Women Mathematicians - Special Topics - A brief course - The history of mathematics

The history of mathematics: A brief course (2013)

Part VII. Special Topics

The ordering of material that we have used up to now—first, by cultures and, within each culture, roughly chronological—becomes useless after the beginning of the eighteenth century. From that point on, there is essentially only one mathematical culture, a world-wide one, with a broad consensus as to methods, although some specialties are more concentrated in one geographical area than another. As for chronology, so much mathematics has been produced every year, and mathematics has been advancing along so many broad fronts, that a chapter devoted to a single decade in the eighteenth century or a single year in the twentieth would be prodigiously long. As the time period grew shorter and the chapters grew longer, all perspective would be lost. For that reason, this final part of the history, except for Chapter 35, which discusses women mathematicians in the late nineteenth and early twentieth centuries, consists of chapters, each of which is devoted to the development of a single subject area.

In an effort to convey as much mathematics as possible in this book, we have slighted some other questions of sociological and political interest, such as the increasing democratization of mathematics that accompanied the increase in prosperity after the industrial revolution, its opening up to people from working-class backgrounds. Especially important in that democratization was the gradual involvement of women in the mathematical world. We shall devote the first chapter in this final part to that subject. For lack of space, we are forced to omit other interesting subjects, such as the influence of the Nazi and Communist regimes on mathematics in Germany and the Soviet Union and the impact of the Cold War on mathematical research in the United States.

We ended our narrative of the development of different mathematical subjects at different points. We left the story of both algebra and geometry at the point they had reached around the beginning of the seventeenth century, and we left the story of calculus and its outgrowths in the nineteenth century. Certain prominent parts of mathematics, such as probability, mathematical logic, set theory, and modern number theory have hardly been mentioned at all. While the enormous literature generated by these subjects in the modern era makes the task of summarizing them nearly impossible, we can at least make a grand sweep of each of them to provide some measure of completeness to our coverage of the world of mathematics. These last eleven chapters will fill in some of these gaps. These chapters, much more than those that have preceded, are written in the style that we called heritage in Chapter 1. That is, they aim to show how certain familiar features of modern mathematics arose rather than to describe objectively what mathematical life was like in the past.

Contents of Part VII

1. Chapter 35 (Women Mathematicians), as mentioned above, discusses women mathematicians in the late nineteenth and early twentieth centuries.

2. Chapter 36 (Probability) traces the history of probability from the Renaissance to the nineteenth century.

3. Chapter 37 (Algebra from 1600 to 1850) discusses the development of algebra up to the mid-nineteenth century.

4. Chapters 38–40 (Projective and Algebraic Geometry and Topology, Differential Geometry, Non-Euclidean Geometry) describe, as their titles indicate, the development of projective and algebraic geometry, differential geometry, and non-Euclidean geometry, respectively, to the end of the nineteenth century.

5. Chapter 41 (Complex Analysis) is devoted to complex analysis.

6. Chapters 42 and 43 (Real Numbers, Series, and Integrals; Foundations of Real Analysis) describe the parts of real analysis as branches and roots of calculus, pursuing the analogy introduced in Chapter 34.

7. Chapter 44 (Set Theory) discusses the origin and development of set theory from the 1880s through the early twentieth century.

8. Chapter 45 (Logic) discusses mathematical logic and the philosophy of mathematics from the mid-nineteenth century through the mid-twentieth century.

Chapter 35. Women Mathematicians

The history of women's participation in mathematical research has become an area of considerable interest over the past few decades. The movement of women into mathematics blossomed enormously during the late twentieth century, the result of a long and arduous struggle by brave and determined pioneers. Unfortunately for those who write the history of mathematics, almost all of this movement occurred after the time period one can reasonably cover in a single semester. In the preceding chapters, only three women—Hypatia, Maria Gaetana Agnesi, and Sof'ya Kovalevskaya—are prominent enough to merit mention. Hypatia was primarily a philosopher, and the details of her mathematical activity are not known. What Agnesi and Kovalevskaya did is well understood and appreciated. However, they enter the picture, as we have seen, near the end of the time period we are covering. To make up in some degree for these omissions, we discuss here three of the women who, in the years between 1850 and 1935, made a mark on the mathematical world in their time, overcoming prejudice and personal hardship in many cases in order to do so.

