The laundry list of obstacles that have faced female mathematicians is long: limited access to education, discouragement by teachers, tenure difficulties, and a variety of other discriminations. At this point, however, putting together a summary of grievances would do little to fix the problems of the past. A better approach is to give current and future generations the inspiration, confidence, and tools for success. This requires setting examples, discussing the facts, and presenting alternatives, telling the tales of the women who approached the mathematical wall, hiked up their skirts, and clambered over it. It requires a "complex analysis," if you will, of the lives of female mathematicians, and Complexities, a collection of accounts from and about women from the 1800s to the present, does just that.
A virtual handbook for women in mathematics, the collection contains biographies, analytical essays, research findings, and even a few expository papers. It is ideal for graduate students looking for encouragement to pursue their passion and to learn about their options for the future, from career choices to childbirth timing. Some pieces read a bit like encyclopedia entries (dates of birth, death, and major academic accomplishments), but many are highly insightful. Women will find the book inspiring for its portrayal of their barrier-breaking predecessors, but both genders can find value in its factual and anecdotal portrayal of life as a mathematician, the consequences of the choices we make, and the possibilities for the future.
Collecting so many stories in one place allows for an interesting comparison between female mathematicians, spanning generations, races, and motivations. From Sofia Kovalevskaia to Karen Uhlenbeck, the women actually share fewer similarities than one might think. There were women who persevered through sub-par early education. Others were high-society women like Emmy Noether and Grace Chisolm Young, who were expected to spend their time helping the poor. Some had access to Ivy League universities. There were women who never considered graduate school but were badgered into it by a well-meaning mentor, and there were women who looked anxiously toward graduate school, only to be discouraged by a well-meaning mentor. There were women who devoted their entire beings to mathematics, their choices dependent only on their mathematical opportunities, and there were women who balanced research with a family, some accepting separation from academia and others pushing for strenuous solutions to the two-body problem. There were women who saw teaching as the best use of their abilities, while others charged into research. A desire to do mathematics is one of the few things that unites the women in this book, and while many were attracted to the field by a strong role model, whether it be a famous father or an arithmetically proficient older sister, others fell in love with its beauty on their own.
Because there are so many stories, nearly everyone, male or female, can find a contributor with whom they identify, making the book a source for inspiration and perspective. The profiled women turned to mathematics not because of where it could take them — they faced the risk of never being paid to do mathematics for a day in their lives — but because of a deep love of the subject. The book contains biographies of women like Sophie Germain and Sofia Kovalevskaia, whose need to do mathematics was so strong that they would stay up late at night and read textbooks by candlelight, after their families had hidden their learning materials in an attempt to hamper their "unfeminine" tendencies.
One of the most inspiring stories is that of Elizaveta Fedorovna Litvinova, a Russian woman who organized sympathetic professors and fellow students to teach themselves the mathematics they would need to participate in university classes, even though the universities were closed to women. In St.Petersburg in the 1840s, the disparity between the high school education of males and females left young women starkly underprepared, yet the dedicated and intrepid Litvinova was eventually able to learn enough mathematics to study at a university in Switzerland, where she had to brave risqué stereotypes of Russian women. So strong was her desire to learn mathematics that she chose to continue her studies abroad in spite of her government's demand that she return home, forfeiting her right to ever teach mathematics in Russia at the university level.
In a more modern tale, the energy in the passage by Ingrid Daubechies is particularly enchanting. She describes how her mathematical reasoning and creativity shone through as a child when she delighted in reading Robinson Crusoe and designing doll clothes. Daubechies, who says that she turned to physics because "making a hologram of oneself seemed much more fun than complex analysis," acknowledges that she may have had an advantage by being educated at an all-girls school, where she never heard that boys were better than girls at math and science.
Aside from inspiration, the book also provides a comprehensive look at the different life paths of mathematicians. Aptly described by Nancy Kopell as a "biased random walk," each of these stories is an example of the varied and winding path a mathematician's life can take, both due to and in spite of constraints. It won't do to focus solely on gender, however: this section is valuable to any math student pondering the options ten years in the future. We meet a mathematician who devoted her life to scientific administration within the government, another who dons a hard hat and coveralls to do her math on oil rigs, and others who work in aerospace, biotechnology, communications, and cryptography. We meet women who left their mark on a variety of top research institutions, and women who found happiness at a small liberal arts college.
Career choices are accompanied by a host of personal considerations, from family planning to treatment in the workplace, and one section of the book spotlights approaches to these issues. Given the fact that 80% of women mathematicians are married to scientists, the two-body problem is an important consideration, and several essays address raising children while being a graduate student and working toward tenure. In "Are Student Ratings Unfair to Women?" Neal I. Koblitz examines the idea that because students expect female teachers to be motherly, women receive poor evaluations in intellectually demanding classes that require a "get tough" strategy.
Looking ahead is another key aspect of the book. "Voices from Six Continents" offers a way to compare the past and the future by learning about cultures at different stages from our own. In snippets of perspective from places such as Denmark, Brazil, England, and the Ivory Coast, women tell of the prevalence of female mathematicians in their regions and share their opinions on the causes and solutions. Many mention familial concerns as a source of discrimination: women are not encouraged to study science because they have responsibility in the household, and women are often unable to leave their children so that they can do graduate work abroad if no suitable institutions exist at home.
In addition to comparisons, another benefit of having so many mathematically based stories in one place is the collection of interesting and amusing quotations from famous mathematicians through time. According to Augustus DeMorgan, mathematical problems are difficult for women to solve because "the very great tension of mind which they require is beyond the strength of a woman's power of application." To this, we can now reply with the words of David Hilbert, who defended Emmy Noether's application for a faculty position at the University of Göttingen. "Gentlemen, I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse."
While as a whole the book can be slightly difficult to get through because of the variation in its content and writing styles, this 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. As Margaret Murray says, "The degree of the constraint is as much determined by the limits of our imagination as by the reality of our circumstances."
Lisa DeKeukelaere is a graduate student in applied mathematics at Brown University.