia-Schule Augsburg. Augsburg: Wißner-Verlag. Würzburger Generalanzeiger (July 21, 1894). Würzburger Universitäts-Zeitung 8 (1920). Würzburg: C. J. Becker. Zweiter Jahresbericht des Vereins “Frauenheil” (E.V.) für 1899 (1900). Würzburg: Druck der Königl. Universitätsdruckerei von H. Stürtz.

Chapter 3

Women and Mathematics at the Universities in Prague Martina Beˇcváˇrová

Abstract The chapter is devoted to the largely unknown and mostly forgotten history of female candidates of Ph.D. degree in mathematics at the German University in Prague during its whole existence (1882–1945) and at the Czech University in Prague (1882–1920), respectively, at the Charles University in Prague (1920–1945). In the first part, a short description of the historical background (for instance, woman’s studies at the secondary schools and universities in the Czech lands, the rules and regulations of the process for obtaining the doctoral degrees as well as a statistical overview of all Ph.D. degrees in mathematics awarded at both universities in Prague) is given for a better understanding of the situation and some difficulties with the female doctoral procedures. In the second part, a detailed analysis of the successful doctoral procedures of three women graduated in mathematics at the German University in Prague, and the successful doctoral procedures of eight women (one graduated at the Czech University in Prague and seven graduated at the Charles University in Prague) and one unsuccessful doctoral procedure at the Charles University in Prague is presented with showing their families’ background, their life stories, professional activities, mathematical interests and results and fates of their families and relatives. The text is based on the deep and long time studies of various funds of the Archive of the Charles University in Prague, the Archive of the Czech Technical University in Prague, the National Archive of the Czech Republic (Prague), the Archive of the Academy of Sciences of the Czech Republic (Prague), the Prague City Archives, the Jewish Museum in Prague and some special local archives which collected the register books of births, marriages and deaths, respectively, on the studies of the proceedings of secondary schools which are deposited in the special collection in the National Library of the Czech Republic (Prague-Hostivaˇr).

M. Beˇcváˇrová (B) Department of Applied Mathematics, Faculty of Transportation Sciences, Czech Technical University in Prague, Na Florenci 25, 11000 Prague 1, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_3

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3.1 Introduction Based on hitherto unknown archival materials in the Czech Republic, original mathematical works, selected schoolbooks and monographs, diverse secondary literature, and oral communications with family members, this work offers a detailed analysis of the lives, professional activities, and fates of 12 women who defended their doctorates in mathematics at the Faculty of Philosophy or the Faculty of Sciences of the German University in Prague between 1900 and 1945, at the Czech University in Prague between 1900 and 1920, or at Charles University in Prague between 1921 and 1945. I have tried to fill a gap in the available scholarly literature, because no work has yet to focus on these topics. Moreover, I also hope in part to pay homage to the first female doctors of mathematics, whose names are almost forgotten in my country.1

3.2 The First Women to Study at the Universities in Prague In the second half of the nineteenth century, Prague was a typical multicultural town in central Europe, with historical districts and industrial suburbs. It was the center of economics, business, traffic, schools, science, etc. According the census of 1900, it had 474,901 residents with different languages (Czechs 91.35%, Germans 8.57%, others 0.08%), religions (Catholic 89.02%, Jewish 8.72%, Evangelic 1.82%, others 0.35%, and people without religion 0.09%), and habits. There were many secondary schools and high schools, and there were four universities.2 The Prague Technical University was founded in 1707 as the local polytechnical school. From the beginning, lectures were delivered there in German. As of 1864, parallel regular lectures and seminars in mathematics, physics, descriptive geometry, etc. were taught there simultaneously in Czech and German. In 1869, the Prague Technical University was divided into two independent schools, the so-called Czech Technical University in Prague and the German Technical University in Prague.3 The University in Prague, which was founded in 1348 by the king of Bohemia, and after 1355, Holy Roman emperor, is the oldest university in Central Europe. On 28 February 1882 in Vienna, the law on the foundation of two independent universities in Prague was signed, and the Czech University in Prague and the German University in Prague were founded. Let us note that from the end of the eighteenth century, the 1 This

research was supported by the project The Impact of WWI on the Formation and Transformation of the Scientific Life of the Mathematical Community (GA CR 18-00449S). 2 For more information, see, for instance, the entry on Prague in Ott˚ uv Slovník Nauˇcný [Otto’s Encyclopedia]. 3 Here we will not focus on these two universities because they did not award doctorates in mathematics during the first half of the twentieth century. For more information about the Czech Technical University in Prague and the German Technical University in Prague, see Velflík (1906–1925); Jílek and Lomiˇc (1973); Tayerlová et al. (2004); and Birk (1931).

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lectures were taught in German. From 1871 on, Czech and German parallel regular lectures and seminars in mathematics, physics, medicine, laws, history, etc. were offered at the University in Prague. The Czech University in Prague and the German University in Prague were typical European universities of the nineteenth century. There were four faculties (Philosophy, Medicine, Law, and Theology) specialized in educating and preparing teachers, lawyers, physicians, civil servants and officers, priests, etc. There were many obligatory lectures and seminars and only a few optional lectures and seminars.4 It was not easy for women in the Czech lands to study at secondary schools or universities. In the first half of the nineteenth century, higher education for girls and women was almost unheard of. The reason is that a woman was supposed to be a good wife, mother, and patriot—she should bring up children with care, responsibility, and in the spirit of patriotism, thereby ensuring public respect for her family. If necessary, she should help her husband to run his trade. Public educational institutions for women as well as private ones (mostly religious and aristocratic) were rare and conformed to the above idea of women’s mission. A more significant change occurred in the early 1860s, when activities of certain associations started to develop after the downfall of Bach’s absolutistic government, and the network of schools of all types and levels began to expand. In 1865, the entrepreneur, patron of science and philanthropist Vojtˇech (Vojta, Adalbert) Náprstek (original name Fingerhut, 1826–1894) founded—in his house “UHalánk˚u” on the Bethlem Square in Prague—the American Ladies’ Club (Americký klub dam), which became the oldest women’s organization in the territory of the Austrian monarchy. It was a center where women, especially those from the middle class, were educated. Women could use the library and attend lectures on the natural sciences, mathematics, medicine, philosophy, history, arts, politics, and also technology. The lecturers were Czech scientists, travelers, writers, artists, politicians, and others. The club members made visits to various factories, hospitals, social care institutions, an astronomical observatory, and so on. The activities of the club were very popular and trendy at that time.5 In that period, some women from rich families understood that, without education and opportunities for professional employment, they would not have found jobs, and so they would have been dependent on their parents, husbands, or families. In the year 1865, as a result of an initiative by Eliška Krásnohorská (original name Alžbˇeta Pechová, 1847–1926), Sofie Podlipská (née Rottová, 1833– 1897), Johanna Mužáková (née Rottová, alias Karolina Svˇetlá, 1830–1899), and Marie Riegrová (née Palacká, 1833–1891), the first Czech Manufacturing Associˇ ation (Ceský výrobní spolek) was established, followed by a technical school for girls, which was transformed into the Women’s Manufacturing Association (Ženský 4 For more information about Charles University in Prague, see Havránek and Pousta (1997–1998).

On the German University in Prague, see Seibt (1948); and Míšková (2002). For more information about mathematics and the mathematical community at the German University in Prague, see Beˇcváˇrová (2016a). 5 For further information about the American Ladies’ Club, see Secká (2013).

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výrobní spolek) in 1871. Within the framework of this association, women could attend educational courses (in modern languages, economics, economy and civil service, drawing, engraving, nursing, and other subjects) and prepare themselves for practical employment. The first higher school for girls, Vyšší dívˇcí škola, which was intended mainly for girls descending from middle and higher social levels, was opened in Prague in the year 1863. It provided secondary education, though without the possibility of passing a graduation examination. Four years later, the school obtained a building on Vodiˇckova Street, became popular, and was attended by many students. Among others, the school was attended by renowned representatives of Czech culture such as the painter Zdenka Braunerová (1858–1934), the writer Helena Malíˇrová (née Nosková, 1874–1940), the opera singer Ema Destinnová (1878–1930), and the actresses Hana Kvapilová (1860–1907) and R˚užena Nasková (née Nosková, 1884–1960). The American Ladies’ Club and the Women’s Manufacturing Association gave rise to an initiative to promote the right of women to study at university. In the mid1870s, the first three women—young members of the American Ladies’ Club [Anna Bayerová (1853–1924), Bohuslava Kecková (1854–1911), and Julie Kurková]—left for Switzerland in order to pursue their studies because women were not allowed to study at universities in the Austro-Hungarian Empire.6 After much trouble, the first two completed their studies of medicine in Bern (1881) and Zürich (1880), respectively, and the third died shortly before completing her studies of philosophy. After their return to Bohemia, the young female physicians were not allowed to open their practices. Their lives and activities were followed with empathy and hope by the Czech public.7 An important year with regard to women’s education was the year 1878, when women gained the right to pass a graduation examination at traditional secondary schools (Gymnasium for boys). However, there was no institution to prepare them for this examination. In addition, women with secondary school education had limited possibilities of employment in the monarchy. The emancipation of women with regard to education was not easy. In the year 1890, E. Krásnohorská founded the Minerva Association, which had a clearly defined goal: to open a Gymnasium to prepare girls for university studies under the same conditions as boys. In September of 1890—after many petitions, interventions, and much lobbying—the Empire Council in Vienna amended the obsolete legislation and approved Minerva: the first Gymnasium for girls in Central Europe. In 1892, the first Gymnasium for girls in Vienna was established, inspired by the Prague Gymnasium for girls. The first students of Minerva graduated in the year 1895 at the so-called Academic Gymnasium (for boys) in Prague. The examinations were stricter and more demanding for girls than for boys.8 6 It

should be mentioned that, in Switzerland, women were permitted to study at university as early as the 1860s (e.g. at the Technical School and at University of Zürich since 1864). 7 For more information, see Báhenská (2005); and Kopác (1968). 8 Some of the women to be mentioned were students there. For more information, see Uhrová (2012).

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It might seem that after 1878 there was no obstacle for women to study at a secondary school and a university. In the same year, the Ministry of Education and Enlightenment issued a decree that allowed women to attend all “university lectures suitable for women”. However, the reality was quite different. The first five graduates of Minerva who applied for admittance to the Faculty of Medicine in Prague were refused by a decision of the professors in 1895. Complicated negotiations were needed to enable women to study.9 In 1895, the Faculty of Philosophy of the Czech University in Prague10 admitted six Minerva graduates as so-called visiting students, which meant on probation. At the same time, the Faculty of Medicine of the German University in Prague11 allowed the first three Minerva graduates to study. In 1896, the Faculty of Medicine of the Czech University in Prague also allowed women to be admitted to study as visiting students. Starting in that year, the Austro-Hungarian Empire began to recognize the foreign diplomas of women following a demanding recognition procedure at a university of the monarchy. As of 1897, all faculties of philosophy in the monarchy admitted women to regular studies without obstructions and under the same conditions as men. It is interesting that professors of mathematics and natural sciences at both universities in Prague (for example, F. J. Studniˇcka, G. H. W. Kowalewski) were not conservative. They supported women and helped them to study at secondary schools, and they arranged that women could attend their lectures as visiting students. Three years later, women had the right to study at all faculties of medicine in the whole monarchy. In 1900, eight women completed their studies at the Faculty of Philosophy of the Czech University in Prague, where they were prepared for the profession of secondary school teachers in various subjects of humanistic and natural sciences (mathematics, physics, geography, and history).12 Some of them gained a position at the Prague Minerva or the Girls’ Lycée of the Vesna Association in Brno.13 In 1908, 9 See some Czech newspapers from this time and especially the reports from the professors’ meetings

at the Czech University and the German University in Prague, which are held in the archive of Charles University. 10 During the years 1882–1920, the university used the name Ceská ˇ Karlo-Ferdinandova universita v Praze. The university used the name Universita Karlova as of the year 1920, when the act “Lex Mareš” was passed, codifying the mutual relationship of the two Prague universities. Later we will use the abbreviated form Czech University. 11 During the years 1882–1919, the university used the name Nˇ emecká Karlo-Ferdinandova universita v Praze. From 1920 to 1939, it was called Nˇemecká univerzita v Praze, and beginning in 1939 it was known as Nˇemecká Karlova univerzita v Praze. Here we will use the abbreviated form German University. 12 Women were permitted to pass examinations for teaching proficiency since the year 1904. Until the end of the World War I, however, they were allowed to teach at secondary schools for girls only. After the formation of the Czechoslovak Republic, they could teach at secondary schools of all types. 13 The Vesna Association was established in 1870 in Brno as the so-called singers’ union. Later on, it was changed into an educational and manufacturing association for women. In 1886, thanks to Eliška Machová (1858–1926), an association activist and teacher, the association established a Czech school for continuing education for girls. This school soon changed into a technical school

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the first eight women completed their studies of pharmacy and, at the same time, the Association of Academically Educated Women was established. In 1901, the ˇ first two female doctors—Marie Zdeˇnka Baborová-Ciháková (1877–1937; zoology) and Marie Fabiánová (1872–1943; mathematics)14 —graduated from the Faculty of Philosophy of the Czech University in Prague. In 1902, Albína Honzáková (1875– 1940) graduated from the Faculty of Medicine of the Czech University in Prague.15 The German University in Prague was more open with regard to women’s studies, but more conservative with regard to female doctorates; the first women, Hedwig Fischmann (1885–?) and Charlotta Weil (1886–?), were awarded the doctorate at the Faculty of Philosophy of the German University in Prague as late as 1908 (the former in the subject of German language and literature, the latter in chemistry).16 At the time of World War I, the number of women studying increased. Women filled in the gaps left by male soldiers. In 1918, the “Washington Declaration”17 adopted a principle that women are equal to men with regard to politics and social and cultural matters. In 1918, the independent Czechoslovak Republic was formed, which, among other things, gave women suffrage and the right to study at faculties of law as well. Section 106 of the new Czechoslovak constitution of 1920 declared that neither sex is privileged. In the same year, the Czech Technical University in Prague admitted the first 20 regular female students. Since the 1920s, women were able study all university subjects (except for theology).18 In the following, I will only focus on questions related to women studying mathematics in Prague from the beginning of the 1880s to the end of the 1940s.

and a “literature school,” which was gradually expanded into a higher school for girls. In 1891, the number of the schools increased because a traditional boarding school for girls was established. In 1901, the school system was reorganized to a large extent and the following structure became standard: six-class public lycée (preparation for university studies), technical school (preparation for practical life, including a 1-year program and a 2-year program offering a special course for teachers of women’s works at public schools, courses for nurses at nursing schools, courses for cooks and housewives, and occasional courses on lace making, embroidery, hat-making, and ironing for female workers and servants), higher school for girls (preparation for administrative workers, clerks, home teachers, etc. offering education in trade, languages, music, and economics), and a boarding school for girls. For more information, see Kotzianová (1989). 14 See Rostoˇ cilová (1972); and Anonymous (1901). 15 See Anonymous (1902). 16 See Štrbᡠnová (2004). On the problems involved with the educating women of German nationality in the Czech lands, see Horská (1995). 17 The “Washington Declaration” is also known as the “Declaration of Independence of the Czechoslovak Nation”, which came into being in October of 1918. The future Czechoslovak President Tomáš Garigue Masaryk (1850–1937) wrote the declaration and submitted it to the U.S. Government and U.S. President, Thomas Woodrow Wilson (1856–1924). The declaration called for the union of the Czech and Slovak nations and the creation of the Czechoslovak Republic; it formulated basic civil rights and freedoms, internal and foreign policy, and the state system of the future independent Czechoslovakia. 18 For more information on the situation in the Czech lands and later in Czechoslovakia, see Beˇcváˇrová (2016b). For information on the situation in Germany, see Abele et al. (2004); and Tobies (2012). On the situation in the United States, see Green and LaDuke (2009).

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3.3 Doctorates in Mathematics 3.3.1 The German University in Prague During the Years 1882–1945 From the year 1882, when the German University in Prague was established, until the year 1945, when it was closed, there were 43 doctoral degrees awarded in mathematics. Thirty-nine doctoral theses were defended (including those by three females and ten foreigners); three doctorates were recognized by Czechoslovak authorities, one on the condition of passing an additional Ph.D. examination in mathematics; three candidates did not obtain the doctorate; one candidate was rejected during the first stage of the proceedings (however, 3 years later, he submitted a new thesis and was successful); and five international recognitions were denied for formal reasons.19 From the years 1882/1883 to 1912/1913, the Faculty of Philosophy of the German University in Prague awarded 395 doctorates in philosophy, six of which (i.e. 1.5%) were in mathematics. There were no females among those who were awarded doctorates because, even at the beginning of the twentieth century, German professors of mathematics [for example, Karl Bobek (1855–1899), Anton Josef Gmeiner (1862– 1927), and Josef Grünwald (1878–1911)] held very conservative opinions about awarding doctoral degrees to women. From 1882/1883 to 1906/1907, two mathematicians applied for international recognition of their foreign doctoral diplomas. One was refused and the other was approved, although both diplomas were issued by the same German university (Erlangen), which made the two cases quite identical (they had both graduated from a so-called “real school” and not from a classical gymnasium, which disqualified them as candidates for a doctorate at universities in the Austro-Hungarian Empire). In the academic years 1912/1913–1919/1920, the Faculty of Philosophy of the German University in Prague awarded 230 doctorates. Only four candidates, including one female, defended a doctorate in mathematics, which is 1.7%. In the academic years 1920/1921–1938/1939, the Faculty of Science of the German University in Prague awarded 773 doctorates, including 25 doctorates, i.e. 3.2%, in mathematics, including two women. One of the candidates, however, obtained the degree only in the second, remedial proceeding. The other two candidates failed because they did not submit their doctoral theses. The 1930s, when Germany was becoming increasingly fascist and the German intelligentsia of Jewish origin or anti-fascist orientation were forced to emigrate, brought an increase in the number of applications for international recognition of diplomas and studies in foreign countries, a shortening of obligatory studies, and an acceleration of Ph.D. proceedings at the Faculty of Science of the German University in Prague. It is interesting that the German mathematicians in Prague (for example, L. Berwald and K. Löwner) recommended, without any problems, to grant requests 19 This analysis is based on various sources housed in the archive of Charles University in Prague. See the list of archival sources in the bibliography. See also Výborná et al. (1965).

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submitted by their regular as well as extramural students, which enabled them to complete their doctoral studies in a shortened time. At the same time, however, the German mathematicians did not support the international recognition of the diplomas already awarded. From 1920/1921 to 1938/1939, seven applications for international recognition were submitted: three of them were probably denied, three were approved, and in one case an additional doctoral examination was ordered. In the years 1939/1940–1944/1945, the Faculty of Science of the German University in Prague awarded 88 doctorates, including four in mathematics, i.e. 4.5%. One applicant did not—even at the third attempt—pass a subsidiary Ph.D. examination in theoretical physics, and his doctoral proceedings were officially stopped. Let us remark that there were no women among the candidates for doctorates in mathematics, which is not surprising considering the Nazi conception of women’s role in society.20 It may appear strange that the number of doctorates in mathematics awarded at the Faculty of Philosophy of the German University in Prague was less than 2% and that it was only 3–5% at the Faculty of Science, notwithstanding the fact that mathematics was very important at that time and that professors of mathematics did not lack talented students. The explanation for this seemingly paradoxical phenomenon is relatively simple. The doctoral candidates in mathematics usually intended to pursue an academic career. However, the corresponding positions at Austro-Hungarian universities were few, since every larger university or technical university had only two or three (at most) positions for regular or extraordinary professors of mathematics and only one or two positions for regular or extraordinary professors of descriptive geometry. There were not any research institutions focused on mathematics and its classical applications, so some doctors of mathematics found employment in the financial sector (especially in the insurance business), the state administration (especially in national economy statistics), the army (especially as teachers of mathematics), or at secondary schools, which did not require a doctoral degree. The lower interest of German-speaking doctoral candidates in mathematics at the German University in Prague may also have its source in the fact that, in the nineteenth century, this university was not the only institution where a candidate could submit a doctoral thesis in mathematics in the German language and pass the Ph.D. examinations in that language as well.21 Moreover, many mathematicians 20 In the 1930s, the Nazi government was not interested in women studying at universities, and several laws were passed that made it even more difficult for them to study. Many of these laws, however, were already recalled during WWII. In the wartime, women filled the empty seats left by male students. The situation was the same in Prague at the German University. During the war, however, many women did not finish their studies. 21 In the Austrian Empire (or Austro-Hungarian Empire), it was possible to undergo the Ph.D. examination with international recognition at universities in Vienna, Graz, Innsbruck, Budapest, ˇ ˇ Cernovce (Cernovice, Czernowitz) and Kolozsvár (Klausenberg, Cluj, Kluž). Vienna was an especially popular destination for the Germans from the Czech lands. Only with minor difficulties of a purely formal nature, it was possible to obtain doctorates in Germany and France throughout the nineteenth century. The destination of our (German as well as Czech) mathematicians was usually Göttingen, Berlin, Munich, or Hamburg, though Czech mathematicians also went to Paris or Strasbourg.

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regarded Prague as a “provincial university” with a relatively small community of German mathematicians looking to find employment outside of the Czech region. The increase in the number of doctoral candidates in mathematics at the German University in Prague after the year 1920 (when the new Czechoslovak constitution was proclaimed) was partly caused by the fact that the Czechoslovak authorities did not automatically recognize diplomas and academic degrees awarded by foreign schools and made the procedure of international recognition stricter, eventually requiring additional Czechoslovak state examinations. The candidates of German nationality who formerly went to Vienna, Budapest, Berlin, Göttingen, or Munich now remained in Prague. The Faculty of Science at the German University in Prague was a relatively small but significant European institution for natural sciences and pedagogy. The University was attractive for foreign students with Jewish heritage and democratic opinions from Lithuania, Latvia, Ukraine, Hungary, and Poland and— as of the mid-1930s—from Germany as well. This was partly due to the renown and professional achievements of certain professors (such as the mathematicians L. Berwald, K. Löwner, and G. A. Pick; the physicists P. Frank and A. Lampa; the philosopher and mathematician R. Carnap; the chemists A. Kirpal and C. I. Cori; and the botanist E.G. Pringsheim), relatively low tuition fees and living costs, the accessibility of Prague, a multicultural environment, as well as political and religious freedom.22

3.3.2 The Czech University in Prague During the Years 1882–1945 From 1882 to 1939, the doctoral candidates at the C.k. Czech Charles-Ferdinand University (“Charles University”), submitted 159 theses in mathematics (including 12 by women and/or foreigners),23 and 150 doctorates were awarded. All the theses, except for two, were written in the Czech language.24 In the years 1882/1883–1920/1921, the candidates at the Faculty of Philosophy of the Charles University in Prague defended 1118 doctorates in philosophy, including 62 in mathematics (5.5%). In total, sixty-two doctoral theses written in the Czech language were submitted, all of which were accepted and evaluated positively. Three of the candidates were absent for some part of the Ph.D. examination, and as a consequence did not obtain the doctoral degree. The candidates usually took the main Ph.D. examination in mathematics and a subsidiary Ph.D. examination in philosophy. All 22 For more information about mathematics and the mathematical community at the German University in Prague, see Beˇcváˇrová (2016a). 23 The foreigners included six Russians, one Latvian, and one Ukrainian (according to today’s structure of Europe). In the student catalogues or Ph.D. protocols, Russia (the Soviet Union) is given as the state of birth (or origin). They were all citizens of Russian nationality who had left a Russia convulsed by civil war and political problems to settle in the Czechoslovak Republic. 24 This analysis is based on various sources held at the archive of Charles University in Prague. See the list of the archival sources in the bibliography. See also Tulachová (1965).

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59 successful candidates underwent the complete doctoral proceeding. One doctorate was obtained by a woman. To round out this information, one doctoral degree was withdrawn (cancelled) after 16 years based on a decision of Czechoslovak court of justice because its holder had committed a deplorable crime. In the years 1920/1921–1939/1940, the Faculty of Science of Charles University allowed 1088 doctoral defenses in the natural sciences, including 97 in mathematics (8.9%).25 One doctoral proceeding was stopped at the very beginning because the submitted thesis was not accepted. A year later, the candidate submitted a new thesis and succeeded in earning his degree. Candidates submitted 95 theses in Czech and 2 in French. Five candidates did not undergo the prescribed Ph.D. examinations and did not obtain the degree (including one woman). The candidates usually took the main Ph.D. examination in mathematics (specialization: mathematical analysis and algebra, geometry and algebra, geometry and mathematical analysis) and a subsidiary examination in the philosophy of exact sciences26 (experimental physics and analytical mechanics, in a few cases). The number of successfully accomplished doctoral proceedings was 91, including 8 by women. Five of the candidates had to undergo some of the Ph.D. examinations repeatedly (including two women). One candidate submitted his doctoral thesis in the spring of 1939 and in autumn of the same year he passed both Ph.D. examinations, yet he did not graduate until the summer of 1945. Six candidates submitted their theses by the year 1939, each of which was accepted and evaluated positively. The doctoral proceedings began before the closure of the Czech universities, but the candidates did not have enough time to pass all required examinations. Their doctoral proceedings were finally carried out between the years 1945 and 1952. From November of 1939 to the summer of 1945, Charles University did not award any doctorate in mathematics because the university was closed on 17 November 1939 by the Nazi occupiers. University activities resumed only after the liberation, starting with the summer semester of 1945. The number of doctoral proceedings in mathematics at the Faculty of Philosophy was 5%, and at the Faculty of Science almost 8%. What was mentioned above with regard to the German University in Prague also applies to the Czech universities. However, one should note that, for the Czech doctoral candidates in mathematics, Prague was, until 1920, the only place where they could submit their doctoral theses in the Czech language and take the Ph.D. examination in their mother tongue. After the year 1920, this possibility was extended to Brno. However, this did not result in

25 In the school year of 1920/1921, the newly established Faculty of Science of Charles University in Prague started its educational activities. The first 25 doctoral candidates were still registered at the Faculty of Philosophy. In the winter semester of the school year of 1939/1940, the Faculty of Science of Charles University initiated nine doctorate proceedings, though most of them were not completed until after the war. One of the doctoral proceedings was in mathematics. 26 At the Faculty of Science of Charles University in Prague, a 1-hour subsidiary Ph.D. examination in the philosophy of exact sciences replaced the former examination in classical philosophy. This change enabled a deeper connection between philosophy, history, logic, mathematics, and the natural sciences.

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a decrease of interest in doctoral proceedings held in Prague because, after the formation of the Czechoslovak Republic, the chances of the holders of doctorate degrees to find employment increased somewhat. This was caused by the fact that the number of positions for professors, associate professors, and assistants at Czech universities increased (because new schools were founded, the number of the faculties of the Czech Technical University in Prague and the Czech Technical University in Brno increased), the number of positions for mathematics experts in state administration increased (new ministries, insurance institutions, banks, financial administration, etc.), and the network of Czech secondary and professional schools was expanded.27

3.3.3 Doctoral Degrees Awarded in Mathematics at Prague Universities: A Brief Comparison The proportion of all doctorates awarded at the Czech Faculty of Philosophy and the German Faculty of Philosophy was 1118 to 625 (1.8 to 1); the proportion of doctorates awarded in mathematics was 59 to 10 (5.9 to 1); and the proportion of doctorates awarded in mathematics to women was 1 to 1. The proportion of all doctorates awarded at the Czech Faculty of Science and the German Faculty of Science was 1088 to 773 (1.4 to 1); the proportion of doctorates awarded in mathematics was 91 to 25 (3.6 to 1); and that of doctorates awarded in mathematics to females was 8 to 3 (2.7 to 1). During the years 1882/1883–1944/1945, 2206 doctorates were defended at the Czech University in Prague or at Charles University, and 1486 doctorates were defended at the German University in Prague, which means that Charles University awarded approximately 1.5 times more doctorates than the German University in Prague. Comparing the numbers of doctorates awarded in mathematics in the same period, we can see that Charles University awarded 150 doctorates in mathematics (including those started before 1939 but completed only after the war), whereas the German University in Prague awarded 39 doctorates (excluding international recognition). This means that Charles University awarded four times more doctorates in mathematics than the German University in Prague. Charles University had only one regular professor of mathematics until the beginning of the twentieth century, whereas the German University in Prague had, from the start, two professors of mathematics, one regular and one extraordinary professor. It was only from the year 1903 that both universities had two professors of mathematics. Charles University had three professors of mathematics [Karel Petr (1868–1950), Jan Sobotka (1862–1931), and Václav Láska (1862–1943)] as of the year 1911, whereas the German University in Prague usually had two mathematicians [Georg Alexander Pick (1859–1942) and Gerhard Hermann Waldemar Kowalewski (1876–1950), or G. A. Pick and Ludwig Berwald (1883–1942)] in the prewar and interwar period. In the 1930s, the German University in Prague had three professors of mathematics [L. Berwald, Karl Löwner 27 For

more information see Výborná et al. (1965).

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(1893–1968), and Arthur Winternitz (1893–1961)]. Charles University had more faculty members with the right to supervise and evaluate doctoral theses [Bohumil Bydžovský (1880–1969), Václav Hlavatý (1894–1969), Vojtˇech Jarník (1897–1970), Vladimír Koˇrínek (1899–1981), Miloš Kössler (1884–1961), V. Láska, K. Petr, and Emil Schoenbaum (1882–1967)].28 The subjects of doctoral theses in mathematics at the German University in Prague usually reflected more closely the new trends in mathematics (especially modern analysis, differential and affine geometry) and they represented a higher level of expertise.29 Their authors obtained positions at prestigious foreign universities and reached considerable renown.30 It was naturally due to the fact that, compared to Charles University, the German University in Prague had about the same number of teachers educating a smaller number of students and doctoral candidates.31 After Czechoslovakia was formed, the German University in Prague was not abolished but rather became a recognized and respected state university with equal rights, which was not suppressed or financially disadvantaged by the new republic.32 In postwar Europe, which was divided into states on the basis of nationality (above all), this was, in fact, the only official and recognized state university for the so-called national minority. The university retained its position and renown until the beginning of World War II. Let us note that citizens of German nationality were not discriminated against in Czechoslovakia with regard to university studies. On the contrary, according to the population census in February of 1921, 8.761 million people declared to be of Czechoslovak nationality, 3.123 million people of German nationality, 0.745 million people of Hungarian nationality, 0.461 million people of Russian nationality, 0.181 million people of Jewish nationality, and 0.075 million people declared to be of Polish nationality. Around 23.3% of the population was German. In Czechoslovakia of 1921, there existed three Czech (Czechoslovak) universities (Prague, Brno, Bratislava) and two Czech Technical Universities (Prague, Brno), one German university (Prague), and two German technical universities (Prague, Brno). This situation remained unchanged in Bohemia and Moravia until November of 1939.

28 Female doctoral candidates wrote their theses under the supervision of G. A. Pick (two women), A. Winternitz (one woman), F. J. Studniˇcka (one woman), K. Petr (one woman), M. Kössler (one woman), V. Hlavatý (three women), and E. Schoenbaum (three women). 29 See Beˇ cváˇrová (2016a); and Beˇcváˇrová and Netuka (2015). For more information about mathematics at the Czech University in Prague, see Beˇcváˇrová (2008); and Beˇcváˇrová (née Nˇemcová) (1998). 30 For example, we can mention F. A. Behrend, L. Bers, A. Erdélyi, P. Kuhn, E. Lammel, H. Löwig, K. Löwner, M. Pinl, and O. Varga. Their careers and works are discussed in Beˇcváˇrová (2016a); and Beˇcváˇrová and Netuka (2015). 31 See Beˇ cváˇrová (2016a); and Beˇcváˇrová and Netuka (2015). 32 After World War I, the Imperial Russian University in Warsaw was made Polish, the German ˇ university in Cernovce in Bukovina was abolished, the German university in Kolozsvár was made Hungarian, the German university in Dorpat (Jurjev, Tartu) was turned into an Estonian university, and the German schools in Lvov were abolished.

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3.3.4 Information on Doctorates Awarded at Charles University in Prague in the Years 1945–1953 The history of the German University in Prague came to its definitive end on 18 October 1945, when President Edvard Beneš (1884–1948) issued a decree abolishing all German universities and high schools in Czechoslovakia, retroactive from 17 November 1939. This day is symbolic because it was then that Czech universities in the territory of the Protectorate of Bohemia and Moravia were closed for a period of 3 years by a decree issued by the Reichsprotektor Konstantin Hermann Karl, Freiherr von Neurath (1873–1956). The top representatives of the German Reich did not, however, intend to re-open the Czech universities, because they wanted to destroy Czech intelligentsia. On the same day, 9 representatives of students’ movement were executed in Ruzynˇe prison, and almost 1100 students were deported to the concentration camp in Sachsenhausen. The professors were forced to take leave, to retire, or to work in the arms industry. Almost immediately after the liberation, Charles University recommenced its activities and regular education by opening a special summer semester of 1945 so that more than seven grades of secondary-school graduates could be enrolled. During the years 1945–1952, the Faculty of Science at Charles University in Prague started 1047 doctoral proceedings, including 55 in mathematics (i.e. 5.2%). In all, 54 theses were submitted in Czech, 1 in Polish, and 53 Czech citizens, 1 Pole, and 1 Bulgarian undertook doctoral proceedings. A doctorate was awarded to 54 candidates, including five women (9.3%). The facts above indicate that, even after 1945, the number of doctorates awarded in mathematics to women did not significantly increase. A deeper interest in studying mathematics, obtaining a doctoral degree, and pursuing an academic career did not emerge until the beginning of the 1960s.33

3.4 Information on Women’s Doctorates Awarded in Prague Uring the Years 1900–1945 In this section, I will present a short analysis of the successful doctoral proceedings of three women who earned a doctorate in mathematics from the German University in Prague, of eight women who earned the same degree in mathematics from Charles University in Prague, and of one unsuccessful doctoral proceeding. The documents housed in the archive of Charles University in Prague, the archive of the Czech Technical University in Prague, and the National Archive of the Czech Republic indicate the social environments from which these women came and provide information about their cultural, intellectual, and material backgrounds. They show us how their 33 This analysis is based on various sources held at the archive of Charles University in Prague. See

the list of the archival sources in the bibliography.

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families and social events influenced them, how they were motivated by the present circumstances, how they lived, what they dedicated themselves to, what they did, what problems they solved, and how complicated their lives were (in light of the formation and collapse of states, domestic and citizenship issues, the availability of common citizenship documents, anti-Semitism, emigration, war, forced deportation to ghettos and concentration camps, etc.). The documents also reveal much about what took place in our society during the first half of the twentieth century, and they provide new insights into the significance of such things as nationality, national and domestic affairs, entrepreneurship, shifting attitudes towards Jews and education, the advent of economic crisis, views on household modernization, and the development of tourism and medical care. The following section offers a brief summary of the doctoral proceedings in mathematics undertaken by 12 women during the years 1900–1945 (or 1952). The analysis is based on various sources held in the archive of Charles University in Prague, the archive of the Czech Technical University in Prague, and the National Archive of the Czech Republic (a complete list of archival sources can be found in the bibliography). I have also made use of information gathered from oral interviews with relatives of some of the women.

3.4.1 Doctorates Awarded at the German University in Prague Saly Ruth Ramler (1894–1993) defended her Ph.D. thesis in 1919 under the supervision of Georg Alexander Pick and obtained her Ph.D. degree from the Faculty of Philosophy of the German University in Prague. Called Salˇca or Luise by her family, Saly Ramler was a daughter of Gerson (Gustav) Ramler (1863–1930), a Jewish shopkeeper in Galicia (today Ukraine), later in Prague, and Franziska Ramler (née Rosenblatt, 1860–?), who came from a Jewish shopkeeper’s family that settled in Kolomea. Ramler had five brothers and sisters: Fredryk (?–?); Natali (1887–?), called Neché or Necha; Ernestine (1889–?), called Ernestina; Erna or Arnoštka; Rosa (1891–1938), called R˚užena; and Leon (1892– 1942), called Leo or Lev. Natali Ramler studied from 1908 to 1913 at the Faculty of Philosophy of the German University in Prague. In 1913, she defended her thesis in philosophy and obtained a doctoral degree. She became a teacher of German and English at the First German Girls’ Lyceum in Prague. In 1937, she was made a director of that school, and in the winter of 1939 she emigrated to the United States. Ernestine Ramler became a bank clerk in Prague. In the winter of 1939, she emigrated to the United States with her older sister Natali. Their emigration was not so difficult because they knew German, English, and Czech fluently; they had enough money and relatives in Boston; and especially because they prepared for their emigration before the Nazis came to Prague.

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Rosa Ramler became a shopkeeper in Prague (she had a shop with lace work). From the 1930s, she started working as a clerk. In 1938, she prepared her emigration to the United States, but she did not coordinate her activities with her older sisters. In December of 1938, she travelled to Switzerland and drowned in Lake Zürich. Leon Ramler studied at the Faculty of Law of the German University in Prague. In 1920, he obtained his doctorate in law. He worked as a secretary and as a representative of the company Griotte Podniky, a famous company in Prague that exported cherry liqueur all over the world. He was a good lawyer and an astute businessman, who could speak German, Czech, French, English, and Spanish. He believed that Czechoslovakia was the best country for his life and he did not plan his emigration before the Nazis arrived in Prague. In 1940, he started to look for the necessary documents to emigrate to the United States, but it was too late for him. In November of 1941, he was deported to the Jewish ghetto in Terezín. In April of 1942, he was sent to the Piaski (the Jewish ghetto), where he was murdered. In 1914, Saly Ramler passed the so-called “matura” at the First German Secondary Girls’ School in Prague-Vinohrady. From 1914 to 1919, she studied mathematics and physics at the Faculty of Philosophy of the German University in Prague. In 1919, she defended her Ph.D. thesis, which was titled Geometrische Darstellung und Einteilung der Affinitäten in der Ebene und im Raume. Dreiecks- und Tetraederinhalt (reviewers: G. A. Pick and G. H. W. Kowalewski). Her thesis is no longer available. She passed the first (main) oral examination in mathematics in November of 1919, and she took the second (subsidiary) oral examination in philosophy in December of 1919. She obtained her doctoral degree at the graduation ceremony on December 11, 1919. In 1919, she wanted to obtain her first professional position at the German Technical University in Brno. She prepared all the necessary documents, but we have no evidence that she took part in the competition. In 1921, she passed the so-called teacher examinations to become a teacher of mathematics and physics. In 1921, she started her career as a teacher of mathematics at the German Secondary Girls’ School in Prague II. Yet she had higher ambitions and she continued her education. She took part in special sports courses in Germany (July 1921) and she took part in the meeting of German mathematicians in Jena (September 1921). In the spring of 1922, she left to pursue mathematical studies in Luxemburg. One year later, she transferred her studies to Germany. She spoke perfect German, Czech, French, and Italian (and later English). In July of 1923, she married the famous Dutch-American mathematician Dirk Jan Struik (1894–2000). In 1974, Struik recalled his first meeting with his future wife Saly and described her doctoral thesis as follows: “… in July 1923, I married at Prague, in the ancient Town Hall with the medieval clock, Saly Ruth Ramler. She was a Ph.D. in mathematics of the University of Prague, where she had studied under G. Pick and G. Kowalewski. Her thesis was a demonstration of the use of affine reflections in building the structure of affine geometry, a new subject at the time. We had met the previous year at a German mathematical congress. After marriage we settled in Delft.”34 34 D.

J. Struik (1974), p. xiv.

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Let us note that Saly Ramler travelled with her husband to the Netherlands, then to Italy, Germany, and France. In 1926, they immigrated to the United States, where Struik obtained a position as a professor at the Massachusetts Institute of Technology (MIT). The motivation for their travel had a political component, as is shown in the following quotation: From 1924 to 1926, with Struik’s Rockefeller Fellowship, he and his wife travelled to several other European countries and studied, met, and collaborated with many of the great mathematicians and scientists of the twentieth century, including Tullio Levi-Civita, Richard Courant, and David Hilbert. Nevertheless, by 1926, Struik found himself unemployed in Holland and with limited opportunities in Europe. As a long-time mathematical and political friend of Struik, Lee Lorch of York University in Toronto, Canada, understood from him and wrote in an electronic correspondence to us, that Struik’s “political commitments and activities closed European opportunities”. Eventually, however, Struik received two offers, one from Otto Schmidt to go to Moscow and the other from Norbert Wiener to visit MIT. It was a hard choice for him: in the end, he decided to accept the teaching post from Samuel Stratton, the president of MIT.35

In the first decade after her marriage, Saly Ramler Struik travelled with her husband all over Europe. She fascinated his colleagues with her elegance, education, and knowledge. She was interested in mathematics and the history of mathematics, as we can see from the recollections of Struik and C. Davis: “Ruth, working with F. Enriques, published an Italian edition of the tenth book of Euclid’s Elements”.36 “Dirk’s love for the history of mathematics was reawakened when Ruth and he wrote a joint article probing (but not solving) the question of whether A.-L. Cauchy, when he was in Prague (1833–1836), might have met the Prague mathematician Bernard Bolzano”.37 Saly Ramler Struik left mathematics as a young woman, gave up her professional career, and devoted herself to her husband and their daughters (Ruth Rebekka, Anne, and Gwendolyn), although it was a very difficult decision for her, as the following words show: While she was an accomplished mathematician, she was kept out of mathematics by illness for much of her adult life. She struggled with the tension between raising three daughters and wanting to do mathematics. She found it unfair that women cannot have a career and a family, and she resented and suffered from the discrimination bred out of the traditional expectation that a married woman do nothing but attend to the family. However, in later years she became mathematically active again, attending meetings and publishing. The Kovalevskaya Fund at the Gauss School in Peru was endowed in her memory.38

In 1977, Saly Ramler Struik published what is perhaps her first and only article— “Flächengleichheit und Cavalierische Gleichheit von Dreiecken”39 —and its content 35 Powell

and Frankenstein (2001), p. 43.

36 D. J. Struik (1974), p. xiv. F. Enriques published a modern Italian translation of Euclid’s Elements;

see Enriques (1930–1932). et al. (2001), p. 585. The article in question is Struik and Struik (1928). 38 Quoted from http://www.tufts.edu/as/math/struik.html (accessed July 20, 2017). 39 S. R. Struik (1977). 37 Davis

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was later summarized in the journal Zentralblatt für Mathematik und ihre Grenzgebiete40 as well as in the journal Mathematical Reviews.41 Let us note that the article was signed “Saly Ruth Struik, Belmont, Mass., and USA”. It was published thanks to the help of Paul Isaak Bernays (1888–1977), Struik’s family friend. In a note at the end of the article, she wrote: “Mein besonderer Dank gilt Prof. Paul Bernays, dessen Hilfe bei der endgültigen Fassung dieser Arbeit von grossem Wert gewesen ist. Bemerkung der Redaktion: Dies ist die letzte mathematische Arbeit, die Bernays noch betreut hat. Er ist am 18. 9. 1977 in Zürich verstorben”.42 No further mathematical work of Saly Ruth Struik (Saly Ruth Ramler or Saly Ruth Ramler-Struik) has been found. It is interesting that in 1978, Oene Bottema43 published an article titled “Equi-affinities in Three-Dimensional Space” in the journal of the University in Belgrade,44 in which he cited Saly Ruth Ramler’s forgotten Ph.D. thesis as an inspiring source. Saly Ruth Struik had interesting hobbies. From her husband’s and her friends’ recollections, it is known that she liked modern dance, travelling, the history of mathematics, and especially her family. D. J. Struik respected his wife greatly, as is evidenced from comments made on the occasion of his 100th birthday. “When asked what he had done to achieve such longevity, he retorted that he simply had not died. At other times he attributed this and his happiness to 3 M: Marriage, Mathematics, and Marxism. “Mathematicians grow very old, it is a healthy profession. The reason you live long is that you have pleasant thoughts”. Asked on his 100th birthday what he missed most, he replied: “My wife”.45 Hilda Falk (1897–1942) defended her Ph.D. thesis in 1921 under the supervision of G. A. Pick and obtained her doctoral degree from the Faculty of Science at the German University in Prague. Hilda Falk was a daughter of Otto Falk (1862–1899), who was a famous Prague lawyer, and Elisabeth Falk (née Grätz, 1873–1906). Hilda Falk had one older sister, Margareth (1896–1942), who became a teacher at the elementary school in Most. In 1916, Hilda Falk passed the so-called “matura” at the First German Secondary Girls’ School in Prague. From 1916 to 1921, she studied mathematics and physics at both the Faculties of Philosophy and Science of the German University in Prague. In 1920, she passed the so-called teacher examinations to become a teacher of mathematics and physics. There is no evidence, however, that she ever worked as a teacher. In 1921, Hilda Falk defended her Ph.D. thesis Beiträge zur äquiformen Flächentheorie [the reviewers were G. A. Pick and Adalbert Prey (1876–1950), a professor of 40 See

https://www.zbmath.org/?q=ai:struik.s-r, ZBL 0367.50004 (accessed April 12, 2017). review MR0513833 at http://www.ams.org/mathscinet (accessed February 5, 2017). 42 S. R. Struik (1977), p. 143. 43 Oene Bottema (1901–1992) was a Dutch mathematician who defended his Ph.D. thesis Figuur van vier kruisende rechte lijnen at the University in Leiden in 1927 under the supervision of Willem van der Woude (1876–1974) and taught at the Technical University in Delft. 44 See Bottema (1978), pp. 9–10. 45 See http://www.tufts.edu/as/math/struik.html. (accessed 21 July 2017). For more information on Saly Ruth Ramler Struik see Beˇcváˇrová (2018). 41 See

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physics]. She passed the first (main) oral examination in mathematics and theoretical physics in April of 1921. She underwent the second (subsidiary) oral examination in philosophy in May of 1921. She was awarded her doctoral degree in natural sciences at the graduation ceremony on 6 May 1921. Her Ph.D. thesis is not available at the archive of Charles University in Prague. She never married and had no children. From 1921 until 1922, she worked as an assistant of physics at the German Technical University in Prague. In 1923, she became an assistant of physics at the Medical Institute of the German University in Prague. In 1924, she obtained a position as a secretary at the company Klatze a Lorenz in Prague. After some years, she was appointed a higher clerk at the same company. In 1939, as a Jew, she lost her professional position and later her civil rights. In 1941, she tried to emigrate to Shanghai in China, where a large Jewish community from central Europe lived, but she failed to acquire her visa and passport for emigration. In 1942, she and her sister were sent to the Jewish ghetto in Terezín and then to the Jewish ghetto in Riga, where they were murdered by fascists. As a student, Hilda Falk had scientific ambitions and wanted to become a researcher at the university. Her dream was impossible at that time in Prague because there were not enough job opportunities in the sciences for men and women alike. She did not write any articles. As we know from her passports and the official records kept in the National Archive of the Czech Republic, Hilda Falk was interested in car driving, travelling abroad, and modern sports (skiing and tennis). Josefine Mayer (née Keller; 1904–1986) defended her Ph.D. thesis in 1933 under the supervision of Arthur Winternitz and obtained her doctoral degree from the Faculty of Science of the German University in Prague. She was a daughter of Rudolf Keller (1875–1964), a rich Prague newspaper owner and publisher of the Prager Zeitung, and Helene Keller (née Fischel, 1882–1942), who came from a well-known and rich Prague Jewish family. In 1921, Josefine Keller passed the so called “matura” at the German Secondary Girls’ School in Prague-Vinohrady. From 1921 to 1922, she studied mathematics and physics at the German University in Prague. In June 1923, she married Jan Jindˇrich Frankl (1900–?), a clerk in Prague. They shortly lived in Prague. In the autumn of 1923, they moved to Leipzig, where Josefine wanted to study philosophy. In the summer of 1924, she divorced and married Ernst John, a German citizen and editor of the newspaper Neue Leipziger Zeitung. In 1925, their daughter Sofie was born and the young family moved to Prague. In 1927, Josefine divorced once more. In 1928, she married Alfred Maria Mayer (1899–?), a well-known Prague newspaper owner and publisher of the Prager Zeitung. They had only one son, Petr (1930–1938). From 1925 until 1932, Josefine Keller (or John or Mayer) studied mathematics and physics at the Faculty of Science of the German University in Prague probably for her amusement because she knew that, as a married woman and mother, she could not become a teacher or researcher. She wrote her Ph.D. thesis as a mother of two children. In 1933, Josefine Mayer defended the Ph.D. thesis Zur Axiomatik der ebenen Affinen der Geometrie (reviewers: A. Winternitz and L. Berwald). She passed the first (main) oral examination in mathematics in June of 1933. She underwent the

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second (subsidiary) oral examination in natural philosophy in the same month. She was awarded her doctoral degree in natural sciences at the graduation ceremony on 30 June 1933. Her Ph.D. thesis is not available at the archive of Charles University in Prague. She did not write any mathematics articles, and she never had to work regularly because she came from a very rich Prague family. She liked car driving, travelling abroad, sports (skiing), and organizing social parties. During WWII, her family of Jewish origin had to emigrate from Czechoslovakia to save their lives. They obtained their visas and passports to move to the United States, but we have no information about her time there.46

3.4.2 Doctorates Awarded at Charles University in Prague Marie Fabiánová (1872–1943) defended her Ph.D. thesis in 1901 under the supervision of František Josef Studniˇcka (1836–1903). She was the second woman to obtain Ph.D. degree from the Faculty of Philosophy of the Czech University in Prague. Marie Fabiánová was a daughter of Václav Fabián (1844–?), the chief engineer of the Pardubice—Liberec railway, and Juliena (née Haklová, ?–?), a daughter of an innkeeper and miller. Marie’s younger brother Václav (1877–?) also became an engineer and worked for the same company as his father. Both studied at the Czech Technical University in Prague. From 1890 until 1895, Marie Fabiánová studied at the prominent secondary girls’ school in Prague called Minerva. From 1895 to 1899, she studied history, philosophy, mathematics, physics, and chemistry at the Czech University in Prague. In 1900, she finished her Ph.D. thesis O rozvoji dyperiodických funkcí v rˇady a produkty (“On the Expansion of Doubly Periodic Functions Into Series and Products”), which was evaluated by F. J. Studniˇcka and F. Koláˇcek). She passed the first (main) oral examination in mathematics and physics in December of 1900. She underwent the second (subsidiary) oral examination in philosophy in November of 1901, and she was awarded her doctoral degree at the graduation ceremony held on 13 November 1901. Only her Ph.D. thesis is kept in the archive of Charles University in Prague. Before finishing her Ph.D. studies, she started to teach mathematics and physics as a suplent (substitute teacher) at Minerva. She never married or had children. In 1902, she became a regular teacher of mathematics, physics, geometry, and German at the same school. In 1923, she was named a director of the newly founded secondary girls’ school in Prague named Druhé cˇ eské mˇestské dívˇcí reálné gymnasium v Praze II. She retired in 1929 after a long-lasting and serious illness. She did not write any

46 I was able to piece together this information about Josefine Mayer (John-Frankl-Keller) from official records written in the personal cards of her family, from her passports, and from the police registers of Prague citizens that are kept in the National Archive of the Czech Republic.

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mathematics articles, but she wrote a very short article on the life and works of her colleague and supporter J. Grim, which was published in the reports of Minerva.47 She was the first female member of the Jednota cˇ eských matematik˚u a fysik˚u (Union of Czech Mathematicians and Physicists), which was founded in Prague in 1862 and which became a suitable center of mathematical activities for teachers and students at all levels and recruited new people interested in mathematics and physics. She became its member as a student in 1896, and she was very active. In 1901, she was elected as a secretary for Prague II.48 Miluše Jašková (1905–1975) defended her Ph.D. thesis in 1928 under the supervision of Karel Petr and obtained her Ph.D. degree from the Faculty of Science at Charles University in Prague. She was a daughter of Martin Jašek (1879–1945), a renowned Czech teacher of mathematics, physics, philosophy, and propedeutics at the secondary girls’ school in Pilsen, and Emilie (née Pácalová, 1879– 1969), the daughter of a Prague stonecutter. Her father was interested in the mathematical heritage of Bernard Bolzano (1781–1848). He partly catalogued his manuscripts deposited in Vienna and Prague. He discovered Bolzano’s example of a continuous non-differentiable function, the so-called Bolzano’s function.49 For a long time, Martin Jašek collaborated with Saly Ramler, who helped him with reading and making a list of Bolzano’s manuscripts. Miluše Jašková had one younger brother, Miloš (1911–1944). He studied mathematics and chemistry at the Faculty of Science of Charles University in Prague, where he became a doctor of chemistry in 1934. He was a specialist in explosives and worked for the Czechoslovak criminal police. He was executed by the Nazi because of his illegal activities. From 1915 until 1923, Miluše Jašková studied at the popular secondary girls’ school in Pilsen called Dívˇcí lyceum (later the Czech State Girls’ Reform Realgymnasium), where her father was a regular teacher of mathematics. From 1923 to 1924, she studied mathematics and physics at the Faculty of Philosophy of the University in Vienna, where her family lived. From 1924 to 1928, she studied mathematics, physics, philosophy, history, Czech, and French at Charles University in Prague. In 1928, she defended her Ph.D. thesis Rozvoj Euler-Maclaurin˚uv (“Euler-Maclaurin Expansion”, evaluated by K. Petr and B. Bydžovský). She tried to pass the first (main) oral examination in mathematical analysis and algebra in June of 1928, but she failed. On her second attempt, she passed the main oral examination in December of 1928. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in May of 1928, and she was awarded her doctoral degree in natural sciences at the graduation ceremony held on December 14, 1928. Her Ph.D. thesis is not available at the archive of Charles University in Prague. After finishing her Ph.D. thesis, Miluše Jašková lived in Pilsen, where she relaxed after her studies and devoted herself to sports (especially tennis and swimming). 47 See

Fabiánová (1918). J. Grim helped her at the beginning of her career by advising her in how to teach, how to organize her lectures, which books she could buy, etc. 48 For more information on the activities of the Union, see Beˇ cváˇrová (2008). 49 For relevant mathematical and historical commentary, see Hykšová (2003).

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In 1929, she married a Russian engineer named Vsevolod Greˇcenko (1898–1948) and took care of their only son Alexander (born 1930), who became a professor of machine engineering. As we know from her son’s recollections, she never worked regularly and had no professional activities. She was not interested in mathematics and she did not write any mathematical articles. She liked sports, travelling, reading books, and going to theatres and concerts. Helena Navrátilová (1907–?) defended her Ph.D. in 1932 under the supervision of Emil Schoenbaum and obtained her Ph.D. degree from the Faculty of Science of Charles University in Prague. She was a daughter of Florian Navrátil (1864–?), who was the chief financial officer for the police in Telˇc, and Marie (née Krejˇcová, 1876–?). From 1916 to 1925, Helena Navrátilová studied at the “vyšší reálka” (higher real school) in Telˇc, that is, at a special secondary school where usually boys studied and prepared for their future careers as technical specialists. From 1925 to 1926, she studied at the commercial academy in Prague to become a secretary, but she decided not to perform this work. From 1926 to 1928, she studied at the Czech Technical University in Prague, where she graduated from a 2-year special program in financial mathematics, statistics, banking, financial operations, and insurance. She wanted to become a higher bank clerk. During her studies, she changed her decision entirely. From 1927 to 1932, she studied mathematics and philosophy at Charles University in Prague. In 1932, Helena Navrátilová defended her Ph.D. thesis Zákon rˇídkých zjev˚u a jeho aplikace na kolektivy pojistných událostí (“The Law of Rare Events and Its Application to Collections of Insurance Events”, evaluated by E. Schoenbaum and M. Kössler). She passed the first (main) oral examination in mathematical analysis and algebra in November of 1932. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in December of 1932, and she obtained her doctorate in natural sciences at the graduation ceremony held on December 19, 1932. Her Ph.D. thesis is not available at the archive of Charles University in Prague. In 1930, she passed the so-called teacher examinations to become a teacher of mathematics and sports. It is likely that she became a teacher of mathematics and physical education at a secondary school for girls, but we have no information about her later life. Jarmila Šimerková (1910–1975) defended her Ph.D. thesis in 1933 under the supervision of Miloš Kössler and obtained her Ph.D. degree from the Faculty of ˇ ek Šimerka Science of Charles University in Prague. She was a daughter of Cenˇ (1871–?), who was a physician and surgeon in Pilsen, and Pavla (née Klenka, ?–?), a daughter of the head of surgery at the hospital in Brno. From 1920 to 1928, Jarmila Šimerková studied at the popular secondary girls’ school in Pilsen called Dívˇcí lyceum (later the Czech State Girls’ Reform Realgymnasium), where Martin Jašek was a regular teacher of mathematics. From 1928 to 1930, she studied at the Czech Technical University in Prague, where she graduated from a 2-year special program in financial mathematics, statistics, banking, financial

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operations, and insurance. From 1930 to 1933, she studied exclusively mathematics at Charles University in Prague to improve her knowledge. In 1931, as a student, she married Boˇrivoj Iglauer (1901–?), a clerk at an insurance company in Prague. She took care of her family, her daughters Pavla (born 1932) and Jana (born 1936). In 1933, Jarmila Šimerková defended her Ph.D. thesis Zavedení libovolných funkcí v poˇctu pravdˇepodobnosti (“Introduction of Random Functions in Probability Calculus”, reviewed by E. Schoenbaum and M. Kössler). She passed the first (main) oral examination in mathematical analysis and algebra in June of 1933. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in November of 1933. She obtained her doctoral degree in natural sciences at the graduation ceremony held on 24 November 1933. Her Ph.D. thesis is not available in Prague. Jarmila Šimerková never worked regularly and had no professional activities. As a student, she was interested in mathematics and the activities of the Czech mathematical community. From 1931 to 1932, she was a regular member of the Jednota cˇ eských matematik˚u a fysik˚u (Union of Czech Mathematicians and Physicists). She did not write any articles on mathematics. She liked travelling abroad and especially motoring.50 ˇ Vˇera Cechová (1910–1990) defended her Ph.D. thesis in 1933 under the supervision of E. Schoenbaum and obtained her Ph.D. degree from the Faculty of Science ˇ of Charles University in Prague. She was a daughter of Eduard Cech (1876–?), who was a teacher at the industrial school in Pilsen (later he became a higher officer at the Ministry of National Enlightenment in Prague), and Božena (née Tutoviˇcová, ?–?), a daughter of a school inspector and director of a municipal school. ˇ Vˇera Cechová had two older brothers, Jan (1907–?) and Eduard (1908–1979), ˇ and one younger sister Blanka (1912–?). Jan Cech studied at the Faculty of Law of Charles University in Prague, where he graduated in 1930 as a doctor of law. Eduard ˇ Cech studied at the Faculty of Philosophy of Charles University in Prague, where he graduated in 1937 as a doctor of philosophy. He became a teacher of German and Czech at the prominent commercial academy in Prague (the so-called Academy on Ressl’s street). He was a well-known philologist and teacher who wrote many textbooks for secondary schools. ˇ From 1920 to 1923, Vˇera Cechová studied at the popular secondary girls’ school called Mˇestské reálné gymnasium dívˇcí “Krásnohorská” in Prague II, where Marie Fabiánová taught mathematics and physics. From 1923 to 1928, she studied at the secondary girls’ school called Druhé cˇ eské mˇestské dívˇcí reálné gymnasium in Prague II, where Marie Fabiánová was a director. From 1928 to 1930, she studied at the Czech Technical University in Prague, where she graduated from a 2-year special program in financial mathematics, statistics, banking, financial operations, and insurance. From 1930 to 1932, she studied exclusively mathematics at Charles University in Prague to improve her knowledge. She did not write any articles on 50 This small amount of information on Jarmila Šimerková Iglauerová stems from official records written in the personal cards of her family, from her passports, and from the police registers of Prague citizens that are housed in the National Archive of the Czech Republic.

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mathematics and she was not interested in the activities of the Czech mathematical community. ˇ In 1933, Vˇera Cechová defended her Ph.D. thesis Teorie risika (“Risk Theory”, reviewed by E. Schoenbaum and M. Kössler). She passed the first (main) oral examination in mathematical analysis and algebra in June of 1933. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in November of 1933, and she obtained her doctoral degree in natural sciences at the graduation ceremony held on 15 November 1933. Her Ph.D. thesis is not available at the archive of Charles University in Prague. ˇ In 1936, Vˇera Cechová began her professional career as a bank clerk—as a specialist at an insurance company in Prague. In 1946, she married her schoolmate Dr. Otto Fischer (1909–1975), a Czechoslovak mathematician and ˇ Fischerová worked all her life as an insurance specialist in statistics.51 Vˇera Cechová specialist and took care of her family. Her son Jan (born 1951) became a distinguished specialist in statistics and economics and an important Czech politician.52 Ludmila Illingerová (1908–1974) defended her Ph.D. thesis in 1934 under the supervision of Václav Hlavatý and obtained her Ph.D. degree from the Faculty of Science of Charles University in Prague. She was a daughter of Karel Illinger (1871– ?), who was a higher post-officer in Havlíˇck˚uv Brod and later in Prague, and Milada (née Pospíšilová, ?–?), the daughter of a train conductor in Pardubice. From 1925 to 1927, Ludmila Illingerová studied at the Czech State Gymnasium in Prague XVI. From 1927 to 1932, she studied mathematics, psychology, and history at the Faculty of Science of Charles University in Prague. From 1928 to 1931, she also studied at the Czech Technical University in Prague, where she attended special lectures on descriptive and synthetic geometry. In 1932, she passed the so-called teacher examinations to become a teacher of mathematics and descriptive geometry. In 1934, Ludmila Illingerová defended her Ph.D. thesis Loxodromická geometrie (“Loxodromic Geometry”, evaluated by B. Bydžovský and V. Hlavatý). She passed the first (main) oral examination in geometry and mathematical analysis in October of 1934. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in October of 1934. She obtained her doctoral degree in natural sciences at the graduation ceremony on 16 November 1934. Her Ph.D. thesis is not available at the archive of Charles University in Prague. 51 Otto Fischer studied mathematics and physics at the Faculty of Science of Charles University in Prague. In 1933, he defended his Ph.D. thesis in mathematics and became a doctor of natural sciences. In 1935, he obtained a position in the Ústˇrední psychotechnický ústav in Prague (Central Psychotechnical Institute in Prague) and devoted himself to statistical research. In September of 1943, he was deported to the Jewish ghetto in Terezín. In the horrendous conditions of the ghetto and under the permanent Nazi terror, he taught Jewish children mathematics, history, literature, and arts and tried to improve their lives. He also organized cultural activities. From Terezín, he was sent to the concentration camp in Auschwitz-Birkenau. He survived because he was sent as a young and strong man to work under inhuman conditions in the concentration camp in Gross-Rosen. After WWII, he worked at the Mathematical Institute of the Czechoslovak Academy of Sciences. For more information, see Vondráˇcek and Šidák (1976). 52 I obtained information on Vˇ ˇ era Cechová Fischerová from archival material held in the National Archive of the Czech Republic and from an interview with her son Jan Fischer.

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As of 1934, she was a teacher of mathematics, drawing, and descriptive geometry at a secondary school. She taught in many places in the Czech lands as well as in Slovakia. She was very active as a secondary school teacher (a member of various committees for social help and support of poor students, trips for students, sports, health education, enlightenment, etc.). In 1935, she married Alois Mˇestka (1904–?), a teacher at several industrial schools. Their lifestyles clashed and they were not able to live together. During the war period, they separated. Then, Ludmila IllingerováMˇestková worked as a director of a secondary school in Prague and took care of her son Ivo (1936–1972).53 Ludmila Illingerová was interested in mathematics, and she published five articles. The first of them, “Pˇríspˇevek k neeuklidovské geometrii” (“A Contribution to Non-Euclidean Geometry”),54 was a seminar thesis that originated in V. Hlavatý’s special seminar on the philosophy of mathematics in 1931. Illingerová explained the “apparent” difference between Poincaré’s and Klein’s models of non-Euclidean geometry of the plane. V. Hlavatý discussed her work in the journal Jahrbuch über die Fortschritte der Mathematik.55 Illingerová participated in the Second Congress of Mathematicians of the Slavic Countries, which took place in Prague in 1934. There she gave a talk titled “Loxodromická geometrie” (“Loxodromic Geometry”), whose German abstract appeared under the title “Die loxodromische Geometrie”.56 One year later, she sent a short ˇ abstract of her Ph.D. thesis to the Czech mathematical journal Casopis pro pˇestování 57 matematiky a fysiky. It was published under the same title and contained only some basic information on the content of Illingerová’s thesis. The publication of an abstract was necessary for a successful doctoral proceeding. While her first articles were research articles, her later publications dealt with didactical issues. In 1935, Illingerová published a very short mathematical note titled “Poznámka k cˇ lánku p. Jos. Kopeˇcného: Über die Bestimmung der Summe der Winkel im ebenen Dreieck” (“A Remark on the Article by Jos. Kopeˇcný …”),58 in which she showed that it is impossible to use a hyperbolic and elliptic plane in the regular constructive proof of the theorem on the sum of angles in the plane triangle.59 Under the name Ludmila Mˇestková-Illingerová, she published only one article, “Nˇekteré znaky dˇelitelnosti” (“Some Criteria of Divisibility”),60 in which she explained the criteria of dividing the numbers 7 (and 49), 13, 17, 19, 37, 99, and 101 for students and secondary school teachers from the point of view of object 53 We

have only modest information about Ludmila Illingerová-Mˇestková and her relatives thanks to documents preserved in the National Archives of the Czech Republic. 54 Illingerová (1933). 55 See Jahrbuch über die Fortschritte der Mathematik, JFM 59.0553.02, and the French abstract in the journal Zentralblatt für Mathematik und ihre Grenzgebiete, ZBL 0006.17806. 56 Illingerová (1935a). 57 Illingerová (1936). 58 Illingerová (1935b). 59 See the review in Jahrbuch über die Fortschritte der Mathematik, JMF 61.0967.03. 60 Mˇ estková-Illingerová (1942).

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teaching. She tried to explain and simplify the notes contained in the popular Czech textbook Aritmetika pro IV. tˇrídu stˇredních škol (“Textbook on Arithmetic for the Fourth Class of Secondary Schools”) by B. Bydžovský, S. Teplý, and F. Vyˇcichlo.61 Jiˇrina Frantíková (1914–2000) defended her Ph.D. thesis in 1937 under the supervision of E. Schoenbaum and obtained her Ph.D. degree from the Faculty of Science of Charles University in Prague. She was a daughter of Matouš Frantík (1884–1955), who was a teacher of Czech and German at secondary schools in ˇ Prague and Ceská Tˇrebová, and Marie (née Mikulíˇcková, 1894–1972), a daughter of a senior postal officer. Jiˇrina had one younger brother, Vratislav (1917–2007), who studied at the Czech Technical University in Prague and became an engineer. From 1924 to 1929, Jiˇrina Frantíková studied at the Czech State Real School in ˇ Ceská Tˇrebová. From 1929 to 1931, she studied at the Czech State Real School in Prague-Smíchov. During her secondary studies, she wanted to become a teacher of Czech and German like her father, but after the “matura”, she changed her decision. From 1931 to 1933, she studied at the Czech Technical University in Prague, where she graduated from a 2-year special program in financial mathematics, statistics, banking, financial operations, and insurance. From 1933 to 1936, she studied mathematics, philosophy, and the Czech language at Charles University. In 1936, Jiˇrina Frantíková finished her Ph.D. thesis Úrokový problém pro d˚uchody životní s malou poznámkou pro prémiové reservy smíšeného pojištˇení (“An Interest Income Problem for Life Pensions with a Brief Remark About Premium Reserves of Mixed Insurance”, reviewed by E. Schoenbaum and M. Kössler). She passed the first (main) oral examination in mathematical analysis and algebra in November of 1936. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in December of 1936. She obtained her doctoral degree in natural sciences at the special graduation ceremony on 7 June 1937, which was attended by the president of the Czechoslovak Republic. Her Ph.D. thesis is not available at the archive of Charles University in Prague. In 1937, she started her professional career as a clerk at the Central Social Insurance Company in Prague. After WWII, she worked as a financial specialist at the Ministry of Finance (on the issues of the formation of the state budget, pensions, and insurance). She also collaborated with E. Schoenbaum as a scientific secretary for the journal Aktuárské vˇedy: Pojistná matematika – Matematická statistika. In 1947, thanks to that collaboration and a scholarship paid by the OSN, she was able to visit the United States. In 1948, she married František Chytil (1908–?), a lawyer. She worked all her active life at the ministry and she also took care of her only son, Ivo. She liked foreign languages (she spoke German, French, and English fluently), travelling, and car driving.62 61 Bydžovský

et al. (1940). The problem is on page 7. This textbook was used in Czechoslovakia from the beginning of the 1930s until the end of the 1950s. 62 I obtained this modest information about Jiˇrina Frantíková Chytilová from archival documents held in the National Archive of the Czech Republic and from personal correspondence with her niece.

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In 1937, a short abstract of her Ph.D. thesis63 was published in the journal Spisy vydávané pˇrírodovˇedeckou fakultou Karlovy university, which specialized in publishing articles of this kind. For more information, see the review in the journal Zentralblatt für Mathematik und ihre Grenzgebiete.64 Jiˇrina Frantíková published an article titled “Some Approximate Formulas”.65 It was reviewed by K. Löer from Göttingen in the journal Jahrbuch über die Fortschritte der Mathematik 66 and by W. Simonsen from Copenhagen in the journal Zentralblatt für Mathematik und ihre Grenzgebiete.67 Libuše Kuˇcerová (1902–1987) started her Ph.D. proceedings in 1937 under the supervision of V. Hlavatý. Despite many problems during WWII and the postwar changes in Czechoslovak society, she successfully finished her doctorate in 1952 and obtained her Ph.D. degree from the Faculty of Science of Charles University in Prague. Libuše Kuˇcerová was a daughter of Václav Kuˇcera (1866–?), who was a teacher at the municipal school in Sadská (a small town in the central Czech lands), and Albína (née Bˇreská, ?–?), a daughter of a teacher at the village school. From 1915 to 1921, Libuše Kuˇcerová studied at the Czech State Real School in Nymburk. From 1921 to 1923, she studied at the Czech Technical University in Prague. She attended lectures on mathematics, physics, descriptive geometry, geology, chemistry, drawing, statistics, mechanics and civil industry, architecture, civil laws, and English and French. She wanted to become a civil engineer, but she did not finish her studies. From 1923 to 1925, she studied only descriptive geometry and mathematics at the same school. From 1922 to 1926, she also studied as an extraordinary student at the Faculty of Science of Charles University in Prague. She attended lectures there on mathematics, history, Czech, pedagogy, and art history. In 1926, she passed the so-called teacher examinations to become a teacher of mathematics and descriptive geometry. In 1926, she became a suplent (substitute teacher) at the secondary school in Kralupy nad Vltavou. Two years later, she was appointed a regular teacher of mathematics and descriptive geometry at secondary schools. She taught mathematics, drawing, and descriptive geometry at several places in the Czech lands (Liberec, Prague, Duchcov, Nymburk, and again Prague). She was a very active teacher; she prepared mathematical competitions, sports tournaments, and two long day trips for her students.68 In 1937, Libuše Kuˇcerová finished her Ph.D. thesis Geometrie cˇ tyrrozmˇerného Minkowskiho prostoru M 4 v souvislosti s trojrozmˇernou cyklografíí (“Geometry of

63 Frantíková

(1937). Zentralblatt für Mathematik und ihre Grenzgebiete, ZBL 0018.15903. 65 Frantíková (1936/1937). 66 See Jahrbuch über die Fortschritte der Mathematik, JFM 63.1122.04. 67 See Zentralblatt für Mathematik und ihre Grenzgebiete, ZBL 0016.31601. 68 She taught mathematics at various secondary schools in Prague even after her retirement age. In fact, my husband was one of her students. 64 See

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the Four-Dimensional Minkowski’s Space M 4 in Connection with ThreeDimensional Cyclography”, evaluated by V. Hlavatý and B. Bydžovský). She tried to pass the first (main) oral examination in geometry and mathematical analysis in January of 1951 (i.e. 14 years after finishing her Ph.D. thesis and 25 years after finishing her university studies), but she was not successful. Finally, she passed the main oral examination in June of 1952. She underwent the second (subsidiary) oral examination in the philosophy of exact sciences in March of 1952. Let us note that she had to pass her subsidiary oral examination according to the “new” communist ideology. She obtained her doctoral degree in natural sciences at the graduation ceremony held on 28 March 1952. Her Ph.D. thesis is not available at the archive of Charles University. In 1943, she married the engineer Josef Tuháˇcek (1903–?), her schoolmate from the Czech Technical University in Prague, who became an officer of the Czechoslovak army. They had no children.69 As a student in 1922, Libuše Kuˇcerová became a member of the Jednota cˇ eských matematik˚u a fysik˚u (Union of Czech Mathematicians and Physicists). She was interested in mathematics as well as the activities of the Czech mathematical community. Libuše Kuˇcerová wrote three articles (two short notes and one short abstract from her Ph.D. thesis). In 1933, she published her note “Poznámka ke Cliffordovým rovnobˇežkám” (“A Remark on Clifford’s Parallels”).70 V. Hlavatý, who was Kuˇcerová’s teacher and doctoral supervisor, reviewed the work in the journal Jahrbuch über die Fortschritte der Mathematik.71 We can note that Kuˇcerová became interested in the problems of four-dimensional geometry under the beneficial and strong influence of V. Hlavatý. She investigated a similar problem in her article “Poznámka k stejnoúhlým rovinám cˇ tyˇrrozmˇerného prostoru” (“A Remark on Isocline Planes in FourDimensional Spaces”).72 She took her inspiration from foreign as well as Czech monographs and articles. Her knowledge of classical as well as modern mathematical literature was excellent. The last known mathematical work by Kuˇcerová is “La géométrie de l’espace à quatre dimensions de Minkowski en connexion avec la cyclographie à trois dimensions: Laboratoire pour la philosophie des mathématiques”.73 It is an abstract of her Ph.D. thesis.74 Vˇera Kofránková (1909–1996) started her Ph.D. thesis in 1937 under the supervision of V. Hlavatý, but she never finished her major examination in mathematics. She was a daughter of Josef Kofránek (1878–1951), who was a civil engineer (a specialist in the construction of sluice gates and the regulation of rivers), and Marie (née Svobodová, 1884–1962), the daughter of a businessman. Vˇera Kofránková had 69 I obtained this modest information about Libuše Kuˇ cerová Tuháˇcková from documents held at the National Archives of the Czech Republic and from interviews with her pupils and students. 70 Kuˇ cerová (1933). 71 See Jahrbuch über die Fortschritte der Mathematik, JFM 59.0554.01. 72 Kuˇ cerová (1936). 73 Kuˇ cerová (1938). 74 See Jahrbuch über die Fortschritte der Mathematik, JFM 64.0661.05.