Women first began to break into the intellectual world of modern Europe in the eighteenth century, mingling with the educated society of their communities, but not allowed to attend the meetings of scientific societies. The struggle for a woman's right to be a scientist or mathematician was very much an obstacle course, similar to running the high hurdles. The first hurdle was to get the family to support a scientific education. That hurdle alone caused many to drop out at the very beginning, leaving only a few lucky or very determined women to go on to the second hurdle, gaining access to higher education. The second hurdle began to be crossed in the late nineteenth century. On the continent, a few women were admitted to university lectures without being matriculated, as exceptional cases. These cases established a precedent, and the exceptions eventually became regularized. In Britain, the University of London began admitting women in the 1870s, and in the United States there were women's colleges for undergraduate education. The opening of Bryn Mawr College in 1885 with a program of graduate studies in mathematics was an important milestone in this progress. Once a woman had gotten past the second hurdle, the third and highest of all had to be faced: getting hired and accepted by the mathematical community, and finding time to do mathematics in addition to the heavy familial responsibilities laid on women by society. The three pioneers we are about to discuss had to improvise their solutions to this problem. The fundamental societal changes needed to provide women with the same assured, routine access that men enjoyed when pursuing such a career required many decades to be recognized and partially implemented.

35.1 Sof'ya Kovalevskaya

Most of the early women mathematicians came from a leisured class of people with independent incomes. Only such people can afford both to defy convention and to spend most of their time pursuing what interests them. However, merely having an independent income was not in itself sufficient to draw a young woman into a scientific career. In most cases, some contact with intellectual circles was present as well. Hypatia was the daughter of a distinguished scholar, and Maria Gaetana Agnesi's father encouraged her by hiring tutors to instruct her in classical languages. In the case of Sof'ya Kovalevskaya, the urge to study mathematics and science fused with her participation in the radical political and social movements of her time, which looked to science as the engine of material progress and aimed to establish a society in accordance with the ideals of democracy and socialism.

She was born Sof'ya Vasil'evna Kryukovskaya in Moscow, where her father was an officer in the army, on January 15, 1850 (January 3 on the Julian calendar in effect in the Russia of her day). As a child she looked with admiration on her older sister Anna (1843–1887) and followed Anna's lead into radical political and social activism. According to her Polish tutor, she showed talent for mathematics when still in her early teens. She also showed great sympathy for the cause of Polish independence during the rebellion of 1863, which was crushed by the Tsar's troops. When she was 15, one of her neighbors, a physicist, was impressed upon discovering that she had invented the rudiments of trigonometry all by herself in order to read a book on optics; he urged her father to allow her to study more science. She was allowed to study up through the beginnings of calculus with a private tutor in St. Petersburg, but matriculation at a Russian university did not appear to be an option. Thinking that Western Europe was more enlightened in this regard, many young Russian women used a variety of methods to travel abroad. Some were able to persuade their parents to let them go. Others had to adopt more radical means, either running away or arranging a marriage of convenience, in Sof’ya's case to a young radical publisher named Vladimir Onufrevich Kovalevskii (1842–1883). They were married in 1869 and soon after left for Vienna and Heidelberg, where Kovalevskaya studied science and mathematics for a year without being allowed to enroll in the university, before moving on to Berlin with recommendations from her Heidelberg professors to meet the man who was to have the dominant influence on her professional life, Weierstrass. At Berlin also, the university would not accept her as a regular student, but Weierstrass agreed to tutor her privately.

Although the next four years were extremely stressful for a number of personal reasons, her regular meetings with Weierstrass brought her knowledge of mathematical analysis up to the level of the very best students in the world (those attending Weierstrass' lectures). By 1874, Weierstrass thought she had done more than enough work for a degree and proposed three of her papers as dissertations. Since Berlin would not award the degree, he wrote to the University of Göttingen and requested that the degree be granted in absentia. It was, and one of the three papers became a classic work in differential equations, published the following year in the most distinguished German journal, the Journal für die reine und angewandte Mathematik.

