"It's a wonderful life!"
Reflections on the career as a mathematician
Is life really so wonderful? Off and on, we all have our doubts about it, including me. Just when I started to prepare my talk, I had caught someone cheating in a midterm exam. Though I had iron-clad proof of this, nevertheless all the hassle that goes with it brought about these thoughts.
How did I come up with my title? There is a 1946 movie with that title. George Bailey, the character played by Jimmy Stewart, reflects on his life in a crisis situation. There was one scene which captured my imagination: the Guardian Angel shows George Bailey how the world would have been without him. There are flashbacks starting with his childhood up to the current point in his life. Looking back on it, he realizes that life even with all its dark sides is wonderful in the end.
Personally, I never had much need to know how the world would have looked without me. However, all other things equal, how would life have been if I had lived in a different time and place, would be something of interest to me. This is the stuff of fairy tales. But at least it is possible to play this as an intellectual game, looking at actual and potential role models over the centuries. In its essence, it got me back to the roots of what shaped my life.
I was born and raised in Germany before WWII. After getting my Ph.D. in 1962, I married a fellow mathematician and we immigrated to the US in 1963, where we have been teaching at a university ever since, first at Ohio State, since 1968 at Binghamton. The first notable woman in mathematics was the legendary Hypatia, who lived from about AD 370 to 415 in Alexandria Egypt. "Hypatia" was the code name I once chose as a student in a written exam way back in Germany. That was how close I ever came to having her as a role model. Not much is known about her, but more than you might think. She was the daughter of Theon, also a philosopher at the Academia in Alexandria, a forerunner of a modern university, which had seen Euclid several hundred years earlier as one of its members. Her works, now lost, included commentaries on Diophantus, Appollonius and Ptolemy. She had many pupils, and her letters to one of them, Synesius, a later bishop, have come upon us. From all what we know about her and her contemporaries, she was a scientist, who happened to be a woman. Ironically, being a scientist contributed to her untimely death. Christians at the time equated science with paganism. A new bishop in Alexandria started to persecute the members of the Academy. In 415 AD, a mob instigated . by the bishop barbarously murdered Hypatia. Her death marked the beginning of the end of the Academy in Alexandria.
Almost fourteen hundred years pass until we find again female mathematicians. At least I could not dig up any names. Perhaps some nun in the middle ages pursued her math behind the walls of a monastery. My sister used to say, if we would have lived in those times we would have been accused as witches and burned at the stakes. Why this long gap? Some names come up in the 18th century like Maria Gaetana Agnesi (1718-1799), a child prodigy, who, in 1750 at the University of Bologna, became the first woman ever to occupy a chair of mathematics. Let's look instead at Madam Dorothea Schlozer, Ph.D, a name you may have never heard before.
Dorothea was born in Gottingen in 1770 and died in Avignon, Southern France, in 1825, a contemporary of Beethoven. Her Ph.D. is in philosophy, not in mathematics, and the result of an experiment, as strange as it seems. Her story sheds light on why there were no successors to Hypatia for such a long time. It was the age of enlightenment, starting up many debates in academia about things which had been taken for granted for centuries. One of them was the belief that women's brains were not capable of learning to the extent necessary to become a scholar, a scientist, to eventually get a Ph.D. One proponent of the theory that women are entirely capable of doing this was August Ludwig Schlozer, a professor in Gottingen and the father of Dorothea. He was tired of these academic debates and he proposed an experiment: have a female go through the rigors of academic training and see what happens. The guinea pig was his daughter Dorothea, and as you saw from what I mentioned earlier, the experiment succeeded. There was a small glitch at the end: Any candidate passing successfully the exam had to take an oath. But at that time in Germany, females were disqualified to take an oath. A way out was found. The next day Dorothea paid a visit to the Rector ( =president of the university) confirming by a hand shake that she would adhere to whatever male candidates promised in their oath. Did she live happily ever after? Not quite. Dorothea was just 17 when she got her degree. What would a male Ph.D. have done at the time? Become a private teacher for some prince or the like and work his way into a university as a faculty member. The experiment was over. A career was out of the question. At the age of 22 she got married off. Her husband, Matthaus Rodde, was a widower with three children, 15 years her senior. He was from a patrician family, supposedly rich. Not being as rich as he claimed turned out to be a problem in their marriage later on. Overall, Dorothea led an ordinary, somewhat unhappy life, perhaps typical for a female at the end of the 18th century. In our next episode we enter the 19th century and we meet a mother and a daughter, who, in their own eyes and those of their contemporaries, were mathematicians. The social setting in which it takes place is completely different: ladies of nobility. As customary, many in those circles pursued music and some of them grew beyond the status of amateur and made a name for themselves. Just to mention Frederic the Great for one and his sister Amalie. These people had private teachers and the very best. Making a living was never a question. Just in the same way as some of their peers became eminent musicians, these two ladies became mathematicians, being taught by the best teachers of their time.