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one younger brother, Ivo (1911–1994), and one younger sister, Marie (1912–2000). Ivo Kofránek studied at the Faculty of Medicine of Charles University in Prague. In 1936, he became a physician. Marie Kofránková studied at the Faculty of Law of Charles University in Prague. In 1936, she became a lawyer. From 1920 to 1928, Vˇera Kofránková studied at the Czech State Real Gymnasium in Brandýs nad Labem. From 1928 to 1932, she studied descriptive and synthetic geometry at the Czech Technical University in Prague. From 1928 to 1935, she studied mathematics, natural philosophy, Czech, English, and Italian. In 1933, she passed the so-called teacher examinations to become a teacher of mathematics and descriptive geometry. In 1937, Vˇera Kofránková completed her Ph.D. thesis Kˇrivky, jejichž polomˇer kˇrivosti je lineární kombinací polomˇer˚u kˇrivosti koneˇcného poˇctu daných kˇrivek, aplikace (“Curves Whose Radius of Curvature Is a Linear Combination of the Radii of the Curvature of a Finite Number of Curves: Applications”, evaluated by V. Hlavatý and B. Bydžovský). She passed only the second (subsidiary) oral examination in the philosophy of exact sciences in May of 1937. Her Ph.D. thesis is not available at the archive of Charles University in Prague. In 1934, she started her professional career as a substitute teacher at a secondary school in Prague. In 1935, she married the Czech mathematician Zdenˇek Pírko (1909– 1983), her schoolmate.75 As we know from her daughter’s and granddaughter’s recollections, their marriage was not ideal. They had to live in different places where they taught. In 1948, they amicably divorced. She taught mathematics, drawing, and descriptive geometry at several secondary schools in Central Bohemia and later in Prague. Throughout her life, she took care of her only daughter Ivana (born 1945), who became a gynecologist. She liked playing piano, crocheting, needlework, and solving rebus puzzles. As a student and beginning teacher, she was a member of the Jednota cˇ eských matematik˚u a fysik˚u (Union of Czech Mathematicians and Physicists). In 1936, she published a short abstract of her Ph.D. thesis with a French resumé in the journal Spisy vydávané pˇrírodovˇedeckou fakultou Karlovy university.76 The article was reviewed in the journals Jahrbuch über die Fortschritte der Mathematik 77 and Zentralblatt für Mathematik und ihre Grenzgebiete.78

75 For

more information on Zdenˇek Pírko, see Jarkovský (1984). (1936). 77 See Jahrbuch über die Fortschritte der Mathematik, JFM 62.1467.07. 78 See Zentralblatt für Mathematik und ihre Grenzgebiete, ZBL 0016.04204. 76 Kofránková

3 Women and Mathematics at the Universities in Prague Table 3.1 Social background of the candidates

101

Father’s profession

German female doctors

Czech female doctors

Secondary school teacher

0

3

Engineer

0

2

Clerk

0

2

Physician

0

1

Lawyer

1

0

Municipal school teacher

0

1

Entrepreneur

1

0

Retailer

1

0

3.5 Comparison All of these female doctoral candidates were, at that time, Czechoslovak citizens79 of Czech or German nationality. The candidates of Czech nationality studied exclusively at Charles University in Prague, whereas the candidates of German nationality studied exclusively at the German University in Prague. At the time of their studies, all the female candidates declared themselves to be religious believers, as was more or less usual in the Czech lands. All the German candidates were Jewish, though one of them converted to another religion during her studies, even twice. The Czech female candidates were Roman Catholics, though one of them converted to the Bohemian Brethren (Czech Brethren or Bohemian Protestants) during her studies, and one of them left the Roman Catholic Church in the 1950s. The majority of the female candidates (except for one) descended from socially well-situated “middle-class” families, which valued education and supported the educational, cultural, athletic, and other general activities pursued by their daughters. Their fathers descended mainly from socially lower but financially secure levels of society (farm or manor administrators, farmers, craftsmen, lower school teachers). This provided the female candidates with financial means and a sufficient intellectual background (Table 3.1).80

79 Because of the origin of her parents, one female candidate had to apply for Czechoslovak citizenship in the administrative procedure. She was born in Galicia. Her parents came to Prague in the 1890s and they were Jews of German-Ukrainian nationality with no roots or relatives in the Czech lands. 80 The majority of male candidates for doctoral degrees in mathematics also descended from socially well-situated families (until the 1930s). To compare our statistics with the situation in other countries, see Tobies (2008), where the author also points out and analyses different influential factors on the careers of women.

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Table 3.2 High-school education of the candidates

German female doctors

Czech female doctors

German gymnasium for girls

3

0

Czech real school

0

3

Czech real gymnasium 0 for girls

2

Czech reformed real gymnasium for girls

0

2

Minerva school for 0 girls (“Krásnohorská”)

2

The German doctoral candidates studied, at various times, at the same German gymnasium for girls in Prague II. The Czech candidates studied at Czech secondary schools (real gymnasiums, real reformed gymnasiums, and real schools). Only the first Czech doctoral candidate studied at the time when the schools for girls did not have the same rights as those for boys, and that is why she had to pass an additional graduation examination at the traditional gymnasium for boys on Štˇepánská Street in Prague. None of the candidates studied at a traditional gymnasium, which emphasized Latin, Greek, history, and geography and which was usually preferred by students of the humanities (Table 3.2). Another matter of interest is the duration of the studies of the individual female doctoral candidates in mathematics. The average duration was nine semesters, which was in accordance with the requirement of the time, since the minimum duration of doctoral studies was prescribed to be eight semesters of university study (a shorter length of study than eight semesters had to be allowed by the professorial council). The shortest duration of university studies of a candidate was 5 semesters, and the longest was 14. If we also include the duration of the studies of the Czech candidates at a technical school or the duration of the studies of the German candidates abroad, the shortest duration of their studies was 9 semesters, whereas the longest was 22 semesters (Table 3.3).81 It is also of interest that the Czech candidates, in particular, studied not only mathematics but also modern foreign languages (French, English, and Italian), history, philosophy, and the arts. Even physical training was not omitted. The scope of their interests was very wide.82

81 Let us add that male doctoral candidates in mathematics usually studied only at a university; they

did not attend special lectures at a technical university. 82 The scope of interest of male doctoral candidates in mathematics was not as wide; they did not stray from their professional interests and they preferred to specialize more deeply in mathematics or physics.

3 Women and Mathematics at the Universities in Prague Table 3.3 The duration of studies of the individual female doctoral candidates

103

Number of semesters German female of university studies doctors

Czech female doctors

5

0

1a

6

0

2b

7

0

0

8

0

2c

9

1

1d

10

1

2e

11

0

0

12

0

0

13

1f

0

14

0

1g

a At the same time, the candidate also studied four semesters at the

Czech Technical University in Prague at the same time, both the candidates also studied four semesters at the Czech Technical University in Prague c One candidate also studied two semesters at the University in Vienna d At the same time, the candidate also studied eight semesters at the Czech Technical University in Prague e One candidate also studied four semesters at the Czech Technical University in Prague, the other also studied for six semesters at the same school f The candidate probably also studied two semesters at the University of Leipzig g Mostly at the same time, the candidate also studied eight semesters at the Czech Technical University in Prague b Mostly

All the German female doctoral candidates properly submitted their doctoral theses, which were accepted, and they were successful at the first defense of both Ph.D. examinations. Two of the Czech candidates failed at the first attempt at the main Ph.D. examination in mathematics; however, they were successful at the remedial examination. One of the Czech candidates missed the main Ph.D. examination in mathematics and did not obtain her doctorate. All the German and Czech candidates passed an examination in philosophy (later changed to the philosophy of exact sciences) within the framework of the subsidiary Ph.D. examination. The majority of the female doctoral candidates in mathematics (except for one) submitted their doctoral thesis, passed both Ph.D. examinations, and obtained their doctorate within 2 years after the completion of their studies, most often in the last year of their studies. This means that they did not extend their studies or postpone the submission of their theses and their examinations, as is typical today. Let us complete

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Table 3.4 Duration of studies Number of years from the completion of studies to graduation

German female doctors

Czech female doctors

0 yeara

2

5

1 year

0

2

2 years

1

0

25 years

0

1

a The

zero indicates that the candidate submitted her doctoral thesis in the last year of her studies, passed both the Ph.D. examinations, and her doctoral graduation took place in the same calendar year

this information with the fact that many of them, apart from their doctoral proceedings, also passed the demanding examinations for teaching proficiency (two of the three German female doctors,83 six of the nine Czech female doctors84 ) (Table 3.4). The first doctorate in mathematics awarded to a woman—a Czech candidate— happened as early as the year 1901. Afterwards, however, there was a pause of almost 30 years at Charles University (i.e. Czech University), whose reasons are unknown. Let us note that women studied at the university (almost all specializations). They defended their Ph.D. theses in languages, history, philosophy, chemistry, physics, zoology, etc. but not in mathematics. Further Czech female doctors in mathematics appeared only in the late 1920s and especially in the first half of the 1930s. We can assume that the Czech candidates of the 1930s at least knew each other, given that they always went to the same lectures and seminars at the same university or technical university.85 Two of the candidates already knew each other from their school days because they had attended the same secondary school for girls in Pilsen. One of the female candidates studied at the school, where the first Czech female doctor in mathematics worked as a secondary school teacher and headmistress. The first doctorate in mathematics awarded to a woman by the German University in Prague happened in 1919, i.e. almost 20 years after the first doctorate in mathematics was awarded at the Czech University. The second doctorate in mathematics was awarded in 1921, and the third in 1934 (Table 3.5).

83 Both German doctoral candidates in mathematics passed teaching examinations in the subjects of mathematics/physics. 84 Four of the Czech doctoral candidates in mathematics passed teaching examinations in the subjects of mathematics/descriptive geometry, one candidate in mathematics/physics, and one in mathematics/physical education. 85 Four of the nine Czech candidates studied insurance mathematics at the Czech Technical University in Prague, and the other three studied “a course for prospective teachers of descriptive geometry”. I should mention that the study of insurance mathematics was, in the 1920s and 1930s, relatively popular among women because it enabled them to be prepared for various professions in the fields of insurance, banking, and accounting. The course was especially popular because it lasted only 2 years.

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Table 3.5 Dates of awarded doctoral degrees Years in which doctoral degrees ere awarded

German female doctors

Czech female doctors

1900

0

1

1919

1

0

1921

1

0

1928

0

1

1932

0

1

1933

0

2

1934

1

1

1936

0

1

1952a

0

1

a The

last candidate submitted her doctoral thesis in 1937 but did not pass her Ph.D. examinations until the early 1950s

Let us conclude by summarizing the types of employment that the female doctors and doctoral candidates in mathematics pursued after completing their doctoral proceedings. It is no surprise that almost half of them worked as secondary school teachers (5 of 12). A quarter of them became housewives (3 of 12); one of them was a housewife and worked as a mathematician only occasionally; one of them worked at the Ministry of Finances; one of them worked at an insurance company in Prague. The employment history of one of the female doctors could not be ascertained; most probably, she worked as a secondary school teacher. Let us emphasize that the profession of high school teacher was one of the few where women frequently found positions. This job required studying at university, but a doctoral degree was not a prerequisite for it.86

3.6 Conclusion Let us say that, although our 12 women did not devote themselves to mathematical research after obtaining their doctoral degrees, for various reasons they proved that they could compete with men provided they had the motivation and courage to do so at a time when the public regarded female mathematicians with distrust.

86 Male

doctors and doctoral candidates in mathematics had many more professional opportunities than their female colleagues. They could work as assistants and later as private docents or professors at universities; they could become secondary school teachers (and later directors of secondary schools or higher schools, school inspectors, clerks at the Ministry of Education etc.); and they could become senior officials at banks, insurance companies, or businesses or work in the research ˇ departments of large industrial enterprises (such as CKD in Prague or Škoda in Pilsen).

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ˇ Protokol o pˇrísných zkouškách k dosažení hodnosti doktora na Filozofické fakultˇe c.k. Ceské univerzity v Praze [Register of Doctoral Candidates, Faculty of Philosophy, Czech Karl-Ferdinand University in Prague]. Protokol o pˇrísných zkouškách k dosažení hodnosti doktorské na Pˇrírodovˇedecké fakultˇe Univerzity Karlovy v Praze, Rigorosa. Protokol. PˇrF UK, I, 1–802, 1920–1935 [Register of Doctoral Candidates, Faculty of Science, Charles University in Prague]. Protokol o pˇrísných zkouškách k dosažení hodnosti doktorské na Pˇrírodovˇedecké fakultˇe Univerzity Karlovy v Praze, Rigorosa. Protokol. PˇrF UK, II, 803–2136, 1935–1953 [Register of Doctoral Candidates, Faculty of Science, Charles University in Prague]. Posudky disertaˇcních prací, Filozofická fakulta Univerzity Karlovy v Praze [Reviews of Doctoral Thesis, Faculty of Philosophy, Charles University in Prague]. Posudky disertaˇcních prací, Pˇrírodovˇedecká fakulta Univerzity Karlovy v Praze [Reviews of Doctoral Thesis, Faculty of Science, Charles University in Prague]. ˇ Matriky doktor˚u Ceské Karlo-Ferdinandovy univerzity v Praze [The Book Registers of Doctors at the Czech Charles-Ferdinand University in Prague]. Matriky doktor˚u Univerzity Karlovy v Praze [The Book Registers of Doctors at Charles University in Prague]. ˇ Zkoušky uˇcitelské zp˚usobilosti – Ceská zkušební komise [Teachers’ Examinations – Czech Commission].

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National Archive of the Czech Republic Fond Pˇr II – EO, Policejní ˇreditelství Praha II – evidence obyvatelstva [Pˇr II – EO, Police Direction Prague II – Register of Citizens]. Osobní evidenˇcní karty [Personal Registration Documents]. Fond Policejní ˇreditelství Praha II – všeobecná spisovna – 1914–1920 [Police Direction Prague II – General Register – 1914–1920]. Folder F 10/148 Falková Hilda, box 7151. Folder F 10/64 Falková Margarethe, box 7151. Fond Policejní ˇreditelství Praha II – všeobecná spisovna – 1921–1930 [Police Direction Prague II – General Register – 1921–1930]. Folder F 1243/7 Franklová – Kellerová Josefina 1904, box 712. Folder J 676/18 Jašková Miluše 1905, box 1325. Folder J 1543/22 Johnová Josefina 1904, box 1390. Folder K 3432/37 Kuˇcerová Libuše 1902, box 1906. Folder R 170/26 Ramler Gerson 1863, box 2970. Folder S 7181/18 Struiková – Ramlerová Salˇca Ruth 1894, box 3605. Fond Policejní ˇreditelství Praha II – všeobecná spisovna – Pˇr 1931–1940 [Police Direction Prague II – General Register – Pˇr 1931–1940]. ˇ Folder C 306/9 Cechová Vˇera 1910, box 5107.

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Folder F 839/12 Fischer Otto 1909, box 5750. Folder I 35/1 Illingerová Ludmila dr. 1908, box 6931. Folder K 2264/12 Kofránková Vˇera 1909, box 7603. Folder M 298/1 Mayer Alfred 1899, box 8597. Folder P 1853/15 Pírko Zdenˇek 1909, box 9658. Folder R 217/2 Ramlerová Arnoštka 1890, box 9975. Folder R 217/5 Ramlerová Natalie 1887, box 9975. Folder R 217/3 Ramlerová R˚užena 1891, box 9975. Folder T 1482/6 Tuháˇcek Josef 1903, box 11314. Folder T 2366/17 Tuháˇcková Libuše, box 11780. Fond Policejní ˇreditelství Praha II – všeobecná spisovna – 1941–1950 [Police Direction Prague II – General Register – 1941–1950]. Folder F 22/10 Fabianová Marie 1872, box 1973. Folder F 65/3 Falková Hilda dr. 1897, box 1983. Folder F 63/21 Falková Margarethe 1896, box 5629. Folder F 1421/1 Frankl Jan (Hanuš) 1900, box 2296. Folder F 1442/3 Frantíková Jiˇrina 1914, box 2300. Folder G 806/2 Greˇcenko Vsevolod 1898, box 2577. Folder G 806/3 Greˇcenková Miluše 1905, box 2577. Folder Ch 176/28 Chytil František 1908, box 4070. Folder Ch 177/27 Chytilová Jiˇrina 1914, box 4071. Folder I 12/6 Iglauer Boˇrivoj 1910, box 4170. Folder I 12/5 Iglauerová Jarmila 1910, box 4170. Folder K 1484/8 Keller Rudolf 1875, box 5065. Folder K 1502/11 Kellerová Helena 1882, box 5068. Folder M 425/11 Mayerová Josefina 1904, box 6928. Folder M 2231/9 Mˇestka Alois 1904, box 7370. Folder M 2233/1 Mˇestková Ludmila dr. 1908, box 7370. Folder R 259/3 Ramler Leo 1892, box 9293.

Secondary Literature Abele, A., Neunzert, H., & Tobies, R. (2004). Traumjob Mathematik! Berufswege von Frauen and Männern in der Mathematik. Basel: Birkhäuser Verlag. ˇ Anonymous. (1901). Druhá promoce ženská na Ceské universitˇe Karlo-Ferdinandovˇe v Praze (The Second Female Doctoral Graduation Ceremony at the Czech University in Prague). Ženské listy, 29, 242–246. Anonymous (1902). První lékaˇrská promoce ženská v Praze (The First Female Doctoral Graduation Ceremony in Prague at the Faculty of Medicine). Ženské listy, 30, 69–73. ˇ Báhenská, M. (2005). Poˇcátky emancipace žen v Cechách: Dívˇcí vzdˇelávání a ženské spolky v Praze v 19. století (The Origins of the Emancipation of Women in Bohemia: Girls’ Education and Women’s Associations in Prague in the Nineteenth Century). Prague: Libri. Beˇcváˇrová [née Nˇemcová], M. (1998). František Josef Studniˇcka (1836–1903). Prague: Prometheus. ˇ Beˇcváˇrová, M. (2008). Ceská matematická komunita v letech 1848–1918 (The Czech Mathematical Community from 1848 to 1918). Prague: Matfyzpress. Beˇcváˇrová, M. (2016a). Matematika na Nˇemecké univerzitˇe v Praze v letech 1882–1945 (Mathematics at the German University in Prague from 1882 to 1945). Prague: Karolinum. Beˇcváˇrová, M. (2016b). Women and Mathematics at the Universities in Prague in the First Half of the 20th Century. Antiquitates Mathematicae, 10(1), 133–167. Beˇcváˇrová, M. (2018). Saly Ruth Struik, 1894–1993. The Mathematical Intelligencer, 40(4), 79–85.

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Beˇcváˇrová, M., & Netuka, I. (2015). Karl Löwner and His student Lipman Bers: Pre-war Prague mathematicians. Zürich: European Mathematical Society. Birk, A. (1931). Die Deutsche Technische Hochschule in Prag, 1806–1931. Prague: Calve. Bottema, O. (1978). Equi-affinities in three-dimensional space: With a dedication in French to D. S. Mitrinovíc on his seventieth birthday. Univerzitet u Beogradu: Publikacije Elektrotehniˇc Fakulteta – Serija Matematika i Fizika, 603, 9–15. Bydžovský, B., et al. (1940). Aritmetika pro IV. tˇrídu stˇredních škol (Textbook on Arithmetic for the Fourth Class of Secondary School) (7th ed.). Prague: Jednota cˇ eskoslo-venských matematik˚u a fysik˚u. Enriques, F. (Trans.). (1930–1936). Gli elementi d’Euclide e la critica antica e moderna (3 vols.). Rome: Alberto Stock. Fabiánová, M. (1918). Vládní rada Josef Grim. In: XXVIII. výroˇcní zpráva mˇestského reál. gymnasia dívˇcího “Krásnohorská” v Praze II: Za školní rok 1917–18. Prague: N´akladem u´ stavu. 3–4. Frantíková, J. (1936/1937). Some approximate formulas. Aktuárské vˇedy, 6, 102–106. Frantíková, J. (1937). Úrokový problém pro d˚uchody životní s malou poznámkou pro prémiové reservy smíšeného pojištˇení (An Interest Income Problem for Life Pensions with a Brief Note About Premium Reserves). Výtah disertaˇcních prací pˇredložených pˇrírodovˇedecké fakultˇe Karlovy university v Praze v roce 1936–37: Spisy vydávané pˇrírodovˇedeckou fakultou Karlovy university, 154, 11–14. Green, J., & LaDuke, J. (2009). Pioneering Women in American Mathematics: The Pre-1940 PhD’s. London: London Mathematical Society. Havránek, J., & Pousta, Z. (Eds.). (1997–1998). Dˇejiny Univerzity Karlovy, 1348–1990 (History of Charles University, 1348–1990) (Vols. 3–4). Prague: Univerzita Karlova. Horská, P. (1995). Die deutschen Frauenvereine in Böhmen. Germanoslavica, 2(7), 117–121. Hykšová, M. (2003). Karel Rychlík (1885–1968). Prague: Prometheus. Illingerová, L. (1933). Pˇríspˇevek k neeuklidovské geometrii (Contribution to Non-Euclidean ˇ Geometry). Casopis pro pˇestování matematiky a fysiky, 62, 154–163. ˇ Illingerová, L. (1935a). Die loxodromische Geometrie. Casopis pro pˇestování matematiky a fysiky, 64, 193–194. Illingerová, L. (1935b). Poznámka k cˇ lánku p. Jos. Kopeˇcného: Über die Bestimmung der Summe ˇ der Winkel im ebenen Dreieck (A Remark on the Article of Jos. Kopeˇcný …). Casopis pro pˇestování matematiky a fysiky, 64, D133–D134. Illingerová, L. (1936). Loxodromická geometrie (Výtah z disertace) (Loxodromic Geometry ˇ (Dissertation Abstract)). Casopis pro pˇestování matematiky a fysiky, 65, D6–D8. Jarkovský, Z. (1984). In Memoriam Professor Zdenˇek Pírko. Czechoslovak Mathematical Journal, 34, 163–164. ˇ Jílek, F., & Lomiˇc, V. (1973). Dˇejiny CVUT (History of the CTU in Prague) (Vol. 1.1). Prague: ˇ CVUT. Kofránková, V. (1936). Kˇrivky, jejichž polomˇer kˇrivosti je lineární kombinací polomˇer˚u kˇrivosti koneˇcného poˇctu daných kˇrivek, aplikace (Curves Whose Radius of Curvature Is a Linear Combination of the Radii of the Curvature of a Finite Number of Curves). Výtah disertaˇcních prací pˇredložených pˇrírodovˇedecké fakultˇe Karlovy university v Praze v roce 1936: Spisy vydávané pˇrírodovˇedeckou fakultou Karlovy university, 150: 33–37. Kopáˇc, J. (1968). Dˇejiny cˇ eské školy a pedagogiky v letech 1867–1914 (History of the Czech School and Pedagogy During the Years 1867–1914). Brno: Univ. J. E. Purkynˇe. Kotzianová, Z. (1989). Spolek Vesna v Brnˇe v letech 1870–1918 a jeho význam pro rozvoj cˇ eské národní kultury (The Vesna Association in Brno in the Years 1870–1918 and Its Importance for the Development of Czech National Culture). Doctoral dissertation, FF UJEP, Brno. Kuˇcerová, L. (1933). Poznámka ke Cliffordovým rovnobˇežkám (Remark on Clifford’s Parallels). ˇ Casopis pro pˇestování matematiky a fysiky, 62, 231–232. Kuˇcerová, L. (1936). Poznámka k stejnoúhlým rovinám cˇ tyˇrrozmˇerného prostoru (A Remark on ˇ Isocline Planes in Four-Dimensional Spaces). Casopis pro pˇestování matematiky a fysiky, 65, D9–D13.

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Kuˇcerová, L. (1938). La géométrie de l’espace à quatre dimensions de Minkowski en connexion avec la cyclographie à trois dimensions: Laboratoire pour la philosophie des mathématiques. Výtah disertaˇcních prací pˇredložených pˇrírodovˇedecké fakultˇe Karlovy university v Praze v roce 1937–38: Spisy vydávané pˇrírodovˇedeckou fakultou Karlovy university, 161, 19–21. Mˇestková-Illingerová, L. (1942). Nˇekteré znaky dˇelitelnosti (Some Criteria of Divisibility). Rozhledy matematicko-pˇrírodovˇedecké, 21, 49–53. Míšková, A. (2002). Nˇemecká (Karlova) univerzita od Mnichova k 9. kvˇetnu 1945 (vedení univerzity a obmˇena profesorského sboru). Prague: Karolinum (German translation: Die Deutsche (Karls-) Universität vom Münchener Abkommen bis zum Ende des Zweiten Weltkrieges: Universitätsleitung und Wandel des Professorenkollegiums. Prague: Karolinum, 2007). Powell, A. B., & Frankenstein, M. (2001). In Memoriam Dirk Jan Struik: Marxist Mathematician, Historian, and Educator (30 September, 1894–21 October, 2000). For the Learning of Mathematics: An International Journal of Mathematics Education, 21(1), 40–43. ˇ Rostoˇcilová, V. (1972). První promoce absolventek dívˇcího gymnasia ‘Minerva’ na Ceské univerzitˇe v Praze (First Graduation Ceremonies of the Graduates of the Girls’ Gymnasium ‘Minerva’ at the Czech University in Prague). Acta Universitatis Carolinae, Historia Universitatis Carolinae Pragensis, 12, 241–245. Secká, M. (2013). Americký klub dam: Kr˚ucˇ ek k ženské vzdˇelanosti (The American Ladies’ Club: A Step Towards Women’s Education). Prague: Národní muzeum. Seibt, F. (1948). Die Deutsche Universität in Prag: Ein Gedenken anlässlich der 600. Jahrfeier der KU in Prag. Munich: Gräfeling. Štrbáˇnová, S., et al. (2004). Women Scholars and Institutions: Proceedings of the International Conference, Prague, June 8–11, 2003. Prague: Výzkumné centrum pro dˇejiny vˇedy. Struik, D. J. (1974). A Letter from Dirk Struik. In: Cohen, R. S. et al. (Eds.), For Dirk Struik: Scientific, Historical, and Political Essays in Honor of Dirk J. Struik (pp. xiii–xvii). Dordrecht: Reidel. Struik, D. J., & Struik, S. R. (1928). Cauchy and Bolzano in Prague. Isis, 11, 364–366. Struik, S. R. (1977). Flächengleichheit und Cavalierische Gleichheit von Dreiecken. Elemente der Mathematik: Zeitschrift zur Pflege der Mathematik und zur Förderung des mathematischphysikalischen Unterrichts, 32(6), 137–143. ˇ ˇ Tayerlová, M., et al. (2004). Ceská technika (Czech Technical University). Prague: CVUT. Tobies, R. (2008). “Aller Männerkultur zum Trotz”: Frauen in Mathematik, Naturwissenschaften und Technik (2nd ed.). Frankfurt am Main: Campus. Tobies, R. (2012). German graduates in mathematics in the first half of the 20th century: biographies and prosopography. In: Rollet, L. & Nabonnand, P. (Eds.), Les uns et les autres: Biographies et prosopographies en histoire des science (pp.387–407). Nancy: Presses universitaires de Nancy. Tulachová, M. (1965). Disertace pražské university 1882–1953 (Dissertations at the Prague University, 1882–1952). Prague: Universita Karlova. Uhrová, E. (2012). Anna Honzáková a jiné dámy (Anna Honzáková and Other Ladies). Prague. Velflík, A. V. (1906–1925). Dˇejiny technického uˇcení v Praze (History of the Technical University ˇ in Prague). Prague: Ceská matice technická. Vondráˇcek, J., & Šidák, Z. (1976). Vzpomínka na RNDr. Otto Fischera (A Recollection of RNDr. Otto Fischer). Aplikace matematiky, 21(5), 393–394. Výborná, M., et al. (1965). Disertace pražské university, vol. II (1882–1945) (Dissertations at the Prague University, vol. II (1882–1945)). Prague: Universita Karlova.

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Martina Beˇcváˇrová (born in 1971) studied mathematics and physics at Charles University in Prague, where she received her Ph.D. in the history of mathematics (1997). In 2016, she became a professor at Charles University in Prague. From 1998 until now, she teaches mathematics at the Faculty of Transportation Sciences of the Czech Technical University in Prague. Her research interests include the history of mathematics, the history of scientific societies and schools and the history of education in mathematics. She published as an author or co-author seventeen monographs and several research papers (in Czech, English, German, Italian, Polish, Croatian and Bulgarian).

Part II

Couples

Introductory Reflections on Couples Nicola M. R. Oswald Shaping “Collaboration”: From Teams to Couples It is perfectly natural in science to operate in teams of researchers. Working in laboratories and projects of applied sciences often demands a coordinated team of investigators with diverse skills and knowledge. A publication in natural sciences is nowadays quite often authored by more than a dozen persons. Additionally, the role of collaborating scientific networks comes more and more to the fore.1 This development of shared expertise and, consequently, merging of different positions into teams of researchers gained importance when new branches of sciences, especially concerning technological and industrial research, developed. Comprehensive studies as well as case studies show that around the end of the nineteenth and, moreover, in the first half of the twentieth century the arising need of qualified employees allowed educated women certain careers in academic and industrial branches. Although a necessary precondition was given by the ever-increasing equal access to education for women and men; this did however not automatically imply an approximately equal salary or level of position. In her ground-breaking work Women Scientists in America. Struggles and Strategies to 1940, Margaret Rossiter analysed the situation of women scientists in the United States in detail. She also put a special focus on research in teams. Concerning the development of astronomy, the author describes, for example, that its rapid growth in combination with “certain competitive forces within the field itself” (Rossiter [1984], p. 55) led to an acceptance of female assistants working at the Harvard College Observatory in the 1880s and thereafter. Rossiter furthermore explained that those assistance jobs, considered as “women’s work”, were explicitly downgraded 1 Scientific

collaboration is in itself an interesting field of research. A rather fascinating example of a social network structure in science (astrophysics) is considered in the case study Heidler (2011).

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and “feminized”, which is tantamount to the fact that they had to be as cheap as possible.2 Another interesting example is given by American women in chemistry in the 1930s. As a result of the “growth and bureaucratization” of the chemical industry since World War I as well as of the growing number of publications in chemistry journals, female chemists could find positions in so-called “hybrid positions such as “chemical librarians”, “chemical secretaries”, bibliographers and abstractors” (Rossiter [1984], p. 253). Again specific “women’s work” was established for female scientists, who “could not be promoted above a certain minimum level” (Rossiter [1984], p. 254).3 And also, in Europe the event of World War I marks a certain change in the hiring strategies for scientists in industrial laboratories. In their profound introduction of the collective book Women in Industrial Research, Renate Tobies and Annette Vogt, explain that “[d]uring that war and in the 1920s, [...] women scientists were needed in the laboratories of chemical and pharmaceutical companies.” (Tobies & Vogt [2014], p. 7). Certainly, those examples show only a fraction of the diligent research on women in industrial and academic research. However, they illustrate that some branches of sciences fostering research in collaborating teams could at least deliver access to certain opportunities for women scientists. Is there a similar need for collaboration in mathematics? Mathematics itself is traditionally considered a rather lonely field of research done by individualists. The image of a lonesome mathematical genius is a classical and repeatedly used presentation. Particularly in popular scientific publications, like the mystified biography of Èvariste Galois The French mathematician (Petsinis [1998]), the description of Carl Friedrich Gauß as a quirky character in the fiction novel Measuring the World (Kehlmann [2007]) or the bestseller-biography The Man who loved only numbers (Hoffman [1999]), such a cliché is consolidated and reread with pleasure. This reflects social beliefs concerning mathematicians, which might be difficult (or not wanted) to be fulfilled by women. With respect to this traditional image, Laura Fainsilber concludes in her Report on the discussion on “role models”4 : Certainly the stereotypical image of the absorbed, unkempt, asexual man, is quite foreign to most of us, and we do not recognize it as a model.

Although mathematical research certainly demands a high level of concentrated mental effort and although there were and are individual mathematicians who are firmly against partner work,5 the described secluded individualism does only reflect a single aspect of practicing mathematics. Historical documents form a much more multifaceted picture: biographical notes show that the ritual of common “mathematical walks” has been one possibility to share and discuss ideas, deepening the 2 See

Rossiter (1984), pp. 54–56. Rossiter (1984), pp. 252–254. 4 Published in EWM (1991), p. 20. 5 For example, Leopold Kronecker (1823–1891) expressed that partner work on a specific topic even hindered the progress of mathematics (Kronecker [1892]). 3 See

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understanding and development of theories; several exchanges of letters between mathematicians illustrate that there had been an intense interchange of ideas and that results were fostered through mutual exchange.6 And François Lê even suggests the sociological definition of a common “cultural system” to characterize a group of mathematicians working with and on geometric equations in the second half of the nineteenth century (Lê [2016]).7 The Concept of Couples in Sciences A quite concrete way of sharing expertise and collaborating ideas is characterized by the concept of “Couples in the Sciences”. This “unit of analysis” (Abir-Am et al. [1996], p. 5) can be considered as a structuring tool shaping the investigation of gendered history. It has already been taken into account by several approaches in the history of sciences in general.8 The basis is formed by the definition of those Couples in Sciences as two partners of researchers, both having a higher education, either working and publishing together or at least “creating a nurturing environment for the other’s scientific work, even if the scientific work itself did not involve joint research and publications” (Slack [2012], p. 272). This definition may be understood as friction to induce discussions and to detect “patterns of collaboration”. Its aim is to shed light “not only [on] the history of women in science but also [on] the history of scientific collaboration” (Abir-Am et al. [1996], p. 3). In this part of the book, the focus is on three married couples of heterosexual persons, although the analysing concept could be extended to other kinds of co-workers. The integration of research into a family context is not an idea of the past century, its nature has changed with respect to contemporary circumstances. Whilst at the beginning of the nineteenth century “entire households, [were] engaged in the enterprise of science” (Abir-Am et al. [1996], p. 4),9 the ever-increasing institutionalized academia of the late nineteenth and twentieth century demanded women with a higher education. Examples in the history of sciences show that some women had had opportunities to take part in the current scientific discourse with the indirect or direct support of their families. Certainly, the parental influence was very strong on the scientific ambitions of young female researchers. Renate Tobies pointed out that more than 58% percent of female mathematicians, achieving their PhD between 1908 and 1945 in the German-speaking area, had a father with an academic background (Tobies [2008], 6 Famous

examples are the correspondence in Marin Mersenne’s network (see, e.g. Goldstein [2013]), the encounter between Leonhard Euler and Christian Goldbach (see, Lemmermeyer & Mattmüller [2015]) or the letter exchange between Sophie Germain and Carl Friedrich Gauss (see Del Centina & Fiocca [2012]). 7 The concept of mathematical schools has been experiencing a professionalization since the nineteenth century at the latest. A famous example of such a “school”, which unites mathematicians in one subject area, is that of Emmy Noether (1882–1835) (see Koreuber [2015]). 8 For example, see the diversified views on Collaboration Couples in the Sciences in Abir-Am et al. (1996) and Lykknes et al. (2012). 9 Ann B. Shteir introduced the term “family firm” for English botanical couples in Shteir (1987).

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p. 33).10 Famous examples like Emmy Noether (1882–1935) and Iris Runge (1888– 1966)11 show the direct influence of their fathers’ work as scientists. In Hermann Weyl’s eulogy for Noether one can read: But is it not a lovely thought to imagine that after this life on earth the souls recognize each other again, and how then the soul of your father would meet you? Has a father ever found such a great independent succession in his daughter?12

Besides the background of the own family, the interaction of married researchers formed a fruitful basis, a “unit of analysis” for historical work. Hereby the approach can vary widely and the analysis of the studies might be undertaken with different foci. One example is given by the study of the couple Hilda Geiringer (1893–1973) and Richard von Mises (1883–1953) by Reinhard Siegmund-Schultze (2017).13 The author placed a strong focus on the co-working of the mathematicians concerning the development of applied mathematics. Their interaction as a married couple formed the framework of his studies. He concluded: In spite of some subliminal but outwardly covered conflicts between the two, the overall outcome of the collaboration of the mathematicians’ couple was positive, both on the individual, subjective side and with respect to the development of applied mathematics as a whole.14

Three Examples: A Journey of Collaboration Through Time and Space The second part of the book deals with three mathematician couples: Grace Chisholm (1868–1944) and William Henry Young (1863–1942), Emma Shadkhan (1893– 1968) and Wladimir S. Woytinsky (1885–1960), Stanislawa (1897–1988) and Otton Marcin Nikodym (1887–1974). Although all of them had quite different ways of collaboration, each of them can be analysed from the point of view as a Couple in Science. Elisabeth Mühlhausen discusses a dispute between Grace Chisholm Young and the mathematician Max Dehn (1878–1952) which took place in an intense exchange of letters in 1906. As the relationship itself, the division of labour as well as the family duties of the couple Young has been analysed in detail in former publications of the author,15 the focus here is centred on this exemplary event reflecting the role 10 Tobies included 750 male and 116 female scientists into her study. Also, around 45% of the male mathematicians had fathers with academic background. Considering both, women and men, this reproduction of social classes is still nowadays a grave obstacle for the equality seeking society in Germany, see Kemper & Weinbach (2009). 11 See Tollmien (1990) on Emmy Noether and Tobies (2010) on Iris Runge. 12 “Aber ist es nicht ein lieblicher Gedanke, sich vorzustellen, daß doch nach diesem Erdenleben sich die Seelen noch einmal erkennen, und wie dann die Seele Deines Vaters Dir begegnen w¨urde? Hat je ein Vater in seiner Tochter so große selbstständige Nachfolge gefunden?” Excerpt of Weyl’s eulogy, 17.04.1935, quoted from Roquette (2007), p.19. 13 For biographical details of the eventful lives of Geiringer and von Mises see Siegmund-Schultze (1993). 14 Siegmund-Schultze (2017), p. 126. 15 See a.o. Mühlhausen (1993, 2016).

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of the mathematician and spouse of Wilhelm Henry Young. Several historians have thrown light on the well-documented interaction of the Youngs. After visiting Girton College providing higher education for female students and completing the mathematics tripos in Cambridge, Grace had obtained her PhD supervised by Felix Klein (1849–1925) at the University of Göttingen. Although she gave up any ambitions for an academic career after the birth of her first child, she never stopped practicing mathematics: She not only supported the work of her husband by editing his publications, she furthermore continued doing research herself: When Grace Chisholm Young published a joint paper with her husband in the Proceedings of the London Mathematical Society in 1910, she was just the fourth woman to have published there since the journal’s creation in 1865.16

The correspondence with Max Dehn illuminates her deep involvement in the development of mathematical theories. The reason why she is only mentioned in a few of her husband’s works was a quite reasonable decision: William Henry Young had to find a permanent position to earn a good living for the whole family. Since the employment opportunities for women were not profitable enough to obtain an appropriate income, the Youngs put all the stakes, which means all their mathematical results and work, on the academic career of William Henry. In Mühlhausen (2016), it is pointed out that Grace’s decision was not only consciously made, moreover, correspondence proves that she did indeed have a multifaceted and self-determined life. Still, [...] Grace’s experience shows how gendered ideas of genius within mathematics served to exclude women---and cause them to exclude themselves---from its highest reaches.17

In the subsequent chapter, the co-working of a couple who had to overcome a series of political and social obstacles is considered. The eventful lives of Emma and Wladimir Woytinsky took place in difficult political conditions, and yet they always revolved around the development of applied statistics. Whilst their famous book series “Die Welt in Zahlen” was created from 1925 to 1928 in Berlin under his sole authorship, their public collaboration underwent a certain change in the 1940s and 1950s in the United States of America. The author Annette Vogt analyses the different conditions of their life stages and illustrates common activities, namely the joint research on applied statistics and the engagement for social welfare. The third chapter is dedicated to Stanislawa Nikodym, the first female Polish doctoral student in mathematics, and to her husband Otton Nikodym, among others well-known for the Radon–Nikodym theorem. Including numerous contemporary documents, the author Danuta Ciesielska analyses vividly the mutual support of both mathematicians. The pair had largely independent careers, working on different subjects and the search for permanent positions led them through several stages in Europe and the USA. All three examples show in a remarkable manner that the analysis of Couples in Sciences allows insights into different aspects of mathematical co-working. In this context, Annette Vogt points out that the role in the scientific relationship reflects 16 Jones 17 Jones

(2009), p. 143. (2009), p. 206.

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the social and economic role of women in the society. This is implicitly underlined by the different characterizations used for those couples: The Youngs are considered as “Mathematical Union”.18 Annette Vogt underlines the fact that the Woytinskys were publishing together and explicitly distinguishes them from a so-called “Couple of Scientists”. This aspect is also taken up by Danuta Ciesielska describing the Nikodyms as a “typical collaborating couple” due to the independence of their scientific careers.

References Abir-Am, P., Pycior, H. M., Slack, N. G. (1996). Creative couples in the sciences (Lives of Women in Science). Rutgers University Press. Del Centina, A. & Fiocca, A. (2012). The correspondence between sophie germain and carl friedrich gauss. Archive for History of Exact Sciences, 66, 585–700. EWM (1991). Report on the fifth annual EWM meeting. European women in mathematics, CIRM, LUmony, France, December 9–13, 1991. Goldstein, C. (2013). Routine controversies: Mathematical challenges in Mersenne’s correspondence. Revue d’histoire des Sciences, Tome, 66(2), 249–273. Grattan-Guinness, I. (1972). A mathematical union: William henry and grace chisholm young. Annals of Science, 29(2), 105–186. Heidler, R. (2011). Cognitive and social structure of the elite collaboration network of astrophysics: A case study on shifting network structures. Minerva, 49(4), 461–488. Hilbert, D. (1921). Adolf Hurwitz. Math. Annalen Bd., 83(Nr. 3/4), 161–168. Hoffman, P. (1999). The man who loved only numbers: The story of Paul Erdos and the search for mathematical truth. Hachette Books. Kehlmann, D. (2007). Measuring the world. Riverrun. Kemper, A. & Weinbach, H. (2009). Klassismus. Eine einführung. Münster. Koreuber, M. (2015). Emmy noether, die noether-schule und die moderne algebra: Zur geschichte einer kulturellen bewegung. Reihe: Mathematik im Kontext. Berlin: Springer. Kronecker, L. (1892). In: Jahresbericht der DMV, 1890/91, Teubner Verlag, 1: 24. Lê, F. (2016). Reflections on the notion of culture in the history of mathematics: The example of ‘Geometrical Equations’. Science in Context, 29(3), 273–304. Lemmermeyer, F. & Mattmüller, M, (2015). Correspondence of leonhard euler with christian goldbach. Birkhäuser. Lykknes, A., Opitz, D. L., & van Tiggelen, B., (Eds.). (2012). For better or for worse? Collaborative couples in the sciences. Basel: Birkhäuser. Mühlhausen, E. (1993). Grace chisholm young ‘I liked being incognito to the outside world’. In T. Weber-Reich (Ed.), Des kennenlernens werth--bedeutende frauen göttingens. Göttingen. 195–211. Mühlhausen, E. (2016). Die mathematikerin und mehrfache mutter grace emily chisholm young (1868–1944). In A. Blunck, R. Motzer, & N. Oswald (Eds.), Tagungsband mathematik und gender, 5, 13–22. Petsinis, T. (1998). The french mathematician. Penguin Books Ltd. Roquette, P. (2007). Zu Emmy noethers geburtstag. Einige neue Noetheriana. Mitteilungen der DMV 15: 15–21.

18 This

description refers to the title of (Grattan-Guinness, 1972).

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Rossiter, M. W. (1984). Women scientists in america. Struggles and strategies to 1940. The John Hopkins University Press (Paperbacks edition). Siegmund-Schultze, R. (2017). Hilda geiringer (1893–1973)---the overall successful development of a female mathematician under male dominance and in spite of conditions adverse to women’s emancipation. Oberwolfach Reports (ISSN: 1660-8933), 17(1), 125–127. Siegmund-Schultze, R., & Geiringer-von Mises, H. (1993). Charlier series, ideology, and the human side of the emancipation of applied mathematics at the university of berlin during the 1920s. Historia Mathematica, 20, 364–381. Shteir, A. B. (1987). Botany in the breakfast room, women and early nineteenth-century British plant study. In P. Abir-Am & D. Outram (Eds.), Uneasy careers and intimate lives: Women in science, 1789–1979. New Brunswick, NJ: Rutgers University Press. 31–43. Slack, N. G. (2012). Epilogue: Collaborative couples---past, present and future. In A. Lykknes, D. L. Opitz & B. van Tiggelen (Eds.), For better or for worse? Collaborative couples in the sciences. Basel: Birkhäuser. 271–304. Tobies, R. (2008). Aller männerkultur zum trotz---Frauen in mathematik, naturwissenschaften und technik. campus Verlag. Tobies, R. (2010). Iris runge---A life at the crossroads of mathematics, science and industry. Birkhäuser. Tollmien, C. (1990). Sind wir doch der meinung, daß ein weiblicher Kopf nur ganz ausnahmsweise in der mathematik schöpferisch tätig sein kann...---eine biographie der mathematikerin Emmy Noether (1882–1935) und zugleich ein beitrag zur geschichte der habilitation von frauen an der universität göttingen. Göttinger Jahrbuch, 38, 153–219.

Chapter 4

Grace Chisholm Young, William Henry Young, Their Results on the Theory of Sets of Points at the Beginning of the Twentieth Century, and a Controversy with Max Dehn Elisabeth Mühlhausen Abstract In 1903 a publication by the English mathematician William Henry Young (1863–1942) was reviewed in the “Jahrbuch über die Fortschritte der Mathematik.” The mathematician Grace Chisholm Young (1868–1944) was motivated to contact Max Dehn (1878–1952) and discuss his critical remarks on her husband’s work. This was the beginning of an interesting correspondence which not only encompassed mathematics. The ensuing controversy reveals in detail how the couple Young worked together. At that time, Grace lived with their children in the center of mathematics, Göttingen, and her husband abroad as a lecturer and examiner in Cambridge and Liverpool. They enjoyed an intensive mathematical correspondence and even more lively discussions when they were together in Göttingen or elsewhere.

In the “Quarterly Journal of pure and applied mathematics” of 1903, an article by William Henry Young (1863–1942) was published entitled “On the Analysis of Linear Sets of Points.”1 Ivor Grattan-Guinness characterizes this contribution in his mathematical bibliography of the couple Young as “one of the comparatively few papers on Cantor’s theory of coherences and adherences.”2 W. H. Young and his wife Grace Chisholm Young (1868–1944) published 214 mathematical papers and four books. According to the annotated bibliography by Grattan-Guinness, this is number 23. Immediately following the introduction to this paper, the author referred to Cantor’s theory of point-sets in connection with open and closed sets. Furthermore, he quoted Arthur Schoenflies (1853–1928), who pointed out that up to this point, there had been very few results and no applications to open sets. This then became the starting point for the derivation of a number of theorems on the “mutual 1 Young,

W. H. (1903). (1975), esp. p. 48.

2 Grattan-Guinness

E. Mühlhausen (B) Dammstr. 2, 37434 Krebeck, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_4

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properties of the ‘adherences’ and ‘coherences’ introduced at the various stages of Cantor’s process of dealing with an open set, to one another, and to the corresponding elements of the set obtaining by closing the original set.”3 Concerning his methods, the author explicitly stated: “In no part of the present paper is any use made of Cantor’s numbers.”4 This publication was reviewed in the same year in the “Jahrbuch der Fortschritte der Mathematik” by Max Dehn (1878–1952). His review reads as follows: The author intends to give proofs for Cantor’s theorems about sets of points (Acta Math. 7) without using Cantor’s order types of countable well-ordered sets. But it seems to the reviewer that the proof of Theorem 7: “The set of derived and deduced sets which are distinct from one another is as most countable infinite”, indeed even the possibility, to give meaning to this statement necessarily implies the introduction of Cantor’s concepts. The author further explains Cantor’s method to construct adherences and coherences for open sets and gives several examples. The above also applies to Theorem 10 analogously to Theorem 7. At the end the following theorem is proved: Each adherence consists entirely of points which are limiting points of every adherence preceding it in the natural order.5

This Jahrbuch appeared two years later, as was customary at the time, in 1905. When William Young learned of it, he was very upset. He himself did not contact the reviewer but left this task to his wife. In January 1906 she wrote a letter to Max Dehn demanding that he explain the reasons for his critical remarks in the Fortschritte.6 That was the beginning of an interesting correspondence not only concerning mathematics. Nonetheless, Grace Chisholm Young was disappointed by Max Dehn’s replies to her questions, and finally, she brought it to an end in the following way: I have not a good impression from your letters and do not believe that a further correspondence will serve any purpose. For this reason I see no necessity to excuse the impoliteness, mentioned above, and will decline from any further correspondence. If you do not inform me otherwise I assume that you have nothing against my showing the letters to other people. Yours faithfully Grace Chisholm Young7 3 See

Young, W. H. (1903), p. 103. Young, W. H. (1903), p. 103. 5 „Verf[asser] beabsichtigt, Beweise für die Cantorschen Sätze über Punktmengen (Acta Math.7) ohne die von Cantor eingeführten Ordnungstypen von abzählbaren, wohlgeordneten Mengen zu benutzen. Es scheint aber Ref[erent], als ob der Beweis des Theorems 7: „Die Menge der derivierten und deduzierten Mengen ist höchstens abzählbar unendlich“, ja schon die Möglichkeit, dieser Aussage einen Sinn zu verleihen, diese Einführung mit Notwendigkeit involviert. Verf[asser] setzt ferner die von Cantor geschaffene Methode der Kohärenz- und Adhärenzbildung für nicht abgeschlossene Mengen auseinander und gibt hierfür eine Reihe von Beispielen. Das oben Bemerkte trifft ebenfalls das dem Theorem 7 analoge Theorem 10. Zum Schluß wird noch der Satz bewiesen: Jede Adhärenz besteht ausschließlich aus solchen Punkten, die Grenzpunkte jeder im Konstruktionsmodus vorangehenden Adhärenz sind.“ In: Jahrbuch über die Fortschritte der Mathematik, Band 34, Jahrgang 1903 [published 1905], p. 530. 6 Liverpool University Archives, D. 140/30/2. Correspondence between Grace Chisholm Young and Max Dehn. „Brief 1. von Grace Chisholm Young enthält eine Forderung an Herrn Dehn die Gründe seiner Kritik in den Fortschritten der Mathematik anzugeben“. 7 „Aus ihren Briefen habe ich keinen guten Eindruck bekommen und ich glaube nicht, dass aus weiterer Korrespondenz irgendwelcher Nutzen zu ziehen wäre. Aus diesen Gründen sehe ich mich 4 See

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The story, however, was not over yet. Another person was definitely involved. It was Giulio Vivanti (1859–1949) in Messina to whom Grace Chisholm Young sent a copy of the entire correspondence. As he was an expert and a friend as well, his opinion was important to her. Thus we have now introduced the main actors in this drama. All of them had already published on set-theoretic problems: Max Dehn on applications of set-theory to elementary geometry and Giulio Vivanti on the theory of sets per se and on the historical development of set-theory. Otherwise he was mainly concerned with analysis. The Youngs had written by far the most set-theoretical papers, all of which had been worked out together but published only under W. H. Young’s name. Before they got married in 1896, his wife was Grace Emily Chisholm, in fact, Dr. Grace Emily Chisholm. She had obtained her Ph.D. degree “magna cum laude” in 1895 for a dissertation on the algebraic groups of spherical trigonometry.8 This subject had been evidently suggested by Klein, who was so interested in the problem that he discussed it at length. He referred to her treatment of it twelve years later in Elementary mathematics from an advanced standpoint, in which he also designated her as the first woman in Prussia to pass the regular examination for the doctor’s degree. Things had already moved a long way since 1874, when the Russian mathematician Sofja Kowalewskaja (1850–1891) took the degree in Göttingen in absentia, having been refused permission to attend Weierstrass’ lectures.9 Miss Chisholm entered Girton College in Cambridge in 1889 and studied successfully there. Since there was no possibility of a fellowship for women, she was advised by her coaches and lecturers to go to Göttingen. In her obituary for Felix Klein (1849–1925) for “The Times,” she described the situation at that time: Klein’s connexions with England were intimate. He was an honorary member of our most distinguished learned societies, and received the De Morgan Medal of the London Mathematical Society in 1893, an honour he much appreciated. When in that year he and others succeeded in opening the doors of the University of Göttingen to women, it was, I think, a real pleasure to him that the first woman to take the degree of D. Phil. should do so under his auspices, and should be a Girton girl who had sat at the feet of his revered friend Cayley.10

In winter 1893, she arrived in Göttingen, at the same time as the Americans Mary Frances Winston (1869–1959) and Margaret Eliza Maltby (1860–1944). With the

nicht genötigt, die obengenannte Unhöflichkeit zu übersehen, und werde auf weitere Korrespondenz verzichten. Wenn Sie mir nicht anders schreiben, so nehme ich an, dass Sie Nichts dagegen haben, wenn ich die Briefe irgend Jemandem vorlege. Ihre ergebene Grace Chisholm Young“, Liverpool University Archives, D.140/30/2, Grace Chisholm Young to Max Dehn, letter from March 16, 1906. 8 Chisholm (1895). 9 Cartwright (1944), esp. pp. 187–188. 10 Liverpool University Archives, D. 140/3/3.2 Newspaper cutting, The Times, Thursday, July 9, 1925. The professor of mathematics Arthur Cayley (1821–1895) gave lectures at the University of Cambridge, which the students of Girton College were allowed to hear with special permission. Felix Klein and Arthur Cayley were friends and correspondents. The mathematical terms ‘Cayley-Klein metric’ and ‘Cayley-Klein model of hyperbolic geometry’ are named after them.

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support of Felix Klein, the three women finally obtained special permission to go to lectures and seminars.11 In a letter to the Mathematical Club at Girton College, Grace Chisholm describes Felix Klein’s seminary quite vividly: […]; it takes place every Wednesday at 11 o’clock, and lasts about two hours, and the members make ‘Vortrag’s on their special subjects on different Wednesdays. The students who have been here some time, and some of the new students who came from other Universities, have already got their special subjects; for the others, Prof. Klein has always suggestions as to special lines of work which they might take up, generally in connection with the lectures. Miss Winston made her Vortrag on the last Wednesday before the Christmas holidays. It would be nervous work in any case to make a Vortrag before an audience of about a dozen men, half of whom are Doctors, and one Prof. Klein; but the strain considerably increased by having to speak German. There are about a dozen of us in our lectures; we are a motley crew: five are Americans, one a Swiss-French, one a Hungarian, and one an Italian.12

The so-called “Pentagon” photography shows five of Felix Klein’s students around 198313 : Mary Winston, previously mentioned, attained her doctorate with Klein in 1897. She translated the 1900 lecture by David Hilbert (1862–1943), presenting his famous problems to be published in the Bulletin of the American Mathematical Society in 1902. Charles Jaccottet (1872–1938) wrote his dissertation under Klein in 1895 and later taught at the École industrielle cantonale in Lausanne in Switzerland. Poul Heegaard (1871–1948) from Denmark received a stipend to support a year’s study abroad from the University of Copenhagen and went to Paris and Göttingen where this contact to Felix Klein became very influential for his future work. Here he got the idea for his dissertation and, in 1898, was awarded a doctorate in Copenhagen. Heegaard included an important mathematical contribution in his survey article together with Max Dehn in 1907, in which they set out the foundations of combinatorial topology. Gino Fano (1871–1952) had graduated from the University of Turin and went to Göttingen for further studies under Felix Klein. As an undergraduate student of Corrado Segre (1863–1924) and a correspondent of Felix Klein, Fano had translated Klein’s Erlanger Program into Italian. Later he held a professorship in Messina and then in Turin where he worked mainly on projective and algebraic geometry. At that time W. H. Young was a fellow of Peterhouse College in Cambridge where he was lecturing and examining in mathematics. He was busy preparing and pushing students for the Tripos-examinations and was not motivated or possessed of ideas at all for research work.14 In 1888 he was an appointed lecturer in mathematics at Girton College. In addition to his fellowship at Peterhouse and in Grace Chisholm’s last year there, he was her coach. 11 See

Mühlhausen (1993). (1984), p. 4. 13 This photography is privately owned by the American mathematician Sylvia M. Wiegand, a granddaughter of the Youngs. It was first published in Green & LaDuke (1987), esp. p. 14. 14 Grattan-Guinness (1972), esp. p. 113. 12 Chisholm

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After her success in Göttingen, she returned to Cambridge and gave him a copy of her dissertation, and obviously, they became closer and began a relationship— not only concerning mathematics. A year later, they got married. Together with his wife, William Young would have liked to write a textbook of astronomy because his pet subject required for the Tripos was teaching mathematical astronomy.15 This project had not yet started when she, on the other hand, inspired him to focus on mathematical research, as she wrote later: “At the end of our first year together, he proposed, and I eagerly agreed, to throw up lucre, go abroad, and devote ourselves to research.”16 Grace Young claims that her husband suggested the move, but it’s obvious that this idea came from her, based on her own already established record of research done abroad.17 However, in autumn 1897, the Youngs and their three-month-old baby left for Göttingen. Encouraged by Felix Klein and supported by his wife, William Young wrote his first publication on a geometrical problem at the age of 35.18 It was published in the Proceedings of the London Mathematical Society, of which he had been a member since 1894 under the title “On systems of one-vectors in space of n dimensions.”19 Joint research had begun, but somehow William Young wasn’t satisfied with the situation in Göttingen. His wife describes the problems with the landlady and the servant girl in their little home situated closely to the house of the Klein family in her memoirs.20 The situation could only be improved by moving to another place, but William Young had different plans: Göttingen had opened my husband’s eyes to a wider and truer intellectual life, but it had not satisfied his mathematical cravings. His proposal was that we should again move off, and this time to Italy, to obtain a grasp of geometry: real geometry, not glorified Euclid, geometry such as Italy understood and Italy alone.21

So they moved to Italy and traveled through Tuscany and Umbria; their little son Frankie—called affectionately Bimbo by the Italians and the parents was always with them. In winter 1898, they studied for some months at the University of Turin with Professor Corrado Segre, the correspondent of Felix Klein. In the field of algebraic geometry in higher-dimensional spaces they wanted to get onto the latest research level.22 The result of their studies were two contributions, written in Italian and 15 Hardy

(1942), esp. pp. 219–220. (1942), p. 221. 17 This assumption is expressed by the British mathematician Mary Lucy Cartwright (1900–1998) in her obituary for Grace Chisholm Young, see Cartwright (1944), p. 188. 18 Grattan-Guinness (1972), p. 133. 19 Young, W. H. (1898). 20 Grattan-Guinness (1972), p. 132. 21 Grattan-Guinness (1972), p. 133. 22 Compare Jones (2009), p. 102. Corrado Segre (1863–1924) in 1883 published a dissertation on quadrics in projective space. Since then he had been exchanging letters with Felix Klein. From 1888 on until his death he held a professorship for higher geometry in Turin. 16 Hardy

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delivered in April 1899 at the Accademia Reale delle Science di Torino.23 Grace Young constructed upon the latest results of Eduard Study (1862–1930) and Segre in spherical geometry,24 while William Young dealt with the results of Enrico D’Ovidio (1842–1933) and Hilbert concerning quadratic relations.25 In autumn 1899, they returned to Göttingen and made their permanent home there. In the following years, they had five more children: Cecily *1900, Janet *1901, Helen *1903, Laurence *1905, and Patrick *1908. As William Young couldn’t get a job in Göttingen, the basic pattern for their relationship was quite unusual for the next decades. William Young traveled a lot to earn money as a coach and lecturer. He started in Cambridge, went later to Liverpool, and was only in the family home during holidays. Grace Young brought up the children and taught them languages and to play an instrument.26 Besides that, she studied medicine, wrote children’s books,27 and always kept contact with the mathematical community. Both of them worked intensely on mathematics. The first time in Göttingen went by without a plan until Felix Klein suggested “You want a new research topic in mathematics? What about set-theory?”28 This is quoted from the obituary Grattan-Guinness wrote for Cecily Tanner (1900–1992), second child and eldest daughter of the Youngs. She was born shortly before Klein’s advice was offered and had Klein’s daughter Luise as her “deputy godmother.” Later on, she became a mathematician and was also interested in the history of mathematics. She put much effort into sorting and cataloging the mounds of her parents’ letters and files and in transcribing their correspondence. The collective legacy was transferred to the Archives of Liverpool University.29 One year before, Felix Klein had advised Arthur Schoenflies, an active worker in the field of set-theory, to write an article on the state of set-theory for the Encyclopaedia, and soon thereafter, Schoenflies published a much longer report “Die Entwicklung der Lehre von den Punktmannigfaltigkeiten” in the journal of the Deutsche Mathematiker-Vereinigung,30 the association of German mathematicians founded on Cantor’s instigation in 1980.31 For all his organizational genius, Klein could not have foreseen the consequence of his advice: the development of set theory, and especially its applications to problems in mathematical analysis, was the great contribution to mathematics that Grace and Will [Young] were to 23 Grattan-Guinness

(1975), p. 47. G. C. (1899). 25 Young, W. H. (1899). 26 Compare Wiegand (1987). 27 See Mühlhausen (2004). 28 Grattan-Guinness (1993). 29 See Grattan-Guinness (1993), pp. 10 and 13. 30 Schoenflies (1899). Arthur Moritz Schoenflies (1853–1928) hold a professorship for applied mathematics 1891–1899 in Göttingen, then took a chair at the University of Königsberg. Besides set-theory he is known for his applications of group theory to crystallography and also wrote about projective geometry in Klein’s encyclopaedia. 31 See Grattan-Guinness (1972), p. 140. 24 Young,

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make during the next twenty-five years and so put themselves among the leaders in an actively developing field. They worked on other areas of mathematics also—in particular, they continued to develop their ideas on geometry—but in set theory and its applications they found their mathematical home.32

They hadn’t got this far by 1903 when the controversial memoir appeared. They were in the middle of an exciting phase, at least in the foundational and set-theoretic aspects, in which distressing things happened such as the appearance of antinomies which threatened to bring down the whole beautiful theory. There was much to understand, to discuss, and to continuously improve. The concepts had to be formulated more precisely after yet another carefully constructed example showed gaps or confused thought. This is very visible to the reader when one struggles to read the publications of the Youngs. Also the contribution of Arthur Rosenthal (1887– 1959) in the 1920s to the “Enzyklopädie der mathematischen Wissenschaften” on the subject of point-sets gives a lively picture of this intense discussion of all those involved in the relatively new subject.33 The publications of the Youngs appeared in rapid succession, many of them related directly to their predecessors and interdependent. For the assessment of the quality of these papers, I would like to quote the Cambridge mathematician Godfrey Harold Hardy (1877–1947) in his obituary for William Young: He makes astonishingly few mistakes, and the critical passage will almost always be found to be accurate and clear; but his repetitions are sometimes rather trying to the reader anxious to dig out the kernel of what he has to say. A theorem will be proved, in varying degrees of generality, in half a dozen different papers, with continual cross-references, and promises of further developments not always fulfilled. It is not surprising that a good many of Young’s theorems should have been missed and rediscovered. At his best, however, he can be as sharp and concise as any reader could desire;…34

Before going on to the context, I would like to describe what the controversy between Dehn and Young looked like mathematically. At the beginning of 1906, Grace Chisholm Young had sent a letter to Max Dehn asking him to explain his criticism more closely. Dehn answered that he could not give quite precise information as the journal with the publication was not accessible to him. He repeated his objection and added that this did not refer to the axiomatic side that doubted the rigor of the proof.35 He wrote 32 See

Grattan-Guinness (1972), p. 140.

33 Rosenthal (1924). Arthur Rosenthal (1887–1959) did research concerning geometry, in particular

the classification of polyhedrons and Hilbert’s axioms. He also made contributions in analysis and measure theory. 34 See Hardy (1942), p. 223. 35 „… Leider habe ich hier das Quart. Jour. nicht zur Verfügung und kann Ihnen deshalb keine ganz genaue Auskunft geben. Soweit, ich mich erinnere, schien es mir damals, als ob die Aufstellung des Begriff der successiven Ableitungen einer Punktmenge und die Ableitung der für ihn geltenden Hauptsätze, die Theorie der abzählbaren wohlgeordneten Mengen involviere, zu gewissermassen eine geometrische Veranschaulichung dieser Theorie liefere. Das ist der Einwand, der die axiomatische Seite, soviel mir scheint gar nicht trifft. Es soll durchaus keine Verdächtigung der Beweisstrenge sein…“, Liverpool University Archives, D. 140/30/2. Max Dehn to Grace Chisholm Young, letter from February 3, 1906.

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from Münster because after his studies at Freiburg and Göttingen, he had become an assistant professor (Privatdozent) at the University of Münster in 1901. Next, Grace Chisholm Young sent him an offprint asking him to read the publication more carefully. She went on: Now, Herr Doktor, you must excuse me if I say that either your command of the English language is not sufficient to have taken this review, or you have not studied it carefully enough. I know you to be a competent mathematician who is entirely capable to understand the line of the thought.36

After she had furthermore informed him that she was happy to consider his criticism, she wrote: The ‘Fortschritte’ is not there to print the personal opinion of a critic but should represent the content of the work in hand and everywhere in the world it is understood in this sense.37

Her husband had asked her to explain the attacked Theorems 7 and 10 and their proofs in detail in German for Max Dehn. Consequently, she added a supplemental, mathematical manuscript. To verify the proof of Theorem 7, she defined the terms derivation and deduction again and showed that this definition and also the proofs do not require previous knowledge of Cantor’s ordinal numbers. In his reply, Max Dehn didn’t mention this manuscript at all. He wrote that he had reread the original publication again, but only until Theorem 8, which is about a third of the whole paper. In that part, he pointed out that the use of the terms “subsequently” and “continue” already demonstrate to him the clandestine use of Cantor’s transfinite numbers. He finished his letter with the sentence: Maybe we can meet each other, and talk it over, because next week I will be in Göttingen for a visit.38

Grace Young was not sure if she should meet him and wrote to her husband in Liverpool that she would first like to hear Hilbert’s and Zermelo’s statement on the matter. She had already planned a synopsis concerning the controversy to their set-theory textbook, the book they had been working on for about three years.

36 „… Nun, Herr Doktor, Sie müssen es mir nicht übel nehmen, wenn ich sage, entweder sind Sie in der englischen Sprache nicht genug bewandert um diese Recension angenommen zu haben, oder Sie haben es nicht gewissenhaft studiert. Denn ich kenne Sie als tüchtiger junger Mathematiker, der vollkommen fähig ist, den Gedankengang zu begreifen…“, Liverpool University Archives, D. 140/30/2. Grace Chisholm Young to Max Dehn, letter from February 17, 1906. 37 „… Eigentlich soll man sich nicht über falsche Kritik aufregen, aber die ‚Fortschritte’ hat nicht den Zweck, die persönliche Meinung des Kritikers abzudrucken, sondern soll den Inhalt der Werke charakterisieren, und wird in diesem Sinne überall in der Welt gebraucht.“, Liverpool University Archives, D. 140/30/2, Grace Chisholm Young to Max Dehn, letter from February 17, 1906. 38 „… Übrigens können wir vielleicht, wenn es Ihnen recht ist, uns auch mündlich über den Gegenstand verständigen, da ich Anfang nächster Woche nach Göttingen komme.”, Liverpool University Archives, D. 140/30/02 Max Dehn to Grace Chisholm Young, letter from February 28, 1906.

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You see I write to you as if Dehn was wrong, & I do think he is wrong. After all we want is not to prove the argument in the Analysis [of Linear Sets of Points] right, but to find out the truth.39

The available sources unfortunately do not verify whether she talked to Hilbert, Zermelo, or Dehn. Disappointed that Dehn hadn’t read the complete publication and furthermore ignored her German manuscript, she wrote back in English: As I am anxious to finish this business as soon as possible, I have written out categorically the theorems which are used, & avoided the expressions to which you take exception… I think, as I took some trouble before in sending you a manuscript in German with a sketch of the line of argument, you ought to have referred to this in your letter & pointed out exactly where you thought you had detected a flaw.40

She repeated her request to read the manuscript and added another one she calls synapsis, to explain step by step the prerequisites for Theorem 7, including Cantor’s theorem of deduction and the construction of a certain set K (E). Some days later, Dehn wrote back that from his point of view, the construction was wrong and so he wouldn’t read any further. He then sent the manuscripts back.41 Grace Young was quite upset about this letter, remarking that he had not provided proof for his objection and now she would have to send the whole correspondence to somebody else42 —and then she sent copies of the seven letters to Giulio Vivanti in Messina. According to Vivanti’s view, Dehn’s cautiously formulated remark in his review of the paper doesn’t take away value and also does not make the work less attractive to readers. He could not find any weak point in the proof of Theorem 7. He considers it useful to include a comment in the set-theory book, already in print, just in order to eliminate any doubts. He advised her not to mention Dehn and their correspondence. And he expressed very clearly that there had been a lack of objectivity from the onset, thus depriving the dispute of a basis of any scientific discussion.43 Grace Young accepted this critique and her three-page-long footnote in the book contains the synapsis without mention of Dehn and his destructive remarks.44 But then she couldn’t resist adding: Lebegues, like the present authors, regards the theory of derivation and deduction as the natural basis for the study of the transfinite numbers, not vice versa.45 39 Liverpool

University Archives, D. 140/6/932, Grace Young to Will Young, letter from March 2, 1906. 40 Liverpool University Archives, D. 140/30/2, Grace Chisholm Young to Max Dehn, letter from March 9, 1906. 41 Liverpool University Archives, D. 140/30/2, Max Dehn to Grace Chisholm Young, letter from March 14, 1906. 42 Liverpool University Archives, D. 140/30/2, Grace Chisholm Young to Max Dehn, letter from March 16, 1906. 43 Liverpool University Archives, D. 140/6/939, Guilio Vivanti, Messina, to Grace Young, Göttingen, letter from May 27, 1906. 44 See Young and Young (1906), pp. 284–286. 45 Young and Young (1906), p. 286.

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In this controversy, the mathematical truth lay in the middle (if I may express myself imprecisely). The transfinite numbers are not actually used explicitly but are implicit in the well-ordering of the set of derived sets of a point-set. The search for explanations for the vehemence of this argument remains more important for me than the decision as to who was right here. Could the Youngs not just have ignored the short remark of Max Dehn in his review? This they clearly could not have done for the following reasons: First of all, the controversy was over keeping to various rules in the mathematical community. For the Youngs this included, e.g., maintaining respect in the eyes of a review journal in which, in their opinion, only a short description of the publication without any judgment should appear. Grace Young accused Max Dehn of not sticking to this rule. Max Dehn did not answer this accusation but, interestingly enough, did not review any further publication of the Youngs. This was done thereafter by Giulio Vivanti, among others. Secondly it was also a question of the mathematical content. William Young must have felt that the entire content of the publication had been put in doubt because the main goal stressed in the introduction was to prove the properties of point-sets without the use of Cantor’s numbers. If Max Dehn had been correct with his statement, the publication would have been superfluous and worthless. Young could not accept this. This leads us to the third reason for doing something: one’s reputation in the mathematical world. He could not allow Dehn to persevere because he was at the beginning of his career and relatively old. When in 1903 Cambridge University awarded him the D.Sc. on the basis of the quality of the twenty papers which had appeared up to that time, he was already forty years old. If he had wanted to earn a professorship, he would have had to make a greater impact. For this purpose, a textbook on set-theory, for which he had held a contract since 1903 with the Cambridge University Press, would have been useful.46 At the time of the controversy, this book with the title “The Theory of Sets of Points” was almost finished. Since this book also contained his own results on set-theory, he had concluded the paper in question on pages 53–63 without any change. And, of course, one does not like to be criticized by a mathematician fifteen years younger than oneself, especially if one has already proved one’s competence by a highly regarded habilitation thesis. The matter had to be clarified, but why he didn’t do this himself? Why did he leave this to his wife instead? There is a very clear explanation for this. This couple, William and Grace Young, had a kind of contract with each other. They did research together, but the results were published under his name so he could get a professorship as soon as possible. Grace couldn’t work as a mathematician anyway, and there was, of course, the child-project as well!47 She adhered to this agreement and supported her husband very effectively. In 1906 she was in the final phase of their book-project and rewrote parts of it, filled in proofs, corrected mistakes, sent finished parts and proof sheets to other mathematicians, and so on. Therefore, she was in contact with

46 See 47 See

Grattan-Guinness (1972), p. 141. Grattan-Guinness (1972), p. 142.

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Felix Bernstein (1878–1956), Philip Jourdain (1879–1919), Oswald Veblen (1880– 1960), and of course Giulio Vivanti during this time. She was well acquainted with the subject and she could write letters without consulting her husband. Dealing with criticism was her responsibility. As the managing part of the couple, she opposed Max Dehn. Her husband, working far away at Liverpool University, had no time, and his German was not as good as hers. So the controversy with Dehn became her own controversy. Finally, she was able to find a way out by breaking off the correspondence altogether and looking for allies. Luckily there was Vivanti, who brought her back to her senses with his calm, well thought out evaluation of the situation. Grace Young’s loss on a social level was low: Dehn was neither a friend nor a correspondent. I could imagine that Dehn was relieved when Grace Young finally left him alone with her long, demanding letters. He had not given much trouble to his reasoning anyway and did not take the matter very seriously. In addition to this, since his review in the Fortschritte, he had changed to a different working field. He had then already met up with the Danish mathematician Poul Heegaard in 1904 and in collaboration with him was writing the first comprehensive article on topology. At that time, it was called “Analysis Situs,” and it appeared in the first edition of the Encyclopaedia in 1907. The dispute with Dehn was a gain at the mathematical level for Grace Young because she thus had to prove the paper again very carefully. This led to the footnote in their joint book “The Theory of sets of Points,” which was ready to be published in May 1906. In the second edition in 1972 by Rosalind Tanner and Ivor Grattan-Guinness, a letter from Georg Cantor to Grace Young written in January 1907 is quoted: I hope that the recognition that you will earn from this meritorious and laborious enterprise, containing so many beautiful contributions of your own to the theory, will console you for the minor oversights you write of.48

It was the first textbook in English on this subject and introduced set-theory outside the continent where it had started with Cantor’s ideas. The book has a lot of references and footnotes. I wonder what could be divulged from them concerning other conflicts or simply looking into the Göttingen “network” and its position in the world at that time. This was definitely a network that Grace Chisholm Young used extensively.

Bibliography Cartwright, M. L. (1944). Grace Chisholm Young. Journal of the London Mathematical Society, (1) 19, 185–192. Chisholm, G. (1984). Extracts from a letter to the Mathematical Club. The Girton Review, 37, 1–4. 48 Tanner,

R. C. H. Preface to the second edition of Young and Young (1906). Reprint New York 1972, p. V.

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Chisholm, G. (1895). Algebraisch-gruppentheoretische Untersuchungen zur sphärischen Trigonometrie. Göttingen. Grattan-Guinness, I. (1972). A mathematical union: William Henry and Grace Chisholm Young. Annals of Science, 29, 2, 105–186. Grattan-Guinness, I. (1975). Mathematical bibliography for W. H. and G. C. Young. Historia Mathematica, 2, 43–58. Grattan-Guinness, I. (1993). Obituary: Cecily Tanner 1900–1992. Newsletter of the British Society for the History of Mathematics, (23), 10–15. Green, J., & LaDuke, J. (1987). Woman in the American Mathematical Community: The pre-1940 Ph.D.’s. The Mathematical Intelligencer, 9(1), 11–23. Hardy, G. H. (1942). William Henry Young. Journal of the London Mathematical Society, (1) 17, 218–231. Jones, C. G. (2009). Femininity, mathematics and science, 1880–1914. Basingstoke. Mühlhausen, E. (1993). Grace Emily Chisholm Young ‘I liked being incognito to the outside world’. In: Weber-Reich, T. (Ed.), “Des Kennenlernens werth” – Bedeutende Frauen Göttingens (pp. 195–211). Göttingen. Mühlhausen, E. (2004). Können Mathematiker Kinderbücher schreiben? – Die Mathematikerin Grace Chisholm Young (1868–1944) als Kinderbuchautorin. In: Roloff, H. & Weidauer, M. (Eds.), Wege zu Adam Ries, Tagung zur Geschichte der Mathematik, Erfurt 2002 (pp. 285–294). Augsburg. Rosenthal, A. (1924). Neuere Untersuchungen über Funktionen reeller Veränderlicher. Enzyklopädie der mathematischen Wissenschaften. Schoenflies, A. (1899). Die Entwicklung der Lehre von den Punktmannigfaltigkeiten. Jahresberichte der deutschen Mathematiker-Vereinigung, 8 (printed 1900) 2, 1–251. Wiegand, S. M. (1987). Grace Chisholm Young (1868–1944). In: Grinstein, L. S. & Campbell, P. J. (Eds.), Women of mathematics: A bibliographic sourcebook (pp. 247–254). New York. Young, G. C. (1899). Sulla varietà razionale normale M34 di S6 rappresentante della trigononometria sferica. Atti della Accademia Reale delle Scienze di Torino, 587–596. Young, G. C., & Young, W. H. (1906). The theory of sets of points. Cambridge. Reprint New York 1972. Young, W. H. (1898). On systems of one-vectors in space of n dimensions. Proceedings of the London Mathematical Society, (1) 29, 478–487. Young, W. H. (1899). Sulle Sizigie che lagano le Relazioni Quadratiche fra le Coordinate die Retta in S4 . Atti della Accademia Realew delle Scienze di Torino, 596–599. Young, W. H. (1903). On the analysis of linear sets of points. Quarterly Journal of Pure and Applied Mathematics, 35, 102–116.

