#### by Claudia Henrion

Mary Ellen Rudin’s1 life presents a fascinating example of how radically expectations can change in the course of one generation. Her story illustrates some of the dramatic shifts that have taken place in women’s vision of themselves, their work, and their families.2 Mary Ellen is a kind, generous person and a very talented mathematician. She married a mathematician, had four children, and was actively involved in mathematical research. Though she is now a full professor at the University of Wisconsin, for most of her working life she did not have a standard professorship at a research institution. She describes herself as an “amateur mathematician” — a label that is perhaps misleading, for it does not convey the fact that she has published more than eighty articles in set-theoretic topology, and is internationally recognized as a first-class mathematician. In fact, her reputation in the mathematics community became so strong that the University of Wisconsin was sufficiently embarrassed by her status as a lecturer that they suddenly promoted her from lecturer to full professor in 1971. She was forty-seven years old.

At the University of Wisconsin, Mary Ellen became a kind of mathematical mother, helping to establish a strong and flourishing logic and set theory community. People from all over the world would travel there in order to be part of the stimulating, exciting, and supportive environment. She continues to be productive in publishing articles in new and difficult areas of set-theoretic topology. However, for Mary Ellen, being a mother and wife is just as important as being a mathematician. As she says, she came from the “housewives’ generation.” There was no question in her mind that she would have children, and that she would follow her husband wherever his career would take them.

Indeed, Mary Ellen feels that her situation was in many ways ideal — she had all the advantages of a mathematical life: a professional community, the option to teach when she wanted to, the stimulation of graduate students, the resources of a first-rate library, and an office. Yet she was not weighed down with administrative responsibilities and time-consuming meetings. The informality of her position allowed her to shape her work around her family. She could adjust to the ever-changing needs of her children and devote whatever free time she had to research. “I didn’t have to prove to anybody that I was a mathematician, and I didn’t have to do all the grungy things that you have to do in order to have a career as a mathematician. The pressure was entirely from within. I did lots of mathematics, but I did it because I wanted to do it and enjoyed doing it, not because it would further my career.”3

Mary Ellen was able to integrate these different roles of her life smoothly, which enabled her to continue to be productive as a research mathematician when most other women of her generation did not. But as she points out in an interview published in More Mathematical People, the situation for young women today is quite different: they see themselves as professionals.

If you do something professionally, it’s harder for you to quit. You stay with it in hard times. The young women today are much more professional. We were amateurs…. We did what we did because we loved it. And some of us were lucky. For us, things worked out, and we did very well. Some were not so lucky, and they just dropped out along the way. But the young women mathematicians today are thinking in terms of a career from the beginning. It’s true that they want a full-time job. They want to do all the research in the world. They want to have a husband and children. They want to have a home. We wanted everything too — in spades — but the one thing we didn’t demand, in fact it never occurred to us, was a career. The fact that they are thinking in terms of a career means that when it’s a question of washing the socks or doing mathematics, they will often do mathematics. I think it was easier to quit doing mathematics in our day.4

Given Mary Ellen’s vision of herself as a mother and housewife, how did she find her way into mathematics? And what kept her going?

#### Early training

Mary Ellen Rudin had a rather unusual childhood and early training in mathematics. In both cases she was quite isolated from mainstream culture. She grew up in a little town in southwest Texas. It was a very isolated community, at an altitude of about 3,000 feet, tucked in a canyon formed by the Frio River and mountains on all sides.

In those days you entered the town by going fifty miles up a dirt road — you had to ford the river seven times to get there…. My father was there to build a new road, but the Depression hit and the State Highway Department never completed the road while we lived there…. It was a real mountain community. Many kids came to school on horseback. We had a well and an electric pump, which gave us running water, but most people didn’t have running water or any of the things you think of as being perfectly standard. Yet it was wonderful. There were miles of wild country and beautiful trees along the river. Everywhere there was a beautiful view.

We had a very good school, and there were some very bright kids. We also had a lot of time to develop games. We had few toys. There was no movie house in town. We listened to things on the radio. That was our only contact with the outside world. But our games were very elaborate and purely in the imagination. I think actually that that is something that contributes to making a mathematician — having some time to think and being in the habit of imagining all sorts of complicated things.5

