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Sophie Germain taught herself mathematics by candlelight, huddled in her bedclothes. Ada Byron Lovelace anticipated aspects of general-purpose digital computing by more than a century. Cora Ratto de Sadosky advanced messages of tolerance and equality while sharing her mathematical talents with generations of students.
This captivating book gives voice to women mathematicians from the late eighteenth century through to the present day. It documents the complex nature of the conditions women around the world have faced—and continue to face—while pursuing their careers in mathematics. The stories of the three women above and those of many more appear here, each one enlightening and inspiring. The earlier parts of the book provide historical context and perspective, beginning with excursions into the lives of fifteen women born before 1920. Included are histories of collective efforts to improve women's opportunities in research mathematics. In addition, a photo essay puts a human face on the subject as it illustrates women's contributions in professional associations.
More than eighty women from academe, government, and the private sector provide a rich mélange of insights and strategies for creating workable career paths while maintaining rewarding personal lives. The book discusses related social and cultural issues, and includes a summary of recent comparative data relating to women and men in mathematics and women from other sciences. First-person accounts provide explicit how-tos; many narratives demonstrate great determination and perseverance. Talented women vividly portray their pleasure in discovering new mathematics. The senior among them speak out candidly, interweaving their mathematics with autobiographical detail. At the beginning of a new century, women at all stages of their careers share their outlooks and experiences.
Clear, engaging, and meticulously researched, Complexities will inspire young women who are contemplating careers in mathematics and will speak to women in many fields of endeavor and walks of life.
"A definitive work, very carefully written, Complexities will inspire a wide range of women mathematicians and scientists for a long period of time. . . . By far this is the most important study of women in mathematics that even a giant amongst men mathematicians will find himself reading with sheer pleasure."—Current Engineering Practice
"[T]he variation in [the book's] content and writing styles . . . is exactly its strength—it is both an excellent reference for a professor wishing to provide a student with a few inspiring gems and a comprehensive overall picture of the life of women in mathematics. Its lessons are gleaned from the trials and tribulations of a specific group, but the advice is universal."—Lisa DeKeukelaere, MAA Online
"The collection documents the complex nature of the conditions women have faced while pursuing their careers in mathematics. It shows the pleasure women had in discovering new mathematics, and energy to do a good job!"—Silke Göbel, Zentralblatt
"As a female mathematics student, I found that reading this book increased my appreciation for the courage and determination of the women who entered mathematics before me, while also building my personal confidence in the prospect of finding a rewarding and fulfilling life in the mathematical community."—Gwen Spencer, Math Horizons
The women appearing in this chapter, inspirations for many currently active mathematicians, received their doctorates before 1960. Their stories look back through time, arranged in reverse chronological order of the women's births. It is appropriate to begin with reminiscences of Olga Taussky Todd, namesake of the conference that spurred the compilation of this book. Her charmingly informal remarks are augmented by excerpts from a memorial article about her. Later in the book, two articles give more formal descriptions of her life and work.
Julia Robinson's mathematics led to her election as the first woman mathematician in the National Academy of Sciences (NAS); she was also the first woman elected president of the American Mathematical Society (AMS). In 1980 she was the second woman to present the prestigious AMS Colloquium Lectures. The first woman to do so, fifty-three years earlier in 1927, was Anna Johnson Pell Wheeler. Among the few women who earned a doctorate in mathematics in the first half of the twentieth century, Pell Wheeler was singular in receiving such recognition for her mathematics. She is likely best known for her active mentoring of other women. While chair of the mathematics department at Bryn Mawr College, shebrought two refugees to her institution: Emmy Noether, then an established mathematician, and also, to work with Noether, the young Taussky Todd-the women whose lives bracket this chapter. Noether is indisputably among the greatest mathematicians of the twentieth century. In recognition of the importance of her work, the Association for Women in Mathematics (AWM) has sponsored since 1980 the annual Noether Lecture series at the Joint Mathematics Meetings (JMM). Robinson and Taussky Todd are two of the outstanding senior women mathematicians who have delivered these lectures.
Marjorie Lee Browne is the third African-American woman currently known to have earned a Ph.D. in mathematics; two of the women profiled in "A Dual Triumph," III.1, cite her as an inspiration. The accomplishments of Cora Ratto de Sadosky, not only an excellent mathematician but also a fiery advocate of the need for social justice, and of Mabel Barnes, a remarkable survivor with a fifty-year career in teaching, deserve also to reach a wider audience.
