## Things Are Changing in the World of Mathematics

**June 11, 2008**

Sonia Kovalevsky, one of the women mathematicians profiled in the book under review. In 2003, the AWM and SIAM established the Sonia Kovalevsky Lecture, to be presented annually at the SIAM Annual Meeting. Dianne O’Leary will give the 2008 lecture in San Diego.

**Book ReviewPhilip J. Davis**

**Traumjob Mathematik! Berufswege von Frauen und Männern in der Mathematik (Dream Jobs in Mathematics! Employment Possibilities for Women and Men [in Germany]).** *By Andrea E. Abele, Helmut Neunzert, and Renata Tobies, Birkhäuser, Berlin, 2004, 192 pages, $39.95. *

Yes, things are changing in the mathematical world, and they always have been. Driven by the mindsets and social experiences of the groups that have been major players in the creation of mathematics and its applications, the very nature and stuff of mathematics have changed substantially over the centuries. History displays, among others, a classical Chinese mathematical culture, and Babylonian, Greek, Indian, Arabic cultures, each with its own concerns, methods, and outlooks. And I shouldn't forget to mention the mathematics of more primitive cultures, as detailed in books on ethnomathematics, such as one by Marcia Ascher.*

The Western world once went to sleep, mathematically speaking, for a thousand years. It could happen again---the theory of trans-infinite sub-retractive non-Riemannian partial structures might lie molding, neglected except as studied by Archeologists of Ideas. In the deep past, mathematics was not hung up on theorems and proofs. Early on, numbers and theorems shared the creative stage with numerology, mysticism, magic, and with noble but fruitless attempts to extract the future from the stars and extract gold from base materials---all with an assist from mathematics. Napier, of logarithmic fame, used numbers to predict the date of the Apocalypse. In 1964, deploring current tendencies, Carl Ludwig Siegel wrote to Louis J. Mordell (both were brilliant number theorists):

"I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction . . . cannot be blocked up."

Senseless abstraction may still be around, but what was really not anticipated years ago was the new and radical dimension lent to mathematics by the digital computer that has made many directions passé.

In the world of mathematics, things are also changing as regards social groups. Whereas Jews were very prominent as mathematicians in, say, the hundred years from 1870 to 1970, this seems to be the case rather less now. Brilliant students from Asia may soon dominate the field, and assertions parallel to the eight stereotypes listed below will probably emerge.

Things are also changing rapidly gender-wise. When I was a graduate student, one could count the number of women in the courses I took on the toes of a sparrow. Currently, of about fifty graduate students in the Division of Applied Mathematics at Brown, 30% are women.

I could have said "no," for ample reasons, when asked to review this book. In the first place, my German is not sufficiently strong to pick up nuances of meaning. In the second, my knowledge of German educational structures and of women's mathematical associations in Europe is weak. Lastly, dealing with feminist issues is for me like walking on eggs, and I fear that if I don't watch it, the Political Correctness Police may knock at my door. Nonetheless, nothing ventured, nothing gained. Besides, as Goethe wrote at the very end of Faust, "Das ewig-Weibliche zieht uns hinan," which I like to translate very loosely as: Women constantly engage our thoughts.

Inspired, then, by *Traumjob Mathematik*, my intention here is to free-associate a bit on its themes: employment and educational statistics culled from German archives from 1900 to the present; a gallery of distinguished women mathematicians (most of them little known in the USA); analyses of the conditions and attitudes that have led to the statistics; and, at the bottom line, upbeat encouragement for women to go into mathematics. "Studying mathematics pays off," the authors say. The dismal percentages seem to belie the statement, which is nonetheless made plausible by moments of optimism in the lives detailed in the book; the women portrayed have all worked creatively at the highest levels of scientific research and are now presented as role models.

Over the years, I've known and worked productively with women mathematicians on four continents and in many different types of mathematical activity: Irene Stegun, Ida Rhodes, Olga Taussky Todd, Rosemary Canning, Emilie Haynsworth, Kathleen Shannon, Christa Binder, Julie Gainsberg, Kay O'Halloran, to name just a few. I have interacted with these women entirely as individuals, admittedly paying little attention to the larger psychologic–economic–social issues involved. In the years when I had a government job, however, I used to say that the government was a much better place for women mathematicians than the universities.

I have known or known of five of the women whose professional lives are profiled in some detail in Traumjob: Hilde Geiringer, Emmy Noether, Grace Chisholm Young (whose granddaughter, Sylvia Wiegand, is a past president of the Association for Women in Mathematics), Irmgard-Flügge Lotz, and, of course, the widely known Sonia Kowalewski, who diluted her mathematical activities by writing novels and espousing political nihilism. I can think of a dozen English-language books that foster employment breakthroughs for women, covering more or less the same general ground as the book under review. All you have to do is to visit the Web site of the AWM and you will be overwhelmed with books, databases, talks, meetings, newspaper articles, employment tips, etc.

At the beginning of *Traumjob*, the authors list eight prejudices [*Vorurteile*], or stereotypical views, about the position of women in mathematics, all of which, they grant, contain some truth. A search of the feminist literature would surely reveal several dozen more such generalizations, both true and false.

Later in the book, the authors give their list a second reading. At the risk of tedium, I cite here the list (in my own translations, which are not word for word) and comment briefly on each one.

