+ 7 +
Gender and Age in Mathematics
In Chapter 5 we quoted Hardy’s dictum, “Mathematics is a young man’s game,” which has become a catch phrase. In this chapter we consider both aspects of that catch phrase: “young” and “man.” What happens to a mathematician as he or she gets older? Do women have the same career opportunities as men? Do they follow the same rhythms and patterns in their work as their male colleagues? In the second half of the chapter, we consider age and aging of mathematicians. In the first half, we survey the past and present status of women mathematicians.
Women in Mathematics
By the expression “a young man’s game” Hardy did not mean to exclude women. The famous British analyst Mary Cartwright was his student. In 1941, when Hardy wrote his Apology, normal usage was to write “man” either in the sense of “masculine” or in the sense of “human.” Nowadays, of course, we say “young person” if we mean to include both sexes. Hardy was not only a pacifist and an atheist but can even be counted as an early feminist, as attested by his active support for the English-American mathematician-turned-biophysicist Dorothy Wrinch.1
Here we will give a brief survey of women in mathematics, in the past and today. We will see that until recent decades it was indeed overwhelmingly a “man’s game,” because women were excluded. Not primarily by mathematicians—but first, by parents’ demands to conform to social expectations, and then by exclusionary policies of university administrations. Full equality or equity is still a goal to be struggled for.
We begin with short accounts of the lives of three great female mathematicians who overcame tremendous obstacles in order to become mathematicians: Sophie Germain, Sofia Kovalevskaya, and Emmy Noether.
Marie-Sophie Germain (1776–1831)
In chapter 1 we told how at the age of 13 in revolutionary Paris Germain became fascinated with Archimedes and with mathematics, despite the fierce opposition of her parents. After they realized they could not defeat Sophie, her father funded her research and supported her efforts to break into the community of mathematicians. For many years this was the only encouragement she received. Like many other women who practiced mathematics in later times, she devoted herself solely to her profession and never married.
In 1794 the École Polytechnique opened in Paris, reserved for men. Sophie assumed the identity of a former student, Antoine-August Le Blanc. Although M. Le Blanc had left, the Academy continued to print lecture notes and problems for him. Sophie obtained Le Blanc’s notes and problems and submitted solutions under Le Blanc’s name to the supervisor of the course, Joseph-Louis Lagrange. But Lagrange noticed that M. Le Blanc’s solutions showed remarkable improvement, and Germain was forced to reveal her identity. Lagrange was astonished and became her mentor and friend.
Germain wrote to Legendre about problems suggested by his 1798 Essai sur le Théorie des Nombres and the subsequent Legendre-Germain correspondence became virtually a collaboration. Legendre included some of her discoveries in a supplement to the second edition of the book. Several of her letters were later published in her Oeuvres Philosophiques de Sophie Germain.
However, Germain’s most famous correspondence was with Carl Friedrich Gauss (1777–1855). She had developed a thorough understanding of the methods presented in his 1801 Disquisitiones Arithmeticae. Between 1804 and 1809 she wrote a dozen letters to him, initially adopting again the pseudonym “M. LeBlanc” for fear of being ignored because she was a woman. During their correspondence, Gauss gave her number theory proofs high praise, an evaluation he repeated in letters to his colleagues. Germain’s true identity was revealed to Gauss only after the 1806 French occupation of his hometown of Braunschweig. Recalling Archimedes’ fate and fearing for Gauss’ safety, she contacted a French commander who was a friend of her family. When Gauss learned that the intervention was due to Germain, who was also “M. LeBlanc,” he wrote to her with delight: “When a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius.”2
She became interested in Fermat’s Last Theorem and adopted a new approach to the problem. She considered those prime numbers p such that 2p + 1 is also prime. (Germain’s primes include 5, because 11 = 2 × 5 + 1 is also prime, but not 13, because 27 = 2 × 13 + 1 is not prime.) For values of n equal to these Germain primes, she proved that if xn + yn = zn, then x, y, or z must be a multiple of n. In 1825 Johann Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre independently proved that the case n = 5 has no solutions. They based their proofs on Sophie Germain’s work. This remained the most important result on Fermat’s Last Theorem from 1738 until the work of Kummer in 1840.
After Gauss turned from number theory to applied mathematics, Germain also stopped working on number theory and instead took up a major challenge in the theory of elasticity. In 1808 the German physicist Ernst F. Chladni had visited Paris where he exhibited the so-called Chladni figures that are produced by a layer of sand on top of a vibrating plate. The Institut de France set up a prize competition with the following challenge: “Formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence.”3 Lagrange said that the mathematical methods available were inadequate to solve it, but Germain nevertheless spent the next decade attempting to derive a theory of elasticity, competing and collaborating with some of the most eminent mathematicians and physicists. She first submitted a manuscript in 1811, in which she originated the concept of mean curvature. Even though she was the only entrant in 1811, she did not win the award. Lagrange was one of the judges, and he corrected errors in Germain’s calculations and came up with an equation that he believed might describe Chladni’s patterns. The deadline was extended by 2 years, and again Germain submitted the only entry. She showed that Lagrange’s equation did yield Chladni’s patterns in several cases, but she could not give a satisfactory derivation of Lagrange’s equation from physical principles. For this work she received an honorable mention. Germain’s third attempt in the reopened contest of 1815 finally won the prize, a medal of 1 kilogram of gold. As a result, she became the first female mathematician to attend sessions at the French Academy of Sciences.
To public disappointment, she did not appear at the award ceremony, perhaps because Poisson, her chief rival on the subject of elasticity, was a judge of the contest. He had sent a laconic and formal acknowledgment of her work but avoided any serious discussion with her and ignored her in public. Later, when others built upon her work, and elasticity became an important scientific topic, she was closed out.
She was stricken with breast cancer in 1829 but, undeterred by that and the fighting of the 1830 revolution, she completed papers on number theory and on the curvature of surfaces (1831). In a eulogy her friend Count Libri-Carducci, wrote of her “unfailing benevolence, which caused her always to think of others before herself . . . in science, never thinking of the advantages that success procures. She rejoiced even when she saw her ideas made fruitful on occasion by other persons who adopted them.”4
Sofia Vasilyevna Kovalevskaya (1850–1891)
We have already met Kovalevskaya twice: first, in chapter 1, describing her childhood fascination with mathematics, stimulated by the wallpaper in her nursery; and then in chapter 5, describing her friendship with her mentor Karl Weierstrass. Now we will fill in the missing parts of her biography.
When Sofia started studying mathematics under a tutor, Y. I. Malevich, she “began to feel an attraction for my mathematics so intense that I started to neglect my other studies.”5
So her father ordered a stop to the math lessons. But she found a copy of Bourdeu’s Algebra and read it at night when her parents were asleep. A year later, a neighbor, a Professor Tyrtov, gave her family a physics textbook he had written. When Sofia didn’t understand the trigonometric formulas, she tried to decipher them, and rediscovered the way in which the notion of sine was historically developed. So Professor Tyrtov tried to convince Sofia’s father to let her study mathematics. But he refused for several more years.
When Sofia reached the age of 18, she wanted to go to a university. But Russian universities didn’t admit women. In fact, Russian women weren’t even allowed to live apart from their families without written permission from a father or husband! Her father had provided her with tutoring, including calculus at age 15, but he would not give her permission to go abroad. So she escaped by making a “marriage of convenience” with an amenable young student of paleontology, Vladimir Kovalevskii. They left Russia with her sister, Anyuta. For the next 15 years, frequent quarrels and misunderstandings between husband and wife resulted in exasperation, tension, and sorrow.