Elisabeth Mühlhausen born in 1950 in Hanover, studied mathematics and biology in Hanover, Brunswick and Berlin. She taught as high school teacher (in a German “Gymnasium”) in Berlin and changed to the Felix-Klein-Gymnasium in Göttingen in 2000. From 1990 to 1993 she was given a time off from teaching and studied History of Exact Sciencies and Technology with Prof. Knobloch at the TU Berlin. During this period she did research on the topic ‘Women in Mathematics’ at the University Archives of Liverpool, Cambridge and Göttingen. Nowadays she lives in retirement in an idyllic village near Göttingen.

Chapter 5

Emma S. and Wladimir S. Woytinsky: An Unusual Couple in Statistics Annette B. Vogt

Abstract First, I discussed the concept of Couples in Science or Couples in Statistics as well as the concept of Couples of Scientists. And statistics means the field between statistics as the science of the state and economic statistics as well as mathematical statistics. In the history of science the concept of Couples in Science was developed first by Helena M. Pycior, Nancy G. Slack, and Pnina G. Abir-Am (see Pycior et al. (1996)), 16 years later another volume was published (see Lykknes et al. (2012)). Inspired by the two volumes, I’m giving my “definition” of Couples in Science. The significant element is the fact that both are working and publishing together, that they are collaborating and publishing together. Second, I described important aspects of the life and work of Emma S. (1893–1968) and Wladimir S. Woytinsky (1885–1960) (also Vojtinskij). Both were socialists, political activists, and Russian Jews. They had to live in exile from 1920 onwards, first in Germany, then in France and Switzerland, from 1935 on in the USA, mostly in Washington, D. C. They were unusual statisticians because of the lack of continuous academic training in this field before 1924. In Berlin they were working closely together but the result of this collaboration, the seven volumes “Die Welt in Zahlen” (The World in Figures, Woytinsky (1925–1928)) was published under his name only. Between 1947 and 1959 another period of close collaboration on statistics followed. Now they were working together and publishing together. They did famous and highly acknowledged work compiling large data collections on statistics on world population and production, and on world commerce and trade (Woytinsky and Woytinsky 1953, 1955). Third, I investigated their practices as statisticians, their collaboration, and the division of labor in their work in the 1920s as well as in the 1950 s. In contrast to the years in Berlin now they were working together and publishing together. Finally, I analyzed and compared the different circumstances and their working conditions in Berlin and in Washington D. C., and I discussed why Emma S. and Wladimir S. Woytinsky were so unusual—as a couple and as a couple in statistics.

A. B. Vogt (B) Max Planck Institute for the History of Science, Boltzmannstr. 22, 14195 Berlin, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_5

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“How can I describe and evaluate those five years of work on The World in Figures? We were at the library by nine in the morning and worked without interruption until it closed at half-past five, at lunchtime munching dry homemade sandwiches … We had neither a typewriter nor a calculating machine. Wolik drew all the charts, and I colored them with seven different pencils. … Yet we loved it for the new horizons it opened to us, for the fascination of our expanding knowledge of the world, but most of all for keeping us close to each other in every thought, every interest, for making us one not only emotionally but also intellectually.”1 With these emotional words Emma S. Woytinsky (1893–1968) described her collaboration with her husband Wladimir S. Woytinsky (1885–1960), called Wolik, on a special statistical oeuvre that was published about forty years before her autobiography. Between 1925 and 1928 all in all seven volumes were published in Berlin. These books titled “The World in Figures” (Die Welt in Zahlen) were a data compilation of nearly all available statistical data on populations, states, economy and finance, agriculture, trade and traffic, cultural and social life from all over the world. The author was Wladimir S. Woytinsky, and each volume was dedicated “To my wife, my faithful collaborator and comrade” (“Meiner Frau, der treuen Mitarbeiterin und Weggefährtin”). About thirty years later two similar volumes of huge data compilations were published, in 1953 and in 1955, now in the USA, and now both authors were mentioned on the cover: W. S. and E. S. Woytinsky.2 One reviewer of the book “World Population and Production: Trends and Outlooks” (1953) called them: “The Woytinskys, a husband-wife-team,”3 friends called them research partners, and quite often they were referred to as the Woytinsky couple. Emma S. and Wladimir S. Woytinsky were an unusual couple in many respects, concerning their life and fate, their engagement in politics and left-wing activities, their emigration paths from Russia through Europe to the USA, their activities during WW II in the USA, and last but not least their later career as an acknowledged couple in statistics and on lecture tours throughout Asia, India, and Latin and South America.

5.1 Couples in Science In the history of science, the concept of Couples in Science was first developed in the 1990s by Helena M. Pycior, Nancy G. Slack, and Pnina G. Abir-Am.4 They differentiated between married scientists who were working together and (at least partly) publishing together—the Couples in Science by definition—and mixed couples in laboratories who were working together but were not married. In their case studies, which were published in 1996, they described especially couples in sciences who 1 Woytinsky,

E. S. (1965), p. 110. and Woytinsky (1953, 1955). 3 Kiser (1955), p. 203. 4 See Pycior et al. (1996). 2 Woytinsky

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became famous and were awarded the prestigious Nobel Prize, like Marie (1867– 1934) and Pierre (1859–1906) Curie in 1903 and Irène (1897–1956) and Frédéric (1900–1958) Joliot-Curie in 1935. Only sixteen years later, another volume was published in which the authors on the one hand tried to develop further the concept of Couples in Science, and on the other hand they presented new case studies.5 This is still an ongoing process, research and investigations remain to be done on couples in science in all fields of mathematics and the sciences, the humanities and the social sciences, technology, and medicine. Inspired by the two volumes, my “definition” of Couples in Science is the following: both partners had studied science (or technology, medicine, the humanities), and both finished their studies with an academic degree, at least partly because of the long history of the exclusion of female students from university studies and examinations. The crucial point is that both are working, collaborating, and publishing together. In contrast to the Couples in Science we have to distinguish those Couples of Scientists in history of science and mathematics who studied and finished their studies with an academic degree, but worked in different scientific disciplines and they did not publish together. Such Couples of Scientists became more important in the history of science and mathematics in the twentieth century. Here I would like to use the concept of Couples in Science or Couples in Statistics to investigate a special couple of statisticians, who worked together in the first half of the twentieth century in several countries, first in Germany, then in France and Switzerland, respectively, and finally in the USA. Statistics means the field between statistics as the science of the state and economic statistics as well as mathematical statistics.6 Until recent time, in the historiography of statistics, only a few female statisticians were investigated and mentioned in the literature, none of them was married or worked together with a male partner.7 Compared with other scientific disciplines like biology or mathematics, the underrepresented number of female statisticians seems surprising. One of the reasons for this was the general development of science and education until the late nineteenth century and the exclusion of women from almost all academic professions in nearly all countries.8 A second reason could be the special field of statistics and its development as state statistics.9 In statistics, the situation changed only in the mid twentieth century. For example, in the volume entitled “Past, Present, and Future of Statistical Science,” all in all fifteen of the fifty authors were female statisticians, and some of them served as Presidents of the Statistical

5 See

Lykknes et al. (2012). the history of statistics see Gorroochurn (2016); Porter (1996); Salsburg (2002); and Stigler (1986; 1999). 7 See Johnson and Kotz (1997); and Heyde and Seneta (2001). 8 See Vogt (2007). 9 On the history of statistics in France, Great Britain, and Germany see Schwebber (2006); Desrosières (1998); and Tooze (2007). 6 On

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Societies in Canada and the USA.10 Compared with the situation at the beginning of the twentieth century this represents an enormous progress. Furthermore, we are convinced that, by looking back and studying in detail the publications of statisticians we will find more female statisticians in history of statistics than have been supposed.

5.2 Emma S. and Wladimir S. Woytinsky: From Vitebsk and St. Petersburg to Washington, D. C The life and work—politically and scientifically—of Emma S. and Wladimir S. Woytinsky (also transliterated as Vojtinskij) was fascinating, surprising, and finally very successful. Their life looks partly like a historical movie in which private and political, social, and economical elements were interwoven, closely linked with the most influential world affairs of the twentieth century. In the beginning, both were less interested in a scientific career than in expanding their political activities in Russia. Both were socialists, left-wing and political activists, furthermore, both were Russian Jews. Their political activities as well as their Jewishness forced them to go into exile several times. Because they were in conflict with the Bolsheviks they emigrated in 1922 to Germany. Living mostly in Berlin at that time, they became involved in serious scientific work, and it was in Berlin that they established the basis of their later scientific careers as acknowledged statisticians. Because of the Nazi regime, they had to flee again, first to Paris. After an intermezzo in Geneva, they emigrated in 1935 to the USA where they lived mostly in Washington, D. C. Here they collaborated together closely again and now they also published together. After all, the USA became their homeland, after a long journey through Europe, from Vitebsk and St. Petersburg to Berlin, via Irkutsk and Tbilisi, from Berlin to Paris and Geneva, from Marseille to New York, and finally to Washington. Whereas Wladimir S. Woytinsky belongs to the well-known active participants in three Russian revolutions—in 1905, in February of 1917 and in October (November, respectively) of 1917—the political activities of Emma S. Woytinsky are less known. She herself kept secret most of the details of her life before 1916. From 1917 on, after their marriage, both were living together most of the time, they were working together, traveling together, and they were very active in various socialist movements. Thanks to Emma Woytinsky’s engagement both of them escaped the Nazi persecution in 1933, and thanks to her they emigrated to the USA already in 1935. Quite remarkable is the fact that both published autobiographies, his autobiography came out posthumously in 1961, her autobiography was published in 1965. Thus, in addition to their scientific publications (books and articles) and the huge archival material we have another type of source, i.e., their own descriptions of their life and work. Studied with the necessary carefulness these sources provide us with useful information, especially some background information about their life, their fate, and their work, including their collaboration in statistics. We learned from their 10 See

Lin et al. (2014).

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own descriptions how this collaboration was organized, what kind of division of labor they preferred, and how they shared their findings and conclusions. Her autobiography begun with the following kitschy sentence: “My life began on March 14, 1916—the night when, for the second time, I met Wolik, for whom I had been waiting all my youth.”,11 thus to hide the facts of her earlier life. This first sentence, published five years after his death, successfully hid all biographical background information, including the date of her birth, any informations about her parents and her family, her personal development, or her political activities until 1916. One will ask immediately how and why a Jewish girl from a partly wealthy middleclass family from Vitebsk (also Witebsk, today in Belarus) came to Siberia? What did she do there? Why did she meet a young man who was in exile in Siberia after some years in prison because of his political activities against the Russian Empire? Reading her autobiography carefully, one gets some additional biographical information. When she described two of her girlfriends at college in Polotsk (also Polatsk, near Vitebsk), we learn about her school education and her academic training. Remarkable was her ambitious aim what she would do in her later life, she called it her dreams as a college girl: I had not found my niche but wanted it to be as big as the world.12

This aim was indeed a very conscious one. Emma S. Woytinsky (née Shadkhan) was born on April 19, 1893, in Vitebsk to a wealthy middle-class Jewish family with eight children.13 Little is known about her family. She only mentioned that in 1912 her family moved to Siberia because her father had to work in Irkutsk. Vitebsk was a town in the Russian Empire with a total population of 65,900 people (according to the Russian census of 1897) at that time. Among these, 34,400 were Jewish people (ca. 52%), i.e., the majority of inhabitants were Jewish. One of the most famous habitants of Vitebsk was the painter Marc Chagall (1887–1985). Because of the location near two rivers, the town was an old trade and commerce center with some industry and with railway connections to St. Petersburg and Warszaw and to Moscow and Riga. From 1906 until 1912/13 Emma Shadkhan attended a girl’s school (comparable to a lyceum or gymnasium) in Polotsk (not far from Vitebsk). She finished this school in 1912/13 with a Gold Medal. Because of the anti-Jewish discrimination in the Russian Empire only a Gold Medal allowed her as a Jewish college girl to attend further academic courses. In the Russian Empire Jewish students were excluded from higher education, among the exclusion rules was the one that only 1% of male Jewish students were allowed to study at the Imperial universities. As a woman Emma could not study regularly at universities, but in St. Petersburg— the capital of the Russian Empire—special women’s courses had been established 11 Woytinsky,

E. S. (1965), p. 1. E. S. (1965), p. 152. 13 See the date of birth on her identity card in Germany (1932) and France (1934). In: Archive IISH Amsterdam, Vojtinskij Papers, no. 30. 12 Woytinsky,

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already in 1878, the Higher Women’s Courses (comparable to the Women’s Colleges in Great Britain and the USA). The first director was the historian Nikolaj BestuzhevRjumin (1829–1897), therefore the institution was also called Bestuzhev Courses (as Emma Woytinsky called it in her autobiography). Many professors from the university of St. Petersburg and members of the Imperial Academy of Sciences supported this project, even when this first Russian institution of women’s higher education was closed by an order of the Czar between 1886 and 1889. It reopened again in 1890, in 1906 it got more autonomy, and from 1910 on the Higher Women’s Courses were recognized as equivalent to University courses, and the certificate of graduation was awarded as a diploma. The graduates were then allowed to become women teachers in high schools for girls. Many graduates were active in revolutionary and social movements, and many Jewish women studied at these Bestuzhev Courses or Higher Women’s Courses in St. Petersburg. The Bestuzhev Courses were the first institution of women’s higher education in Russia, similar courses later were established in Moscow and Odessa. The Bestuzhev Courses in St. Petersburg possessed an own building, a chemical laboratory and an observatory, and an excellent library. The teachers were mostly professors or Privatdozenten (lecturers) from the Imperial University of St. Petersburg, i.e., they were some of the best academic teachers in the whole Russian Empire. As in St. Petersburg, in Odessa too, many Jewish women attended the courses, and they became involved in revolutionary and social movements. Among these Jewish women was Sof’ja (Sofya) Aleksandrovna Janovskaja (Yanovskaya), née Nejmark (1896–1966) who attended the Higher Women’s Courses in Odessa and studied mathematics. Later she became a professor in Moscow and the head of the department of logic.14 The famous female mathematician and historian of mathematics, Pelageja (Pelageya) Jakovlevna (Yakovlevna) Kochina (1899–1999), was a graduate of the Bestuzhev Courses in St. Petersburg. She described the Higher Women’s Courses, as they were called when P. Ja. Kochina attended them, in her memories.15 Emma Shadkhan became a student of these famous Bestuzhev Courses in St. Petersburg. From about 1912/13 to 1916/17 she studied literature, modern languages (in particular German and French), but also mathematics and sciences like physics and geography. It was probably in St. Petersburg that Emma also became involved in political activities, i.e., she became interested in socialist ideas and participated in circles to study Marxist literature and to help political prisoners and exiled social democrats in Siberia. In her autobiography she described in detail that they collected money, bought food and medicine, and sent packages to political prisoners. It seems obvious that therefore she traveled regularly to Siberia each summer while in winter she studied at the Bestuzhev Courses in St. Petersburg. Thanks to her studies in St. Petersburg Emma Woytinsky got an excellent academic education, including some courses and training in mathematics. After receiving the certificate (the diploma) she was working as a teacher. After her marriage in

14 See 15 See

Minc and Nenarokov (1982), pp. 81–124, esp. p. 116. Kochina (1988), pp. 45–47.

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July of 1917 in Irkutsk, Siberia,16 she was nearly always together with Wladimir S. Woytinsky—partly working with him, serving as his assistant, secretary, translator, research partner, and “guardian angel,” and last but not least always his comrade. Compared to her excellent academic education, his education was more complicate. Born on November 12, 1885, in St. Petersburg to an intellectual family (his father was an engineer and a teacher of mathematics), he studied law and economics at the University of St. Petersburg. But he was not able to finish his study and to earn an academic degree because of his participation in the first Russian Revolution of 1905. He became a member of the Russian Social Democratic Party and worked illegally until he was arrested in 1908. He was in prison from 1908 to 1912, and in 1912 he was sent to Siberia where he lived in exile near Irkutsk. But before he had to go the long way of a Russian revolutionary through prisons and labor camps. In 1906 he published his first academic investigation on economics, the book “Rynok i ceny” (Market and Prices), thus becoming one of the pioneers of political economy in Russia. Sixty years later his book was translated into English and posthumously published in 1964.17 While living in exile near Irkutsk—no longer in prison but still facing many restrictions (he was not permitted to visit certain cities or have jobs, he had to go to the police office every week, etc.)—he began, like many other revolutionaries, to study as an autodidact classical literature on history, law, and economy, and all classical Marxist literature. Without any regular courses he became a highly educated scholar. His profound knowledge in each of these disciplines helped him to become an extraordinary scholar, being able to change the scientific fields, and who always remained interested in the history of society and in revolutionary and social movements. From their educational background, it is clear that they were unusual statisticians because of their lack of continuous academic training in this field before 1924. From a formal perspective, Emma Woytinsky had an even better academic education than her husband, because she received a diploma from an academic institution, whereas Wladimir Woytinsky was just a former student of law and economics without any degree at all. Neither she nor he had clear plans for their future when they arrived in Germany as political emigrées in 1922. But very soon they were deeply involved in a statistics project in Berlin, and from this time onward she supported him as best as she could. Why they became important and well-acknowledged statisticians and researchers in the world? Some special circumstances in Berlin were responsible for this development, among them Wladimir’s old interest in economics and statistics, which went back to his book in 1906. Furthermore, he had the idea to use the emigration in Berlin to do scientific work. And Berlin was then a good place to do this with its general interest in economic research. Berlin was also a special center of statistical institutions—three statistical offices were here, the municipal, the Prussian, and the Imperial Statistical Offices—with good libraries. Moreover, both of them received 16 See the entry about the marriage in the special passport (Verbanntenpass) of the exiled Wladimir Woytinsky, in: Archive IISH Amsterdam, Vojtinskij Papers., no. 30. 17 See Dmitriev (2016); and Woytinsky (1964).

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a special education and training as statisticians in Berlin by a system that one could describe as “learning by doing,” because they got support and private lessons in statistics from Ladislaus von Bortkiewicz (1861–1931) who was one of the leading statisticians at that time.18 In their autobiographies both of them described in detail how they came into contact with L. von Bortkiewicz (thanks to the publisher Hans Lachmann-Mosse (1885–1944)), how they were supported by him, how the division of labor was organized between them, what they learned from Bortkiewicz, and how they finally became friends with this strict German professor who was born in St. Petersburg too and commanded the Russian language. As Wladimir Woytinsky pointed out: Bortkiewicz was probably the best statistician in Europe, and I had much to learn from him. We discussed the outline of each chapter. When the chapter was completed, he read the Russian draft and commented on it, usually in writing. Then he read the German text, occasionally correcting the style and watching the terminology. … The World in Figures represented the kind of statitsics he liked. He did not have the slightest resentment at seeing such statistics produced by a younger man who lacked his erudition and experience. As time went on, our relations became less formal, Emma and I frequently met him and his sister socially, and we became good friends.19

About the beginnings of this collaboration Emma Woytinsky remembered: On returning to Berlin, we organized our work. The best international statistical library was in the Prussian Statistical Office. Its librarian put a large table at our disposal, introduced us to his staff, and we began the task that kept us nailed to that table for the next five years. From the beginning, the title of the project was Ves’ Mir v Tsifrakh (The World in Figures).20

And about “Professor Bortkiewicz” she wrote, that they learned that he was called the “Pope of statistics,” and “that it was extremely difficult to satisfy him, to reach his level.”21 The result of this close collaboration on statistics was the publication of the seven volumes “Die Welt in Zahlen” (The World in Figures) which were published by the Mosse publishing house in Berlin between 1925 and 1928. Only Wladimir S. Woytinsky was named as its author, the editor of the series was Ladislaus von Bortkiewicz (it would be his only editorial work), and each volume was dedicated to Wladimir S. Woytinskys wife. Emma Woytinsky pointed out: “For German scholars, the fact that Bortkiewicz for the first time had associated his name with a project was a guarantee of both its exceptional quality and the scholarly status of its author.”22 One could assume that this was the reason why Emma Woytinsky had agreed that her name was hidden. She was interested in Wladimir’s career as a scholar, and therefore she supported him as best as she could. Furthermore, she obviously knew quite well that her chances to make a career as a professional female statistician were 18 See

Gumbel (1931, 1968); and Härdle and Vogt (2015). W. S. (1961), p. 453. 20 Woytinsky, E. S. (1965), p. 108. 21 Woytinsky, E. S. (1965), p. 109). 22 Woytinsky, E. S. (1965), p. 109; and Woytinsky (1925–1928). 19 Woytinsky,

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small—as a foreigner, a woman, and a Jewish woman. Both worked more than five years in the library of the Prussian Statistical Office, and thus she got an impression of the marginal role of female statisticians. And perhaps L. von Bortkiewicz had told her about the situation of women in the world of academia and their limited career chances. His sister Helene von Bortkiewicz (1870–1939) had studied mathematics at the University of Göttingen. After having been a school teacher, she worked in a bank in St. Petersburg from 1914 to 1917, before she was able to leave Soviet Russia and from then on lived with her brother in Berlin. As far as we know she was never able to do any scientific work of her own. The volumes of “Die Welt in Zahlen” (The World in Figures) were enthusiastically reviewed, Wladimir S. Woytinsky became known worldwide, and he was able to get a paid position as a statistician. In 1929 he became the head of the newly established statistical department of the ADGB (Allgemeiner Deutscher Gewerkschaftsbund, the trade unions organization). Emma Woytinsky reported proudly on this time and the success of Wolik, quoting one of the reviews of the book written by the wellknown statistician Robert R. Kuczynski (1876–1947) in his journal “Finanzpolitische Korrespondenz” (Financial-Political Correspondence): When one sees how a single scholar has created an international encyclopedia that by far overshadows what has been produced in this field by the International Statistical Institute or the League of Nations with their much larger means, skepticism (sic) again revives about “organized” scientific production. Of course, it would have been contrary to the habits of those institutions to draw a genius like Woytinsky into their work.23

R. R. Kuczynski knew the work of the International Statistical Institute as well as the work of the statisticians in Geneva, who were working for the League of Nations. And it seems that he had foreseen that Wl. S. Woytinsky will get no position in Geneva just few years later. Perhaps therefore Emma Woytinsky had chosen to quote this review instead of others in her autobiography. Regarding Kuczynski’s remark about “a single scholar,” one has to take into account that, all in all, three scholars had been involved in the project, besides Wladimir and Emma Woytinsky the editor L. von Bortkiewicz, but nevertheless R. R. Kuczynski was right to esteem the work highly. After quoting him, Emma Woytinsky wrote: Wolik’s reputation spread first in Germany and German speaking countries – Austria and Switzerland – and also in Czechoslovakia, The Netherlands and the Scandinavian countries. It continued to spread throughout Europe and even reached Japan. Up to the present time, I have never met a German statistician or economist who has not at least heard of The World in Figures and Wolik’s name.24

When Wladimir S. Woytinsky became the head of the new statistical department of the ADGB in 1929 their close collaboration was finished for the next ten to fifteen years. She was working on her own projects, among them was a study of the social conditions and municipal policy of Berlin entitled “Sozialdemokratie und Kommunalpolitik. Gemeindearbeit in Berlin,”25 and she became his translator from 23 Quot.

in Woytinsky, E. S. (1965), p. 114. E. S. (1965), p. 114. 25 See Woytinsky, E. S. (1929). 24 Woytinsky,

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Russian into German. While Wladimir S. Woytinsky was working in the ADGB he wrote his memoirs on his time in Russia, on the revolution of 1905 and the following years. About his work in the ADGB and for the German trade union organizations he later wrote only these four sentences: My work with the ADGB originally focused on labor statistics. I reorganized union statistics of unemployment and collective agreements and developed the statistical section in the annual reports. In addition, I lectured and wrote articles for labor magazines. Contrary to my expectations, there was not much politics and very little fighting in that work.26

One could imagine that he had not been satisfied with this situation, and obviously he had enough time to write his memoirs, which were translated by Emma (and Friedrich Schlömer) into German and published in Berlin in 1931 and in the spring of 1933.27 In 1933 the Woytinskys had to go into exile immediately, because they were active socialists and Jews. Moreover, Wladimir S. Woytinsky was not only known as the head of the statistical department of the ADGB, but also as one of the three speakers of the “WTB plan” of 1931/32, i.e., a plan to counter and overcome the deep crisis—the Great Depression—after 1929. WTB were the initials of Wl. Woytinsky, Fritz Tarnow (1880–1951) and Fritz Baade (1893–1974) who demanded another economic policy to defeat the crisis. Unfortunately, their plan failed because of the resistance from the leaders of the ADGB and of the SPD. An anti-WTB-plan was developed in 1932 by the director of the Imperial Statistical Office and the director of the Institute for Business Cycles, Ernst Wagemann (1884–1956), but this plan also failed. Although a member of the Nazi Party NSDAP since 1933, Ernst Wagemann would help in 1940 the former collaborator in Woytinskys statistical department and member of the SPD, Bruno Gleitze (1903–1980), to defend his thesis (on crime statistics during the economic crisis of 1929 and the following years) at the University of Berlin. After the capitulation of Nazi Germany Bruno Gleitze belonged to the key persons in Berlin East to establish a new statistical office (Zentralverwaltung für Statistik). In 1946 he became the first dean of the newly established Economics Faculty at the University of Berlin. However he never mentioned Wladimir S. Woytinsky as his chief or mentor, neither in the Nazi time (of course not) nor during his time in Berlin East nor even later when he was in Berlin West (from December 1948 onwards) or in the 1950s in the Federal Republic of Germany. As of 1946, around the time when Bruno Gleitze became the dean of the Economics Faculty in Berlin East, Emma and Wladimir Woytinsky started again their close collaboration in statistics. Emma and Wladimir Woytinsky escaped the Nazi persecution via Geneva and Paris to New York and Washington. After a brief stay in Paris and in Geneva where he was unsuccessful looking for a job, they emigrated in 1935 to the USA. During their second emigration in Paris they found it difficult to find any position. Wladimir S. Woytinsky was contributing to a project for the ILO (International Labour (sic) Office, associated with the League of Nations) in Geneva, while Emma Woytinsky 26 Woytinsky, 27 See

W. S. (1961), p. 461. Woytinsky, W. S. (1931a, 1933).

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attended courses at the École des Hautes Études for postgraduates, her special interest were courses on the economic situation in the USA after the Great Depression. When it became clear, that the ILO in Geneva would not offer a job to her husband, she convinced him to emigrate to the USA. To be better prepared for this new exile, they took private English lessons with a teacher in Oxford from September to November in 1935. At the end of 1935, they arrived in the United States where they were living mostly in Washington, D. C. In the USA Wladimir S. Woytinsky became an expert on social welfare and worked in different boards to introduce social welfare as part of the New Deal Policy of President Franklin D. Roosevelt (1882–1945). To this day, he is still remembered as one of the “fathers” of the U.S. social welfare system: “Wladimir became one of the architects of the Social Security system and was the principal economist of the Social Security Board, where he served until 1947.”28 He was able to bring together the experiences of the European and German social system with the demands of the U.S. society, and he was highly in favor of Roosevelt’s New Deal Policy, even more so after his negative experiences with the WTB planning discussions in Germany before the Nazi’s had come to power. Meanwhile he had experience as a statistician and an economist, and a socialist scholar. He met several colleagues in the USA again who had also to emigrate from Nazi Germany and later from Europe under Nazi occupation. During WW II both actively participated in the alliance against Nazi Germany. During this time Emma S. Woytinsky worked in the Economic Warfare Board, later in the Foreign Economic Administration. Here she studied the economy of Nazi Germany, being the only female expert in different teams.29 After the Allied victory both of the Woytinskys became freelance scholars and were funded by grants from various organizations and foundations, including the Rockefeller Foundation, the John Hopkins University, and the Twentieth Century Fund. Between 1947 and 1959 a second period of close collaboration on statistics followed. Now they were working together and publishing together. They did famous and highly acknowledged work compiling large statistical data collections on world population and production (1953), and on world commerce and trade (1955). These two volumes are comparable to the seven volumes published in the 1920s. Again the authors had a special focus on the development of economics and society from an international and comparative perspective. One has to have in mind that this work was done without modern computers; the first modern computers were available for civil research projects only from the mid-1950s. Emma Woytinsky was in the USA as well acknowledged as Wladimir Woytinsky. Therefore it seems obvious that now both became authors of the two volumes, her authorship has not been hidden anymore. As it was customary at this time, only initials were used for their first and middle names. 28 See https://lsa.umich.edu/…/wladimir-s--and-emma-s--woytinsky-fellowship-fund.html. It is the

website of the Wladimir S. and Emma S. Woytinsky Fellowship Fund at the University of Michigan (accessed May 13, 2019). 29 See Woytinsky, E. S. (1965), p. 187 and p. 188.

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W. S. and E. S. Woytinsky was used only in 1959 when their last booklet “Lessons of the Recessions” came out in Washington D. C. It was a small book (102 pages) and a kind of legacy of their work and their convictions. After the successful publication of the two volumes in 1953 and 1955, both were able to find other activities that allowed them to combine their interest in economic and social developments with their curiosity. Supported by the U.S. State Department, they went on several lecture and research tours together, from 1955 to 1956 to the Far East, including India and Japan, and from 1957 to 1958 eight months to Latin and South America. After their experience with translators and the problems during their first lecture tour, Emma Woytinsky decided to learn Spanish before their second tour. It worked well with the exception of Brazil (where Portuguese is the language), and she was not only his translator or interpreter but also gave several lectures of her own at universities. After the death of Wladimir Woytinsky on June 11, 1960, in Washington D. C., Emma Woytinsky took care of his legacy. She donated his manuscripts and correspondence to two famous archives: to the Hoover Institution Library and Archive in Stanford (California) and to the Archive of the International Institute of Social History (IISH) in Amsterdam. In 1962 she edited a special volume in his honor, she asked friends and colleagues to participate with an article, and she published in this volume a complete bibliography of all his writings, including newspaper articles in different languages.30 The basic bibliography of W. S. Woytinsky, made by himself, is kept in the IISH Amsterdam (Vojtinskij Papers). Furthermore, she established the Wladimir S. and Emma S. Woytinsky Fellowship Fund for graduate students preparing their Ph.D. in economics or statistics at the Department of Economics of the University of Michigan.31 And she kept here in the special collection of the university library Wl. Woytinsky’s writings from 1905 to 1960. Since 1965 the Department of Economics at the University of Michigan is celebrating the W. S. Woytinsky Lecture, the first speaker was the later Nobel Prize Winner in Economics (the winner of the Nobel Memorial Prize in Economic Sciences) in 1992, Gary Becker (1930–2014), who gave a talk on “Human Capital and the Personal Distribution of Income: An Analytical Approach.” The Woytinsky Lectures are taking place mostly every two years, on September 12, 2016, the speaker was Hal R. Varian, Chief Economist at Google, who talked on “Google Tools for Data,” on November 7, 2018, Esther Duflo gave a talk “Every Child Counts: Transforming education systems around the world.”32 Shortly before her death she published another book, now on her own research topics. It was partly a summary of her work and scientific interest for more than thirteen years, beginning in Paris when she attended special courses on the economy 30 See

Woytinsky, E. S. (1962), pp. 229–272. the homepage of the University of Michigan, the Wladimir S. and Emma S. Woytinsky Fellowship Fund, https://lsa.umich.edu/…/wladimir-s--and-emma-s--woytinsky-fellowship-fund. html (accessed May 13, 2019). 32 See the homepage of the University of Michigan, W. S. Woytinsky Lectures https://lsa.umich.edu/ econ/alumni-friends/w-s--woytinsky-lecture.html (accessed May 13, 2019). In 2019 Esther Duflo (b. 1972) received the Nobel Prize in Economics, together with her husband and colleague Abhijit Banerjee (b. 1961) and Michael Kremer (b. 1964). 31 See

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in the USA.33 The foreword was written by an old friend and colleague, the wellknown “U.S. Labor Official” Ewan Clague (1896–1987), whom she had known since the 1940s. Ewan Clague was the director of research and statistics for the Social Security Board, where Wladimir Woytinsky worked until 1947, and from 1946 to 1965 he served as the Commissioner of Labor Statistics for the Department of Labor of the U.S. government.34 Emma Woytinsky died on April 13, 1968, in her home in Washington D. C., after a heart attack. Obituaries in some newspapers reported on her death, and she was named the “noted economist, author, and statistician.”35 Thanks to her activities commemorating the life and work of Wladimir S. Woytinsky he is better known than she is, but her legacy lives on in the Wladimir S. and Emma S. Woytinsky Fellowship Fund for graduate students at the University of Michigan.

5.3 Emma S. and Wladimir S. Woytinsky—An Unusual Couple in Statistics I investigated their practices as statisticians, their collaboration, and the division of labor in the 1920s in Berlin as well as their collaboration in the 1950s in the USA. In contrast to their years in Berlin when they collaborated but did not publish together, in the USA they were working, collaborating, and publishing together all in all four books.36 The dedication of each volume to Emma S. Woytinsky in the 1920s was a hint that she had contributed more to these seven volumes than the “normal” or “usual” work of a secretary. Thanks to both their autobiographies and the description of this collaboration in Berlin between 1922 and 1927/28 we learn how much she contributed to these books. W. S. Woytinsky reported: Emma and I worked together. It was hard work – all day in the library, without taking time out for lunch, munching sandwiches at a desk covered with books and papers. We had no secretary, no draftsman, no typist, not even a calculating machine. We shared the work of assembling the material; Emma made most of the computations and also did our housekeeping (sic). I wrote the text and drew the charts. We worked sixty to sixty-six hours a week for five years, but the work was fascinating. I was making up the time spent on politics, in prisons, and in exile, and I particularly appreciated the opportunity to work with Bortkiewicz.37

As already mentioned, probably she supported the work without official acknowledgment as a co-author because she knew that in the male-dominated field of statistics 33 See

Woytinsky, E. S. (1967). the obituaries in New York Times, 15.4.1987, and in The Washington Post, 15.4.1987. 35 See for example the obituary in: Iowa City Press, 15.4.1968. 36 See Woytinsky and Woytinsky (1943, 1953, 1955, 1959). 37 Woytinsky, W. S. (1961), p. 452. 34 See

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her chances as a woman would be much less than his chances as a man. Furthermore, they were emigrées, foreigners, and outsiders in more than one sense. Thus she concentrated her activities to help him to find official acceptance and finally a position, and this strategy was successful. When he had received a position she did her own research, and her book “Sozialdemokratie und Kommunalpolitik. Gemeindearbeit in Berlin” was published in 1929. Here she was interested in the question whether a social democratic party would be able to fulfill its social program if given the opportunity to do so. Furthermore, Emma Woytinsky was also active in circles of Russian emigrées to help political prisoners in the Soviet Union. She had done similar work like in St. Petersburg, she and her friends collected money, bought food and medicine and had sent parcels to the political prisoners, which was still possible until 1932.38 Wladimir Woytinsky and Emma Woytinsky have described their collaboration in detail in their autobiographies, i.e., how they were working together, how they shared common views on the role of statistics, and how they organized the division of labor, but hers is more detailed than his. She made it clear that they had divided their labor according to their specific skills. Obviously, she had better language capacities, so it was her duty to study the literature in different languages. In Berlin she also prepared the graphic presentations and colored them. The choice of the graphic presentations was the usual one at the time. We know from a review written by Wladimir Woytinsky in 1931 that he was very skeptical concerning the new graphic methods and elements (the pictograms) that had been developed by Otto Neurath (1882–1945) and his colleagues.39 It is quite remarkable that the Woytinskys were able to synthesize a product of data collections which is still valuable—as a source as well as a type of data collection. Regarding the philosophy behind these data collections Wladimir Woytinsky pointed out that they learned it from Bortkiewicz: Bortkiewicz had a photographic memory and knew the literature on practically any topic of economics and statistics. He had the rare ability to visualize a statistical series as an expression of a continuous economic or historical process. And he had a philosophy of statistics that he had never developed in his writings. For him statistics was not a body of mathematical formulas and techniques but the art of quantitative thinking. … To him the essence was to use measurement to obtain a better understanding of facts of life. … The World in Figures represented the kind of statistics he liked.40

In 1954 Wladimir Woytinsky made a similar statement toward the end of his article on the limits of mathematics in statistics: “I feel that in the broad realm of human knowledge there is room for two disciplines which have the same name: statistics as a branch of applied mathematics; and social and economic statistics as the art and science of quantitative thinking on human affairs.”41 38 See

Woytinsky, E. S. (1965), p. 139. Woytinsky, W. S. (1931b); and Sandner (2014), p. 191. 40 Woytinsky, W. S. (1961), p. 453. 41 Woytinsky, W. S. (1954), p. 18 (emphasis original). 39 See

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When the Woytinskys were working together again in the 1940s and 1950s, they used similar approaches like in the 1920s in preparing their two books on world population and production, and world commerce and governments. As before, the division of labor was organized according to their specific capacities. About this second close collaboration Wladimir Woytinsky reported: This project was the first major literary venture Emma and I had undertaken on the basis of complete equality. In Germany, she had helped me in my statistical and economic work but did not try her hand at doing part of it on her own. Now she took all responsibility for entire sections of the study, mainly those requiring extensive work in libraries (…), while I worked on sociological and political sections (…) and general world surveys. This division of labor was in line with our personal inclinations. We had no assistants and did all technical work ourselves. Emma prepared computations and supervised typing, I took care of charts and maps. The project took almost seven years …42

And Emma Woytinsky wrote about this collaboration: Of all the work we did jointly or separately, we liked best the two international volumes World Population and Production and World Commerce and Governments, a project … Financed by the Rockefeller Foundation and the Twentieth Century Fund. … In a way, this work took us back to the years when we worked on the The World in Figures in Germany. … As soon as the war ended, we began … The final product consisted of two volumes comprising nearly 2,200 pages, some 800 tables, and more than 500 charts and maps.43

Both of these volumes are still cited in the literature as sourcebooks containing relevant data on various economic figures before 1955. When analyzing and comparing the different circumstances and working conditions in Berlin and in Washington, one sees that the question of co-authorship depended deeply on the social situation of women, of women statisticians, and on the acceptance or non-acceptance of female statisticians in the world of academia and in the public sphere. In this regard, the situation for Emma Woytinsky was much better in the U.S.A. than it had been in Germany, better in the 1950s than in the 1920s. Why were Emma S. and Wladimir S. Woytinsky so unusual, as a couple in general and as a couple in statistics? They were an unusual couple in statistics because of their lack of any academic training in this field, and because she received an academic degree while he did not. But they both received an intensive academic training in statistics in an unusual way, from one of the leading statisticians at that time. Emma and Wladimir Woytinsky contributed to the development of statistics in the twentieth century in three directions: First, they contributed to statistics through their publications (the seven volumes in the 1920s and the two volumes in the 1950s). Second, they contributed to the development of statistics through their approach and their three-dimensional perspective: They prepared the data compilations from a socialist (or Marxist) perspective (economics is the most important factor in societies), from an international perspective (as it is clear from the title “The World in Figures”), and they prepared the statistical data compilations from a historical perspective (i.e., a long-durée perspective). Third, their aims were always to better 42 Woytinsky, 43 Woytinsky,

W. S. (1961), p. 519. E. S. (1965), p. 189.

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understand how things developed (“to obtain a better understanding of facts of life” as Wladimir Woytinsky called it) and to predict the future. Here again they followed the philosophy of statistics that they had learned from L. von Bortkiewicz in the 1920s which was still innovative when they were working on the new data compilations in the 1950s. Their interest in the historical development of the economy and society and their curiosity gave them the power to go on two long lecture tours in 1955/56 and in 1957/58. They wanted to study the new directions of economic growth in India and Japan as well as in South and Latin America. While on tour, they encountered the future as well as the past. In her autobiography Emma Woytinsky recounts an amazing and surprising meeting on a small airport in South America in 1957/58 with a former reader of the “World in Figures”: We were leaving Quito, Ecuador, for Santiago, Chile, at about five in the morning. Shortly before the departure of the plane, the United States cultural attaché arrived and shouted, “Woytinsky, Woytinsky!” Suddenly a man who was taking care of passengers at a little airport counter left his customers, ran to Wolik, and bubbling from excitement, said, “Are you the author of Die Welt in Zahlen? I never dreamed I would meet you. I am so glad, so glad. I still have all your seven volumes. I wouldn’t think of leaving them to Hitler!” He embraced the surprised Wolik and ran back to his customers.44

Emma Woytinsky was just as proud as her husband that more than thirty years after its publication they will meet a reader of their work on the other side of the world who saved it from the Nazi’s.

Bibliography Archival Sources International Institute of Social History (IIHS) Amsterdam, The Vojtinskij Papers, 1919–1960, 43 boxes. University Uppsala, Dept. of Manuscripts and Music, Uppsala University Library, L. von Bortkiewicz Papers (41 boxes, there is no material of or about the Woytinskys).

Secondary Literature Desrosières, A. (1998). The Politics of Large Numbers: A history of statistical reasoning. Harvard University Press. Dmitriev, A. L. (2016). V. (sic) S. Woytinsky and Mathematical School in Political Economy: First Steps. (To the 110th Anniversary of the Publication of the Book “Market and Prices”). Vestnik of Saint Petersburg University, Series 5, Economics, Issue 3, 95–108 (in Russian). Gorroochurn, P. (2016). Classic topics on the history of modern mathematical statistics. From Laplace to more recent times. Hoboken, New Jersey: Wiley. 44 Woytinsky,

E. S. (1965), p. 114.

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Gumbel, E. J. (1931). Nachruf auf Ladislaus von Bortkiewicz. Deutsches statistisches Zentralblatt, 23(8), Spalten 231–236 (Bibliographie von LvB, Spalten 233–236). Gumbel, E. J. (1968). Bortkiewicz, Ladislaus von. International Encyclopedia of the Social Sciences. New York: Macmillan Publishers (Vol. 2, pp. 128–131) (Bibliography LvB, 130–131). Härdle, W. K., & Vogt, A. B. (2015). Ladislaus von Bortkiewicz – Statistician, Economist and a European Intellectual. International Statistical Review, 83(1), 17–35. Heyde, C. C., & Seneta, E. (Eds.). (2001). Statisticians of the Centuries. New York et al: Springer. Johnson, N. L., & Kotz, S. (Eds.). (1997). Leading personalities in statistical sciences: From the seventeenth century to the present. New York: Wiley. Kiser, C. V. (1955). “Review: World population and production by W. S. Woytinsky and E. S. Woytinsky” (1953). The Milbank Quarterly, 33(2), 203–206. Kochina, P. J. (1988). Nauka, Ljudi, Gody. Vospominanija i vystuplenija. Moskva: Nauka (in Russian; Memoirs). Lin, X., Genest, C., Banks, D. L., Molenberghs, G., Scott, D. W., & Wang, J.-L. (Eds.). (2014). Past, Present, and Future of Statistical Science. Boca Raton, London, New York: CRC Press. (52 articles of 50 authors, 622 pages). Lykknes, A., Opitz, D. L., & van Tiggelen, B. (Eds.). (2012). For better or for worse? Collaborative Couples in the sciences. Heidelberg, New York and London: Birkhäuser. Minc, I. I., & Nenarokov, A. P. (Eds.). (1982). Zenchiny-revoljucionery i uchenye (Women revolutionery and scholars). Moskva: Nauka (in Russian). Porter, T. M. (1996). Trust in Numbers: The pursuit of objectivity in science and public life. Princeton: Princeton University Press. Pycior, H. M., Slack, N. G., & Abir-Am, P. G. (Eds.). (1996). Creative couples in the Sciences. New Brunswick/New Jersey: Rutgers University Press. Salsburg, D. (2002). The Lady Tasting Tea. How statistics revolutionized science in the 20th century. New York: A. W. H. Freeman & Henry Holt Company. Sandner, G. (2014). Otto Neurath. Eine politische Biographie. Wien: Paul Zsolny Verlag. Schwebber, L. (2006). Disciplining Statistics: Demography and vital statistics in France and England, 1830–1885. Duke University Press. Stigler, S. M. (1986). The History of Statistics. The measurement of uncertainty before 1900. Cambridge, MA: Harvard University Press, reprint 1990. Stigler, S. M. (1999). Statistics on the Table. The history of statistical concepts and methods. Cambridge et al: Harvard Universuty Press, reprint 2002. Tooze, A. (2007). Statistics & German State 1900–1945: The making of modern economic knowledge. Cambridge (Studies in modern economic history, vol. 9). Vogt, A. (2007). Vom Hintereingang zum Hauptportal? Lise Meitner und ihre Kolleginnen an der Berliner Universität und in der Kaiser-Wilhelm-Gesellschaft. Stuttgart: Franz Steiner Verlag, Pallas & Athene Bd. 17. Woytinsky, E. S. (1929). Sozialdemokratie und Kommunalpolitik. Gemeindearbeit in Berlin. Berlin: B. Laubsche Verlagsbuchhandlung. Woytinsky, E. S. (Ed.). (1962). So much alive. The life and work of Wladimir S. Woytinsky. New York: The Vanguard Press, Inc. Woytinsky, E. S. (1965). Two lifes in one. New York, Washington: Frederick A. Praeger Publisher. Woytinsky, E. S. (1967). Profile of the U. S. economy. A survey of growth and change. New York, Washington: Frederick A. Praeger Publisher. Foreword by Ewan Clague. Woytinsky, W. S. (1925–1928). Die Welt in Zahlen (The World in Figures). 7 volumes. Berlin: Rudolf Mosse Buchverlag. (Editor: L. v. Bortkiewicz, Reihe: Serie populärer statistischer Bücher (Series of popular books on statistics); only these 7 volumes were published). Woytinsky, W. S. (1931). Der erste Sturm. (The first Storm. Memoirs of the Russian Revolution of 1905). Berlin: Büchergilde. Woytinsky, W. S. (1931). Rezension (review of) Otto Neurath “Gesellschaft und Wirtschaft.” Die Arbeit. Zeitschrift für Gewerkschaftspolitik und Wirtschaftskunde, 8, 158–161.

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Woytinsky, W. S. (1933). Wehe dem Besiegten! (Woe to the Defeated). Berlin: Büchergilde (in spring 1933). Woytinsky, W. S. (1954). Limits of Mathematics in Statistics. The American Statistician, February 6–10, 18. Woytinsky, W. S. (1961). Stormy Passage. A personal history through two russian revolutions to democracy and freedom: 1905–1960. New York: The Vanguard Press, Inc. Woytinsky, W. S. (1964). Market and Price. New York: Columbia University Press. Woytinsky, W. S., & Woytinsky, E. S. (1943). Employment and wages in the United States. Woytinsky, W. S., & Woytinsky, E. S. (1953). World Population and Production: Trends and Outlook. New York: Twentieth Century Fund. Woytinsky, W. S., & Woytinsky, E. S. (1955). World Commerce and Governments: Trends and Outlook. New York: Twentieth Century Fund. Woytinsky, W. S., & Woytinsky, E. S. (1959). Lessons of the Recessions. Washington, D. C.: Public Affairs Institute.

Annette B. Vogt is a historian of science and mathematics; she is affiliated to the Max Planck Institute for the History of Science in Berlin, until 2018 as senior research scholar, now as emeritus. Until the present she is teaching history of statistics and history of economic thought at the Economics Faculty of the Humboldt University Berlin where she was appointed honorary professor in 2014. In 2012 she was elected a Corresponding Member, in 2016 a Full Member of the International Academy of History of Science (Académie Internationale d’Histoire des Sciences) in Paris. Her research topics are history of science, history of mathematics and history of statistics in the 19th and 20th centuries; and history of Jewish scientists and history of women scientists. She published several books, together with Renate Tobies, she edited a book on women scientists in industrial research (2014). Furthermore, she published ca. 200 articles in books and journals on the history of mathematics and statistics, on women scientists and Jewish scientists.

Chapter 6

Stanisława and Otton Nikodym Danuta Ciesielska

Abstract Stanisława Nikodym was the first Polish women to obtain a PhD in mathematics. She earned this degree from the University of Warsaw in 1925. Three years later, she presented a talk in the section on analysis and topology at the International Congress of Mathematicians in Bologna. She was the wife of Otton Nikodym. The couple was an example of a “collaborating couple” who supported each other in scientific research and academic life, but they had independent scientific careers. He began his scientific research at the age of 36, shortly after marrying Stanisława, who was then 26 and had just finished her studies. Otton Nikodym is one of the most renowned mathematicians of Polish origin, whereas her results in topology were interesting and cited on occasion. Unfortunately, she is almost unknown in Poland and abroad. The main goal of this paper is to present a very complicated history of Stanisława as a young woman in the Russian Empire, as an educated scholar in Poland, and as an emigrant to the United States who sorely missed Poland. It will also outline the story of Otton’s intellectual development and Stanisława and Otton’s marriage.

6.1 Introduction Women entered the world of scientific research during the first decades of the twentieth century. At that time, many Polish mathematicians became world-famous, and the “Polish Mathematical School” attracted international attention. During the interwar period, a married couple of Polish scholars began to conduct their research in mathematics. It was a hard time for them. Now, he is world-famous, whereas she is almost unknown. Her name cannot be found in any dictionary of Polish mathematicians. How is this even possible? After all, she shared a last name with her husband Otton Nikodym, whose accomplishments are related in nearly every reference work on mathematics. I will not attempt to answer this question. Instead, my aim is to D. Ciesielska (B) ´ Institute for the History of Science, Polish Academy of Sciences, Nowy Swiat 72, 00-330 Warsaw, Poland e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_6

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present the story of a special “mathematical couple” in Poland and the United States and to discuss her mathematical findings and their reception.

6.2 Childhood and School Years Stanisława Dorota Nikodym (née Liliental) was born on July 2, 1897. She was born in Warsaw, which was then part of the Kingdom of Poland (itself part of the Russian Empire). She was brought up in the family of Regina (née Eiger, 1877–1924) and Natan (1868–1927). It was a family of assimilated, Polish-speaking Jews. Regina studied in Warsaw at an informal high school called the “Flying University.”1 During her studies, she began to conduct pioneering research on the folklore and literature of Polish Jews. The most important works by Regina are a book about children in the Jewish culture and a series of three articles on Jewish festivals.2 Regina was also interested in literature and she translated Jewish poetry into Polish. Stanisława had one brother, named Antoni.3 Stanisława was educated in Warsaw.4 For 6 years, she was a pupil in Helena Szalay’s school.5 For 7 years after that, she was a student at the private Polish school for young women that was run by the well-known Polish educator Karolina Kochanowska.6 Otton Marcin Nikodym (1887–1974) was born on August 13 in Demycze, a suburb of the small Galician city of Zabłotów (now Zablotiv, Ukraine),7 to a Roman Catholic family. His parents died when he was very young. His father Otton Bogusław, whose family came from the Czech city of Žlutice, was killed in an accident. His grandparents were of French, Italian, Czech, and Polish origins. Otton was brought up by his grandparents. His grandfather Marcin Cyprian (Cipriani) was an architect. After a short stay in Vienna, where Otton started his education, the whole family returned to Lvov in 1897. Here, Otton finished gymnasium of the real type. As a result, he had to pass external examinations in Greek and Latin to start his university studies.

1 Some

years earlier, Marie Curie and her sisters Helena and Bronisława had also studied there. (1927, 1908, 1919); for information about Regina Liliental see Gr˛acikowski (2013). 3 Antoni Liliental (1908–1940) graduated in chemistry from the Warsaw University of Technology and he was an assistant there. He was also an officer in the Polish Army. During WWII, he was captured by the Soviets and killed (with 4,400 other Polish officers) in Katy´n in 1940. See “Rozpoznane ofiary katy´nskie” (1943) and Liliental (2015). 4 Warsaw was the capital city of the Kingdom of Poland, which was created in 1815 by the Congress of Vienna. Until 1915, the Kingdom of Poland was under Russian occupation, with the Emperor of Russia functioning as the Polish King. 5 Helena Szalay (Szalajowa, née Skłodowska, 1866–1961) was a sister of Marie Curie. 6 This school, which was called “Pensja,” was a private high school for young women. 7 Galicia (formal name: Kingdom of Galicia and Lodomeria) was part of the Austro-Hungarian Empire from 1772 to 1918. After the reforms of 1867 in Austria, it became an ethnic Polishadministered autonomous country in the Empire with the capital city of Lvov. 2 Lilientalowa

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6.3 Studies and Life Before Marriage 6.3.1 Otton’s Studies Otton studied at the University of Lvov.8 As of 1873, Galicia was an autonomous province of the Austro-Hungarian Empire, with Polish as its official language. Otton studied in the Polish language, which was obligatory in schools and universities. During the time of autonomy, many Polish scientists decided to move to Lvov and Kraków to study and to work at the Polish university. Among them were Wacław Sierpi´nski,9 who moved from Warsaw (in the Russian Empire), and Marian Smoluchowski,10 who moved from Vienna. Nikodym attended many lectures and seminars on mathematics, physics, and chemistry. The most important for Otton were two seminars: Sierpi´nski’s higher seminar on mathematics and Smoluchowski’s seminar on theoretical physics.11 Nikodym also took part in lectures by Józef Puzyna,12 Jan Łukasiewicz,13 and Kazimierz Twardowski.14 Franciszek Leja,15 a friend of Otton, studied mathematics and physics at the University of Lvov at the same time as Otton. Leja was not satisfied with his studies there, as he recounted in his memoires:

8 During

the time of his studies, the formal name of the university was the “Imperial and Royal Franz I University in Lvov.” In the interwar period, it was called the “John Casimir University,” and now it is the “Ivan Franko National University of Lviv.” 9 Wacław Sierpi´ nski (1882–1969), who held a PhD in mathematics from the Jagiellonian University in Kraków, had also studied in Göttingen. He was a professor of mathematics at the Universities of Lvov and Warsaw and a leader of the “Warsaw School of Mathematics.” 10 Marian Smoluchowski (1872–1917), a physicist, earned a PhD and habilitation from the University of Vienna. He studied in Glasgow, Paris, and Berlin as well. Smoluchowski is best known from the Einstein–Smoluchowski relation, the Feynman–Smoluchowski ratchet, and his interpretation of Brown motions. He was a professor at the University of Lvov and at the Jagiellonian University in Kraków. 11 For details, see Prytula (2015). 12 Józef Puzyna (1856–1919) held a PhD in mathematics from the University of Lvov and had studied in Berlin under L. Fuchs, K. Weierstrass, and L. Kronecker. He was a professor and rector at the University of Lvov. 13 Jan Łukasiewicz (1878–1956), a logician, held a PhD from the University of Lvov and was a member of the “Warsaw-Lvov School of Logic.” He was a professor in Lvov, Warsaw, and Dublin, and he invented the so-called “Polish notation.” 14 Kazimierz Twardowski (1866–1938), who earned a PhD and habilitation in philosophy from the University of Vienna, was the leader of the “Warsaw-Lvov School of Logic.” He was a professor at the University of Lvov. 15 Franciszek Leja (1885–1979) held a PhD in mathematics from the Jagiellonian University and had also studied at the Sorbonne. He was a professor at the Technical University in Warsaw, the University of Warsaw, and the Jagiellonian University, and he was the founder and leader of the “Kraków School of Complex Analysis.”

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Fig. 6.1 Stanisława Liliental (ca. 1919). Archive of the University of Warsaw (AUW RP 1583)

During my studies, the University in Lvov had two professors of mathematics: J. Puzyna and J. Rajewski,16 and the latter was on long-term health leave. There was no assistant professor or assistant to tutor students. Under these conditions, mathematical studies must have been rather poor, since not every aspect of mathematics could have been taught. The government in Vienna probably was not interested in education in Galicia. […] Lectures on theoretical physics were conducted by a young assistant professor, Marian Smoluchowski, who had graduated from the University of Vienna a few years before. From his lectures I learned that a missile fired from the ground upwards may not get back to earth when its initial speed exceeds 11 km per second.17

Leja’s opinion may have been critical, but even he pointed out that Smoluchowski’s influence on students was significant. Smoluchowski was 28 years old when he started working in Lvov, and in those days he was the youngest professor (with chair) in the 16 Jan Rajewski (1857–1906) earned a PhD in mathematics from the University of Lvov, where he later worked as a professor. 17 Leja (1979): “W czasie moich studiów Uniwersytet Lwowski miał dwóch profesorów matematyki: J. Puzyn˛e i J. Rajewskiego, z których drugi był wówczas na dłu˙zszym urlopie zdrowotnym. Nie było przy tym z˙ adnego docenta ani asystenta do prowadzenia c´ wicze´n. W tych warunkach studia matematyczne musiały kule´c, bo nie wszystkie działy matematyki mogły by´c wykładane. Władze wiede´nskie nie interesowały si˛e zapewne zbytnio szkolnictwem w Galicji. […] Wykłady fizyki teoretycznej we Lwowie prowadził wówczas młody docent Marian Smoluchowski, który przed kilkoma laty uko´nczył studia w Uniwersytecie Wiede´nskim. Na wykładzie tym dowiedziałem si˛e po raz pierwszy, z˙ e pocisk wystrzelony z Ziemi pionowo w gór˛e mo˙ze nie wróci´c na ziemi˛e, gdy pocz˛atkowa jego szybko´sc´ przekroczy 11 km na sekund˛e.”

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Habsburg Monarchy. He was very well educated in Vienna (under J. Stefan18 and F. Exner19 ), Glasgow (in Lord Kelvin’s laboratory), Paris (in Lippmann’s laboratory20 ), and Berlin (in Warburg’s laboratory21 ). Smoluchowski had a great influence on his students and colleagues not only as a scientist but also as a man. His opinions on many different matters were widely appreciated. In 1912, during a meeting of the Scientific-Literature Society in Lvov, Smoluchowski delivered a lecture in which he discussed the role of women in the sciences in the past, present, and future.22 The talk was published in 1917 and reprinted in Smoluchowski’s collected works in 1928 by the Polish Academy of Science and Fine Arts. The talk ends with the words: Those women who enter the scientific path should have their vocation facilitated; any and all external obstacles ought finally to be removed: those funny superstitions, those outdated views which obstruct women’s access to some scientific institutions, rendering it difficult for them to study, pursue scientific work, obtain university chairs. May the free competition principle govern in this (as well as on any other) field. Let us wish for the competition to be as animated as practicable .23

In 1910 in Lvov, Otton passed the government exam for future teachers of mathematics and physics at the high school level, and he started working as a teacher the next year.

6.3.2 Stanisława’s Studies Stanisława (Fig. 6.1) lived in the Russian Empire, and after finishing school the most important problem for her was the fact that, as a women, she was not allowed to study at a university. Some of the most-determined women went abroad for study. Those who wanted to study mathematics, physics, and astronomy left for France or Germany. At the end of the nineteenth century, Göttingen was starting to become a “Mecca of mathematicians.” From 1890 to 1920, the following women arrived in Göttingen from the Russian Empire to study mathematics: Nadeschda Nikolaevna 18 Joseph 19 Franz

Stefan (1835–1893), a physicist, was a professor at the University of Vienna. S. Exner (1849–1929), a mathematical physicist, was a professor at the University of

Vienna. 20 Gabriel Lippmann (1845–1921), a Nobel laureate in physics, was a professor at the Sorbonne. 21 Emil Warburg (1846–1931) was a professor of physics in Strassburg, Freiburg, and Berlin. 22 Smoluchowski’s position in support of women was special and direct but it was not common in the Polish scientific community in 1912 (or later). An English translation of his talk appeared in a special issue of the journal Acta Poloniae Historica 117 (2018), which is dedicated to the role of women in the sciences; see Smoluchowski (2018). 23 Smoluchowski (1928): “Kobietom, które wst˛ epuj˛a na drog˛e naukow˛a, powinno si˛e ułatwia´c ich powołanie; powinny nareszcie znikn˛ac´ wszelkie zewn˛etrzne przeszkody, owe s´mieszne przes˛ady, owe przestarzałe pogl˛ady, które zamykaj˛a dost˛ep kobietom do niektórych instytucyj naukowych, które im utrudniaj˛a kształcenie si˛e, prac˛e naukow˛a, dost˛ep do katedr uniwersyteckich. Niech tu (jak na ka˙zdem innem polu) panuje zasada wolnej konkurencji. Oby ta konkurencja była jak naj˙zywsza.” English translation Smoluchowski (2018).

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von Gernet (1877–1943), Ljubov Nikolaevna Zapolskaya (Sapolsky, Sapolski, 1871– 1943), Vera Miller-Lebedeva (1880–1970), Helene v. Bortkevitsch (1870–1939),24 Alexandrine von Stebnitzky (1868–1928),25 Tatyana Afanasyeva (-Ehrenfest, 1876– 1964), Miss Potylizyn,26 and Maria Wassilievna Joukovsky.27 Alexandrine von Stebnitzky, Helene v. Bortkevitsch, and Ljubov Nikolaevna Zapolskaya were members of the St. Petersburg Mathematical Society. We should note that the most famous woman who went abroad from the Kingdom of Poland to study was the double Nobel laureate Marie Curie.28 In 1914, the Great War broke out. Stanisława was living in the Russian Empire, which was a member of Triple Entente. Otton was living in Kraków in the AustroHungarian Empire, which was a member of the Central Powers. At the beginning of the war, the Western Front was rather stable, but the war continued in East Europe. By November of 1914, the Russian army invaded Galicia and East Prussia. In May of 1915, the Central Powers achieved a remarkable breakthrough on the Galician southern frontier. By mid-1915, the Russians had been expelled from central Poland and Galicia. On August 5, 1915, the German army captured Warsaw. Before that, the Russian Imperial University of Warsaw had been evacuated to Rostov-on-Done.29 Central Poland was under German occupation. Regarding the story of the University of Warsaw in those times, let us quote from its official webpage: The German authorities gave their permission for the creation of the University of Warsaw with the Polish language as the language of instruction. The solemn inauguration of the University took place on November 15, 1915 in the presence of the German governor, general Hans Hartwig von Beseler. […] Also women were admitted to study at the University for the first time.30

In 1916, Stanisława started studying mathematics in Warsaw (for the first page of her transcripts, see Fig. 6.2). Who were the staff members at the university? In 1915, Stefan Mazurkiewicz (1888–1945) was appointed to the chair of mathematics at the University of Warsaw. He was born and educated in Warsaw but he passed his matura in Kraków, after which he studied in Munich, Göttingen, and Lvov, where he earned 24 Her Polish name was Helena Bortkiewicz, and she was a sister of Władysław Bortkiewicz (Ladislaus von Bortkewitsch, 1868–1931), a professor of statistics in Berlin. 25 In Polish: Aleksandra Stebnicka. She was a daughter of the Russian general and scientist Hieronim Stebnicki (1832–1897) and a sister of Olga, who was the mother of the Nobel laureate in physics Peter Kapitsa (1894–1984). Alexandrine worked at the Pulkovo Astronomical Observatory. 26 It is possible that she was a daughter of Alexander Potylyzyn (1845–1905), a professor of chemistry at the Imperial University of Warsaw. 27 For further details, see Tobies’ chapter in this book, Tobies (2020). 28 During her time studying at the Sorbonne, she was known as Maria Skłodowska. Maria graduated in mathematics and physics from the Sorbonne. Some years earlier, her sister Bronisława Dłuska (1865–1939) had graduated in medicine from the same university. 29 The evacuation meant that almost all professors, books, and equipment had to be transported to Rostov-on-Done. From 1915 to 1917, the university was called the Imperial Warsaw University in Rostov-on-Don, and the journals published by the university were printed under that name. 30 “Facts and Figures,” University of Warsaw: www.mianowski.waw.pl/foundation/history/?lang=en (accessed September 12, 2017).

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Fig. 6.2 Stanisława Liliental: photograph, signature, transcript. Archive of the University of Warsaw (AUW RP 1583)

a PhD in mathematics in 1913. Some years later, Zygmunt Janiszewski (1888–1920) was appointed to the Chair of mathematics at the university. Janiszewski was born in Warsaw. He passed his matura in Lvov and studied in Zurich, Göttingen, and Paris, where he obtained a PhD in mathematics in 1912. After WWI, Wacław Sierpi´nski joined the university staff. Likewise born and educated in Warsaw, he graduated from the Russian Imperial University of Warsaw in 1904 and then moved to Kraków, where he obtained a PhD in mathematics from the Jagiellonian University. It is significant that these three men were born in Warsaw (that is, in the part of Poland ruled by Russians) and went to Polish schools or universities in Galicia for further education. In 1918, Poland reemerged in Europe. Formation of the new state was held in 1917 and 1918. In 1917, Kasa im. Mianowskiego (The Mianowski Fund) sent a questionnaire to Polish scholars about the needs of Polish science.31 Among others,

31 Kasa im. Mianowskiego (The Mianowski Fund) was an organization of Polish scientists and intellectuals. The organization had been operating since 1881 in the Kingdom of Poland and its main goal was to support Polish scholars active in the humanities and sciences. In the mathematical

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Zaremba,32 Mazurkiewicz, and Janiszewski responded.33 In his reply, Janiszewski proposed an idea for creating a mathematical school in Poland. His response is well known.34 Mazurkiewicz’s response was not so revolutionary; he just pointed out some specific needs, among them a mathematical library,35 a system of grants for opportunities to study abroad, and more positions for educated mathematicians at Polish universities. Let us now return to Stanisława and her studies at the University of Warsaw. The mathematicians who taught Stanisława were Samuel Dickstein (1851–1939), Wacław Sierpi´nski, Kazimierz Kuratowski (1896–1980), Stefan Kwietniewski (1874–1940), Stanisław Le´sniewski (1886–1939), Antoni Przeborski (1871–1941), Witold Pogorzelski (1895–1963), Jan Łukasiewicz, and Juliusz Rudnicki (1881– 1848). This is an impressive list. There is no doubt that Janiszewski, Mazurkiewicz, Kuratowski, and Sierpi´nski had a huge impact on Stanisława’s mathematical education. In 1919, she attended two courses by Janiszewski: selected topics in topology and a seminar on topology. It must have been Janiszewski who introduced her to the problem of continua disconnecting a plane, the problem of her doctoral dissertation. In 1921, Mazurkiewicz lectured on Jordanian continua, which was also one of her interests. It is interesting that Sierpi´nski, Le´sniewski, and Łukasiewicz taught Otton in Lvov and Stanisława in Warsaw. This is part of the history of the reemergence of Polish universities and the scientific environment in Poland. During her studies, Stanisława, who was also artistically talented, started her career as a watercolor painter (see Figs. 6.8, 6.9), writer, and poet.36 Most of her poems and dramas have not yet been published.37 In 1918, Stanisława postponed her studies for a while and volunteered to serve the Polish army, for which she taught illiterate Polish soldiers how to read and write.38 sciences, over 50 titles were sponsored by The Mianowski Fund, including the well-known periodical Prace Matematyczno-Fizyczne. For more information about Kasa Mianowskiego, see Hübner et al. (2017). 32 Stanisław Zaremba (1863–1942) graduated from the Saint Petersburg State Institute of Technology and the Sorbonne, where he received a PhD in mathematics. He was a professor at the Jagiellonian University in Kraków. 33 All the replies were published in the newly founded journal entitled Nauka Polska: Jej potrzeby, organizacja i rozwój [“Polish Science: Its Needs, Organization, and Development”]. 34 Janiszewski (1919). See also Duda (2013): “[…] in 1918 he published a program which can be summarized in a few points: concentrating all active mathematicians in the country in one area of mathematics, presumably a new one (where there is no long tradition to learn nor needed are extensive libraries); founding a journal to support the group (the journal should be devoted specifically to the chosen area and should publish only in internationally recognized languages); working atmosphere was to be that of cooperation and common assistance.” 35 Mazurkiewicz (1919). After WWII, the Mathematical Institute of the Polish Academy of Sciences was created in Warsaw. Now the “Central Mathematical Library” operates there, just as Mazurkiewicz hoped it once would. 36 On the collections of watercolor paintings held in Polish archives, see Wódz (2014). 37 A poem titled “Jak ro´slina wyrwana z korzeni” [“Like an Uprooted Plant”] and its English translation were published in Ciesielska (2017). 38 In the Kingdom of Poland, Russian was a language of school instruction.

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Fig. 6.3 The bench with the sculpture of Otton Nikodym and Stefan Banach (sculpture by Stefan Dousa, photograph by D. Ciesielska and K. Ciesielski)

Fig. 6.4 First Congress of Mathematicians of Slavic Countries, Stanisława (in a hat, bottom row, Otton behind her), Mathematical Archive of the IM PAS Warsaw

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Fig. 6.5 The Nikodyms’ apartment in Kenyon. From left to right: Otton Nikodym, Stanisława Nikodym, William R. Transue, Dan Silverman, unknown. Courtesy of Virginia Transue

6.3.3 Otton’s Life up to 1924 After his studies, Otton Nikodym lived in Kraków until 1924. He was a teacher of mathematics and physics at a gymnasium. In 1911, Nikodym became a teacher at the gymnasium from which Stefan Banach (1892–1945) had graduated in 1910. One of his colleagues at the school was Leja, his friend from Lvov. In fact, this period in Kraków was superb for young people interested in mathematics. Among them were Banach, Witold Wilkosz (1891–1941), Franciszek Leja, Alfred Rosenblatt,39 and Antoni Hoborski.40 These young and well-educated men used to meet and talk about mathematics and physics. Banach, Wilkosz, and Nikodym had a habit of walking together in the evening and discussing mathematics. Wilkosz attended some mathematical courses in Turin and passed some exams, but WWI put an end to his studies and he had to return to Kraków. Banach, who was interested in mathematics but considered it to be a nearly complete science to which very little could be added,

39 Alfred Rosenblatt (1880–1947) held a PhD in mathematics from the Jagiellonian University and had also studied in Vienna and Göttingen. He was a professor at the Jagiellonian University in Kraków and at St. Marco University in Lima, Peru. 40 Antoni Hoborski (1879–1940) held a PhD in mathematics from the Jagiellonian University and had also studied in Paris and Göttingen. He was a professor of mathematics at the Jagiellonian University and at the Mining Academy in Kraków, where he served as the first rector.

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Fig. 6.6 Stanisława Nikodym, act of naturalization in the United States, PIASA Archives

decided to study engineering in Lvov.41 At the outbreak of WWI, Banach returned to Kraków. He attended some mathematical lectures at the Jagiellonian University and enriched his mathematical knowledge with independent studies. Inspired by Wilkosz, he was interested in problems that later led him to the idea of the abstract functional space.42 Nikodym and Leja were school teachers who occasionally gave lectures at the Jagiellonian University. Sometimes accidental events have a huge impact on history. One such event took place in 1916 in Kraków. During an evening stroll, the 29-year-old mathematician Hugo Steinhaus made a certain “mathematical discovery.”43 Steinhaus later became a famous mathematician, but then he was just a well-educated independent scholar. During his evening walk in the Planty Gardens in the center of Kraków, Steinhaus overheard the words “Lebesgue integral.” At the time, this was a recent idea known 41 There was no technical university in Kraków. The Technical University in Lvov was the nearest one. 42 Banach (1922) wrote: “M. Wilkosz et moi, nous avons certain résultats (que nous nous proponous publier plus tard) sur les opérations dont les domaines sont des esambles de fonctions duhameliennes, c’este-à-dire, qui sont les dérivées leurs fonctions primitives.” 43 Hugo Steinhaus (1887–1972) earned a PhD in mathematics under Hilbert in Göttingen. A professor of mathematics at the University of Lvov and at Wrocław University, he was the founder and one of the leaders of the “Lvov School of Mathematics.”

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Fig. 6.7 Otton’s grave. The mosaic was designed by Stanisława (photograph by Janusz Łysko)

almost exclusively to specialists. Steinhaus was intrigued. He joined the conversation between two young men, who turned out to be Stefan Banach and Otton Nikodym.44 Later, Steinhaus, an author of many important papers, used to say that his best mathematical discovery was his “discovery” of Stefan Banach. Yet there is a flip side to this story: Otton Nikodym remained undiscovered for the next 10 years! Otton was a brilliant school teacher. Among his students were mathematicians and physicists, two of whom deserve special mention: Marian Mi˛esowicz45 and Stanisław Krystyn Zaremba.46 About Otton, Mi˛esowicz recalled: “He was able to 44 On October 14, 2016 in Kraków, a small monument commemorating this event was unveiled. See Fig. 6.3 and, for further information, Ciesielska & Ciesielski (2017). 45 Marian Mi˛ esowicz (1907–1992) earned a PhD in physics from the Jagiellonian University and went on to become a professor at the Mining Academy in Kraków and a member of the Polish parliament. 46 The son of Stanisław Zaremba, Stanisław Krystyn Zaremba (1903–1990) was a Polish, Canadian, and British mathematician and climber. He held a PhD in mathematics from the University of Vilnius.

6 Stanisława and Otton Nikodym Fig. 6.8 S. Liliental: “Town Hall in Sandomierz” (1933). Photograph by Marek Banaczek, Regional Museum in Sandomierz, inv. no. MS-433/s

Fig. 6.9 S. Nikodym: “Orchard in the Sandomierz Area” (1979). Photograph by Marek Banaczek, Regional Museum in Sandomierz, inv. no. MS-441/s

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evoke the students’ appreciation, admiration, and enthusiasm for his precision and elegance of expressing physical laws in a strict mathematical form.”47 Mi˛esowicz used to cite Otton’s opinion about the role of calculus in mathematics: “Calculus is for mathematics what Beethoven’s Ninth Symphony is for music.” Let us also cite the opinion of E. Tarnawski, another of Nikodym’s students: Otton Nikodym sticks in my memory as a rather uncommon personality. […] I see him as slim and dark-haired with a beard. He was in his late twenties, but different from others of his age; almost impersonal as if unchanged with age, isolated and distant […] aesthetically neutral, physically flimsy, not raising his voice but always unimpassioned, however audibly. […] His lectures were interesting because of their content. […] He presented science as it was, without incorporating his own personality, which disappeared from view.48

Nikodym was very interested in the theory of education. He lectured on the foundations of teaching mathematics and physics (referred to today as the didactics of mathematics and physics). His first paper, “O sposobie ponumerowania wszystkich ułamków” [“On the Methods of Numbering All Fractions”],49 was a very short essay published in 1912 in the Polish journal Wektor, which sought to popularize mathematics. Until 1925, this was Otton’s only publication.

6.4 Common Life and Work During the Interwar Period Stanisława spent the summer holiday of 1923 at a mountain resort in Zakopane. There she met many Polish mathematicians, among them Otton Nikodym. On July 30, 1923, she wrote the following to her mother: “Yesterday I met professor Ruziewicz50 from Lvov. I spent the entire evening enjoying sweets and a lively discussion in the company of professors of mathematics.”51 The back of the postcard also contains Before WWII, he worked at Vilnius and at the Jagiellonian University, and after the war he worked in Great Britain, Canada, and the United States. 47 Quoted from Maligranda (2003). For the original Polish, see Mi˛ esowicz (1980) and Derkowska (1983): “Umiał wzbudza´c u uczniów uznanie, a u niektórych zachwyt i zapał, do doskonało´sci i elegancji wyra˙zania praw fizyki w s´cisłej matematycznej formie.” 48 Quoted from Maligranda (2003). For the original Polish, see Derkowska (1983): “Otto Nikodym tkwi w mej s´wiadomo´sci jako do´sc´ niezwykła indywidualno´sc´ […]. W pami˛eci widz˛e go jako szczupłego, ciemnego bruneta z brod˛a. Był dopiero dwudziestoparoletni, ale ró˙zny od innych w tym wieku. Wyobcowany i daleki. […] Ascetycznie obj˛etny, o w˛atłej strukturze fizycznej, nie podnosił głosu, ale mówił stale głosem beznami˛etnym, niemniej wyra´znym […]. Wykład był interesuj˛acy przez konsekwencj˛e uporz˛adkowania tre´sci […]. Otton Nikodym dawał prezentacj˛e wiedzy, tak˛a jaka ona jest, bez wł˛aczania własnej osobowo´sci.” 49 Nikodym, O. (1912). 50 Stanisław Ruziewicz (1889–1941) held a PhD in mathematics from the University of Lvov and had also studied in Göttingen. He was a professor at the University of Lvov and at the Academy of Foreign Trade in Lvov, where he also served as rector. 51 Quoted from the Polish Institute of Arts and Sciences—Stanisława and Otton Nikodym Papers: “Wczoraj poznałam jeszcze prof. Ruziewicza ze Lwowa. Cały wieczór sp˛edziłam w towarzystwie prof. matematyki w cukierni w´sród b. o˙zywionych dyskusji.”