Familial connections abound in this chapter: both Ratto de Sadosky and Barnes had daughters who became mathematicians, while Ratto de Sadosky, Taussky Todd, and Robinson had mathematician husbands. Women with sisters will find much that resonates in Robinson's story, told by her sister Constance Reid, herself a well-known biographer of mathematicians. Noether's family included several mathematicians (her father, Max, was an analyst, while she was an algebraist), so that a mention of the surname is often followed by the question, "Which Noether?" Emmy's mathematician brother Fritz was the father of Gottfried, statistician and late husband of her biographer Emiliana Pasca Noether. Browne was also influenced at home by the keen interest in mathematics of both her father and brother, though neither was a professional mathematician. Going beyond the biological family, Browne inspired and assisted in practical ways students who became close to her and to each other.
In Her Own Words
Based on a talk at the panel "Centennial Reflections on Women in American Mathematics," organized by Judy Green and Jeanne LaDuke, AMS Centennial Meeting, Providence, 1988 as reported in the AWM Newsletter 18(6), 1988, 10-11. Taussky Todd (1906-1995) was then a professor emerita of mathematics, California Institute of Technology.
At any time a mathematician's life is not an easy one, and mine began at a particularly critical time in history. I graduated from the university in Vienna, Austria, in 1930 and had then a variety of jobs connected with mathematical research, mostly unpaid or underpaid, at various institutions, with an assortment of bosses and pupils.
The fact that I studied and worked in several countries made me able to observe a number of facts about the behavior and treatment of women. Now we live with "Women's Lib," and it has not only changed the opportunities for women, but also their behavior toward each other. Women are now more supportive of their women colleagues. This was not always the case. Even the great and kind Emmy Noether was no exception. She was convinced that men had greater strength and that women ought not to attempt to work like men, that regular appointments ought to go to men so that they could support a family. Women ought to look for marriage.
I came to the U.S. for the first time in 1934, and this is where the story really ought to begin. I came to Bryn Mawr College, where the department chairman was Anna Johnson Pell Wheeler. She had gone through hard times. Her first husband, Pell, became very ill, and she had to look for work. But she really made it. She was the first woman to give the AMS Colloquium Lecture series. However, at the age of about fifty her health seemed to break down. One day I discussed with her the problem of women in academic life and bemoaned their poor strength. She predicted that women would become more athletic in the future. Well, if she saw our young women at Caltech nowadays, dressed in short outfits on the coldest days, full of strength, she would see that her prediction has come true.
The next time I turned up in the U.S. was after World War II, which we (my husband and myself ) had spent in Great Britain, with a truly tough time. We gave up academic life and joined the British Civil Service, and I particularly had to leave my favorite subject. This second visit was by invitation of the U.S. National Bureau of Standards, where we had hard and new work (exploitation of computers), but no more war. I think women were treated as well as the men there. We stayed there until 1957, when Caltech offered both of us very attractive positions. I myself was the first woman to teach in this real men's place, a fact that is not true any longer. So there is progress!
In 1958 I was invited to give the one-hour lecture at an AMS meeting, the first woman since Emmy Noether in 1934. The lecture, titled "Integral Matrices," was published in the AMS Bulletin and is cited in the book by Curtis and Reiner. At such an occasion the chairman usually says a few kind words by way of introduction. I trained myself to say "thank you for your kind words." However, he only mentioned my name and Caltech, and I almost thanked him for his "kind words."
Soon there will not be occasions for a woman to be "a first." I cannot help wondering what would happen nowadays if a woman chairman had to fill an opening and a woman and man of fairly identical qualifications applied for it. Difficult situations will, of course, still occur in many ways. Blame it all on Adam and Eve.
Some predictions for women in university departments:
universities will employ more women; departments will employ more women; however, in the same department some men colleagues will be more jealous of the achievements of women colleagues than of men colleagues and even persecute them if possible.
And women will always be different from men.
Linear Algebra and Its Applications 280(1), Elsevier, is a special issue honoring Olga Taussky Todd. It gives an account of Olga's life and work, with full references.
Remembering Olga Taussky Todd
Based on "Remembering Olga Taussky Todd," AWM Newsletter 26(1), 1996, 7-9, and "Postscript to Olga Taussky Todd," AWM Newsletter 26(2), 1996, 28, which were adapted for reprinting in the Mathematical Intelligencer 19(1), 1997, 15-17, by permission of AWM and the author. Copyright © 1997 Springer-Verlag Inc. Davis was then a professor of mathematics, University of Toronto, where he is currently a professor emeritus.