Sofia went to Heidelberg to study mathematics and natural science, while her sister went to Paris. But the University of Heidelberg did not admit women! Eventually Sofia convinced the authorities to let her attend lectures unofficially if she got permission from each lecturer. Sofia immediately attracted attention with her uncommon mathematical ability. After 2 years she went to Berlin to study with Karl Weierstrass. She had to study privately with him, as the university in Berlin would not allow women even to attend class sessions.
By the spring of 1874, Kovalevskaya had written three papers any of which Weierstrass deemed worthy of a doctorate, one each on partial differential equations, Abelian integrals, and Saturn’s rings. The first of these was published in Crelle’s Journal in 1875.6 In 1874 the University of Göttingen awarded Kovalevskaya a doctorate, summa cum laude without examination and without her having attended any classes at that university. Sofia Kovalevskaya was the first woman in Europe to earn a doctorate in mathematics. Her doctoral dissertation is today called the Cauchy-Kovalevskaya theorem.
She returned to Russia and settled in St. Petersburg, her husband’s hometown. Despite the doctorate and strong letters of recommendation from Weierstrass,7 Kovalevskaya was unable to obtain an academic position. Her best job offer was to teach arithmetic to elementary school girls, and she remarked bitterly, “I was unfortunately weak in the multiplication table.”8 For the next 6 years, she devoted herself to her family (her daughter Sofia Vladimirovna, nicknamed Fufa, was born in 1878), to scientific journalism, and to promoting women’s right to higher education. She also wrote fiction, including a novella, Vera Barantzova, which was translated into several languages. (Sofia’s interest in literature went back to her childhood, when Dostoyevsky was a regular guest in her family’s home.) In 1890 she wrote in a letter, “It seems to me that the poet must see what others do not see, must see more deeply than other people. And the mathematician must do the same.”9
In St. Petersburg she was visited by a fellow student of Weierstrass, the Swede Gosta Mittag-Leffler. He later wrote, “When she speaks her face lights up with such an expression of feminine kindness and highest intelligence that it is simply dazzling. Her manner is simple and natural, without the slightest trace of pedantry or pretension. She is in all respects a complete ‘woman of the world.’ As a scholar she is characterized by her unusual clarity and precision of expression. . . . I fully understand why Weierstrass considers her the most gifted of his students.”10
She came back to mathematics in 1880, when Chebyshev and Mittag-Leffler invited her to speak at an international congress in St. Petersburg. In that year she returned to Berlin and visited Weierstrass, and took up the study of light propagation in anisotropic media. In 1882 she wrote three articles on the refraction of light.
In the spring of 1883, Vladimir committed suicide. The couple had been separated for 2 years. To escape from guilt, Kovalevskaya immersed herself in mathematics.
In 1883 Mittag-Leffler overcame opposition and obtained a position for her as privat docent at the University of Stockholm. In June 1889, after surviving an extremely hostile atmosphere, she was awarded a lifetime professorship, the first woman to hold a chair at a modern European university.
At Stockholm she taught the latest topics in analysis, became an editor of the new journal Acta Mathematica, took responsibility for liaison with mathematicians in Paris and Berlin, and helped to organize international conferences. She again wrote reminiscences and dramas, as she had when she was young. In 1886 the French Academy of Sciences announced the topic of the Prix Bordin: significant contributions to the motion of a rigid body. Kovalevskaya entered and won the prize with a paper, Mémoire sur un cas particulier du problème de le rotation d’un corps pesant autour d’un point fixe, où l’intégration s’effectue à l’aide des fonctions ultraelliptiques du temps. In recognition of the brilliance of this paper, the award was increased from 3000 to 5000 francs. In 1889 her further research on this topic won a prize from the Swedish Academy of Sciences, and she was elected a corresponding member of the Russian Imperial Academy of Sciences. The Russian government had repeatedly refused her a university position, but the rules at the Imperial Academy were changed to allow a woman to be elected. Two years later, early in 1891, Kovalevskaya died of influenza and pneumonia, at the height of her mathematical powers and reputation.
Emmy Amalie Noether (1882–1935)
Next we turn to Emmy Amalie Noether, who persisted in the face of tremendous obstacles to become one of the greatest algebraists of the 20th century. Emmy was the eldest of four children. Her father, Max, was a distinguished mathematician and a professor at Erlangen. Her mother, Ida Kaufmann, came from a wealthy Cologne family. Her mother taught her to cook and clean. She was sent to the Höhere Töchter Schule in Erlangen, a kind of young ladies finishing school, where she studied German, English, French, and arithmetic. She took piano lessons and loved dancing at parties with the children of her father’s university colleagues. After high school and further language study, she was certified to teach English and French in girl’s schools in Bavaria.
But Emmy Noether never became a language teacher. At the age of 18, she decided to take classes in mathematics at the University of Erlangen, where her father was a professor and her brother, Fritz, was a student. She could not be a regular student. In 1898 the academic senate of the University of Erlangen had resolved that admitting female students would “overthrow all academic order.” But Emmy was permitted to audit classes. After 2 years she went to the University of Göttingen and attended lectures by Blumenthal, Hilbert, Klein, and Minkowski. Then in 1904, without ever having been a regular undergraduate student, she passed the qualifying exam to matriculate in mathematics at Erlangen as a doctoral student. In 1908 she received a doctorate summa cum laude for a dissertation entitled “On Complete Systems of Invariants for Ternary Biquadratic Forms,” written under the direction of her father’s colleague Paul Albert Gordan, whom she had known since childhood. Gordan’s lifework was calculating explicit algebraic formulas for the invariants. Following this constructive approach of Gordan, Noether’s thesis listed systems of 331 covariant forms. Later she scornfully rejected this dissertation as Formelsgestrupp—a jungle of formulas.
She could not teach at Erlangen because they did not hire women professors. But she helped her father, teaching his classes when he was sick. Because of his disabilities, he was grateful for his daughter’s help. And soon she started publishing papers on her own work. Ernst Fischer (known for the Riesz-Fischer theorem) had succeeded Gordan in 1911 and influenced Noether away from Gordan’s calculational approach and toward Hilbert’s abstract approach. Noether’s reputation grew quickly. In 1908 she was elected to the Circolo Matematico di Palermo, then in 1909 to the Deutsche Mathematiker-Vereinigung.
In 1915 Hilbert and Klein invited Noether back to Göttingen. Although her specialty was algebra, not mathematical physics, Emmy’s first accomplishment in Göttingen was a central result in theoretical physics, now known as Noether’s theorem. It states a correspondence between differentiable symmetries and conservation laws and led to new formulations for several concepts in Einstein’s general theory of relativity. The American theoretical physicist Lee Smolin recently wrote, “The connection between symmetries and conservation laws is one of the great discoveries of twentieth century physics. But I think very few non-experts will have heard either of it or its maker—Emily Noether, a great German mathematician. It is as essential to twentieth century physics as famous ideas like the impossibility of exceeding the speed of light.”11
Hilbert and Klein convinced her to remain at Göttingen. In order to allow Noether to lecture, Hilbert announced her courses under his name. For example, in the catalogue for the winter semester of 1916–1917, there appears: “Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from 4–6, no tuition.”
Hilbert and Klein had a long fight to have her officially included in the faculty. It was war time, and her opponents asked, “What would the country’s soldiers think when they returned home and were expected to learn at the feet of a woman?” Hilbert answered, “I do not see that the sex of the candidate is an argument against her admission as a Privat dozent. After all, the university senate is not a bathhouse.”12
Emmy was a pacifist and hated the war. In 1918 the Kaiser surrendered and Germany became a republic. Then Noether and all women received the vote, and a year later, at the age of 37, she finally became a privat dozent—an unpaid faculty member, entitled to try to collect fees from her students. Three years later she was given a higher title, “unofficial associate professor,” but still she remained unpaid, and as long as she remained at Göttingen, this never changed. There were several reasons for this. Not only was she a woman, she was also a Jew, a Social Democrat, and a pacifist. However, she did act as thesis adviser for a number of Göttingen Ph.D. students.