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the following note: “Pozdrowienia dla Antosia od Prof. Nikodyma” [“Greetings to Anto´s52 from Prof. Nikodym”]. On April 2, 1924, Stanisława and Otton got married in Warsaw.53 It is commonly believed that it was Sierpi´nski who encouraged Otton to start publishing his results, but I am certain that is not the truth. The correspondence of the newly married couple with Stanisław’s family presents the story in the different way. In my opinion, Stanisława and some friends from Kraków had far more impact on Otton than Sierpi´nski did. The couple lived in Otton’s small apartment on Kochanowskiego Street 23 in Kraków. Many mathematicians visited them there to discuss mathematical problems. Furthermore, Stanisława had just finished her studies, and she wanted to become a scientist. Otton had worked as a school teacher from 1911, and even though he was an excellent mathematician, he did not publish any of his results then. From April of 1924 to June of 1925, the couple had a very busy time. Stanisława and Otton decided to apply for doctoral degrees at the University of Warsaw. In May of 1924, Regina wrote the following about them in a letter to Natan: “Often they sit together studying algebra, and Otton told me that she is extremely talented—that she has the mind of a man despite being such a feminine and childish person.”54 There is no doubt that Otton and Stanisława supported each other in their mathematical research, but Stanisława was a highly independent mathematician who did her own research on topics different from (though related to) Otton’s. Moreover, it was she who encouraged him to conduct research and—more importantly—persuaded him to publish his mathematical results. He was a husband who supported his wife’s aspirations. Was he influenced by Smoluchowski’s idea of the equality of men and women in scientific work? This is certainly a possibility! The Nikodyms were very special couple with individual careers in mathematics and academic life. It must be kept in mind that the marriages of many Polish mathematicians were quite different in those days. Many wives of Polish mathematicians supported their husbands in their work; some of the wives worked as scientific secretaries for their husbands, while others edited their papers. Some of those who had a job served as teachers, without a degree. One of the oddest stories concerns an unusual method used by a mathematician from Lvov to choose his wife. In the late 1920s, Juliusz Schauder55 decided to get married, as his friend Henry Schaefer recalled: His [Schauder’s] approach to important matters of life was unusual. Once, when we were taking a walk in a park, he told me: “I am now approaching the age of thirty. This is a serious age; it is time for me to marry and to settle down. I would like to marry a girl studying mathematics, preferably one who knows some foreign languages, so that she can help me to prepare my results for publication. Among your classmates, I like the looks of only two 52 “Anto´s”

is a nickname for Antoni, Stanisława’s brother. must have been at this point when Stanisława converted to Catholicism. 54 Quoted from the Polish Institute of Arts and Sciences—Stanisława and Otton Nikodym Papers: “Nieraz siedz˛a razem nad algiebr˛a i powiedział mi Otton, z˙ e jest ona ogromnie zdolna, ma m˛ezki (sic!) umysł przy tak wybitnej kobieco´sci i dzieci˛eco´sci.” 55 Juliusz Paweł Schauder (1899–1943) held a PhD in mathematics from the University in Lvov and had also studied in Leipzig and Paris as a fellow of the Rockefeller Fundation. He lectured at Lvov Univeristy and was a professor at the Ukrainian University in Lviv. 53 It

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girls: Miss X and Miss Y, but I have not met either of them. Tell me, which one of two do you think I should marry.” I replied: “Marry Miss X.” This he did, after first confirming my recommendation with S. Banach.56

Stanisława presented a completely different picture in her letters to Regina. On June 7, 1924, she wrote to her mother: “However, I have gained a lot in Kraków. I owe much to Ottonek,57 who is able to see the essence of things and who grasps each part of mathematics very deeply. […] Now I must be on time, and I want to leave you on a positive note without digging up any annoying memories and wrong answers.”58 And on October 26, 1924, she wrote: Ottonek says that we must learn to fight against all the obstacles that fate sends our way and appreciate our victory, when we will receive the necessary diplomas for which we are striving and embark upon a new creative life. Ottonek believes that I will accomplish very nice things, for he sees how I am going about mathematics. Really, I have benefited a lot recently and I owe a great deal to Ottonek. He said that he senses all the happiness coming to us together, including good health and his doctorate. It seems that his scientific success is constantly growing. Sierpi´nski wrote that he communicated one of Ottonek’s results to Professor Alexandrov (in Moscow),59 who really admired it. […] Sierpi´nski wrote that Professor Fréchet, a very well-known French mathematician, asked him for Ottonek’s address, because he wanted to send him an article in which his name is cited many times. Ottonek obtained many results that were later achieved by Professor Fréchet, but he hid them in a drawer and communicated only some of them to Wa˙zewski,60 who cited Otton in an article published in Comptes rendus [des séances de l’Académie des Sciences 176 (1923), pp. 69–70] (Paris scientific journal). In this way, Fréchet was able to base his work on one of Ottoneczek’s61 nice ideas.62 56 Schaefer

(1993). is a nickname for Otton. 58 Quoted from the Polish Institute of Arts and Sciences—Stanisława and Otton Nikodym Papers: “Sporo jednak skorzystałam tu w Krakowie. Du˙zo bardzo zawdzi˛eczam Ottonkowi, który umie przejrze´c istot˛e rzeczy i uj˛ac´ gł˛eboko ka˙zd˛a parti˛e matematyki. […] Teraz to zd˛az˙ y´c musz˛e, a zda´c chc˛e dobrze, z˙ eby znów nie wlec za sob˛a wspomnie´n przykrych i złych odpowiedzi.” 59 Pavel Sergeyevich Alexandrov (1896–1962) was a Soviet mathematician. He was a professor at Moscow University and at the Stieklov Institute in Moscow. 60 Tadeusz Wa˙ zewski (1894–1974) held a PhD in mathematics from the Sorbonne and was a professor of mathematics at the Jagiellonian University. 61 “Ottoneczek” is another nickname for Otton. 62 Quoted from the Polish Institute of Arts and Sciences—Stanisława and Otton Nikodym Papers: “Ottonek mówi, z˙ e te wszystkie przeszkody los nam zsyła, by´smy umieli je zwalcza´c i tem wi˛ecej cenili zwyci˛estwo własne, gdy osi˛agn˛awszy dyplomy potrzebne, do których d˛az˙ ymy, zaczniemy nowe twórcze z˙ ycie. Ottonek wierzy, z˙ e b˛ed˛e robiła b. ładne rzeczy, bo widzi jak mi idzie praca nad matematyk˛a. Istotnie, ostatnio moc skorzystałam, zawdzi˛eczaj˛ac oczywi´scie strasznie du˙zo swemu Ottonkowi, Ottonek mówi, z˙ e czuje i˙z wszystkie rado´sci zbieraj˛a si˛e razem i zdrowie r˛eki i mój i Jego doktorat. Zdaje si˛e z˙ e Ottonka sukcesy naukowe stale rosn˛a. Sierpi´nski pisał, z˙ e zakomunikował jeden rezultat Ottonka profesorowi Aleksandrowowi (w Moskwie), któremu si˛e on b. podobał. […] Prócz tego pisał prof. Sierp., z˙ e prof Fréchet, znany matematyk francuski, zapytywał prof. Sierpi´nskiego o adres Ottonka, bo chce Mu przesła´c swoj˛a prac˛e, w której Jego nazwisko wielokrotnie cytuje. Ottonek miał wiele rezultatów Prof. Fréchet’a wcze´sniej, ale schował je do szuflady, cz˛es´c´ tylko zakomunikował kiedy´s w kawiarni Wa˙zewskiemu, który zacytował Ottona w Comptes Rendus (paryskie pismo naukowe), a Fréchet w ten sposób oparł si˛e nast˛epnie na b. ładnym pomy´sle mojego Ottoneczka.” 57 “Ottonek”

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Stanisława missed her mother dearly, and from the day of her wedding she was trying to move back to Warsaw. Otton wanted to make her wife happy, and so finally, in November of 1924, the couple left Kraków and moved to Regina and Natan’s apartment in Warsaw (Koszykowa Street 53). Stanisława was quite happy living with her family, but the happiness did not last long. On December 4, 1924, her mother died owing to complications after surgery. Their life in Warsaw was not so easy. They continued to work diligently on their theses; Stanisława learned a lot, and Otton was writing articles. Neither of them had a permanent job. On June 26, 1925, Stanisława and Otton received their PhDs from Warsaw University.63 Stanisława’s supervisor was Mazurkiewicz, and Otton’s was Sierpi´nski. In the dissertations she discussed topological problems, he discussed some properties of topological analytic sets. Hers was entitled O rozcinaniu płaszczyzny przez zbiory spójne i kontinua [“On Disconnecting the Plane with Connected Sets and Continua”].64 His thesis was entitled Przyczynek do teoryi zbiorów A [“A Contribution to the Theory of A Sets”].65 Otton’s results in mathematics are very well known and have been described at length,66 while Stanisława’s results are almost forgotten. She was interested in topology. Her main interest was continua, so let us provide a short introduction to continua and to the problem of continua disconnecting the plane. Here, we are following the excellent “History of Continuum Theory” by Janusz Jerzy Charatonik (1934–2004).67 A plane continuum on the Euclidean plane is a compact and connected non-empty set, and in set-theoretical topology it is a non-empty compact connected metric space. In 1913, studying Jordan’s definition of a curve and the problem of sets homeomorphic to the closed interval, Hans Hahn (1879–1934) and Mazurkiewicz independently obtained a characterization of local connectedness.68 Phenomena related to the topology of the plane, in particular those concerning the structure of continua disconnecting the plane, began to be studied very early in the history of topology. One of the earliest results in this area was the Brouwer–Phragmén theorem.69 Related results were obtained by Béla von Kerékjártó (1898–1946), Kuratowski, Robert Lee Moore (1882–1974), Gordon Thomas Whyburn (1904–1969), and Raymond Louis Wilder (1896–1982). Disconnecting the plane by continua was Janiszewski’s subject of interest. In 1913, he proved two theorems in his habilitation thesis, which are now referred to as Janiszewski’s first and second theorems. The theorems state: 63 See

Ksi˛ega dyplomów doktorskich and Piotrowski (2012). as Nikodym, S. (1925a). 65 Published as Nikodym, O. (1925b). 66 See Derkowska (1983); Maligranda (2003); and Szyma´ nski (1990). 67 Charatonik (1987). 68 Recall that a space is said to be locally connected if each of its points has a neighborhood basis consisting of open connected sets. 69 In 1885, Lars Edvard Phragmén (1881–1966) proved that the boundary of an open bounded subset of the plane contains a nondegenerate continuum. In 1910, L. E. J. Brouwer (1863–1937) showed that the boundary of each bounded component of the complement of each continuum K in the plane is itself a continuum (a generalization to any n -dimensional space was due to P. S. Alexandrov). 64 Published

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Theorem 1 (Janiszewski’s First Theorem) If the intersection of two planar continua, neither of which disconnects the plane, is connected, then their union also does not disconnect the plane. After 1913, the term “Janiszewski space” has been used to designate a locally connected continuum X having the property that, for every two of its subcontinua A and B with a non-connected intersection, there exist two points in X that are separated by the union A ∪ B. Thanks to this definition, Janiszewski’s Second Theorem may be formulated as follows: Theorem 2 (Janiszewski’s Second Theorem) A two-dimensional sphere is a Janiszewski space. In her first paper, “Sur les coupures du plan faites par les ensembles connexes et les continus,”70 Stanisława stated a very interesting theorem: Theorem 3 If C is a bounded continuum, S is connected, and S and C disconnect the plane, then the following conditions are equivalent: (1) S ∪ C cuts the plane between external point of (\S) ∪ C, (2) S ∩ C is disconnected. In the last paragraph of the paper, Stanisława presented examples showing that both assumptions of the theorem are necessary. This paper was cited in the extensive historical study on history of continuum theory by Charatonik. He wrote: “Further generalizations and modifications of these two theorems of Janiszewski are contained in [195], [313], [355], [395], [528], [537], [624] and [625]. A strengthening of these theorems was proved in the middle of the forties by R. H. Bing (1914–1986) in [57] and [58].”71 For the academic year 1925/26, Stanisława and Otton received a government grant to study at the Sorbonne. The next mathematical problem came to Stanisława during a visit by an American mathematician. In 1926, John Robert Kline (1891–1955) visited the Jagiellonian University and posed the problem of finding a “necessary and sufficient condition for a proper subcontinuum C of a Jordanian continuum J to be Jordanian. This condition should use only the set J/C.” In a paper from 1928 entitled “Sur une condition necessaire et suisante pour qu’un sous-continu d’un continu jordanien et plan soit lui-meme jordanien,”72 Stanisława stated this condition as follows: Theorem 4 A proper subcontinuum C of the Jordanian continuum J is Jordanian if any point of the set C ∩ J \C is accessible by a proper arc from J\C. In the same year, she published a paper entitled “Sur quelques propriétés des ensembles partout localement connexes,”73 in which she offered a generalization of some results by Kuratowski and Knaster. Here she stated: 70 Nikodym,

S. (1925). (1987). The papers cited here are Eilenberg (1935); Jones (1955); Knaster and Kuratowski (1924); Mullikin (1922); Kuratowski and Straszewicz (1928); Nikodym, S. (1925a); Straszewicz (1923); and Straszewicz (1925). 72 Nikodym, S. (1928a). 73 Nikodym, S. (1928b). 71 Charatonik

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Theorem 5 If the sum and intersection of two closed sets A and B are Jordanian continua, then so are A and B. In 1927, Otton received his habilitation degree from the University of Warsaw and became a “private docent” there. From 1928 to 1931, Otton lectured on mathematics at the Jagiellonian University and he spent some time working in the private mathematical library owned by Samuel Dickstein.74 Stanisława and Otton took part in many important conferences in Poland and abroad. They were delegates at the First Congress of the Polish Mathematical Society (Lvov, September 1–10, 1927). He presented a talk on mathematical education, while she did not present any paper. In 1928, the International Congress of Mathematicians took place in Bologna. Stanisława and Otton were delegates at the Congress, representing Kraków, and both of them presented lectures. Stanisława presented her earlier results about the topological properties of a plane (“Sur une propriété topologique du plan euclidien”).75 In his talk entitled “Sur le fondement des raisonnements locaux de l’analyse classique,” Otton proposed a new idea of local argumentation in classical analysis.76 In 1929, Stanisława and Otton participated in the First Congress of Mathematicians of Slavic Countries in Warsaw (for a photograph of the participants, see Fig. 6.4). Here in Warsaw, Otton Nikodym posed the following research problem: Is it possible to decompose an open disc into open pairwise disjoint arcs? In two later papers, Stanisława provided a positive answer to this problem.77 In 1935, they took part in the Congress of Applied Mathematics in Cluj in Romania and delivered talks there. Stanisława presented the paper “Une propriété topologique de la surface partout localement homéomorphe au plan euclidien,”78 and Otton’s was entitled “Sur le principe du minimum.”79 Before 1936, Stanisława became an assistant to Franciszek Leja, then a professor of mathematics at the Faculty of Chemistry in Warsaw University of Technology (from 1924 to 1936). He wrote in his memoires: The Chair of Mathematics, which I held at the Department of Chemistry, had one assistantship. It was occupied by undergraduates of mathematics from the University of Warsaw: firstly by Kazimierz Zarankiewicz80 for about 8 years, secondly by Stanisława Nikodymowa, the wife of my friend from our studies at the University in Lvov, Otton Nikodym, who later became a docent and a professor of mathematics at universities in Poland and abroad in the USA.81 74 Samuel Dickstein (1851–1939), a Polish mathematician, graduated from the Imperial Warsaw University and was a professor at the University of Warsaw. 75 Nikodym, S. (1930c). 76 See Nikodym, O. (1930a). 77 Nikodym, S. (1930a, 1930b). 78 Nikodym, S. (1935a). 79 Nikodym, O. (1935b). 80 Kazimierz Zarankiewicz (1902–1959) was a professor of mathematics and mechanics at the Technical University of Warsaw. 81 Leja (1979): “Katedra matematyki, któr˛ a kierowałem na Wydziale Chemii, miała jeden etat asystenta. Etat ten zajmowali absolwenci matematyki Uniwersytetu Warszawskiego: najpierw

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In 1936, Leja moved to Kraków,82 where he became the chair of mathematics at the Jagiellonian University. As a result of Leja’s decision, Stanisława lost her job. Until WWII, the couple stayed in Warsaw. She and her husband had no permanent job, though she was probably working for a publishing house and he lectured at the University of Warsaw. By 1939, she had published seven papers83 and two books,84 while he had published 33 papers and 4 books (one of the books was a joint work by the couple).85 She participated in an important project led by Maria Loria,86 the purpose of which was to make a list of all scientific papers published by Polish women published up to 1929. Stanisława prepared the mathematical section of project.87 Most of the mathematical papers cited in the book are Stanisława’s results, the others being textbooks and papers on logic. During WWII, they stayed in Warsaw, even though a Jewish–Catholic couple could not have felt safe in a city occupied by Germans. Derkowska wrote about the help they received: “At the beginning of the occupation period, a young man [Aleksander Gorczy´nski] risked his life and stole Nikodym’s files from the City Hall.”88 Stanisława and Otton lost their belongings, among them unpublished results in mathematics, in the Warsaw Uprising (August 1–October 2, 1944).

6.5 Emigration to the United States After WWII, Otton was appointed to the newly created Gliwice Technical University, which was located in Kraków. From April 1, 1945 to December 31, 1946, he was a professor at the Mining Academy in Kraków, where he held the chair at the Faculty of Transportation.89 . In the late autumn of 1946, the couple went to Belgium, where Otton participated in a congress on applied mathematics. Before the departure a Kazimierz Zarankiewicz przez około 8 lat, a nast˛epnie Stanisława Nikodymowa, z˙ ona mojego kolegi jeszcze z Uniwersytetu Lwowskiego Ottona Nikodyma, pó´zniejszego docenta i profesora matematyki Uniwersytetów w Polsce i za granic˛a w U.S.A.” 82 This was the result of a scandal. Two professors at the Polytechnic were running a chemical factory in Silesia. This factory created tremendous pollution, and a local community sent a letter to Franciszek Leja, then a dean of the faculty at the Polytechnic, but the letter was stolen from the post office. Some days later, Leja learned the whole story and he wanted to fire them. Leja was from Galicia, whereas these professors and most of the senate members of the Polytechnic were from the Russian part of Poland, and they decided to cover the problem up. Leja left his position in Warsaw and moved to Kraków. 83 Nikodym, S. (1925a, 1928a, b, 1930a, b, c, 1935a). 84 Nikodym, S. (1934, 1936). 85 Nikodym & Nikodym (1936). 86 Maria Loria (?–1937) was a doctor of medicine and a bacteriologist. She studied in Vienna, Berlin, and in the United States. For details, see Dadej (2019). 87 Nikodym, S. (1934). 88 Derkowska (1983): “Na pocz˛ atku okupacji, z nara˙zeniem z˙ ycia w tajemnicy przed profesorem młody człowiek wykradł z ratusza kart˛e Nikodyma.” 89 See Tutajewska (2017).

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reprint of Stanisława’s problem book appeared,90 and the couple prepared two books for publishing.91 In 1948, the Nikodyms arrived in the United States. Otton and Stanisława found positions at Kenyon College in Gambier, Ohio (for a photograph of the Nikodyms at Kenyon, see Fig. 6.5). The local academic journal announced her hiring as follows: Dr. Stanislawa Nikodym as visiting Assistant Professor of Mathematics. Dr. Nikodym was born in Moscow and studied at the Gymnasium of Latin in Warsaw and the University of Warsaw where she was granted her Doctor’s degree in Mathematics. She was an instructor in Higher Mathematics for six years at the Higher Polytechnical School in Warsaw and has delivered lectures on topology to the International Congress of Mathematicians at Bologne and Turna-Sevrin. She is the author of numerous textbooks and papers, printed in Polish, Italian, and French.92

In a note about Otton’s election to the Academy of Sciences we read: “Professor Nikodym, a native of Poland, came to Kenyon in 1948. He holds the doctor’s degree from the University of Warsaw and is a member of the Mathematical Societies of Poland, France, Belgium. He recently lectured at the Institute for Advanced Studies at Princeton.”93 In 1954, the International Congress of Mathematicians took place in Cambridge, Massachusetts. Otton was a delegate, but Stanisława was not on the list. During her time at Kenyon, Stanisława published only two papers, both jointly with Otton.94 In the first of the two, “Some Theorems on Divisibility of Infinite Cardinals,” it is mentioned that the authors’ research had been supported by the “Office of Ordnance Research, U.S. Army.” Stanisława’s last academic article appeared in 1957.95 In it, she addressed some problems concerning the properties of cardinal numbers in field theory. In the referee’s report published in the Zentralbatt für Mathematik, Ðuro Kurepa (1907–1993) wrote: “The authors prove several more or less known statements concerning ordinal and cardinal numbers, using a particular terminology suitable for field theory.”96 It is possible that this unfavorable opinion discouraged Stanisława from doing future work. The couple traveled often, and The Kenyon Collegian reported the following about their trip to Europe in 1959: Prof. Nikodym spoke on some results of his recent research in modern abstract mathematics and their application to theoretical quantum-physics. At the Belgium Mathematical Center, his topic was a canonical representation of maximal normal operators in Hilbert-Hermite space. Before the Belgian Mathematical Society, of which he is an honorary member, he spoke on Dirac’s Delta-function. In Paris, Prof. Nikodym was welcomed by the president of the French Academy of Sciences on the proposal of Prof. Louis de Broghe (!), Nobel 90 Nikodymowa

(1946). & Nikodymowa (1947, 1948). 92 “Prexy Announces Appointments, New Promotions” (1948). 93 “Prof. Nikodym Elected To Science Academy” (1949). 94 Nikodym & Nikodym (1955, 1957). 95 Nikodym &Nikodym (1957). 96 Kurepa (1957). 91 Nikodym

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prizewinner in physics. His lectures in that city were delivered at the Institut Henri Poincare and at a mathematical seminar at the Sorbonne. The seminar was under the direction of Prof. P. Lelong, whom President De Gaulle has named scientific attaché. In West Germany, Prof. Nikodym spoke at the Mathematical Institute of the University of Heidelberg.97

His lectures on measure theory in 1965 at the University of Naples (Italy) are some of his best remembered. The Nikodyms stayed in Gambier until their retirement in 1966, when they moved to Utica, New York. After WWII, he wrote more than 50 research papers and lectured at many American and European universities. Stanisława, as already mentioned, published only two. In 1966, Otton published an extensive study (almost a thousand pages) on the mathematical apparatus of quantum theory.98 He dedicated this work to Stanisława: “[…] to my wife Dr. Stanisława Nikodym.” It is possible that his wife supported him in his work. Otton died on May 4, 1974 in New Hartford (near Utica). He was buried in the “Polish cemetery for the meritorious” in Doylestown, Pennsylvania (see Fig. 6.7).99 Stanisława and Otton became citizens of the United States (for Stanisława see Fig. 6.6). After Otton’s death, Stanisława visited Poland in the 1970s and 1980s. She died on March 26, 1988 in Je˙zewo Stare (Podlachia, Poland) and was burried in Tykocin.100

6.6 Conclusion Otton Martin Nikodym is far better known abroad than in Poland. I hope that the unveiling of the bench with the statue of him and Banach in Kraków in 2016 will change this and that Otton will become just as recognizable in Poland as his famous interlocutor. Moreover, I hope that Stanisława will become better known both in Poland and abroad.101 97 “Professor

Nikodym Lectures in Europe” (1959).

98 This book, The Mathematical Apparatus for Quantum-Theories: Based on the Theory of Boolean

Lattices, was published by the Springer Publishing House in 1966. It was intended to be the first of two volumes, but the second volume remains unpublished. 99 For more information about Otton Nikodym’s life and work, see Derkowska (1983); Maligranda (2003); Piotrowski (2014); and Szyma´nski (1990). 100 Stanisława Nikodym Death Certificate. 101 The author wishes to thank Witold Liliental (Canada), Dorota Liliental (Warsaw), Krystyna Kuperberg (Auburn University, USA), Dominik Abłamowicz and Bo˙zena Ewa Wódz (Regional Museum in Sandomierz, Poland), Anna Olszewska (Registry Office, Tykocin, Poland), Iwona Dadej (Freie Universität, Berlin), Galina Sinkievich (Saint-Petersburg State University of Architecture and Civil Engineering), Dominik Woł˛acewicz (PIASA, New York), Piotr Gr˛acikowski (University of Wrocław, Poland), Lech Maligranda (Luleå Technical University, Sweden), Walerian Piotrowski (Institute of Cardiology, Warsaw), Yaroslav Prytula (Ivan Franko University in Lviv), Edward Tutaj (Jagiellonian University in Kraków), Janusz Łysko (USA), John Greczek (USA), Virginia Transue (USA), and Mirosława Gołub (Tykocin, Poland).

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Bibliography Archival Sources Archive of the University of Warsaw: “Stanisława Liliental.” AUW RP 1583. Central Mathematical Library, Mathematical Archive, Mathematical Institute PAS. Ksi˛ega dyplomów doktorskich [Record of Doctoral Diplomas]. Archive of the University of Warsaw: “Stanisława Nikodym” (AUW KDD/WF–89), “Otton Nikodym” (AUW KDD/WF–90). Stanisław Nikodym Death Certificate, Registry Office in Tykocin, sygn. 2002123/00/AZ/1988//416854. The Polish Institute of Arts and Sciences: Stanisława and Otton Nikodym Papers. PIASA Archives, Fond 77, sygn. 1, 2, 21, 22.

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Kurepa, D. (1957). Reviewer’s report on the paper, “Some Theorems on Divisibility of Infinite Cardinals,” by O. M. Nikodym and S. Nikodym. Report no. Zbl 0086.04401. Liliental, W. (2015). Anto´s Liliental Who Died in Katy´n. Jewish Historical Institute (January 12). Retrieved October 30, 2016, from http://www.jhi.pl/en/blog/2015-01-12-antos-liliental-whodied-in-katyn. ´ eta z˙ ydowskie w przeszło´sci i tera´zniejszo´sci. Rozprawy Lilientalowa, Regina. (1908–1919). Swi˛ wydziału filologicznego Polskiej Akademii Umiej˛etno´sci w Krakowie, 45(1908), 52 (1914), 58 (1919). Lilientalowa, Regina. (1927). Dziecko z˙ydowskie: Prace Komisji etnograficznej. Kraków: Polska Akademia Umiej˛etno´sci. Leja, Franciszek. (1979). Dawniej było inaczej. Unpublished manuscript. Kraków: Mathematics Institute, Jagiellonian University. Mazurkiewicz, Stefan. (1919). O potrzebach matematyki w Polsce. Nauka Polska: Jej potrzeby, organizacja i rozwój, 2, 1–3. Maligranda, Lech. (2003). Otton Marcin Nikodym. MacTutor. Retrieved January 10, 2017, from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Nikodym.html. “Members of the St. Petersburg Mathematical Society in the Period 1890–1899”. Retrieved September 12, 2017, from http://www.mathsoc.spb.ru/history/mem1890-1899.html. Mi˛esowicz, Marian. (1980). Mój nauczyciel, wspomnienie Profesora Nikodyma. Matematyka, 3, 182–183. Mullikin, Anna M. (1922). Certain theorems relating to plane connected point sets. Transactions of the American Mathematical Society, 24(1922), 144–162. Nikodym, Otton. (1912). O sposobie ponumerowania wszystkich ułamków. Wektor, 2, 14–17. Nikodym, Otton. (1925a). Sur une propriété de l’opération A. Fundamenta Mathematicae, 7, 149– 154. Nikodym, Otton. (1930a). Sur le fondement des raisonnements locaux de l’analyse classique. Atti del Congresso Inter. dei Mathem. Bologna, 3, 215–222. Nikodym, Otton. (1930b). Sur une généralisation des intégrales de M. J. Radon. Fundamenta Mathematicae, 15, 131–179. Nikodym, Otton. (1935a). Sur le principe du minimum. Mathematica, Cluj, 9, 110–128. Nikodym, Otton, & Nikodym, Stanisława. (1936). Wst˛ep do rachunku ró˙zniczkowego: dla u˙zytku samouków, absolwentów szkół s´rednich oraz słuchaczy elementarnego kursu matematyki wy˙zszej. Warszawa: Nasza Ksi˛egarnia. Nikodym, Otton & Nikodymowa, Stanisława. (1947). Wst˛ep do rachunku ró˙zniczkowego. Pozna´n: Ksi˛egarnia W. Wilaka. Nikodym, Otton, & Nikodym, Stanisława (1955). Sur l’extension des corps algébriques abstraits par un procédé généralisé de Cantor. Atti Accad Naz Lincei VIII Ser Rend Cl Sci Fis Mat Nat, 17, 334–339. Nikodym, Otton, & Nikodym, Stanisława. (1957). Some theorems on divisibility of infinite cardinals. Archiv der Matematik, 8, 96–103. Nikodym, Stanisława. (1925b). Sur les coupures du plan faites par les ensembles connexes et les continus. Fundamenta Mathematicae, 7, 15–23. Nikodym, Stanisława. (1928a). Sur une condition nécessaire et suffisante pour qu’un souscontinu jordanien et plan soit lui-meme jordanien. Fundamenta Mathematicae, 12, 160–187. Nikodym, Stanisława. (1928b). Sur quelques propriétés des ensembles partout localement connexes. Fundamenta Mathematicae, 12, 240–243. Nikodym, Stanisława. (1930c). Sur la décomposition du cercle ouvert en arcs simples ouverts II. Fundamenta Mathematicae, 16, 7–16. Nikodym, Stanisława. (1930d). Sur la décomposition du cercle ouvert en arcs simples ouverts. Fundamenta Mathematicae, 15, 263–270. Nikodym, Stanisława. (1930e). Sur une propriété topologique du plan euclidien. Atti Congresso Bologna, 2, 229–235.

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Nikodym, Stanisława. (1934). Matematyka. In M. Loriowa (Ed.), Materjały do bibljografji pi´smiennictwa kobiet polskich (do roku 1929): Nauki matematyczno-przyrodnicze i nauki stosowane. Drukarnia Polska: Lwów. Nikodym, Stanisława. (1935b). Une propriété topologique de la surface partout localement homéomorphe au plan Euclidien. Mathematica. Cluj, 9, 103–109. Nikodym, Stanisława. (1936). Wybrane zadania z analizy matematycznej. Warszawa: Koło Chemików Studentów Politechniki Warszawskiej. Nikodym, Stanisława. (1946). Wybrane zadania z analizy matematycznej. Kraków: Ksi˛egarnia Lingwistyczna. Nikodym Otton, & Nikodymowa, Stanisława. (1948). Wzory i krótkie repetytorium matematyki. Pozna´n: Ksi˛egarnia W. Wilaka. Piotrowski, Walerian. (2012). Doktoraty z matematyki i logiki na Uniwersytecie Warszawskim w latach 1915–1939. Antiquitates Mathematicae: Annales Societatis Mathematicae Polonae (Series VI), 6, 97–131. Piotrowski, Walerian. (2014). Jeszcze w sprawie biografii Ottona i Stanisławy Nikodymów. Wiadomo´sci Matematyczne, 50(2), 69–74. “Prexy Announces Appointments, New Promotions”. (1948). The Kenyon Collegian 84/1 (October 8). “Prof. Nikodym Elected To Science Academy”. (1949). The Kenyon Collegian 73/22 (May 20). “Professor Nikodym Lectures in Europe”. (1959). The Kenyon Collegian 86/1 (October 2). Prytula, Yaroslav G. (2015). Otton Marcin Nikodym. Retrieved May 20, 2017, from http://www. mmf.lnu.edu.ua/index.php/istoriia/vydatni-osobystosti/item/1067-otton-marcin-nikodym.html. “Rozpoznane ofiary katy´nskie”. (1943). Nowy Kurjer Warszawski 105. Schaefer, H. M. (1993). My Memories of Juliusz Schauder. Topological Methods in Nonlinear Analysis, 1(2), 15–19. Smoluchowski, M. (1928). Kobiety w naukach s´cisłych: Odczyt wygłoszony w Zwi˛azku NaukowoLiterackim we Lwowie w r. 1912-tym. In: Pisma Marjana Smoluchowskiego (Vol. 3.1, pp. 138– 152). Kraków: Polska Akademia Umiej˛etno´sci. Smoluchowski, Marian. (2018). Women in exact science a Lecture deliverd at the scientifict-literaty association in Lwów in the the year 1912, Acta Poloniae Historica 117, 231–240. Straszewicz, Stefan. (1923). Über eine Verallgemeinerung des Jordan’schen Kurvensatzes. Fundamenta Mathematicae, 4(1923), 128–135. Straszewicz, Stefan. (1925). Über die Zerschneidung der Ebene durch abgeschlossene Mengen. Fundamenta Mathematicae, 7(1925), 159–187. Szyma´nski, Wacław. (1990). Who Was Otto Nikodym? The Mathematical Intelligencer, 12(2), 27–31. Tobies, Renate. (2020). Internationality: Women in Felix Klein’s Courses at the University of Göttingen (1893–1920). In: this book, chapter I.1. Tutajewska, Maria. Kierownicy katedr (od momentu ich utworzenia). Retrieved MAy 30, 2017, from http://www.matematyka.pk.edu.pl/HISTORIA.pdf. Wódz, Beata Ewa. (2014). Sandomierskie w szkicach malarskich Stanisławy Nikodymowej z Lilientalów. Zeszyty Sandomierskie, 20(38), 32–36.

Danuta Ciesielska is a mathematician and an historian of science. She studied mathematics at the Jagiellonian University in Kraków, from which she received PhD in pure mathematics in 2002. She is a researcher in the Institute for the History of Science of the Polish Academy of Science in Warsaw. Her research interest includes history of mathematics, especially history of analytic, projective and algebraic geometry, and history of scientific societies. She is an Associate Editor of the “Quartely Journal of the History of Sciences and Technology,” she also served as a Section Editor of the journal “Antiquitates Mathematicae” of the Polish Mathematical Society, which is devoted to the history of mathematics. She is a member of the Commission on the History of Science of the Polish Academy of Arts and Sciences. Her work for raising public awareness in mathematics includes the placing of a bench with the figures of Banach and Nikodym in the city center of Kraków in 2016.

Part III

Approaches

Introductory Reflections on approaching Women in Mathematics Nicola M. R. Oswald The challenge How can we talk about and work on gender issues? At first sight this question might seem rather free of conflict, perhaps one might have in mind an alleged sober and objective empirical approach counting women and men in certain positions and levels of their careers in mathematics. However, a closer look reveals rather quickly that the handling of gender issues bears unexpected hurdles with respect to subjectivity. To illustrate these challenges, we want to by way of example refer to a revealing secondary aspect of a survey, which was commissioned by the German Association of University Professors and Lecturers (Deutscher Hochschulverband, DHV) in 2016.1 Among other questions, 674 professors at universities were asked whether they have the impression that female junior researchers have either fewer or better (or equal) chances of receiving a position at a university than their male equally qualified colleagues. Among the male professors, 92% evaluated the chances as better or equal, only 4% thought that chances were fewer. Interestingly, the interviewed female professors painted a completely different picture: 48% gave the answer “better or equal chances” and 44% voted for “fewer chances”. Without going into the actual content of the interview question, the gendered difference in answering reveals how strong the perception is influenced by own experiences.

1 See

Allensbacher Archiv, IfD-Umfrage Nr. 7244 (Oktober 2016). A summary of the results was published in the journal “Forschung & Lehre” (No. 1/17, 2017, p. 31).

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Turning to the historiography of women in mathematics: In particular concerning the historical reappraisal of biographies, one mostly has to rely on descriptions and self-descriptions. Naturally, their interpretations play a decisive role. When, for example, Sofia Kowalewskaja (1850–1891), the first female professor of mathematics, is characterized as “das Weibgenie mit dem Männerhirn”2 in Laura Marholm’s bestseller Das Buch der Frauen. Zeitpsychologische Porträts (English translation: Women’s Book) of 1894, one can interpret and use this colorful description in various ways. The reader may either deduce the ingenuity of the mathematician or focus on her (negative or positive?) masculine attribution or perhaps even both. Of course, a detailed contextualization, including Marholm’s own biography as well as contemporary circumstances, sharpens the understanding of the background; however, it is certainly impossible to achieve objectivity taking into account this description. The book Writing About Lives in Sciences (2016) is dedicated to this challenge. Here, the “special relation” between narrator and research object is taken into focus, more precisely: “The role that the gender of the biographer, and that of his or her biographee, may have in the process of writing a biography.”3 It is underlined, that “when we write, and not just when we write biographies, we are writing about ourselves too.”4 In this context, in 1953, the historian of science Eduard Jan Dijksterhuis (1892– 1965) had already formulated questions for reflection to each historian: Do you really think that you would succeed in eliminating the influence of your own person, your education, your individual interests, your time, your social class on the study and interpretation of the material, even if you had a wealth of sources at your disposal which you trusted with good reason?5

Although it must not be the main goal to overcome this self-relatedness, though it is important to develop a certain awareness of it. “Narrators” should search for appropriate methods of retelling a story, respectively, analyzing working attitudes and conditions, not only but certainly in particular when they write about a personal subject such as gender. Sociology: Shaping Categories Connected to Individual Strategies Altogether the difficult objectivity of historical research in combination with gender issues demands a careful handling. An idea of a possible approach can be found with the help of the so-called “qualitative content analysis”. Such a technique has been defined by the sociologist and psychologist Philipp Mayring:

2 Marholm

(1895), p. 188. & Franceschi (2014), p. 7. 4 Govoni & Franceschi (2014), p. 30. 5 Translated from Dijksterhuis’ inaugural address at the University of Utrecht on October, 26 of 1953. 3 Govoni

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This technique is aiming to filter a certain structure from the given material. This structure is in the form of a category system applied to the material. All textual components, which are related to the categories, are systematically extracted from the material.6

In the case of historical work, the “material” is mostly given by (auto-)biographies and related documents. Certainly, it might be difficult to apply Mayring’s technique in its full rigor to historical documents in general. However, the search for appropriate categories related to gender may be rather instructive. Therefore, we firstly have to clarify against which background our analysis takes place. We return to Marholm’s characterization of Kowaleskaja’s “Männerhirn”. Indeed, it is striking that at various points female mathematicians (here: in Europe) are “equipped” with male attributions—often in a well-meaning manner. Another example is contained in Hermann Weyl’s warm-hearted eulogy from 1935 for Emmy Noether (1882–1935): The power of your genius seemed to have gone beyond the boundaries of your gender in particular. That’s why in Göttingen we usually called you, in reverent derision, the Noether.7

Here, Noether is given a male article in the German language (“den” instead of “die”), because she had pushed the boundaries of her female sex, respectively, gender.8 Furthermore, the term “genius” is again related to her seemingly unfeminine mathematical skills. Such descriptions prove a repetitive factor in the history of female mathematicians. Still today, the combination of femininity and mathematics is sometimes considered as a curiosity. This gains an odd dimension, when, for example, the first female mathematician awarded with the Fields Medaille (in 2014), Maryam Mirzakhani, is irritatingly in the focus of online discussions and comments because of her short (supposed more male than female) haircut.9 Certainly, we cannot cover all of the many involved social facets included in those descriptions of and behavior towards women, however, we can surmise that the core of this seemingly oddity of mathematics and femininity is connected to our socially learned patterns respectively behaviors which are valued as normal. Culture [is] an intricate system of more or less formalized ways of thinking, feeling, and acting, which, being learned and shared by a plurality of people, serves both objectively and symbolically to constitute these people as a particular and distinct group.10

6 Mayring

(2010), p. 82. Macht Deines Genies schien insbesondere die Grenzen Deines Geschlechts gesprengt zu haben. Darum nannten wir Dich in Göttingen meist, in ehrfürchtigem Spott, den Noether.” Excerpt of Weyl’s eulogy, 17.04.1935, quoted from Roquette (2007), p. 19. 8 At this point, one has to take into account that the German word “Geschlecht” does not explicitly distinguish between the biological “sex” and the social “gender”. 9 We believe that it is not necessary to repeat any of this unqualified comments at this point. 10 Rocher (1968), p. 111, quoted from Lê (2016), p. 276. 7 “Die

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European (and, of course, not only European) culture is certainly characterized by a patriarchic11 structure. A superior genius and implicitly mathematical skills are traditionally attributed to the “dominant” male gender. So-called gender roles, The role or behaviour learned by a person as appropriate to their gender, determined by the prevailing cultural norms.12 have a stubborn resistance. As a consequence of these cultural norms, a certain femininity is denied to women in mathematics—successful or not. This gets, in an unprofessional and maybe even abusive way, to the personal and intimate core of single individuals. Although if this cannot be taken as an intended affront to the personality, this “formalized way of thinking” has not uncommonly provoked and still provokes a protective defensive attitude. Female scientists want to be perceived as self-determined and, in particular, not be considered as victims. On the one hand they are often denied special treatment, while on the other hand developed strong “individual strategies” underlining the alleged individual independence in the system (academia). Such a mechanism can be identified in current as well as also in historical studies. In her interdisciplinary study Nicht als Gleiche vorgesehen. Über das „akademische Frauensterben“ auf dem Weg an die Spitze der Wissenschaft, published in 2015, sociologist Heike Kahlert analyzed and deduced factors which lead female scientists to the cancelation of their career at a university, in particular after their Ph.D. Among others, she focused on the highly influential and culturally favored aspect of self-criticism and self-selection: International research considers a combination of foreign selectivity and self-selectivity as responsible for the decrease of women in this period of scientific career, which consists of hard factors as the framework of employment relationships and soft factors in the form of aspects of profession and organization culture. [...] a career in science is in the gatekeepers’ opinion the result of individual conduct of the aspirants and not (also) the result of professional leadership with respect to promoting young academics. Also, the questioned Ph.D. students and postdocs consider their possible career in science furthermost as an individually created product and don’t except hardly any support of the university teachers. Instead the majority assumes that they are self-responsible for the development of their further career inside and outside of science.13

Kahlert did not only underline the significant role of self-reflection of female scientists, furthermore, but she also pointed out the strong tendency to a conscious individualization of the own success. The private career advancement or failure is highly

11 From

the definition of “Patriarchy” in Oxford living Dictionary: “A system of society or government in which men hold the power and women are largely excluded from it.” https://en.oxforddic tionaries.com/definition/patriarchy (accessed April 13, 2019). 12 From the definition of “gender roles” Oxford living dictionary https://en.oxforddictionaries.com/ definition/gender_role (accessed April 13, 2019). 13 Kahlert (2015), pp. 61, 73.

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considered as detached from external circumstances.14 Having in mind the wellknown, and nowadays well-discussed effects of direct and indirect sexism and gender roles, this might appear paradoxical. However, interestingly, those behavioral aspects seem to be anchored by a long-lasting tradition. In comprehensive studies of historical conditions of female scientists, we can find similar results. The American historian of science Margaret Rossiter identified in her quantitative and comparative work Women Scientists in America: Struggles and Strategies to 1940 principles reflecting those aspects. She pointed out that women. [...] accepted inequity in scientific employment as something that they could not change [...] but that some might overcome by hard work (such as by earning more degrees than the men) and by stoicism (such as suffering any unemployment in silence). [...] Thus rather than continuing to identify and fight discriminatory conditions, the women scientists of the 1920s and 1930s began to internalize the double standards [...].15

And furthermore, Rossiter shaped the scientific discourse by characterizing a phenomenon, the so-called Marie Curie strategy. It describes the fact that when women were compared and compared themselves to the twice selected Nobel Laureate, the threshold to their success was forced up “to almost unattainable heights” (p. 127). The author explained that numerous female scientists could not withstand this, often self-imposed, enormous pressure. Let us turn to another qualitative case study: The autobiography Braun (1990) of the mathematician Hel(ene) Braun (1914–1986), scientific assistant to number theorist Carl Ludwig Siegel (1896–1981), fourth woman in Prussia to receive a “Habilitation”16 and later professor at the universities of Göttingen and Hamburg, describes with a striking openness her search for independence. She gave personal and profound insights into her career as well as her very subjective perception of social and political circumstances. Here, for example, by a concise assessment about women in mathematics: I say it over and over again that mathematicians are always enthusiastic about every woman who is able to write a pretty integral sign on a blackboard. [...] And the whole thing about women have to achieve double as much as men to reach a certain position? Well, this only shows how much some living beings do overestimate themselves.17

The way Hel Braun relativizes here and in numerous other observations her experiences was and is a typical phenomenon for women in sciences: She definitely chose an individual strategy and refused to be recognized as part of a discriminated 14 In the study Hendrix et al. (2013), p. 327, with 1.555 interviewed professors (393 female, 1.162 male) about the most conducive factors on the way to a professorship, “high self-motivation: personal initiative, independence, ambition, engagement” achieved rank 3 (19,6%), whilst “successful publications” was put on rank 7 (11,2%); their male colleagues answered the other way round: rank 7 for high “self-motivation” (13,6%) and rank 3 for “successful publications” (18,7%). 15 Rossiter (1984), p. 130. 16 See https://www.mathematik.de/images/Presse/Presseinformationen/20000000_SN_TUC_Hab ilitationFrauen_2000.pdf (accessed March 21, 2019). (The “Habilitation” officially qualifies for self-contained university teaching. This certificate was and still often is the key for access to a professorship in German universities.) 17 Braun (1990), p. 72.

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minority. However, through her clear dissociation, she implicitly gave a description of the mathematical community and behavior towards female mathematicians. In this sense, the tracking of intended or generated strategies of selected female mathematicians might be quite informative and, respectively, more tangible than “accidental” biographies. Although those individual strategies complicate the search for common categories studying women in science on the one hand, the consideration of the strategies themselves in the course of time can function as connecting element on the other hand. Individual strategies and, historically conditioned, the individual reception deliver a scope of the study and are put together by individual components (categories). Content: Focussing the Focus There might be different ways of a meaningful reception of these “individual strategies”. From a historiographic perspective, the question remains: How can those strategies of individual scientists be examined, respectively, on behalf of which foci should they be considered? Each selection of a methodical approach is decisively dependent on the underlying material. This is illustrated by an example in Marta Cavazza’s study concerning Laura Bassi (1711–1778),18 the first female professor at a university in Europe. Cavazza in analyzing “Rhetorical Strategies” in selected biographies of Laura Bassi in the eighteenth century points out that “Terms like ‘miracle’, ‘marvel of her sex’, and ‘monster of intelligence’ were frequently used” (Cavazza [2014], p. 78), and comes to the conclusion that these eighteenth-century biographies were written “[...] with the explicit goal of minimizing and controlling change” (p. 80). On this basis, the author invites to renegotiate gender roles taking place in the eighteenth century in Italy and, in general, Europe. Cavazza is therewith adding to the biographical presentation of Bassi’s life a component of external perception through her choice of method, which allows conclusions for the situation in a wider setting. She chose this meta level to derive knowledge about the general role of women in sciences. In this sense, and having the basic approach of Philipp Mayring in mind, the biography of Bassi, respectively, the biographies written about her life serve as “material” and categories are inductively generated by focussing on rhetorical strategies. This search of a leading strategy to derive meaningful statements on the basis of biographical material is also reflected by the three chapters of this part of the book: Firstly, the focus lies on the French scientist Sophie Germain (1776–1831), an autodidact and certainly an important female pioneer in mathematics of the early nineteenth century, whose life had been decorated with various legends in numerous references. The author Jenny Boucard chooses a historiographic approach analyzing some of these receptions. Thereby particularly focussing on the purposeful use of her biography and works during the time of the French Third Republic. In the second chapter, Marjorie Senechal adopts a very different position in her presentation of the 18 see

“The Biographies of Laura Bassi” in Govoni & Franceschi (2014), pp. 67–86.

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mathematician Dorothy Wrinch (1894–1976). She contrasts the mathematician’s life by reference to the development of an independent branch of mathematics: Crystal Geometry. This works out in a particularly interesting manner since the author herself was an assistant to Wrinch and interweaves her own experiences into her study. An even closer approach is selected by Lisbeth Fajstrup, Anne Katrine Gjerløff, and Tinne Hoff Kjeldsen. The authors decided to directly interview four Danish female professors of mathematics and analyzed their life stories. The interviews were semistructured and constructed on the basis of a set of research questions and categories. The authors decidedly underline the fact that they did not write a “heroine story”, however, they allow an insight into experienced reality. Those three rather different approaches reflect the multifaceted methodological options as well as the required diligence in analyzing the careers of female mathematicians.

References Braun, H. (1990). Eine frau und die mathematik 1933–1940. Der beginn einer wissenschaftlichen laufbahn. (Ed. Max Koecher). Springer-Verlag. Govoni, P. & Franceschi, Z. A. (2014). Writing about lives in sciences. (Auto) biography, gender and genre. V&R unipress. Hendrix, U., Hilgemann, M., Beate, K., Niegel, J. (2013). Gender-Report 2013— Geschlechter(un)gerechtigkeit an nordrhein-westfälischen hochschulen. In Studien Netzwerk Frauen- und Geschlechterforschung NRW, Nr. 17. Kahlert, H. (2015). Nicht als gleiche vorgesehen. Über das, akademische frauensterben auf dem wan die spitze der wissenschaft. Beiträge zur hochschulforschung, 37. Jahrgang, 3/2015: 60–78. Lê, F. (2016). Reflections on the notion of culture in the history of mathematics: The example of ‘Geometrical Equations’. Science in Context, 29(3), 276. Marholm, L. (1895). Das buch der frauen. Zeitpsychologische porträts. Paris und Leipzig: Albert Langen. Mayring, P. (2010). Qualitative inhaltsanalyse. Grundlagen und techniken. Beltz Verlag (11th ed.). Rocher, G. (1968). Introduction à la sociologie géenérale. Vol. 1. L’action sociale. Montréal: HMH. Roquette, P. (2007). Zu Emmy noethers geburtstag. Einige neue Noetheriana. Mitteilungen der DMV, 15: 15–21. Rossiter, M. (1984). Women scientists in america. Struggles and strategies to 1940. The John Hopkins University Press (third printing).

Chapter 7

Arithmetic and Memorial Practices by and Around Sophie Germain in the 19th Century Jenny Boucard

Abstract Sophie Germain (1776–1831) is an emblematic example of a woman who produced mathematics in the first third of the nineteenth century. Self-taught, she was recognised for her work in the theory of elasticity and number theory. After some biographical elements, I will focus on her contribution to number theory in the context of the mathematical practices and social positions of the mathematicians of her time. I will then analyse some receptions and uses of Germain’s life and scientific work under the French Third Republic.

7.1 Introduction Biographical studies of women scientists considered as pioneers are an important part of the historical writings dedicated to the question of “women in mathematics”: in particular, they highlight the range of the necessary adaptations, attitudes and reactions of the various actors involved in a given place and time in order to join traditionally male scholarly spheres.1 More generally, work on the relationship between women and science is part of questions developed since the 1970s about the production of science from a social and cultural perspective: they take into account scientific practices and discourses on science, non-institutionalised scholarly networks, and the circulation of knowledge or memorial practices. In this paper, I chose to focus on a particular woman in mathematics: Sophie Germain (1776–1831). In the history of mathematics, Germain is known for her work in elasticity theory and number theory. Many features of her life also contributed to

1 See,

for example, Tobies et al. (2001), Gardey (2000), and Introduction.

J. Boucard (B) Faculté des sciences et des techniques de Nantes, Centre François Viète d’histoire des sciences et des techniques, 2, Rue de La Houssinière - BP 92 208, 44 322 Nantes, France e-mail: [email protected]

© Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_7

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J. Boucard

her becoming a pioneer as a woman who produced mathematics.2 Note for example that she became the first woman to be invited to the sessions of the Paris Académie des sciences as a mathematician (and not as a wife).3 Germain’s life and work have already been the subject of many academic studies. In this chapter, after recalling some biographical elements of Germain, I first discuss how Germain was able or not to infiltrate scholarly networks and collaborate with other mathematicians to research in number theory. For this purpose, I rely on A. Del Centina, R. Laubenbacher and D. Pengelley’s studies of Germain’s published and unpublished number-theoretical writings, published or not.4 I situate Germain’s work in the context of number theory practices5 and contemporary mechanisms of mathematical circulation, using a corpus of number theory texts based on a systematic study of mathematical journals from the first third of the nineteenth century.6 I can thus question the specificities of her work from the point of view of her gender and the status of number theory in France at the time. Then I study how Germain’s life and capability to do mathematics alone or in collaboration with male mathematicians was used in France in debates on the women question mainly from the 1870s to the 1910s. This period is particularly interesting for several reasons: new receptions of Germain’s philosophical and mathematical works, the importance of scholarly commemorations under the Third Republic, the rise of feminist movements,7 reflections and reforms on secondary education and higher education (especially for women), the development of new social sciences and the growing importance of science in French society (let us recall the slogan of the Association pour l’avancement des sciences, created in 1872: “For science, for the homeland”8 ), and,

2 She

appears systematically in the biographical collections of women mathematicians and is presented as one of the first women to have done independent mathematical work. She is also one of the few women mathematicians, with Sophia Kovalevskaya (1850–1891), to have a stamp with her effigy (released March 18, 2016). More generally, she has also be included in books about great women in history: for example, she is one of the eighteen “exceptional women” according to Jean Haechler (2007). The historical reliability of some of these biographical collections is sometimes questionable, but it is interesting to note the place of Germain in them. She is also one of the only women to appear in current high school mathematics French textbooks, most often to define so-called Germain prime numbers. 3 The term “mathematician” was rarely used in the first half of the nineteenth century: it is the term “geometer” that appeared most often in this period. Following the comment of one of the reviewers of this paper, I nevertheless chose this first term for the whole period studied here in order to avoid misunderstandings. 4 See Laubenbacher and Pengelley (1999); Del Centina (2005, 2008); Laubenbacher and Pengelley (2010); and Del Centina and Fiocca (2012). 5 By the study of mathematical practices I mean the description and analysis of the mathematical activities carried out by actors from the historical traces that are available to us. It is not the study of concepts or mathematical theories but that of an activity that articulates several elements (tools and mathematical objects, calculations, procedures, theorems, proofs, methods, etc.). 6 See Boucard (2011, 2015). 7 See Offen (2000). 8 “Pour la science, pour la patrie”.

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more generally, debates on the place of women in science in general and mathematics in particular. Several of these references will illustrate different representations of women in mathematics through the construction of historical narratives around Germain.

7.2 Biographical Elements of Germain: A Pioneer? Two texts published in the nineteenth century are usually used for the biography of Sophie Germain: Notice historique (Libri 1832) written by one of Germain’s close friends, the mathematician Guglielmo Libri (1803–1869), who knew her well and to whom she related her childhood; and Étude sur la vie et les œuvres de Sophie Germain by the journalist Hippolyte Stupuy (1832–1900), published for the first time in 1879 when Stupuy’s edition of Germain’s philosophical work appeared. Much additional information can also be found in her recent work on the elastic theory of surfaces9 and in books and articles about Germain’s life.10 The aim of this section is to highlight the main events of her life told in most stories about Germain and that contributed to the construction of Germain’s image as a pioneer in mathematics. Indeed, the stories existing about Germain’s life particularly emphasise two aspects of her self-taught education that contribute to the making of her exceptional status. First, passionate about mathematics from an early age, she had to face her family, who forbade her to study mathematics (she hid to study the books found in her father’s library). Then, not allowed to participate in post-revolutionary institutions of higher education, she had to pretend to be a man to access to ressource of the École Polytechnique teaching and to communicate with teachers.

7.2.1 A Self-taught Mathematical Formation Sophie Germain was born in 1776 in a bourgeois family, “liberal and educated”,11 on rue Saint-Denis in Paris. Her father, Ambroise-François Germain (1726–1821), belonged to the corporation of clothiers and bonnetiers in Paris and was representative of the Tiers-État to the États généraux, then deputy to the Assemblée législative between 1790 and 1791 (the Constituants).12 He was known for his analysis of economic questions. He and his wife Marie-Madeleine Gruguelu (?– 1823) had three daughters: Marie-Madeleine (1770–1804), Sophie and Angélique

9 See

Bucciarelli (1980). Boyé (2017); and Musielak (2015). 11 “bourgeoise libérale et instruite”, Stupuy (1879), p. 3. 12 See Szramkiewicz (1974). Szramkiewicz indicates that Germain’s father is often wrongly presented as director of the Banque de France. 10 See

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Ambroise (1779–?). In 1790, the eldest married the notary Charles Lherbette (1752– 1836), recent widower of Angélique Ambroisine Germain (1765–1787), cousin of Ambroise-François. They had two children, including Amand Jacques (1791–1864). Close to his two aunts, Amand Jacques was the witness of the second marriage of Angélique Ambroise and published a philosophical text written by Sophie after her death. In 1809, Angélique Ambroise first married René-Claude Geoffroy (1767– 1831), a doctor from a family of pharmacists, then Henri Dutrochet (1776–1847), physician and botanist, elected to the Académie des sciences in 1828. Little additional information is known about the Germain family. It is nevertheless interesting to note that Germain’s father may have established relationships with scholars during his political activities: for example, among the deputies of the États généraux for the generality of Paris were the astronomer Jean-Sylvain Bailly (1736–1793) the doctor Joseph-Ignace Guillotin (1738–1814) (Tiers-État) and the mathematician Achille Pierre Dionis du Séjour (1734–1794) (Noblesse).13 Similarly, the two marriages of the youngest daughter suggest new links with families of doctors and naturalists. Access to education for women was very limited under the Ancien Régime and was an issue under discussion at the end of that century.14 But more generally, between 1792 and 1793, the existing schools were shut down, with the banning of religious congregations and the subsequent suppression of academies, military schools, universities. Since Germain had only sisters and we have no evidence to the contrary, she was unlikely to have had private preceptorship.15 According to Libri, she became passionate about mathematics by reading Montucla’s famous Histoire des mathématiques, and especially by the story of Archimedes’ life. She would have started studying several mathematical books in her father’s library, starting with Étienne Bézout’s Cours de mathématiques, then an important reference textbook.16 Her parents then worked hard to dissuade her from immersing herself in mathematics, considered useless or even dangerous for women. The scientific literature then intended for amateurs and especially for women, which had been booming for several decades, only exceptionally included calculations or mathematical tools, favouring a qualitative approach to astronomical or physical knowledge.17 In general, before the Revolution, the share of the mathematical and physical sciences was extremely small in the colleges; the situation changed with the creation of the École polytechnique in 1794. Nevertheless, Germain, then aged eighteen, could not be admitted to this new institution reserved for men. Using Le Blanc, the name of a Polytechnic student apparently resigning, she managed to obtain the content of the lessons of some teachers, including Antoine-François Fourcroy (1755– 1809) and Joseph-Louis Lagrange (1736–1813). She even succeeded in establishing a form of exchange with Lagrange, relying on the polytechnic custom that students 13 See

Lemay (1991). Fayolle (2012). 15 In some cases, a woman could benefit from private lessons given to a brother to acquire a learned culture. See Peiffer (1991), p. 198. 16 See Alfonsi (2008). 17 See, for example, Peiffer (1991). 14 See

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passed on written notes on their lessons to teachers.18 Impressed by the quality of the comments of the student Le Blanc, Lagrange wanted to meet him and thus learned the true identity of his interlocutor. This event allowed Germain to enter into contact with several of the most recognised mathematicians of that time.

7.2.2 First Exchanges with the Academic Community and Germain’s Mathematical Work This first meeting with Lagrange allowed the introduction of Germain into the learned community: many scholars of the time wanted to meet and support her. Traces remain from exchanges with Jacques Antoine Joseph Cousin (1739–1800), author of popular Leçons de calcul différentiel et de calcul intégral, the physician and agronomist Henri-Alexandre Tessier (1741–1837), the astronomer Jérôme Lalande (1732–1807), and the Hellenist Jean-Baptiste Gaspard d’Ansse de Villoison (1750–1805).19 Her status as a young, self-taught woman in mathematics, regularly emphasised by her interlocutors, played an important role in these different relationships. In connection with several recognised mathematicians, including Adrien-Marie Legendre (1752–1833), Carl Friedrich Gauss (1777–1855), Libri and Joseph Fourier (1768–1830), Germain joined into scholarly networks while remaining unmarried and without the assistance of a man from her family. This was very unusual for a woman at the time, noticed by some of her contemporaries. The consideration of this specific trait of Germain as a woman mathematician was for example reflected in the statement by Jean-Baptiste Biot (1774–1862), when Germain’s first memoir was read as part of the Académie des sciences award in 1812: “The opening of the sealed note, made known the name of a woman, M.lle Germain, probably the person of her sex who penetrated the deepest in mathematics, without excluding M.me Du Chatelet; because here there was no Clairault.”20 Germain participated actively in the construction of her image as an autonomous woman mathematician, far from the image of the “learned woman” as she wrote in one of her letters to Gauss for example, when she told him that she was Le Blanc: […] this circumstance determines me to confess to you that I am not so completely unknown to you as you think; but that, fearing the ridicule attached to the title of learned woman, I formerly borrowed the name of Mr. Le Blanc to write to you, and to give you notes which, undoubtedly, did not deserve the indulgence with which you were kind enough to answer them.21 18 See

Stupuy (1879), and Bologne (2004). Bucciarelli (1980), Chap. 2. 20 “L’ouverture du billet cacheté, fit connoître le nom d’une femme, M.lle Germain, probablement la personne de son sexe qui ait pénétré le plus profondément dans les mathématiques, sans en excepter M.me Du Châtelet; car ici il n’y avait point de Clairault.” 21 “[…] cette circonstance me détermine à vous avouer que je ne vous suis pas aussi parfaitement inconnue que vous le croyez: mais que, craignant le ridicule attaché au titre de femme savante, j’ai autrefois emprunté le nom de Mr. Le Blanc pour vous écrire et vous communiquer des notes 19 See

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It was precisely the aristocratic women of the salons (participants, organisers) who had links with science under the Ancien Régime. Next to the image of “learned woman” generally negatively connoted, worldly science (“science mondaine”) was also strongly criticised during the revolutionary period.22 Germain also reacted vividly to those who tried to flatter her, like Lalande when he wished to send her one of the copies of his Astronomie pour les dames, one of his books devoid of mathematical formulas. Germain had two areas of predilection in mathematics: theory of elasticity and number theory. On the theory of elasticity, she was rewarded by the Académie des sciences in 1816. Nevertheless, Germain had to submit three different versions of her memoir initially submitted in 1812 because academicians criticised her mathematical errors, her formulation of analytical methods and some of them were maybe against the fact that she did not adopt the molecular approach. Analysis was taught at the École polytechnique, and Germain’s weakness in this domain seemed to have been a cause of the errors contained in her early work on the theory of elasticity.23 She began to work on number theory in the early nineteenth century and rapidly mastered the most recent arithmetical writings, by Legendre and Gauss. Even if she corresponded with Gauss about number theory as early as 1804, Germain only published one short note on the subject, in Crelle’s Journal. One of her theorems was also included in one of Legendre’s works on number theory in 1823. She also wrote some texts in philosophy that were recognised by Auguste Comte (1798–1857) and published by her nephew after her death. She died in 1831 of cancer as a “rentier” according to her death certificate.

7.3 Sophie Germain and Number Theory, 1800–1830: Results and Practices It is well-known that, beyond solitary study, the scholarly production of a scientist depends on the scholarly and social networks in which he or she evolves.24 In the case of Germain and number theory, it is thus important to know who produced and published number theory in the early nineteenth century; what mathematical training was needed; what role could discriminatory institutional mechanisms have in number-theoretical work; what were the possible practices and modes of circulation for number theory25 ; and how can Germain’s work be analysed in this context.

qui, sans doute, ne méritaient pas l’indulgence avec laquelle vous avez bien voulu y répondre.” Del Centina and Fiocca (2012), p. 669. 22 See Chappey (2004). 23 See Bucciarelli (1980), and Dahan Dalmedico (1987). 24 See, among others, Govoni (2000). 25 This series of questions is inspired by Gardey (2005), pp. 31–32.

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7.3.1 A Panorama of Number Theory from the 1800s to the 1830s The early nineteenth century was a transitional period for publications. Indeed, in 1805, the available publication outlets for mathematics were few and far between. Academic periodicals were difficult to access for non-academics, and scholarly journals such as the Journal des sçavants did not contain any articles on number theory. Books were also expensive to publish, and sales depended on limited specialised sellers. For example, the number of publications including mathematical reasonings about congruences,26 introduced by Gauss in 1801 in his Disquisitiones arithmeticae (DA), was limited: 30 texts between 1801 and 1825, many of which were actually algebra treatises, as those by Sylvestre-François Lacroix (1765–1843) and Peter Barlow (1776–1862),27 in which congruences were used very briefly. The number of publications including congruences increased from 1825 when new mathematical and scientific journals were created: 224 texts between 1826 and 1850 of which only 17 were not included in periodicals.28 In fact, Germain seems to be the only woman who published on number theory in the first half of the nineteenth century. At the turn of the nineteenth century, two treatises on number theory appeared: Legendre’s Essai sur la théorie des nombres29 and Gauss’s DA.30 Three points should be highlighted here. First, Legendre and Gauss had divergent opinions on the definition of number theory. Legendre identified number theory with indeterminate analysis. Gauss explicitly distinguished between these two domains, proposing number theory as being the domain where integer and rational numbers were considered, and not limited to equations. Second, in his book, Gauss gave a coherent presentation of number theory by organising it around two fundamental objects: congruences and quadratic forms. He gave two different proofs of the quadratic reciprocity law and a method to resolve the binomial equation x p = 1 algebraically by reindexing the roots with a primitive root of p, insisting on the links existing between different parts of his work and different mathematical domains, such as algebra and number theory. Third, Legendre’s and Gauss’s books were fundamental references for anyone planning to study number theory. Indeed, at the time, French teaching programmes were focused on engineering, especially with the École polytechnique, and number theory was not taught at all. That is why if someone, male or female, wanted to study this domain, he or she had to read former publications, and especially Legendre’s and Gauss’s books.

26 Two numbers a and b are said congruents modulo a third number p if a–b is divisible by p. Gauss

noted this relation a ≡ b(mod p). The purpose of taking congruences as a marker for studying number theory in the first half of the nineteenth century and the constitution of the corpus from which these results were obtained is explained in Boucard (2011), Chap. 1. 27 See Lacroix (1804), and Barlow (1811). 28 See Boucard and Verdier (2015). 29 See Legendre (1798). 30 See Gauss (1801).

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Apart from several memoirs on Gauss sums, reciprocity laws and complex integers published by Gauss after 1801, DA were mostly used for the algebraic resolution of binomial equations before 1825. Between the 1820s and the 1860s, new scholars read Gauss’ DA and published arithmetical papers linked to it. In addition, evolutions in other mathematical areas, such as the use of complex numbers, Fourier analysis or elliptic functions, were applied in number theory. Historians C. Goldstein and N. Schappacher showed that a research domain that they called Arithmetic Algebraic Analysis was then developed by an international network of scholars.31 But, at Germain’s time, the use of analysis in number theory was marginal and Germain’s potential weakness in analysis did not constitute a significant limitation. Within French number-theoretic production, there was multiform activity based on a strong link between equations and congruences. Specific problems were discussed such as the imaginary roots of congruences (by Louis Poinsot (1777–1859), Victor-Amédée Lebesgue (1791–1875), Évariste Galois (1811–1832), Germain), the number of integer roots of a congruence (by Libri, Lebesgue) or Fermat’s Last Theorem (FLT) (by Legendre, Libri, Germain).32 These publications had common roots with Lagrange’s and Legendre’s arithmetical approach and integrated Gauss’s objects and methods in varying proportions.

7.3.2 Germain’s Arithmetical Traces: Publications, Manuscripts and Correspondence As Gauss observed in his correspondence, Germain was precisely one of the first mathematicians who mastered the contents of Gauss’s DA and who applied the theory of congruences to her number-theoretical work. Moreover, after she impressed Lagrange with her mathematical skills, she became progressively close to mathematicians such as Gauss, Legendre, Augustin Louis Cauchy (1789–1857), Poinsot or Libri. Germain also met Carl Gustav Jakob Jacobi (1804–1851) and Galois. These mathematicians were the main authors who published on number theory at her time. Yet Germain only published one arithmetic note in Crelle’s Journal in 1831 on cyclotomy (Germain 1831), as a result of Legendre’s and her work on FLT in particular. This note tells us that Germain was familiar with some of the notation and vocabulary used by Gauss. Nevertheless, it gives us very little information about her arithmetic practices. Another published trace of Germain’s number-theoretic contribution can be found in a memoir about FLT presented by Legendre in 1823 at the Académie des sciences and published in 182733 : Legendre attributed three propositions to “Miss Sophie Germain”34 concerning some specific prime factors θ = 2kn 31 See

Goldstein and Schappacher (2007). Boucard (2015). 33 See Legendre (1827). 34 “Mlle Sophie Germain” Legendre (1827), p. 17. 32 See

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+ 1 when studying the equation x n + yn + zn = 0, with n is an uneven prime and k an integer. We must, therefore, turn to unpublished sources to better understand Germain’s work in number theory: mostly her correspondence with Gauss (we actually know ten letters from Germain and four from Gauss, written between 1804 and 1829) and some of her manuscripts, including those entitled “Remarques sur l’impossibilité de satisfaire en nombres entiers à l’équation x p +yp +zp ” and “Démonstration de l’impossibilité de satisfaire en nombres entiers l’équation z2 (8n±3) + y2 (8n±3) = z2 (8n±3)”.35 From this corpus of texts, we get a good idea of her readings in number theory. Not only did Germain seems to read Legendre’s and Gauss’s books as soon as they were published and to quickly appropriate their content, but she also studied the few arithmetic publications as they appeared, or before: she took notes about a treatise by Barlow36 , she received memoirs by Gauss (see below) or Poinsot.37 I also found notes by Germain entitled “De l’équation x 5 +y5 = 2a z5 ” about a memoir by Gabriel Lamé (1795–1870) which was apparently never published but might have been presented to the Académie des sciences.38 She studied mostly themes developed in DA—quadratic, cubic and biquadratic residues, cylclotomy and quadratic forms—and worked on FLT. Germain thus had access to and studied extensively arithmetic publications from the beginning of the century. These allowed her to acquire the necessary level of expertise to produce original research in number theory. As will be highlighted in the next subsection, these readings shaped her arithmetic work. This and the known exchanges she had with Gauss, Legendre, Libri or Poinsot show that she was well integrated into the arithmetic networks of her time. The fact that she worked in number theory in the 1810s and 1820s was also recognised by several scholars. Antoine-Augustin Cournot (1801–1877) for example mentioned her in his account of Legendre (1827) published in 1828 in the Bulletin de Férussac.

35 The manuscripts are held in the Bibliothèque nationale de France in Paris and the Biblioteca Moreniana in Florence. Several letters can also be found in those libraries and in the Staats- und Universitätsbibliothek in Göttigen. The correspondence between Germain and Gauss is reproduced in Del Centina and Fiocca (2012) and some letters from Germain to Libri, Poinsot, Lagrange and Legendre are transcribed in Del Centina (2005). Germain’s arithmetic manuscripts are partially or fully reproduced and commented in Laubenbacher and Pengelley (1999), Del Centina (2008), and Laubenbacher and Pengelley (2010). 36 See Barlow (1811). 37 In a letter to Poinsot dated July 2, 1819, Germain thanked him for a memoir he sent to her before its publication. See Poinsot (1820). 38 On her notes, she added “Notes prises du mémoire de M. Lamé” (Bibliothèque nationale de France, Département des manuscrits, Manuscript FR 9114, f. 156–161). Lamé’s memoir may be the work he submitted on the occasion of the Académie prize concerning FLT in 1818. See Goldstein (2009).

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7.3.3 Epistolary Exchanges Between Two Mathematicians in Number Theory: Gauss and Germain (1804–1829) In her first letter to Gauss, dated November 21, 1804, Germain showed that she was familiar with the concepts, notations and methods of Gauss. She quoted DA several times, used Gauss’ notation for congruences and cyclotomy and proposed several number-theoretic propositions: a generalisation of a theorem stated by Gauss in the 7th section, a new proof for the quadratic character of 2 and a statement about a particular case of FLT. But from the first answer by Gauss, June 16, 1805, we find an imbalance in the correspondence between the two mathematicians: he warmly congratulated Germain for her new proof of the quadratic residue behaviour of 2 but he did not comment on her other arithmetic propositions. Gauss emphasised the fact that he had no time to pursue his research in number theory, monopolised as he was by his astronomical occupations. Table 7.1 shows that Gauss and Germain exchanged mathematical results, proofs and methods, as well as memoirs, books, and personal news. Here I chose three examples to highlight the nature of the mathematical exchanges between Gauss and Germain. Table 7.1 List of known letters between Germain and Gauss (from Del Centina (2005); Del Centina & Fiocca (2012)). Complete bibliographical references indicated in the table can be found in the bibliography Date

Sender

Recipient

Main discussions about Number theory

1804/11/21

Le Blanc

Gauss

Sec. 7 of Gauss (1801), FLT (exp. p – 1, p = 8 k + 7), 2 as a quadratic residue, variety of proofs

1805/06/16

Gauss

Le Blanc

2 as a quadratic residue, No time, Bookseller

1805/07/21

Le Blanc

Gauss

Quadratic forms, power residues, generalisations

1805/08/20

Gauss

Le Blanc

Sending Gauss (1799)

1805/11/16

Le Blanc

Gauss

Quadratic forms

1807/02/20

Germain

Gauss

Cyclotomy, discussion about her identity

1807/04/30

Gauss

Germain

Counterexample to explain an error by Germain, No time, Results on cubic and biquadratic residues to prove

1807/06/27

Germain

Gauss

Proofs of Gauss’ theorems & generalisations

1808/01/19

Gauss

Germain

Göttingen, Congratulations, Work on cubic and biquadratic residues, sending Gauss (1808)

1808/03/19

Germain

Gauss

Proof of a result of Gauss (1808), Forwards Legendre’s thanks

1809/05/22

Germain

Gauss

Thanks for Gauss (1811), 2 as a biquadratic residue, questions about possible generalisation of reciprocity law

1809/05/26

Germain

Gauss

Thanks for his astronomical treatise

1819/05/12

Germain

Gauss

Sending of Poinsot (1820), Comments on Gauss (1818), Poinsot (1820), cubic and biquadratic residues, FLT

1829/03/28

Germain

Gauss

Thanks for Gauss (1828)

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In 1807, Gauss learned of Germain’s true identity. In the letter dated February 20, 1807, Germain justified her use of a pseudonym and added an arithmetic note containing four results on numbers of the form x 2 + ny2 , numbers that had been the subject of previous research by Leonard Euler (1707–1783), Lagrange, Legendre and Gauss for example. In his reply of April 30, 1807, Gauss noticed that some of her conjectures were false, including the following: if the sum of two n-th powers is of the form h2 + nf 2 , then the sum of the two numbers is of that form as well.39 Gauss illustrated with a counterexample: 1511 + 811 = 8658345793967 = 15958262 + 11.7453912 but 15 + 8 = 23 = x 2 + 11y2 . First, Gauss expressed the pleasure he had reading Germain’s note and was careful not to upset her: The learned notes, of which all Your letters are so richly filled, have given me a thousand pleasures. I have studied them attentively, and I admire the ease with which you have penetrated all the branches of Arithmetic, and the sagacity with which You have been able to generalise and perfect them. I beg You to consider as a proof of this attention, if I dare to add a remark to a place of Your last letter.40

It is interesting that Gauss took the time to find a non-trivial large counterexample: he used several methods he developed in DA in order to calculate what seems to be the smallest counterexample for n prime and two co-prime numbers for the n-th powers.41 Gauss also explained to Germain why another of her propositions was not true by using his theory of quadratic forms. In this same letter, he also announced his progress on the theory of cubic and biquadratic residues. Gauss gave the statement of three propositions on the cubic residue behaviour of 2 for prime numbers of the form 3n ± 1, on the biquadratic character of 2 or for prime numbers of the form 8n ± 1 and on quadratic residues without any proof. Germain responded on June 27, 1807, and proposed a proof for each statement sent by Gauss. For cubic and biquadratic residues, she used a method she found in the 7th section of DA. She also submitted to Gauss new similar statements. When Gauss answered to Germain in January 1808, he did not comment precisely on her work, but he had written in July 1807 to Wilhelm Olbers (1758– 1840), enthusiastic after he read Germain’s letter: Lagrange still shows great interest in astronomy and higher arithmetic; he considers the two theorems (for which primes 2 is a cubic or a biquadratic residue) which I told you about some time ago, ‘what one can have that is most beautiful and difficult to prove’. But Sophie Germain has sent me the proofs of them; although I have not yet been able to look carefully through them, I think they are good, at least she has approached the problem in the right direction; they are only somewhat longer than they need to be ?.42 la somme des puissances n-ièmes, de deux nombres quelconques est de la forme h2 + nf 2 la somme de ces deux nombres eux-mêmes sera de la même forme.” 40 “Les notes savantes, dont toutes Vos lettres sont si richement remplies, m’ont donné mille plaisirs. Je les ai étudiées avec attention, et j’admire la facilité avec laquelle vous avez pénétré toutes les branches de l’Arithmétique, et la sagacité avec laquelle Vous les avez su généraliser et perfectionner. Je Vous prie d’envisager comme une preuve de cette attention, si j’ose ajouter une remarque à un endroit de Votre dernière lettre.” Del Centina and Fiocca (2012), p. 672. 41 See MacKinnon (1990), and Waterhouse (1994). 42 “Lagrange interessirt sich noch mit vieler Wärme für die Astronomie und höhere Arithmetik; die beiden Probe-Theoreme (in welchen Primzahlen 2 ein kubischer oder ein biquadratischer Rest ist) 39 “si

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So Gauss appreciated the arithmetic exchanges he had with Germain, with whom he could discuss on the main arithmetic subjects in DA and in his further research: quadratic, cubic and biquadratic residues, quadratic forms and cyclotomy. On the other hand, although Germain referred to FLT as early as 1804, it seems that Gauss never gave her any feedback on this question. A possible explanation of this nonreaction is that Gauss was simply not interested by Fermat’s equation, as he wrote to Olbers in 1816 when FLT was the subject of the Grand prix at the Académie des sciences: I am very much obliged for your news concerning the [newly established] Paris prize. But I confess that Fermat’s theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.43

Some general features emerge from this correspondence. Through her studies and her connections with mathematicians, Sophie Germain had a great mastery of Gauss’ arithmetic work and, more generally, of the number-theoretic research of her time. Gauss admired her arithmetic skills very much and seemed to consider her as a full-fledged colleague in their exchanges, although he never had much time to pursue his arithmetical research or to write to her at greater length. Let us note that Gauss is not the only mathematician who asked Germain to prove an arithmetical claim: according to one of Germain’s manuscript, Lagrange also asked her to prove a result about triangular numbers.44

7.3.4 Germain’s Arithmetical Practices and Program: The Case of Fermat’s Last Theorem Germain’s known result about FLT is the one published in the memoir of Legendre, Recherches sur quelques objets d’analyse indéterminée et particulièrement sur le théorème de Fermat presented to the Académie des sciences in 1823 and published in 1827: Theorem 1 For an odd prime exponent p, if there exists an auxiliary prime θ such that there are no two nonzero consecutive P -th powers modulo θ , nor is P itself a P die ich auch Ihnen vor einiger Zeit mittheilte, hält er für “ce qu’il peut y avoir de plus beau et de plus difficile à démontrer”. Aber die Sophie Germain hat mir die Beweise derselben geschickt; noch habe ich sie zwar nicht durchgehen können, ich glaube aber, dass sie gut sind; wenigstens hat sie die Sache von der rechten Seite angegriffen, nur etwas weitlaüfiger sind sie als nöthig sein wird.” Del Centina and Fiocca (2012), p. 627. 43 “Für Ihre Nachrichten, die Pariser Preise betreffend, bin ich Ihnen sehr verbunden. Ich gestehe zwar, daß das Fermat’sche Theorem als isolirter Satz für mich wenig Interesse hat, denn es lassen sich eine Menge solcher Sätze leicht aufstellen, die man weder beweisen, noch widerlegen kann.” The translation is from Laubenbacher and Pengelley (2010). 44 See Alexanderson (2012).