The next eight years may well be described as Kovalevskaya's wandering in the intellectual wilderness. She and Vladimir, who had obtained a doctorate in geology from the University of Jena, returned to Russia; but neither found an academic position commensurate with their talents. They began to invest in real estate, in the hope of gaining the independent wealth they would need to pursue their scientific interests. During this time, Kovalevskaya gave birth to a daughter, Sof'ya Vladimirovna Kovalevskaya (1878–1951). Soon afterward, their investments failed, and they were forced to declare bankruptcy. Vladimir's life began to unravel at this point; and Kovalevskaya, knowing that she would have to depend on herself, reopened her mathematical contacts and began to attend mathematical meetings. Recognizing the gap in her résumé since her dissertation, she asked Weierstrass for a problem to work on in order to reestablish her credentials. While she was in Paris in the spring of 1883, Vladimir (back in Russia) committed suicide, leading Sof'ya to an intense depression that nearly resulted in her own death. When she recovered, she resumed work on the problem that Weierstrass had given her. Meanwhile, Weierstrass and his student Gösta Mittag-Leffler (1846–1927) collaborated to find her a teaching position at the newly founded institution in Stockholm.1 At first she was Privatdozent, meaning that she was paid a certain amount for each student she taught. After the first year, she received a regular salary. She was to spend the last eight years of her life teaching at this institution.

In the mid-1880s, Kovalevskaya made a second mathematical discovery of profound importance. Mathematical physics is made complicated by the fact that the differential equations used to describe even simple, idealized cases of physical laws are extremely difficult to solve. The obstacle consists of two parts. First, the equations must be reduced to a set of integrals to be evaluated; second, those integrals must be computed. In many important cases, such as the equations of the three-body problem, the first is impossible using only algebraic methods. When it is possible, the second is often impossible if only elementary functions are to be used. For example, the equation of pendulum motion can be reduced to an integral, but that integral involves the square root of a cubic or quartic polynomial; it is known as an elliptic integral. Such is the case in the phenomenon studied by Kovalevskaya, the motion of a rigid body about a fixed point.

The six equations of motion for a rigid body in general cannot be reduced to integrals using only algebraic transformations. In Kovalevskaya's day only two special cases were known in which such a reduction was possible, and the integrals in both cases were elliptic integrals. Only in the case of bodies satisfying the hypotheses of both of these cases simultaneously were the integrals elementary. With Weierstrass, however, Kovalevskaya had studied not merely elliptic integrals, but integrals of completely arbitrary algebraic functions. Such integrals were known as abelian integrals after Abel, the first person to make significant progress in studying them. She was not daunted by the prospect of working with such integrals, since she knew that the secret of taming them was to use the functions known as theta functions, which had been introduced earlier by Abel and his rival in the creation of elliptic function theory, Jacobi. All she had to do was reduce the equations of motion to integrals; evaluating them was within her power. Unfortunately, it turns out that the completely general set of such equations cannot be reduced to integrals. But Kovalevskaya found a new case, much less symmetric than the cases already known (due to Euler and Lagrange), in which this reduction was possible. The algebraic changes of variable by which she made this reduction are quite impressive, spread over some 16 pages of one of the papers she eventually published on this subject. Still more impressive is the 80-page argument that follows to evaluate these integrals, which turn out to be hyperelliptic, involving the square root of a fifth-degree polynomial. This work so impressed the leading mathematicians of Paris that they decided the time had come to propose a contest for work in this area. When the contest was held in 1888, Kovalevskaya submitted a paper and was awarded the prize. She had finally reached the top of her profession and was rewarded with a tenured position in Stockholm. Sadly, she was not to be in that lofty position for long. In January 1891 she contracted pneumonia while returning to Stockholm from a winter vacation in Italy and died on February 10.

35.1.1 Resistance from Conservatives

Lest it be thought that the existence of such a powerful talent as Sof'ya Kovalevskaya removed all doubt as to women's ability to create mathematics, we must point out that minds did not simply change immediately. Confronted with the evidence that good women mathematicians had already existed, the geometer Gino Loria (1862–1954) rationalized his continuing opposition to the admission of women to universities as follows, in an article in Revue scientifique in 1904:

As for. . .Sonja Kowalevsky, the collaboration [she] obtained from first-rate mathematicians prevents us from fixing with precision her mathematical role. Nevertheless what we know allows us to put the finishing touches on a character portrait of any woman mathematician. She is always a child prodigy, who, because of her unusual aptitudes, is admired, encouraged, and strongly aided by her friends and teachers. In childhood she manages to surpass her male fellow-students; in her youth she succeeds only in equalling them; while at the end of her studies, when her comrades of the other sex are progressing vigorously and boldly, she always seeks the support of a teacher, friend, or relative; and after a few years, exhausted by efforts beyond her strength, she finally abandons a work which is bringing her no joy.

Loria could have known better. Six years before Loria wrote these words Felix Klein (1849–1925) was quoted by the journal Le progrès de l'est as saying that he found his women students to be in every respect the equals of their male colleagues.