But-Oh! ye lords of ladies intellectual Inform us truly, have they not hen-pecked you all?
Byron: Don Juan I, XXII.
These lines were inspired by Byron's wife, Lady Byron, nee Annabella Milbanke (1792- 1860). They were married in early 1815, soon afterwards they were estranged and separated in 1816. In her own eyes and that of her husband, she was a mathematician, the favorite pupil of William Frend. On December 10, 1815, Lady Byron gave birth to a daughter, Ada Augusta, who, at the age of 19, married Lord King, the Earl of Lovelace, eleven years her senior. She had two sons and a daughter, Annabella, who also pursued mathematics.
Byron's daughter died of cancer at the age of 36. Ada studied mathematics under the guidance of Augustus De Morgan, the first Professor of Mathematics at the University College in London. Here is what Mrs. De Morgan has to say about her:
... her mathematical studies ... were carried farther than her mother's had been. I well remember accompanying her to see Mr. Babbage's wonderful analytic engine. While other visitors gazed at the working of the beautiful instrument with the sort of expression, and I dare say the sort of feeling that some savages are said to have shown on first seeing a looking-glass or hearing a gun - if indeed, they had as strong an idea of its marvelousness - Miss Byron, young as she was, understood its working, and saw the great beauty of the invention."
Charles Babbage's Analytical Engine, a forerunner of the modern computer, was at the center of her investigations and she has a publication on the topic to her credit which appeared 1846 in Taylor's Scientific Memoirs, 3, 666-731.
There was an extensive correspondence between De Morgan and Lady Lovelace, up to 50 letters have survived. After the death of Lady Lovelace, De Morgan writes to her mother:
"But I feel bound to tell you that the power of thinking on these matters which Lady L. has always shown from the beginning of my correspondence with her, has been something ... utterly out of the common way for a beginner, man or woman .... Had any young beginner, about to go to Cambridge, shown the same power, I should have prophesized first that his aptitude at grasping the strong points and the real difficulties of first principles would have very much lowered his chance of being a senior wrangler; secondly that they would have certainly made him an original mathematical investigator, perhaps of first rate eminence.
As long as I can remember, I had three goals in life: become a scientist ( originally a physicist, since mathematics was so boring in school), have a career and have a family. In this pursuit, I had very few role models. It was not exactly easy to reach those goals, but all this pales compared to the obstacles Sonya Kovalevskaya had to overcome. Her persistence, her tenacity to break out of her cage, the sacrifices she had to make are unbelievable. Sofiya Vasilevna Kryukovskaya was born in January 1850, the daughter of a high ranking Russian general. Her family was well-to-do and she and her older sister Anyuta received the best education two girls at the time could expect and more, perhaps a little bit short on science and math. But she got books on these subjects from her father's library and she studied it on her own. A friend of the family, a physics professor at one of the military academies, gave her the text book he had written. All this generated a burning desire in Sonya to study at a university and become a scientist and mathematician, an impossible goal in 19-th century Russia.
Sonya's life is well documented, in particular through letters to her sister in the early years. (I always wonder, what people will do a hundred years from now who want to give a talk about us. Phone calls and e-mail messages leave not many records.) Anyone interested in more what I can say here should read a biography about her. At the end I will give a short list of some of the biographies I consulted.