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-th power modulo θ , then in any solution to the Fermat equation zp = x p + yp , one of x, y, or z must be divisible by P 2 . Legendre explicitly credited Germain with this theorem and the table where she verified the existence of an auxiliary prime for all primes p less than 100. For numerous p, this theorem gives a way to prove the first case of FLT, namely to prove there is no integer and nonzero solutions prime to p. This theorem constituted the first published attempt for a general result concerning FLT. Indeed, before Ernst Kummer (1810–1893)’s work in the 1840s, only three cases (for p prime) exponents were proved: 3 (Euler), 5 (Legendre and Dirichlet) and 7 (Lamé).45 From Germain’s manuscripts and the letters she exchanged with Gauss,46 we know that Germain had a more ambitious plan to prove FLT, initiated at least in 1804. She tried to construct a proof for whole families of exponents—contrary to Legendre or Johann Peter Gustav Dirichlet (1805–1859), who obtained proofs for a single exponent—and she imagined a plan to prove FLT in general. In 1819, she explained the “metaphysics” of her method to Gauss: The order in which the residues (powers equal to the exponent) are distributed in the sequence of natural numbers determines the necessary divisors which belong to the numbers among which one established not only the equation of Fermat, but also many other analogous equations.47

Congruences and roots of unity were fundamental in her plan which utilised her precise knowledge of Gauss’s work. In the above quotation, she also insisted on the importance of the order “in which the residues are distributed”. She attributed this “metaphysics” of the method to another French mathematician, Poinsot, who presented his paper about algebra and number theory Mémoire sur l’application de l’algèbre à la théorie des nombres at the Académie des sciences in 1818. As mentioned before, she also wrote a letter to Poinsot in 1819 to thank him for sending her his memoir—which was later published, in 1820—and to congratulate him for his use of imaginary roots of congruences and for his reflections on the different orders that can be obtained from the consideration of the roots of certain congruences. This highlights yet again that Germain had access to some of the latest number-theoretical research, which she studied attentively. Germain’s method is indeed based on an analogy between equations and congruences and on the order of the residues of p-th powers modulo a prime θ. If the equation x p + yp = zp (p prime) has a nonzero solution, then for every prime θ, the congruence x p + yp ≡ zp (mod θ ) also has a nonzero solution. But, if x is not divisible by θ, there exists an inverse of x modulo θ, namely a number k such that 1 + (ky)p ≡ (kz)p (mod 45 On

the history of FLT, see for example Edwards (1977), and Corry (2010). mentionned before, a detailed study of Germain’s work on FLT is given in Laubenbacher and Pengelley (2010) and Del Centina (2008). 47 “L’ordre dans lequel les résidus (puissances égales à l’exposant) se trouvent placés dans la série des nombres naturels détermine les diviseurs nécessaires qui appartiennent aux nombres entre lesquels on établit non seulement l’équation de Fermat, mais encore beaucoup d’équations analogues à celles-là.” Del Centina and Fiocca (2012), p. 690. 46 As

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θ ). In that case, a new congruence can be found and it follows that there must exist some nonzero consecutive p-th power residues. Germain’s idea to prove FLT for a given p was to show that there exists an infinity of θ: it would mean that x(or y, or z) is divisible by an infinity of prime numbers, which is impossible. But, as Germain soon observed, her plan could not succeed because some of the p’s are such that there is not an infinitude of θ s. But, although Germain’s plan did not succeed, she showed really impressive skills in calculation, obtained general results on FLT, and managed to show that the potential solutions of certain cases should be of a very big size, as she wrote to Gauss in 1819: I have never been able to arrive at infinity, although I have pushed back the limits quite far by a method of trials too long to describe here […] You can easily imagine, Monsieur, that I have been able to succeed at proving that this equation is not possible except with numbers whose size frightens the imagination […]48

These different facts underline several interesting points regarding Germain’s status in number theory at that time: the content of her work lay both in the line of arithmetic work on indeterminate equations and based on a contemporary analogy between equations and congruences (Legendre, Poinsot, Libri or Cauchy) and integrated tools, methods and results of DA. She was indeed quite familiar with those publications and connected to the mathematicians who published them. So if, as a woman, she could not be taught or attend scientific institutions, the marginal status of number theory at the beginning of the nineteenth century and the fact that it was not taught in the École polytechnique, for example, meant that her gender stigmatised her less than other mathematical domains. Every mathematician wishing to learn number theory had to study the same texts as Germain (Gauss’ and Legendre’s writings mostly) and the minuscule number of amateurs in number theory certainly allowed her to have privileged contact with Gauss. Thus, the exclusion of women from academic spheres seems to have been less disabling for studying number theory: the specific mathematical culture transmitted in an institution such as the École polytechnique seems to have been less of a problem for Germain in the case of number theory than for the theory of elasticity. Furthermore, being a woman from the bourgeoisie certainly facilitated her access to scholarly networks compared to a man of a lower social class who did not attend institutions such as the École polytechnique. Nevertheless, Germain never directly published her results on FLT. Maybe it was because she knew that she did not succeed in achieving her grand plan. As a woman, she did not have access to some institutions that made it easier to publish mathematical research, as for example the École polytechnique and its Journal. But, more generally, the possibilities of publishing an article in number theory outside of an academic framework were relatively small until the end of the 1820s, or even the 1830s. Indeed, another mathematician of her time, Lebesgue, whose work was mainly concerned 48 “Je n’ai jamais pu arriver à l’infini quoique j’ai reculé bien loin les limites par une méthode de tâtonnement trop longue pour qu’il me soit possible de l’exposer ici. […] Vous concevrez aisément, Monsieur, que j’ai dû parvenir à prouver que cette équation ne serait possible qu’en nombres dont la grandeur effraye l’imagination […]” Del Centina and Fiocca (2012), p. 691. Translation is from Laubenbacher and Pengelley (2010).

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with number theory but who was neither a polytechnician nor an academician, was only able to regularly publish his arithmetical memoirs as of 1836, in the Comptes rendus de l’Académie des sciences and in the Journal de mathématiques pures et appliquées, created in 1835 and 1836, respectively.

7.4 Germain’s “Second Life”: Some Appropriations of Germain’s Life and Work After Her Death If Germain was periodically quoted for the content of her mathematical and philosophical work, her status as a woman mathematician was also often mobilised to nourish the debates on the woman question under the French Third Republic. This part is a first attempt to analyse how the figure of Germain entered the history of science and was used in the discourses on science and women in France from her death in 1831 to the 1910s. For this, I collected some two hundred references to Germain in French publications published from the death of Germain until the early 1930s and available on several digital databases.49 These sources show multiple images of the life and work of Germain mainly in France under the Third Republic. Through Germain, we can see how a woman mathematician, a professional scholar who was different, even opposite to the models of the woman then in force, was used as an example or a counterexample to fuel the debates on women, their education and their position in French society.50 Beyond the reception of her work, Germain was mentioned very regularly in the revival of debates on the question of women from the 1840s at least. Indeed, as early as the 1820s, many authors came back to the social demands of the Enlightenment, especially those concerning the place of women in society. In a context of significant literacy, the development of state education and instruction, the beginning of industrialisation (and therefore, the emergence of the figure of the working-class woman), and the question of creating public education for girls (thus countering the Church’s monopoly) was regularly debated. During this period, Utopian movements, such as the Saint-Simonians, began to claim rights for women. It was also the century of

49 I used mainly Google Books, Gallica, Numdam and the Jahrbuch database. By the nature of these databases and the way the texts are digitalised, the result cannot be considered an exhaustive and precise quantitative study. But this study of Germain’s receptions has the merit of highlighting several uses of Germain’s narrative in contemporary debates on women and mathematics in particular. 50 Compared to The Many Lives and Deaths of Sofia Kovalevskaya by Katharina Rowold (2001), I also use the references to Germain to study discourses on the woman question but in a different national context, from different initial features of the woman mathematician (Germain was not able to get a higher education or an academic position, she was single her whole life and the institutional and scientific contexts were different). Thanks to digital databases and recent secondary literature such as Kalifa et al. (2011) and Offen (2018), I also was able to take into account the writings published in the daily and specialised press.

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the “Civilisation of the journal”51 : the rise of newspapers was particularly important in the nineteenth century, in all areas. I already mentioned scientific and mathematical journals. This was also the case for the press dedicated to specific categories of the population. For example, the women’s journal La Femme nouvelle was created in 1832 on the initiative of several Saint-Simonians. First feminine then feminist press developed throughout the century. Nevertheless, women’s rights advocates practically never claimed absolute equality between men and women; they defended the complementarity of the sexes, which would not necessarily be hierarchical and associated with separate private spheres (feminine) and public spheres (masculine). Moreover, in post-revolutionary France, family prevailed as a basic sociopolitical unit in society. This postulate of a fundamental difference between the sexes was generally shared throughout Europe by the defenders and detractors of women’s rights during the nineteenth century. An emblematic illustration of this was the theory of Ernest Wilfred Legouvé (1807–1903), summed up by the motto “equality in difference”. It was the leitmotif of the Republican movement of the late 1840s and Republican politics under the Third Republic in France.52

7.4.1 Mathematical, Philosophical and Material Memories of Germain After Germain’s death in 1831, Libri’s eulogy and the publication of her philosophical work by her nephew, some biographies can be found in the 1830s and 1840s in biographical dictionaries. She was then described as one of the creators of mathematical physics; there were sometimes references to her philosophical writings but her work in number theory was generally hardly mentioned. For several reasons, it was mostly from the 1870s that Germain’s number-theoretical and philosophical achievements were regularly quoted. Before 1870, occasional references to her scientific research were made in scholarly periodicals. An example can be found in the Nouvelles annales de mathématiques, a French mathematical journal published from 1842 to 1927, dedicated to teachers and students (candidates to polytechnic and normal schools at first). It contained mathematical articles, mathematical questions and answers sent by the editors and readers, but also biographical and bibliographical reviews and texts. As chief editor of the journal, Olry Terquem (1782–1862) mentioned Germain at least three times between 1857 and 1861. In 1857, below the answer to a mathematical question by a certain “Mlle Adolphine D”, he noted that five French women succeded in mathematical studies: Marie Crous (17th century), Jeanne Dumée (1660–1706), “la célèbre marquise du Châtelet”, Hortense Lepaute (1723–1888) and Germain, “laureate of the Académie des sciences for the very difficult question of the vibrating 51 See 52 See

Kalifa et al. (2011). Offen (1986, 2018).

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plates”.53 After Joseph Bertrand (1822–1900) announced the deposit of Germain’s manuscripts to the Académie des sciences,54 Terquem published in 1860 a 4-page biography of Germain where he mentioned her “epistolary exchange with Gauss”55 and Germain’s new theorems in number theory published by Legendre.56 The purpose of this biography was to introduce his review of Germain’s philosophical work, which was to be published in 1861. As early as 1860, he highlighted the importance of finding and editing the correspondence between Gauss and Germain. Epistolary exchanges between Gauss and Germain were then mainly rediscovered from the end of the 1870s, as Schering was preparing a volume of Gauss’s collected papers.57 The content of their correspondence was vividly commented, especially to reconstitute the chronology of Gauss’s work on cubic and biquadratic residues. Nevertheless, the arithmetical aptitudes of Germain were also recognised. For example, she was qualified as a “conscientious and competent reader” by the Belgian mathematician Paul Mansion (1844–1919): The admiration expressed by Gauss, at the end as at the beginning of his letter, should not surprise us. Aside from Sophie Germain, nobody seemed to take an interest in the Disquisitiones, nor to dedicate to these the attention they deserved. It was natural that the young Hanoverian mathematician, still not very well known at that time, felt and expressed a strong pleasure at having found in Sophie Germain such a conscientious and competent reader.58

It seems that, from this period, Germain’s work entered the history of number theory more vividly. If we take the example of two important books about the history of number theory, there was no citation of Germain’s arithmetical writings in the Report on the Theory of Numbers published by Henry J. S. Smith (1826–1883) between 1859 and 1865,59 whereas they were mentioned several times in the History of the Theory of Numbers published by Leonard E. Dickson (1874–1954) between 1919 and 1923, in connection with FLT60 and the factors of an ± bn (where a, b, n are integers).61 Smith and Dickson did not have the same perspective when writing those texts. On the one hand, Smith compiled his report at the request of the British 53 “lauréat

de l’Académie des sciences pour la question très-difficile des plaques vibrantes”. rendus hebdomadaires des séances de l’Académie des sciences, vol. 49, p. 45, 1859. 55 “commerce épistolaire avec Gauss”. 56 “En arithmologie, elle démontra des théorèmes nouveaux, que Legendre a admis dans son ouvrage.” Terquem (1860), p. 12. 57 References to Germain’s work in connection with the publication of Gauss’s works and the discovery of their correspondence are studied in Del Centina and Fiocca (2012). 58 “L’admiration que Gauss exprime ainsi, à la fin comme au début de sa lettre, ne doit pas nous étonner. À part Sophie Germain, personne ne semblait, en effet, s’occuper des Disquisitiones ni leur accorder l’attention dont elles étaient dignes. Il est tout naturel que le jeune Géomètre hanovrien, encore peu connu à cette époque, éprouvât et exprimât un vif contentement d’avoir trouvé, en Sophie Germain, un lecteur consciencieux et compétent.” Mansion (1880), p. 219. Quotation and translation quoted from Del Centina and Fiocca (2012), p. 597. 59 See Smith (1859–1865). 60 See Dickson (1919–1923), vol. 2, pp. 732–735. 61 See Dickson (1919–1923), vol. 1, pp. 382–383. 54 Comptes

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Association for the Advancement of Science and organised it according to the theory of congruences and the theory of forms, that is, to say according to Gauss’s DA structure.62 On the other hand, for his History of the Theory of Numbers, Dickson studied an impressive number of publications and manuscripts. He had some difficulties finding money for the publication for his massive compendium, which was finally funded by the Carnegie Institution of Washington.63 In his History, Dickson adopted an organisation centred on divisibility and diophantine analysis (which is the topic of the massive second volume). Beyond the facts that Germain’s number-theoretical work was then better known and that Dickson was able to read some of her manuscripts and letters to Gauss, her results fitted better in Dickson’s presentation. From the 1880s, Germain’s number-theoretical work was also discussed from her correspondence with Gauss by several mathematicians such as Angelo Genocchi (1817–1889) and Mansion, for example, who promoted number theory in scholar societies as the Association française pour l’avancement des sciences or mathematical journals as Nouvelle correspondance mathématique. They belonged to a cluster identified in Goldstein and Schappacher (2007) where the authors were mostly engineers and high school teachers, such as Genocchi or Édouard Lucas (1842–1891). They worked mainly on arithmetic questions already studied in Legendre’s and Gauss’s work. Like Germain, they approached these questions without using analysis: Germain’s research was, therefore, close to their interests. During the twentieth century, mathematicians also followed Germain’s path and developed new results about FLT by using her results or methods, sometimes without knowing it. Her arithmetic work is now quoted in most books about the history of number theory and, as mentioned before, it has been the focus of whole studies by historians of mathematics. Germain’s philosophical writings also enjoyed renewed interest in the late 1870s. The positivist journalist Stupuy edited them in 1879, with an Étude sur la vie et les œuvres de Sophie Germain. Not only was Germain’s philosophical thought debated by many philosophers and scholars, especially positivists, but her precise biography was put back on the scene. Indeed, according to Stupuy, the historical character of Sophie Germain was constructed especially from the last quarter of the nineteenth century. In 1879, he commented: A painful confession must be made. While so many women have found fame in frivolous writings, the only French woman who has succeeded in hard work, was esteemed by mathematicians, who, moreover, do not understand an entire aspect of her genius, is scarcely known to the public.64

But, less than twenty years later, for the new edition of her philosophical works, Stupuy highlighted a radical change: 62 See

Goldstein and Schappacher (2007), p. 55. Fenster (2007). 64 “Il faut en faire l’aveu pénible. Tandis que tant de femmes ont trouvé la célébrité dans les écrits frivoles, la seule femme française qui ait réussi dans les travaux sévères, estimée des géomètres, auxquels d’ailleurs tout un aspect de son génie échappe, est à peine connue du public.” Stupuy (1879), “Étude sur la vie et les œuvres de Sophie Germain”. 63 See

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Sophie Germain has taken up this second life, which is made of the memory of the living: her work and her biography have conquered their place in the memory of humans; her tomb, abandoned, was found and restored; the Paris City Council, always concerned with Parisian glories, gave the name of Sophie Germain to one of the streets of the capital, to one of its girls’ high schools, and her bust, reconstituted from the phrenological head preserved at the Museum of Natural History, adorns the main courtyard of this school. In addition, a commemorative plaque was placed on the house where she died.65

Indeed, with the (re)editions of Germain’s scholarly work and the vivid debates concerning the role of women in society, Germain became a regular reference for scholars, journalists, etc. The publication of her philosophical memoirs was quoted and discussed in numerous articles. For example, Louis Liard (1846–1917), a philosopher known for the reform of the French university in 1896 named after him, published a review of Germain’s Œuvres philosophiques in the Revue philosophique de la France et de l’étranger 66 ; Doctor Second commented on the publication of Germain’s philosophical writings in the positivist journal La Philosophie positive edited by Émile Littré (1801–1881) in October 1879. He also referred to Germain’s philosophy in La Nouvelle revue in an article about biology teaching.67 In April 1879, the architect and art historian Eugène-Emmanuel Viollet-Le-Duc (1814–1879), known for his rationalist approach to ornamental art and his writings with positivist tendency, published a text entitled “Sophie Germain”, celebrating Stupuy’s edition, in the daily Republican newspaper Le xixe siècle. Considered a moderate Republican in the 1870s, Viollet-Le-Duc promoted an alliance between science and art as part of a “national, political and social regeneration”.68 In his article about Germain, ViolletLe-Duc selected quotes from her work to underline the fundamental importance of close relations between art and science rather than a reductive specialisation. Beside comments and precise writings about Germain’s philosophical and number-theoretical work by scholars, mentions of Germain can be found in very different contexts. In the 1880s, she was the subject of discussions about naming places and buildings, especially in the Paris city council. As a councillor of the ninth arrondissement of Paris from 1884 to 1893, Stupuy may actually have played a role in building a civic legacy of Sophie Germain in Paris. For example, in the 1880s, after numerous debates, the Paris city council voted to give the name of Sophie Germain to a Parisian street. In 1882, the first école primaire supérieure for girls, in 1882, was named after Germain, also after discussions in the Paris City Council. The Paris City Council also ordered a marble bust of Germain for the Sophie-Germain school. 65 “Sophie Germain a repris cette seconde vie qui est faite du souvenir des vivants: son œuvre et sa biographie ont conquis leur place dans la mémoire des hommes; son tombeau, délaissé, a été retrouvé et restauré; le Conseil municipal de Paris, toujours soucieux des gloires parisiennes, a donné le nom de Sophie Germain à l’une des rues de la Capitale, à l’une de ses écoles supérieures de jeunes filles, et son buste, reconstitué d’après la tête phrénologique qui existe au Museum d’histoire naturelle, orne la cour principale de cette école. De plus, une plaque commémorative a été aposée sur la maison où elle est morte.” Stupuy (1896). 66 See Liard (1879). 67 See Second (1879). 68 See Baridon (2010).

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The traces of associated discussions can be found in the daily and weekly press.69 These debates show that Germain had gradually become a reference as a woman mathematician. These facts of her life were indeed used after her death, and especially from the 1870s, in the context of various subjects. This is the purpose of the following subsections.

7.4.2 Germain, a Woman Among Women in Mathematics and Science As a woman, Germain was also very often mentioned as a pioneer among other pioneers. Thus, references to Germain were very often made to celebrate remarkable achievements by a woman in mathematics: we already mentioned Terquem’s text when he commented on the fact that a woman—Adolphine D.—answered to a question in the Nouvelle annales de mathématiques. Germain’s biography was also summed up in numerous articles published at the time when Kovalevskaya attended the sessions of the Académie,70 won the Borda prize at the end of the 1880s,71 and died in 1891.72 The doctoral thesis obtained by Dorothea Klumpke (1861–1942) in 1893 was followed by articles on women and mathematics and the growing number of women studying in university. For example, in Le xixe siècle, André Balz (1845– 1931), agrégé and docteur ès lettres, published a text entitled Les femmes et les astres in which he was ironic about the scope of Klumpke’s astronomical research, “at a sidereal distance from contemporary utilitarianism”.73 He also noted that if there already were several women who had a doctoral thesis in medicine, Klumpke was the first woman with a mathematical thesis that “has another scope and also this superiority that it can serve absolutely nothing”.74 The debates about Marie Curie

69 For example, beside the Bulletin de la ville de Paris, traces of debates about naming a Parisian street after Germain can be found in: Le xixe siècle (July, 30, 1882), Le Petit Parisien (July, 30, 1882), Le Petit journal (July, 30, 1882), Le Figaro (August, 28, 1882), Le Radical (August, 30, 1882) Journal des débats politiques et littéraires (August, 8, 1882), Le Passant (August, 5, 1882), etc. 70 For example, Journal des débats politiques et littéraires (June 29, 1886). 71 Journal des débats politiques et littéraires (December 11, 1888), Le Petit journal (December 29, 1888), or the bimonthly La Femme (vol. 11, no. 3, 1889), created in 1879 and whose subtitile was sometimes “organe des institutions féminines, chrétiennes, sociales et de l’Union nationale des amies de la jeune fille”. 72 Among other references: Le Petit journal (February 17, 1891), Le Petit Parisien (February 18, 1891), La Croix (February 18, 1891). 73 “à une distance sidérale de l’utilitarisme contemporain”. 74 “a une autre envergure et aussi cette supériorité qu’il ne peut servir absolument à rien”.

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(1867–1934)’s possible election at the Académie des sciences was also the occasion for short notes and articles containing references to Germain in various journals.75 From the end of the nineteenth century, several collections of biographies of women in mathematics or women in science appeared in parallel with the rise of feminist movements (and anti-feminist reactions). Among them, we can cite Les femmes dans la science by Alphonse Rebière (1842–1900) in 1894 and 1897, Weibliche wissenschaftliche Leistungen in den Gebieten Mathematik, Astronomie und Nautik by Georg Weyer in 1897, Les femmes dans la science in 1909 by Maurice d’Ocagne (1862–1938), or Women in Science by H. J. Mozans (1851–1921) in 1913.76 More generally, in France, if mistrust of the activities of literate women under the Third Republic was slightly less than under the previous regime, the debates on (ab)normality of scholarly women were continuing. Articles, books on women writers, and some anthologies during the 1870s and 1880s were common. The first anthology edited by a woman seems to have been published in 1893, but a collection of biographies of notable women, under the title Les femmes dans l’histoire, appeared in 1888 by Henriette de Witt (1829–1908), daughter of the former minister of education François Guizot.77 De Witt used Germain and Mary Sommerville (1780–1872) as evidences in her arguments against the usefulness of higher education for women, as we will see later. A work dedicated to women in science was introduced by the mathematics teacher Rebière in 1894. He was also involved in the contemporary debates on female education: in the 1880s, following the Camille Sée law (1880), which introduced public secondary education for young girls, Rebière had analysed the place of mathematics in the different classes of female secondary education.78 In 1894, he gave a lecture in front of the Saint-Simon circle79 entitled “Les femmes et la science” in which he presented the portrait of six women of science (Hypatia, du Châtelet, Maria Gaetana Agnesi (1718–1799), Germain, Sommerville, Kovalevskaya) and announced a more ambitious work on the same subject. The second edition of Les femmes et la science appeared in 1897 and contained nearly 650 portraits of women engaged in some way in science and a collection of historical and contemporary quotes on women’s ability 75 For example: Le Figaro (January 3, 1911), La Paix sociale (February 2, 1911), L’action féminine. Bulletin officiel du conseil des femmes françaises (February, 1910) or Le Gaulois (February 8, 1922). 76 See Tobies et al. (2001). 77 See Offen (2018). 78 See Hulin (2008). 79 According to the first volume of the Bulletin of the Saint-Simon circle, the purpose of this foundation, created in 1883, was to bring together men who “have the cult of science and the mind” (“ont le culte de la science et des choses de l’esprit”) by choosing history as “common ground” (“terrain commun”) and thus to set up a “vast association inspired by the love of science and homeland” (“vaste association inspirée par l’amour de la science et de la patrie”). This positioning and the reference to Saint-Simon suggest, among other things, a position favorable to sex equality (in the sense that this may take in the 1880s). It should be noted that scientists like Paul Bert, an actor in the reforms for girls’ education, were among them. The subject of Rebière’s conference seems perfectly adapted to his audience.

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in science. Note that apart from a summary of the different opinions on the question,80 Rebière did not give his own opinion. Nevertheless, these women of science had been much in the news. The two books by Rebière were the subject of numerous reports and the starting point of articles on the question of women in science, in which his portraits were reproduced or summarised. Some authors used their review of Rebière’s work to develop their own ideas on the issue; the reading of their writings reveals some trends of the time. Émile Fage (1822–1906), a member of the Société des lettres, sciences et arts de la Corrèze, used his review of Rebière (1894) to give his opinion on the life of women mathematicians. Thus, if Germain is one of the women who were “less light than the Marquise du Chatelet and less tormented than Sophie Kowalevski”, the conclusion remained unchanged: “the learned women, even the happiest, have ignored the true happiness”.81 For Fage, a learned woman could not be happy because she was not able to find essential love, a spouse and a family.82 The journalist (and future Republican deputy) Émile Cére (1863–1932) stated an opposite opinion from Rebière (1894). He wrote in Le petit journal (22 September 1896) about his “women’s academy of science” (“académie des sciences féminine”), of which Germain would be the president: A woman who studies mathematics is she necessarily a bad housewife? No, she is not. Most of the scholars whom we spoke of were accomplished wives, excellent mothers, and those who will follow them in the career of the sciences will, we hope, have their domestic virtues, with their scientific merit.83

These words refer to a central question debated at least since the beginning of the century: the fact that women’s happiness was based on their domestic life as a wife and mother and that the practice of science risked driving them away. During the 1890s, feminism was everywhere: numerous articles about feminism and women’s rights were published in the daily press and in popular and specialised journals.84 In 1897 and 1898, the term “feminism” was contentious for several authors reviewing85 : authors used Germain among other women to prove their point.86 In his review, Jacques Boyer (1869–?) announced his goal: “Since everything is feminism, the moment seems favourable to sketch in broad strokes this little-known part of its history.”87 He then used Rebière (1897) to conclude: 80 See

Rebière (1897), pp. 289–290. moins légères que la marquise du Châtelet et moins tourmentées que Sophie Kowlevski”, “[…] les savantes, même les plus heureuses, ont ignoré le vrai bonheur.” Fage (1894), pp. 254–255. 82 See Fage (1894), p. 253. 83 “Une femme qui étudie les mathématiques est-elle forcément une mauvaise ménagère? Non pas. La plupart des savantes dont nous avons parlé furent des épouses accomplies, d’excellentes mères de famille et celles qui les suivront dans la carrière des sciences auront, nous l’espérons bien, leurs vertus domestiques, avec leur mérite scientifique.” Cére (1896). 84 See Offen (2018). 85 See Rebière (1897). 86 See, for example, Le Temps (June, 5); and Boyer (1898), p. 600. 87 “Puisque tout est au féminisme, le moment paraît favorable pour esquisser à grands traits cette partie peu connue de son histoire.” 81 “[…]

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[…] we met few women initiators […] Now, to establish these findings, we rely only on the experience of centuries [without using contemporary medical research] […] That, by the way, could not be due to a lack of education, but to [the] emotional nature [of the woman], to her sometimes quick but rarely deep mind that does not seem to predispose women to play an important role in science.88

Still in 1897 and 1898, a series of six articles entitled Les femmes dans la science was published in the Christian feminist journal La femme by the French feminist Marie Abbadie d’Arrast (1837–1913).89 She summed up several biographies of Rebière (1897) by noticing that Rebière did not position himself in relation to the various comments about women reported in his book.90 She also supplemented this with other examples, including Charlotte Angas-Scott, who belonged to the “United States School of Women’s Science”,91 and argued for sex equality in the sciences. Another example of Rebière’s reader is the mathematician d’Ocagne who wrote several historical summary articles, especially on calculation and calculating machines.92 In a conference entitled Les femmes dans la science held on November 30, 1908, at the Université des Annales in Paris, an institution of higher education for young girls. Like Rebière, d’Ocagne listed short biographies of scientific women and some quotations on women in science. However, he introduced his text with two questions: “Are women, in general, as talented as men for the study of sciences, and if so, is it desirable that they take care of the study of sciences and in which way?”93 He noticed that among women in science, the majority were women mathematicians. He gave several reasons to justify this fact: mathematics did not require “any material work”94 ; mathematics relied first on a form of divination, of intuition before the phases of rigorous reasoning; as such, could be thought of in analogy with poetry or music. Referring to Henri Poincaré (1854–1912), d’Ocagne also stated that mathematicians needed sensitivity to understand mathematical beauty. In the background, mathematics relied, among other things, on qualities that were most often given to women. From the testimonies of mathematics teachers who 88 “[…]

nous avons rencontré peu d’initiatrices […] Or, pour établir ces constatations, nous nous basons uniquement sur l’expérience des siècles [sans utiliser les recherches médicales contemporaines] […] Cela, du reste, ne saurait tenir à un défaut de l’éducation mais à [la] nature émotive [de la femme], à son esprit vif quelquefois mais rarement profond qui ne semblent nullement prédisposer la femme à jouer un rôle important dans la science.” Boyer (1898). 89 See d’Abbadie d’Arrast (1897–1898). A few years later, she chaired the Legislation section in the Conseil national des femmes françaises, created in 1901, and encouraged international collaborations on issues on women. See Offen (2018). 90 See d’Abbadie d’Arrast (1897–1898), vol 20, no 5, p. 35. 91 “l’École scientifique féminine des États-Unis” d’Abbadie d’Arrast (1897–1898), vol 20, no. 5, p. 37. Being part of the faculty members in Bryn Mawr college since its creation in 1885, Angas Scott played a central role in the implementation of mathematics programs in this institution of education and research reserved for women. See Parshall (2015). 92 Maurice d’Ocagne is known for his work on graphical calculus: see Dominique Tournès work. 93 “les femmes sont-elles, en général, aussi douées que les hommes pour l’étude des sciences, et, dans ce cas, est-il souhaitable qu’elles s’en occupent et dans quelle mesure?” d’Ocagne (1909), p. 64. 94 “Aucune besogne matérielle” d’Ocagne (1909), p. 81.

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taught women—Felix Klein (1849–1925) in Göttingen and Carl Vogt (1817–1895) in Geneva—and continuing his analogy with music, d’Ocagne concluded that even if there were some outstanding women like Kovalevskaya, it seemed that most women were able to understand but not to invent mathematics. D’Ocagne then answered his second question: while it may be useful to encourage the few women with special skills for science, the science education for most women should remain general and superficial. Substantial science education would divert women from their family duties. On October 24, 1925, on the occasion of the public meeting of the Institut de France, d’Ocagne held another conference about women in mathematics, Quelques figures de mathématiciennes. If he noticed that the number of female scientists had become important, the point was similar. The general conclusion he uttered after Kovalevskaya’s end of life narrative was even more pessimistic: “This lamentable epilogue of such a prestigious story seems well done to illustrate Madame de Stael’s disillusioned word about the glory that, often, ‘could only be for a woman the bright mourning of happiness’”.95 From these first references, the main points of debate concerning women in mathematics (and in science more generally) arise: appropriate level and nature of education for women; compatibility of the practice of mathematics or science with happiness and the domestic life, then considered natural and indispensable for women; compatibility of the (natural or socially constructed) qualities of women with the practice of mathematics. Germain was part of the list of women mathematicians presented and used by some authors to develop certain arguments and by others to develop opposing arguments. Subsequently, we see how specific characteristics of Germain’s life and work were mobilised to fuel comments about women in mathematics in writings focused on the question of women’s education and social sciences publications.

7.4.3 Which Education for Women? Germain as an Argument to Develop or Restrain (Scientific) Teaching for Girls Throughout the nineteenth century, whether for boys and then for girls, education was not a route to social elevation in France: primary education was dedicated to the working-class and separated from secondary education, defined to educate the bourgeoisie. Thus, for girls, both forms of teaching had to have very distinct objectives: manual labour and skills to work in the factory for the working classes; arts and skills to become a good mother and a good wife for girls from the bourgeoisie. The majority of the debates concerned the teaching of girls from the bourgeoisie and in this context, education of girls was gradually considered as an issue of society: 95 “Ce lamentable épilogue d’une si prestigieuse histoire semble bien fait pour illustrer le mot désabusé de Mme de Staël sur la gloire qui, bien souvent, ‘ne saurait être pour une femme que le deuil éclatant du bonheur’” d’Ocagne (1925). With this slightly odd phrase, de Staël suggested that accomplishment would ruin women’s happiness.

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wives, mothers, educators, and women were fundamental for the regeneration or, on the contrary, degeneration of the nation.96 In the early 1830s and in the wake of the Guizot Law on primary education for boys, there was a progressive development of pedagogical and moral literature on the education of girls.

7.4.3.1

Germain as a Model of Vocation and Perseverance in Education and Moral Literature

In a biography published after the death of Germain, Libri recalled in great detail the childhood obstacles that Germain had to overcome to train in mathematics. This trait of her character was included in books of morality and education for schools, families and children, when the development of youth literature was important, following the laws Guizot (1830s) and Falloux (1850) on primary education. Le gymnase moral des jeunes personnes97 included biographies of famous women. The goal of the writer Jean-Baptiste-Joseph Champagnac (1798–1858) was to provide models for young girls: By offering our young [girl] readers a certain number of emulators capable of electrifying them by their examples, we wanted to open to them a kind of arena, where they could come to fight for merit and virtues with the models presented to them.98

A virtue was associated with each biography: for Germain, it was dedication (“vocation” in French), and her biography focused on her childhood and her perseverance to study mathematics despite her parents’ hostility.99 It is possible that Champagnac also wrote the Dictionnaire historique d’éducation published under the name of M. Delacroix,100 an inspired version of the work of the same name by Jean-Jacques Filassier (1771). Germain was introduced there as one of the “astonishing examples of dedication and sacrifice which can lead to the love of science”,101 which is “such an extraordinary vocation in a woman”.102 Germain’s childhood was also described and illustrated in one of the first journals for children, containing many illustrations, La semaine des enfants, created in 1857.103 The author of the article is likely Théodore-Henri Barrau (1794–1865), who was a pedagogue and promoter of primary education in France. Barrau also published 96 See

Mayeur (1979). Champagnac (1837). 98 “En offrant à nos jeunes lectrices un certain nombre d’émules capables de les électriser par leurs exemples, nous avons voulu leur ouvrir une sorte d’arène, où elles pussent venir lutter de mérite et de vertus avec les modèles qui leur sont présentés.” Champagnac (1837), p. 6. 99 See Champagnac (1837), pp. 52–68. 100 See La Littérature française contemporaine, 1827–1844, vol. 3, Louandre, C. and Bourquelot, F., eds. (1848), p. 296. 101 “exemples étonnants du dévouement et des sacrifices auxquels peut conduire l’amour des sciences” Delacroix (1837–1838), p. 105. 102 “une vocation si extraordinaire dans une femme” Delacroix (1837–1838), p. 108. 103 See Barrau (1860). 97 See

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the Livre de morale pratique (1872), where Germain appeared in the chapter entitled “Duties of man to himself”104 : “A woman, by her love of studying, managed to be among the first mathematicians of the nineteenth century”.105 As we will see later, in comments by some authors on the ability of women to create mathematics, Germain’s perseverance was not cited as a positive quality but more as proof of her lack of genius.

7.4.3.2

Contents and Goals of Higher Instruction for Women

In the context of the debates on the education of women differentiated from that of men, Legouvé developed his work concerning women. For him, the failure of the French Revolution could be explained precisely because of the injustice that was then reserved for women. His principle of “equality in difference” postulated a feminine specificity which made it possible to propose a form of emancipation for women while preserving the ideal of the patriarchal family and the bourgeois-style social order in France.106 The theory of Legouvé was still central in the debates at the end of the Second Empire, when Minister Victor Duruy (1811–1894) tried to create a first public secondary education for girls. It was still fundamental in the Third Republic, during the implementation of the Camille Sée law and the many debates that followed the first feedback. Indeed, the form, content and purpose of this secondary education had to be decided. The place of science and mathematics in the curricula was minimal107 : science was most often considered to wither young girls’ minds, and mathematics, which was mainly used in preparatory classes for the grandes écoles, was particularly considered useless beyond the elementary arithmetic.108 In these debates, references to Germain were mobilised to argue for or against a high level of instruction for women or in a more specific way to discuss the possibility and the utility of a scientific or mathematical education for women. This last question was already ancient in the 1860s during the first reforms to create a public secondary education for women. In the fundamental Histoire morale des femmes, Legouvé wrote for example that women were incapable of generalisation, genius and invention in every intellectual and artistic field, including mathematics: “No mathematical discovery, no metaphysical theory is due to a woman.”109 For Republicans, women’s education was fundamental. In his speech on equality of education on April 1870, Minister Jules Ferry (1832–1893) already said: “Bishops know it well: the one who holds the woman, that one holds everything, first because 104 “Devoirs

de l’homme envers lui-même”.

105 “Une femme, par son amour pour l’étude, parvint à se placer parmi les premiers mathématiciens

du dix-neuvième siècle.” Barrau (1872), p. 73. Offen (1986). 107 See Hulin (2008). 108 See Mayeur (1979). 109 “Aucune découverte mathématique, aucune théorie métaphysique n’est due à une femme.” Legouvé (1849), p. 372. 106 See

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he holds the child then because he holds the husband […] That’s why the church wants to keep the woman, and that’s also why democracy has to take her away from the church”.110 At this time, many publication projects on teaching and pedagogy were born. This is, for example, the case of the Dictionnaire de pédagogie edited by the philosopher and Republican Ferdinand Buisson (1841–1932), on which many authors like Legouvé collaborated. Debates on education that took place during the revolutionary period were fundamental examples in this literature. For example, George Dumesnil (1855–1916), in his Pédagogie révolutionnaire (1883), analysed Condorcet’s writings at length and referred to Germain to justify the mathematical skills of women. Moreover, with the establishment of a secular moral education programme under the Third Republic, the essential role of women in the family continued to be central to pedagogical writings. Several moral works were written by women. Among these, Francinet by G. Bruno111 was a real bestseller for children aged 11 to 13 from its first edition in 1869.112 In this book, the author insisted that a mother’s influence was a central factor in a child’s education. She nevertheless suggested possible mathematical capacities for girls when telling Germain’s story through a teacher, Mr. Edmond. One of the three children who were listening, a girl, reacted: Thank you, sir, said Aimée, for telling us this interesting story. I am very proud to know that there was a girl as intelligent and as learned. I did not think women were able to understand anything about mathematics.113

Statistical surveys and teachers’ testimonies multiplied in the 1880s, with the first French laws on public secondary education for young girls, the development of women’s education in Europe and discussions about the model of American coeducation. More generally, with the general rise of statistics, increasing numbers of studies on the place of women in society were being conducted. On this point, the book led by the American journalist Théodore Stanton (1851–1925), The Woman Question in Europe is an emblematic example. It was published in 1884 from the results of an investigation launched by Stanton in 1878, following the first congress on women’s rights organised in France by Léon Richer and Marion Deraisme. Education for girls played an important role in this book composed of seventeen chapters, one each for seventeen different countries.114 The naturalist Nicolas Joseph Joly (1812–1885), a professor at the Science faculty in Toulouse, referred in particular to Stanton in his 110 “Les évêques le savent bien: celui qui tient la femme, celui-là tient tout, d’abord parce qu’il tient

l’enfant, ensuite parce qu’il tient le mari […] C’est pour cela que l’église veut retenir la femme, et c’est aussi pour cela qu’il faut que la démocratie la lui enlève”. 111 Under the pseudonym G. Bruno was actually hiding the wife of Alfred Fouillée (1838–1912), a philosopher whose works included psychological studies. 112 See Offen (2018). 113 “Je vous remercie bien, monsieur, dit Aimée, de nous avoir raconté cette intéressante histoire. Je suis toute fière de savoir qu’il y ait eu une jeune fille aussi intelligente et aussi savante. Je ne pensais pas que les femmes fussent capables de comprendre quelque chose aux mathématiques.” Bruno (1869), p. 505. 114 See Offen (2000).

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article “Dans l’espèce humaine à égalité d’instruction les intelligences sont égales chez les deux sexes” (“In the human species with equal education the intelligences are equal in both sexes”) (1885). Based on the principle of Legouvé, Joly thus defended the equality of intelligence (and not identity) between the two sexes, while defending the existence of mathematical abilities for women: Like so many others, Mr. Legouve at their head, I had for a long time shared the opinion of believing that women have no aptitude for the mathematical sciences; but I had to face the facts when I saw the sister of Mr. Theodore Stanton follow fruitfully the courses of transcendental mathematics of our eminent and regretted colleague Despeyrous.115

Joly also cited Kovalevskaya and Germain to support his remarks. Two years later, Legouvé also returned to his statement on women and mathematics, after observing the performance of girls in this domain. If Germain was regularly referenced to justify the usefulness of girls’ education, sometimes even in science, her example could also serve as a contrary argument. Indeed, many authors argued that too much education for women could denature them or distort them from their natural familial functions. Feminists could rely on the existence of recognised women scientists to counter this view. Nevertheless, De Witt, quoted previously, relied on Germain to justify, on the contrary, the uselessness of higher education: […] Mlle. Sophie Germain had excited by her work the surprise and admiration of M. de Lagrange. But neither of these great [women] mathematicians [Germain and Sommerville] had received special and higher education. […] The keys, often useless, were not forged beforehand for doors which they did not bother to open, as professors too often do not always remember. The purpose of women’s education is to make women worthy of the name, natural, effective and useful companions of men to whom God has placed them.116

These debates, surveys and other studies were also regularly discussed in the daily or weekly press. This was, for example, the case of the journalist Henri de Parville, known for his writings on science and industry117 : he summed up several recent publications on women to discuss their abilities in science on the occasion of Klumpke’s doctoral thesis, in an article published on December 20, 1893, in the Journal des débats politiques et littéraires.118 115 “Comme

tant d’autres, M. Legouvé à leur tête, j’avais longtemps partagé l’opinion qui consiste à croire que les femmes n’ont aucune aptitude pour les sciences mathématiques; mais j’ai dû me rendre à l’évidence quand j’ai vu la sœur de M. Théodore Stanton suivre avec fruit les cours de mathématiques transcendantes de notre éminent et regretté confrère Despeyrous.” Joly (1885), p. 150. 116 “[…] Mlle Sophie Germain avait excité par ses travaux la surprise et l’admiration de M. de Lagrange. Mais ni l’une ni l’autre de ces grandes mathématiciennes [Germain et Sommerville] n’avait reçu un enseignement spécial et supérieur. […] On ne leur avait pas forgé d’avance les clefs, souvent inutiles, pour des portes qu’elles ne se souciaient pas d’ouvrir, comme le font actuellement trop souvent les professeurs, qui ne se rappellent pas toujours que le but de l’éducation féminine est de faire des femmes dignes de ce nom, compagnes naturelles, efficaces et utiles des hommes auprès desquels Dieu les a placées […]” Witt (1888), p. 394. 117 See his obituary in the Journal des débats politiques et littéraires of July 12, 1909. 118 See de Parville (1893).

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As mentioned earlier, many writings on feminism were also published at the turn of the twentieth century. The results of the surveys on women’s education were also used in this context. This is, for example, the case of the article written by MarieThérèse de Solms-Blanc (1840–1907) under the pseudonym Thérèze Bentzon: it was published, in 1902 in the journal Le Carnet and was part of a series of texts entitled “Survey on Feminism”. Bentzon had been collaborating since the early 1870s in the Revue des deux mondes. An expert in American literature, she travelled in the United States on behalf of the journal in the 1890s and was, therefore, well aware of the situation of women across the Atlantic.119 Thus, she reported the words of Bryn Mawr College President, Martha Carey Thomas (1857–1935): the proportion of women students was continually increasing and young women graduates, in addition to being able to find suitable work if needed, became very good mothers. According to Bentzon, the situation in Europe was, of course, different because there were fewer job opportunities for qualified females. Nevertheless, an educated woman was, according to her, a better mother, and the unmarried women could thus benefit from a “spiritual and moral maternity” which was equal to biological motherhood.120 In addition, many women had previously demonstrated their literary or scientific excellence, with Germain and Kovalevskaya being cited for science. If Bentzon defended the feminine specificity, she also affirmed the importance for unmarried women at least to be able to exercise a profession and at pay equal to men. The position of another author, a few years later, was quite different: Emma Angot (1850–?), in her article “A bit of feminism”, published in Le Correspondant in 1909, affirmed that women were inferior to men in several fields and that even if some jobs could be adapted to women, it was useless, even dangerous, to add tension to the job market. Dangerous because it could further move women away from marriage. The aim of girls’ education was, therefore, to develop their feminine qualities and not to inculcate in them a useless science. For Angot, Marie Curie was the dangerous example par excellence “because she turned many heads”,121 and Germain did not provide proof of the usefulness of women in science. But the role of women must above all be moral: And when intellectual progress is assured, will there be moral gain? No, certainly. Then there is nothing done. France does not decline for lack of knowledge, it collapses for lack of principles; and if women try to awaken national self-esteem and elevate private dignity, it is not the card of solidarity or even the degree of license that will help them.122

These remarks by Angot referred to a shared fear at the beginning of the twentieth century of national degeneration and depopulation. Like the prospect of unmarried women, women’s work was considered by anti-feminists (and some feminist 119 See

Offen (2018), pp. 189–190. spirituelle et morale qui vaut bien l’autre” Bentzon (1902), p. 435. 121 “car il a tourné bien des têtes” Angot (1909), p. 956. 122 “Et quand le progrès intellectuel serait assuré, en sortira-t-il un gain moral? Non, certainement. Alors il n’y a rien de fait. La France ne décline pas faute de savoir, elle s’affaisse faute de principes; et si les femmes essayent de réveiller l’amour-propre national et de relever la dignité privée, ce n’est point le brevet de solidarité ou même le diplôme de licence qui les y aidera.” Angot (1909), p. 968. 120 “maternité

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currents) as a threat to motherhood and the national birth rate.123 To the arguments of doctors affirming maternity as natural thus followed the idea of maternity as a patriotic duty.124

7.4.4 The Question of Women Through the New Social Sciences: Germain as a (Counter-)Example Since the seventeenth century, reason and natural law were considered by scholars and philosophers as fundamental to any enlightened society. Natural and biological laws were thus used to deduce social laws.125 The use of science to study the “nature of women” and in particular to justify their inadequacy to all intellectual activity was more and more frequent from the second half of the eighteenth century. A new model was also gradually imposed: that of the incommensurability of the sexes. It was well illustrated by the words of Rousseau: “in what they have in common, they are equal; in that they are different, they are not comparable”.126 In the 1750s, several successful books were devoted to the study of the “nature of women” (Roussel 1775 for example). With this incommensurable sex model, there were two bodies, totally infused with the masculine or feminine.127 From the second half of the eighteenth century, demonstrations of the natural difference existing between women and men were made from the consideration of their skeletons: if this difference was observable in bones, which did not have a procreative function, it could prove that there was a difference beyond just genitalia between men and women. Using skeletons to justify natural differences between man and woman was far from innocuous. The skeleton was then thought of as an ideal object of study for the analysis of the body. In addition, the advent of palaeontology at the turn of the nineteenth century, and with it the validation of the importance of fossil bones, contributed to enhancing the study of skeletons for the study of the human. At the same time, starting in the nineteenth century, measurements were made on different parts of the skeleton, and especially the skull to measure intelligence and its evolution within the human species. This idea that the mind and intelligence could be based on the material study of the brain and nervous system was developed throughout the nineteenth century.128 It was in this context that phrenology was developed by Franz-Josef Gall (1758– 1828), who postulated relationships between human faculties and the shape of the skull, considered as formative for the brain. If phrenology was criticised, many treatises were dedicated to this science in the nineteenth century. Phrenological work 123 See

Offen (2000); and Bologne (2004). Knibiehler (1976). 125 See Offen (1986), p. 460. 126 “en ce qu’ils ont de commun, ils sont égaux; en ce qu’ils ont de différent, ils ne sont pas comparables.” 127 See Laqueur (1992). 128 See Peyre and Wiels (1995). 124 See

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was based on an important iconography, consisting of portraits of ancient men, skulls or preserved casts and studies of living people. In several publications, phrenologists base their remarks on a collection of portraits. Many artists also practiced phrenology.129 From the 1840s at least, the skull of Germain was regularly cited to illustrate some intellectual faculties such as “habitativity” and “concentrativity”.130 For example, the French painter Jean Hippolyte Bruyères (1801–1855), son-in-law to one of Gall’s collaborators, published in 1847 a book entitled La Phrénologie. Le geste et la physionomie démontrés par 120 portraits, sujets et compositions gravés sur acier, in which some of Germain’s intellectual and behavioural faculties are analysed from observations of her skull: “Plate no 7 contains an example of great development of the organ attributed to the penchant of the habitativity: it is a bust, molded on nature after the death, of miss Sophie Germain, known by her talent in mathematics: she was sedentary and homeless, and for a great number of years she did not leave her room: her occupations helped to give strength to her home instinct […] She was original and aimed at singularity: self-esteem and firmness are very pronounced on her head […] the organ of calculation is very marked. In the hypothesis of an organ of concentrativity, the considerable development of the region of habitativity on this head can make it possible to admit that the two organs exist together to a high degree. And, in fact, Mademoiselle Germain was exclusively engaged in her calculating work with a great strength of concentration of mind.”131

It is easy to see here direct links between this description and some biographical aspects of Germain: for example, firmness is described later in the book as a characteristic of “obstinate and stubborn children”132 which probably referred to her childhood. Bruyères also distinguished computational abilities with more general mathematical faculties, citing in particular prodigious calculators that showed mathematical incompetence. This opposition between computer and mathematician would be a regular argument to minimise women’s achievements in mathematics as we will see below. In the 1860s, the question of women returned to the scene and was debated particularly by actors of the new humanities such as sociology, anthropology and psychology. Many authors defended the inferiority of women through scholarly publications. They developed scientific studies on the sizes of skulls and brains, and using Darwin’s and Spencers’ theories of evolution. Exceptions like Germain were 129 See

Baridon (2003). was also taken as an example in Cubí i Soler (1858); and Rengade (1881). 131 “La planche no 7 contient un exemple d’un grand développement de l’organe attribué au penchant de l’habitativité: c’est un buste, moulé sur nature après la mort, de mademoiselle Sophie Germain, connue par son talent en mathématiques: elle était sédentaire et casanière, et, pendant un grand nombre d’années, elle n’a point quitté sa chambre: ses occupations contribuaient à donner de la force à son instinct casanier […] Elle était originale et visait à la singularité: l’estime de soi et la fermeté sont très prononcées sur sa tête […] l’organe du calcul est très marqué. Dans l’hypothèse d’un organe de la concentrativité, le développement considérable de la région de l’habitativité sur cette tête peut faire admettre que les deux organes existent ensemble à un haut degré. Et, en effet, mademoiselle Germain était exclusivement livrée à ses travaux de calcul avec une grande force de concentration d’esprit.” Bruyères (1847), pp. 46–47. 132 “enfants opiniâtres et têtus” Bruyères (1847), p. 131. 130 Germain

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generally justified by their lack of femininity. Thus, the philosopher Etienne Vacherot (1809–1897), Republican and defender of democracy against the imperial regime and religion under the Second Empire, wrote in a chapter on the psychological method that scientific thought was a virile thought. Women who had distinguished themselves in mathematics, like Germain, “who calculated like a geometer and thought like a metaphysician” had “lost the spirit” of the woman and “acquired in exchange a virile spirit.”133 Proponents of women generally responded with an ancient argument, but used more and more frequently: female inferiority was socially constructed.134 These debates were obviously not unrelated to those on women’s education mentioned above. In 1879, the doctor and anthropologist Gustave Le Bon (1831–1941) used the measurement of skulls to affirm that the inferiority of all women, some social classes and certain races was accentuated with the progress of civilisation.135 He concluded that girls’ education had to be very limited, because it could be dangerous for women, and therefore, to family and society more generally. Several feminist Republicans opposed such ‘scientific’ discourses by regularly using the results of the first surveys of women’s education in the United States and France. Some also answered with scientific arguments: the anthropologist Léonce Manouvrier (1850–1927) published an article in the Revue scientifique in 1882 in which, in response to the theory of Le Bon, he showed that the size of the brain in men and women was proportional to body mass. Nevertheless, Manouvrier did not deduce an absolute equality of the physical and intellectual faculties of men and women: “the muscular superiority of males and the gestational requirements of females necessarily resulted in a sexual division of labour that accentuated secondary sexual differences.”136 These scientific-medical arguments were regularly reproduced in the press or used in pedagogical debates. Professor Joly, in his article above mentioned on gender equality, exposed his fears concerning this type of speech: “But, even today, unkind philosophers and physiologists or relying on inaccurate data, pose, as an axiom, the intellectual inferiority of the woman in parallel with the male sex.”137 Those debates were also taken in hand in socialist and feminist literature by authors who vividly criticised the so-called scientific arguments showing women’s intellectual inferiority in feminist journal such as Le Droit des femmes or socialist périodicals like La Revue socialiste. In the latter, the feminist and socialist Léonie Rouzade (1839–1916) published a text entitled “Les Femmes devant la démocratie” to defend, against anthropological and other scientific arguments, the idea that women’s inferiority lay in social and cultural causes and that education should resolve the problem. She argued for equal rights and general access to work in order to assure emancipation and independence for women. She explicitly aligned herself with socialist thought, using the socialist formula “à travail égal, salaire égal” (“Equal pay for equal work”) 133 “qui calculait comme un géomètre et pensait comme un métaphysicien”, “perdu l’esprit”, “acquis

en échange un esprit viril” Vacherot (1869), p. 258. Offen 2000, 2018). 135 See Knibiehler (1976). 136 See Offen (2018), p. 82. 137 See Joly (1885), p. 131. 134 See

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and promoting education for all people, men and women.138 In the same order of ideas, the socialist Johannes Sagnol (1863–1948) published in the same periodical “L’égalité des sexes”, where he also defended physical sex equality—because men’s general physical superiority was balanced by women’s superior functions in reproduction—and intellectual sex equality. According to him, every question regarding nature or society was analysed by science but as science was not precise enough, the objectivity of scientific claims was more than questionable.139 In his article, Sagnol proposed a socialist advocacy in favour of women rights. Indeed, according to Sagnol, despite the limitations that had always been imposed on women by men, especially to join the liberal professions, women regularly showed their intellectual superiority, like Germain “who, at age 14 was the strongest mathematician of her time”.140 For Sagnol, the physical, intellectual and moral equality of men and woman should induce social equality: “Now, since [the woman] is equal to [the man], she is entitled like him, with the same independence and the same social benefits.”141 For Rouzade and Sagnol, equality did not mean identity: they both agree on the complementarity between men and women, the former having, in particular, the physical strength and the latter, the fertility. But they both advocated that physical strength lost its importance in a more and more mechanised society and that in industrialised civilisations, women and men could perform equally in numerous jobs. They also promoted instruction for both men and women, from all social classes, because the lack of instruction, in particular scientific, was the reason for the supposed intellectual inferiority of women. While both Rouzade and Sagnol placed the family as a central element of society, they also advocated a major shift in family structure, in which women would be more independent: with this model, intellectual women should not be regarded as dangerous, on the contrary. By the end of the 1880s and the 1890s, demands for women’s rights were growing, as more and more women entered secondary or higher education and had a profession. Feminist publications were still developing; two international congresses for women’s rights were organised in 1889. The question of women was still debated with scientific arguments: the knowledge war was revived by anti-feminist writers around the issue of women’s mental capacity.142 Books that quickly became best-sellers were published. The biologists Patrick Geddes (1854–1932) and John Arthur Thomson (1861–1933) thus published in 1889 The Evolution of Sex, based on Darwin’s evolutionary theory; they argued that intellectual and affective differences between the sexes were essential.143 Shortly after, the works of Cesare Lombroso (1835–1909) and his son-in-law historian Guglielmo Ferrero (1871–1942) were at the heart of the debate. Lombroso, considered the founder of criminal anthropology, had already 138 See

Rouzade (1887). Sagnol (1889), p. 685. 140 “qui, à 14 ans était la plus forte mathématicienne de son temps” Sagnol (1889), p. 694. 141 “Or, puisque [la femme] est l’égale de [l’homme], elle a droit comme lui, à la même indépendance et aux mêmes avantages sociaux.” Sagnol (1889), p. 697. 142 See Offen (2000; 2018). 143 See Offen (2000). 139 See

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published L’uomo delinquente in 1876. Merging statistics, anthropology, and evolutionary theory, he argued that delinquency was a natural disposition and that criminals were beings that had not completed their evolution.144 In 1893, La Dona delinquente, the prostituta e la donna normal by Lombroso and Ferrero appeared. In Part I, devoted to the “normal woman”, Chap. 9 dealt with intelligence. That of the woman was then characterised by “the absence of any creative power”: even if illustrious women had existed, like Germain in mathematics, they were much less frequent than men and remained far from “the power of male geniuses”.145 Lombroso and Ferrero criticised Sagnol’s thesis: for them, the inferiority of women could not be explained by “social conditions”146 since even when they received an education similar to that of men (as in the French aristocracy of the eighteenth century), they showed no trace of genius. They said that women with higher intellectual abilities were actually male. Finally, if they were incapable of creativity or synthesis, women knew how to assimilate and reproduce, which explained the good or even better success of girls in studies.147 This book, strongly criticised, was nevertheless a great success, and was partially translated into English in 1895, and completely in French in 1896. Many French scientists, like Manouvrier, strongly opposed Lombroso’s theories. Between 1896 and 1914, several scientists gathered around Alexandre Lacassagne (1843–1924) to develop criminal anthropology while denouncing the overly simplistic position of Lombroso and his Italian colleagues on the born criminal: for them, the social milieu also played a fundamental role in revealing delinquency.148 They admitted, however, a natural basis of crime. Similarly, for Lacassagne, there were natural differences between the sexes. Thus, Lacassagne wrote in the section devoted to the sexes of his Précis de médecine légale (1906): The real difference is in brain function. In women, feelings predominate, in men intelligence. Women are occipitals, men are frontal. Hence their inability in science. Except Sophie Germain and Sophie Kowalewsky, not one woman was superior in mathematics. The grand generalisations of physics and biology are not within their reach.149

Those scientists, often close to Republican circles, applied the famous principle of “equality in difference”. Alfred Fouillée (1838–1912), in psychology, published in the Revue des deux mondes in 1893 an article entitled “La psychologie des sexes et ses fondements physiologiques”, then reproduced in a more general work in 1895.150 He

144 See

Kaluszynski (1989). Lombroso and Ferrero (1893), p. 166. 146 See Lombroso and Ferrero (1893), p. 166. 147 See Lombroso and Ferrero (1893), p. 172. 148 See Kaluszynski (1989). 149 “La véritable différence se trouve dans les fonctions cérébrales. Chez la femme prédominent les sentiments, chez les hommes l’intelligence. Ce sont des occipitales, les hommes sont des frontaux. De là leur inaptitude aux sciences. Sauf Sophie Germain et Sophie Kowalewsky, pas une femme n’a été supérieure en mathématique. Les grandes généralisations de la physique, de la biologie ne sont pas à leur portée.” Lacassagne (1906), p. 130. 150 See Fouillée (1895). 145 See

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began by recalling the importance of biological considerations in educating psychological, moral and social relations between the sexes. He objected, however, to overly simplistic methods, such as the weighing of skulls, and based his argument on recent discoveries concerning reproduction. In his chapter on women and men’s intelligence, Fouillée positioned himself against Le Bon and Lombroso, relying in particular on the writings of Manouvrier to affirm the equality of brain development in both sexes. Nevertheless, he defended the idea of the sexual specificities of intellect and supported the fact that “the functions which aim at the propagation and the nutrition of the species are in antagonism with too much expenditure of the brain”151 : “ with Fouillée, the man remained more adapted to generalisation, whereas the woman had intuition and sagacity for particular facts. Against Le Bon, who had postulated that women were incapable of scientific reasoning, Fouillée gave examples of women who showed exceptional scientific skills, such as Germain. But these women still had to remain exceptions: “A force and an expenditure of intelligence which, if they were general among the women of a society, would lead to the disappearance of this very society, must be considered as an attack on the natural functions of sex.”152 Hence a need to offer an education to a woman to eventually learn “the professions that are related to the capabilities and dignity of her sex” but especially prepare “for domestic life, her role as a wife, as a mother, and educator.”153 Theories like that of Fouillee met their opponents. For example, the sociologist Jacques Lourbet, in La femme devant la science contemporaine (1896), explicitly attacked “misogynist authors” like Fouillée and their problematic use of science: “Science! What a beautiful passport today for paradoxes!”154 On the contrary, he insisted on “the influence of the environment on the woman”,155 taking away from Fouillée’s arguments about insects: “We see that in these insects a certain environment awaits the larvae and makes, according to the case, workers or queens; in this way the sovereign power of the milieu is clearly affirmed”.156 Lourbet thus considered it remarkable that, in the conditions of subordination in which women had been locked up for centuries, so many women (including Germain in mathematics) “have been illustrious in the sciences, the humanities, the arts, and even in war, and who [have] become supreme leaders of the people!”157 151 “les

fonctions qui ont pour but la propagation et la nutrition de l’espèce sont en antagonisme avec une trop forte dépense du cerveau.” Fouillée (1893), p. 415. 152 “Une force et une dépense d’intelligence qui, si elles étaient générales parmi les femmes d’une société, amèneraient la disparition de cette société même, doivent être considérées comme une atteinte aux fonctions naturelles du sexe.” Fouillée (1893), p. 420–421. 153 “aux professions qui sont en rapport avec les capacités et avec la dignité de son sexe”, “à la vie domestique, à son rôle d’épouse, de mère et d’éducatrice.” Fouillée (1893), p. 425. 154 See Lourbet (1896), p. 67. 155 See Lourbet (1896), p. 75. 156 “On voit par là que chez ces insectes un certain milieu attend les larves et en fait, selon le cas, des ouvrières ou des reines; ainsi s’affirme d’une manière éclatante la puissance souveraine du milieu.” Lourbet (1896), p. 82. 157 “qui se sont illustrées dans la sciences, les lettres, les arts, la guerre même, ou qui sont devenues les chefs suprêmes des peuples!” Lourbet (1896), p. 85.

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Another author who used Legouvé’s principle, “equality in difference”, was the philosopher Henri Marion (1846–1896). A Republican reformer of the Third Republic, Marion was the author of several books and pedagogical articles. Elected to the Council of Public Instruction in 1878, he participated in the organisation of secondary education for girls. He taught his course on the “science of education” at the Sorbonne from 1883 to 1896, a course published in two volumes posthumously. In the first volume, Psychologie de la femme (1900), Marion proposed a science of psychology-based education to determine the “nature of women”.158 Marion thus proposed an analysis of the specificities of the intelligence of the woman: “The woman is as intelligent as the man, she is so only otherwise. Let us look more closely, faculty by faculty, at what are the characteristic differences”,159 namely an ability to store information rather than reason personally, an “omnipresent” imagination but lacking “power and fertility”,160 a “futile curiosity” rather than “broad, selfless, truly intellectual.”161 According to Marion, the woman could nevertheless excel in the mathematical sciences, as Kovalevskaya and Germain, for example: “The proof is made, it seems to me, that the woman can be a great mathematician and that there is no incompatibility radical between her natural gifts and the highest scientific culture, at least in the order of the exact sciences”.162 On the other hand, according to Marion, women lacked the capacity for abstraction and generalisation in the natural sciences. In any case, learned men and philosophers were sufficient to advance knowledge: it was much more useful for women to perform their own functions: “maternity”, “softening social life” and the preservation of traditions and morals.163 At the turn of the twentieth century, the neurologist Paul Julius Möbius (1853– 1907) published a book on the foundations of mathematics (1900) and applied his neurological theory to express his opinion on the abilities of women in this field. Taking up the idea that women had no capacity for abstraction or creativity, he defined women mathematicians as the result of a process of degeneration. This idea joined Lombroso’s theories and more generally the literature on the intermediate sex types of the time, which resulted from abnormalities during the development of individuals.164 His argument was also based on social and historical facts. On the one hand, he noted that the number of women with mathematical skills was extremely low (he announced one woman in a million…) and that a woman practicing mathematics was unnatural. On the other hand, according to him, the women who left their names in the history of mathematics did not introduce anything innovative165 : 158 See

Mosconi (2012). Marion (1900), p. 197. 160 See Marion (1900), p. 205. 161 See Marion (1900), p. 208. 162 See Marion (1900), p. 212. 163 See Marion (1900), p. 220. 164 See Rowold (2001). 165 On that subject, he also criticised vividely Rebière’s too wide acceptation of the notion of “women in the science”. 159 See

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It can be said that a female mathematician is unnatural; it is in a certain sense a hermaphrodite […] Sophie Germain has the appearance of a man; Kovalevsky proves that a woman can hardly possess genius and health […] It is an exaggeration to speak of mathematical genius of the woman; none has found anything essential, none has devised new methods. They were good students, nothing more. […] The most original was Germain.166

Germain was described by Möbius as original precisely because she worked alone her whole life; she did not have any father, brother or husband to assist her. Möbius’s writings were taken up and discussed by many authors. Thus, the controversial Otto Weininger (1880–1903), relied notably on Möbius (the latter accused him of plagiarism) in his book Geschlecht und Charakter. Eine prinzipielle Untersuchung (1903), which also became a huge success. He used his principle of intermediate sex forms to study the capacity of women to be independent. To become the internal equal of the man, “to attain his intellectual and moral freedom, his interests and creative power”,167 the woman had to develop a degree of masculinity. Even in this case, women remained intellectually far inferior to men: Germain was one of those women who were recognised only because of gender and were not as intelligent as men of the fifth or sixth rank of genius. This intellectual inferiority of women was due to feminine psychology: No woman has any real interest in science, even though she may successfully pretend that she has, both to herself and to many good men who are bad psychologists. We can be certain that behind every woman who has been able to claim an independent scientific achievement that was not totally insignificant (Sophie Germain, Mary Sommerville, etc.) there was always a man, to whom she was trying to get closer in that way.168

Some extracts of Möbius’ text were also translated into French by the Italian mathematician and historian of mathematics Gino Loria (1862–1954) and concluded his article Les femmes mathématiciennes published in 1903 in La Revue scientifique. Loria’s purpose was to answer the following question: Do the notions of feminine psychology that we possess lead us to consider as probable, or even as simply possible, that the woman is destined to give to the future, to science, contributions comparable to those that will be transmitted to the most distant the glorious names of Pythagoras and Newton, Archimedes and Leibniz, Descartes and Lagrange?169 166 “Man

kann also sagen, daß ein mathematisches Weib wider die Natur sei, in gewissen Sinne eine Zwitter […] Von den Mathematikerinnen sieht besonders Sophie Germain männlich aus. Die Kowalewsky zeigt, daß Gesundheit und hervorragendes Talent beim Weibe schwer zusammenbestehen […] Es ist eine Uebertreibung, wenn vom mathem. Genie bei Weibern gesprochen wird. Niemand wird bezweifeln, daß die Mathematik sich ebenso günftig entwickelt haben würde, wenn die aufgezählten weiblichen Mathematiker nicht gelebt hätten. Keine hat etwas Wesentliches geleistet, neue Methoden erdacht. Sie waren gute Shülerinnen, nicht mehr […] Um originellsten scheint die Germain gewesen zu sein.” Möbius (1900), p. 85–86. 167 See Weininger (1903), p. 58. 168 “Kein Weib hat wirkliches Interesse für die Wissenschaft, sie mag es sich selbst und noch so vielen braven Männern, aber schlechten Psychologen, vorlügen. Man kann sicher sein, daß, wo immer eine Frau irgend etwas nicht ganz Unerhebliches in wissenschaftlichen Dingen selbständig geleistet hat (Sophie Germain, Marie Sommerville etc.), dahinter stets ein Mann sicher verbirgt, dem sie auf diese Weise näher zu kommen trachtete […]” Weininger (1903), p. 168. 169 “les notions de psychologie féminine que nous possédons conduisent-elles à considérer comme probable, ou même comme simplement possible, que la femme soit destinée à donner à l’avenir, à

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Loria announced that he was going to use statistics, “this cold and implacable investigator of the principles that govern human actions”,170 that is to say the historical data he had about women in mathematics. The question was fundamental to him because its answer should be able to guide political and individual decisions about the adequacy of women for mathematics. According to him, women could advantageously do complex calculations, especially in astronomy, thanks to their “patience” and their “tact”.171 But There is a profound difference between the intellectual work of the astronomer who observes stars and performs mathematical calculations by applying now classical formulas, and the original work of one who does mathematics and researches the properties of figures.172

What about women who actually did mathematics? He used biographical elements of Germain’s life in order to show that her accomplishments were not so original but laborious: Her anonymous correspondence with Gauss, the “princeps mathematicorum” of the Germans, places her at the head of scholars who appreciated the inestimable value of new methods and became familiar with the handling of these delicate and powerful methods by which the immortal analyst gave a new and solid foundation to higher arithmetic. The manner in which Sophie Germain treated a question put to the competition—on the proposition of Napoleon I.—by the Institut de France, gave proof of an extraordinary perseverance, of a singular tenacity, rather than an exceptional analytical skill.173

He concluded by recalling the words of Möbius, which confirmed his theory. This publication was followed by exchanges with Josephine Joteyko (1866–1928) in this same journal.174 She was a Polish psychologist who studied child development to build a modern and rational pedagogy.175 She first studied in Geneva and obtained her doctoral thesis in Paris in 1896. In her answer to Loria, Joteyko highlighted contradictions in Loria’s arguments and insisted on the importance of the debate, in relation to the one about the possibility for women to enter universities. la science, des contributions comparables à celles que transmettront à la postérité la plus éloignée les noms glorieux de Pythagore et Newton, d’Archimède et Leibniz, de Descartes et Lagrange?” Loria (1903). 170 “cette froide et implacable investigatrice des principes qui gouvernent les actions humaines”. 171 See Loria (1903). 172 “il y a une différence profonde entre le travail intellectuel de l’astronome qui observe des astres et effectue des calculs mathématiques en appliquant des formules désormais classiques, et le travail original, de celui qui fait des mathématiques et recherche les propriétés des figures.” Loria (1904). 173 “Sa correspondance anonyme avec Gauss, le “princeps mathematicorum” des Allemands, la place à la tête des savants qui surent apprécier l’inestimable valeur des méthodes nouvelles et se familiariser avec le maniement de ces méthodes délicates et puissantes grâce auxquelles l’immortel analyste donna une base nouvelle et solide à l’arithmétique supérieure. La façon dont Sophie Germain traita une question mise au concours—sur la proposition de Napoléon 1er—par l’Institut de France, donna la preuve d’une persévérance extraordinaire, d’une singulière ténacité, plutôt que d’une habileté analytique exceptionnelle.” Loria (1903). 174 See Joteyko (1904); and Loria (1904). 175 See Wils (2005).