35.2 Grace Chisholm Young

Klein began taking on women students in the 1890s. The first of these students was Grace Chisholm, who completed the doctorate under his supervision in 1895 with a dissertation on the algebraic groups of spherical trigonometry. Her life and career were documented by her daughter and written up in an article by I. Grattan-Guinness (1972), which forms the basis for the present essay.

She was born on March 15, 1868, near London, the fifth child of parents of modest but comfortable means and the third child to survive. As a child she was stricken with polio and never completely recovered the use of her right hand. She was tutored at home and passed the Cambridge Senior Examination in 1885. She attended Girton College and met the prominent algebraist Arthur Cayley (1821–1895). Her impressions of him were not flattering. To her he seemed to be a lumbering intellectual dinosaur, preventing any new life from emerging to enjoy the mathematical sunshine. In a colorful phrase, she wrote, “Cayley, unconscious himself of the effect he was having on his entourage, sat, like a figure of Buddha on its pedestal, dead-weight on the mathematical school of Cambridge” (Grattan-Guinness, 1972, p. 115).

In her first year at Cambridge, she might have been tutored by William Young (1863–1942), who later became her husband, except that she had heard that his teaching methods were ill-suited to young women. She found that Newnham College, the other women's college at Cambridge, had a much more serious professional atmosphere than Girton. She made contacts there with two other young women who had the same tutor that she had. With the support of this tutor and her fellow women students, she began to move among the serious mathematicians at Cambridge and prepare to take the Tripos Examination.2 In particular, she made friends with a student named Isabel Maddison (1869–1950) of Newnham College, who was being tutored by William Young. In 1890, after reading a few names of the top Wranglers, the moderator—W. W. Rouse Ball (1850–1925), the author of a best-selling popular history of mathematics—made a long pause to get the attention of the audience, then said in a loud, clear voice, “Above the Senior Wrangler: Fawcett, Newnham.” The young woman, Philippa Fawcett3 of Newnham College, had scored a major triumph for women's education, being the top mathematics student at Cambridge in her year. No better role model can be imagined for students such as Isabel Maddison and Grace Chisholm. They finished first and second, respectively, in the year-end examinations at Girton College the following year. That fall, due to the absence of her regular tutor, Chisholm was forced to take lessons from William Young. In 1892 she ranked between the 23rd and 24th men on the Tripos, and Isabel Maddison finished in a tie with the 27th. (The rankings went as far as 112.) As a result, each received a First in mathematics. That same year they became the first women to attempt the Final Honours examinations at Oxford, where Chisholm obtained a First and Maddison a Second. This achievement made Chisholm the first person to obtain a First in any subject from both Oxford and Cambridge.4

Unfortunately, Cambridge did not offer Grace Chisholm support for graduate study, and her application to Cornell University in the United States was rejected. As an interesting irony, then, she was forced to apply to a university with a higher standard of quality than Cornell at the time, and one that was the mathematical equal of Cambridge: the University of Göttingen. There, thanks to the liberal views of Felix Klein and Friedrich Althoff,5 she was accepted, along with two young American women, Mary Frances (“May”) Winston (1869–1959) and Margaret Eliza Maltby (1860–1944). In 1895, Chisholm broached the subject of a Ph.D. with Klein, who agreed to use his influence in the faculty to obtain authorization for the degree. It turned out to be necessary to go all the way to the Ministry of Culture in Berlin and obtain permission from Althoff personally. Fortunately, Althoff continued to be an enthusiastic supporter, and her final oral examination took place on April 26 of that year. She passed it and was granted the Ph.D. magna cum laude. She herself could hardly take in the magnitude of her achievement. More than two decades had passed since the university had awarded the Ph.D. to Sof'ya Kovalevskaya in absentia. Grace Chisholm had become the first woman to obtain that degree in mathematics through regular channels anywhere in Germany. She and Mary Winston were left alone together for a few minutes, which they used “to execute a war dance of triumph.” Her two companions Mary Winston and Margaret Maltby also received the Ph.D. degree at Göttingen, Maltby (in physics) in 1895 and Winston in 1896.6