Here I will report on one episode in her life: the breaking-out of the cage. In 19-th century Russia, women were not allowed to study at universities. But this was possible in Germany. So somehow Sonya had to find a way to go to Germany. It was not simply a matter of getting a passport and a train ticket. At that time, daughters were the wardens of their parents, and after marriage, of their husbands. Sonya's parents did not permit her to go abroad. The only way out was a fictitious marriage. As strange as it may seem, such things were not unheard of in Russia at the time. Three years before Sonya made her plans, her sister and her friend Mariya schemed to find a fictitious husband for Mariya. Through intermediaries they found Vladimir Kovalevsky, but two hours before the marriage was to take place, the deal fell apart. Three years later, when Sonya looked for a husband, Vladimir and Sonya agreed to a fictitious marriage. Her parents at first do not give their permission. Vladimir had a brisk publishing business at the time but he also was a revolutionary. But that was not what the general and his wife had envisioned for their daughter. The true reasons were unknown to them. So Sonya faked eloping with her fiance and wrote from his place that she never would return home. The parents gave in and the two got married in September 1868 and moved to St. Petersburg, where Sonya, escorted by her husband and a cousin, attended lectures in physiology by a famous professor who had given his permission. In April 1869 the couple leaves for Germany. Vladimir studies in Jena, and gets his Ph.D. in biology there in 1872, returning to Russia afterwards. Sonya studies in Heidelberg, Berlin and Gottingen, where she gets her Ph.D. in mathematics in 1874.
During those years their marriage is fictitious. Sonya in her letters to her sister calls Vladimir "brother". After her return to Russia the pair drew closer to each other and in 1878 their daughter was born. But all those years their marriage was troubled by financial problems. Money problems and an impending fraud trial drove Vladimir to suicide. This sounds like a movie script and I don't know if her life story ever made it into a movie, but it certainly would be worth it
This brings me to the generation of my grandmother. What if I would have lived during her time? I think of two people I am familiar with: My grandmother and Emmy Noether, a towering legend among mathematicians, both born in 1882.
My grandmother, Erna Thie nee Sandow, born into a family of musicians in Berlin, was a budding concert pianist when an injury to one of her fingers forced her to switch careers. She became an elementary school teacher, the only career open to girls at that time. I am sure, nowadays she would have ended up somewhere in academia. As every female teacher of her generation, her contract contained a clause, saying that she would have to quit her job if she got married. That was what she had to do in 1909 when she married my grandfather. She did not give in. She founded her own private elementary school and only shortly before WWII she was forced to close it. At the end her school had become the haven for many Jewish children who were forced out of the public schools. When I was growing up, I saw nothing out of the ordinary in the fact that my grandmother had a career. I thought that was what women did. Only later I realized how unusual she was for her time.
Emmy Noether, one of the great mathematicians of this century, had a different career. She prevailed in getting a university education and obtained a Ph.D. at the University of Erlangen in 1907, a place where I was a student 50 years later. Though not smooth, this was not impossible at the time. The main obstacle was getting her "Habilitation" , the permission to become a "Privat Dozent", a university lecturer. Emmy Noether, now in Gottingen, was told that her Habilitation was impossible because of "unmet legal requirements", meaning Habilitation could only be granted to males. Hilbert tried to get an exception from the government, but to no avail, after all, this was a university and not a bathing establishment. In 1933 Emmy Noether was forced to emigrate to the US. Many of her fellow colleagues in the same situation, but of lesser stature, ended up in Princeton and places like that. Emmy Noether ended up at Bryn Mewr, for the great benefit of some of the students there. Unfortunately, Emmy Noether died two years later prematurely, so we don't know if she would have triumphed at the end.
Briefly, two role models out of the generation of my mother: Ruth Moufang and Hanna Neumann, both mathematicians. Ruth Moufang was not permitted to get her Habilitation during NS-times but could do so after WWII. I met her in the fifties in Frankfurt. The first time I saw her lecturing when I visited Frankfurt, escaping Erlangen for a short time because I just had split up with a boyfriend. Seeing her gave me the confidence that succeeding in academia was not impossible.