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Finally, note that these debates were reproduced, recalled and discussed in a wide variety of publications. Thus, Joteyko’s answer was also published in the journal La Femme. Loria’s words were partially reproduced in the journal La France médicale in 1903 and Möbius’ ideas were evoked in the Journal de la jeunesse in 1905. The lack of feminity of women mathematicians, such as Germain, was also used in some articles of the daily press, as in Le Petit parisien (January 6, 1906): Another forgotten, Sophie Germain, whose name disappears between these two dates: 1776– 1831, on her gravestone, has fixed the attention of our city officials, who have recently dedicated to her one of their schools. She was the least woman of women: she was thirteen when she became passionate about mathematics, she studied alone, limiting herself to submit from time to time by letters her observations to scholars who took her for a student of the École polytechnique. An academic award, won by her about a horribly abstruse question, forced her to say who she was. As a true mathematician, she did not seek any other title than that of Miss X…176

7.5 Conclusion As a woman, Germain had to be a self-taught mathematician because she had no access to mathematical instruction and more generally to institutions. She also had to overcome prejudices about the inadequacy of women for mathematics. Moreover, it was difficult for her, as a single woman, to build relationships with male mathematicians. Nevertheless, the political and scholarly proximity of her family to several social circles certainly played a role in her passion for mathematics and her possibilities for meeting scholars. Furthermore, as a woman practicing mathematics, and therefore, as an exception, she had specific opportunities to meet and discuss with several scholars, even if some of them did not consider her as a mathematical equal as she wished. Otherwise, number theory seemed then to be a domain where being excluded from institutions like the École polytechnique was less limiting than for other kinds of mathematics: not taught in educational institutions, number theory was a mathematical domain where self-taught individuals could more easily stand out; in the early nineteenth century, analytical methods were far from being systematically used in number theory, so Germain did not have the same difficulties as with her study of the theory of elasticity. Number theory also appeared as a marginal domain in France and was not a field as competitive as analysis, for example. Germain was definitely one of the first mathematicians to master Gauss’ DA and to propose some generalisations and new applications of Gauss’s results and methods. She was then able to exchange technical communication on number theory with Gauss, Legendre, 176 “Une

autre oubliée, Sophie Germain, dont le nom s’efface entre ces deux dates: 1776–1831, sur sa pierre tombale, a fixé l’attention de nos édiles, qui lui ont récemment dédié une de leurs écoles. Celle-ci fut bien la moins femme des femmes: elle avait treize ans lorsqu’elle se prit de passion pour les mathématiques, qu’elle étudia seule, se bornant à soumettre de temps en temps par lettres ses observations à des savants qui la prenaient pour un Élève de l’École polytechnique. Un prix académique, remporté par elle dans une question horriblement abstruse, l’obligea à dire qui elle était. En vraie mathématicienne, elle n’ambitionnait d’autre titre que celui de Mlle X…”

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Libri and others and could be a bridge between some of them (Gauss and Legendre at least). Germain was aware of most of the recent publications in number theory. She was connected to most of the contemporary mathematicians studying number theory who valued her arithmetical contribution. In connection with most of the mathematicians studying number theory at her time, Germain was valued as a number-theorist among them. The fact that her work was not published extensively can be understood by the limited editorial space, especially for authors who did not belong to institutions such as the Académie des sciences and the École polytechnique, especially for number theory. Germain was thus able to evolve in limited but existing alternative spaces, modelled by her gender but also by personal, social and mathematical factors. This indicates the complexity of women’s possibilities in science.177 As we saw, the reception of Germain’s achievements in number theory depended partially on an editorial project of Gauss’s collected papers from the 1860s and a period during which several mathematicians promoted number theory in the context of teaching and science diffusion. At the same time, her philosophical work was re-edited and printed with a precise biography of her by Stupuy. During her life, Germain deployed strategies to hide her sex and especially to avoid being considered as a “learned woman”. Nevertheless, after her death, and especially under the Third Republic, her name and biography were used to feed the debates on women in mathematics and more generally on the woman question. This recourse to women recognised for their intellectual, and especially scientific successes was indeed frequent, in the daily press and in general and specialised periodicals. References to Germain were thus mobilised for the question of education for women, their social and economic rights, and to defend or attack feminist doctrines. I could also have taken the later example of the discussions on women’s suffrage: for example, after having published an article on “The Vote of Women” in La nouvelle revue in 1922,178 Senator Louis Martin listed Germain among many accomplished women in defence of his proposed law “to give women the right to vote and eligibility” to the Senate in 1923. The range of types of arguments to discuss these questions expanded during the nineteenth century with the creation and development of natural and social sciences, such as anthropology, sociology, psychology, physiology and statistics. Beyond the evocations of Germain in lists of names, several characteristics of her personality and her works were also employed to make a point. Her perseverance in the study of mathematics despite the constraints imposed by her family and by the post-revolutionary French society was in turn presented in youth literature as an example to be followed or, on the contrary, as a proof of her lack of genius. Some authors noted Germain’s independence from learned men to describe her as a fullfledged mathematician; others, on the contrary, used her epistolary exchanges with Gauss as a proof of her lack of originality, or her tendency to imitation. Similarly, the fact that she was unable to access educational institutions was seen as proof of the need for secondary or even higher education for women (she would have obtained much more original results and educated women could participate in the 177 On 178 See

this issue for mathematics in another period of time, see in particular Goldstein (2003). Martin (1922).

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progress of knowledge) or, on the contrary, as an indication of its uselessness (a truly capable woman in mathematics did not need institutionalised higher education). Generally, apart from the mathematical journals, the substance of Germain’s mathematical research was not discussed. It is the moral virtues of the woman mathematician that was debated and their (in)adequacy within French society. Here we find a situation that is to a certain extent analogous to that of texts celebrating women in early modern France,179 for example: Germain’s virtues and moral behaviour were used as a model or counterexample for other women. She could nevertheless also be mobilised to prove (to men) the interest for the society of a better situation for women. Germain’s figure was used extensively to feed discussions concerning women mathematicians or women scientists. Were women capable of mathematics? According to the authors, the answer was often a categorical one: the feminine qualities—patience, assimilation, intuition—were not compatible with the practice of mathematics, which required masculine qualities like creativity, and capacities of abstraction and generalisation. At the end of the nineteenth century, following the first surveys on secondary and higher education for women, this radical position was sometimes softened: women showed certain skills for mathematics, in their learning or even in the first phase of creativity, without showing the rigorous and abstract spirit necessary. In this case, what could explain the case of Germain and other women mathematicians? Either they were considered as second-class mathematicians, who had not actually invented mathematics, or they were described as virile, abnormal or even degenerate. Moreover, Germain’s celibacy was sometimes used to show the danger of women practicing mathematics and science: it removed them from their natural roles as wives and mothers and made them unhappy. Defenders of women’s rights and proponents of the intellectual equality of the sexes, on the contrary, often used Germain and other women mathematicians to show that, despite all the limits that women faced, some were still capable of science: the famous intellectual inferiority of women was thus socially constructed. As we have seen, the positions of these feminists on the usefulness of education for women were very diverse: for many of them, the fundamental role of women remained embedded in the family. “Equality in difference” remained the order of the day and Germain was one of the recurring examples to illustrate it, in one way or another. The examples given above show how women mathematicians, like Germain, could be used as historical paradigms or counterexamples to inform debates on education and more generally on the place of women in society. To follow these various receptions of Germain thus appears to be fruitful for analysing the question of women in mathematics and society under the Third Republic in France and the ways in which it was treated according to the various factors considered. Acknowledgements I warmly thank Eva Kaufholz and Nicola Oswald for their editorial work, and Deborah Kent for improving the English of this paper.

179 See

Goldstein (2003).

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Bibliography d’Abbadie d’Arrast, Marie. (1897–1898). Les femmes dans la science. La Femme 19(23, December 15), 181–183, 20(1, January 1, 1898), 3–5, 20(2, January 15, 1898), 10–11, 20(3, February 1, 1898), 18, 20(4, February 15, 1898), 27–31, 20(5, Mars 1, 1898), 33–37. Alexanderson, Gerald L. (2012). About the cover: Sophie Germain and a problem in number theory. Bulletin of the American Mathematical Society, 49(2), 327–331. Alfonsi, Liliane. (2008). Étienne Bézout: Analyse algébrique au siècle des Lumières. Revue d’histoire des mathématiques, 14(2), 211–287. Angot, Emma. (1909). Un peu de féminisme. Le Correspondant, 81, 951–971. Balz, A. (1893). Les Femmes et les Astres. Le xixe siècle (December 17), 1. Baridon, L. (2003). Du portrait comme une science: phrénologie et arts visuels en France au xixe siècle. In C. Bouton, V. Laurand, & L. Raïd (Eds.), La Physiognonomie. Problèmes philosophiques d’une pseudo-science (pp. 143–170). Paris: Éditions Kimé. Baridon, L. (2010). Viollet-Le-Duc, Eugène-Emmanuel. In P. Sénéchal & C. Barbillon (Eds.), Dictionnaire critique des historiens de l’art actifs en France de la Révolution à la Première Guerre mondiale. Paris, website of the Institut national d’histoire de l’art. Retrieved May 15, 2018, from https://www.inha.fr/fr/ressources/publications/publications-numeriques/dictio nnaire-critique-des-historiens-de-l-art/viollet-le-duc-eugene-emmanuel.html. Barlow, Peter. (1811). An elementary investigation of the theory of numbers: With its application to the indeterminate and diophantine analysis, the analytical and geometrical division of the circle, and several other curious algebraical and arithmetical problems. London: Johnson. Barrau, Théodore-Henri. (1860). Récits historiques. Sophie Germain. La Semaine des enfants, 4(173), 121–122. Barrau, Théodore-Henri. (1872). Livre de morale pratique ou choix de préceptes et de beaux exemples destiné à la lecture courante dans les écoles et dans les familles. Paris: Hachette. Bentzon, T. (1902). Enquête sur le féminisme. Le Carnet, XI, 433–438. Bologne, Jean Claude. (2004). Histoire du célibat et des célibataires. Paris: Fayard. Boucard, J. (2011). Un “rapprochement curieux de l’algèbre et de la théorie des nombres”: études sur l’utilisation des congruences en France de 1801 à 1850. PhD Thesis, Université Pierre et Marie Curie, Paris. Boucard, Jenny. (2015). Résidus et congruences de 1750 à 1850: une diversité de pratiques entre algèbre et théorie des nombres. In C. Gilain & A. Guilbaud (Eds.), Les sciences mathématiques 1750–1850: continuités et ruptures (pp. 509–540). Paris: CNRS Éditions. Boucard, Jenny, & Verdier, Norbert. (2015). Circulations mathématiques et congruences dans les périodiques de la première moitié du xixe siècle. Philosophia Scientiæ, 19(2), 57–77. Boyé, Anne. (2017). Sophie Germain, une mathématicienne face aux préjugés de son temps. Bulletin de l’APMEP, 523, 231–243. Boyer, Jacques. (1898). Les femmes dans la science. La Revue des revues, 26, 600–615. Bruno, G. (1869). Francinet. Principes généraux de la morale, l’industrie, le commerce et l’agriculture. Paris: Belin. Edition used: 1882. Bruyères, J. H. (1847). La Phrénologie. Le geste et la physionomie démontrés par 120 portraits, sujets et compositions gravés sur acier. Paris: Aubert. Bucciarelli, Louis L. (1980). Sophie Germain: An essay in the history of the theory of elasticity. Boston: Reidel. Del Centina, Andrea. (2005). Letters of Sophie Germain preserved in Florence. Archive for History of Exact Sciences, 32, 60–75. Del Centina, A. (2008). Unpublished manuscripts of Sophie Germain and a reevaluation of her work on Fermat’s last theorem. Archive for History of Exact Sciences, 62, 349–392. Del Centina, Andrea, & Fiocca, Alessandra. (2012). The Correspondence between Sophie Germain and Carl Friedrich Gauss. Archive for History of Exact Sciences, 66, 585–700. Cére, Émile. 1896. “Les femmes de science.” Le Petit journal (September 22): 1.

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Champagnac, J. -B. -J. (1837). Le Gymnase moral des jeunes personnes ou nouvelles anecdotiques relatives à des femmes célèbres de notre siècle. Paris: Lehuby. Edition used: 1856, Paris: Ducrocq. Chappey, Jean-Luc. (2004). Enjeux sociaux et politiques de la ‘vulgarisation scientifique’ en révolution (1780–1810). Annales historique de la Révolution française, 338, 11–51. Corry, Leo. (2010). On the history of Fermat’s last theorem: Fresh views on an old tale. Math Semesterber, 57, 123–138. Cubí i Soler, Mariano. (1858). Leçons de phrénologie scientifique et pratique. Paris: Baillière. Dahan Dalmedico, Amy. (1987). Mécanique et théorie des surfaces: les travaux de Sophie Germain. Historia Mathematica, 14, 347–365. Delacroix, M. (1837–1838). Dictionnaire historique d’éducation, ou choix d’exemples et de faits puisés dans l’histoire ancienne et moderne, propres à former et à enrichir toutes les facultés du cœur et de l’esprit, d’après M. l’abée Filassier, (Vol. 2). Paris: Angé et Cherest. Edition used: 1847, Paris: Librairie des écoles. Dickson, L. E. (1919–1923). History of the theory of numbers, (Vol. 3). Washington: Carnegie Institue of Washington. Dumesnil, Georges. (1883). La Pédagogie révolutionnaire. Paris: C. Delagrave. Edwards, H. M. (1977). Fermat’s last theorem. A genetic introduction to algebraic number theory. New York: Springer-Verlag. Fage, É. (1894). Chronique des livres et des revues. 1. Les Femmes dans la science […], par A. Rebière. Bulletin de la Société des lettres, sciences et arts, 16, 245–255. Fayolle, C. (2012). L’éducation est-elle un instrument de l’égalité? Les débats sur l’éducation des femmes à la période révolutionnaire et post-révolutionnaire. In É. Viennot (Ed.), Revisiter la ‘querelle des femmes’. Discours sur l’égalité/inégalité des sexes, de 1750 aux lendemains de la Révolution (pp. 97–107). Saint-Étienne: Publications de l’Université de Saint-Etienne. Fenster, D. (2007). Gauss goes West: The reception of the Disquisitiones Arithmeticae in the USA. In C. Goldstein, N. Schappacher, & J. Schwermer (Eds.), The shaping of arithmetic after C. F. Gauss’s Disquisitiones arithmeticae (pp. 463–479). Berlin: Springer. Fouillée, Alfred. (1893). La Psychologie des sexes et ses fondements physiologiques. Revue des deux mondes, 119, 397–429. Fouillée, Alfred. (1895). Tempérament et caractère selon les individus, les sexes et les races. Paris: Félix Alcan. Gardey, Delphine. (2000). Histoire de pionnières. Travail, genre et sociétés, 2(29–34), 42. Gardey, Delphine. (2005). La Part de l’ombre ou celle des Lumières? Les sciences et la recherche au risque du genre. Travail, genre et sociétés, 14, 29–47. Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Helmstedt: C. G. Fleckeisen. Gauss, Carl Friedrich. (1801). Disquisitiones arithmeticae. Leipzig: Fleischer. Gauss, Carl Friedrich. (1808). Theorematis arithmetici demonstratio nova. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (Commentationes Mathematicae), 1, 69–74. Gauss, C. F. (1811). Summatio quarundam serierum singularium. Commentationes societatis regiae scientiarum Gottingensis recentiores, 1. Gauss, Carl Friedrich. (1818). Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 4, 3–20. Gauss, Carl Friedrich. (1828). Theoria residuorum biquadraticorum, commentatio prima. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 6, 27–56. Germain, S. (1831). Note sur la manière dont se décomposent les valeurs de y et z dans l’équation 4(x p – 1)/(x – 1) = y2 ± pz2 , et celles de Y’ et Z’ dans l’équation 4(x pˆ2 – 1)/(x – 1) = Y’2 ± pZ’2 . Journal für die reine und angewandte Mathematik, 27, 201–204. Goldstein, C. (2003). Weder Öffentlich Noch Privat: Mathematik im Frankreich des frühen 17. Jahrhunderts. In T. Wobbe (Ed.), Zwischen Vorderbühne Hinterbühne. Beiträge zum Wandel der

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Geschlechterbeziehungen in der Wissenschaft vom 17. Jahrhundert bis zur Gegenwart (pp. 41– 72). Bielefeld: Transcript. Goldstein, Catherine. (2009). Gabriel Lamé et la théorie des nombres: ‘une passion maheureuse’? Bulletin de la SABIX, 44, 131–139. Goldstein, C., & Schappacher, N. (2007). A book in search of a discipline (1801–1860). In C. Goldstein, N. Schappacher, & J. Schwermer (Eds.), The shaping of arithmetic after C. F. Gauss’s Disquisitiones arithmeticae (pp. 3–65). Berlin: Springer. Govoni, Paola. (2000). Biography: A Critical tool to bridge the history of science and the history of women in science. Nuncius, 15(1), 399–409. Haechler, Jean. (2007). Les Insoumises: 18 portraits de femmes exceptionnelles. Paris: Nouveau Monde éditions. Hulin, N. (2008). Les Femmes, l’Enseignement et les Sciences. Un long cheminement (xixe – xxe siècle). Paris: L’Harmattan. Joly, N. J. (1885). Dans l’espèce humaine à égalité d’instruction les intelligences sont égales chez les deux sexes. Preuves nombreuses à l’appui de cette assertion. Mémoires de l’Académie des sciences, inscriptions et belles-lettres de Toulouse, VIII.7, 131–158. Joteyko, Jòzefa. (1904). À propos des femmes mathématiciennes. Revue scientifique, 21, 12–15. Kalifa, Dominique, Régnier, Philippe, Therenty, Marie-Ève, & Vaillant, Alain (Eds.). (2011). La Civilisation du journal: histoire culturelle et littéraire de la presse française au xixe siècle. Paris: Nouveau Monde éditions. Kaluszynski, Martine. (1989). Les Congrès internationaux d’anthropologie criminelle (1885–1914). Mil neuf cent, 7(1), 59–70. Knibiehler, Yvonne. (1976). Le discours sur la femme: constantes et ruptures. Romantisme, 6(13), 41–55. Lacassagne, Alexandre. (1906). Précis de médecine légale. Paris: Masson. Lacroix, Sylvestre-Fran çois. (1804). Complément des élémens d’algèbre (3rd ed.). Paris: Courcier. Laqueur, Thomas Walter. (1992). La Fabrique du sexe: essai sur le corps et le genre en Occident. Paris: Gallimard. Laubenbacher, Reinhard, & Pengelley, David. (1999). Mathematical expeditions: Chronicles by the explorers. New York: Springer. Laubenbacher, Reinhard, & Pengelley, David. (2010). ‘Voici ce que j’ai trouvé’: Sophie Germain’s grand plan to prove Fermat’s last theorem. Historia Mathematica, 37, 641–692. Legendre, A. –M. (1798). Essai sur la théorie des nombres. Paris: Duprat. Legendre, Adrien-Marie. (1827). Recherches sur quelques objets d’analyse indéterminée et particulièrement sur le théorème de Fermat. Mémoires de l’Académie royale des sciences de l’Institut de France, 6, 1–60. Legouvé, Ernest Wilfred. (1849). Histoire morale des femmes. Paris: Gustave Sandré. Lemay, Edna Hinie. (1991). Dictionnaire des Constituants (1789–1791), 2 vols. Paris: Universitas. Liard, Louis. (1879). Analyses. Sophie Germain. Œuvres philosophiques. Revue philosophique de la France et de l’étranger, 8, 426–431. Libri, G. (1832). Notice sur Mlle Sophie Germain. Journal des débats (May 18), 1–2. Lombroso, C., & Ferrero, G. (1893). La donna delinquente, la prostituta e la donna normale. Torino: Roux. French translation by Louise Meille: La Femme criminelle et la prostituée. Paris: Félix Alcan, 1896. Loria, Gino. (1903). Les femmes mathématiciennes. La Revue Scientifique, 20, 385–392. Loria, Gino. (1904). Encore les femmes mathématiciennes. La Revue Scientifique, 21, 338–340. Lourbet, Jacques. (1896). La Femme devant la science contemporaine. Paris: Félix Alcan. MacKinnon, Nick. (1990). Sophie Germain or was Gauss a feminist? The Mathematical Gazette, 74(470), 346–351. Mansion, Paul. (1880). Bibliographie. Nouvelle Correspondance Mathématique, 6, 407–408. Marion, Henri. (1900). Psychologie de la femme. Paris: Armand Colin. Martin, Louis. (1922). Le Vote des femmes. La Nouvelle Revue, 44, 137–148. Mayeur Frano, Ise. (1979). L’Éducation des filles en France au xixe siècle. Paris: Hachette.

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Möbius, Paul Julius. (1900). Ueber die Anlage zur Mathematik. Leipzig: Johann Ambrosius Barth. Mosconi, Nicole. (2012). Henri Marion et ‘l’égalité dans la différence’. Le Télémaque, 41(1), 133– 150. Musielak, Dora E. (2015). Prime mystery: The life and mathematics of Sophie Germain. Bloomington: Authorhouse. d’Ocagne, Maurice. (1909). Les Femmes dans la science. Revue des Questions Scientifiques, 15, 64–91. d’Ocagne, Maurice. (1925). Quelques figures de mathématiciennes. Paris: Institut de France. Offen, Karen. (1986). Ernest Legouvé and the doctrine of ‘equality in difference’ for women: A case study of male feminism in nineteenth-century french thought. The Journal of Modern History, 58(2), 452–484. Offen, K. (2000). European feminisms 1700–1950. A political history. Stanford: Stanford University Press. French translation by Geneviève Knibiehler: Les Féminismes en Europe, 1700–1950. Rennes: Presses universitaires de Rennes, 2012. Offen, Karen. (2018). Debating the woman question in the French third republic, 1870–1920. Cambridge (GB): Cambridge University Press. Parshall, Karen Hunger. (2015). Training women in mathematical research: The first fifty years of Bryn Mawr college (1885–1935). The Mathematical Intelligencer, 37(2), 71–83. De Parville, H. (1893). Revue des sciences. Journal des débats politiques et littéraires (December 20), 1–2. Peiffer, J. (1991). L’Engouement des femmes pour les sciences au xviiie siècle. In D. Haase-Dubosc & É, Viennot (Eds.), Femmes et Pouvoirs sous l’Ancien Régime (pp. 196–222). Paris: éditions Rivages. Peyre, Évelyne, & Wiels, Joëlle. (1995). De la ‘nature des femmes’ et de son incompatibilité avec l’exercice du pouvoir: le poids des discours scientifiques depuis le xviiie siècle. In É. Viennot (Ed.), La Démocratie à la française’ ou les femmes indésirables (pp. 127–157). Paris: Presses de l’université de Paris VII. Poinsot, Louis. (1820). Mémoire sur l’application de l’algèbre à la théorie des nombres. Journal de l’École polytechnique, 11, 342–410. Rebière, Alphonse. (1894). Les Femmes dans la science. Paris: Librairie Nony et Cie. Rebière, Alphonse. (1897). Les Femmes dans la science (2nd ed.). Paris: Librairie Nony et Cie. Rengade, J. (1881). La Vie normale et la Santé. Traité complet de la structure du corps humain. Paris: Librairie illustrée. Roussel, P. (1775). Système physique et moral de la femme ou Tableau philosophique de la constitution, de l’état organique, du tempérament, des mœurs, & des fonctions propres au sexe. Paris: Vincent. Rouzade, Léoni. (1887). Les Femmes devant la démocratie. La Revue socialiste, 5, 519–534. Rowold, Katharina. (2001). The many lives and deaths of Sofia Kovalevskaia: Approaches to women’s role in scholarship and culture in Germany at the turn of the twentieth century. Women’s History Review, 10(4), 603–627. Sagnol, Joannès (1889). “L’Égalité des sexes.” La Revue socialiste 9: 685–697; 10: 82–98. Second, L.-A. (1879). Enseignement de la biologie. La Nouvelle revue, 1, 263–282. Smith, H. J. S. (1859–1865). Report on the theory of numbers. In Report of the British association for the advancement of science (1859, pp. 228–267; 1860, pp. 120–169; 1861, pp. 292–340; 1862, pp. 503–526; 1863, pp. 768–786; 1865, pp. 322–337). Oxford: Oxford Clarendon Press. Repr. In J. W. L. Glaisher (Ed.), The collected mathematical papers (Vol. 1, pp. 38–364). Oxford: Clarendon Press, 1894. Stupuy, Hyppolyte. (1879). Œuvres philosophiques de Sophie Germain. Paris: Ritti. Stupuy, H. (1896). Préface à la nouvelle édition. In Œuvres philosophiques de Sophie Germain. Paris: Firmin Didot. 2nd Edition. Szramkiewicz, Romuald. (1974). Les Régents et les censeurs de la Banque de France nommés sous le Consulat et l’Empire. Genève: Droz.

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Terquem, Olry (1860). “Biographie. Sophie Germain.” Nouvelles annales de mathématiques (Supplément: Bulletin de bibliographie, d’histoire et de biographie mathématiques) I–19: 9–13. Tobies, R., Gispert, H. (trans. and comm.), &Peiffer, J. (trans. and comm). (2001). Femmes et Mathématiques dans le monde occidental, un panorama historiographique. Gazette des mathématiciens, 90, 26–35. Vacherot, Étienne. (1869). La Religion. Paris: Chamerot & Lauwereyns. Viollet-Le-Duc, E. -E. (1879). Sophie Germain. Le xixe siècle (April 30), 5, 114. Waterhouse, William C. (1994). A Counterexample for Germain. The American Mathematical Monthly, 101, 140–150. Weininger, O. (1903). Geschlecht und Charakter. Eine prinzipielle Untersuchung. Wien & Leipzig: Wilhelm Braumüller. English translation by Löb Ladislaus: Sex & character: an investigation of fundamental principles, S. Daniel & M. Laura (Eds.). Bloomington: Indiana University Press, 2005. Wils, Kaat. (2005). Le Génie s’abritant sous un crâne féminin? La carrière belge de la physiologiste et pédologue Iosefa Ioteyko. In J. Carroy, N. Edelman, A. Ohayon, & N. Richard (Eds.), Les Femmes dans les sciences de l’homme ( xixe –xxe )(pp. 49–67). Paris: Seli Arslan. de Witt, Henriette. (1888). Les Femmes dans l’histoire. Paris: Hachette.

Jenny Boucard is a lecturer in the history of mathematics at the Centre François Viète for Epistemology and the History of Science and Technology at the University of Nantes. Her research focuses mainly on the history of number theory, the notion of order in science, philosophy and art in the nineteenth century, and the circulation of mathematical knowledge through journals in the contemporary era.

Chapter 8

Dorothy Wrinch, 1894–1976 Marjorie Senechal

Abstract Dorothy Wrinch, born in Rosario, Argentina and raised in Surbiton, England, was educated at Girton College, Cambridge, and mentored by Betrand Russell, G. H. Hardy, and D’Arcy Wentworth Thompson. Determined to determine the atomic structure of proteins, she spent her long career on the borderlines between mathematics, physics, chemistry and biology—and on the margins of those professions too. Brilliant and feisty, she controversially galvanized the field of protein chemistry. Today she is remembered less for her for her bitter feud with the chemist Linus Pauling than for her remarkable early work in logic and her novel insights into the mathematical problems of crystal structure determination.

8.1 Meeting Dorothy Wrinch In 1968, when I first met Dorothy Wrinch, she was a “retired” research professor of physics at Smith College (Fig. 8.1). After a long career in the borderlines between mathematics, physics, chemistry, and biology, she still worked in her office all day every day, writing a book on the geometry of crystal structure. I had recently joined the math department. Crystal geometry was a subject I was eager to learn, and she was looking for someone to make models and draw illustrations. What better tutorial could I hope for? We quickly settled into a first-name friendship despite the 45-year difference in our ages. Dorothy1 never finished a draft of the book but I treasured my time with her, learned the basics of crystal geometry, and benefitted from her deep knowledge and broad expertise. I also admired her wide circle of friendships among the artists, musicians, and social scientists at Smith, and I relished her acerbic comments on people, places, and things. Smith College was, by intent, a gracious place, where praise was freely given and rigorous criticism rare. Dorothy was generous with 1 To

call Dorothy “Wrinch” throughout this essay would be uncomfortably formal.

M. Senechal (B) Clark Science Center, Smith College, 01063 Northampton, MA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_8

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Fig. 8.1 Dorothy Wrinch with Smith College students, circa 1950

suggestions, information, materials, and time, but not with praise. She called out every hint of sloppy thinking. I was grateful for that. Though less so for her imperial style, as in “Make me a model of this by tomorrow,” and in her refusal to call a cube a cube. But, as usual, she was right: it is a regular hexahedron.2 Dorothy herself was polyhedral, and showed different people her different sides. My children remember the nice lady who gave them cookies. A Smith artist saw a scientist immersed in literature, music, and visual arts. Scientists with long memories saw a stubborn fighter. The poet Sylvia Plath, a Smith alumna who returned to teach, saw a self-described misunderstood genius. A senior member of my department worried that I was wasting my time to the detriment of my research. My colleague didn’t know of Dorothy’s remarkable mathematical career, and her many firsts, including first-class honors in Cambridge University’s formidable Mathematical Tripos, the first woman to teach mathematics to Cambridge men, the first woman hired to teach mathematics at the co-educational University of London, and the first woman to earn a D.Sc. from Oxford. And my time with Dorothy was research, in more ways than I then understood. Not only research in crystal geometry, but also research in how science is really done, the struggles and sorrows behind the stories we tell and teach. I learned that mostly indirectly. Colleagues filled me in a little, and I gleaned more later from the notes, correspondence, and other papers she left to Smith, and letters to, from, 2 But she could, when it mattered, bend to custom, as when she titled a paper “Multiple cube systems

in mineralogy.”

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Fig. 8.2 Surbiton High School students, early 1900s. Dorothy Wrinch is second from left

and about her in other archives. In this essay, I outline her journeys, from Rosario, Argentina to Northampton, Massachusetts; from mathematics to biology and chemistry and back to mathematics again. It is thus a précis of—and also postscript to—my biography I Died for Beauty: Dorothy Wrinch and the Cultures of Science.3

8.2 From Rosario to London Dorothy Maud Wrinch was born on September 13, 1894 in Rosario, Argentina, the first child of British ex-patriates Hugh Wrinch, a waterworks engineer, and Ada Minnie (Souter) Wrinch. On the family’s return to England, when Dorothy was three, Hugh was appointed manager of the Chelsea Waterworks Company plant in Surbiton, near London. Their second child, Dorothy’s sister Muriel, was born there in 1899. “The tyke will be a mathematician,” Dorothy’s father told the headmistress of Surbiton High School, on enrolling his impish four-year-old in that strict Anglican school for girls (Fig. 8.2). But what did “be a mathematician” mean? A schoolteacher, as Ada Minnie had been before she married? An accountant? Hugh’s father had been

3 See

Senechal (2013).

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a corn, coal, and malt merchant. Hugh himself held the title M. I. Mech. E.4 As far as I can determine, no one in either family had attended university. But Surbiton High School’s headmistress, Alice Proctor, knew that a mathematician’s path ran through Cambridge, which meant, if they were female, through Girton or Newnham, its “degree-level” colleges for women. (Cambridge did not officially recognize its women’s colleges until 1948!) Dorothy’s parents concurred. Our “grandparents were way ahead of their time,” Muriel’s daughters told me. “They were strong advocates of women’s rights and sent both their daughters to university.” Miss Proctor had been a history student at Girton in 1890, the year that Phillipa Fawcett, a Newnham student, scored higher on the Mathematical Tripos than any of the men. Miss Proctor, a militant feminist, instilled her passions for education and women’s rights in her young charges, together with the requisite fear of hellfire. She also modernized the school curriculum and abolished marks and prizes (competition only on the playing field!). Thus she did and didn’t prepare Dorothy for Cambridge: mathematics was Cambridge’s most competitive sport. Dorothy Wrinch didn’t set Cambridge records, but she was Girton’s only Wrangler (winner of first-class honors on the Mathematical Tripos) in her third and final year there, 1916. Though the Wranglers were no longer ranked, Wrangler status was still the gateway to good jobs—for men. But Wranglerhood did not confer a living on women and Dorothy’s career choices were limited. She could, of course, teach— Girton College was founded, in 1869, in part to provide teachers for girls’ schools. But Dorothy didn’t want to be a teacher, or a school administrator like Philippa, nor did she want to crunch numbers in a lab. She made the same choice I would many years later—to keep on studying—for the same reason: the alternatives were less attractive. But what would she study, and where and with whom? Dorothy chose logic, not least because she admired its evangelist, Bertrand Russell. In her first year at Girton, she had attended his lectures on “Our Knowledge of the External World” and read some of his papers on her own. She also admired his outspoken opposition to the Great War. But Russell’s pacificism had cost him his position at Trinity College and his rooms there. Any teaching he did now would be private, and in London (he had moved in with his brother). He agreed to take Dorothy as one of four students (and the only woman). Girton made this possible financially by giving her a stipend to study for the Moral Sciences Tripos. And so her life-long friendship with Bertrand Russell began. Bertie was, she told his archivist half a century later, the greatest beneficial influence on her life, both personally and professionally. The small group met weekly with Russell for a year; their text was Whitehead and Russell, the weighty Principia Mathematica. Dorothy’s first publications reflect her immersion in and deep understanding of his ideas.5 Logic didn’t hold her interest, but the Russell–Whitehead program of logical atomism—Euclid-style building of

4 Member

of the Institution of Mechanical Engineers. full chronological list of Dorothy Wrinch’s publications can be found on my website, http:// www.marjoriesenechal.com/.

5A

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all mathematics from a few key notions—did, assuming different forms as her career twisted and turned. The next year Dorothy tackled a research problem in set theory under the joint supervision of Bertrand Russell and G. H. Hardy. Hardy was still at Trinity College; he had not yet moved from Cambridge to Oxford. A staunch advocate of both Russell and women in mathematics, he became Dorothy’s mentor.6 The paper she wrote, on transfinite numbers, won Girton’s Gamble Prize for alumnae in 1918 (it was published later in several parts7 ). The Wrinch–Hardy relationship was not wholly harmonious.8 He judged her work to be “good, solid, intelligent stuff,” but worried that she lacked focus. In his view, a mathematician should find a fertile field and cultivate it. Dorothy was drawn to many fields and insisted on grazing in them all. By 1929, when she received a D.Sc. from Oxford, she had published 55 papers in logic, set theory, asymptotic expansions, relativity theory, hydrodynamics, electrostatics, aerodynamics, the philosophy of scientific method, and a pseudonymous book on childrearing.9 By 1929, she had also taught mathematics at the co-educational University College London, while earning M.Sc. (1920) and D.Sc. (1921) degrees there; had married (1922) her former Girton tutor and UCL thesis supervisor John Nicholson (a mathematical physicist at Balliol College, Oxford); and directed the mathematics studies at Oxford’s five women’s colleges. And she had given birth to a daughter, Pamela, in 1927, without skipping a beat. Dorothy had also learned that female research mathematicians of modest backgrounds lived from fellowship to fellowship. The women’s colleges offered shortterm grants, but no lifetime sinecures. Most of her fellowship and grant applications were unsuccessful, but she stayed afloat.10 Girton’s Yarrow Fellowship supported her from 1920 to 1923, when she moved to Oxford to join her husband.

8.3 From Logic to Birds’ Tails Dorothy’s interests in the 1920s spanned philosophy, applied mathematics, biological form, and motherhood. Here are three examples.

6 For

more on Hardy and Russell, and Hardy’s support for women mathematicians, see Senechal (2007). 7 See Wrinch (1919). 8 See Senechal (2007). 9 Oxford granted its D.Sc. at that time for a collection of published papers. 10 Fortunately for her biographers, Dorothy saved her fellowship applications—plans of work, letters of support.

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8.3.1 Scientific Method In 1917, Dorothy joined the prestigious Aristotelian Society for the Systematic Study of Philosophy. (Founded in 1880, it’s still going strong today.) For the next few years, she lectured to it, off and on, and wrote papers on “scientific method” alone and with her friend and fellow-mathematician Harold Jeffreys.11 Science evolves through a ladder of tasks, she said (I paraphrase here). The ladder is Russellian–Euclidean to the core. 1. 2. 3. 4.

Science begins by gathering “brute facts.” In the next stage, facts are sorted into classes. Then theories are proposed to link the classes and explain the facts. Now the logician finds relationships among the theories and deduces their logical consequences. 5. Finally, these relationships and consequences are organized into Euclid-style axioms and theorems. When the science is fully mature (e.g., geometry, physics), the sequence is turned on its head: the axioms come first. Now the subject is ready to teach to the young. Surely this is how Euclid had constructed The Elements! Wrinch and Jeffreys focussed on stage 4 above: how do scientists choose among competing theories? Their solution: a rule of thumb they called the Simplicity Postulate: scientists opt for simplicity. To make this precise, they took “theory” to mean a quantitative law, which in turn is described by a differential equation of finite order and degree, with integral coefficients. The sum of the order, degree, and coefficients is a whole number and thus theories can be ranked on a number line. In practice, scientists don’t tally such sums and choose their theories accordingly. Yet they opt for simplicity! Why? Because, said Wrinch and Jeffreys, they believe the simplest theory is the one most likely to be true. The simplicity line is the a priori probability scale! Identifying simplicity with probability led them to clarify the notions of probability itself. Though the axioms they proposed are a footnote to history now, their simplicity postulate did at least as much for probability theory as for simplicity, and brought Bayesian statistics into scientific discourse.12

8.3.2 Sponges and Asymptotic Expansions Though sponges are porous blobs, some species have rigid skeletons, which are bundles of spikes called spicules. Their geometry varies from species to species. Early twentieth century spongeologists asked why. Do the shapes of the spicules confer some evolutionary benefit to the sponge? Or are they shaped by their environment? 11 See, 12 See

for example, Wrinch (1920), and Wrinch and Jeffreys (1921). Senechal (2017).

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Consider the species Latrunculia, whose spicules are knobby. To John Nicholson (and his late colleague, Arthur Dendy) the knobs looked like the stationary points in a vibrating string. Could they be formed by the vibration of seawater? Adapting Lord Rayleigh’s work on the physics of sound, John measured the spicules carefully and found that the knobs appeared where their theory predicted.13 In this work, he had modeled the spicules as vibrating rods fixed at both ends.14 He gave the important case of rods fixed at one end only to his student, Dorothy. She expanded the relevant Bessel and hypergeometric functions asymptotically15 and, together with her father, computed their values (by hand).16 This may be the only non-fictional father–daughter mathematical collaboration on record. These were the first and last tables Dorothy computed, though she was (at the time) a member of the British Committee on Mathematical Tables, the first female member in its 50-year history. Hardy urged Dorothy to write about asymptotic expansions for the distinguished series Cambridge Tracts, which he edited. But the referee judged the manuscript she submitted to be idiosyncratic (nonstandard notation, eclectic selection of material) and Hardy turned it down. (It appeared later in the American Journal of Mathematics.17 )

8.3.3 Parenthood Motherhood spurred Dorothy intellectually as well as emotionally and physically, prompting a brief detour on her sister Muriel’s professional turf (child development). The book The Retreat from Parenthood, which Dorothy wrote under the pseudonym Jean Ayling, bewails the trend in Britain’s educated classes not “to breed.” Indeed, why would any woman eager for a life of her own bring children into this world of outmoded homes, poor sanitation, and nonexistent social services? “Miss Ayling” proposed a detailed, all-encompassing national Child Rearing Services organization to assist parents on these and other fronts.18

8.3.4 On Growth and Form In July 1918, at a joint session in London of the Aristotelian Society, the British Psychological Society, and the Mind Association, Dorothy gave a brief talk “On 13 See

Dendy and Nicholson (1917). Nicholson (1917). 15 See Wrinch (1922). 16 See Wrinch and Wrinch (1923, 1924, 1926). 17 See Wrinch (1928). 18 See Ayling (1930). 14 See

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the Summation of Pleasures,”19 treating such thorny questions as determining the degree of pleasure we get from eating chocolates at the opera if we know the degrees of those pleasures separately. She also served as Russell’s eyes and ears among the philosophers at that meeting; he could not attend because he’d been jailed for his pacifist activities. The main event of the weekend was a public debate by two famous scientists on the eternal question, “Are physical, biological, and psychological categories irreducible?” Can life be reduced to physics and chemistry, and they in turn to mathematics? Is the body a machine, and does the body explain the mind? Debating were the physiologist John Scott Haldane and the naturalist D’Arcy Wentworth Thompson, who’d been friends since childhood. Both rejected the notion of vitalism, an extra-scientific “spark of life.” But Haldane also rejected reductionism: in his experience, simple models in physiology dissolved into complexity on close inspection. D’Arcy,20 on the other hand, was agnostic on this issue. Mathematics and physics have much to tell us, he argued; we should listen to them first. Take the mysterious subject of heredity, for example. “I for my part look forward, in faith and hope, to the ultimate reduction of the phenomena of heredity to much simpler categories, to explanations based on mechanical lines… that the special science which deals with it has at least found, in Mendel, its Kepler, and only waits for its Newton.”21 Young Dorothy Wrinch, in the audience, would follow that clarion call and never turn back. The Wrinch–Thompson correspondence in the archives of the University of St. Andrew spans twenty years. “Dear Professor Thompson,” she began in September 1924. It is clear that they were already acquainted. I wonder if I might bother you with a question? I am working out some new wing profiles suitable I hope for aeroplanes and I want to find out if there is any bird whose wing profile is of a similar type. (Fig. 8.3) Aerodynamics is peppered with albatross wing sections and the Herring Gull etc. I want to find some type of wing profile sufficiently similar for the loan of the name of the bird to the section to be suitable. As so often happens in maths it’s the maths itself which decides what may be worked and what may not! And I have manage to get these sections done not because I set out to do them, but because the matter in question allows the solution and now I want a nice name for them. Such is the powerlessness of the mathematician. I had hoped perhaps to see you at the Reading meeting. We should be absolutely delighted if you could ever find time to look us up when you are in the south, in London or in Oxford.

“As to your Birds’ wings,” D’Arcy replied, “I should like to take you to Leadenhall Market: where, any day of the week, you may find birds of many sorts, and study their wings to your heart’s content. The curves you draw seem to me excellent curves, but not a bit like the profiles of any real bird’s wing…. The profile that you draw—while it does not remind me in the least of the ‘profile’ of any bird’s wing—does remind me of the section of a bird’s body, or (what is the same thing) of the horizontal section of a yacht…. There are heaps of questions I should like to ask you about stream-lines, 19 See

Wrinch (1917). called him D’Arcy, except Wikipedia, a product of our time. 21 See Thompson (1918). 20 Everyone

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Fig. 8.3 Dorothy Wrinch, right; companion and date unknown

and various bird and fish-forms: in fact, you are going to have a few such questions before long. There must, I suppose, be a simple answer to the question why so many birds’ tails end in a long fork,—swallows, albatrosses, etc., etc…” Sponge spicules were a case study in D’Arcy’s broad field of morphology.22 His 770-page masterpiece, On Growth and Form,23 had appeared in 1917, the same year as the Nicholson–Dendy papers. Its chapter headings suggest the reach of his program; here are a few: magnitude, growth rates, the form and structure of living cells, the forms of tissues, the shapes of sponge spicules, and spiral shells, and horns and teeth, and eggs. On Growth and Form is a mathematical call to arms: While I have thought to shew (sic) the naturalist how a few mathematical concepts and dynamical principles may help and guide him, I have tried to shew (sic) the mathematician a field for his labor—a field which few have entered and no man has explored.

D’Arcy became Dorothy’s mentor, and she the mathematician he’d been hoping to find. Start with cell division, he urged her. “Investigate the field of force within the dividing egg (or other cell); the conditions of stability of its surface-equilibrium and the point at which the unstable equilibrium breaks down, and is replaced by the new stability of the divided cell.” All her work up to now, electrostatics, hydrodynamics, aerodynamics, scientific method, and her talents and drive and ambition, would find their focus.

22 A

term coined by Goethe to mean the forms of living things. Thompson (1917).

23 See

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In On Growth and Form, Dorothy marked up D’Arcy’s chapter on the structure of the cell, noting with approval his remarks on electrical theories of mitosis.

8.4 From Electrostatics to Proteins The first few years after Pamela’s birth were a time of great personal stress for Dorothy. She managed to manage her staggering workload and write her pseudonymous book. But her home was a hell far worse than those she described in it. John Nicholson’s behavior became increasingly erratic and violent. In 1930, he was hospitalized for alcoholism; soon afterward he was permanently committed to Warneford Hospital for “lunacy.” (He died there in 1955.) Eager for a change of scene, Dorothy applied for grants to study abroad. “Nothing is more wanted in biology than that mathematicians of first-class standing should interest themselves in biological problems,” D’Arcy wrote to the University of London in May 1931, hoping (in vain) to tap its Dixon Fund for her. “I do not know of anyone better qualified—or even so well qualified—as Dr. Wrinch to undertake the task. I strongly recommend that she be helped and encouraged to apply herself to it.” Other applications were unsuccessful too, but Girton awarded her its Hertha Ayrton Fellowship and Oxford’s Lady Margaret Hall added to it. In August 1931, Dorothy and Pamela set off for Vienna, a mecca for biologists and mathematicians. There she could visit The Vivarium, a famous biological laboratory led by Hans Przibram, a specialist on the growth and form of insects, and attend the lively mathematics colloquium led by Karl Menger, a leading light in the new field of topology.24 Kurt Gödel and Olga Taussky were also colloquium members at the time. Gödel had announced his Incompleteness Theorem the year before, shattering the Principia program of reducing all mathematics to formal logic. But his discovery did not shake Dorothy’s faith in D’Arcy’s program of rebuilding biology on mathematics and physics, or in her ladder of scientific evolution. Dorothy was eager to learn topology, which was little taught in England. D’Arcy hoped this flexible geometry would be the language of growth and form. Topology could describe the changing shape of the mitotic cell, the branching veins in insect wings and leaves of plants. “I think I have got something about venation in leaves at last!” she wrote to him from Vienna. And in the spring, from Oxford: “I have at last got my ‘biological hypothesis’. I am so thrilled. Without your wonderful book I should never have got it…. Enclosed is first draft of cytology paper…. I do think morphology is electrostatic, don’t you?”25 Whether he agreed with her, we can never know. Her cytology paper has disappeared, and with it her thoughts on venation and her biological hypothesis. 24 See

Menger (1994).

25 I took these quotes from Dorothy’s handwritten postcards to D’Arcy, in his papers at the University

of St. Andrews.

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In November 1933, Ada Minnie Souter Wrinch was hit and killed by a bus. Hugh Wrinch died a few months later. Dorothy curtailed her trips abroad and set about filling gaps in her knowledge of biology, chemistry, and botany. In a letter to the Principal of Lady Margaret Hall, she explained that she’d be “gathering the data needed to test my theories as to the applicability of potential theory to chromosome mechanics, testing various conclusions as to the influence of electrical factors to which I have been led during the last few years…. My urgent need is to study certain problems of electrodynamics.” In an application to the Leverhulme Trust, we see her ladder program. “My idea,” she wrote, “is to discover and make precise what postulates as to the forces involved are required, if the known biological facts are to be deducible.” Then she would look for a law that predicted these special cases, a law describing “the mechanism by which genetic changes in the cell involve morphological changes in the organism.” Each “genetic character,” whatever the phrase might mean (DNA was not yet in the picture), corresponded to a specific potential field in the cell. How do these characters combine? This was the question she’d raised in her London talk in 1918 on the summation of pleasures. After solving the problem in this new context, she would apply it, mapping the changes in the organism as she moved the conductors around. “In the course of this work the problems of the arrangement of genes on chromosomes, in particular the cases of ring chromosomes, present themselves,” she noted in passing. The Trust turned her down but Dorothy wasn’t daunted. The problem of the arrangement of genes on chromosomes had “presented itself” to her, and she turned her attention to that. Discussions with a small group of like-minded young scientists (John Desmond Bernal, Joseph Needham, Dorothy Needham, Conrad Waddington, and Joseph Woodger) grew into the Theoretical Biology Club, remembered today in histories of molecular biology for its pioneering vision.26 Ladder-building was their program too: atoms at the base, then molecules, then cells, then tissues, then organs, then us. D’Arcy Thompson had posed the challenge in On Growth and Form: “The biologist, as well as the philosopher, learns to recognize that the whole is not merely the sum of its parts. It is this, and much more than this. For it is not a bundle of parts but an organisation of parts, of parts in their mutual arrangement… this is no merely metaphysical conception, but is in biology the fundamental truth…” But where in this hierarchy does inorganic become organic? When does the nonliving become life? The members of the club worked in tandem, each in his or her own idiosyncratic style. Joseph Woodger wrote a Principia Biologica in unreadable Russellian symbols. Dorothy chose the chromosome, unveiling her model in a lecture “On the Molecular Structure of Chromosomes” in 1935, at the University of Manchester, after trying it out on the club.27 A number of new sciences have passed from the embryonic stage. Discarding description as their ultimate purpose, they are now ready to take their places in the world state of science… 26 See 27 See

Peterson (2017). Wrinch (1936).

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I visualize the animal and plant morphology of the future, and particularly cytology, as a member of the world state.

Dorothy envisioned the chromosome as a molecular fabric: proteins laid parallel, like a warp; nucleic acids threading under and over them, like a woof. The two ends of each acid, hanging from this hypothetical loom, linked up in the living cell, she said. The chromosome is a woven sheath. She put the “genes” not in potential fields but in the chromosome’s characteristic protein pattern. “The genetic constitution of a chromosome is to reside not only in the nature of the [amino acid] residues of which it is composed, not only on the proportions of these different residues, but essentially and fundamentally in their linear arrangement in a sequence of sequences.” “More might have been made of Wrinch’s sequence hypothesis had not her molecular model of the chromosome come under attack,” says historian Robert Olby.28 The chief problem seems to have been that the warp-woof right-angles implied optical properties for the chromosomes that were opposite those actually observed. Dorothy’s insight was forgotten, along with her chromosome model. Francis Crick rediscovered the sequence hypothesis in the late 1940s. But the Rockefeller Foundation noticed at the time. The journal Nature announced the Foundation’s five-year grant to Oxford: “This grant is to enable Dr. Wrinch to continue and develop her researches into relationships between chromosomes and protein aggregates which have been the subject of several notable contributions…. Her recent work on the structure and behaviour of chromosomes in relation to protein aggregates is a new field of inquiry from which further results of high importance may be confidently anticipated.” But Dorothy prioritized proteins. The failure of her chromosome model had taught her, she said, that the widely accepted linear polypeptide chain model for proteins was probably wrong too. Instead, she proposed, the protein chains loop into hexagons, and the hexagons join to make a lacey fabric. Protein folding could be explained: the fabric folded into polyhedral cages, like origami, and unfolded again. Her cyclol model, as she called it, also accounted for the proportions of the different amino acid residues in proteins (an accepted experimental fact that later proved false) (Fig. 8.4). The visual beauty of her cyclol model, its simplicity, and its explanatory power made it, and their maker, scientific sensations on both sides of the Atlantic. But not all scientists excused her inattention to chemical detail. Her loops linked up by a chemical bond—she called it the cyclol bond—that had never been observed. Linus Pauling, author of The Nature of the Chemical Bond, declared that it could not exist. And thought she had filled the gaps in her scientific knowledge, her views of scientific method remained unchanged. “Her idea of science is completely different from theirs,” said Pauling. And so the scientific controversy (sometimes called the Protein War) of the 1930s began. Not all the weapons were principled. What was a mathematician doing here? Indeed, what was a woman doing here? Especially one who didn’t know her place. Nobel prize winners lined up pro and con; the battle lasted for years. Suffice it to say (here) that despite its fatal flaws, her model sparked imaginations and catalyzed the field of protein structure. 28 See

Olby (1974).

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Fig. 8.4 Dorothy Wrinch’s “cyclol” model of protein structure. Made and photographed in Niels Bohr’s laboratory, 1938

“She influenced many, including [the biochemist] Joseph Needham in England and, in America, Ross Harrison, the great embryologist at Yale, and Irving Langmuir, the physical chemist. I believe that her influence has been vastly underestimated.”29 (Carolyn Cohen, structural biologist) “Her value to me, to all of us, is that her hypothesis produced so much interest in the structure of protein that now, about 40 years later, there are close to 100 structures known30 … it was a great attempt, produced tremendous scientific activity, and I think we owe a great deal of gratitude to anybody who can start the acquiring of such a vast body of knowledge.”31 (David Harker, protein crystallographer)

29 Cohen

(1980).

30 In 2017, 40 years after Harker made this remark, the number of known protein structures exceeded

125,000. 31 Harker (1978).

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8.5 From Oxford to Smith In the spring of 1939, her Rockefeller grant nearing its end, Dorothy applied to Oxford’s Somerville College for its prestigious Lady Carlisle Research Fellowship.32 D’Arcy Thompson wrote a discerning letter in support: One can hardly pass a fair judgment on D. W. without knowing her sad history. You doubtless know that she made an unhappy, even a tragic, marriage; from the consequences of which she still suffers acutely. A certain excitability, a certain forced gaiety, which one sees in her and which is apt to jar on one, all comes of frayed nerves, under circumstances which few women would have come through unharmed. What she wants is the chance of settling down to work, to work of a high class, without worry and anxiety for the next few years. I think it well worth while to help her to do so.

The committee voted unanimously to appoint her. The fellowship permitted its holder to spend one year away from Oxford. As war loomed over Britain, Dorothy took Pamela to the United States. David Harker, then in the chemistry department of The Johns Hopkins University in Baltimore, arranged a temporary position for her there. “She taught a very nice course on the mathematics connected with organic structural chemistry,” he remembered, “and she, of course, told us all about her ‘cyclols’, as she called her models for protein molecules.” But, under her fellowship’s terms, four of the five years had to be spent at Somerville. Dorothy would not bring Pamela back to wartime England, so she resigned it. Johns Hopkins gave her another unpaid year and the Rockefeller Foundation gave her a one-year extension but, they told her, it would be her last. She cast about for an academic job. Karl Menger, then at Notre Dame, agreed to keep a lookout: “If I should hear of any opening for a lady-instructor in mathematics I certainly shall think of you.” Professor Otto Charles Glaser, a biologist at Amherst College who admired Dorothy’s work, tried to place her at the Institute for Advanced Study in Princeton. He asked a trustee friend to propose this to the IAS mathematicians. This is not the place for her, they told him, though “they know of Dr. W’s work and have great respect for it.” Their budget for mathematics was tight, she explained to a friend, and could not be stretched to proteins.33 But Professor Glaser had a plan B. Why not move here? Amherst was then still a men’s college, but there were two women’s colleges nearby. She could influence a rising generation of scientists. He proposed that she be appointed as a visiting professor at Amherst and the two women’s colleges, Smith, and Mount Holyoke, for the 1940–1941 academic. Though no woman had ever taught a course at Amherst, the three college’s presidents agreed to this experiment in intercollege cooperation. In fact, Professor Glaser had a plan B + . F. B. Hanson, a Rockefeller official, noted in his diary in August 1941, in Woods Hole, “G. married Dorothy Wrinch in the office of the Director of the MBL on the day after W completed her work under 32 Like all of the fellowships offered by Cambridge and Oxford women’s colleges, the Lady Carlisle Research Fellowship was open only to women. 33 See Wrinch (1941).

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NS grant.” Her good friend John Fulton, a Yale neurophysiologist and historian of medicine, gave the bride away. “God bless thee,” wrote D’Arcy Thompson, “and he will bless thee!—as I once heard a great man say in much the same circumstances. I never met your Otto, but I’ve known his work for many years. He has been generous in sending me his papers, and they have a box to themselves—you only have half-a-box—in my room at College.” Today, Five Colleges, Inc., the consortium of Amherst, Smith, Mount Holyoke, and Hampshire colleges and the University of Massachusetts, is a national model for intercollegiate cooperation.34 But in 1940, Hampshire College had not been imagined and the University of Massachusetts (then called Mass Aggie), Amherst, Smith, and Mount Holyoke had no formal connections. Dorothy’s joint appointment was the first in local history. She accepted with pleasure. Her course on molecular biology, the first of its kind to be given in any center of higher education, was a cross-disciplinary reading seminar, taught at each college each week. She also gave three lectures for the general public: “Patterns in Biology” at Mount Holyoke, “Patterns in Chemistry” at Smith, and “Patterns in Medicine” at Amherst. “She was delightful and dramatic,” a biologist who’d been there told me, “and she had a great accent. But we weren’t convinced she was right.” Though the experiment was judged a success by all concerned, it could not be repeated. The United States had entered World War II and the colleges moved to a war footing, curtailing their usual curricula to train recruits in practica. But Gladys Anslow, a nationally known and widely respected physicist at Smith, arranged for her to join Smith’s physics department. And so Dorothy found a satisfying and permanent academic home at last. Otto Glaser died in 1951, the year that Dorothy’s cyclol bond was discovered in the family of compounds called ergot alkaloids (among which, LSD). The cyclol bond has never been found in proteins, but cyclol chemistry is a respected field today. Otto never knew of her triumph. Nor did Dorothy at first: she learned of it when a Smith colleague came across the paper a few years later. After Otto’s death, Dorothy moved to Northampton, serving for many years as Faculty Resident in one of Smith’s home-like red-brick dormitories. When I knew her she had an apartment of her own a block from the campus.

34 See

https://www.huffingtonpost.com/entry/open-access-education-students-win-when-schoolsteam (accessed January 20, 2018).

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8.6 Working with Dorothy The Dorothy I knew was a passionate evangelist for symmetry. Symmetry in all its manifestations: ornamental patterns, snowflakes, mathematical groups, molecular structures, crystals, music, and dance. In February 1973, I co-organized a pandisciplinary Symmetry Festival in her honor, but Dorothy was ill and could not attend.35 Crystals took center stage for Dorothy, playing two different roles at once. First, crystals were the tool with which she hoped to validate her protein model. That proteins are molecules, not structureless colloids, was a new idea in the mid1930s, when Dorothy first pondered them. X-ray diffraction was the one, and only, way to determine their structures. That crystals diffract x-rays had been discovered in 1912; by 1915, the structures of simple inorganic crystals had been deduced from photographs of the scattered spots. Proteins do crystallize, but the process is tricky. And deducing their structures from diffraction patterns was, back then, slow, tedious, error-prone, and far more painstaking than computing tables of Bessel functions by hand. “Most crystallographers in those days thought the protein structure problem was hopeless,”36 but eventually their hard work paid off with a long string of Nobel prizes, beginning with the 1962 chemistry prize (to Max Perutz and John Kendrew) for the structure of myoglobin. Dorothy addressed theoretical questions instead. Does the pattern of diffracted rays contain all the information needed to determine the crystal structure? And if it does, how can you extract that information from the pattern? Her 1946 book on these questions, Fourier Transforms and Structure Factors, may be her most important and lasting contribution to applied mathematics (Fig. 8.5).37 But she also saw crystals as a rung in the ladder of life. in 1937, she reported to the Rockefeller Foundation on her visit to the famous crystallographer Paul Niggli in Zurich. He had “very kindly applied his great knowledge of crystal forms to my cyclol molecules and suggested many relevant facts, etc., which fit in very well,” she wrote. “He really does agree with me in thinking that practically all the strange things which characterise the cyclol structures have exact precedents in the inorganic world—even the cage-like form of the molecules.” I think she intended Whole Number Geometry and the Angstrom World, the book on crystal geometry she was writing when I met her, to describe that rung. Above it was a structure she’d already sketched in two dense monographs. “This is it, she wrote of them, an account which satisfies me of how the protein problem can be pictured and how with this picture we can see the integration into a single whole of many different issues and theses, in physiology, normal and abnormal, in morphology, in pharmacology, cytology, enzymology, immunology, high polymer chemistry and organic synthetic chemistry… crystallography, etc.” 35 See

Senechal and Fleck (1974). In 2008 I co-organized a year-long Festival of Disorder. Crick (1990). 37 See Wrinch (1946). 36 See

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Fig. 8.5 Dorothy Wrinch’s lantern slide shows two sets of scatterers (above) and their optical diffraction patterns (below)

This tapestry wove her own strands together too: her discipleship in logic, her expertise in mathematical physics, her theory of scientific method (still intact), her immersion in chemistry and biology, her mastery of diffraction geometry, her studies of crystal symmetry, her life-long search for patterns. On a long walk in the English countryside in 1917, Bertrand Russell had asked his young student, Dorothy Wrinch, what she most hoped for in life. Something to which I can devote all my energies, she replied. She found it in the protein problem. I like to think they spoke of her long and thorny path to it, in their old age, on his visits to Northampton.

Bibliography Ayling, Jean. (1930). The retreat from parenthood. London: Kegan Paul Trench Trubner & Co. Cohen, C. (1980). Deciphering protein designs. In: M. Senechal (Ed.), Structures of matter and patterns in science. Cambridge, MA: Schenkman Press. Crick, F. (1990). What mad pursuit: A personal view of scientific discovery, Basic Books. Dendy, Arthur, & Nicholson, John W. (1917). On the influence of vibrations upon the form of certain sponge-spicules. Proceedings of the Royal Society of London. Series B, 89(622), 573–587. Harker, D. (1978). Colored lattices. In M. Senechal, (op. cit.). Menger, K. (1994). Reminiscences of the vienna circle and the mathematical colloquium. L. Golland, B. McGuinness, & A. Sklar. Kluwer Academic Publishers. Nicholson, John W. (1917). The lateral vibrations of bars of variable section. Proceedings of the Royal Society of London Series A, 93(654), 506–519. Olby, R (1974). The path to the double Helix. University of Washington Press, Dover, 1996.

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Peterson, E. L. (2017). The life organic: The theoretical biology club and the roots of epigenetics. University of Pittsburgh Press. Senechal, Marjorie. (2007). Hardy as mentor. The Mathematical Intelligencer, 29(7), 16–23. Senechal, Marjorie. (2013). I died for beauty: Dorothy Wrinch and the cultures of science. New York: Oxford University Press. Senechal, M. (2017). The simplicity postulate. In R. Kossak & P. Ording (Eds.), Simplicity: Ideals of practice in mathematics and the arts (pp. 79–82). Springer. Senechal, M., & Fleck, G. (Eds.). (1974). Patterns of symmetry. University of Massachusetts Press. Thompson, D. W. (1917). On growth and form. Cambridge University Press. Wrinch, D. (1917). On the summation of pleasures. Proceedings of the Aristotelian Society, New Series, v. 18(1917–1918), 589–594. Wrinch, Dorothy. (1919). On the exponentiation of well-ordered series. Proceedings of the Cambridge Philosophical Society, 19, 219–233. Wrinch, Dorothy. (1920). On the structure of scientific inquiry. Proceedings of the Aristotelian Society, 1920–1921, 181–210. Wrinch, D. (1922). On the lateral vibrations of bars of a conical type. Proceedings of the Royal Society of London, Series A, 101, 493–508. Wrinch, Dorothy. (1928). On the asymptotic evaluation of functions defined by contour integrals. American Journal of Mathematics, 50, 269–302. Wrinch, Dorothy. (1936). On the molecular structure of chromosomes. Protoplasma, 25(4), 550– 569. Wrinch, D. (1941). Dorothy Wrinch to Eric Neville (March 29, 1941), Wrinch Papers, Sophia Smith Collection, Smith College. Wrinch, D. (1946). Fourier transforms and structure factors. Asxred Monograph, no. 2, American Society for X-ray and Electron Diffraction. Wrinch, Dorothy, & Jeffreys, Harold. (1921). On certain fundamental principles of scientific inquiry. Philosphical Magazine, 42, 369–390. Wrinch, H. E. H., & Wrinch, Dorothy. (1923). Table of the Bessel Function I n (x). Philosphical Magazine, 45, 846–849. Wrinch, H. E. H., & Wrinch, D. (1924). Tables of Bessel functions. Philosphical Magazine, 47(Ser. 6), 62–65. Wrinch, H. E. H., & Wrinch, D. (1926). The roots of hypergeometric functions with a numerator and four denominators. Philosphical Magazine 1(Ser. 7), 273–276.

Marjorie Senechal graduated from the University of Chicago and received her PhD in mathematics from the Illinois Institute of Technology. She met Dorothy Wrinch early in her career at Smith College, where she is now the Louise Wolff Kahn Professor Emerita of Mathematics and History of Science and Technology. Senechal has written widely on tessellations and patterns, both periodic and aperiodic, and on the history of silk. She is the author or editor of twelve books, including Crystalline Symmetries; Quasicrystals and Geometry; Shaping Space, and I Died for Beauty: Dorothy Wrinch and the Cultures of Science, and is Editor-in-Chief of the Mathematical Intelligencer.

Chapter 9

Living by Numbers: The Strategies and Life Stories of Mid-Twentieth Century Danish Women Mathematicians Lisbeth Fajstrup, Anne Katrine Gjerløff, and Tinne Hoff Kjeldsen

Abstract This chapter is based on interviews with four women mathematicians who made a research career in mathematics in Denmark from the mid-twentieth century on. Through semi-structured research interviews, we try to capture and pass on glimpses of strategies and experiences in their lives. The four interviewees represent the very few women who achieved faculty positions in a male-dominated field at the universities in Denmark. Their personal stories, in addition to being admirable examples of academic achievement, become stories of how they navigated and succeeded in a society without apparent or fixed solutions for ambitious working women and mothers—and of how they later in life reflected on their choices and options. Their stories are both about women receiving an atypical education in a male-dominated field and about pursuing and succeeding in having a career and a family during a period of change in social values and possibilities for women. In the chapter, we focus on their stories about school life, on the milieu at the universities during their studies, their career choices, gender biases, and on their descriptions of family life and relationships. We coin the concept of “implicit girl,” which was revealed in the interviews, a girl who is created implicitly in our educational system and thus situated in the culture of our society at large.

L. Fajstrup Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark e-mail: [email protected] A. K. Gjerløff Natural History Museum of Denmark, University of Copenhagen, Gothersgade 130, 1133 Copenhagen K, Denmark e-mail: [email protected] T. H. Kjeldsen (B) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK 2100 Copenhagen Ø, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_9

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9.1 Introduction Many articles and official reports1 have pointed out the lack of women among the permanent faculty in mathematics departments around the world. The absence of women becomes more and more pronounced the further we move north.2 It seems that Northern Europe is unable to attract the growing number of women mathematics students into pursuing an academic research career in mathematics. Why is that? At the first Nordic Summer School for women PhD students in mathematics in Luleå, Sweden, in 1996, interviews with some women mathematicians who had made it into the mathematics departments were presented. One of them described her move from South America to Northern Europe as moving from a first-world country to a third-world country when it came to the life of a woman in mathematics. Sociological constructs such as the leaking pipe line, the glass ceiling, the cooling-out effect, and tacit gender biases have been developed and brought forward in attempts to grasp the phenomenon of the lack of women in academia in science and mathematics in many countries.3 There are many studies of such kinds, and this chapter is not one of them. It is neither a contribution to statistics nor to the theoretical development of sociology and gender studies about women in science and mathematics. Nor is it a heroine story. Rather, it is a piece of writing in which we try, by means of semi-structured research interviews, to capture and pass on glimpses of experiences in the lives and strategies of four women who made a research career in mathematics in Denmark from the mid-twentieth century on. How was it? Why and when did they choose to become mathematicians? Were they interested in mathematics at an early age? Was there encouragement from family members or teachers? What was their path to university positions? What motivated them? What drove their interests in mathematics? How and why did they choose their research area? How was the daily life in the mathematics department? Did they have a career strategy? Did they experience gender biases? How did they balance work and family life? What kind of reactions did they experience—being one of few? These are some of the issues that we were curious about. The main part of the chapter is a presentation and analysis of the life stories of the four women as they unfold in the interviews, but first we present our methodological approach and situate our four women interviewees in a broader historical context. We finish with some concluding remarks in which we draw parallels to the current situation and reflect upon its significance for women pursuing a career in academia.

1 See,

for instance, Alper (1993), Hanna (1996), Hobbs and Koomen (2006), She Figures (2015), and AMS (2015). 2 See https://womenandmath.wordpress.com/past-activities/statistics-on-women-in-mathematics/ (all websites cited in this article were last accessed on November 3, 2018). 3 See http://eige.europa.eu/sites/default/files/-/garcia_working_paper_5_academic_careers_g ender_inequality.pdf; Kahlert (2015), Alper (1993), MacLachlan (2014), and Cech and Blair-Loy (2010).

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9.2 Methodological Reflections Our investigation is based on semi-structured interviews with Bodil Branner from the Technical University of Denmark, Inge Henningsen and Gerd Grubb from the University of Copenhagen, and Eva Vedel Jensen from Aarhus University (born in 1943, 1941, 1939, and 1951, respectively). Among them, they cover quite a broad spectrum in the sense that (1) they are the majority of the very few women who made a research career in the mathematical sciences at universities in Denmark in the twentieth century; (2) they worked in different (and different kinds of) institutions; (3) together, their scope of interests and endeavors cover pure and applied mathematics/statistics, developments of new research areas, and classical analysis. However, we are not striving for representativity, and we are not searching for something that can be generalized in a strict scholarly sense. Such aspects might become visible in the stories of the four interviewees, but our focus has been on their experienced reality as a basis for how they coped with becoming and being one of the very few research mathematicians with two X-chromosomes in Denmark in the twentieth century. The interviews were constructed around a set of research questions (see Fig. 9.1 for a selection of questions and a sketch of our interview guide),4 which we categorized into three themes: (1) Growing up in Denmark in the 1940s, 1950s, and 1960s: personal background with focus on the role of childhood, school, and family in relation to educational and career choices; (2) being a student and creating a career and a life in academia: career opportunities and choices related to the subject matter of mathematics, professional institutions, and balancing work and life; (3) gender aspects: gender-related obstacles and opportunities with respect to pursuing a research career and life in and with mathematics. We chose the semi-structured interview format5 because we wanted to capture the interviewees’ experiences in life: its looseness allowed the interviewees to tell stories, and its structure helped us to stay on our chosen tracks. The interviews were guided by these three themes. Our empirical data consist of four two-hour-long, semi-structured interviews, one with each of the four interviewees. We began with theme 1 in all four interviews, but aspects of all three themes were, in practice, interwoven with each other. Three of the interviews took place at the private home of the interviewees, while the fourth one took place at her office at the university. Two of us were present during the interviews, being responsible for various parts of them. All the interviews were audiotaped and transcribed. Our analyses of the interviews are performed from a gender perspective anchored in the concrete experiences of various chosen life episodes as they were remembered, constructed, and presented by the four interviewees in response to our questions. In recent years, the interview has evolved into a widely used methodology for constructing and telling life stories. It is sometimes regarded as a social practice that 4 Regarding 5 See

interviews as qualitative research methods, see Kvale and Brinkmann (2009). Brinkmann and Tanggaard (2010), pp. 37–42.

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Fig. 9.1 Selection of questions and a sketch of our interview guide

mediates the subject’s relationship with itself.6 An interview is always situated in a specific social and cultural context that influences not only the questions asked but also the construction of answers. However, we have used these interviews as sources of information for phenomena that took place outside of them—as (tinted) “windows” into the life world of the four interviewees—in order to learn something about how they experienced past events and social practices, always keeping in mind that the interview in itself is a social episode that mediates the construction of the experience. This methodological dilemma is inherent in the interview approach to knowledge. In this particular project, we consider it to be a strength because it indirectly gives us information about the situation for women in mathematics today through our questions and through the way in which our interviewees reflect back on their experiences. In our analyses of the interviews, we maintained this double focus on the intended and unintended mediated experience of the past and the present. Life as a woman mathematician (and as a male mathematician, for that matter) is personal; it is an experience of the subject, and as such our point of departure is descriptive but of course also interpretive in our design of the interview guide, in our 6 Brinkmann

and Tanggaard (2010), p. 30.

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choice of questions, and in our interpretation of the interview transcripts. We try to capture the life experiences of being a woman mathematician through personal stories from the four interviewees. In this sense, our investigation relies on an empirical phenomenological approach based on subjective and selective memories and concrete stories about pieces of life experiences and strategies related to becoming and being a woman mathematician in Denmark in the twentieth century. One of the strengths of an oral-history approach is that history becomes alive, and it is possible to dig into issues for which there are no other records available. However, there are also pitfalls that cannot be avoided. An interviewee does not give a description of past events; it is rather a reconstruction. She speaks from memory, which is not always reliable. The present is always present in memories. Memories are selective and subject to change. One also needs to be aware of the self-image that is being constructed and conveyed, consciously or unconsciously, by the interviewee. There might be events that the interviewee will keep to herself for reasons of self-preservation or for the sake of others. Answers need to be critically assessed with respect to such issues and with respect to the questions asked in the context of the interviews. Because, as Soraya de Chadarevian has emphasized,7 both the interviewee and the interviewer are participants in the construction of the historical record that an interview constitutes. It is a co-production in which the interview is shaped by the way the questions are asked and by the way the answers are interpreted. Despite its success in the general public, recollections or individual biographies have not enjoyed a high reputation as a genre in the modern academic history of science. Massimo Mazzotti,8 in his short account of scientific biographies, gives several reasons for this at various times in history from enlightenment historiography, with its focus on the rational progress of the discipline, to the new ways of doing social history, with an emphasis on micro-analysis that was embraced by the historiography of science in the 1960s and 1970s (somewhat later in the historiography of mathematics9 ). He points out that the end of the twentieth century witnessed “the beginning of a sophisticated debate on the nature and the role of biography in professional history of science,”10 with reference to such works as Telling Lives in Science: Essays on Scientific Biographies.11 One of the issues is the relevance of the personal for understanding scientific change and practices. We will not go further into this debate and its methodological quarrels. We are not writing a scientific biography of four women mathematicians. We will not dig into their mathematical merits, and we will not provide a historical interpretation of their mathematical practices; rather, we will only touch upon their mathematical work superficially and on a nontechnical level. We are using these mini-autobiographies as windows into a life in mathematics as experienced by these four women seen from a gender perspective, because we think, as Paola Govoni phrased it in her report “Biography: A Critical 7 Chadarevian

(2011). (2014). 9 See, for instance, Richards (1995). 10 Mazzotti (2014), p. 122. 11 Shortland and Yeo (1996). 8 Mazzotti

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Tool to Bridge the History of Science and the History of Women in Science,” that, “through the individual or collective biography of women scientists, women may be more easily attracted to science and the scientific professions, providing them with encouraging reference models to measure themselves against, and raising their level of professional expectations.”12 Lyndall Gordon, an academic biographer, claims that “the real subject of biography is always going to be yourself, some aspect of your personality, some reflection of what’s happing in your life at the time you’re writing the book.”13 Here she touches upon another issue with biographies: the relation between the person one is writing about and oneself. We are not going to undertake a thorough analysis of this, but we are three women academics in Denmark writing in the twenty-first century about women mathematicians in Denmark of the twentieth century. Two of us are mathematicians conducting research between pure and applied mathematics and on the history of modern mathematics, respectively, and one of us is a historian whose research focuses on the history of science and gender. Sure enough, then, our own professional life situations are related to the lives of the four interviewees. The same research questions could also have been asked about women mathematicians working in the twenty-first century, and they reflect our own concerns and curiosity about the paucity of women in the field. Our agenda is to explore the life of our predecessors from a gender perspective, an approach that originated from our concern about the very few women research mathematicians in Denmark today. We have taken this double perspective into account in our analyses because we think that exploring and understanding life as a woman mathematician in Denmark in the twentieth century can also provide insights into why the number of women research mathematicians is still so low. On a more progressive note, moreover, it can also show how women have lived rewarding lives as research mathematicians and how they organized their lives to do so. Our chapter ends with some concluding remarks about a concept that we would like to call “the implicit girl” and about the current situation.