Growing up, she was not interested in mathematics (or any one
particular thing) more than anything else; she certainly had no idea
that she would become a mathematician. When she began college at the
University of Texas, Austin, therefore, she simply followed her
parents’ guidance and decided to pursue a general education; in her
first term she took a broad range of courses. However, as fate would
have it, on her first day at registration she wandered over to the
mathematics table, where there were very few people, and began talking
with “an old white-haired gentleman” who asked her a lot of
questions involving the use of terms such as *if* and
*then*, *and* and *or*, to see if she used them
correctly from a mathematical point of view. As it turned out, the man
she spoke with was
R. L. Moore,
a mathematician
famous for his unconventional teaching style, and for the number of
mathematicians he produced. That conversation was to shape the course
of her life. When she walked into her calculus class the next day, it
was Moore who was teaching it. She went on to take many more courses
with him, and she ultimately majored in mathematics. In fact, Moore
was the only mathematics professor she studied with until her senior
year. However, even then she still had no idea what she would do with
her mathematics degree — being a high school teacher was the only
thing she could imagine, and it was not what she wanted to do. But
Moore had other plans in mind. She was invited to stay at the
University of Texas for graduate school. She accepted the offer, and
R. L. Moore became her graduate advisor.

I’m a mathematician because Moore caught me and demanded that I become a mathematician. He schooled me and pushed me at just the right rate. He always looked for people who had not been influenced by other mathematical experiences, and he caught me before I had been subjected to influence of any kind. I was pure, unadulterated. He almost never got anybody like that.6

Moore’s methods were unusual and controversial. He had a very competitive and adversarial style, which was simultaneously meant to weed out the weaker members and boost the confidence of the survivors.

I was always conscious of being maneuvered by him. I hated being maneuvered. But part of his technique of teaching was to build your ability to withstand pressure from outside — pressure to give up mathematical research, pressure to change mathematical fields, pressure to achieve non-mathematical goals. So he maneuvered you in order to build your ego. He built your confidence that you could do anything. No matter what mathematical problem you were faced with, you could do it. I have that total confidence to this day.7

In addition to developing their confidence, Moore taught his students to be highly self-sufficient. In class, he would simply put definitions on the board and give students potential theorems to work on. Some of the theorems were true, others false, but the students had to discover which were which on their own. They were discouraged from reading journals or consulting other people; they were expected to work completely independently. In fact, Moore had his own unique mathematical language, which made it even more difficult for his students to learn from outside sources. He wanted them to focus on their own ideas, their own insights, their own intuitions — in a sense creating mathematics from scratch. At the time Mary Ellen wrote her thesis, she had never seen a single mathematics paper.

Though the kind of independence they developed was a significant asset, there were certainly problems with his methods as well. Many students were turned off by the extreme degree of competition. Others grew frustrated by the fact that their mathematical training made them so isolated from what was going on in the field more generally. In fact, Mary Ellen does not use his methods in her own classes. When asked how she felt later about her mathematical education, she said:

I really resented it, I admit. I felt cheated because, although I had a Ph.D., I had never really been to graduate school. I hadn’t learned any of the things that people ordinarily learn when they go to graduate school. I didn’t know any algebra, literally none. I didn’t know any topology. I didn’t know any analysis — I didn’t even know what an analytic function was. I had my confidence built, and my confidence was plenty strong. But when my students get their Ph.D.’s, they know everything I can get them to learn about what’s been done. Of course, they’re not always as confident as I was.

Although Moore’s primary goal was to instill confidence and independence in his students, what developed in the process was a very strong community among his students, which proved to be tremendously helpful in their careers and mathematical development. “We have all been very close to each other for our entire careers. That is, we were a team. We were a team against Moore and we were a team against each other, but at the same time we were a team for each other. It was a very close family type of relationship.”8

This mathematical family played an important role in practical ways: helping her to get jobs, grants, and recognition. Indeed, when Mary Ellen finished graduate school, she never applied for a job; she was simply informed by Moore that she had a position at Duke, and it was assumed that she would take it. “Moore simply told me that I’d be going to Duke the next year. He and J. M. Thomas, who was a professor at Duke, had been on a train trip together. Duke had a women’s college which was sort of pressuring them to hire a woman mathematician. So Moore told Thomas, ‘I’ve got the very best, and I’ll ship her to you next September.’ ”9 He did, and Mary Ellen went.

In a similar vein, near the end of her position at Duke, one of her fellow Moore students, R. L. Wilder, arranged for her to go to the University of Michigan and even applied for a grant for her. He was impressed with her mathematical ability and was hoping to work with her there. However, Mary Ellen decided at the last moment to do something different, namely to marry Walter Rudin, who was then at the University of Rochester. In accord with the traditions of the time, Mary Ellen never questioned that she would follow her husband. Wilder, therefore, simply arranged for her grant to be transferred to the University of Rochester.

#### Marriage, motherhood, and mathematics

Over the course of the next decade, Mary Ellen Rudin had four children. Two of them came in the first two years after she got married. “I was very busy and very happy as a mother, and certainly expected to be a mother, and wanted to be a mother very much.” It would not have been unusual to stop doing mathematics at that point — raising four children was a full-time occupation, and there was no incentive to have a career outside of the home. What, then, enabled her to continue on this path, even when many other even very gifted women did not?