Later on when I met Hanna Neumann, I got reassured that success in academia and raising a family by a couple who are both mathematicians is not impossible. Hanna and her husband Bernhard were one of the first pioneers in this respect.
Now to my generation. Born and raised in Germany in a very dark and austere time made you appreciate what you had. It created a hunger for learning. Nevertheless, my childhood memories are brightly lit scenes. The end of WWII finds me in a one-room school house in a small village. You learned your lessons and those of the older students too. I became a voracious reader, in particular old school texts of my uncle. I dreamed of visiting Yellowstone Park. I became a teachaholic: helping in the nursery school, teaching my playmates and siblings. But there was also the afternoons in the fields, making hay, picking apples, harvesting potatoes.
Finally, after WWII, I could enter the gymnasium with a two-year delay. I remember fondly the "Kohlenferien" , weeks off because schools could not be heated. Once a week we came to school in our coats and got lots of homework. Learning was with high intensity: Prime numbers! Fascinating! My father, out of work, had time for his family. We made chlorine gas out on the porch. Chemistry was my first love.
I finished the gymnasium in 1955 and got my Abitur in Diisseldorf. Looking back to that time, it was essential that it was an all-girls high school. Here a quote from the first female four-star general in the US army: "Limitations are not part of our vocabulary". This was our attitude. In my school, our intellectual development was not hampered by the social aspects dominating coed schools in the US. I was ready when entering the university: there were no barriers, only what you did counted, not who you were.
How would have things been if I would have been raised in the US? Yes, I would have made it with my "no limitations" attitude. A recent Ph.D. student of mine went to an all-girls high school. Her "no limitations" attitude shows. So does it for my son's wife who first went to Bryn Mawr, then got a Ph.D. in Physical Chemistry from the University of Chicago. Another case in point is a colleague of mine in biology who combines academia and family. She went to Smith College.
Would I have continued in academia in Germany after the Ph.D.? Yes! But being married to a fellow mathematician? I doubt it. This was unheard of 50 years ago and appears to have not changed now. Recently I read in a German publication "Mathe - nichts fiir Madchen?" (Math is not for girls?) that there was an increase of female Ph.D.'s in the fifties and sixties but that this was not followed by an increase of positions in academia held by females. They say they don't know why. I can tell them. Here is one of them! For that reason, my husband and I immigrated to the US in 1963. First, we were at Ohio State and then at Binghamton since 1968. In this country, raising a family and having a career for the parents is not impossible, but not a cake-walk either. We made it. We raised two sons and each of us had a career. We were lucky.
Combining academia with raising a family is hard for males too. But it is harder for women: the tenure clock ticks, the biological clock ticks. Equality is on paper. Men don't have to bear children! Formerly they hired the husband and they took the wife in stride. Now often things are reversed. But that is not right either. Should hiring be done blindfolded, as if not married? That is also not right. So far there is no patent solution. But I have it better than an experimental scientist. I am not tied to the lab. I can do math while doing my housework. I love to do my dishes, I love to do my laundry, because I can do my math with it. I hate to do house cleaning, because I can't do my math with it. I hired a cleaning lady. My advice: get all the help you can. You need it and can afford it.
Whatever you do, there will be some years your career will be dormant. You are on automatic pilot. I got back to it about 30 years ago when my sons were in their last years in high school. My research output accelerated. At that time I started out the best thing I ever did: working with Ph.D. students. Fourteen have finished. It is something which gives me tremendous satisfaction. You can make a difference is some one's life! You cannot ask for more in life! It is indeed a wonderful life!
 Auguste Dick "Emmy Noether, 1882-1935", Birckhauser Verlag Basel, 1981.
 Barbel Kern, Horst Kern, "Madame Doctorin Schlozer", C.H. Beck Verlag, Miinchen, 1990.
 Pelageya Kochina, "Love and Mathematics: Sofya Kovalevskaya", Mir Publishers, Moscow, 1985.
 B.H. Neumann, "Byron's daughter", The Mathematical Gazette, 62 (1973), 94-97.
 Encyclopedia Britannica.