After 1919 Noether developed a new point of view in algebra: focus on simple, general, axiomatic properties shared by many algebraic structures. In this way she produced a new theory of “ideals” that helped turn the algebraic specialty called ring theory into a major mathematical topic. In Idealtheorie in Ringbereichen (1921) she proved the fundamental result: in any commutative ring with an ascending chain condition, the ideals can be decomposed into intersections of primary ideals. (In chapter 4, in our discussion of Grothendieck, we explained the meaning of “ideal” in a ring.)
Beyond its importance as a basic theorem in algebra, this work gradually changed the way mathematicians think. Noether’s conceptual approach to algebra led to a body of principles unifying algebra, geometry, linear algebra, topology, and logic. “She taught us to think in simple, and thus general, terms . . . homomorphic image, the group or ring with operators, the ideal . . . and not in complicated algebraic calculations,” said her Russian coworker P. S. Aleksandrov.
After spending the year 1924 studying with Noether, the Dutch mathematician B. L. van der Waerden wrote his very influential book Moderne Algebra. Most of the second volume consists of Noether’s work. In 1926 André Weil visited Göttingen. In subsequent decades, his influential Bourbaki group adopted Noether’s axiomatic style as the correct mode for all of pure mathematics.
From 1927 on, Noether collaborated with Helmut Hasse and Richard Brauer on noncommutative algebras. Along with Emil Artin and Helmut Hasse, she founded the theory of central simple algebras.
Her keen mind and infectious enthusiasm made Emmy Noether an effective teacher for those students who could keep up with her. The ones who caught on to her fast style became loyal followers known as “Noether’s boys.” Much of her work appeared in papers written by colleagues and students rather than under her own name.
What was Emmy like as a person? “Warm, like a loaf of bread,” wrote Hermann Weyl. “There radiated from her a broad, comforting, vital warmth. She was fat, rough and loud, but so kind that all who knew her loved her.” Her students were like her family, and she was always willing to listen to their problems.
In 1933 the Nazis came into power in Germany and demanded that all Jews be thrown out of the universities. In April 1933 she was denied permission to teach. Hermann Weyl wrote, “Her heart knew no malice; she did not believe in evil—indeed it never entered her mind that it could play a role among men. This was never more forcefully apparent to me than in the last stormy summer, that of 1933, which we spent together at Göttingen . . . her courage, her frankness, her unconcern about her own fate, her conciliatory spirit, were in the midst of all the hatred and meanness, despair and sorrow, surrounding us, a real solace.”13
Noether’s brother, Fritz, was fired from his position teaching mathematics at the Technical University in Breslau, on complaints that his presence contradicted the Aryan principle. He was offered a teaching position at Tomsk in Siberia and moved there with his family in 1934. In 1937 he was arrested as a German spy and sentenced to 25 years in prison. While in prison, he was accused of “anti-Soviet propaganda,” and he was executed at Orel on September 10, 1941. Over 40 years later, after the death of Stalin, his son was informed of his father’s complete “rehabilitation.”14
Friends tried to get Emmy a position at the University of Moscow. Then came an offer of a professorship at Bryn Mawr College. There was a difficulty, because Noether had no interest in undergraduate teaching and Bryn Mawr had no prospect of creating a permanent position without undergraduate teaching. (Their graduate program in mathematics had only four faculty and five students.) But the Rockefeller Foundation and the Emergency Committee to Aid Displaced German Scholars paid her first year’s salary, and her supporters, including Birkhoff, Lefschetz, and Wiener, succeeded in persuading the college to extend her appointment.
Emmy took four of the students under her wing and taught them in a mixture of German and English. She also lectured at the Institute for Advanced Study in Princeton. The grant was renewed for two more years in 1934. This was the first time that Emmy Noether was ever paid a full professor’s salary and accepted as a full faculty member. For the first time, she had colleagues who were women. Anna Pell Wheeler, the head of the mathematics department at Bryn Mawr, became a great friend of hers. Emmy told people that these were the happiest years of her life.
In 1935 she developed a uterine tumor and scheduled surgery during a college break at Bryn Mawr. She died during or shortly after the surgery from sudden and unexplained complications. Her death surprised nearly everyone, as she had told only her closest friends of her illness. Noether had never married, and she had no relatives in the United States. Her ashes are buried in the cloisters of Thomas Great Hall on the Bryn Mawr campus. A coed gymnasium in Erlangen specializing in math has been named after her.
Contemporary Women Mathematicians
What is it like today to be a woman in mathematics? Certainly things have changed a lot since the times of Germain, Kovalevskaya, and Noether. At the beginning of the 20th century, women in many countries could not take university classes for official credit. They could audit only with permission of the lecturer. Some women experienced skepticism or even opposition from their teachers when expressing interest in mathematics during their early school years.
Many universities continued to discourage women applicants in the decades before World War II. Even after such policies became illegal, attitudes toward women interested in mathematics remained prejudicial. Pregnant women, and women with small children, found it hard to continue their studies. It took the help of committed teachers to encourage female students to persist. The ensuing psychological injury has healed slowly.
Since World War II, the number of women mathematicians has increased. As women started to share their experiences through meetings, caucuses, and the establishment of the Association for Women in Mathematics, they were able to develop a diversity of solutions to the challenge of balancing family and work lives.
For younger women today, whose careers started in more recent decades, the path has been less difficult. The goal of equal opportunity and active effort for equal representation in all academic fields has made mathematical life more welcoming for women. In the United States, “affirmative action” actually requires all university departments, including mathematics, to actively seek females for faculty positions. Among math students, women are now respectably represented.
Women and minorities are entering graduate programs in increasing numbers and are making major contributions. In the last two decades, there has been a steady growth in female participation in mathematics, in both undergraduate and graduate programs. One-third of Ph.D.s in mathematics are now granted to women, nearly doubling the numbers since the eighties.15
Today there are scores of women in tenured positions in university mathematics departments, and many of them, more than one can count on both hands, are internationally recognized. While sexist attitudes linger, women are assuming leadership positions in larger numbers in national and departmental organizations.
Critical to the successful outcome of each woman’s story are supportive colleagues in the mathematics community. Margaret Murray found that nearly all the women she interviewed had had at least one college teacher who had encouraged and influenced them, even in the days when women in mathematics challenged the prevalent social norms.
But many, probably most of these women, had to resist and overcome prejudice and discrimination on their way to success. And even with equal opportunity and affirmative action in place, the number of women in leading positions in mathematics remains small, much smaller than in other fields of science like biology. Among faculty, they are still a small minority. In the most prestigious departments, the crème de la crème of the Ivy League, there are practically none.
In her excellent book Women in Mathematics, Claudia Henrion asks, “First, why is it that women continue to be significantly underrepresented in mathematics, particularly at the highest levels? And second, why is it that even the most successful women in mathematics, those who have already made it by standard measures of success, often continue to feel (to varying degrees) like outsiders in the mathematics community?”16 In the course of considering graduate programs herself, Henrion writes, she was advised to look elsewhere by the chair of a mathematics department that had a large, prestigious program: “You’re too normal; you wouldn’t fit in here.”17 Most of the women Henrion interviewed reported variations on that experience at various times in their careers.