*1. Mathematicians--whether men or wom-en--are unworldly. They derive their enjoyment from their work and not from social relationships. And therefore mathematics is "against the nature of women."*I would agree mildly with this, while pointing out that the "nerdish" mathematical characters, of which there are plenty, get far more publicity in books and in the media than we dull "normals" do.

2.

*Women are not interested in mathematics.*

With some few exceptions, I believe this to be correct.

*3. Women are less productive mathematically than men.*

If by "women" one means women mathematicians and if by "productive" one considers the whole range of professional activities, I would disagree.

*4. Even if a woman's productivity and interest happen to equal those of a man, women lack confidence in what they have produced and need bolstering.*

I have no opinion on this one.

*5. Thematically, women are less flexible than men.*

Restricting attention again to women mathematicians, I would agree mildly.

*6. Women who are interested in mathematics for the most part select mathematical education and reject other possibilities.*

By and large, I think this is true, although it's possible that academic opportunities are simply not there. My European mathematical friends write me that the public does not like the universities, while university mathematicians do not like teaching---preferring research in certain favored fields, such as scientific computing. Funds are not forthcoming, and there are fewer academic positions either for men or for women. Compounding the difficulty, in many universities, and despite "gender awareness" guidelines, male networks still work against women. Positions in industry and other work are more plentiful.

*7. Women interest themselves less in applied mathematics or the scientific applications of mathematics.*

This also is true, by and large.

*8. Women mathematicians have a harder job reaching the higher positions in the field than men. *True.

Notice that these eight points do not face head on the assertion that not so long ago toppled Harvard president Larry Summers: On average, women are not "hardwired" for science and mathematics. In paraphrasing Summers, I use "being hardwired" as a picturesque substitute for "having the inborn ability." My reactions to the points made by the Traumjob authors are based on what I have observed over the years, bolstered by a recent communication from a friend who is a neurological anatomist. He wrote me that:

"(a) Men's and women's brains are different, and the differences seem to begin at the end of about 2 months of development in the womb.

(b) Various psychological studies show inborn differences between the ways that men and women approach problem solving.

(c) At the prodigy level, ‘prodigy-ness' [for both men and women] seems to be inborn."

My friend added that "these questions will be argued beyond our lifetimes."

Yet, despite all the obstacles, more and more women are taking up mathematics. Now outward changes, of whatever sort, affect what is considered to be mathematics. In her provocative book *Pythagoras' Trousers*,† radical-feminist Margaret Wertheim, trained in mathematics, blasted the "oppressive patriarchy" that has created contemporary science and wondered what qualities a science and mathematics practiced equally by men and women (or perhaps even dominated by women) might have. Wertheim conjectured

"that if more women were in mathematics and science (particularly in physics), then they would create an environment in which one could pursue the quest for mathematical relationships in the world around us, but within a more human ethos. . . .

The issue is not that physics is done by men, but rather the kind of men who have tended to dominate it. . . ."

If there were more women in the business, would there be less subjugation of the soul by the tyranny of numbers? Would the loss of independence from the strictures of computer logic be abated? Would fewer applications of mathematics be deleterious to society? Hmmm. Or is it the case that mathematics by itself, and of its own nature, is "hardwired" to have a serious down side when humans---who created it---put it to work?

Returning to the intriguing title of the book under review, one might ask: What are the Dream Jobs (insofar as such things exist) that professionals often chase with the madness of Captain Ahab pursuing Moby Dick, the White Whale? I should say that first of all a Dream Job provides one with a decent standard of living. ("Grub first, then ethics": Bertolt Brecht,‡ or "then aesthetics": PJD.) Secondly, a Dream Job is a job that one really enjoys, despite the inevitable bumps in the road. (A recent Gallup poll found that about 77% of Americans hate their jobs.) Thirdly, a Dream Job should offer freedom to shift intellectual gears as one grows in experience. There's no doubt that luck also enters the picture. Speaking of myself, I would say that I've been lucky and have had a Dream Job for years.

The very last sentence in *Traumjob* asserts that Dream Jobs in mathematics are possible for women. The Supremes agree: Check out the lyrics of the song "Things Are Changing." Who could disagree?

*Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at* [email protected].

*See "Is Mathematics a Unified Whole?" (SIAM News, March 2003; http://www.siam.org/news/news.php?id=301).

†See "Deus Ex Hominibus: Science, Theology, and Masculinity," *SIAM News*, March 1998; http://www.siam.org/news/news.php?id=816.

‡ "Erst das Fressen, dann die Moral."

**Call for Nominations: The Association for Women in Mathematics invites nominations for the following prizes:**

The 2009 Kovalevsky Prize Lecture. This lecture, established jointly with SIAM in 2003, is given annually at the SIAM Annual Meeting. Nominations must be received by November 1, 2008.

*The Alice T. Schafer Mathematics Prize.* This prize honors an undergraduate woman for excellence in mathematics. Nominations must be received by October 1, 2008.

*The 2010 Emmy Noether Lecture.* This lecture is given annually at the Joint Mathematics Meetings (Fan Chung has been named the 2009 lecturer). Nominations must be received by October 15, 2008.

Inquiries about the nomination process for each prize can be sent to [email protected].