Olga Taussky is remembered by many for her lectures. One was AWM's Noether Lecture in 1981; this had a special resonance, for she had known Emmy Noether both at Göttingen and at Bryn Mawr. Others remember Olga as author of some beautiful research papers, as teacher, as collaborator, and as someone whose zest for mathematics was deeply felt and contagious. The field she is most identified with-which might be called "linear algebra and applications," though "real and complex matrix theory" would be preferred by some-did not have autonomous existence in the 1930s, despite the textbook by C. C. MacDuffee. Her stature in that field is the very highest, as was palpable in the standing ovation after her survey talk at the second Raleigh conference in 1982.
It is amusing to hear the story of a job interview where a member of the committee asked her, with motivation we can imagine, "I see you have written several joint papers. Were you the senior or the junior author?" Another member of the committee was G. H. Hardy, who interjected, "That is a most improper question. Do not answer it!" At another interview she was asked, "I see you have collaborated with some men, but with no women. Why?" Olga replied that that was why she was applying for a position in a women's college! It is less amusing to learn that the senior woman mathematician insisted that women students not do their theses with Olga, even when male colleagues considered her the most suited to the projected research, because it would be damaging to their career to have a woman supervisor.
In 1938, while both were working at the University of London, Olga Taussky and John Todd married. Jack's scientific background was rather different-classical analysis-and his background was different-Presbyterian northern Irish. But their ensuing collaboration over fifty-seven years was close and extraordinarily fruitful. There were few joint papers, but they talked everything over, and everything either did was influenced by the other.
Olga went into applied work for the Ministry of Aircraft Production during the war. The problems included analysis of aircraft designs for their stability properties. The tools were the localization of eigenvalues, stability analysis (testing whether the real parts of all eigenvalues are < 0, or anyway not too far above 0), and numerical computation. The Todds' war work coincided with the start of the great expansion of number-crunching technique; Jack did, but Olga did not, keep always adept at the most powerful computational methods. Don't imagine Olga uttering only abstract notions and Jack only results of machine computations. Her curiosity extended to the details of numerical examples; his encompassed the theory. A good example is the Hilbert matrix, a passion they shared.
At the end of the war they moved to the U.S. National Bureau of Standards, first in Washington and then in Los Angeles. This was the period when, stimulated by the coming of peace and the computer revolution, the new matrix theory community was being established. What now look like fundamental theorems of matrix theory-Gaussian elimination, the Cauchy interlacing theorem, the Cayley-Hamilton theorem, Sylvester's inertia theorem, the Smith and Jordan forms, Perron-Frobenius theory, the variational principle for eigenvalues-were known and had not been entirely forgotten. They weren't taught much: there is an "introductory linear algebra course" everywhere now, but then nowhere; as a consequence, when I began graduate study in physics in 1946, four different courses I took began with about six weeks on "vectors." What happened in the following decade was the recognition of matrix theory as a body of doctrine and as a necessary toolkit for the scientist. Simultaneously it recognized itself as a "field" of research; recognition by others took longer.
It had been several years since the Todds had taught. Olga had grown up in a world where women-even Emmy Noether-might be barred from university professorships. It was most welcome when Caltech invited her and Jack to join the faculty in 1957. The offer was (as was usual at the time) for the husband to become professor and the wife research associate, but their offices were adjacent and the same size, and Olga was welcome to conduct seminars and supervise theses. The anomaly in their status ceased to look ideal when, in 1971, a very young assistant professor of English was glorified by the press as the first woman ever on Caltech's faculty. The first, indeed! What about Olga? I saw no sign that Olga held this against the young woman herself, but it did rub her the wrong way; she went straight to the administration and had her rank changed to professor.
Olga Taussky always wished to ease the way of younger women in mathematics and was sorry not to have more of them to work with. She said so, and she showed it in her life. Marjorie Senechal recalls giving a paper at an AMS meeting for the first time in 1962, and feeling quite alone and far from home. Olga turned the whole experience into a pleasant one by coming up to Marjorie, all smiles, introducing herself, and saying, "It's so nice to have another woman here! Welcome to mathematics!"
Being Julia Robinson's Sister
Julia Robinson, 1919-1985. Based on the after-dinner talk "Being Julia Robinson's Sister," delivered at the 1996 Robinson Celebration, MSRI, Berkeley, as reported in the AWM Newsletter 26(5), 1996, 22-28. Reid is a noted writer on mathematics and mathematicians who resides in San Francisco.
When I was asked to speak tonight, I could not refuse. The Julia Robinson Celebration of Women in Mathematics is a truly celebratory occasion, and I feel that as Julia's sister I should be here. Yet I find myself in a very difficult position. Here I am to speak about Julia, and being spoken about is the last thing Julia would want. As a mathematician, as was done earlier in the meeting-yes. But as a person-no.