In the popular mind, mathematics is still a male occupation. Fewer girls than boys choose to take advanced math courses or to major in math. And even among those women who do graduate work in math and receive doctorates, there is still a lower rate of promotion and research productivity. As we saw earlier in writing about the AWM in chapter 6, there are three distinct major issues that keep down the number of female mathematicians. First, women have greater difficulty in establishing the connections and support systems that are essential for scientific success; second, the desire and need to have children forces women to interrupt their research careers in ways that are hard to compensate for; and third, the lingering stereotypes and social expectations of gender create self-doubt and hinder the active fighting for success and recognition that successful male mathematicians often display.
It often takes intense effort to create support groups for women, while men, it seems, are largely free to get on with doing mathematics. Female graduate students of mathematics report difficulty in connecting to the study circles and conversation groups of their male fellow students. If there are no other females in her particular mathematical specialty, a woman in graduate school needs to have exceptional tact or perseverance to be accepted into male social circles.
As Henrion says in her first chapter, “It is so much a part of the normal course of events for men to be tied into the mathematical community that it is easy for them to take it for granted, not even recognizing the way in which community is invisibly operating behind the scenes: in helping them to get jobs, in brainstorming sessions with colleagues about ideas, in the exchange of news about recent theorems in bars or bathrooms, in the suggestion of each other’s names as speakers or journal editors.”18
One of the most famous of the female mathematicians interviewed by Henrion is Karen Uhlenbeck, a professor at the University of Texas in Austin and a winner of the prestigious “genius” MacArthur award. As a girl, Uhlenbeck was an extreme tomboy, who thought she might become a forest ranger as an adult. Her mathematical style has always been utterly individualistic, finding surprising connections between ideas in theoretical physics and abstract geometry or nonlinear partial differential equations. Her career path through graduate school and faculty positions seems to have developed without much conscious planning on her part, but rather by other people noticing her work and being impressed by it. Even with a MacArthur fellowship and membership in the National Academy of Sciences to her credit, Uhlenbeck says “[I was] not able to transform myself completely into the model of the successful mathematician because at some point it seemed like it was so hopeless that I just resigned myself to being on the outside looking in.”19
A second issue is the question of family life. A century ago, the few women who aspired to academia expected to remain unmarried. Neither Germain, Kovalevskaya, nor Noether had the standard husband-and-children family life. Today, that limitation is not accepted by most female mathematicians. But having children is almost bound to disrupt a mother’s professional activities. Even pregnancy can already be problematic. Childbirth and child care in early infancy usually cause a sharp break in one’s study or one’s research unless one is lucky enough to have an exceptionally supportive husband, mother, or mother-in-law. The myth of the young, energetic mathematician contributes to the pressure young women (and men) feel, despite the fact that there are many examples of prominent mathematicians who did excellent work in their later years. If the focus were not so much on the young mathematician, it would be more effective to design programs with women in mind.
Figure 7-1. Karen Uhlenbeck, a MacArthur award recipient and leading researcher in mathematical physics. Courtesy of Dirk Ferus.
Different female mathematicians have given different answers to the question: Is it best to have children in your last year of grad school? (as did the logician Lenore Blum and the combinatorialist Fan Chung Graham). Or to have children after college and defer graduate study? (the path taken by the topologist Joan Birman of Barnard College–Columbia University, who received her Ph.D. at the age of 41). Or to have them much later, in your late thirties or even forties, after you have achieved tenure?
Chung decided to forego maternity leave during her second year at Bell Labs and had her second child during her 4-week vacation (during which she wrote a paper). Her manager wondered if she would quit because she was having a child. He was unaware that she already had a child who was 2 years old. It was crucial that she had full-time child care and a supportive husband, Ron Graham. When one of Lenore Blum’s professors saw her with her 4-month old baby, he said “Where did that child come from? Whose is it?”20 He had been oblivious to her pregnancy and giving birth.
Joan Birman is an outstanding researcher in topology, specializing in braids and knots. She earned her doctorate at the Courant Institute as a student of Wilhelm Magnus, after years working at engineering firms and raising three children. She had returned to graduate school at New York University later in life, starting part-time in a master’s program. She points out that since many women have children during the very years that traditionally are considered the most productive years for a mathematician, it would be more appropriate to look at their productivity over a longer time span, recognizing that women may need to enter the mathematical research pipeline later in life. Birman reported a relatively smooth path to a Ph.D. and eventually a position as a respected research mathematician. But many women find themselves having children at about the same time that they come up for tenure, “and the men are working like lunatics on their mathematics, and the women, they see this choice, and if you give it a little less effort, your research is dead.”21
Figure 7-2. Fam Chung, American combinatorialist. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.
In the linear trajectory from graduate school through postdoctoral study, assistant professor, associate professor, and full professor, any deviation is suspect. Taking a couple of years off from mathematics makes it very difficult to return. There are very few reentry points. Some colleagues may still perceive you as being “not fully serious about your work” if you have a child. The silent message—“You’re either a mathematician or a mother, you can’t be both”—goes with the assumption that men do the math and women do the mothering. To a young woman who wants to become a mathematician, there may seem to be no favorable period for having children. The mathematical community needs to find ways of dealing with this conflict.
Figure 7-3. Joan Birman, topologist at Columbia University. Photograph by Joseph Birman.
One outstanding success at reentering research after a long hiatus was Lenore Blum, who has recently collaborated with Steve Smale and Mike Shub on theoretical numerical analysis using real numbers rather than integers. Early in her career she accepted a job at a women’s college in Berkeley, Mills College, after a promised position at the University of California, Berkeley, mysteriously failed to materialize. At Mills she developed and created an exemplary math program and became deeply involved in increasing female participation in mathematics. She helped organize and was president of the AWM. Then, after 13 years devoted to this work, at the age of 40, she returned to research and managed to start a highly successful research career.
The third often mentioned negative factor is psychological. Every mathematician is subject to self-doubt. Am I smart enough? Can I make it? Can I still do it? It seems that such self-doubt is more severe or more dominant for many women. The social stereotype that mathematics is a male domain is involuntarily internalized. It may take many years for a woman to feel at ease and accepted by her mathematical community.
Many young mathematicians grow up in environments where their interest and ability are not recognized. There is a sense of vulnerability from which many women continue to suffer. In Complexities, a volume by and about women mathematicians, Katherine Socha writes of such self-doubt and the difference caring mentors make: “I worried about being good enough and only after several years of graduate school did I develop the perspective and confidence to ask ‘good enough by whose standards?’ ”22 Socha comments, “It was only my stubborn determination to prove my intellectual mettle that kept me enrolled in mathematics classes.”23
Judy Roitman said of herself, “I think I tend to be a little pushy and strident because I had to be in graduate school to be heard. So I am generally not a very good colleague. I don’t think I socialize correctly or work well with other people. Basically, I’m too sensitive to not being taken seriously, and I tend to simply fade away and give up.”24
The widely publicized studies of Benbow and Stanley (1980), reporting male superiority in tested performance, further contributed to gender stereotypes. But since then, meta-analyses of large databases show a steady shrinking of gender differences in mathematical achievement.25
Figure 7-4. Manuel, Lenore, and Avrim Blum. All three are now professors at Carnegie-Mellon University. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.
Claudia Henrion conducted extensive interviews with half a dozen prominent female mathematicians. She found a remarkable range of different ways of achieving success within the male-dominated mathematical structure.
Mary Rudin, whom we quote extensively in chapter 8, spent most of her mathematical career as a self-styled “amateur,” holding part-time jobs while devoting equal time and effort to running a household and raising four children. She was placed in her first jobs by her mentor, R. L. Moore, with no effort on her part. Her husband, Walter, was a well-known tenured professor specializing in a totally different field of mathematics. In 1971, when it became embarrassing that one of their best known mathematicians was a female and a part-time instructor, the math department at Wisconsin in one single step elevated her to full professor with tenure, a promotion she appreciated but had neither asked for nor expected. She was 47 years old.