Grace Chisholm sent a copy of her dissertation to her former tutor William Young, and in the fall of 1895 they began collaboration on a book on astronomy, a project that both soon forgot in the pleasant fog of courtship. They were married in June 1896. They planned a life in which Grace would do mathematical research and William would support the family by his teaching. Grace sent off her first research paper for publication, and William, who was then 33 years old, continued tutoring. Circumstances intervened, however, to change these plans. Cambridge began to reduce the importance of coaching, and the first of their four children was born in June 1897. Because of what they regarded as the intellectual dryness of Cambridge and the need for a more substantial career for William, they moved back to Germany in the autumn of 1897. With the help of Felix Klein, William sent off his first research paper to the London Mathematical Society. It was Klein's advice a few years later that caused both Youngs to begin working in set theory. William, once started in mathematics, proved to be a prolific writer. In the words of Grattan-Guinness (1972, p. 142), he “definitely belongs to the category of creative men who published more than was good for him.” Moreover, he received a great deal of collaboration from his wife that, apparently by mutual consent, was not publicly acknowledged. He himself admitted that much of his role was to lay out for Grace problems that he couldn't solve himself. To the modern eye he appears too eager to interpret this situation by saying that “we are rising together to new heights.” As he wrote to her:

The fact is that our papers ought to be published under our joint names, but if this were done neither of us get the benefit of it. No. Mine the laurels now and the knowledge. Yours the knowledge only. Everything under my name now, and later when the loaves and fishes are no more procurable in that way, everything or much under your name. [Grattan-Guinness, 1972, p. 141]

Perhaps the criticism Loria made of Sof'ya Kovalevskaya for obtaining help from first-rate mathematicians might more properly have been leveled against William Young. The rationalization in this quotation seems self-serving. Yet, the only person who could make that judgment, Grace Chisholm Young herself, never gave any hint that she felt exploited, and William was certainly a very talented mathematician in his own right, whose talent manifested itself rather late in life. And one cannot deny that, given the state of society at the time, the situation William Young is describing was very likely the best option for both of them.

In 1903 Cambridge University Press agreed to publish a work on set theory under both their names. That book appeared in 1906; a book on geometry appeared under both names in 1905. Grace was busy bearing children all this time (their last three children were born in 1903, 1904, and 1908) and studying medicine. She began to write mathematical papers under her own name in 1913, after William took a position in Calcutta, which of course required him to be away for long periods of time. These papers, especially her paper on the differentiability properties of completely arbitrary functions, added to her reputation and were cited in textbooks on measure theory for many decades.

Sadly, the fanaticism of World War I caused some strains between the Youngs and their old mentor Felix Klein. As a patriotic German, Klein had signed a declaration of support for the German position at the beginning of the war. Four years later, as the defeat of Germany drew near, Grace wrote to him, asking him to withdraw his signature. Of course, propaganda had been intense in all the belligerent countries during the war, and even the mildest-mannered people tended to believe what they were told and to hate the enemy. Klein replied diplomatically, saying that, “Everyone will hold to his own country in light and dark days, but we must free ourselves from passion if international cooperation such as we all desire is to assert itself again for the good of the whole” (Grattan-Guinness, 1972, p. 160). If only other scholars, in other countries, had been as magnanimous as Klein, German scholars might have had less justification for complaining of exclusion in the bitter postwar period. At least there was no irreparable breach between the Youngs and Klein. When Klein died in 1925, his widow thanked the Youngs for sending their sympathy, saying, “From all over the world I received such lovely letters full of affection and gratitude, so many tell me that he showed them the way on which their life was built. I had him for fifty years, this wonderful man; how privileged I am above most women. . .” (Grattan-Guinness, 1972, p. 171).

All four of their children eventually obtained doctoral degrees, and the pair had good grounds for being well-satisfied with their married life. When World War II began in September 1939, they were on holiday in Switzerland, and there was fear that Switzerland would be invaded. Grace immediately returned to England, but William stayed behind. The fall of France in 1940 enforced a long separation on them. The health of William, who was by then in his late 70s, declined rapidly, and he died in a nursing home in June 1942. Grace survived for nearly two more years, dying in March 1944. Grattan-Guinness (1972, p. 181) has eloquently characterized this remarkable woman:

She knew more than half a dozen languages herself, and in addition she was a good mathematician, a virtually qualified medical doctor, and in her spare time, pianist, poet, painter, author, Platonic and Elizabethan scholar—and a devoted mother to all her children. And in the blend of her rôles as scholar and as mother lay the fulfillment of her complicated personality.