To some extent her early mathematical training contributed to this commitment, not only by helping her to develop a highly autonomous work style and providing a community which propelled her along a mathematical path, but also through the underlying messages conveyed by R. L. Moore about what constitutes success and failure.

Moore had warned Mary Ellen about his earlier women students. One wrote a “fantastic thesis” but then immediately went off to China as a missionary. The other was “very influential as a teacher and administrator,” but did not continue in research. Mary Ellen says that Moore “viewed his two earlier women students as failures and he didn’t hesitate to tell me about them in great detail so I would realize that he didn’t want to have another failure with a woman.” In the context of other conversations he would subtly work on her — steering her to a life of mathematics. On one occasion when she mentioned something about having a husband someday, he responded, “Husband! But, Miss Estill, I thought that you were going to be a mathematician.” But as Mary Ellen says, “Although he may have had his doubts, I never saw any contradiction in being both a housewife and a mathematician — of the two I was more driven to be a housewife.”10

But other factors also played an important role in her continued commitment to mathematics. Like Fan Chung, Mary Ellen had significant help in raising her children. The Rudins’ nanny, Lila Hilgendorf, was in many ways a second mother to the children:

She cared for the children and did some housecleaning. When I would walk into the house, she would walk out; and when I had to go, she was there. We had that relationship until she died this year. She was absolutely the best mother I ever saw, and my children just adored her. One thing that she did for me was absolutely fantastic. The first weekend after my retarded child was born, she said, “Oh, I just have to have him for the weekend!” And he went to her house for the weekend for the rest of her life. Actually he lived at her house the last several years. So when people ask me how it is, in my position to have four children, I have to say that when you’ve got Lila, it’s easy. I am afraid that few women will ever have such an easy, non-pressured career as mine.11

Of course, central to this arrangement was the fact that they had the economic resources to hire such a nanny. They were able to live comfortably on Walter’s salary, even with domestic help. And Mary Ellen did not need to work to make ends meet; hence she could devote her free time to research.

But equally important was the fact that both Mary Ellen and her husband felt comfortable hiring domestic help. As Mary Ellen points out. “I think there are a fair number of women in my day who had the potential to be mathematicians, but who didn’t become mathematicians because it didn’t seem right to have someone help you with your children.” Caring for the children was, after all, seen as a wife’s primary responsibility. And even if a woman felt comfortable with such arrangements, often her husband did not. Walter, however, both understood and supported Mary Ellen’s desire to do research. Hiring help seemed like a natural solution.

In Ravenna Helson’s study of creative women mathematicians, there were very few things that the different women she interviewed had in common. One unexpected finding, however, was that most of the women she studied were married to Central European Jews. Many of these men, like Walter, had had nannies when they were young. Hence domestic help seemed normal; they recognized that “the person who changes your diapers doesn’t necessarily determine what you do in life. It doesn’t determine one’s values.” Furthermore, they came from a cultural milieu that celebrated intellectual wives — having a wife who was a mathematician was therefore seen as an asset, not a problem.

Mary Ellen and Walter’s marriage was a significant factor in enabling her to stay actively engaged in mathematics in other ways as well. Because Walter is a highly respected mathematician, it has been fairly easy for Mary Ellen to integrate into the mathematical community. Both the University of Rochester and the University of Wisconsin were glad to have her teach; she could use libraries and other resources, and she was given an office and invited to seminars. Certainly Mary Ellen’s talent and her ties to the Moore family helped, but if she had had no direct connection to these universities, it is unlikely that they would have given her such a warm reception. This kind of community support has been quite important in sustaining her mathematically.

The whole community support — family and other mathematicians’ support — are needed in order to be a mathematician. You may be able to get along without some of it, up to a point, but it certainly makes it more probable that you will be an effective mathematician if you have it. It allows you to focus on your mathematics problems. If you look at the mathematicians from the nineteenth century, like Grace Chisolm Young, Emmy Noether, Sofia Kovalevskaia, they were people who had to completely upset the community around them. They were strong enough to do it. But I’m sure there were infinitely many more potential mathematicians who would have been glad to pursue mathematics if they had the opportunity to go to school, and social acceptance for what they were doing. It was hard; I think none of them were really happy women in some sense. Each of them had a certain kind of support, which perhaps was all they cared about. But their community of support was in the middle of a hostile environment. It is helpful if the environment isn’t hostile.