The sharpest contrast with Rudin’s story is that of Vivienne Malone-Mayes, an African-American mathematician whom we have met in chapters 1 and 5, and will meet again in chapter 9 in connection with her denial of admission to the course taught by Robert Lee Moore. Mayes earned her bachelor’s and master’s degrees at Fisk, in Nashville, Tennessee. Fisk is a traditionally black institution (TBI) comparable to Howard University in Washington D.C. as one of the highly ranked, prestigious TBIs. At Fisk she was inspired by two teachers, Evelyn Boyd Granville, who was one of the only two African-American females who had earned a math Ph.D. at that time, and Lee Lorch, a white professor who was a legendary fighter for civil rights and equality for Black Americans. After graduating from Fisk, she taught at Paul Quinn College and at Bishop College, where, like her mentors Granville and Lorch, she sought out talented Black students and encouraged them to aspire to do advanced graduate work for a doctorate. As a consequence, she herself was challenged to go back to school for a Ph.D. She had grown up just a few blocks away from Baylor University, but when she applied for admission to their graduate program, she was told that Blacks were not admitted. In the early 1960s the University of Texas in Austin had been forced to give up its whites-only admission policy. Mayes applied and was admitted to the graduate program at Austin, a path that many in her community thought was absurd for a Black woman, and certainly not practical for getting a job. By the time she had completed her Ph.D. at Austin, Baylor had reversed its policy and offered her a faculty position! She taught at that overwhelmingly white institution from 1966 to 1994, and in 1971 the Baylor student congress elected her outstanding faculty member of the year.
However, Mayes’ successful teaching career at Baylor came at the price of virtually complete isolation from the rest of the faculty. In fact, during her whole time there she was paid much less than comparable white instructors. When she asked about teaching summer classes, she was told by the department chair, “I haven’t even offered those to the white male faculty yet.”26 Throughout her career Mayes was sustained by her religion and by her goal of serving her students and her African-American community.
In all these very different stories, three elements of success are present: having persistent self-confidence, connecting with mentors and collaborators, and being in the right place at the right time, when a situation opened up that permitted continued mathematical work and advancement.
What Can Be Done?
Contributors to Complexities write of the impact of math and computer camps for high school students, of special women’s mentoring programs at prestigious institutions, and of networking and support groups including the AWM. Most important are the encouragement provided by caring mentors and the group interaction of students drawn from previously marginalized groups.
Joan Birman told Henrion, “I guess that if I could see any solution to the non-participation of women in mathematics, it would be, first of all, if women were able to think about going back to mathematics at a later point, and if there was a practical way for them to do this . . . and if the whole community was ready to accept this as another option—all of these things would help.”27 Joan Birman would not have been able to get a Ph.D. at Columbia, where she is now a professor, because she needed to start graduate school as a part-time student, and Columbia does not allow part-time graduate students in mathematics.
But at Smith College the Ada Comstock program allows older women who left school in order to raise a family to finish their bachelor’s degree. Certain graduate programs, like the one at New York University, are receptive to older students or those who have taken some time off. The National Science Foundation has a program for women in mathematics who are returning to research.
There should be ways for mathematicians to have a part-time status during certain periods of their careers, perhaps in graduate school or as professors. This is one way of allowing people to have children and yet remain professionally active, even if it is at a reduced pace for a few years. For extenuating personal circumstances, such as having children, the tenure-track period could be lengthened. Many colleges and universities are beginning to institute such policies. Day care at mathematics meetings, flexible teaching schedules, and regular day care at colleges and universities are important. And of course, a change in attitude in the mathematics community. Attitudes can be more important than formal policies in determining whether women return to mathematics. The mathematics community must convey a clear message that having children is not in conflict with a career in mathematics.
Aging Mathematicians
We will report on a survey in which mathematicians told us about their experiences and feelings about aging, and we will present a critique by Louis Mordell of Hardy’s views on the subject.
In chapter 2 we quoted Hardy’s beautiful and melancholy essay, A Mathematician’s Apology. At 60, says Hardy, he is too old to have new ideas because “mathematics is a young man’s game.” So he is reduced to writing books instead of doing a mathematician’s proper business, discovering and creating new mathematics. First, as to “young.” In this very same essay Hardy says that his own best time for mathematical creation was in his forties, in collaboration with John Littlewood and Srinivasa Ramanujan. And the long productive life of his collaborator Litlewood is a striking refutation of Hardy’s “young man” theory. Littlewood outlived Hardy by 30 years and was over 70 when with Mary Cartwright he wrote one of his most intricate and important papers, dealing with Van der Pol’s equation and its generalizations. It is 110 pages of hard analysis. He called that paper “the Monster” and said: “It is very hard going and I should not have read it had I not written it myself.” Years later it was recognized as an early breakthrough in the discovery of chaos. (See chapter 2.) Over a decade later, at age 84, he published, in the first issue of the Journal of Applied Probability, “very precise bounds for the probability in the tail of the binomial distribution. The bounds he gave are still the best.”28 Littlewood’s last paper appeared in 1972, when he was 87.
Hardy’s Apology was challenged by Hardy’s successor at Cambridge, Louis Joel Mordell (1888–1972). Mordell’s life is another counterexample to Hardy’s pessimism. Mordell had arrived at Cambridge in 1906 as the wunderkind son of a poor Hebrew scholar in Philadelphia. He retired in 1953, at age 75, but almost half of Mordell’s 270 publications appeared after his retirement! Mordell is reported to have said modestly, “I did work in my 70s many a younger man would have been proud to have done.”29 (Nowadays, Mordell is most often mentioned in connection with his famous conjecture in number theory.)30 After he retired, Mordell visited universities around the world, reaching a total of 190. In 1971, well into his eighties and still traveling enthusiastically, he attended a number theory conference in Moscow, went on an Asian tour, and lectured in Leningrad. A few months later he died at home in Cambridge.
A year earlier, in October 1970, Mordell published his critique of Hardy’s Apology. To begin with, he disputes Hardy’s self-belittling dictum, “The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he and other mathematicians have done.”31
Figure 7-5. Louis Joel Mordell and Gábor Szegö. Courtesy of Dolph Briscoe Center for American History, The University of Texas at Austin. Identifier: di_05555. Title: Mordeel and Szego. Date: 1958/08. Source: Halmos (Paul) Photograph Collection.
The real function of a mathematician is the advancement of mathematics. Undoubtedly the production of new results is the most important thing he can do, but there are many other activities which he can initiate or participate in. Hardy had his full share of these. He took a leading part in the reform of the mathematical tripos some sixty years ago. Hardy was twice secretary and president of the London Mathematics Society. . . . We all know only too well that with advancing age we are no longer in our prime, and that our powers are dimmed and are not what they once were. Most of us, but not Hardy, accept the inevitable. We can perhaps find pleasure in thinking about some of our past work. We can still be of service to younger mathematicians.32
Mordell mentions Littlewood, Chapman, Besicovitch, and Davenport, all still active in their sixties, seventies, and eighties.