35.3 Emmy Noether

Sof'ya Kovalevskaya and Grace Chisholm Young had had to improvise their careers, taking advantage of the opportunities that arose from time to time. One might have thought that Amalie Emmy Noether was better situated in regard to both the number of opportunities arising and the ability to take advantage of them. After all, she came a full generation later than Kovalevskaya, the University of Göttingen had been awarding degrees to women for five years when she enrolled, and she was the eldest child of the distinguished mathematician Max Noether. According to Dick (1981), on whose biography of her the following account is based, she was born on March 23, 1882 in Erlangen, Germany, where her father was a professor of mathematics. She was to acquire three younger brothers in 1883, 1884, and 1889. Her childhood was quite a normal one for a girl of her day, and at the age of 18 she took the examinations for teachers of French and English, scoring very well. This achievement made her eligible to teach modern languages at women's educational institutions. She decided instead to attend the University of Erlangen. There, she was one of only two women in the student body of 986, and she was only an auditor, preparing simultaneously to take the graduation examinations in Nürnberg. After passing these examinations, she went to the University of Göttingen for one year, again not as a matriculated student. If it seems strange that Grace Chisholm was allowed to matriculate at Göttingen and Emmy Noether was not, the explanation seems to be precisely that Emmy Noether was a German.

In 1904 she was allowed to matriculate at Erlangen, where she wrote a dissertation under the direction of Paul Gordan (1837–1912). Gordan was a constructivist and disliked abstract proofs. According to Kowalewski (1950, p. 25) he is said to have remarked of one proof of the Hilbert basis theorem, “That is no longer mathematics; that is theology.” In her dissertation, Emmy Noether followed Gordan's constructivist methods; but she was later to become famous for work done from a much more abstract point of view. She received the doctorate summa cum laude in 1907. Thus, she overcame the first two obstacles to a career in mathematics with only a small amount of difficulty, not much more than faced by her brother Fritz (1884–1941), who was also a mathematician. That third obstacle, however, finding work at a university, was formidable. Emmy Noether spent many years working without salary at the Mathematical Institute in Erlangen. This position enabled her to look after her father, who had been frail since he contracted polio at the age of 14. It also allowed her to continue working on mathematical ideas. For nearly two decades she corresponded with Gordan's successor in Erlangen, Ernst Fischer (1875–1954), who is best remembered for having discovered the Riesz–Fischer theorem independently of F. Riesz (1880–1956). By staying in touch with the mathematical community and giving lectures on her discoveries, she kept her name before certain influential mathematicians, namely David Hilbert (1862–1943) and Felix Klein,7 and in 1915 she was invited to work as a Privatdozent in Göttingen, the same rank originally offered to Kovalevskaya at Stockholm in 1883. Over the next four years Klein and Hilbert used all their influence to get her a regular appointment at Göttingen; during part of that time she lectured for Hilbert in mathematical physics. That work led her to a theorem in general relativity that was highly praised by both Hilbert and Einstein. Despite this brilliant work, however, she was not allowed to pass the Habilitation needed to acquire a professorship. Only after the German defeat in World War I, which was followed by the abdication of the Kaiser and a general spirit of reform in Germany, was she allowed to “habilitate.” Between Sof'ya Kovalevskaya and Emmy Noether there was a curious kind of symmetry: Kovalevskaya was probably aided in her efforts to become a student in Berlin because many of the students were away at war at the time. Noether was aided in her efforts to become a professor by an influx of returning war veterans. She began lecturing in courses offered under the name Dr. Emmy Noether (without any mention of Hilbert) in the fall of 1919. Through the efforts of Richard Courant (1888–1972) she was eventually granted a small salary for her lectures.

In the 1920s she moved into the area of abstract algebra, and it is in this area that mathematicians know her work best. Noetherian rings became a basic area of study after her work, which became part of a standard textbook by her student Bartel Leendert van der Waerden (1903–1996). He later described her influence on this work (1975, p. 32):

When I came to Göttingen in 1924, a new world opened up before me. I learned from Emmy Noether that the tools by which my questions could be handled had already been developed by Dedekind [Richard Dedekind (1831–1916) and Weber [Heinrich Weber, 1842–1913)], by Hilbert, Lasker [Emanuel Lasker (1868–1941)] and Macaulay [Francis Sowerby Macaulay (1862–1937)], by Steinitz [Ernst Steinitz (1871–1928)] and by Emmy Noether herself.