So I decided my subject would be simply "Being Julia Robinson's Sister." That is the one subject connected with Julia that I can talk freely about-because it's my life, not Julia's. But in the course of the evening, talking about our sisterhood-from not so much a personal point of view as from what one might call "a point of view pertaining somewhat to mathematics"-I can tell you something about Julia, some things that will not violate her desire for personal privacy, and something also about the feelings that she expressed to me on the subject of her other sisters-all the women here and the others who are mathematicians.
Julia was born twenty-three months after I was, essentially two years-the worst possible difference in age for siblings, in my opinion-close enough for the younger to almost catch up with the elder-who is nevertheless always just a little bit ahead. I have to confess that as children we fought almost all the time. My earliest memory of Julia is of her tearing the hair off my doll while I poked the eyes out of hers. We were not close. In addition to age and sibling rivalry separating us, there was also a serious illness that was to keep Julia away from home for a year and out of school from the time she was nine until she was thirteen. It was to affect her entire life-to prevent her from having the children she very much wanted and to make it physically impossible for her take on the rigors of a full-time professional position at Berkeley.
While I could tell you something about these early years, I prefer to concentrate on that longer period in our lives that extended up to Julia's death, when we were very close. That period began in 1950, when I married and moved to San Francisco and Julia returned to Berkeley after a year at the RAND Corporation in Santa Monica. At that time she had been married since 1941 to Raphael Robinson, who had been her number theory teacher at Berkeley; she had got her Ph.D. in 1948 under Alfred Tarski with an important result in a combination of logic and number theory, and during the year that she had just spent at RAND she had solved an important problem in game theory. She had also begun to work on Hilbert's tenth problem.
Excerpted from Complexities Copyright © 2005 by Princeton University Press. Excerpted by permission.
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|1||From the twentieth century||3|
|In her own words||4|
|Remembering Olga Taussky Todd||6|
|Being Julia Robinson's sister||8|
|Euphemia Lofton Haynes||18|
|Marjorie Lee Bowne||19|
|Cora Ratto de Sadosky||24|
|Fifty years in mathematics||27|
|2||From earlier time||38|
|My grandmother, Grace Chisholm Young||39|
|Like mother, like daughter||46|
|Charlotte Angas Scott||48|
|Elizaveta Fedorovna Litvinova||54|
|Ada Byron Lovelace||60|
|Christine Ladd Frankln and Mary Fairfax Somerville||74|
|Women and mathematical ability||79|
|AWM's first twenty years : the presidents' perspectives||80|
|Activities and awards||98|
|AWM in the 1990s||105|
|1996 : women preside||115|
|Affirmative action : what is it and what should it be?||116|
|Women invited as speakers at ICMs||122|
|ICM activities on women in mathematics||124|
|International views on education||128|
|Crossing ocean and equator||131|
|Voices from six continents||134|
|III||Choices and challenges||149|
|Pathways in mathematics||151|
|1||A dual triumph||177|
|Black and female||178|
|How I decided to pursue a Ph.D. in mathematics||184|
|A double dose of discriminatin||186|
|Prejudice and isolation, or cooperation and support?||188|
|2||Inside the academy||191|
|Moment maps in stable bundles||193|
|Honors and awards||195|
|Country school to grad school||202|
|The real world of the 1930s||204|
|Increasing minority representation in mathematics||208|
|Research and teaching in Liberal Arts Colleges||212|
|Sustaining a research program||213|
|Are student ratings unfair to women?||215|
|Rules for academic success||218|
|3||Outside the academy||221|
|Government and administration||222|
|National security agency||236|
|4||Having a life||243|
|How I became a mathematician||244|
|Is geography destiny?||248|
|Making a choice||251|
|Universities and the two-body problem||253|
|The two-city existence||257|
|Spatial separation in family life||258|
|Tenure track, mommy track||260|
|Problems, including mathematical problems, from my early years||267|
|Looking back ... looking ahead||272|
|Olga Taussky and class field theory||281|
|Numbers, matrices, and communtativity||292|
|The Taussky Todd celebration||308|
|A mathematician at NIST today||314|
|What use is statistics for massive data?||328|
|Math, with an attitude||340|
|V||Into a new century||347|
|Biased random walk : a brief mathematical biography||349|
|Mathematics in "my century"||356|
|Outreach and variety||361|
|Demographic trends and challenges||364|
|Me, a mathematician?||370|
|My path toward mathematics||372|
|A cautionary tale||381|
|Role models and mentors||385|
|An energetic career||388|
|For the love of mathematics||390|
|Mathematics : mortals and morals||393|