Mordell even turns Hardy’s notion of mathematical beauty against him. Hardy’s words have often been quoted: “A mathematician, like a painter or poet, is a maker of patterns and these must be beautiful. Beauty is the first test: there is no permanent place in the world for ugly mathematics.” But, says Mordell, “I do not think that Hardy’s work is characterized by beauty. It is distinguished more by his insight, his generality, and the power he displays in carrying out his ideas. Many of the results that he obtains are very important indeed, but the proofs are often long and require concentrated attention, and this may blunt one’s feelings even if the ideas are beautiful.”33
Mordell ends by refuting Hardy’s dictum in which he said, “Mathematics is not a contemplative subject.” Replies Mordell: “What does he mean when he says mathematics is not a contemplative subject? Many people can derive a great deal of pleasure from the contemplation of mathematics, e.g., from the beauty of its proofs, the importance of its results, and the history of its development. But alas, apparently not Hardy.”34
Opinions about aging in mathematics vary broadly. Following are a few examples. Members of the Bourbaki collective withdrew from active group participation at 50. Albert Einstein said, “A person who has not made his great contribution to science before the age of thirty will never do so.”35 The French number theorist André Weil wrote, “There are examples to show that in mathematics an old person can do useful work, even inspired work, but they are rare and each case fills us with wonder and admiration.”36 At Felix Klein’s 50th birthday party, he whispered to Grace Chisholm Young: “Ah, I envy you. You are in the happy age of productivity. When everyone begins to speak well of you, you are on the downward road.”37
In an article in The New Yorker, “Mathematics and Creativity,” Alfred Adler wrote: “consuming commitment can rarely be continued into middle and old age, and mathematicians, after a time, do minor work. In addition, mathematics is continually generating new concepts which seem profound to the older men and must be painstakingly studied and learned. The young mathematicians absorb these concepts in their university studies and find them simple. What is agonizingly difficult for their teachers appears only natural to them. The students begin where the teachers have stopped, the teachers become scholarly observers.”38
L. E. J. Brouwer (1881–1966), one of the founders of algebraic topology and the great leader of the intuitionist school on the foundations of mathematics, was 24 when he wrote about older academics, “There are others who do not know when to stop, who keep on and on until they go mad. They grow bald, short-sighted and fat, their stomachs stop working, and moaning with asthma and gastric trouble they fancy that in this way equilibrium is within reach and almost reached. So much for science, the last flower and ossification of culture.”39
On the other hand, Joseph Dauben, the biographer of the Israeli-American logician Abraham Robinson, wrote, “[Robinson] was always pleased to dispel the myth that the best mathematicians were under thirty and that a mathematician did her or his best work early, at the very start of one’s career. As a striking counterexample, Robinson’s best mathematics was only beginning to reap the benefits of his wide experience when, suddenly at the age of fifty-five, he died.”40
Abraham de Moivre (1667–1754) found his presumably most important result when he was 66: “the local central limit theorem.” It is reported in all seriousness that de Moivre correctly predicted the day of his own death. Noting that he was sleeping 15 minutes longer each day, de Moivre surmised that he would die on the day he would sleep for 24 hours. A simple mathematical calculation quickly yielded the date, November 27, 1754. He did indeed pass away on that day.41
The English algebraist J. J. Sylvester pointed out that Leibniz, Newton, Euler, Lagrange, Laplace, Gauss, Plato, Archimedes, and Pythagoras all were productive until their seventies or eighties. “The mathematician lives long and lives young,” he wrote. “The wings of the soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life.” Sylvester himself was in his 82nd year, in 1896, when he “found a new enthusiasm and blazed up again over the theory of compound partitions and Goldbach’s conjecture.”42
A Survey on Mathematical Aging43
Who is right, Hardy or Mordell? The question intrigued Reuben Hersh who decided to check up on whether mathematics really is a young man’s game. From the American Mathematical Society membership directory, he chose 250 names of men and women he had known somewhere, at some time.
They were mostly Americans but also included were a few Canadian, Swedish, French, Israeli, and Japanese mathematicians who had spent time in the United States. Reflecting his training and experience, the mailing list was heavy on workers in differential equations: theoretical (both ordinary and partial), applied, numerical, or stochastic. There were also stochastic processors, and a scattering of logicians, algebraists, topologists, geometers, and statisticians. Mathematical specialty should not make much difference for the questions. Of course, the anonymity of these responses was respected.
There were several clusters of questions. Some dealt with the value of the respondents’ early work in contrast with later achievements. Did they experience less enthusiasm or a sense of loss as a consequence of aging? Did the respondent shift emphasis from research to teaching or administration? How did aging affect the status of the respondents in their departments and in the larger mathematical community? Did the mathematicians who answered have suggestions for individuals or institutions when confronted with the challenges of aging?
There were 66 replies, which is considered a good response rate. They came from 23 states, 3 Canadian provinces, Sweden, and Israel. Ages ranged from 60 to 92. There is no claim that this choice of 250 was typical, let alone random. And the 66 of 250 who answered are certainly not typical. They are biased toward people who answer questionnaires, who like to hear from an old acquaintance, who are willing to consider some possibly painful issues, and who aren’t too unhappy or ashamed of their lot in mathematical life. The people who don’t respond to questionnaires are like the dark matter of the cosmos; we know they are out there, but we can only guess what they look like.
Two respondents knew of earlier unpublished surveys by famous mathematicians. George Mackey of Harvard did a study of 50 leading mathematicians and concluded that on average their best work was done in their late thirties. The prominent topologist Gail Young, a student of Robert Lee Moore, did a study of people who matured very young in mathematics. He found that they generally burned out early. Young felt there was a fairly constant period during which a person could do very creative work. Some had their period earlier; others had it later.
Hersh’s questionnaire invited recipients to reveal as much as they liked about their current and past situations. Their answers yield a glimpse of how this sample of mathematicians view their lives in mathematics. Such responses don’t submit readily to tabulation, much less to statistical analysis. These reflections are of interest in their diversity and how they depict individual points of view.
Most of the respondents were satisfied with their life situations. Relevant is a report by S. S. Taylor on retirees (not restricted as to field) at the universities of New Mexico and Rhode Island in the United States and Bath and Sussex in England. Reassuringly, perhaps surprisingly, 98 percent of the New Mexico retirees, 97 percent of the Rhode Island retirees, and 84 percent of the English retirees told Taylor they were “reasonably satisfied” or even “very satisfied” with retirement. Two-thirds of the American respondents told Taylor they received the same or a higher income as before retirement.44 Most of Hersh’s respondents say they continue research after retirement. Some think their recent research is their best ever. Some say they’re doing what they’re interested in, uninhibited by the judgment of the math community.
Survey Results from Hersh’s Questionnaire
The responses are organized into six groups. Excerpts from some answers appear in more than one group.
1. Mathematicians report losing their edge once past youth.
2. Mathematicians may be as good or better in later years.
3. Symptoms of age and coping strategies.
4. Penalties for aging and for following one’s own bent.
5. Advice for aging mathematicians.
6. Advice for mathematics departments.
Many responses are hard to classify. For example, one mathematician is reported to have just published a very good paper at age 75, though he complains bitterly about not being able to do good research any longer. Does this belong in the first group, the second group, or both? Some painful experiences in the fourth group contradict some advice in the fifth group. Many respondents say, “Follow your own bent, regardless of outside pressure”; but many respondents report penalties for doing that. Some respondents don’t give their age; for a few, Hersh was unable to identify their geographical location. The following excerpts are taken from longer messages.
1. Mathematicians Report Losing their Edge Once Past Youth
There are some reports of sad or difficult experiences. A 72-year-old analyst in California confessed, “My zest is fine, but capacity much diminished before age 56. Age and alcohol and depression.” A California geometer explained, “The field of mathematics moves very fast. The pace has been quite extraordinary in the past 50 years. Just trying to keep up in one’s specialty requires many hours of effort. One doesn’t feel comfortable doing the same old thing. Some great mathematicians have been unable to handle this. When a decent problem comes along which seems accessible, I’m eager to jump in. The trouble is that the frontier is moving so fast. It’s not that we give up research mathematics, research mathematics gets away from us.”