9.3 Decades of Change The decades from the late 1930s to the early 1970s—during which the interviewees were born, raised, educated, and created families of their own—were years of considerable social and political change in Denmark.14 The years before World War II witnessed the gradual disappearance of traditional bourgeois values and family structures. Girls and women from the middle classes gained a much broader spectrum of possibilities in life, and even more so if they had support from their family. 12 Govoni

(2000), p. 409. from Govoni and Franceschi (2014), p. 11. 14 General outlines of the social and family history of Denmark can be found in Jacobsen and Løkke (1986). Statistics on gender, education, and work can be found in Danmarks Statistik (2015). 13 Quoted

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A surplus of unmarried daughters, new positions in education and administration, and a growing acknowledgment of gender equality opened the public sphere and labor market for young women around 1900. As teachers, nurses, operators of telegraphs and telephones, and so on, they could build a life independent of men, though the ideal and norm was still that women should be married and live off their husbands’ income as housewives. Political changes were slow but obvious. Women received the right to vote in 1915, and married women became legally independent of their husbands in 1925 (except in matters of taxation and name changes). During the 1930s and 1940s, the acceptance and need for educated working women grew alongside the expansion of the welfare system (and its associated institutions of health care, schools, and other children’s institutions) and a growing labor market in the cities, where jobs as secretaries, accountants, and so on were deemed appropriate for women. During this same period, the ratio of women in the educational system soared as women gained access to public high school and thus did not have to attend private schools to receive the secondary education necessary for further education at university level.15 The primary school system also supported possibilities for different sorts of academic and occupational training and education through new exams such as the “realeksamen” and “mellemskoleeksamen,” which became increasingly popular among girls (from the middle class) to the extent that the number of girls who graduated with these exams began to exceed that of boys as of the 1960s.16 During the 1950s and 1960s, the number of housewives fell dramatically and it became perfectly normal and accepted for middle and even upper-class women to have educations, jobs, and careers of their own. From 1960 to 1970, the percentage of working women rose from 22 to 42% of all married women, and in 1978 it was 80% (though many had part-time jobs).17 Increased living standard and consumerism also contributed to the fact that two incomes gradually became necessary for most families. The relationship between spouses and the responsibility for children and housekeeping changed following this trend, but much more slowly, given that the nuclear family was the ideal norm and women had the main responsibility for childcaring, cooking, housekeeping, and shopping. These burdens were only partially relieved by the introduction of washing machines, electric kitchen equipment, and so on, and by the fact that family planning became possible and each couple eventually only had two to three children. Naturally, this created challenges for working mothers, since welfare institutions like nurseries and kindergarten did not expand as fast as the number of working women, even though the state made heavy investments in the childcare sector during the 1970s to encourage women to work. Still, many families had private childcare, part-time work for the mother, or women chose to stop working for good or for a period of time until their children reached school age and became less dependent. Moreover, the 1960s and 1970s had different expectations

15 Gjerløff

and Jacobsen (2014), vol. 3, pp. 74ff. et al. (2014), vol. 4, p. 339. 17 Jacobsen and Løkke (1986), pp. 64, 71. 16 Gjerløff

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and concerns from today’s in regard to necessary supervision, control, and responsibilities for children. Many children literally took care of themselves from an early age and were given chores to perform to help their mother (even including looking after very young siblings). Today, only a small percentage of children in Denmark are not placed in nurseries and kindergartens from the age 1 and up,18 and the public childcare system is huge. But in the 1950s–1970s families needed to work out solutions and responsibilities at a much more individual level based on their opportunities: where they lived, what work they had, if they had family who could help, or if they had the economic means to pay for help.19 The new possibilities for women also expanded as a consequence of the general economy. Agriculture—traditionally the most important sector in Denmark—gave way to expanding urbanization, industrial production, and a growing service sector with a need for educated workers, women included. In the late 1950s, a government committee (“Teknikerkommissionen,” or Technicians Committee) even published a report advocating that women should be encouraged to receive technical and scientific education to meet the future need for engineers, technicians, and other functions in the growing industrial sector.20 Even though a growing number of women did receive such educations and found jobs as technicians, laboratory assistants, and pharmacists, the number of women engineers and university faculty positions in mathematics and science occupied by women was negligible. The four interviewees thus represent a general trend in Danish social history, but they are at the same time perfectly atypical because they represent the very few women who achieved faculty positions in a male-dominated field at the universities in Denmark. Thus their personal stories, in addition to being admirable examples of academic achievement, become stories of how they navigated and succeeded in a society without apparent or fixed solutions for ambitious working women and mothers—and of how they later in life reflected on their choices and options. Their stories are both about women receiving an atypical education in a maledominated field and about pursuing and succeeding in having a career and a family during a period of change in social values and possibilities for women. In the following, we will focus on their stories about school life, on the milieu at the universities during their studies, their career choices, and on their descriptions of family life and relationships. The goal is not to provide a complete or detailed biography of each of the four individuals but rather to compare and search for common threads that can hint to the existence—or non-existence—of a coherent story about the life and carrier of the first generation of women mathematicians in Danish academia. The interviewees in question are Gerd Grubb, Inge Henningsen, Bodil Branner, and Eva Vedel Jensen (see Fig. 9.2).

18 Danmarks

Statistik (2014), no. 146: https://www.dst.dk/pukora/epub/Nyt/2014/NR146.pdf.

19 On the history of Danish nurseries, kindergarten, the professionalism of childcare and pedagogues,

see the timeline and stories on the webpage www.pædagoghistorie.dk. 20 Betænkning (1959) no. 228; and Gjerløff et al. (2014), p. 293ff.

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Fig. 9.2 The four interviewees. The year indicates when they were born

9.4 Childhood and School Gerd Grubb, Inge Henningsen, and Bodil Branner were all born around 1940 and attended school in the early 1950s, whereas Eva Vedel Jensen was born a decade later, in early 1950s, and attended school in late 1950s and the 1960s. They are all, with each of their individual life stories, examples of the educational trends for women in the post-war period in Denmark. What they have in common is an upper middle-class background and generally supportive families who regarded education for girls natural but not necessarily as the foundation of a career. All except the youngest of them had mothers who worked (though Inge Henningsen’s mother did so only until she had children) or had higher education themselves, and this atypical background may have provided a crucial role model. Also, the fathers in the families are mentioned as being highly supportive. They all have siblings and were raised in families affluent enough to provide the basis for both sons and daughters alike to gain an education. Furthermore, none of the women attended a school in the remote countryside but rather all grew up in areas relatively close to Copenhagen, Aarhus, and Odense (the bigger cities in Denmark), where schools were large and the number of specialized teachers much higher than in more peripheral areas of the country, and where high school education was a logistic and financial possibility for those who had the intellectual abilities and support from their parents. They all had fairly privileged but not upper-class backgrounds in which their

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choices and interests were supported both by relatively open-minded parents and by financial security. According to their own narratives, they were all good pupils; they wanted to learn, they followed rules, and were among the best in their classes. Whereas the intellectual part of school life seemed to have been satisfactory, the social aspect was more complex, given that being both a girl and one of the youngest and brightest in class could make one an outsider, as Gerd Grubb described: “Then they became tired of me, the other children, because I was always the one who raised my finger, was always the first to say “yes, I can do it” […] and I did not know that you should hold yourself back.” All the interviewees lived up to the expectations of well-read and well-mannered girls of the urban (higher) middle class and went to high school after the compulsory nine years of primary school, and they all seem to have thrived off of the intellectual stimulations posed by the expectations in the high school. Bodil Branner recalls that her parents came home from a teacher–parent meeting in high school and told her, much to her surprise, that her physics teacher had asked them if they were aware that she had special abilities for science: “I was completely stunned! Well, I think, one thing was that I could figure it out but… [that there was something special about it], yes, I did not imagine that.” Gerd Grubb remembered her mathematics teacher as being especially demanding and a bit cruel, scorning and smearing all of them when they presented at the black board. She especially recalled one of these episodes: As I remember, it was something about a necessary and sufficient condition, namely the locus of an ellipse. And I said, “An ellipse is one who does such and such and such,” and then he said “Well, how do you characterize an ellipse? What does an ellipse require? You are giving sufficient conditions, but what are the necessary conditions for having an ellipse” And of course I had not thought about it and then I was told I had to go home and really change my attitude and my effort. That was crucial indeed. […] He meant a lot for the sharpness you need in mathematics. He just made me see, well, that it is not enough to say that this leads to that. You should also know what that requires, you must prove both ways, tell what the logical connection is.

However, he was also the one who recognized Grubb’s potential in mathematics in her final year of high school, when he gave her extra teaching materials to work through. Many memoires of dismissive and misogynist male school and high school teachers are recorded from this and earlier periods. Especially Inge Henningsen recalled having had many experiences with peers who remember being told that girls cannot do mathematics: “It’s something that strikes me when I talk to my peers. There are many of them who have a clear memory of the math teacher who said that girls can’t do math. It appears that so many people say that they have had this experience, and fifty years later, they still remember the teacher who said girls can’t do math.” However, this element does not feature in any of the interviewees’ own stories, and this absence is perhaps in part a determining factor in their later choice of education. A point that was also expressed by Henningsen in the interview, namely that not having been exposed to the gender-related bias in school that mathematics

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is not for girls, might have been a sort of “necessary condition” for pursuing a career in mathematics. Like Gerd Grubb, most of them mentioned specific teachers who encouraged their interest in mathematics and other study subjects and parents who also supported their wish to pass a high school exam (“studentereksamen”)—a choice that was still uncommon for women at that time. Yet it is also notable that they all, regardless of their passion for mathematics in school, mentioned that mathematics was not necessarily their one and only interest in high school or even their first choice at university. In Danish high schools during the middle twentieth century, pupils had to choose between specializing in languages or mathematics and the natural sciences. Both fields appealed to the interviewees. As Bodil Branner put it, “I was crazy about Latin. It was in a sense very mathematical […]. One week I wanted to study classical languages, the next week I wanted to do mathematics.” Eva Vedel Jensen likewise remarked: “When I had to decide what field to choose in high school, I actually considered languages instead of mathematics.” Similarly, when it came to choosing a course of study in higher education, philosophy, languages, and other humanistic studies were mentioned as equally interesting and appealing, and mathematics was chosen almost by chance. According to Inge Henningsen, “I wanted to do mathematics but definitely not physics, so if statistics had not been an option, I would have chosen law or philosophy, I think.” And according to Gerd Grubb, “I would actually have liked to study Danish, and I also considered physics, because it was utterly fascinating […] I don’t know what made me decide. Perhaps it was just natural that I chose the mathematical physics field.” A common thread through these narrated life stories seems to be that the main interests of the interviewees were not the natural sciences but rather intellectual analysis and challenges. Their main goals were to use their intellectual abilities, to use logic, and to learn.

9.5 University Students In Denmark, universities opened to women in the late nineteenth century. At that time, there were two universities in Denmark: the University of Copenhagen, founded in 1479, and the Technical University of Denmark, founded in 1829 as the Polyteknisk Læreanstalt.21 The first woman to earn a university degree was Nielsine Nielsen (1850–1916), who graduated in 1885 and became a medical doctor. She was one of the first two women to earn a high school degree—in 1877. Ten years later, in 1895, Thyra Eibe became the first woman to earn a degree (master’s) in mathematics.22 Her

21 See, for instance, http://universitetshistorie.ku.dk/overblik/vigtige_aarstal/ (University of Copen-

hagen: “Vigtige årstal I universitetets historie”). kvindebiografisk Leksikon, Thyra Eibe (1866–1955): http://www.kvinfo.dk/side/597/bio/ 636/origin/170/.

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main academic achievement was the translation of Euclid’s Elements into Danish.23 In 1892, two women, Hanna Adler24 and Kirstine Meyer,25 were the first to graduate with a master’s degree in physics. All three of these pioneering women in science went on to become high school teachers at the same school, which was founded by Adler.26 In 1897, the first women engineers graduated from Polyteknisk Læreanstalt (now the Technical University of Denmark). They went on to work in the private sector.27 In 1903, Kirstine Smith graduated with a degree in mathematics and physics. Her official position after that was as a secretary to the statistician Thorvald N. Thiele, until his death in 1910. She then went on to work with Karl Pearson in London, where she earned her doctorate.28 Until 1921, women had limited access to positions in the public sector, which included academic positions at university.29 Women could get a university degree, but not an academic position. The German mathematician Käthe Fenchel faced another kind of barrier when she came to Denmark in 1933 together with her husband Werner Fenchel to escape persecution in Germany. She had a mathematics degree from Germany and had done mathematical work all her life, but because of nepotism regulations—her husband was hired as a mathematician at the University of Copenhagen—she did not hold an official academic position except as a part-time lecturer at Aarhus University later in life (1965–1970), where she taught algebra and supervised several master theses.30 Vibeke Borchsenius (1921–1999), too, who obtained a master’s degree in mathematics from the University of Copenhagen in 1945, did not make it into the university until 1957, when she was hired by the newly founded mathematics department at Aarhus University, the first mathematics department outside of Copenhagen, where she primarily concentrated on the development of study programs and teaching.31 While these examples of women in mathematics in Denmark before ca. 1960 are certainly not the complete list, they illustrate the barriers that were in place either implicitly or explicitly. Like Kirstine Smith, Käthe Fenchel had a non-academic position at the University of Copenhagen. Borchsenius worked in high school in the

23 The translation is still used and is digitized at https://archive.org/details/euklidselemente00eucl goog. 24 Dansk kvindebiografisk Leksikon, Hanna Adler (1859–1947): http://www.kvinfo.dk/side/597/bio/ 271/origin/170/. 25 Dansk kvindebiografisk Leksikon, Kirstine Meyer (1861–1941): http://www.kvinfo.dk/side/597/ bio/1435/origin/170/. 26 Vibæk (1959). 27 Sveinsdottir (1997). 28 Guttorp and Lindgren (2009). 29 Kvinders adgang til uddannelse og erhverv (1857–1995): http://kvinfo.dk/aarstalslister/kvindersadgang-til-uddannelse-og-erhverv-1857-1995. For an overview of women in Danish academia around 1900, see also Rosenbeck (2014). 30 Høyrup (1987). See also Agnes Scott’s Biographies of Women Mathematicians: https://www.agn esscott.edu/lriddle/women/fenchel.htm. 31 See Vibeke Borchsenius’s from 1999: http://www.au.dk/om/profil/publikationer/nekrolog/199 9vb/.

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twelve years between graduation and her position at Aarhus University—as did Eibe, Adler, Meyer, and many others. Today (the 2010s), Danish women outnumber men as students at the university level, as the number of enrolled students since 2010 has been 56–59% women.32 In 2016, 25% of men and 34% of Danish women had a post-secondary education (from medium-cycle higher education to university degrees—everything from nurses to doctors).33 Even though the majority of university students is women, men still hold far more faculty positions, especially professorships. This situation is the (perhaps) surprising result of a long historical process of discrepancy between the formal acceptance of women in education and their actual possibilities in higher education. This gender imbalance seems to be especially pronounced in mathematics. Despite equal access to public high school and universities from around 1900, up until late twentieth century only a few women in Denmark have had a research or university career in this field. Besides the four interviewees, we can mention Kirsti Andersen (born in 1941), who graduated from Aarhus University in 1967 with a degree in mathematics. She had a research career in the history of mathematics at the department of history for the exact sciences in Aarhus.34 Birgit Grodal (1943–2004) graduated from the University of Copenhagen in 1968 with a master’s degree in mathematics and physics. She was interested in mathematical economics and became a full professor at the economics department at the University of Copenhagen in 1977.35 Dorte Olesen, who graduated with a master’s degree in mathematics from the University of Copenhagen in 1973, became the first female full professor at a mathematics (and physics) department in Denmark, namely at Roskilde University in 1988. She left to become a CEO in 1989.36 When Gerd Grubb, Inge Henningsen, Bodil Branner, and Eva Vedel Jensen entered university, they did not have a female role model in mathematics (neither a fellow student nor a professor). Instead, they mentioned the support of mathematics or physics teachers at high school or other mentors who played a significant part in their choice of study, but the main drive seems to have come from a common interest in finding a field where the urge to use logic and analytical skills could be satisfied. Branner had a very clear memory about her decision to study mathematics, which was due to the charismatic and energetic Svend Bundgaard (1912–1984), who in 1954 became the first professor of mathematics at the newly established Faculty of Science at Aarhus University,37 and who hired Vibeke Borchsenius in 1957. Branner 32 Uddannelses-

og Forskningsministeriet (2017), p. 2. Statistik (2016), no. 266. 34 Kvinfo Ekspertdatabase: http://www.kvinfo.dk/side/634/action/2/vis/447/. 35 Dansk kvindebiografisk leksikon, Birgit Grodal (1943–2004): http://www.kvinfo.dk/side/597/bio/ 958/origin/170/. 36 Dansk kvindebiografisk leksikon, Dorte Olesen (1948–): http://www.kvinfo.dk/side/597/bio/886/. 37 On Svend Bundgaard, see http://www.au.dk/om/profil/historie/showroom/galleri/personer/ svendbundgaard1912-1984/; and the Festschrift Disse fag må lempes til Verden—Oprettelsen og udbygningen af Det Naturvidenskabelige Fakultet ved Aarhus Universitet. Den første periode: Et festskrift i anledning af 50-års jubilæet 2004, which is available 33 Danmarks

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met Bundgaard for the first time when she was in her second year of high school. It was at a career day for high school students: There were three people I wanted to talk to […] a librarian, […] a priest […] a mathematician or someone in the sciences. […] I liked to read books. I don’t remember exactly what the librarian said, probably something about cataloguing. I talked to the priest because I was interested in theology. […] Since I had no intention to become a priest, he thought theology was a bad idea. And then at the table for mathematics was Svend Bundgaard, and there was no one talking to him. I talked to him for an hour, and when I left I was absolutely sure that I wanted to study mathematics. And that never changed.

As we will see below, Bundgaard also played an important role in both her and Gerd Grubb’s choices and opportunities to pursue a research career in mathematics. The specific combinations of scientific fields studied at university also played a part in their choices, such as Inge Henningsen’s choice to study statistics instead of mathematics (in order not to study physics, which was a part of the primary scientific curriculum at the time in Copenhagen). This was not the case at Aarhus University, where Eva Vedel Jensen mentioned that it was possible to study neither physics nor chemistry, while still choosing mathematics. None of the interviewees recall any explicit gender discrimination from teachers or fellow students during their years of university education. Vedel Jensen remarked, “I had no experience of gender differences as a student,” and Henningsen commented: “I had no experience of not being welcome in the group […]. I don’t think I have experienced any specific discrimination.” Henningsen has done some research on gender aspects with respect to education and the mathematical sciences. She attributes the lack of explicit gender discrimination to the hierarchies in the mathematical sciences, noting that “there is a hierarchy, right, within the fields of mathematics. It is clear that statistics is not a high-level subject, and there was an awareness of this hierarchy, which might have had the effect that they took better care of the girls— because of an awareness that you [as a statistician] were pushed a little bit aside, and then you were careful not to be pushing the [fellow] students you had there even further.” Yet women were fewer in number—fewer than 20% of the students were women.38 Many of these dropped out due to such things as marriage or other careers. Such was the experience of Bodil Branner, who had to fight not the mathematical community but rather her mother in order to study mathematics: And then came the fight with my mother. I wanted to study mathematics right after my graduation from high school. And she did not like that. She was in doubt if I should take a higher education […] and secondly, math had nothing to do with people. Thirdly, I had to go to England as an au pair and train to become a housewife and take care of children and learn English. I did not go to England. I began to study mathematics right away. online at http://www.au.dk/fileadmin/www.au.dk/universitetshistorisk_udvalg/filer/-/Henry_Nie lsen__Disse_fag_maa_lempes_til_verden.pdf. 38 The official statistics of the University of Copenhagen count students from mathematics and physics as one group. The percentages of women students with a major in math or physics were 19% in 1947, 14% in 1955, and 17% in 1971. Ellehøj and Grane (1986), vol. 3, p. 61. At the faculty of Mathematics and Natural Sciences as a whole, the percentage of women students grew from 28 to 36% during the same period; see Pihl (1983), vol. 12, p. 86.

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Even though, according to the interviewees, gender differences were perhaps not visible or outspoken in their respective student environments, women were a minority in mathematics departments in Denmark during this period, and female staff members were virtually non-existing. In all four interviews, the years of university studies are described as intellectually stimulating and the environment as mostly welcoming to women students. Also, the period was one of investment in higher education and research, partly caused by the so-called Sputnik-shock and by a general rise in welfare and state investments in education. Gerd Grubb described the feeling of awe caused by the large number of students, the brand-new building, and the joy of becoming one of the scientists: “Niels Bohr himself welcomed us at the first lecture, and a brand-new auditorium was made ready for us […] and we had a lot of lectures […]. I had a lot of admiration and respect for those who did research, but I did not imagine that I was clever enough for that, and I thought that it would be really fantastic, if I could.” Bodil Branner mentioned Käthe Fenchel, whom she met in Aarhus: “Käthe Fenchel, she came every second week and taught algebra. […] I took her course and she was a role model for me. […] She helped her husband Werner Fenchel, typing his manuscripts, taking notes when they were travelling. She went with him to conferences. That she was capable of that mathematically impressed me enormously.” Yet it was also a time of hard work and doubt, as expressed by Eva Vedel Jensen: “It was really, really challenging. After the first year, I was in doubt if I could make it. It was really, really hard and demanding.” The turning point for Gerd Grubb came when she took her exams after the first two years of study—or rather two and a half years, since hardly anyone was able to complete the first two-year program on time. There were eleven finals in December and January, and Svend Bundgaard was one of the external examiners for the oral examinations. He wanted to build the newly established mathematics department in Aarhus on the basis of Stanford University’s in the USA, where he had been visiting. He was a powerful figure, both politically and within the Danish mathematical community. He would spot talented young mathematics students in Copenhagen and offer them jobs as student teaching-instructors at the new department in Aarhus. Grubb was one of these talents: Svend Bundgaard from Aarhus was one of the external examiners and he had simply decided to snatch some students. He had started the mathematics program in Aarhus two years earlier […] and he had his grand master plan which was to have a structure with teaching assistants. He had seen that at Stanford and other American universities and was impressed by their way of teaching […]. He was a talent scout and he did the scouting there [during the oral examinations]. And he knew my mother from their student days, so he called her and asked if it was okay for him to lure me to Aarhus. […] So I got the offer and moved to Aarhus after the first three years of studies.

The experiences of the interviewees during their lives as students contrast with the broader picture from the early to mid-twentieth century, when women first began to be visible as a group in Danish universities.39 And their recollections certainly contrast 39 Examples

of the continuity of early-twentieth-century discriminatory attitudes and opinions of female students and researchers well into the 1950s can be found in Rosenbeck (2014), p. 109ff.

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with the opinion uttered in the current debate that the respectability and scientific level of higher education are threatened by the dominance of women students. Inge Henningsen suggested a possible explanation for this in her interview: Perhaps the mid-1950s to mid-1960s was a narrow window of opportunity during which gifted women could go to university without any social retaliation because they were still too few and too invisible to be considered a threat to the traditionally male-dominated scientific field. Even though they did not experience explicit gender-related biases in their student environments from their professors and fellow students, it still thrived perfectly well in wider society and even manifested itself in the families of some of the interviewees, as we saw above in the case of Bodil Branner, who had to fight with her mother in order to study mathematics.

9.6 Getting a Job: Career Tracks Only Eva Vedel Jensen, the youngest of the four interviewees, was conscious of her wishes to have a research career in academia while she was a student. When asked about the importance of having professional networks and departmental support in her pursuit of a research career, she answered: “But I think I’ve always wanted this [career], so it has not been so important to me. I can certainly see that it means a lot now for the others [the students and postdocs in her research group], and perhaps it also meant something to me, but it has not been so conscious. I’ve just always wanted this.” When asked about competition and gender, she did not consider herself to have been the first choice: “I could see that they often chose men, but they were not quite as persistent as I was. […] There were some at my own age which I think the professors considered to be very interesting, but for one reason or another they chose to work in the private sector […] and then nothing came of it. And then I was still there […].” For the other three, support and actions taken by faculty members on their behalf (most notably by Svend Bundgaard) played a significant role in their pursuit of an academic career. Bundgaard was a strong advocate for letting students go to the USA to take a PhD degree. Until 1989, there was no PhD degree in Denmark. The (roughly equivalent) degree—the “licentiate”—was not used except in theology until 1969. Gerd Grubb explains the impact that Bundgaard’s strategy had on her: The turning point [when I started to believe I could do research] was probably that the Aarhus-people said so. I had a talk with Bundgaard where he asked me: “What do you want to do after graduation?” Well, I supposed I was going to become a high school teacher like everyone else. “Wouldn’t you want to go to the USA and take a PhD-degree?” he asked. They would support me and they helped me getting accepted at a really good place, namely Stanford University. And then […] I took it from there. And I became a researcher.

Rosenbeck argues that a shift in attitudes toward women in academia in Denmark did not occur before around 1970 (ibid., p. 128ff.), but also recounts several instances of women students and researchers being welcomed in their field and having senior male scholars as mentors.

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It was not, however, completely without doubt: I gradually began to believe that I could. But it took a long time, during my PhD study in Stanford. There was also an edge there, and it wasn’t obvious that I could do research. […] I had three years in total, and for the first two years I tried different subjects. My supervisor, he tried to propose some subjects that I worked with and one of the subjects, it’s quite grotesque, he worked on it too together with someone else. When we met at Wednesday-tea, he [my supervisor’s co-worker] said: “Now, how is it going?” and I said “I’ve tried this and this and this” and then he said “we’ve also tried and it doesn’t work.” And I thought that was terrible. […] After it happened a couple of times with that topic, I thought, “I simply don’t want that,” and then I asked [my supervisor] once more if there was something else he could suggest. He suggested something else and it turned out that it had connections back to my [master’s thesis]. […] At that time, Jacques Louis Lions, whom I was also very impressed with, wrote some works together with Enrico Magenes and they were about to come up with something that explained some of the things I would need in order to solve my problem. I had a sketch of something that I sent to Lions, and he wrote back to me: “Ah vous êtes sérieuse de cette chose?” and I got the response back from him that it was interesting, what I did. I do not really think that [my supervisor] himself understood it so well. I never think he really recognized it, but it ended up being a work that was approved as a PhD and led to five research articles in the following years.

Grubb found the culture in the USA quite different from what she was used to, and here she also experienced explicit gender biases toward women and mathematics: “It is best crystallized by a story. The chairman at Stanford […], he was also in the field of partial differentiation. There was a kind of socializing where we students were also invited. At one time he told me that with regard to female mathematicians, it’s like in music, there’s no female Mozart. Then I told him that there is no American Mozart either.” She also felt the gender bias in the community at large: Inside Stanford, it was OK, but I felt that in the society at large, it was considered completely unreasonable to be a woman and have such scientific interests. […] And you can feel it with the women in the AWM [Association for Women in Mathematics]; they are very aggressive, they are very angry. Because there still is a lack of understanding […]. I had a boyfriend who was a law student. At Christmas I went with him to his home in Pennsylvania and met some of his relatives, who said something like: “How can it be that you are a mathematician, you who are…”—something about me looking nice. I said: “I’m from Scandinavia, you know we have free love and we also have free choice in what we are doing.”

Toward the end of her PhD study at Stanford, Hans Tornehave, who was a professor of mathematics in Copenhagen, traveled to the USA. He paid Grubb a visit at Stanford and suggested that she should consider coming to Copenhagen. Grubb did so, and she went directly from her PhD into an assistant, tenure-track position at the mathematics department in Copenhagen. When Bodil Branner’s husband got a job in Copenhagen, Bundgaard knew she was looking for a job there. Her plan was to become a high school teacher but, as in the case of Gerd Grubb, Bundgaard intervened: I was working in my student office and Bundgaard came in and said, “You have to come with me.” Then he knocked on the door to where an oral exam was taking place. I still blush by the thought of us walking in there. I certainly did not initiate that. […] He told them [the two external examiners (from the Technical University)] that he wanted to talk to them and that they had to meet me, and then we left the room. They had an open position at the

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Technical University. I had a brief talk with them and can you believe it: I said that I wanted to take the pedagogical education for high school teachers. They just said: “OK, you can get time off to do that. […] We really need someone for that position.” […] Bundgaard talked to them without my presence, and then they told me to write an application. That is the only application I have ever written.

While Bodil Branner didn’t experience any explicit gender bias during her studies and in getting a job at the Technical University, it was implicitly and explicitly present in various ways at the borderline between traditional views and new ways of living. First, within herself: When she and her husband graduated, he as a chemist, the plan was to go to the USA for him to earn a PhD, and then, with her being there with him, she might as well earn one too. However, when he changed his mind and took a research job with the pharmaceutical company Novo Nordisk in Copenhagen, the PhD degree was somehow off the table, and she never came around to take a PhD degree. Yet since she was hired by the Technical University anyway, she ultimately did not need it. Second, within her family: When they decided to marry, her mother asked Sven (the husband to be) to talk to Branner’s father and ask permission to marry her: “He [Sven] thought that was completely idiotic. Somehow it wasn’t. It just showed my father’s support and concern. […] I’ve only heard about the conversation from Sven, but my father wanted to make sure that Sven would not interfere, that I should be allowed to study and to finish my master’s degree.” Yet it was one thing to be allowed to take a degree and quite another to become an independent working woman. For this, Branner had to have a second fight with her mother: “Then came the fight with my mother again, and my mother-in-law. They had both believed, without us ever talking about it, that I would become a stay-at-home mom and take care of the children.” Branner is the only one of the interviewees who had such fights with her (female) family members. The other three seemed to have had the full support of their parents. Inge Henningsen, for instance, said that her mother had always been of the opinion that “girls should learn to take care of themselves.” She described her path to a university job in Copenhagen as being governed by much the same circumstances as Branner’s: There were jobs and they needed people good enough to be able to teach at that level. She was hired immediately after she graduated in statistics in 1966. As she put it, “They had no one to teach, so quite a lot of people were encouraged to stay on at the university.”

9.7 Work–Life Balance A main topic of conflict—both then and today—is the work–life balance of women scientists (and male scientists as well). The four different life stories give a glimpse of how an individual (family) could cope with the challenges of two careers, children, home, and ambitions in the 1960s and 1970s, when the role of women, academia, and the labor market in general were shifting. Even though private and some public daycare and kindergarten existed in the 1950s in Denmark, it was not common—and certainly not for the middle class—to

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let small children be taken care of outside the home. Many mothers were stay-athome mothers. In 1960, more than 60% of women in the 35–44 age group were registered as housewives. In 1970, it was less than 40%.40 The four interviewees all described their different strategies for coping with managing small children and housework: accepting help from relatives (mostly their mothers), or pursuing a more alternative lifestyle, such as living in a commune where all adults bore responsibility for children and housekeeping. What they all mentioned is the equal partnership they had with their spouses, as Eva Vedel Jensen remarked: “He’s a physicist—it helps!” And certainly all of them had support in pursuing their own careers and maintaining a mutual ideal of equal responsibility for taking care of children, even if careful planning was essential. As Gerd Grubb explained, “[My husband] and I were absolutely equal—and we could work flexible hours both of us—one of us could stay longer at home in the morning, the other come home earlier in the afternoon […]. And we used all the help we could get—day care, nursery and kindergarten.” Bodil Branner moved from Aarhus to Copenhagen when her two children were very young, and they could not find a place for them in a kindergarten. The family thus employed a full-time live-in babysitter, an arrangement that did not go well with the growing awareness of equality: “We got a truly fantastic babysitter the first year […] but simultaneously I began going to feminist group meetings and it bothered me more and more to have other women take care of my children, whom I could not take care of myself. I would not have had the same conflict of consciousness if we had them in a kindergarten, but we couldn’t find one.” The solution was to join a commune. Branner’s family bought a house together with another family, and shared expenses and responsibilities—including childcare. Thus an adult was at home every day to look after the children, with the regular help of a grandparent: It was really amazing. First of all, it was fun to be at home, everything went on at top speed; you had to wash and shop and everything […] but it was fun. […] And then there were four children and often extra children visiting. Secondly, suddenly I could feel that I did not have the sole responsibility. I had felt indirectly and without thinking consciously about it [that] it was my [responsibility] that the house was clean and the shirts ironed and if it were not, people would look at me [and think] that I was not a good housewife or mother. Being four responsible grown-ups, the pressure was gone.

Help from parents (her mother, that is) was essential for Inge Henningsen, who had three children and a husband with a career in politics, which did not go well with equal time and responsibility in the home. Her mother was a huge help, to the extent that it became a joke in her department: “[When I was expecting my third child], professor Hald said to me: ‘But Inge—how will your mother manage that?—she isn’t getting any younger, you know!’” Eva Vedel Jensen also mentioned the invaluable help from her mother in taking care of the children when they were sick and couldn’t go to kindergarten. Another thing that Vedel Jensen pointed out was the structure of academic work, the necessity of going abroad, and the “danger” of becoming totally absorbed in work: 40 Danmarks

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I think the hardest part when I was young was all the travelling. To leave the family. And actually to my surprise, when I came home, they had all had a very nice time. […] It was sort of a pioneer or settler family I came home to. It was not perfect order, but they had had a good time. So it was probably mostly I who had a problem. But I found that hard. Maybe the special thing about this kind of job is that it is so flexible […]. Flexibility is definitely an advantage, but it should be used in a way that does not make it feel like you work all the time. There was a time when I started saying no because it was too much. […] A spring when I had to go travelling four times […]. That was much too much. My daughter was 7–8 years old and it was much worse for me than when she was very young. […] So I simply said no.

Paid maternity leaves were short, but mothers could choose to leave work or perhaps work part-time when their children were small (even though Gerd Grubb remarked that she found herself doing deeper research during a part-time leave). The departments seem to have been quite supportive—or at least relaxed—in this regard: from bringing baby prams to the university, to agreements on maternity leave, and unofficial agreements on half-time teaching at certain points. In Inge Henningsen’s words: “I experienced the university as much more generous then.” Even if this generosity was common, a traditional view on women’s responsibilities shines through, as Bodil Branner recalled: “At my first interview at the department, the professor said that he understood that I had children and family, so he did not expect me to spend as much time at the department. I told him that I should be treated exactly like everybody else.” It seems that the less rigid rules and the personal contacts between university colleagues and superiors played a major role in this occasionally difficult period for the women researchers, who were trying to pursue a career and have children at the same time. The very small communities of the mathematics departments at that time and the personal contacts between students and professors certainly played a major part in creating this flexible attitude.

9.8 Research Careers: Pure and Applied, Mainstream, and Breaking New Ground An academic career comes in many shapes and forms, and this is apparent in the careers of our interviewees as well. The way they chose their paths—their research areas, their work on committees and boards, their interaction with society, and so on—depended on being at the right place at the right time, but also on persistence and their sense to focus on fruitful research topics. Bodil Branner described to us her first experiences with research at the Technical University. In the early years of her employment she was an assistant to Frederik Fabricius-Bjerre, who was a full professor in her department: “He applied for money so that we could hire students to grade homework and then, well, we could do some math…” This quotation illustrates the terms of academic work in Denmark during the period when the student body was rapidly expanding. Teaching was at the forefront when Branner first started out in 1969. Her research career only really took off in the

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1980s, when she became one of the pioneers in the area of holomorphic dynamical systems, a topic she first encountered through her teaching: In the late 1970s, a student wanted to do a project with me, a master’s thesis. He wanted to do dynamical systems, and preferably with an applied flavor. […] A colleague pointed me to an article by Robert May in Nature about iterating second degree polynomials. I saw a remark that it would be interesting if someone would study that for third degree polynomials. And then I told the student, “You know what? That is simply going to be your project!”

In the early 1980s, she heard about the Mandelbrot set: “I went to some talks organized by Predrag Cvitanovic at the Niels Bohr Institute at the University of Copenhagen. And that is where I heard the first talk about the Mandelbrot set. […] In 1983, Cvitanovic organized a workshop called “Chaos.” He phoned me and said, ‘There will be two mathematicians. Could you take care of them?’” One of these mathematicians was John H. Hubbard, a familiar name to mathematicians in this area. Branner recalled her first interactions with him as follows: He had a method for complex systems analogous to what I was using in the real case. […] Then I asked him how he did it. People who know him are familiar with how he does that. He says, “I will show you an example and then, then you will tell me how it works in another example.” He did just that, and then I had to do it. And […] I can only say it was a mystery that I got it right, but I did. I was lucky that since I could do that example, he believed in me.

He told Branner that she would be welcome to visit him at Cornell University, and so she bought a ticket and went on her first research trip abroad that same summer: I have been privileged by the fact that many have believed in me where I myself maybe didn’t, and because they believed in me, I thought: “Well, I have to try and see if I can live up to that.” And the thing we found was that, looking at some special family of third-degree polynomials, we saw a copy of the Mandelbrot set, with some decorations outside. So I told him: “I have found the classical Cantor set construction in this setting, you simply have to cut and leave these intervals out [corresponding to the decorations], what remains gives the Mandelbrot set.” “Well, there is a theorem behind that, we have a result […],” he said.

She arrived home with an invitation to spend a year as a visiting professor at Cornell, which she did in 1984–1985. Later she realized that they had a special program at Cornell University at that time, where the university would pay part of the salary for women guest professors, so it was not as expensive for the department to have women visiting professors. She did not know that at the time, looking back, she recalled that there were in fact several women visiting professors, which she thought was great. Branner has been a prominent figure in the Danish mathematical milieu. She was elected to the Danish Research Council to represent mathematics, and she has served as president of the Danish Mathematical Society. Like Inge Henningsen, she raised her voice regarding her concerns about gender, though through other channels: as one of the founders of the network European Women in Mathematics, as Vice President of the European Mathematical Society (EMS), and as a member of the EMS’s committee for Women in Mathematics. She also initiated and organized the first in a series of Summer Schools for Female PhD Students in Mathematics in Europe.

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Like Branner’s, Gerd Grubb’s choices were also influenced by her husband. Two years into her position at the mathematics department at the University of Copenhagen, she went to the University of Paris for the third year of her adjunct position in 1968–1969 because of her husband: “He was the one going to Paris and that was why I applied to go to France. It was a very good and fun year. […] I got to know a lot of people, some very well, and others have been important acquaintances through all my life. […] My research area was big in Paris. They were very, very good and I think I got more out of that year than [my husband] did.” In the end, as we have already seen, Grubb chose a different problem for her dissertation from that which her supervisor had suggested. Recently, she had an occasion to revisit this work: There was a quite bad survey article in an important journal in 2005 by [two authors], where they referred to this big mysterious problem which had not been solved. And I had solved it long ago [in her thesis]. So I had to tell about that. […] So I wrote a short reply article where I summed up all that was known, answers to what they posed as open problems, and all those good people who had done this and that, including me. […] This was published, but the editor let [one of the authors of the survey] see it. And he was allowed to write a remark, which was framed in black and said something along the lines of how they were not the only ones not knowing Grubb’s results from the 70 s. It was ghastly.

Grubb believes there was a gender aspect to this incident: “I met that author at a conference in the 80 s where he referred to me as ‘sweet little Mrs. Grubb’ and that sort of thing—very condescending.” However, the controversy sparked new interest in her early findings and had an impact on her later research: “That author tried again to include downgrading remarks in another paper, but I saw it in time to contact the editor Malcolm Brown, who removed the remarks. This led to a collaboration with Brown, where we connected my formulation with another formulation by some Russians and got it all tied up with the hard analysis too, originating in my master’s thesis and PhD, which ended up getting a new life in the 2000s.” Even though issues of gender were felt here and there, discrimination does not seem to have been a dominant factor early in Grubb’s career, nor in the careers of the others: “I don’t really recall any discrimination. There are two things that counteract each other: perhaps one has a little less prestige as a woman, on the other hand, [but] one is also so rare that one is noticed.” Except, that is, for when it came to moving up in the hierarchy. When it came to applying for professorships, the “golden boys” turned out to have an advantage. I was frankly passed over for years. […] Colleagues were preferred and they were perhaps from one point of view better, but I always came in second with very nice reviews, but apparently not worth taking […] because I was not the right person’s pet. […] It was the “similar to me” effect [the Huey, Dewey, and Louie effect in Danish], plus gender. […] In one of these cases, my Stanford advisor told me, “Take it easy, he is just ‘the golden boy’.” He also said, “You should remember that you are not being truly recognized by anyone but yourself. You have to bring that recognition yourself.”

When Grubb was passed by again, those words gained renewed relevance: It was very unpleasant, it really was. It took some inner strength to say “Hell no.” There are two options: You can give up and say: “I have had enough and now I just do easy things I choose.” Or you can say: “I know I am good and I know that I am doing really good research,

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so I will continue doing that. Even when it is tough and troublesome.” […] I made it clear that I have to spend, to promise myself to spend a certain amount of my working hours on research, and not let myself get overwhelmed by first year teaching and study board work, which I have in fact worked a lot on. […] It is definitely a struggle to get enough time for research.

When Grubb was chosen for a full professorship at the University of Copenhagen in 1994, it was followed by complaints from some of the other applicants and it took a while before the whole thing was finally settled. For the second time, a woman was made full professor in mathematics in Denmark. In 1988, she became Docteur honoris causa at the Université de Reims in France, and in 2016 she was awarded an honorary doctorate by the University of Lund in Sweden. Inge Henningsen’s main work has been in applying statistics to the social sciences, health, and also to gender and education studies. She has been politically active and is known to the Danish public for speaking about gender issues and for pointing out problems in the way statistics is used. Her first work was theoretical, but as she grew older she became more and more interested in applications: “When you are 18, abstract mathematics is very, very attractive, but then an interest for applications comes when you get a bit older. I have always thought of, well, of mathematics as power. For good and bad. […] As a power I could get and use.” For a time, she worked with an obstetrician on complications during child birth, a collaboration that began because of her personal life: A start was that I, who had my last child when I was 43, read an article claiming that if there was more than 8 years between the births, then the complications for the second childbirth would be equivalent to a first-time birth. There are 8 years between my first and second child, so I thought that sounded strange. […] I contacted the doctor and asked if this really could be true, and that was the beginning of a long collaboration about birth complications, which showed, of course, that the reason was that among those with 8 years between births was a large group of women who had had many abortions in the meantime.

Henningsen’s political mind and her belief in the role of statistics and mathematics as powerful tools shine through the whole interview: “In that way, mathematics and statistics is power and counter-power” (this is a typical statement of hers). She especially has an eye for the need for counter-power, and she mentioned two examples. First, a pamphlet from the Ministry of Education said that women choose more narrowly than men when applying for degree programs: “I am not quite sure about the numbers, but it was approximately that men would choose 50–100 different studies and women only 10. I tried to find out what they meant by that and called various people. One answer was that each of those 50–100 studies was chosen by at least 5% of men!! It turned out they did not really know.” Her second example concerned AIDS research: “In the beginning, they estimated the number of people with AIDS from the number of HIV-positive. They had no idea how many would actually get sick. If this was too low, this meant that the number of HIV positive was estimated way too high, since they knew how many were already sick. Every article on AIDS in the Doctor’s Newsletter said that the epidemic was growing exponentially. I think my work in that area had political impact.”

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Like Henningsen, Eva Vedel Jensen has derived her main inspiration from applications: I think my research is inspired or motivated by applications […]. The kind of mathematics and statistics we do can be applied to measurements on microscopic images, which have a spatial interpretation. What you see in a flat image is not always representative of what is in space […] but then what happens is that we try to see the general picture in it and generalize it—to spaces of other dimensions and so on. The best thing that can happen is then that we insert 3 as the dimension and sometimes some extra methods drop out of it. Then you can go back to the microscope. […] This is the strength of seeing things in a more general framework. […] It is instead of always doing concrete calculations. If you can do something in a coordinate-free manner, it may be easier to see the structure behind.

Her move into interdisciplinary research began to take shape from the very beginning of her career and was primarily caused by Ole Barndorff, a professor at the Department of Mathematics, Aarhus University: What happened was that we have always had a very active professor, [Ole Barndorff], who founded the statistics group at Aarhus University. He had decided that it was a good idea to have better contact with other departments. Together with him, I applied for a three-year PhD scholarship (kandidatstipendium). It was with the Research Council for Health and Medicine. That meant that I got to do serious applications together with researchers in other fields. I came in as this very green little thing and was not at all used to what happened in a lab. […] Then in 1978 a medical doctor, Hans Jørgen Gundersen, who has been a very original researcher, asked if I could try to solve a problem for him within what is called stereology, and I thought that looked very exciting. That was the beginning of my interest in and work in that area, the research area which I have been working in all through my career. I have of course been broadening my interests—looking a bit to the sides and so on during all those years, but stereology has been the primary. The good thing about Gundersen was that he actually understood mathematics. Not many do.

Vedel Jensen remembers very well the first time she became really conscious about the lack of women in her field: “Well, it was remarkable later in my carrier that we were so few. That is when I discovered how few we are. […] I remember a specialist workshop where I walked into the room and there were forty people and I was the only woman. So that made me think whether I had entered the wrong room.” She also recalled an experience at a conference, where she presented a joint paper: “We had a paper at a conference and we were three co-authors, and I was going to give the talk. I was already at the podium with my notes and then the chairman asked for “mister Shoe” (my co-author’s name was Skov). I was standing right next to him but he could not imagine the speaker was a woman.” In contrast to the other three, Vedel Jensen has built a research group around her. She is the youngest of the three and she has experienced the changes in Denmark’s academic infrastructure during the past ten to fifteen years. There has been a focus on building up research groups and applying for external funding for research. Vedel Jensen has been quite successful in that respect, and she attributes some of this success to her experiences as a member of the Danish Research Council: In 2002, I was elected to the Research Council. And that is when I started thinking about building my own group. That was a very different way of thinking than just doing research on your own. […] I was in my early 50 s and I had not been considering this before. Then

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I started applying for substantial funding and was lucky with that. Being on the research council helped me see where I was placed in the Danish landscape of researchers. […] I learned a lot from being on the research council. How if you had good arguments, the others would listen and could change their minds. […] It was a maturing process. […] I have been treated very nicely by the foundations.

Vedel Jensen now has her own research group at Aarhus University, which will continue to develop her work and their own work after she retires in a few years.

9.9 Concluding Remarks: “The Implicit Girl” and Rivalry There was a sense of humbleness when the interviewees were asked about their research. This stands in contrast to the rhetoric used today in Denmark, where competition for external research funding is the rule of the game, thus fostering self-promotional behavior among people and institutions. More than once, the pure mathematicians Bodil Branner and Gerd Grubb attributed their research success to their luck or privilege. They also both emphasized the effect it had on their selfconfidence when someone believed in them and expected something from them. Whether this says more about gender, the culture in pure mathematics, the time period, or the modesty of these two people is hard to tell from this study, but it is a common feature in both interviews. Their experience is consistent with Inge Henningsen’s observation that a common feature among women who succeeded as academic mathematicians might be the fact that they were not directly confronted with the gender-biased assumption that a research career in mathematics is simply not for women. If she is right, a few people can do a lot of damage by “walking around loose-lipped” (as she phrased it) and uttering opinions about women’s (deficient) abilities in mathematics “without anyone contesting this idea.” As long as it is acceptable to make condescending comments and jokes about the inability of women to have an academic career in mathematics, systematic initiatives to encourage women in this direction might have a hard time accomplishing anything. Another factor that was revealed in the interviews is a concept that we would like to call “the implicit girl”—a girl who is created implicitly in our educational system and thus situated in the culture of our society at large. It came up in the interview with Henningsen when she spoke about her experiences with mathematics teachers in high school: That’s what math teachers say, both men and women; they say, “Oh, these girls, if only they dared make some mistakes!” […] And then I tell them that if they [the girls] make mistakes [in the mathematics classroom], you’ll remember it forever. If boys make mistakes, they are just trying out something […]. They are just creative, but when girls make mistakes, there is much more at stake. […] It is much more risky for a woman in these subjects to make mistakes.

This implicit girl comes up here and there in the interviews, in phrases and reflections about what girls can or cannot do in school, in academia, in mathematics, and in life in general. As Gerd Grubb remarked about her school days, “I did not know

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that you should hold yourself back.” When asked about gender-related prejudices, moreover, she responded: “As a man, you are allowed to do whatever you want,” implicitly implying that they, unlike women, can do so without any consequences. When discussing her daily life in the mathematics department, Bodil Branner also mentioned that she was very conscious of the importance of not making mistakes: “At the same time, I also knew that I should not fail mathematically. I could not show any weaknesses. I was very conscious about it, and I knew that it was not very nice, […] but men, they are not [flawless either], but it was only when I attended European Women of Mathematics meetings that I realized that I restricted myself, I did not use myself fully.” For all four women, a pioneering spirit prevailed when they were ready to enter the job market, especially in the mathematics department at Aarhus University but also in Copenhagen. Both universities were faced with the enormous change in the student body and the need for more people to teach the growing cohort of students that entered universities during the late 1960s and early 1970s. It seems that when something new is created and there is a lack of labor, opportunities arise and there is a greater willingness to break down traditional barriers. For better or for worse, it seems as though the decades when only a few women held degrees and positions in mathematics were also decades during which women were apparently not regarded as competitors by their (male) colleagues and during which small professional environments, personal contacts, and less formal university administrations created better conditions for mathematicians who happened also to be women, wives, and mothers. That said, it also became apparent in the interviews that, while gender had not been an explicit issue during their student years, it became a bit more visible as the interviewees began their careers as doctoral students abroad and as faculty members. With heightened competition, the picture seems to change. This brings us to our final point: the situation today. Comparing their own experiences and choices with those of young women scientists today, our interviewees identified one major difference: the fact that they all earned tenure and had a secure foundation for research and employment very early in their careers. This is in contrast to the many years of project-based employment that are common today, now that tenure seems an almost unobtainable goal that few can achieve and only after several years as postdoc at different universities or temporary professor positions. This insecurity creates stress related to academic recognition, finances, and family planning, and these were not issues that the four interviewees had to overcome to the same extent that most young scientists have to now. Paola Govoni, writing in this volume about what she calls the “publish or perish” atmosphere at European universities, has offered the following hypothesis: Conditions are so competitive that they nearly qualify as psychological violence. In these circumstances, the result of a mix of institutional, social, and personal tensions, women are often the first to give up, and not only women with families. To address these issues and support girls in science and mathematics, I believe it is crucial that we bring students into contact with contemporary women who enjoy satisfying careers in both universities and the private sector while maintaining “normal” personal lives. Women who, like many men, can provide positive role models for all young people, not just girls.41 41 See

Govoni’s chapter in this book, Govoni (2020), pp. 326–327.

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We hope that our presentation of the lives of Gerd Grubb, Inge Henningsen, Bodil Branner, and Eva Vedel Jensen—as constructed through the interviews—can serve this very purpose. In our estimation, all four women are excellent role models for girls and boys and for young women and men who might be interested in pursuing a career in the mathematical sciences.

Bibliography Alper, Joe. (1993). The pipeline is leaking women all the way along. Science, 260, 409–411. AMS. (2015). Annual survey of the AMS. http://www.ams.org/profession/data/annual-survey/201 5dp-tableFF4.pdf. Betænkning. (1959). Teknisk og Naturvidenskabelig arbejdskraft: Betænkning afgivet af den af Statsministeriet nedsatte Teknikerkommission. Brinkmann, Svend, & Tanggaard, Lene (Eds.). (2010). Kvalitative metoder—En grundbog. Copenhagen: Hans Reitzels Forlag. Cech, Erin A., & Blair-Loy, Mary. (2010). Perceiving glass ceilings? Meritocratic versus structural explanations of gender inequality among women in science and technology. Social Problems, 57, 371–397. Danmarks Statistik. (2014). Nyt fra Danmarks Statistik, no. 146. https://www.dst.dk/pukora/epub/ Nyt/2014/NR146.pdf. Danmarks Statistik. (2015). Kvinder og mænd i 100 år.—Fra lige valgret mod ligestilling. http:// www.dst.dk/Site/Dst/Udgivelser/GetPubFile.aspx?id=22699&sid=kvind. Danmarks Statistik. (2016). Nyt fra Danmarks Statistik, no. 266. https://www.dst.dk/Site/Dst/Udg ivelser/nyt/GetPdf.aspx?cid=22637. De Chadarevian, S. (2011). An historian’s perspective: Doing and using interviews. Backdoor Broadcasting Company (June 28). http://backdoorbroadcasting.net/2011/06/soraya-de-chadar evian-an-historians-perspective-doing-and-using-interviews/. Ellehøj, S., & Grane, L. (Eds.). (1986). Københavns Universitet, 1479–1979, vol. III: Almindelig historie 1936–979, Studenterne 1760–1967. Copenhagen: G. E. C. Gads forlag. Gjerløff, A. K. et al. (2014). Da skolen blev sin egen 1920–1970: Dansk skolehistorie, Vol. 4. Aarhus: Aarhus University Press. Gjerløff, Anne K., & Jacobsen, Anette F. (2014). Da skolen blev sat i system 1850–1920: Dansk Skolehistorie (Vol. 3). Aarhus: Aarhus University Press. Govoni, P. (2000). Biography: A critical tool to bridge the history of science and the history of women in science. Nuncius: Annali di storia della scienza, 1, 399–409. Govoni, P. (2020). Hearsay, Not-So-Big Data, and Choice: On Understanding Science and Maths by Looking at the Past, Present and Future of Men Who Support Women. In: this book, Part IV. Govoni, Paola, & Franceschi, Zelda A. (Eds.). (2014). Writing about lives in science: (Auto)Biography, gender, and genre. Göttingen: Vandenhoeck & Ruprecht. Guttorp, Peter, & Lindgren, Georg. (2009). Karl Pearson and the scandinavian school of statistics. International Statistical Review, 77, 64–71. Hanna, Gilla (Ed.). (1996). Towards Gender Equity in Mathematics Education: An ICMI Study. Boston: Klüwer. Hobbs, C., & Koomen, E. (2006). Statistics on women in mathematics. https://womenandmath.wor dpress.com/past-activities/statistics-on-women-in-mathematics/. Høyrup, E. (1987). Käthe Fenchel. In L. Grinstein & P. Campbell (Eds.), Women of mathematics: A bibliographic sourcebook. New York: Greenwood Press. Jacobsen, Annette F., & Løkke, Anne. (1986). Familieliv i Danmark. Copenhagen: Systime. Kahlert, Heike. (2015). Nicht als Gleiche vorgesehen: Über das ‘akademische Frauensterben’ auf dem Weg an die Spitze der Wissenschaft. Beiträge zur Hochschulforschung, 37, 60–78.

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Kvale, Steinar, & Brinkmann, Svend. (2009). Interviews: Learning the craft of qualitative research interviewing (2nd ed.). Los Angeles: SAGE. MacLachlan, A. (2014). Alligators and implicit bias: building your career without getting bitten. A paper delivered at the conference Expanding Potential: A Workshop on Navigating the Hurdles Faced by Women in STEM Fields (November 15). For a video of this presentation, visit http:// vimeo.com/groups/expandingpotential/videos/112930356. Mazzotti, Massimo. (2014). Rethinking Scientific Biography: The Enlightenment of Maria Gaetana Agnesi. In P. Govoni & Z. A. Franceschi (Eds.), Writing about lives in science: (Auto)Biography, gender, and genre (pp. 117–137). Göttingen: Vandenhoeck & Ruprecht. Pihl, M. ed. (1983). Københavns Universitet 1479–1979, vol. XII: Det matematisknaturvidenskabelige Fakultet. Copenhagen: G. E. C. Gads forlag. Richards, Joan L. (1995). The History of Mathematics and L’esprit humain: A Critical Reappraisal. Osiris, 10, 122–138. Rosenbeck, Bente. (2014). Har videnskaben køn? Kvinder i forskning. Copenhagen: Museum Tusculanum Press. She Figures. (2015). European Commission: Directorate-General for Research and Innovation Directorate B—Open Innovation and Open Science Unit B.7: Science with and for Society. http://ec.europa.eu/research/swafs/pdf/pub_gender_equality/she_figures_2015-final.pdf. Shortland, Michael, & Yeo, Richard (Eds.). (1996). Telling lives in science: Essays on scientific biographies. Cambridge: Cambridge University Press. Sveinsdottir, S. (1997). De første kvindelige ingeniører Agnes og Betzy—to pionerer. Ingeniørforeningen i Danmark: https://ida.dk/content/de-foerste-kvindelige-ingenioerer-agnes-og-betzypionerer. Uddannelses- og Forskningsministeriet. (2017). Optag 2017, no. 12. https://ufm.dk/uddannelse/sta tistik-og-analyser/sogning-og-optag-pa-videregaende-uddannelser/2017/notat-12-kon.pdf. Vibæk, J. (1959). Hanna Adler og hendes skole. Copenhagen: G. E. C. Gads forlag. www.pædagoghistorie.dk [a website devoted to the history of child care in Denmark].

Lisbeth Fajstrup (born 1960) has a PhD in mathematics from the University of Aarhus (1992). She has been at Aalborg University since then. From 1996 as an associate professor. With a background in core algebraic topology, she is a cofounder of the new mathematical area directed algebraic topology, an area which is motivated by applications in computer science. Her research is published in both mathematics- and computer science journals and is both curiosity- and application driven. Moreover, she has contributed extensively to dissemination of mathematics, in particular through blogging and talks. Anne Katrine Gjerløff (born 1974) has a MA in history and prehistoric archaeology from the University of Copenhagen (1998), and a PhD in history of palaeoanthropology (2005) also from the University of Copenhagen. In post.doc.-positions she has studied human-animal relationships in Denmark in the 19th–20th century and been a part of the Project Dansk Skolehistorie (Danish School History) at University of Aarhus, contributing to vol. 3–4 of the project’s 5-vol. publication (2014). AKG was project manager of the national celebration of the skole200—the bicentennial of compulsory education in Denmark (2012–15). Since 2015 AKG has been a science communicator at The Natural History Museum of Denmark, and from 2019 leader of the Team for Public Engagement and School Service. She has published several peer-reviewed and popular articles on the history and ideas of children, animals, prehistory, nature and hygiene, has edited a number of books and is author of both children’s nonfiction books and chapters in a number of scientific anthologies.

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Tinne Hoff Kjeldsen (born in 1961) has a MA in mathematics from the University of Copenhagen. She received her PhD in the history of mathematics (1999) from Roskilde University, where she was until 2015. From 2003 as an associate professor. From 2015 until now, she is professor of history of mathematics in the Department of Mathematical Sciences at the University of Copenhagen. She has published extensively on history and philosophy of modern mathematics, history of mathematics for the learning of mathematics and history and learning of mathematical modelling in modern science. She has published and edited books for both specialists and for a more general public.

Part IV

Perspectives

Chapter 10

Hearsay, Not-So-Big Data and Choice: Understanding Science and Maths Through the Lives of Men Who Supported Women Paola Govoni Abstract For centuries now, a great deal and large variety of quantitative data from anthropological and biomedical research on women has been conducted, collected, classified and interpreted. How is it that some (male) scientists read this same data and research as indicating that women are intellectually inferior to men while others see a form of ambiguous diversity and still others equality? Following the invitation extended by this project’s editors to build a bridge between women’s past and present in maths, in this chapter I move forward and back in time to discuss how, by interweaving arguments about present-day data with a gendered history of experts in maths (and science), we might achieve a better understanding of both the history of men and women in maths and science, and of maths and science as cultures which are socially constructed. To face the problems lying in wait for humanity, from migration in a climate-changed world to the challenge of providing energy for billions of people, we need good, abundant maths, science and technology embedded in good, abundant politics. It would appear that the only way forward is to train young people—men and women—to reason freely about science as a social culture. And indeed, this is probably the same path that will allow us to overcome sex and race, as well as gender, ethnicity and class, in science. More realistically, young people engaging with science through this approach will be able to appreciate how such elements may affect both the collection and the interpretation of data: a phenomenon we need to keep abreast of given that it is just as relevant for mathematicians and natural scientists as it is for historians and social scientists.

P. Govoni (B) Department of Philosophy and Communication, University of Bologna, Via Zamboni, 38, 40126 Bologna, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6_10

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10.1 Introduction: Betting on Numbers It is well known that, since antiquity, a large amount and variety of quantitative data have been collected and classified regarding women. For many centuries, as scholars have likewise established, the comparative standard used in interpreting those data consisted of the same white, male body that the individuals collecting and processing said data themselves inhabited. From Aristotle’s time (384–322 BCE), when there were only four data points at play to explain women’s difference, or rather inferiority, as compared to men—namely hot and cold, wet and dry1 —to Francis Galton (1822– 1911) and his “law of the deviation of averages”2 or Cesare Lombroso (1835–1909) and his famous book brimming with quantitative data on “criminal” and “normal” women,3 countless (male) natural philosophers and scientists have investigated and measured women’s bodies. We know, furthermore, that the researchers riding this fervent wave of quantification and classification often collected data on women’s cranial circumference and brain weight without normalizing the data in any way, not even in relation to height. Although these episodes were transcended by twentiethcentury science, they left their mark on contemporary culture in a way that seems all but indelible. Even today, there are many humanists and social scientists who imply that this kind of research is still being conducted, for example in neuroscience or endocrinology, without a proper awareness of the social and cultural issues at stake. The reality of what happens in cutting-edge laboratories is, of course, much different. We know that a number of scholars were aware of these critical issues back in the early modern age4 as well as the nineteenth century, as I shall show, and the same is true of contemporary scientists both male and female, such as the geneticist Steve Jones or cognitive neuroscientist Gillian Einstein, to name just a few I shall address in this chapter. Whether in genetics, climate change, cancer research or Alzheimer’s studies, for several decades now an increasing number of scientists and technologists have been carrying out their work in an integrated manner, taking into account the findings of research in the social sciences. I have the impression, however, that the humanities and social sciences have yet to fully grasp these new developments in the way science is conducted. This misunderstanding contributes to nurturing prejudices about science—a term which I understand to include the natural sciences, technologies, medicine and, of course, mathematics—and these prejudices in turn tend to alienate the younger generations and, perhaps, young women in particular. Today, most geneticists are of course aware that individuals’ postal codes—which reflect their economic, social and cultural status, and much more—can be more decisive than DNA in shaping their destiny, for example in terms of life expectancy. 1 For

a discussion of Aristotle in this context, see Sissa (1994), pp. 46–81. (1981). 3 The first English translation, with an introduction by W. D. Morrison, was Lombroso and Ferrero (1895). Lombroso and Ferrero’s book (1st orig. Italian edition 1893) immediately achieved international circulation. William Douglas Morrison (1852–1943) was a criminologist and prison chaplain. For a discussion of Lombroso and Ferrero’s book on women, see Jenny Boucard’s Chap. 3.1. 4 Pomata (2013). 2 Darwin

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And yet, it is also true that many scientists in the past also shared this awareness. We would do well to bear this in mind more often, in an era in which it is easy to blame science, technology and mathematics for the many problems facing humanity: suffice to recall the new, mystical or rhetorical algorithms that often prove so useful for holding the internet responsible for all kinds of social, global or local problems, from controversial political elections and the tendency to induce people to buy an ever-increasing range of products to the problem of stalking. As early as the second half of the nineteenth century, besides data on women’s bodies and their (allegedly backward) position in the history of human evolution, some experts were also collecting data on women’s education, including maths and science training. I believe that the history of that quantitative data on women’s nature, culture and social roles can be useful today, allowing us to investigate a question that remains largely unexplored, even in studies on science, technology and society: Why and how is it that the exact same data and research, including studies on human evolution, led some scholars to conclude that women are inferior to men while other scholars saw an ambiguous form of diversity and still others interpreted the same data as demonstrating equality? Indeed, even today there are sometimes individuals prepared to declare that, if women continue to face difficulties in achieving top professional positions in maths, science and technology even in countries such as the United States that have been supporting female scholars for decades, we simply must accept that the causes lie in their “aptitude”, that is biology. Unfortunately, I am not referring to the so-called locker-room talk. There are well-known examples of this conclusion being reached by a (democratic) economist and President of Harvard in 2005,5 or again in 2017 by a Google software engineer.6 These are people who know how to wrangle data; people who might address the controversial issue of women in science using the same quantitative data and research that the authors of this and other books on women and science use.7 Evidently, in the age of the mystique of big data just as in the times of Aristotle with his four data points and the age of Charles Darwin (1809–1882) when science and sociology were just beginning to employ statistics, quantitative data describing women, their bodies and social roles are imbued with something we might call worldview, social culture or politics. The good news is that this is true for misogynists as well as feminists, for those who care little about this issue or consider themselves “objective”: face to face with quantitative data, all of us have to make a choice, or perhaps a gamble, calling into question—at times without realizing it—both our values and our backgrounds. As the last few decades of the history and sociology of science have shown, this interweaving of science and values impacts 5 Summers

(2005). Another source that is useful for understanding this controversial standpoint is Pinker (2002), Chap. 18, “Gender”, pp. 337–371. On the other hand, see Fine (2017). 6 The 10-page anti-diversity manifesto (2017) by former Google software engineer James Damore is easy to find on the Web. 7 By now the literature on this issue is extraordinarily extensive, I limit my citations to Margaret W. Rossiter’s three volumes: 1982. Women Scientists in America: Struggles and Strategies to 1940. Baltimore: Johns Hopkins University Press; 1995. Before Affirmative Action, 1940–1972; 2012. Forging a New World since 1972.

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both the process through which experts collect and fine-tune data and the process through which these data are interpreted.8 As I seek to show in this chapter, long before historians, philosophers or sociologists of science, it was the best scientific minds themselves who developed an awareness of this fact: suffice to cite the idea of “guesswork” as argued by Richard Feynman (1918–1988), a scientist open-minded enough to admit that, when facing natural phenomena, sooner or later scientists are obliged to engage in guesswork.9 The process of guesswork is, naturally, composed of observation, experimentation, data and deep thinking. Not to mention hearsay, as in the case of male-generated science about women. Typically, in order to understand how society with its values (including genderbased values) interacts with the making of science about women, as historians we are accustomed to recalling the many (male) scientists who have used quantitative data to explain women’s social marginality in biological terms.10 To highlight the political role of the choice we all end up making when engaging data describing the nature and culture of women, in this Chapter I shall instead focus on several evolutionist scientists who opted to support women. I suspect that the gamble that leads me to interpret the abundant, quite polished data demonstrating girls’ disadvantages in maths and women’s challenges in science through a social and cultural lens now, in the twenty-first century, is similar to the move made by the nineteenth-century (male) Darwinian experts which I quote in this chapter: faced with scarce, uncertain data, they wagered in favour of women’s rights and equality with men. In the socalled age of science,11 when women began to make their first, stubborn strides into universities throughout the western world, it is an established fact that many Darwinian scholars asserted women’s inferiority. In fact, although Darwin went on to act inconsistently in his own private life, he is known to have argued that women were intellectually inferior.12 However, both internationally well-known Darwinians such as John Dewey (1859–1952) and much less well-known Italian evolutionists I shall cite here, the passionate supporters of Darwin for both political and scientific reasons, diverged from their hero’s position on “the woman question”. In the first part of this chapter, I assert the importance of asking ourselves every now and then about the risks of projecting our values and personal interests onto the making of science or its history, and also the advantages of such a move: the goal is to keep a close eye on these forms of interference, identifying them in our own and 8 For

a landmark anthology of writings on science studies, see Biagioli (1999). Feynman spoke about this point in one of his well-known 1964 lectures. See Richard Feynman Messenger Lectures, Cornell University, available at http://www.cornell.edu/video/pla ylist/richard-feynman-messenger-lectures (this and following sites were last accessed December 31, 2017). 10 Similarly, early modern historians have often emphasized the anti-feminist role played by medical argument in the so-called Querelle des Femmes. For an important and fresh perspective on the crucial role that medicine played on the pro woman side, see Pomata (2013). 11 MacLeod (2000), Fyfe and Lightman (2007), and Nieto-Galan (2016). 12 See articles by Evelleen Richards, instant classics cited in the following notes, and in particular her definitive Darwin and the Making of Sexual Selection, Richards (2017). These sources are also valuable for the extremely rich bibliography they provide. 9 Richard

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others’ research. Indeed, to successfully curb gender disparities in maths and science, we need a stronger alliance between scientists, social scientists and humanists, not to mention school teachers and policy makers, and this can only be achieved if we first agree on the meaning of the data on girls and maths and women and science. I will thus go on to briefly outline contemporary data on girls, women and mathematics, highlighting certain controversial aspects of the Italian case. I then take a step back in time to the period between the nineteenth and twentieth centuries, when women began accessing higher education and men, on becoming aware of this surprising— and frightening, for many of them—new social phenomena, reacted in various ways: not only by relegating women to the lowest rungs of the evolutionary ladder, but in many cases by explicitly siding with women. Keeping in mind this longstanding alliance between male scientists and women, I conclude by returning to the present to offer some suggestions as to how we might conceive of a future involving gender-free science—a controversial expression that Evelyn Fox Keller coined back in the 1980s but I still find inspiring today13 —by inviting young people to gamble on science and technology as culture. To face the problems lying in wait for humanity, from new migration patterns in a world transformed by climate change to the challenge of providing energy—both food and electricity—for billions of people, we need good, abundant maths, science and technology embedded in good, abundant politics. It would appear that the only way forward is to train young people, and in particular future scientists both male and female, to reason freely about science as a social culture. This is the same pathway that will allow us to overcome sex and gender, along with race and ethnicity, in science and maths. Or, at least, this represents a pragmatic bet, the result of a choice.

10.2 On the Role of Values and Self in Science and Its History The large body of available quantitative data on girls and maths, women and science raise interesting questions, such as: What distinguishes countries in which girls perform as well as boys or even better in maths—as in Sweden, Norway, and Island, Qatar or some parts of China, according to Program for International Student Assessment (PISA) data—from countries in which girls still show persistent difficulties? Do the data allow us to say that, in those contexts in which girls’ maths performance equals that of boys, women in mathematics have the same chances as their male colleagues of reaching the top of the university career ladder? In considering these phenomena, we should seek to avoid not only the risks of misogynist hearsay, obviously, but also the risks of superficial political correctness; indeed, such an approach might give us the advantage in moving beyond the disappointing results achieved to date in both education (girls in maths) and the professional sphere (women in maths and science). Naturally, we must first establish whether or not we agree that, among 13 Keller

(1985).

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human minds developed in comparable social and educational contexts, intelligence tends to be distributed regardless of sex and race. If we can agree on this, then it follows that we must also agree that having departments and research centres full of women discriminated against on the basis of their sex clearly represents a serious problem not just for those women, but for global society as a whole, especially given the enormous challenges we are currently facing. It goes without saying that this argument can only convince those who are ready to accept the fact that the vast amounts of data on girls and women in maths and science collected over the course of the last century demonstrate that their disadvantaged position stems from thousands of years of non-inclusive education and social culture rather than hundreds of thousands of years of biological evolution. Girls underperform in mathematics compared with boys in 37 of the 65 countries and economies that participated in PISA 2012 and it is clear that the more the Global Gender Gap (GGG, the index edited by the World Economic Forum) is reduced, the less divergence there is—judging from the OECD’s PISA data—in boys’ and girls’ performance in mathematics.14 Generally speaking, in countries that have nearly eliminated social discrimination against women as attested by GGG measures, girls perform even better than boys in maths. Yet we know that there are countries in which this rule does not apply: there are countries (such as certain areas of China, for example, or Qatar15 ) where women do not enjoy the rights considered fundamental in the so-called Western countries, but girls still do better in maths than boys. Shifting the focus from girls to women, however, the situation appears to be different. As others have observed, the data I outline here demonstrate that women in academia likewise face significant difficulties in reaching top career positions in almost every country. Female mathematicians occupy a challenging position in Italy, a country which the 2017 GGG ranked 82nd out of 144 countries worldwide,16 but the same is true of countries such as Denmark, which holds 14th place in the same ranking, as Lisbeth Fajstrup, Anne Katrine Gjerløff and Tinne Hoff Kjeldsen discuss in their chapter. In my opinion, if we examine these data in relation to others such as sociology of education or social psychology research on performance anxiety, there is no doubt that girls, just like minorities, have trouble in maths due to sociocultural and psychological factors that could be overcome under the right circumstances.17 Professional female mathematicians and scientists are likely to face hurdles which are not only social and psychological but also institutional and corporative, supranational and long term and, consequently, much more complex: in order to understand this state of affairs, it seems necessary that we complement history with other approaches. 14 World Economic Forum (2017). The Global Gender Gap Report 2017, https://www.weforum. org/reports/the-global-gender-gap-report-2017; OECD (2017). Mathematics performance (PISA), https://data.oecd.org/pisa/mathematics-performance-pisa.htm. 15 OECD (2015). Qatar. Student performance (PISA 2015), see at http://gpseducation.oecd.org/Cou ntryProfile?primaryCountry=QAT&treshold=10&topic=PI. 16 The GGG Report (2017), data on Italy at http://reports.weforum.org/global-gender-gap-report2017/dataexplorer/#economy=ITA. 17 Some of the many studies on this subject, coming from diverse fields, include Ceci et al. (2009), Liben (2015), Carrell et al. (2010), Tomasetto et al. (2015), pp. 186–198; and Hottinger (2016).

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My inquiries into the present, such as the one introduced here and which in my professional life I engage in connection with my teaching, are formulated in the service of the historiographical questions I pose as a historian of science. And vice versa. One particular question I have asked of late is whether the biographies of successful—or unsuccessful—women in math and science are useful tools for attracting girls to and sustaining women in maths and science: the issue at the heart of the project that gave rise to this book. After years of research on the interactions between science and society, I have concluded that it is simply impossible to craft history without adopting the present as a research objective.18 And yet, when political commitments—feminism, in my case—intertwine with research-related concerns, how can we as historians of science overcome the risk of projecting our personal values onto the lives of people in the past? Scientists may be aware of the risks of personal and political issues contaminating their data but nonetheless decide to pursue the best science possible, and evidently the same is true of historians or sociologists. If we hope to curb this phenomenon which so many scholars have interrogated, the first step is openly admitting that it exists. I recently worked on a project about (auto)biography in the history of science. This represented an opportunity to delve a bit deeper into the complexity inherent in working on historical issues that can be traced back to personal—including gender— matters.19 Except for a few examples of experiments in biographical writing between the two world wars, with Virginia Woolf’s work representing the most notable example, autobiography began to be considered a scholarly genre in the 1960s, following second-wave feminism’s discussions of the interrelations between the personal and the social. It was then, and following autobiographical endeavours focused on sex and gender, that scholars began to ask how a biographer might capture the essence of a creative mind in its context. Roughly speaking, this explains why female scholars tend to raise questions about these issues more often than their male counterparts: this represents an example of what is referred to as group cultures.20 All of us decide to or happen to be—or not be—part of a group, but when the network in question intersects with that of female (feminist) scholars, there is always a risk that members of other groups will tend to naturalize these women’s cultural traditions and use them as the basis for marginalization. These considerations led me, in the end, to shift the initial focus I had chosen for this chapter. At first, I had planned to delve into long-term history to identify what women who were able to succeed in maths had in common over the last 300 years. Taking the Italian context as an example to be compared with other national cases, my idea was to look across time from the Enlightenment case of mathematician Maria Gaetana Agnesi (1718–1799),21 a fervent Catholic from a wealthy but not noble family, to the present-day example of Emma Castelnuovo 18 Always

inspiring on this issue are: Bloch (1949); and Carr (1961). and Franceschi (2014). Pnina G. Abir-Am, Evelyn Fox Keller, Georgina Ferry, Paula Findlen, and Londa Schiebinger contributed autobiographical essays to this volume. 20 For a bibliography on this point, see Govoni (2014). 21 Mazzotti (2007). 19 Govoni

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(1913–2014),22 a Jewish educator, maths popularizer, and the daughter and niece of famous (male) mathematicians. This has been a classic approach in the history and sociology of science since the time of Robert K. Merton’s (1910–2003) research on the Royal Society.23 It remains helpful even today for understanding how social and cultural elements interact with science through the lives of experts. And, of course, constructing prosopography helps in granting a voice to those who have been forgotten by history. However, when the people in question are women, the research results are perceived—speaking of the importance of including social psychology in our professional toolbox—differently, and there is a tangible risk that authors will offer biased readers evidence supporting the idea that women are different from men. This risk stems from the persistent tendency to treat the lives and work of male scientists as a point of reference. The concept of “difference” has always been imbued with ambiguity, and I believe this ambiguity continues to complicate women’s conditions in terms of achieving equal rights even today. The conflict between equality and difference in feminism was resolved (theoretically, at least) several decades ago; after all, it is clear that the political notion of equality depends on acknowledging that differences do exist.24 In real life, however, this issue is clearly not easy to resolve.25 Still today, in the vast academic world of hearsay, inside and outside the circle of gender experts, it proves to be complicated. In many cases, simply evoking the concept of difference raises the spectre of ambiguity, as in the case of the abovementioned former Harvard President and his silent but nonetheless very numerous followers. After all, there are not many scholars at this point who still consider women “inferior” (although they do exist and may include Nobel prize-winning scientists, as I shall note below); usually, they settle for calling women “different”. I definitively changed my mind after re-reading an essay which began with a wide international reception and has become a classic of the history of women in maths: a lecture delivered in 1901 by the Italian mathematician Gino Loria (1862–1954) and published soon after, first in French and later in Italian.26 An interesting point in Loria’s analysis is the constant that he identifies in the biographies of women mathematicians over the long term. He argues that a few women who (may) deserve a place in the history of mathematics owe this place to their fathers and brothers, or to the husbands, teachers and colleagues who helped them in their research. Loria uses this common element to demolish the scientific achievements of women in maths,

22 Furinghetti, F. Emma Castelnuovo, MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of St Andrews at http://www-history.mcs.st-andrews.ac.uk/Bio graphies/Castelnuovo_Emma.html. 23 Merton (1938). 24 Scott (1988). 25 Hirsch and Keller, eds. (1990). 26 Loria read the original text at the R. Accademia Virgiliana in Mantova on December 28, 1901. Loria (1903), Loria (1904), and Loria (1936). The English translation of some passages from the lengthy article can be found in Audin (2011), p. 230ff.