Of all the women we have discussed Emmy Noether was unquestionably the most talented mathematically. Her work, both in quantity and quality, places her in the elite of twentieth-century mathematicians, and it was recognized as such during her lifetime. She became an editor of Mathematische Annalen, one of the two or three most prestigious journals in the world. She was invited to speak at the International Congress of Mathematicians in Bologna in 1928 and in Zürich in 1932, when she shared with Emil Artin (1898–1962) a prestigious prize for the advancement of mathematical knowledge. This recognition was clear and simple proof of her ability. Hilbert's successor in Göttingen, Hermann Weyl (1885–1955), made this point when wrote her obituary:

When I was called permanently to Göttingen in 1930, I earnestly tried to obtain from the Ministerium a better position for her, because I was ashamed to occupy such a preferred position beside her, whom I knew to be my superior as a mathematician in many respects. I did not succeed, nor did an attempt to push through her election as a member of the Göttinger Gesellschaft der Wissenschaften. Tradition, prejudice, external considerations, weighted the balance against her scientific merits and scientific greatness, by that time denied by no one. In my Göttingen years, 1930–1933, she was without doubt the strongest center of mathematical activity there. [Dick, 1981, p. 169]

To have been recognized by one of the twentieth century's greatest mathematicians as “the strongest center of mathematical activity” at a university that was second to none in the quality of its research is high praise indeed. It is unfortunate that this recognition was beyond the capability of the Ministerium. The year 1932 was to be the summit of Noether's career. The following year, the advanced culture of Germany, which had enabled her to develop her talents to their fullest, turned its back on its own brilliant past and plunged into the nightmare of Nazism. Despite extraordinary efforts by the greatest scientists on her behalf, Noether was removed from the position that she had achieved through such a long struggle and the assistance of great mathematicians. Along with hundreds of other Jewish mathematicians, including her friends Richard Courant and Hermann Weyl (who was not Jewish, but whose wife was), she had to find a new life in a different land. She accepted a visiting professorship at Bryn Mawr, which allowed her also to lecture at the Institute for Advanced Study in Princeton.8 Despite the gathering clouds in Germany, she returned there in 1934 to visit her brother Fritz, who was about to seek asylum in the Soviet Union. (Ironically, he was arrested in 1937, during one of the many purges conducted by Stalin, and executed as a German spy on the day the Germans occupied Smolensk in 1941.) She returned to Bryn Mawr in the spring of 1934.

Weyl, who went to Princeton in 1933, expressed his indignation at the Nazi policy of excluding “non-Aryans” from teaching. In a letter sent to Heinrich Brandt (1886–1954) in Halle he gave his opinion of Nazi-sympathizing intellectuals like Oswald Spengler and Ludwig Bieberbach9:

What impresses me most about Emmy Noether is that her research has become more and more concrete and profound. Why should this Jewess not work in the area that has led to such great achievements in the hands of the “Aryan” Dedekind? I am happy to leave it to Herrn Spengler and Bieberbach to assign mathematical modes of thought according to cultures and races. [Jentsch, 1986, p. 9]

At Bryn Mawr she was a great success and an inspiration to the women studying there. She taught several graduate and postdoctoral students who went on to successful careers, including her former assistant from Göttingen, Olga Taussky (1906–1995), who was forced to leave a tutoring position in Vienna in 1933. Her time, however, was to be very brief. She developed a tumor in 1935, but she does not seem to have been worried about its possible consequences. It was therefore a great shock to her colleagues in April 1935 when, after an operation at Bryn Mawr Hospital that seemed to offer a good prognosis, she developed complications and died within a few hours.

Questions

Historical Questions

35.1 For what mathematical achievements is Sof'ya Kovalevskaya best remembered?

35.2 What events turned Grace Chisholm Young toward mathematics, and how was she able to fulfill her ambition to become a mathematician?

35.3 What special contribution did Bryn Mawr College make toward the mathematical education of women?

35.4 In what areas of mathematics was Emmy Noether a world leader in research?

Questions for Reflection

35.5 What were the advantages and disadvantages of marriage for a woman seeking an academic career before the twentieth century? How much of this depended on the particular choice of a husband at each stage of the career? The cases of Sof'ya Kovalevskaya, Grace Chisholm Young, and Emmy Noether will be illuminating, but it will be useful to seek more detailed sources than the narratives above.

35.6 How important is (or was) encouragement from family and friends in the decision to study science? How important is it to have a mentor, an established professional in the same field, to help orient early career decisions? How important is it for a young woman to have an older woman as a role model? Try to answer these questions along a scale from “not at all important” through “somewhat important” and “very important” to “essential.” Use the examples of the women whose careers are sketched above to support your rankings.

35.7 How strong are the claims that Loria adduces in his argument against admitting women to universities? Were all the women discussed here encouraged by their families when they were young? Is it really true that it is impossible to “fix with precision” the original contributions of Sof'ya Kovalevskaya? Would collaboration with other mathematicians make it impossible to “fix with precision” the work of any male mathematicians? Consider also the case of the three women discussed above. Is it true that they were exhausted after finishing their education?