And a 79-year-old analyst in Maryland wrote, “At around 56 I had lost whatever originality I once possessed. But not the desire to learn and try.” A 62-year-old friend in Louisiana, also an analyst, wrote, “I used to work late at night, but now I’m too exhausted to do more than make calendar entries and clean up my study.” And a 71-year-old applied mathematician in Rhode Island said, “Clearly at my age I can’t keep up with the best younger people. Some old-timers have looked foolish in their later research efforts. My hope is at least to avoid that.”
The well-known number theorist Ivan Niven was quoted, “As you get older you know too much. You have all these methods, and you try all the combinations and variations you can think of. You’re running down the old tracks and nothing works.”45
And, in explanation of these stories of loss, one respondent, whose age and location are unknown, wrote, “Mathematics tends to be introverted, with unbalanced expenditure of mental energy. As one grows older there is desire for other forms of expression, which dilutes the intensity to solve problems. ‘What does it all mean?’ is asked more often, which also can slow down progress.”
And a famous Swedish analyst, at age 80, explained, “Aging has two sides—your own age, and the age and aging of your subject and your contributions. This aging is brought about by the work of younger competitors.”
On the other hand, some presented a different view.
2. Mathematicians May Be as Good or Better in Their Later Years
This turns out to be the opinion of most women mathematicians. Claudia Henrion interviewed half a dozen leading female U.S. mathematicians,46 and they told her that they had done their best work in their thirties, forties, even fifties. Lenore Blum returned to mathematical research in her forties, after years of teaching at Mills College and involvement in national programs to promote women in mathematics.
Mary Ellen Rudin, who managed to stay professionally active even while raising four children, reports that she is doing some of her best work in her fifties and sixties, now that most of the children are grown. She worked part-time as a lecturer until she was almost 50, when the University of Wisconsin promoted her from a lecturer to a full professor. Rudin said, “I don’t think most people’s best work will be done by the time they’re thirty, and certainly my best work wasn’t done until I was fifty-five years old.”47 Others may follow Mary Ellen Rudin’s path and work part time for a period to balance having children with mathematical research.
The prominent logician Marian Pour-El told Henrion in an interview, “I’ve never felt that you’re over the hill if you’re in your late thirties. I think I’ve done my best work later on, by a long shot.”48
Joan Birman, a topologist at Columbia University–Barnard College, did not get her Ph.D. until she was 40 years old. Birman focused better on math after the issues of marriage were settled, her children were older, etc. “I think doing mathematics when you’re enthusiastic is important—not your age.”49 In fact, most of the women Henrion interviewed found that their work improved as they got older.
A 62-year-old female probabilist in Hersh’s sample wrote, “Men age faster than we girls. It makes up for them being bullies earlier. How to pep them up? I try. . . . People whose lives are fairly stable and satisfactory keep going a lot better. One of my colleagues gave up research at 42 when his marriage broke up. Another similar at 48.”
A male member from the sample emphasized collaboration as a source of long-term productivity. This 71-year-old analyst–applied mathematician in Rhode Island wrote, “Since I became emeritus in July 1995 my research has increased. Most of it is joint with former students and postdocs. The mathematical tools are ones I’ve used before. This is probably typical. It is a great relief to shed 9 years of department chairmanship, too many committees, and obligations to seek external grants. I no longer attend department meetings.”
Then there was the Illinois analyst, aged 69, who told Hersh: “Some of my best work was done after age 47. Possible motivations were a bad spell of drinking and divorce from 1974 to 1977, and prostate cancer treated successfully by radiation and implants. After such trauma I tend to over-accomplish.”
A Colorado applied mathematician thought, “Young guys may luck out but often only when someone older points the way,” and another respondent put it this way: “The young may find gold but cannot read the land; the older have familiarity with the landscape, which guides them to where to dig.” Best of all was the comment from a 70-year-old Ohio numerical analyst: “Recently a friend compared me with Brahms, who turned out great works throughout his life! I hope to live up to the praise.”
3. Symptoms of Aging, and Coping Strategies
In the responses to the survey there were many suggestions to address the challenges of aging. An applied mathematician of age 70 wrote, “As I age my memory declines, making it more difficult to keep in mind all the threads of a complicated situation. Also my computational abilities decline. I take longer to get through a routine calculation, and make more mistakes. I catch mistakes by my sense of what seems right, rather than by repeating computations. On the other hand, I’m more canny in developing effective research strategy, and more daring in carrying it out. . . . I have an intellectual home with a small but active worldwide community of scholars with similar interests.”
And a 74-year-old numerical analyst in California said, “I toy with retiring at the end of this year. I am nervous about it, but clearly recognize the diminishment of my ability to do first-rate research. The main cause is inability to stick with messy detailed manipulations. In the past I could calculate for hours, but now I shy away from such grunge. I still have plenty of things to work on, but I pick them more carefully.”
Another numerical analyst, who shares his time between California and Sweden, bluntly declared:
Getting old is a pain. I still do decent mathematics. However, what I do is very much related to my previous work. I do not jump into a new field, because I have not the same intuition as earlier to ‘know’ it will lead to something. Everything takes much longer to complete and I make more mistakes, or better, I do not know immediately when the result is wrong. So I have to check much more carefully. I have been a good thesis advisor, which I enjoyed very much. Former students still speak to me, and I still work with them. But I have no students anymore, because I cannot be sure I will be around in 4 years. Also, young people should work with young people on “modern” problems. There can be one advantage with old age. If one is lucky and in balance with oneself, one can look at the world as an independent observer.”
A male topologist, 76 years old, of British origin and now living in New York State, analyzed the situation carefully:
The principal obstacles to continuing research are: (a) Research requires energy, and this is in increasingly short supply. (b) Research requires keeping up with the literature, and this becomes difficult as one’s mental and physical energy declines. (c) Good research requires breadth and flexibility, but the tendency as one ages is to concentrate on a narrow path, dominated by what one has always done, and knows well. Collaboration is essential in maintaining research activity. I have tended to collaborate with juniors, since very many of my collaborators have been my students. The younger partner provides energy and awareness of what is currently a “hot topic”; the older provides perhaps greater familiarity with the history of the topic and a larger battery of available methods.
(Gian-Carlo Rota made a similar remark, “At my age, the collaborator is a necessity.”50)
And a 57-year-old logician in Indiana also had an explanation. “There are many useful things someone with mathematical ability can do. But education and research rewards do not encourage people to branch out and explore. They get stuck in the frontiers of their narrow specialty. The going gets rough when they no longer have the ability or willingness for the concentrated effort to do really complicated technical work. I am still able to do this if I get away for a couple of weeks, but at home commitments to family and work preclude that concentration. It does get harder as you get older, from aging but also from accumulation of other responsibilities and interests.”
The shortest and sweetest lines came from an applied mathematician in Rhode Island, aged 71, who described his coping strategy. “My wife and I have been happily married for forty-four years; that’s extremely important. Our garden takes a major part of our time in the growing season.”
4. Penalties for Independence
Several respondents reported penalties for following one’s own path. These comments suggest the need for some soul searching by mathematics departments! Thus, an applied mathematician in British Columbia reported, “My recent work is more interesting and valuable. Math community isn’t interested. Ecology community is.” And a 68-year-old number theorist in Minnesota said, “My best paper was never referred to in the later literature. I tell myself this shows it said the last word on its subject.” And a 57-year-old logician in Indiana admitted, “I did feel a loss when what I was doing was not valued by the mathematical community. It took a while for me to value it for its own sake.”
And a 72-year-old analyst in California commented, “The mathematical community lost interest in my work when fashions changed and I didn’t. After a period as chairman when I was 40, I lost influence in the department.” And still one more replied, “By following my bliss, I gave up my opportunity to get to full professor. My most valuable professional achievements are not appreciated by the leadership in my department.” And finally, in this vein, is the response “Some of my best research has been in recent years, yet I have been getting smaller pay raises and have less influence in my department The situation of some of my contemporaries is even more egregious. Mathematics departments and organizations don’t pay attention to the older members of the profession. My department treats our retirees shabbily: we give them a ‘gold watch’ when they retire, then forget about them.”