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from Hypatia (c. 350/370–415 BCE) to his contemporary Sofia Kovalevskaya (1850– 1891). By themselves, Loria assures us, women could not have achieved anything, because “provident Nature seems to call [women] to other destinies”.27 Beyond the unfounded allegations that led Loria to assert that all of women’s scientific achievements in mathematics must actually be attributed to the men around them, there is a part of Loria’s discourse that I imagine anybody would agree with. In order to succeed in science and mathematics, women just like men must grow up in a family which—at the very least—does not destroy their potential. In addition, or alternatively, they need teachers, friends, colleagues or partners who support them as equals. If they choose to conduct research professionally, women as well as men need to be admitted into that select circle that the founders of the Royal Society called the invisible college. The invisible college is a powerful image, recovered at the beginning of the 1960s by sociologist of science Derek J. de Solla Price (1922– 1983) and analysed in depth by Diana Crane.28 Without being accepted into this college or network—a network which is personal, institutional and political as well as scientific—it is impossible to make science and/or have a place in its history. Roughly speaking, the positive version of the invisible college concept lies at the origins of the Republic of Letters, a supranational space in which eighteenth-century women—and Italian women in particular29 —were able to play a recognized role. Between the nineteenth and twentieth centuries, however, while natural philosophers were evolving into professional scientists, women from the petty and middle bourgeoisie began to access higher education and the labour market. Along with this rise in women’s participation, the women-friendly Republic of Letters evolved into the same old boys’ network that is still in force today. In Loria’s time, women and men in Italy were beginning to compete for the few resources made available for research, a competition many men pursued explicitly after the First World War using any means available, including Lombroso’s already disproven but nonetheless republished science on women.30 It was this competition for resources that brought women in universities south of the Alps to face the same kind of career obstacles that women in northern Europe and overseas were facing in the same period.31 By identifying women as competitors and treating them with a combination of disdain and condescension, male academics were able to relegate women to the margins, a state of affairs that prevailed in university settings between the two world wars and during the Cold War and remains in place today despite the fact that female graduates have outnumbered their male counterparts nearly everywhere since the 1990s. And since Margaret Rossiter’s first 1982 volume on women in science in the United States, this phenomenon has been known as 27 Loria

(1936), p. 465. (1972). 29 The best-known case is that of Laura Bassi (1711–1778), but in reality many Italian women savants achieved international prominence. Findlen (1993), Berti Logan (1994), Cavazza (2009), and Messbarger (2002). 30 In Italy, Lombroso and Ferrero’s volume was reprinted in 1915, 1923 and again in 1927. 31 Govoni (2013). 28 Crane

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backlash.32 If we as scholars aim to use available data to understand the present and perhaps even guide it, to avoid the kind of backlash against women currently occurring in universities all over the world we need to change our perspectives frequently. This is why I decided to put aside the issue of successful women in maths from the Enlightenment to the present and instead focus on uncovering cases in which men from the nineteenth century chose to support women in science. I focus in particular on evolutionist scientists who gambled on women even though doing so went against the claims of the scientist they most esteemed, Charles Darwin, and his equally popular political counterpart, Thomas H. Huxley (1825–1895), who famously claimed that “five-sixths of women will stop in the doll stage of evolution” despite the increase in women’s education occurring at the time and any future efforts in that direction.33 To explore this story, however, we must linger a moment longer in the present, where we live.

10.3 A Snapshot of the Present and the Case of Women in Computer Science For many years, scholars as well as national and international agencies have been producing data on every aspect of the relationship between women and science: their education, research and careers as well as personal and familial aspects. Coming from both sides of the Atlantic, data produced by the National Science Foundation or European Commission have helped to spread an awareness that treating women and men with equal levels of education, commitment and scientific productivity in disparate ways impacts negatively on the production of innovative knowledge.34 Discriminating against women in science is an enormous waste of creativity and, consequently, a waste of money.35 To provide an example, let us consider the data for Italy, where women currently (2017) account for 59.2% of graduates and 52.4% of PhDs. In Italian universities, however, women make up 45.9% of assistant professors, 35.6% of associate professors and 21.4% of full professors.36 If we take into account the fact that this 32 Rossiter

(1984), p. 122. For conditions in other countries, see: Dyhouse (1995), Rowold (2010), Govoni (2015). For a discussion of the fierce resistance mounted against women during 1970 s second-wave feminism, see Malkiel (2016). 33 Huxley to Charles Lyll, March 17, 1860, in Huxley (1901). 34 European Commission (2017). Report on equality between women and men in the EU. Brussels. doi: 10.2838/52591; National Science Foundation (2017). Women, Minorities, and Persons with Disabilities in Science and Engineering, https://www.nsf.gov/statistics/2017/nsf17310/static/dow nloads/nsf17310-digest.pdf. 35 Gaëlle Ferrant & Alexandre Kolev (OECD Development Centre), 2016. The economic cost of gender-based discrimination in social institutions, at the address https://www.oecd.org/dev/develo pment-gender/SIGI_cost_final.pdf. 36 Istat (Istituto Nazionale di Statistica), 2016. Focus Le carriere femminili nel settore universitario, 5, at http://ustat.miur.it/media/1091/notiziario_1_2016.pdf.

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situation is relatively longstanding, given that female graduates overtook male graduates as early as 1991, there is evidently something in the university machine (the same machine that suffers the consequences) which does not function properly. As everyone is probably aware by now, this is a problem for many countries. Ever since the World Economic Forum began providing annual GGG data, there appears to be greater awareness at political and institutional levels throughout Europe that something must be done, although the measures proposed are not yet sufficient and lag decades behind the US’s 1980 Women in Science and Technology Equal Opportunity Act.37 So far, the main consensus seems to consist in the importance of collecting data, and indeed there is no shortage of such data. In general, the most positive data to emerge from the various surveys is widely known: women tend to be more studious than men in all the areas of the world where they have the right to education.38 The negative side is that women all over the world who diligently pursue their studies and dedicate themselves to research end up struggling to reach top career rungs.39 Another global trend shows that women have less interest than men in computer science, a field which for decades now has represented a crucial tool for every area of research, not to mention key markets.40 The presentation of the 2016 GGG reminds us that: Talent and technology together will determine how the Fourth Industrial Revolution can be harnessed to deliver sustainable economic growth and innumerable benefits to society. However, if half of the world’s talent is not integrated—as both beneficiary and shaper—into the transformations currently underway, we will compromise innovation and risk increased inequality.41

The stakes are indeed very high and as usual women, who as social actors remain fragile all over the world including countries with a lesser gender gap, are likely to end up marginalized. The history of the relationship between women and computer science is an interesting case. It is useful both for understanding relationships between men and women in science and for once again providing first-hand proof, if ever it was needed, that the history of women never unfolds in a linear fashion, not even in

37 Public

Law 96-516, 12/12/1980, https://www.gpo.gov/fdsys/pkg/STATUTE-94/pdf/STATUTE94-Pg3007.pdf. 38 Max Roser & Esteban Ortiz-Ospina (2017). ‘Global Rise of Education’. Published online at University of Oxford, OurWorldInData.org. Retrieved from: https://ourworldindata.org/global-riseof-education. 39 For a summary of the latest data from Europe, see European Commission (2017). Eurydice Brief. Modernization of Higher Education in Europe. Academic Staff 2017, doi: 10.2797/806308. 40 2017 Women, Minorities, and Persons with Disabilities in Science and Engineering, pp. 6– 7; EC-DGR, European Commission, Directorate-General for Research and Innovation […], She Figures. 2015, Statistics and Indicators on Gender Equality in Science, Brussels: European Communities, p. 5, available here: https://ec.europa.eu/research/swafs/pdf/pub_gender_equality/she_fig ures_2015-final.pdf. 41 Richard Samans & Saadia Zahida, Preface, World Economic Forum, The Global Gender Gap Report 2016, v, http://www3.weforum.org/docs/GGGR16/WEF_Global_Gender_Gap_Rep ort_2016.pdf.

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historical moments of widespread progress such as the 1960s and 1970s, during the so-called second wave of feminism.42 In the immediate post-war period, many women played a fundamental role in the pioneering phase of information technology. This participation stemmed from a lengthy period in which women worked as human computers, such as the famous group of women at Harvard including astronomer Henrietta Swan Leavitt (1868– 1921), those who participated in the Manhattan Project and ENIAC program, or those who collaborated with NASA during the space race.43 Having been accustomed to work in certain niches of science men disdained as being insufficiently challenging, after the Second World War women acted as entrepreneurs and researchers in computer science. In the United States, women’s enrolment in computer courses grew consistently until 1982 when—judging from National Science Foundation data— there was a sudden collapse.44 It is impossible not to recognize the relationship between women fleeing computer science and the explosion of the male nerd culture characterizing the Bill (Gates) and Steve (Jobs) generation. As historians (not only female but also male) have shown, nerd culture took shape in the public sphere as a masculine culture.45 Although much has obviously changed since then, nerd culture has remained masculine in its public representations, informing academic policies and the business strategies of Silicon Valley firms and even appearing in television series such as The Big Bang Theory. However, there are changes underway in the generation of computer scientists in their 30s, those associated with Facebook, Twitter and so on. In the Silicon Valley, not only women but also Latinos and African Americans have brought cases for discrimination, and this resistance is being enacted using new languages and strategies.46 Beyond labels—such as that of intersectional feminism, which has finally become quite widespread—I have the impression that we ought to seek new ways to support women in science and mathematics in Europe as well, focusing more on new ways of encouraging and advocating for minorities. The new migratory flows reshaping Europe are raising issues of gender and class, ethnicity and culture more generally, issues that have a tangible effect on Europe’s social and cultural physiognomy. I believe it is important to consider women’s rights in general, and the rights of women in science in particular, in the context of these new and profound changes. If not, we run the risk of women losing part of the ground they have gained in favour of new social emergencies judged to be more pressing, as occurred for example after the First and then Second World Wars. Indeed, the history of relations between women and computer science in the United States shows that women might abandon a given field of study very quickly as a result of changing cultural conditions, even

42 Noble

(1993) continues to represent a classic on this long durée issue. several different approaches, see: Light (1999), Misa (2010), Jack (2014), and Ensmenger (2015). 44 Ensmenger (2015), see Fig. 3 on page 63. 45 Ensmenger (2015). 46 Lister (2017). 43 For

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pop cultural ones, even more so than economic changes.47 On the contrary, if the new emergencies introduced by migration flows in Europe are managed by policies which are in turn also informed by an historical consciousness, these flows could instead become a great resource. In Italy, the (currently very low) percentage of foreigners among PhDs is on the rise, going from 2.2% of PhDs in 2004 to 6% in 2010. These degrees are quite evenly distributed between women (46.8%) and men48 : teachers and policy makers should make a point of working on these data using an intersectional approach. In light of this point, another interesting finding emerges from the PISA data regarding assessment in mathematics. How can we explain why girls are better than boys in mathematics in Qatar, a country ranked 130th in the 2017 GGG, as well as in Sweden, a country ranked 5th in the GGG?49 The classic interpretation associating girls’ success in mathematics with social contexts characterized by equality between women and men is clearly not enough to explain these cases. It is undoubtedly true, as we can easily verify by comparing PISA data on girls’ and boys’ performances in maths with those of the GGG, that there is a close correlation between girls’ performance in maths and equal cultural-social contexts, as already mentioned as one of the findings of social psychology research. As usual, however, matters with humans are much more complex: under certain conditions discrimination based on sex and gender, as well as age, racial or social class, may be turned into an opportunity and tool for overturning the status quo in many fields, including mathematics. In contexts characterized by the new migration flows mentioned above, if these phenomena were studied and discussed freely with young people in schools and universities on the basis of concrete data (rather than the empty rhetoric of political correctness that undoubtedly fuels instances of backlash but may also contribute to the rise of populist political movements) it might lead to interesting changes, especially for women and minorities. This is even more true in cultures such as Italy which, as already mentioned, the 2017 GGG ranks 82nd out of 144 countries worldwide. In Italy, the 2015 edition of PISA found that Italian students did not deviate significantly overall from their colleagues as represented by the OECD average. However, Italy is the OECD country with the third highest degree of gender disparity after Lebanon and Austria: boys are 20 points above girls (500 vs. 480), while the average difference internationally is 8 points in favour of boys.50 Besides and consequently, PISA has also found that girls report a statistically significant higher levels of anxiety 47 Brand

(1972), pp. 33–39. (2015). L’inserimento professionale dei dottori di ricerca, p. 2, see at https://www.istat.it/it/ files/2015/01/Dottori-di-ricerca_DEF.pdf?-?title=Inserimento+professionale+dei+dottori+di+ric erca+-+21%2Fgen%2F2015+Testo+integrale.pdf. 49 Math data, PISA (2012), analysed in Education at a Glance 2014, http://www.oecd-ilibrary.org/ docserver/download/9614031e.pdf?expires=-=1483004034&id=id&accname=guest&checksum= 201186F0848C0D63A13BCAF5187BD896; The GGG 2017 Report at http://www3.weforum.org/ docs/WEF_GGGR_2017.pdf. 50 For a detailed analysis of Italian data, see INVALSI. Presentazione Indagine internazionale 2015 OCSE- PISA Principali risultati Italia, at the http://www.invalsi.it/invalsi/doc_evidenza/2016/061 216/Sintesi_Indagine_PISA2015.pdf. 48 Istat

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in relation to mathematics than boys do.51 As many studies have shown, the same is true of the weight, both negative and positive, relationships between girls and female teachers can assume in relation to mathematics performance. In moments of intense migration such as today, it is likely—indeed, desirable—that young male and female second-generation migrants’ aspirations for personal freedom trigger positive competitive processes in the spheres of education and research. This is why I consider it important for those of us dealing with equal opportunities between women and men in science and mathematics to broaden our scope, focusing on gender issues in order to move beyond them. This already occurred in the United States during the first wave of feminism, when women and people of colour came together as allies in the struggle for rights and access to education52 in the so-called age of science.

10.4 Men Supporting Women in the Age of Science From a contemporary perspective, it seems clear that the processes definitively transforming the exclusion of women from education into what we would call a scientific fact—that of women’s mental inferiority—took place in the so-called age of science, the Victorian era or Liberal age in Italy, and in particular with the 1871 publication of another long-awaited work by Darwin, The Descent of Man, and Selection in Relation to Sex. In this book Darwin talks about women and their place in human evolution, a topic that has obviously been the object of a great many publications.53 Darwin’s position is well-known, but it is still worth recalling his specific words: The chief distinction in the intellectual powers of the two sexes is shewn by man attaining to a higher eminence, in whatever he takes up, than woman can attain-whether requiring deep thought, reason, or imagination, or merely the use of the senses and hands.54

Women are distinguished by some cognitive capabilities, Darwin reassures us, such as their “power of intuition, or rapid perception, and perhaps imitation”.55 These traits are, however, “characteristic of the lower races, and therefore of a past and lower state of civilisation”.56 The scientific fact of “women’s inferiority”, Darwin argued systematically if rather weakly, was also supported by the campaign carried out by his supporter and friend Huxley. Not to mention the vast corpus of literature of a more or less philosophical and sociological nature, quantitative data from physiology and anthropology, and an incredible volume of journalistic literature, this latter 51 OECD, The ABC of Gender Equality Education 77, see at https://www.oecd.org/pisa/keyfindings/

pisa-2012-results-gender-eng.pdf. 52 This took place in particular in certain co-educational colleges where science held a crucial place

in the curriculum. See Noble (1993), Chap. 10. 53 Darwin (1981); for the section dedicated to women, see Chap. XIX, vol. II. For an interesting discussion and rich bibliography on this issue, see Richards (2017). 54 Darwin (1981), vol. 2, p. 327. 55 Darwin (1981), vol. 2, p. 326. 56 Darwin (1981), vol. 2, p. 327.

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being the uncontested realm of hearsay as evidenced by the case of Herbert Spencer (1820–1903).57 For many people, the “fact” of women’s inferiority functioned quite tidily to explain and justify a number of social issues and resolve difficulties in the familial sphere as well as in science. For some, there was no end to the need to prove that women, who had been confined to the domestic sphere in every society, were obliged to remain there: naturalistic research on women continued steadily from Galen, Aristotle and Hippocrates up to Darwin, and many readers saw his concept of sexual selection as putting a definitive end to the discussion.58 And yet the trouble with women was that, from Hypatia to Christine de Pizan, Madame du Châtelet to Laura Bassi, Maria Gaetana Agnesi to Mary Somerville, Ada Lovelace and countless other Victorian-age women, their achievements called all of this authoritative naturalistic research into question. In the early and late modern period, the phenomenon of Femmes Savantes was explained using the flexible concept of “exception”: yes, a woman might sometimes be a good natural philosopher or excellent mathematician, but these are the kind of bizarre, outlying phenomena also found in nature. In short, the argument went, these women were monsters. This was in fact the label pinned on Bassi, the first woman worldwide—and only woman for over a century—to hold a paid professorship in a university.59 When a woman managed to engage with Newton, as did Bassi and Châtelet, or with Laplace, as did Somerville, she could be explained away fairly easily as the classic exception that confirms the rule. Periodically the debate would reignite, but for several centuries it remained limited to specific circles of elites; that is, until the question of women’s intellectual capacity become an urgent social issue in 1870s-public debate, when Darwin’s timely book was published. It was during the first wave of feminism that the natural philosophers who had begun to call themselves scientists lost control of the issue: even as scientists put themselves forward as new professional figures at the centre of public debate, they were obliged to face a strange new competitor in women. The women in question were not even aristocrats but, embarrassingly, women of the petty and middle bourgeoisie who wanted to study and do science and mathematics. They did not want to amuse themselves and men in the sitting room but rather to study and work, alongside men, in laboratories and professional scientific societies with an eye to enjoying themselves and earning glory and money. Just like men. How to make sense of this phenomenon in view of the arguments about women presented in the work of Darwin, one of the most insightful theories ever penned about life on earth? Just as today big (or not so big) data enjoys a certain mystique, in those years as well some of the most interesting speeches made in support of higher education for women were based on quantitative research. One of the best known of these is an 1888 investigation of women and the university in Europe conducted by Helene Lange (1848–1930), one of the leading figures of the women’s rights movement 57 Richards (1997), and Richards (1998). Regarding Spencer’s controversial figure and international

impact and appropriation, see Lightman (2016). (2004). 59 Findlen (1993). 58 Schiebinger

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in Germany. The American edition of Lange’s book, supplemented with an essay on female higher education in the United States, enjoyed wide success.60 Lange’s inquiry was a response to the 1878 decree in Germany that allowed women to attend only certain university courses, leaving it up to faculty to decide on a case-by-case basis if they should be admitted.61 Lange’s ambitions went far beyond the national setting, however: her quantitative data were a response to the hostility that had been expressed more or less everywhere at the idea of women studying at university. With her comparative data, Lange showed her readers that, in countries where women studied just like men, the world continued to spin on its axis just as it had before; there was no serious social upset, and the family continued to represent the fundamental core of society. It was these fears that had prayed on the minds of opponents of women’s higher education, aware as they were that, for women just as for men, education constituted a pathway to autonomy including but also going beyond economic independence. We must keep in mind that this was the atmosphere in which important figures from different generations of the international, multifaceted women’s movement for emancipation developed their positions: these included Harriet Martineau (1802– 1876) in Britain, Ellen Key (1849–1926) in Sweden, or the Italian physician and popular writer Gina Lombroso (1872–1944), daughter of Cesare Lombroso. For these and other intellectuals, not suffragists but emancipationists, it was crucial that women’s education and social deliverance be restrained by domesticity, not only to avoid fuelling widespread fears but also to ensure a certain degree of freedom for women.62 The battle against women seeking university educations was more heated in those countries with a more widespread awareness that education represents the key to emancipation for anyone: women, blacks or members of disadvantaged social classes. I believe this is why in the United Kingdom and Germany, unlike Italy, there were long-lasting and fierce battles, fought with legal tools, to keep women—as competitors—out of the university. These struggles were waged to maintain a status quo that had been established in the Middle Ages when universities were founded on a monastic model: deliberately established as worlds without women.63 While efforts to keep women from accessing higher education took on various legal forms in different national contexts, medical and scientific justifications were a constant across the board. The debate over the so-called woman question was a lively, prolific and transnational one that largely revolved around the biological factors which many claimed constituted an insurmountable obstacle to women’s intellectual activity: as the worthy Loria reminded his listeners, pursuing formal education represented a challenge to women’s very nature, a challenge that would lead to sterility, neurosis and social disorder.

60 Lange

(1890). women and higher education in Germany, see Mazón (2003). 62 For a discussion of this issue and further bibliographic resources, see Richards (2017). On the history of feminism, see Offen (2000). 63 Noble (1993). 61 Regarding

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It followed that, if women were to enter labs and university halls, it would have a detrimental effect on educational and research standards. As already mentioned, for many the answer to these and other hotly debated questions appeared in 1871 in Darwin’s The Descent of Man. Many evolutionist scientists—although not all, as I shall show—embraced Darwin’s explanation as to why women’s roles must necessarily unfold in the home. There are several studies on this topic, matched by a body of research asserting the opposite perspective: indeed, it is known that women in a number of countries, including Italy, managed to combine their enthusiasm for evolutionary science with a commitment to women’s rights.64 To my knowledge, however, much less research has been conducted on those male scientists and mathematicians who, while wholeheartedly agreeing with Darwin’s science and Thomas Huxley’s politics (as mentioned, Huxley’s impressive political genius was also expressed in keeping women out of labs and professional societies),65 preferred to make up their own minds about the woman question. After all, for anyone not suffering from an ipse dixit complex, the first few lines cited above sufficed to show that Darwin’s writings on women in The Descent of Man had fallen into the trap of hearsay. That is, it is quite clear that his arguments were political (or moral) rather than scientific. There were many men who supported women in their struggles to obtain the same rights as men. These included not only politicians, such as John S. Mill (1806–1873), who Darwin criticized in The Descent of Man specifically in the section dedicated to women66 or, south of the Alps, republican Salvatore Morelli (1824–1880), who drafted several bills that would have given women the vote, although obviously they failed to pass. The above-mentioned John Dewey wrote an article on women in education that began by posing some interesting considerations about statistical methods applied to the social sciences. Science, a periodical that had already garnered widespread attention in its few years of circulation, published an article by Dewey in 1885 questioning whether women’s studying really caused the host of moral and health problems feared by critics on both sides of the Atlantic. To address this question, the Massachusetts labour bureau had collected data provided by the Association of College Alumnae in relation to 12 institutions that had produced 1,290 female graduates as of 1882. Of these, 705 had responded to a questionnaire designed to ascertain how intellectual labour and college life affected women’s health. The collected data led Dewey to assert in the very first lines of the article: The general conclusion stated in the report is that the health of women engaged in the pursuit of a college education, does not suffer more than that of a corresponding number of other women in other occupations, or without occupations.67

This statement was followed by a presentation of his data analysis; it was only in closing that Dewey acknowledged that these data were unclear in some ways and 64 On

the United States, see Hamlin (2014). On the Italian case, see Govoni (2013). (1998). 66 Darwin (1981), vol. 2, p. 328. 67 Dewey (1885), p. 341. 65 Richards

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that the matter required further investigation. By that point, however, the reader had already been guided in interpreting the data as arguing in favour of educated women, and perhaps Dewey managed to bring some readers, likely undecided individuals and hopefully many women, around to his position. Darwin is known to have had a profound influence on Dewey,68 but evidently Dewey’s choice to support women’s education was dictated by his personal experiences and educational and social ideals; in other words, his political views. The same was true of a number of evolutionist scholars in that period who chose not to share Darwin’s and Huxley’s position on the woman question. Some of the most interesting Italian evolutionist scientists of the Liberal age, whether atheist and anticlerical such as the zoologist Michele Lessona (1823–1894) or Catholic such as the naturalist Paolo Lioy (1834–1911), wrote in support of women’s education or collaborated with female colleagues. In some cases, such as the physicist Pietro Blaserna (1836–1918), men professionally supported some of the first female science graduates. Lessona and Lioy were not only evolutionists, they were actually the first to import Darwin into Italy, translating his books and describing them in glowing terms.69 There is more to discover about this historical moment, but from my initial investigations I can already assert a few facts: these evolutionists bet on women. Physiologist Angelo Mosso (1846–1910) was one of the few Italian scientists of the time whose work enjoyed international visibility. Mosso was unquestionably materialist and positivist, and as early as 1887 he proposed that evolution be included in the school curriculum. He also penned some of the most beautiful writings of the period about women’s freedom in relation to education and the professions. After a conference tour he held in the United States between 1900 and 1901, Mosso wrote that: The greater degree of freedom young ladies enjoy [in the United States] at first seemed to wound my old European sentiments, but afterwards, entering more deeply into the intimacy of family life, I changed my mind; now I am convinced that, without freedom, we cannot master ourselves, and I believe that we must grant absolute independence to woman, to curb and restrain all the impulses that seem to us most terrible.70

Mosso also noted that, while only 25 years earlier woman’s opponents proclaimed that letting them teach would lower the standards […] of teaching; […] now everything has changed. What was forecasted did not come to pass; and the professors want women on school benches and in universities.71 68 Dewey

(1910). (1991). 70 “La maggior libertà che hanno le signorine [negli Stati Uniti] da principio urtava un po’ i miei sentimenti di vecchio europeo, ma dopo, entrando più addentro nelle intimità della vita famigliare, cambiai di parere; ora sono convinto che senza la libertà non esiste la padronanza di noi stessi, e credo che si debba concedere una indipendenza assoluta alla donna, per frenare e moderare tutti gli impulsi che a noi sembrano più temibili”, Mosso (1903), pp. 325–326. 71 “Gli oppositori della donna gridavano che mettendola ad insegnare si doveva abbassare lo standard […] dell’insegnamento; […] ora tutto è cambiato. Le previsioni non si verificarono; ed i maestri desiderano che sui banchi della scuola e nelle università vi siano delle donne”, Mosso (1903), p. 333. 69 Pancaldi

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Of course, matters were not as rosy for American women as they appeared to Mosso’s European eye.72 Nevertheless, his writings on women—only briefly mentioned here—deserve to be considered alongside those of other positivist scientists, Italian and non, and explored in future research. Just as recovering the stories of women in science previously censored by history has proved an essential step in understanding science, its history and its institutions, I believe it is equally important that we uncover the voices of male scientists who cast their lot on the side of women. Telling the stories of men who supported women might hopefully contribute to curbing backlash, helping to smooth tensions with scientists both male and female who even now, for different reasons, prefer to ignore or deny the problem of gender discrimination. These stories help us to understand more about the role cultural and political values play in the processes of constructing or appropriating scientific facts. As for the scientific fact of “woman’s inferiority”, these historical cases are useful for highlighting the role that choosing—or guessing, in Feynman’s words—plays at a certain point in dealing with experimental or mathematical data, because this choice takes on important educational value in the defence of science. This act of choosing shifts responsibility for the idea of “women’s inferiority” from scientists as a professional group and science in general to individual scientists and society in general. In my opinion, reasoning about the contemporary situation on the basis of historical cases such as these can also contribute to supporting women in mathematics who, as the data show, have certainly been discriminated against in the past and still face discrimination today. This discrimination comprises a multiplicity of factors that are social, cultural and anthropological but also economic, with different social actors competing for the same professional territory. Several mathematicians of the past, such as Klein—discussed by Renate Tobies in this volume—wagered on female mathematicians. Over a century later, what challenges do these women face?

10.5 Back to the Present: The Strange Case of Women in Maths in Italy In general, in Italy as in many other European countries, girls decide more often than boys to continue their studies after high school: in 2015, 55.6% of newly graduated girls enrolled in university as compared to 45.0% of boys.73 Regarding enrolment in science in general, including mathematics, the overall situation seems to have improved over time: in 2000–2001, women made up 14.2% of those enrolled, in 2007–2008 they amounted to 17.4%74 and more recently (2015–16 data) they

72 Rossiter

(1984). report, 2016. Focus. Gli immatricolati nell’a.a. 2015/2016 il passaggio dalla scuola all’università dei diplomati nel 2015, 6, http://statistica.miur.it/data/notiziario_2_2016.pdf. 74 Miur data regarding students enrolled in university during the 2007–2008 academic year, http:// statistica.miur.it/Data/uic2008/Gli_Studenti.pdf. 73 Statistical

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Table 10.1 Maths graduates in Italy by sex, 1999–2016. Data kindly provided by DGICASIS, Ufficio Statistica e Studi, Ministero dell’Istruzione, dell’Università e della Ricerca, provided November 17, 2017 by (Statistics Division, Italian Ministry of Education, Universities and Research)

comprised 37.6%.75 If we break down the data, however, the enrolment numbers for computer science courses are worrying. In 2003–2004, Italian graduates in Informatics Science and Technologies programs (in science departments) numbered 506 women and 1,837 men; graduates in Information Engineering programs (in engineering departments) comprised 1,214 women and 5,838 men. In 2015–2016, 25 women and 174 men graduated in Informatics Science and Technologies, while 334 men and 79 women earned degrees in Information Engineering.76 Data on maths graduates also attest to a generalized and serious disinterest among young people in relation to both mathematics and computer science. This is a problem not only for research and innovation but also for Italian schools, as they will be facing a shortage of maths teachers. At the same time, however, the data regarding maths graduates for the last 20 years or so in Italy are debatable (Table 10.1). Unlike engineering, physics and computer science, female graduates outnumber their male peers in mathematics as well as medicine, the humanities and the social sciences (including law). The data I received from the Ministry of Education, Universities and Research (Table 10.1) show that, at the end of the 1990s, women actually made up 73% of graduates in maths. There has been a significant decline in women’s interest in maths in the last few years, while men display increasing interest in this field: considering the role mathematics plays in today’s digital world, it is not 75 Focus. Gli immatricolati nell’a.a. 2015/2016 il passaggio dalla scuola all’università dei diplomati nel 2015 http://statistica.miur.it/data/notiziario_2_2016.pdf (data on p. 5 and p. 11). 76 Ministero dell’Università e della Ricerca—Ufficio di Statistica. Processing of data from the Anagrafe Nazionale degli Studenti Universitari, published at (data updated as of July 4, 2017) http://anagrafe.miur.it/laureati/cerca.php.

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difficult to imagin a future in which women in Italy will follow the trend set by women in computer science in the United States in the 1980s, increasingly distancing themselves from maths. At any rate, the current data show that women in Italian universities not only make up the most high-performing graduates in maths but, even among young people who decide to continue with their education in maths, women occupy an interesting position: 58% of graduates at the master’s level are women. This finding should be interpreted as part of a broader context in which women are more interested in men in seeking education at all levels: the 2015 data confirm that women in Italy constitute 53% of all PhDs, with 63% female graduates in the life sciences, 41% in the basic sciences (a field that includes graduates in mathematics) and, finally, 37% female graduates in engineering.77 Let us examine the situation facing women who continue with their research after obtaining a PhD to pursue an academic career in mathematics. Despite the decades-long trend of female graduates being more numerous and obtaining higher scores than male graduates, in Italian universities there are only 336 women as compared to 432 men working as assistant professors in mathematics. At the level of associate professor, the situation worsens dramatically: 389 women as compared to 720 men. At the top of the career ladder the disparity reaches surreal proportions: men in the position of full professors of mathematics number 654 while women number only 373.78 Given the medium and long-term data on graduates, waiting for a generational changeover does not appear to represent a solution, neither for women in math nor for women in the humanities. In the humanities macro sector (macro sector 11—History, philosophy, pedagogy and psychology), which popular opinion paints as more welcoming to women’s careers than other fields, the situation is as follows: among assistant professors there are 591 women and 509 men; among associate professors there are 790 women and 900 men; among humanities full professors there are currently 373 female full professors while their male colleagues number 652.79 In Italy, women make up 36% of full professors in the humanities: a shocking figure, given that currently (2016 data) women make up 77% of overall humanities graduates and 63% of humanities PhDs.80 Indeed, we must not forget that female graduates in the humanities were more numerous than male graduates long before the Second World War.81 77 Almalaurea.

Indagine Almalaurea 2015 sui dottori di ricerca. Tra performance di studio e mercato del lavoro https://www.almalaurea.it/sites/almalaurea.it/files/docs/info/cs_almalaurea_d ottoridiricerca-ottobre-2015-def.pdf. 78 Source: Miur database at http://cercauniversita.cineca.it/php5/docenti/cerca.php search in the “Macrosector 0/1A—Mathermatics” (search conducted November 14, 2017). 79 Source: Miur database at http://cercauniversita.cineca.it/php5/docenti/cerca.php search in the “Macrosector 11—History, philosophy, pedagogy and psychology” (search conducted November 14, 2017). 80 Miur, Ufficio Statistico (2016). Focus. Le carriere femminili nel settore universitario, see at http:// statistica.miur.it/Data/notiziario_1_2016.pdf. 81 In terms of enrolment, the phenomenon of women overtaking men was already evident in the 1920s: in the academic year 1921–1922, men enrolled in programmes to graduate in literature and

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Let us return to female researchers and professors in mathematics as compared with those in the humanities, conducting a search by birth year. In Italian humanities departments (macro sector 11—History, philosophy, pedagogy and psychology), there are 36 female full professors who were born in 1950 as compared to 64 male ones; female full professors who were born in 1960 number 10 while male ones number 21; there are no female full professors born in 1970, but there are two male ones. In mathematics and computer science (macro sector 01—Mathematics and informatics), 24 of the full professors born in 1950 are men and 10 women; of the full professors born in 1960, 45 are men and six are women; of those born in 1970, six are men and one is a woman.82 These disturbing figures speak for themselves. Judging from the overall data on the numbers of women working as researchers and teachers at Italian universities, I think we can conclude that the situation in Italian universities became significantly worse for women from one academic generation— the post-war generation—to the next. This phenomenon is particularly evident if we focus on the generation of female scholars born between the 1960s and 1970s, women who spent the course of their professional careers in universities in which, in the 1990s, women outnumbered men among both undergraduates and PhDs. After a significant increase in female researchers and professors between the 1970s and 1980s, there was a marked deceleration that succeeded in curbing the push from below.83 In Italy, 2016 marked the 25th anniversary of the year female graduates first outnumbered male graduates. In view of this, as well as the fact that female PhDs also outnumbered men, I would argue that the present situation represents a new backlash against female scholars in Italy. This current backlash is even more apparent than the one mentioned earlier which occurred—in Italy as in other countries—in the period between the wars, in response to women’s achievements in the nineteenth and early twentieth centuries. The shocking data on women in both maths and humanities in Italy testify to the enduring nature of an incredibly complex phenomena that can only be addressed by simultaneously employing multiple tools. Indeed, the problem undoubtedly concerns men and their old boys’ network, but it also has to do with institutions that have proven themselves unable to understand the economic costs of discrimination and hence reacted much more slowly than the rest of society to understand and denounce instances of discrimination. Yet, the problem also concerns girls and women themselves. In Italy, these phenomena are often downplayed by the very women they affect

philosophy numbered 1,547 as compared to 1,300 women, and in the following academic year there were 1,387 women enrolled in the same courses as compared to 1,257 men. Presidenza del Consiglio dei Ministri, Istituto Centrale di Statistica, 1926. Annuario Statistico Italiano, second series, v. IX, years 1922–1925, pp. 97–99. Rome: Stabilimento Poligrafico per l’Amministrazione dello Stato. 82 Source: Miur database at http://statistica.miur.it/scripts/PersonaleDiRuolo/vdocenti1.asp (search conducted November 14, 2017). 83 Istat (2001). Donne all’Università, Bologna: il Mulino; Women and Men in Scientific Careers: New Scenarios, Old Asymmetries, Special issue. In: Polis. Ricerche e studi su società e politica in Italia, 1, 2017. For a contextualizing and long durée approach, see Govoni (2015).

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in that female professionals are concerned that they might end up being marginalized to an even greater extent. Furthermore, women who have reached professional career peaks over the past few decades have a substantial share of responsibility for this state of affairs: having arrived at the top, it is quite rare for female professors in Italy to denounce this situation with the degree of frankness suited to such a grave situation, as women have done for example in the United States.84 What is needed is a new pragmatic alliance among women and men in science, as well as between women and other discriminated minorities. As I have mentioned, this kind of alliance emerged at the dawn of feminism and as early as the late eighteenth century in battles against slavery. The alliance between different social actors then continued in the Victorian and Liberal ages and came to involve a number of evolutionist scientists who made up their own minds about the “woman question” rather than embracing the position supported by official evolutionist thinking. In view of new migratory flows, I believe that such a longstanding alliance—formalized in the 1980s as intersectional feminism—is needed even more urgently than ever.

10.6 Attracting Boys and Girls to Maths and Science as Social Culture Unlike Dewey, we have access to abundant, comparative and long-term data about the cultural, social and psychological aspects of the mathematics performance gap between girls and boys. And there is more. At this point, we finally have access to the findings of integrated scientific research on genetics, human biology, evolutionary psychology, culture and society. For example, there are innovative studies which bring gender into genetics and biomedical research to investigate how the brains and behaviour of gender-diverse individuals react and respond when experiencing sex-specific health conditions, medical treatments or social practices.85 Scientists are exploring how our genes, hormones and phenotypes change when interacting with our education, economic income, stress and much more: I like to think that these scientific findings, which definitively overcome both scientism and radical constructivism, have been achieved in part thanks to decades of laborious dialogue 84 See, for example, some outstanding presentations made as part of the Symposia Leaders in Science and Engineering: The Women of MIT (March 28, 2011—Tuesday, March 29, 2011) and in particular Nancy H. Hopkins, Keynote: The Status of Women in Science and Engineering at MIT, at http:// mit150.mit.edu/symposia/leaders-science-engineering.html. In relation to this point, see Abir-Am (2014). 85 This research has been carried out at the Cognitive Neuroscience, Gender and Health Laboratory coordinated by Gillian Einstein. For a discussion of these issues, see Einstein (2007). Besides, see the recently launched (March 2017) journal Gender and the Genome edited by Marianne J. Legato. For a European approach to the issue, see European Commission (2013). Gendered Innovations: How Gender Analysis Contributes to Research, Report of the Expert Group “Innovation through Gender”, Chairperson: Londa Schiebinger, Rapporteur: Ineke Klinge, at the address https://ec.eur opa.eu/research/science-society/document_library/pdf_06/-/gendered_innovations.pdf.

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(including clashes) between scientists, social scientists and humanity scholars. And the fruits of this dialogue and contestation—mythologized in the phrase ‘science wars’86 —can be found in all areas of research. We have been asking how science and society interact at least since the era of scientists such as Ludwik Fleck (1896– 1961) and sociologists such as Merton, in ways which were sometimes confused but nonetheless effective in opening up new avenues of inquiry. Almost a century after that embarrassing question was first posed, all fields—science and science studies alike—are displaying considerable interest. At the end of the Human Genome Project, Steve Jones, the well-known geneticist referenced above, admitted that humans’ health depends more on our postal code than on our genes. In so doing, he offered us a view—couched in a joke—of the fact that, biologically speaking, we are a social and cultural species. Provocatively, Jones applies this integrated approach to the study of man, in the sense of the male of the species, studying men as woman were studied in the past but doing so by bringing genetics together with anthropology, history, pop culture and more.87 This focus is very similar to that of the historians and sociologists of science who investigate the overlapping of content and context in the making of science.88 Recalling these general interpretative questions helps us to admit that the elements at stake when interpreting data on men’s and women’s brains as well as data on the controversial relationship between girls and maths and women and science include, at least to some extent, a political choice similar to that once made by Dewey, Mosso and other evolutionist scientists who wagered on women. Lawrence Summers, the Harvard Principal mentioned at the beginning of this chapter who declared in 2005 in an educational context that woman have so much trouble reaching top positions in mathematics due to “issues of intrinsic aptitude” likewise made a choice.89 All of us make political choices in dealing with data. These kinds of choices have conditioned several hundred years of natural and sociological research on women, not to mention race and other subjects. Exploring topics such as these in a class with boys and girls, topics that touch on the present but have also been investigated in a historical perspective, requires a great deal of effort, considerable modesty and a substantial dose of courage. In my opinion, however, it has the potential to bring about an essential shift, that is, to attract more young people to science. It can also reveal the controversial reality of science as opposed to its heroic myth, a reality which might be attractive to young people: this reality is made up of cognitive, mathematical, experimental and speculative challenges (leading to extraordinary achievements as well as blind alleys), not to mention political and personal challenges. From climate change to energy production for a human race numbering 7.5 billion, humanity must face challenges that can only be overcome by using science 86 By now the literature on the science wars has become too vast to cite; see Wikipedia, The Free Encyclopedia, “Science Wars”, at https://en.wikipedia.org/wiki/Science_wars. 87 Jones (2002). 88 Regarding this approach, see the always fresh classic: Latour (1987). 89 Summers, Remarks at NBER Conference, at the address http://www.harvard.edu/president/-/spe eches/summers_2005/nber.php. See Abir-Am (2000); and Abir-Am (2014).

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and technology guided by pragmatic and challenging politics, and vice versa. To succeed, in my opinion, we will need to train kindergarten, grade school and university students, especially future scientists and mathematicians, both boys and girls, to be aware that science and its facts are thoroughly permeated by social, cultural and political issues. Since at least the invention of fire we have been immersed in contexts that, in varying proportions and ways, are simultaneously natural, technological, cultural and social. And now science has shown us that our social and cultural network influences our nature as well. I believe that the challenge lies in maintaining an open dialogue among experts—scientists and social scientists—and young people as well as the general public, discussing and debating both the potentialities and dangers of the effects of these interactions, whether we call them epigenetics or actor-network theory. In the process in which the “world […] writes on [our] body”,90 sex, gender and society, including technology, matter a great deal in that they change the context and circumstances. I think that conducting open discussions about these highly complex issues, only briefly referenced here, is the only way to enable young people to cast their bets with greater clarity, following in Dewey’s pragmatic footsteps. The public sphere typically presents a different image of science and mathematics than the culture Feynman describes in his writings. A scientific process calls for the kind of independent approach displayed by scholars such as Dewey and Mosso who, while enthusiastically supporting natural selection, read Darwin’s hearsay about women and said, “no”. The image of science and mathematics as special cultures characterized by certainty tends to drive away individuals who have historically been required to prove themselves, not only women but also social minorities. The report presenting the 2015 PISA assessment (data issued December 6, 2016) implicitly offers an ambiguous definition of science: In the context of massive information flows and rapid change, everyone now needs to be able to “think like a scientist”: to be able to weigh evidence and come to a conclusion.91

Defining science as the ability “to weigh evidence and come to a conclusion” is the result of a consolidated and longstanding tradition, something we can certainly support. Yet, an historian’s work is based on the same ability to weigh evidence in order to come to a conclusion. And I suppose the same is true for a plumber or an art historian, a literary critic or a lawyer. One of the problems that arises when we talk about the troubled relationship between young people and maths, and especially girls and maths, is the idea that science and maths represent special—different— cultures, the only cultures that involve weighing evidence to come to a conclusion: the tragically famous “scientific method”. The image of a science practiced by different people who work in labs sealed off from personal, social, economic and cultural issues,92 not to mention sex and race, tends to drive away girls, as they fall victim of 90 Einstein

(2012). PISA 2015 Results Excellence and Equity in Education, volume I (data published on December 6, 2016), quote on p. 3. 92 Shapin (2012), and Guagnini (2017). 91 OECD,

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the so-called Marie Curie complex, but it drives away boys as well. As I have tried to show, in Italy there are too few graduates in maths and computer science for a world in which everything from finance to romance travels through digital channels or social networks.93 Investigating the social dimension of science facts, instead, would reveal a fascinating side of science that might attract more young people, both men and women.

10.7 Concluding Remarks: Women, from Hearsay to Obstacles in the Labour Market There is more than one explanation for the difficulties girls and women encounter in mathematics and science, of course: this phenomenon is among the most complex we face and, as evidenced by the ample body of medium and long-term data, entails psychological, institutional and social factors.94 It has been established that, if we look at girls who equal or even exceed boys in maths according to PISA data, it is clear that this situation is found in countries where, judging from World Economic Forum data, significant strides have been made in terms of achieving equal opportunity. However, as I have mentioned, matters are complicated all over the world. I began this chapter wondering if the data allow us to say that, in contexts in which girls’ maths performance equals that of boys, women in mathematics have the same chances as their male colleagues of reaching the top of the university career ladder. The answer, at least judging from the Danish case described in another chapter of this book, appears to be “no”: girls in Denmark do not have problems in maths, but women who go on to pursue careers as mathematicians at universities do. It seems obvious, as these cases prove, that the issue is sociocultural and not one of girls’ or women’s “intrinsic aptitude”. And yet, many still believe in the latter explanation even today, and it is periodically re-presented in the public sphere by some authoritative name or other. This position was championed by the controversial scientist James D. Watson, the 1953 co-discoverer of the structure of DNA along with Francis Crick (1916– 2004), Rosalind Franklin (1920–1958) and Maurice Wilkins (1916–2004). Watson is known for his racist and sexist comments, and in a 2007 book he also had his say about the episode mentioned earlier, leading to Summers’ stepping down as Principal at Harvard University. In the book, Watson argues that Summers’ comment about women in science ended up all over the international media, forcing him to resign, because of MIT molecular biologist Nancy Hopkins; indeed, Watson tries to keep Hopkins “in her place” by defining her as “my former student”.95 In describing how

93 Mazzotti

(2017), and Ames et al. (2017). (2014). PISA 2012: Results in Focus What 15-year-olds know and what they can do with what they know, p. 23, http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf. 95 Watson (2007), p. 317. 94 OECD

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Hopkins fled from the lecture hall sickened by Summers’ remarks, Watson asserts that: It did Nancy Hopkins no particular credit as a scientist to admit that the mere hypothesis that there might be genetic differences between male and female brains—and therefore differences in the distribution of one form of cognitive potential—made her sick. Anyone sincerely interested in understanding the imbalance in the representation of men and women in science must reasonably be prepared at least to consider the extent to which nature may figure, even with clear evidence that nurture is strongly implicated. To my regret, Summers, instead of standing firm, within a week apologized publicly three times for being candid about what might well be a fact of evolution that academia will have to live with.96

This passage is definitely interesting, especially for scholars who deal with popularizer scientists and the interactions between science and society. The phrase confirms Watson’s infamous arrogance, but in my opinion what deserves to be highlighted is how roughly scientist and outdated the arguments the geneticist posits as scientific actually are: citing an alleged “fact of evolution” that supposedly explains why women end up blocked at a certain point in their careers is, of course, pure fantasy. This is exactly the kind of disastrous pit—both communicative and educational, not to mention scientific—that has pulled in hundreds of scientists, convinced as they were that scientist argumentation is the one capable of supporting and defending science. It is clear that this kind of approach usually produces the opposite effect in the medium and long term, especially among young people. This statement is not only the fruit of a centuries-old academic misogyny of which Watson represents one of the most authoritative spokesmen (it is known that he himself recounted his controversial relationship with Rosalind Franklin). It is the terrible habit of arguing through hearsay disguised as science and used to defend political beliefs. Many people—not only women and minorities—would balk at entering a profession that promised to involve a perennial war with people of this sort.97 What is worse than the use of such hearsay is the fact that it circulates in the public sphere for decades, centuries or millennia: “Aristotle said it… a Nobel Prize winner said it… so it must be true”. This is how hearsay—repeated dozens, hundreds, thousands of times—sometimes ends up becoming scientific fact, fact deserving of being included in the writings of a scientist as extraordinary as Darwin. Perhaps the science that might be capable of attracting young people is the science of those who assert that: I can live with doubt and uncertainty and not knowing. I think it is much more interesting to live not knowing than to have answers that might be wrong. If we will only allow that, as we progress, we remain unsure, we will leave opportunities for alternatives. We will not become enthusiastic for the fact, the knowledge, the absolute truth of the day, but remain always uncertain…. In order to make progress, one must leave the door to the unknown ajar.98

96 Watson

(2007), p. 318. (1977). 98 Feynman (1981–1982). 97 Keller

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These are the words of the perennially useful, as well as controversial, Feynman. While in educational contexts we share this image of science, we admit that there will never be one single element that explains girls’ difficulties in math once and for all. Instead, we have many studies, including long-term ones, that provide quantitative data on educational, psychological and economic contexts. We also have data from cutting-edge research in genetics and biomedicine. These integrated data confirm that the problem of girls and maths and women in science must be tackled in the educational setting, alongside, obviously, the familial, institutional and social spheres because it is there, not in some fanciful fact of evolution, that the problem arises and is reproduced. As mentioned above, my argument is that, in order to address the problems discussed in this book with any success, we will need an educational system that is much more open to the inclusion of all forms of cultural and social diversity. One move that might aid in supporting such an approach would be to enhance the unimpeded circulation of both idea and people throughout the world.99 This has always been one of the strengths of the culture of science, a strength that today’s restrictive migration policies threaten to undermine. At the same time, such an inclusive education will only be effective if we are able to strengthen the image of science as social culture, highlighting the contradictions inherent in the (political) positions taken by scientists, even one as great as Charles Darwin. Studying the history of scientific “facts” such as the supposed inferiority of woman can help us to understand the political nature of science more generally. As I see it, this kind of inquiry serves both historiographical and political ends by offering a viewpoint that represents an alternative to the ideological barriers that continue to spring up so often in the public sphere and educational institutions alike. These barriers represent an ideological constraint that encumbers the public image of science and keeps at bay various categories of young people, including girls. In my opinion, the only way to move beyond certain a priori positions, either in favour of or against science and technology, is to start from the bottom-up by educating the younger generations. I think scientists and mathematicians should commit to this mission more often, and indeed there are actually a number of institutions they might look at for inspiration, such as the Massachusetts Institute of Technology (MIT). MIT offers programmes on history, anthropology and science, technology and society (HA-STS) which might be particularly useful in providing the kind of education I have in mind. And it is probably no coincidence that a context such as MIT has provided fertile ground for the work of women such as Evelyn Fox Keller and Nancy Hopkins, who are simultaneously scientific professionals and social reformers: I believe it would be highly effective to recount examples such as theirs, combining the lives and work of women in science with the role models provided by capable professors.100 There are many women who have contributed to science and mathematics, and indeed—as the project that gave rise to this volume illustrates—historical accounts from recent decades have granted visibility to hundreds of them. It is certainly important to make 99 Zippel 100 Keller

(2017). (2014).

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their voices heard through articles, books and media and especially digital media, as this is our best tool for reaching boys and girls. At any rate, the subject is delicate, as there is always someone who takes advantage, just as in the past with “heroes of science” narratives, and uses these stories to feed the heroine, martyr and “first woman who…” rhetoric. Too much of the amateur feminist historiography flooding the internet in every language shares this weakness. Myths and rhetoric are harmful to science and, in my opinion, contribute to driving young people (and girls in particular) away: they produce the more or less conscious conviction that women or girls must be exceptionally gifted and prepared for heroism in order to dedicate themselves to science or mathematics. For boys, it is different not only because men have been depicted in a heroic guise from Ulysses to presentday super-heroes. We know all too well that in some countries more than others, it is common to encounter mediocre male scientists and mathematicians in university corridors who have reached their career peaks mainly thanks to their academic genealogies. Of course, this is also the case of some female mathematicians and scientists. When the individuals are women, however, the phenomenon is perceived in a different way because so much has been said about these women. The data instead show that what most frequently occurs—in Europe, but also in the United States with its Equal Opportunity Act and affirmative action—is that many brilliant women do not even attempt to reach top career positions. In 2011, the Proceedings of the National Academy of Sciences of the United States of America (PNAS) published the results of a study and Nature immediately wrote in support of the study’s conclusions.101 With the quantitative data at hand, according to PNAS, the difficulties women encounter when undertaking a science career, especially in the technological and math sectors, can no longer be explained in terms of discrimination in the process of selecting papers for publication, funding or professional assignments. The study instead posited that the factors behind these inequalities were concealed in an “invisible web” deriving from the social and familial organization that structures the daily lives of female scientists. This network of personal and institutional forces leads women, especially mothers—sometimes unintentionally, sometimes deliberately—to invest more time and energy in their families than their male colleagues do, a choice which obviously has repercussions in the professional sphere. In contexts such as the United States in which institutional choices are largely guided by merit and competitiveness and “positive action” has been in place since the 1980s, men and women begin equally and keep pace with each other during the first phases of their careers. In time, however, women’s careers begin to slow down, not because of prejudice and discrimination from inside the community (the classic glass ceiling), says PNAS, but because of this “invisible web” that female scientists apparently actively reproduce, however reluctantly or unintentionally. Communicated in a simplified manner by magazines such as Nature—and let us remember that “Nature said so, so it must be true…”—the conclusions of the PNAS research have become new scuttlebutt: women in science and mathematics do not struggle because of “issues of intrinsic aptitude” (Summers) or a “fact of evolution” 101 Ceci

& Williams (2011), and Dickey Zakaib (2011).

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(Watson), and neither because of an inflexible community or institutions that tend to reward members of the old boys’ network. Women’s difficulties are social and psychological as well, and therefore we must help them: suffice to cite the many so-called empowerment projects in STEM launched in Europe. However, is it really women—who even in countries like Italy already outnumbered men as undergraduates and PhDs more than 25 years ago—who need help? Or should we instead focus on helping those who discriminate against them? At this point, I suspect that to support equal opportunities in science and maths— as in any other research area—we have to work much more on boys and men than on girls and women. Instead, contemporary measures designed to tackle this issue focus on women, such as the European Union’s move to include “gender” in all its calls for funding, thereby draining or even completely undermining its significance and reducing everything to politically correct rhetoric, a situation that might potentially fuel backlash. If we were to broaden our vision to move beyond gender, it might prove productive in many ways, including by offering a better understanding of the data on the relationship between girls and maths, data which cannot be fully understood using only the category of gender. It is possible that the conclusions of the PNAS study hold true in some countries, but they certainly do not describe the countless national contexts addressed in this volume. For example, the Italian data confirm that there is a real glass ceiling in both mathematics and the humanities, a ceiling that is growing thicker rather than more permeable. The more competition there is over resources, the more women are marginalized. You do not need to be a mathematician, I think, to grasp that this situation is very bad for universities and research in general, and not only for women. Matters will not change until everyone, men and women alike, take up data such as those outlined in this book and really think about what they mean. Studies like those published in PNAS do, however, have the advantage of widening the scope of the discussion to address an often-neglected area: that of teaching young people about equal opportunities and equal obligations. As I have argued, we must broaden the policies of inclusion in new contexts characterized by migration and digital communication, integrating gender and race more decisively. If not, I believe there is a risk that women will lose out as they have always lost out throughout history, as the above-mentioned last 50 years of women’s history as (non)participants in computer science shows. In fact, all of this occurs within a more generally alarming context: after their first stage of professional activity in science and mathematics, young people, both men and women, struggle very hard to secure stable positions. Professional conditions are becoming increasingly precarious everywhere, as Nature and Science frequently report.102 The data about female mathematicians’ careers in Italian universities reflect these conditions, showing discrimination and competition over the scarce resources allocated to research. To avoid these kinds of situations, every university might consider setting up an efficient system of watch dogs (on 102 For

one the most recent editorial dedicated to this issue in Nature, see “Many junior scientists need to take a hard look at their job prospects.” In: Nature 550, 429 (October 26, 2017), doi: 10.1038/550429a.

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rotation) to defend not only women but also the quality of research carried out by women, men, foreigners or others. The evaluation system in Italy, which increasingly relies more on algorithms than peer review, feeds deep neuroses on the part of both individuals and institutions. In the quantitative “publish or perish” atmosphere that threatens the quality of research everywhere, conditions are so competitive that they nearly qualify as psychological violence. In these circumstances, the result of a mix of institutional, social and personal tensions, women are often the first to give up, and not only women with families. To address these issues and support girls in science and mathematics, I believe it is crucial that we bring students into contact with contemporary women who enjoy satisfying careers in both universities and the private sector while maintaining “normal” personal lives. Women who, like many men, can provide positive role models for young people, boys and girls alike. That is, the ideal of a sex-, gender-, class-, and race-free science that is behind the old dream of so many scientists and scholars, including Merton and Fox Keller. A dream that, from Galileo to Feynman and beyond, has been controversial and inspirational at the same time: the ideal according to which the only thing that matters in science are the results we achieve face to face with nature and experimental data. In the age of technoscience, it is all too clear that such an approach represents an ideal to aim for; and yet, since Galileo’s time it has been political ideas that have helped to shape the most interesting science facts humanity has developed.103

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is my great pleasure to thank Renate Tobies, Tinne Hoff Kjeldsen, and in particular Nicola Oswald and Eva Kaufholz-Soldat, for inviting me to join this project. The week we spent in January 2017 in the snowy silence of the Oberwolfach Institute was simply perfect, both for the inspiring discussions we shared and our nightly solitary work in the Institute’s dream library. A special thanks to the editors, the scientific committee and the anonymous referees for their generous and useful comments.

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Carrell, S. E., Page, M. P., & West, J. E. (2010). Sex and science: How professor gender perpetuates the gender gap. The Quarterly Journal of Economics, 125, 1101–1144. Cavazza, Marta. (2009). Women and Science in Enlightenment Italy. In P. Findlen & C. M. Sama (Eds.) Italy’s Eighteenth Century. Gender and culture in the Age of the Grand Tour (pp. 275–302). Stanford: Stanford University Press Ceci, S. J., & Williams, W. M. (2011). Understanding current causes of women’s underrepresentation in science. PNAS, 108(8), 3157–3162. https://doi.org/10.1073/pnas.1014871108. Ceci, Stephen J., Williams, Wendy M., & Barnett, S. M. (2009). Women’s underrepresentation in science: Sociocultural and biological considerations. Psychological Bulletin, 135, 218–261. Crane, Diane. (1972). Invisible colleges: Diffusion of knowledge in scientific communities. Chicago: University of Chicago Press. Darwin, C. (1981). The Descent of Man, and Selection in Relation to Sex, with an Introduction by J. T. Bonner & R. M. May. Princeton University Press. 327. (1st orig. edition London 1871). Dewey, John. (1910). The influence of Darwin on philosophy and other essays. New York: Henry Holt. Dewey, John (1885). Education and the Health of Women. In: Science 6, 141 (Oct. 16, 1885), 341–342. Dickey Zakaib, G. (2011). Science gender gap probed. Nature, 470, 153. (Published online February 7, https://doi.org/10.1038/470153a). Dyhouse, Carol. (1995). No distinction of sex? Women in British universities, l870–l939. London: Routledge. Einstein, G. (Ed). (2007). Sex and brain: A reader. Cambridge: MIT Press. Einstein, G. (Ed.). (2012). Situated neuroscience: Elucidating a biology of diversity. In R. Bluhm, H. L. Maibom, & A. J. Jacobson (Eds.), Neurofeminism: Issues at the intersection of feminist theory and cognitive science (pp. 145–174). New York: Palgrave McMillan. Ensmenger, Nathan. (2015). Beards, sandals, and other signs of rugged individualism: Culture & identity within the computing professions. Osiris, 30(1), 38–65. Feynman, R. P. (1981–1982). The pleasure of finding things out. Interview for the BBC. http:// www.bbc.co.uk/programmes/p018dvyg/clips Findlen, Paula. (1993). Science as a career in enlightenment Italy: The strategies of Laura Bassi. Isis, 84, 441–469. Fine, Cornelia. (2017). Testosterone Rex. W. W: Norton. Fyfe, A. & Lightman, B. (Eds.) (2007). Science in the marketplace: Nineteenth-century sites and experiences. Chicago: University of Chicago Press. Govoni, Paola. (2013). The power of weak competitors: Women scholars, ‘popular science’ and the building of a scientific community in Italy, 1860s–1930s. Science in Context, 3, 405–436. Govoni, Paola. (2014). Crafting scientific (Auto)biography. In P. Govoni & Z. A. Franceschi (Eds.), Writing about lives in science: (Auto)biography, gender, and genre (pp. 7–30). Göttingen: V&R Unipress. Govoni, Paola. (2015). Challenging the backlash: Women science students in Italian universities (1870s–2000s). In A. Simões, K. Gavroglu, & M. P. Diogo (Eds.), Sciences in the universities of Europe, 19th and 20th century (pp. 69–88). Boston: Springer. Guagnini, Anna. (2017). Ivory towers? The commercial activity of British professors of engineering and physics, 1880–1914. History and technology, 33(1), 70–108. Hamlin, Kimberly A. (2014). From Eve to evolution. Darwin, science, and women’s rights in Gilded Age America. Chicago: Chicago University Press. Hirsch, Marianne, & Keller, Evelyn Fox (Eds.). (1990). Conflicts in feminism. New York and London: Routledge. Hottinger, S. H. (2016). Inventing the mathematician: Gender, race, and our cultural understanding of mathematics, SUNY press. Huxley, T. (1901). Life and letters of Thomas Henry Huxley. Leonard Huxley (Ed.), vol. 1, New York: Appleton, 228.

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Jack, Jordynn. (2014). Autism and gender from refrigerator mothers to computer geeks. Urbana: University of Illinois Press. Jones, Steve. (2002). Y: The descent of man. London: Little, Brown. Keller, Evelyn Fox. (1977). The Anomaly of a Woman in Physics. In S. Ruddick & P. Daniels (Eds.), Working it out: 23 Women writers, artists, scientists, and scholars talk about their lives and work (pp. 71–79). New York: Pantheon. Keller, Evelyn Fox. (1985). Reflections on gender and science. New Haven: Yale University Press. Keller, Evelyn Fox. (2014). Pot-holes everywhere: How (not) to read my biography of Barbara McClintock. In: Govoni, P. and Franceschi, Z. A. (eds.) Writing about lives in science: (Auto)biography, gender, and genre. Göttingen: V&R Unipress, 33–42. Lange, H. (1890). Higher education of women in Europe, translated and accompanied by comparative statistics by L. R. Klemm. New York: D. Appleton and company (1st orig. edition. Berlin, 1888). Latour, B. (1987). Science in action: How to follow scientists and engineers through society. Milton Keynes: Open University Press. Liben, Lynn S. (2015). The STEM gender gap: The case for spatial interventions. International Journal of Gender, Science, and Technology, 7, 133–150. Light, Jennifer S. (1999). When computers were women. Technology and Culture, 40(3), 455–483. Lightman, Bernard (Ed.). (2016). Global Spencerism: The communication and appropriation of a British evolutionist. Boston and Leiden: Brill. Lister, J. (2017). Crossing boundaries: The future of science education. In: Scientific American, August 15. Lombroso, C., & Ferrero, W. (1895). The female offender, with an introduction by W. D. Morrison. New York: Appleton (in the United Kingdom, Fisher Unwin, London). Loria, Gino. (1903). Les femmes mathématiciennes. Revue scientifique, 4, 385–892. Loria, Gino. (1904). Encore les femmes mathématiciennes. Revue scientifique, 5, 338–340. Loria, G. (1936). Donne matematiche. In: Id., Scritti, conferenze, discorsi sulla storia delle matematiche, ed. by sezione Ligure della società Mathesis. Padova: Cedam, 1936, 447–466. MacLeod, Roy. (2000). The “Creed of Science” in Victorian England. Aldershot: Ashgate. Malkiel, N. W. (2016). “Keep the damned women out”: The struggle for coeducation. Princeton: Princeton University Press. Mazón, P. (2003). Gender and the modern research university: The admission of women to German higher education, 1865–1914. Stanford: Stanford University Press. Mazzotti, Massimo. (2007). The world of Maria Gaetana Agnesi, mathematician of God. Baltimore: Johns Hopkins University Press. Mazzotti, M. (2017). Algorithmic life. In Los Angeles Review of Books. 22 January 2017. https:// lareviewofbooks.org/article/algorithmic-life/#!. Merton, Robert K. (1938). Science, technology and society in seventeenth century England. Osiris, 4, 360–632. Messbarger, Rebecca. (2002). The century of women: Representations of women in eighteenthcentury Italian public discourse. Toronto: University of Toronto Press. Misa, Thomas J. (Ed.). (2010). Gender codes: Why women are leaving computing. Hoboken, N. J.: John Wiley & Sons. Mosso, Angelo. (1903). Mens sana in corpore sano. Milano: Treves. Nieto-Galan, Agustí. (2016). Science in the public sphere: A history of lay knowledge and expertise. London: Routledge. Noble, David F. (1993). A world without women: The Christian clerical culture of western science. New York: Knopf. Offen, Karen. (2000). European feminism, 1700–1950: A political history. Stanford: Stanford University Press. Pancaldi, Giuliano. (1991). Darwin in Italy: Science across cultural frontiers. Bloomington: Indiana University Press. Pinker, Steven. (2002). The blank slate: The modern denial of human nature. New York: Penguin.

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Pomata, G. (2013). Was there a Querelle des femmes in early modern medicine? Arenal. Revista de Historia de las Mujeres, 20(3), 213–241. Richards, Evelleen. (1997). Redrawing the boundaries: Darwinian science and Victorian women intellectuals. In B. Lightman (Ed.), Victorian science in context (pp. 119–142). Chicago: Chicago University Press. Richards, E. (1998). Huxley and woman’s place in science: The ‘woman question’ and the control of Victorian anthropology. In J. R. Moore (Ed.), History, humanity and evolution: Essays for John C. Greene (pp. 253–284). Cambridge: Cambridge University Press. Richards, Evelleen. (2017). Darwin and the making of sexual selection. Chicago: Chicago University Press. Rossiter, Margaret W. (1984). Women scientists in America: Struggles and strategies to 1940. Baltimore: Johns Hopkins University Press. Rowold, Katarina. (2010). The educated woman: Minds, bodies, and women’s higher education in Britain, Germany, and Spain, 1865–1914. London: Routledge. Schiebinger, L. (2004). Nature’s body: Gender in the making of modern science. New Brunswick: Rutgers University Press (1st edition 1993). Scott, Joan W. (1988). Deconstructing equality-versus-difference: Or, the uses of poststructuralist theory for feminism. Feminist Studies, 14, 32–50. Shapin, Steven. (2012). The Ivory Tower: the history of a figure of speech and its cultural uses. The British Journal for the History of Science, 45(1), 1–27. Sissa, G. (1994). The sexual philosophies of Plato and Aristotle. In P. Schmitt Pantel (Ed.), History of women in the west, volume I: From ancient goddesses to Christian saints. Transl. by Arthur Goldhammer, Series edited by Georges Duby and Michelle Perrot. Harvard University Press. 46–81. Summers, L. H. (2005). Remarks at NBER conference on diversifying the science & engineering workforce. Cambridge, Mass. January 14, 2005. http://www.harvard.edu/-/president/speeches/ summers_2005/nber.php. Tomasetto, C., Mirisola, A.O, Galdi, S., & Cadinu, M. (2015). Parents’ math-gender stereotypes, children’s self-perception of ability, and children’s appraisal of parents’ evaluations in 6-year-olds. Contemporary Educational Psychology, 42. Watson, James D. (2007). Avoid boring people: Lessons from a life in science. Oxford: Oxford University Press. Zippel, Kathrin. (2017). Women in global science: Advancing academic careers through international collaboration. Stanford: Stanford University Press.

Paola Govoni is Associate Professor of the History of Science at the University of Bologna. Her publications deal with science and society in the 19th and 20th centuries. Focusing on the roles of men and women in science since the diffusion and appropriation of Darwinism, Paola is now exploring how history, science, and science studies can contribute together to a better understanding of “gendered science”.

Epilogue: Mathematics—Still a Male Domain? Andrea Blunck

Andrea Blunck is Professor of Mathematics and Gender Studies at the Departement of Mathematics, University of Hamburg, since 2004. Ph.D. in mathematics at the University of Hamburg, habilitation in mathematics 1997 at TU Darmstadt. 1999–2001 research stay at the Institute of Geometry, Vienna University of Technology; 2001–2004 visiting professor at the University of Hamburg. Research areas: incidence geometry; mathematics and gender. In the contributions to this book, one can see that in history there have been more women in mathematics than one might have thought. However, the vast majority of mathematicians have been men. This is mainly due to the fact that women were excluded from higher education and from scientific institutions for a long time. The known female mathematicians from the time before 1900 are often described as “exceptional” and thus unsuitable as role models for women from subsequent generations. This changed gradually when women eventually obtained the legal admission to university. Today, many students of mathematics are female. In Germany, in 2017/18 the percentage of female students of mathematics was 42% and even 62% for the mathematics teacher students.1 Interestingly enough, already in the 1920s and early 1930s in Germany, mathematics was a field of study that female students chose disproportionately often; for example, in 1934 there were 16% females among all students in Germany but 22% among the students of mathematics.2 The proportion of women decreases when it comes to higher steps of the career ladder. In Germany, in 2018, around 25% of all university staff in mathematics are female;3 among the professors, the percentage is much lower. The same applies to many other countries. However, there are also interesting differences between countries: The proportion of women among doctoral graduates in mathematics and statistics in 2016 was 20.5% in France, 24.8% in Germany, 26.5% in the UK, 38.9%

1 Statistisches

Bundesamt. et al. (2004), pp. 7–8. 3 Statistisches Bundesamt. 2 Abele

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in Spain, and 65.2% in Portugal.4 In the last decade, in southern Europe there has been a higher percentage of female professors of mathematics than in northern Europe.5 These differences may have various reasons that one should study in detail. On the one hand, the different university systems (with many or few permanent positions), the high or low salaries and the high or low esteem of university professors may be the reasons for women (and men) to pursue a career in academia or to refrain from it. On the other hand, there might be cultural differences concerning the image of mathematics and how this fits together with fields of activity that traditionally are ascribed to women. Teaching may be such a field of activity. This may be a reason for the fact that in Germany among the mathematics teacher students there are much more women than among the “ordinary” mathematics students. Is mathematics a male discipline? How can one answer this question? There are two dimensions that constitute the “masculinity” of mathematics: the proportion of women in mathematics and the image of and stereotypes about mathematics. Both are depending on place and time, and of course, they are connected: If there are only few women (visible) in mathematics, the public considers it a male subject. Conversely, a “male” image of mathematics may prevent women from entering the field. In this context, math educator Paul Ernest (1995) speaks of a “vicious cycle”. Mathematics in fact has a male image in many countries. Picker & Berry (2010) in their study used the “draw a mathematician” test.6 They let 215 children of different ages in the USA, Great Britain, Finland, Sweden, and Romania draw a mathematician. Almost all pictures showed a white middle-aged man with baldhead or messed-up hair. If we want to overcome the masculinity of mathematics and to get more women into mathematics, we should try to change its “male” image. Female mathematicians should become much more visible. Today this is an important concern of many mathematics departments and professional societies. The London Mathematical Society (2017) published “Advice on Diversity at Conferences and Seminars” in order to help finding more female speakers (and more speakers from other minority groups). The DMV (German Mathematical Society) just decided to adopt this idea. More women as plenary speakers, more women on program committees, more women on editorial boards may help to change the culture in mathematics. Showing the diversity of mathematicians may attract a broader range of students, among them also more women. The question how to get more women into mathematics and, in particular, how to keep them in mathematics, was also addressed in two German studies initiated and conducted by Irene Pieper-Seier, professor of mathematics in Oldenburg and pioneer of gender-related research with respect to mathematics, and her collaborators. In the 4 European

Commission (2019). the statistics at http://www.demo.meligrafi.com/images/pdf/history/womeninmathineurope. pdf, gathered by European Women in Mathematics and the Women in Mathematics Committee of the European Mathematical Society. The data are from 2005 and before, but this trend seems to be unbroken also today. 6 This is a variation of the well-known “draw a scientist” test. See, e.g., Chambers (1983). 5 See

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first study, Curdes et al. (2003) made a survey of mathematics students trying to find out (among other things) why less women than men do go on to make a PhD. It turned out that female students were less confident regarding their mathematical abilities; they needed, but not always got, encouragement. Moreover, in contrast to their male peers, they perceived writing a PhD thesis a high risk. In the second study, Flaake et al. (2006) interviewed almost all female professors of mathematics in Germany about their careers and their experiences as a woman in mathematics. In contrast to many of the female students just mentioned, these successful women did obtain encouragement early in their career; they had been integrated into a research group from early on, and thus knew about how research goes on. So in order to keep female students in mathematics, it is important to encourage them. This is what university teachers can do. Ask good female students to work as a tutor or to assist in a research project. Role models as well may be important. Female university teachers themselves are role models! In addition, one should present women mathematicians, not just as role models for female students, but to let all students see that mathematics is diverse. Invite female speakers. Tell your students about female mathematicians in your lectures. These might be women from the past but also women working in mathematics today. Getting to know about women in mathematics may increase the motivation of female students, as one can see from the following quote of a female student of business mathematics from the University of Hamburg:7 Also ja, Emmy Noether zum Beispiel, die hat in Algebra mit dem noetherschen Ring … Da habe ich mich dann zu Hause auch mal hingesetzt und ein bisschen was über sie gelesen, weil ich es halt interessant fand und auch gut fand, dass da auch mal eine Frau war und nicht immer nur Männer.8

I think this quotation shows that learning about successful women in mathematics can be highly motivating for female students. More generally, getting to know many different women mathematicians from different countries, with different family backgrounds, different ways of life help in understanding how diverse mathematics is and thus may contribute a lot to creating a more positive—and more realistic—image of mathematics. One can find inspiring women mathematicians in the exhibition Women in mathematics throughout Europe9 by Sylvie Paycha and her collaborators. And, of course, one can also find many interesting women mathematicians in this book, whose lives and works are well worth being discovered by mathematicians or mathematics students!

7 From an interview taken in the course of the study “Prozesse des Doing Gender in der Mathematik”

conducted by Anina Mischau in April of 2003/04. Emmy Noether for example, in algebra with the Noetherian ring, … At home I read a little bit about her, because I found it interesting and I liked it that at least once there was a woman and not always only men.” 9 http://womeninmath.net; see also Paycha et al. (2016). 8 “Well,

Bibliography

Abele, Andrea E., Neunzert, Helmut, & Tobies, Renate. (2004). Traumjob Mathematik! Berufswege von Frauen und Männern in der Mathematik. Basel: Birkhäuser. Chambers, David W. (1983). Stereotypic images of the scientist: The draw-a-scientist test. Science Education, 67(2), 255–265. Curdes, Beate, Jahnke-Klein, Sylvia, Lohfeld, Wiebke, & Pieper-Seier, Irene. (2003). Mathematikstudentinnen und -studenten: Studienerfahrungen und Zukunftsvorstellungen. Norderstedt: BoD. Ernest, Paul. (1995). Values, gender and images of mathematics: A philosophical perspective. International Journal of Mathematical Education in Science and Technology, 26(3), 449–462. Commission, European. (2019). She-Figures 2018. Luxembourg: Publications Office of the European Union. Flaake, K., Hackmann, K., Pieper-Seier, I., & Radtke, S. (2006). Professorinnen in der Mathematik. Berufliche Werdegänge und Verortungen in der Disziplin. Bielefeld: Kleine. London Mathematical Society. (2017). LMS Advice on Diversity at Conferences and Seminars.https://www.lms.ac.uk/adviceondiversityatconferencesandseminars. Paycha, S., Georgescu, M., & Azzali, S. (2016). Women of mathematics throughout Europe. A gallery of portraits. Berlin: Verlag am Fluss. Picker, Susan, & Berry, John. (2000). Investigating pupils’ images of mathematicians. Educational Studies in Mathematics, 43, 65–94.

© Springer Nature Switzerland AG 2020 E. Kaufholz-Soldat and N. M. R. Oswald (eds.), Against All Odds, Women in the History of Philosophy and Sciences 6, https://doi.org/10.1007/978-3-030-47610-6

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