Next, consider that universities select the top students in high school classes for admission, so that a student who excelled the other students in high school might be able at best to equal the other students at a university. Further selections for graduate school, then for hiring at universities of various levels of prestige, then for academic honors, provide layer after layer of filtering. Except for an extremely tiny elite, those who were at the top at one stage find themselves in the middle at the next and eventually reach (what is ideally) a level commensurate with their talent. What conclusions could be justified in regard to any gender link in this universal process, based on a sample of fewer than five women? And how can Loria be sure he knows their proper level when all the women up to the time of writing were systematically locked out of the best opportunities for professional advancement? Look at the twentieth century and see what becomes of Loria's argument that women never reach the top.

Finally, examine Loria's argument in the light of the cold facts of society: A woman who wished to have a career in mathematics would naturally be well advised to find a mentor with a well-established reputation, as Sof'ya Kovalevskaya did. A woman who did not do that would have no chance of being cited by Loria as an example, since she would never have been heard of. Is this argument not a classical example of Catch-22?

35.8 The primary undergraduate competition for mathematics majors in the United States is the Putnam Examination, administered the first weekend in December each year by the Mathematical Association of America. In addition to its rankings for the top teams and the top individuals, this examination also provides, for women who choose to enter, a prize for the highest-ranking woman. (The people grading the examinations do not know the identities of the entrants, and a woman can enter this competition without giving her name or the name of her university to the graders.) Is this policy an important affirmative-action step to encourage talented young women in mathematical careers, or does it “send the wrong message,” implying that women cannot compete with men on an equal basis in mathematics? If you consider it a good thing, how long should it be continued? Forever? If not, what criterion should be used to determine when to discontinue the separate category?

Notes

1. It is now the University of Stockholm.

2. The Tripos Examination was a venerable tradition at Cambridge, dating back to Medieval times. A high-quality performance merited a First degree, lower quality a Second. Those who gained a First were called Wranglers. With modifications, the system continues at the present time.

3. Philippa Garrett Fawcett (1868–1948) was the daughter of a professor of political economy at Cambridge. Her mother was a prominent advocate of women's rights, and her sister was the first woman to obtain a medical degree at the University of St Andrews in Scotland. Philippa used her Cambridge education to go to the Transvaal in 1902 and help set up an educational system there. From 1905 to 1934 she was Director of Education of the London County Council.

4. Isabel Maddison was awarded the Bachelor of Science degree at the University of London in 1892. She received the Ph.D. at Bryn Mawr in 1896 under the supervision of Charlotte Angas Scott (1858–1931, another alumna of Girton College and a student of Cayley). She taught at Bryn Mawr until her retirement in 1926.

5. Althoff (1839–1908) was the Prussian Under-Secretary of Education and Cultural Affairs during the time of Kaiser Wilhelm II.

6. Margaret Maltby taught at Barnard College (now part of Columbia University in New York) for 31 years and was chair of physics for 20 of those years. Mary Winston had studied at Bryn Mawr with Charlotte Angas Scott. She had met Felix Klein at the World's Columbian Exposition in Chicago in 1893 and had moved to Göttingen at his invitation. After returning to the United States she taught at Kansas State Agricultural College, married Henry Newson, a professor of mathematics at the University of Kansas, bore three children, and went back to teaching after Henry's early death. From 1921 to 1942 she taught at Eureka College in Illinois.

7. Klein wrote to Hilbert, “You know that Fräulein Noether is continually advising me in my projects and that it is really through her that I have become competent in the subject.” (Dick, 1981, p. 31)

8. There was no chance of her lecturing at Princeton University itself, which was all-male at the time.

9. Oswald Spengler (1880–1936) was a German philosopher of history, best known for having written Der Untergang des Abendlandes (The Decline of the West). His philosophy of history, which Weyl alludes to in this quote, suited the Nazis. Although at first sympathetic to them, he was repelled by their crudity and their antisemitism. By the time Weyl wrote this letter, the Nazis had banned all mention of Spengler on German radio. Ludwig Bieberbach (1886–1982) was a mathematician of some talent who worked in Berlin during the Nazi era and edited the Party-approved journal Deutsche Mathematik. At the time when Weyl wrote this letter, Bieberbach was wearing a Nazi uniform to the university and enthusiastically endorsing the persecution of non-Aryans.