An applied mathematician in Alberta reported that he helps out with administrative chores. “The value of my research = quite high, the interest by others = quite low. The math community doesn’t pay attention to most mathematicians’ work. . . . I am called on a lot to do diplomatic or administrative jobs. . . . I am not a very able administrator, but compared to the great majority of mathematicians, I am an administrative genius.”
But here are a couple of comments on the plus side of the story. An 80-year-old very well-known measure theorist in California said, “I have been treated well. I still have my office, 10 years after retirement.” And similarly, a 66-year-old Swedish numerical analyst reported, “My department has treated me well. I still have an office and they pay me a small amount for looking after some graduate students. My research is worth more to me than to the department, so there is no strong reason why they should actively support it.”
5. Advice to Aging Mathematicians
Many respondents had strong advice for their fellow retirees. The great analyst-topologist Isadore (Izzy) Singer was quoted by one of the respondents: “Keep the pencil moving.” This theme resonated in other responses, such as that from the 68-year-old Swedish analyst who said, “Keep working, do not hide behind administrative duties.” And a 67-year-old logician wrote, “Work hard, and have several problems to work on.” And a 65-year-old geometer in New York advised: “Don’t stop. Once you do, it’s hard to get back. It’s not just the field that changes, but you change.” “First and foremost, you need a deep love of the subject” (Analyst, Alberta, age 60.)
There was also more practical and specific advice. “It is important to have an office,” said a Swedish numerical analyst, age 66. And an applied mathematician in Utah said, “Stay away from administration. It wears away your creativity, and is a real plague.” A 59-year-old probabilist in Illinois claimed that “Hardy advised people to do research in a prone position, so that more blood flows to the head.”
A 71-year-old harmonic analyst in Sweden, who had been one of Hersh’s professors when he took advanced calculus at the Courant Institute in 1957, wrote, “Maintain contact with younger colleagues and students. . . . Whenever anyone asks you a mathematical question, devote at least 15 minutes to it, even if it’s ‘not [your] field.’ . . . Try to maintain high ethical standards in this competitive profession. I like to think of mathematics as a collective enterprise. We contribute even by being attentive spectators and consumers of the constant outpouring of new ideas. (In the opposite view, a career in mathematics is an ego trip like downhill skiing, reserved for the youngest and strongest, where only those who break records matter.)”
Finally, we are reminded of the basics. “Always remember, research should be fun. If it becomes too competitive and loses its pleasure, give it up. Don’t take your research or yourself too seriously! I have been blessed with a good sense of humor, but how could one suggest this to others?” (Numerical analyst, British Columbia, age 54.)
“I do what I can do, and enjoy every minute of it. The mathematical community has as little awareness of me as I of them. Constantly learn new things! Do mathematics just for the fun of it!” (Number theorist, New York, age 77.) “Stop growing older. Keep having fun. And have a beer, on me!” (Analyst, California, age 83.)
Now we go to the last category of responses to our questionnaire.
6. Advice for the Mathematics Community
A 74-year old applied mathematician from California proposed a radical shake-up in the usual career path for mathematicians. “I have believed for a long time that a lifetime appointment to research in mathematics, with incidental teaching, is a mistake,” he wrote. “A person’s abilities, skills, and interests change. I’ve often talked about a career path involving research at an early age, say 25–35, writing and teaching at a non-research university thereafter, perhaps with involvement in pre-university mathematics such as teacher training or high school teaching. . . . Shorten the time to get a Ph.D. so that people can start research earlier, as in England. Shorten the undergraduate school time for talented people.”
A 77-year-old number theorist in New York had a very sharp criticism of universities. “For me, a big issue is why retirement is synonymous with severance from all academic activities. The University is the one organism that consciously believes there is nothing to learn from the past.”
“Encourage individuals to take chances and follow their true interests,” said an applied mathematician in Montana (age 70). “I don’t think anyone should tell us when to give up,” said a Californian, age 72. And “a good library, some stimulating colleagues and freedom from too many onerous chores” were all that a 60-year-old Alberta, Canada, analyst required.
Shabby treatment of aging professors is not special to mathematics. After retiring from the Columbia economics department, William Vickrey got a Nobel Prize for work on transportation economics. A New York Times reporter found him in a tiny office far from his department. Vickrey was grateful that after retirement Columbia had allowed him any office at all. Perhaps after being written up in the Times he would have been granted a better office, but, sadly and unexpectedly, he died a few days later.
A fuller account is Bollobás’ description of Littlewood’s retirement at Cambridge:
In 1950 at the statutory age of 65, he retired and became an Emeritus Professor. The Faculty Board realized that it would be madness to lose the services of the most eminent mathematician in England, so they wrote to the General Board: “Professor Littlewood is not only exceptionally eminent but is still at the height of his powers. The loss of his teaching would be irreparable and it is avoidable. Permission is requested to pay a fee of the order of 100 pounds for each term course of lectures.” The response: 15 pounds per term, the fee paid to an apprentice giving his first course as a try out to a class of 2 or 3. So he gave courses at 15 pounds for 4 years. He tried to stop once but there was a cry of distress. At the same time he was declining lucrative offers from the United States.”51
Conclusions
The responses varied. There are tremendous differences in how mathematicians age. Until we find a consensus about which advances are “major,” we can’t refute Hardy’s claim that no major advance has been made by a mathematician over 50. But his slogan, “Mathematics is a young man’s game,” is misleading, even harmful. So far as it may discourage people from mathematics when they’re no longer young, it’s unjustified and destructive.
Some of these answers are advice to aging mathematicians. Most important, the respondents say, “Don’t stop. Keep working.” Some of them found greater pleasure in their work after retirement, when they felt free to work on problems not considered important or prestigious or urgent by other people. Many of them were isolated from former colleagues, but some succeeded in developing new collaborations, especially with younger people. As expectations about age change, many mathematicians find new problems and interests past the arbitrary dividing line of retirement.
Some of the comments we quoted are important for department policy. Many of them make clear the importance of strong mathematical communities. When older or retired mathematicians are cut off from their departments, this amounts to making insiders into outsiders. It weakens the departmental community.
Older and retired mathematicians are an underutilized resource for the mathematics community. Departments can always use extra hands. Undergraduate advising is often understaffed. Has anybody asked, Are there emeriti who enjoy advising? If there’s no librarian on duty in the math library, is there an emeritus who would serve? There’s always too much committee work. Is there an emeritus with years of service on the undergraduate committee or the master’s exam committee? Might he or she have something to contribute there?
Although aging brings losses in memory and computing ability to most (but not all) mathematicians, these may be compensated for by a broader perspective and mature judgment. Older and retired mathematicians are certainly a valuable part of the mathematical community, and they should be recognized as such.
Individuals differ greatly in the ways in which they respond to the challenges of aging and retirement. But interestingly, there are differences between men and women in the trajectory of their productive lives. While many men see their thirties and forties as a particularly productive period and one in which they focus keenly on their work to the exclusion of other interests and responsibilities, women develop more slowly and hit their stride past their childbearing years. In both groups, aging is most successful for those who have established a balance among work, personal life, and broader interests. One of the difficulties of aging is the isolation from colleagues and former students. Lack of institutional support and outmoded departmental practices frequently contribute to such isolation. Men and women who are able to reach out to others, who continue to be valued by their community, and who partner with younger colleagues express the greatest satisfaction with their lives past retirement. This continued engagement of older mathematicians enriches the field and provides a sense of mathematical life in all its fullness to the younger members of the profession.
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