Being married to a mathematician had other benefits as well. The Rudins could travel around the world together to attend conferences, or give talks, and these trips were invariably both mathematically stimulating and exciting vacations. It was not unusual for one of them to give a lecture in the morning, and then be out on the beach in the afternoon. The walls of their house are covered with pictures of the their trips to places such as Hawaii, New Zealand, and China. They think of their colleagues, friends, and students as their mathematical family — one that extends across the globe.

Although being married to Walter helped create the kind of supportive environment that Mary Ellen points out is quite helpful, she and Walter do not actually do mathematics together, or even talk with each other about their results. They are in different fields, and have different work styles. Nor do they feel the need to share that part of themselves with each other. “Everybody should have something [of their own]. I have plenty of things to share with people. I don’t have to share all of my mathematics with them, too.” In addition, she and Walter have very different ways of thinking about mathematics and working on problems. Mary Ellen describes herself as a problem solver rather than a theory builder; it is a style that lends itself to her field of topology. Walter, on the other hand, is more concerned with building structures, a skill that is particularly useful in his field of analysis.

Moreover, Mary Ellen’s way of doing mathematics is highly individualistic. She likes working alone because she thinks quite differently than most other mathematicians she knows — she is very visually oriented. As she says, she cannot follow a proof or an argument without paper and pencil in hand; she relies on pictures to help visualize an idea. “I don’t learn mathematics with my ears.” As a result, there are very few colleagues or students with whom she can sit down and do mathematics spontaneously. More typically, a student will come in with a problem, they will discuss it briefly, she might make a suggestion or two, but then they go off and work on it separately. The next time they come together, she will have a few more suggestions. Both her natural style and her early training contribute to her preference for working alone.

Ironically, while Mary Ellen is accustomed to working alone in one sense, she is often far from alone in another. It is not unusual for her to work at home — not in a study with her door closed, but rather sitting on her couch in the middle of the living room, surrounded by children.12

It’s a very easy house to work in. It has a living room two stories high, and everything else sort of opens into that. It actually suits the way I’ve always handled the household. I have never minded doing mathematics lying on the sofa in the middle of the living room with the children climbing all over me. I like to know, even when I am working on mathematics, what is going on. I like to be in the center of things, so the house lends itself perfectly to my mathematics…. I feel more comfortable and confident when I’m in the middle of things, and to do mathematics you have to feel comfortable and confident.

For some people, the distractions of home make it impossible to do creative mathematical work. But Mary Ellen has an amazing ability to work even in the midst of domestic activity. Part of what enables her to do this is the grit and determination that are a natural part of her work style. She is the kind of person who simply cannot let a problem go once it has taken hold of her. She describes what she does when she gets stuck on a problem:

I am apt to hit it directly on the head, again and again and again and again. I find it very difficult to give it up. I don’t try to find an easier problem elsewhere. I know mathematicians who do that. Some people are very good at going sideways, or going for a partial solution, but I never want to turn aside. I want to go forward and go for the gold. I struggle with myself to put a problem away, when I have looked at it a thousand times and drawn the same picture a thousand times and gotten no idea. But I find it very painful to put something away. I can’t let go.

This kind of intense determination and focus is extremely useful in maintaining the drive to do research even through the inevitable struggles and frustration that arise in working on mathematics.

Clearly, then, there were many factors that contributed to Mary Ellen’s ability to continue as an active research mathematician, even as she raised four children and created a home life. In the end, however, Mary Ellen argues that while all the external factors such as economics, domestic help, and community and family support are important, they alone will not sustain you. You should do mathematics only “if you really want to, and only if you enjoy working hard on hard problems, and only if you find it tremendously satisfying to solve difficult questions.”

#### Is this a viable model today?

Some would argue that while Mary Ellen Rudin’s situation was in many ways ideal, it is no longer a viable option for most women entering mathematics today. First, few families can survive on one person’s income. The model of the husband as breadwinner and wife as homemaker is no longer an economic possibility for many families. Mary Ellen’s arrangement is even less an option for women who are single or self-supporting.

Second, many women find it difficult to be accepted by the mathematics community if they are not seen as professional. While the label “amateur” mathematician may have been an honorable one when women were not expected to do anything other than domestic work, it is no longer a label that would meet with such acceptance. Such a designation would now have negative connotations, and would make it difficult, for example, to receive grants, awards, jobs, and recognition.

Third, it is not only how women are perceived by their colleagues that is at stake, it is also a matter of how women perceive themselves in relation to their work. Women are raised with the expectation that they will be just like men in the commitment to their work — they expect to have equal opportunities and equal treatment. Hence, many women would feel uncomfortable defining themselves as amateurs. They see themselves, and expect to be seen, as professionals.

Nonetheless, Mary Ellen’s model is an important one, and should not be dismissed too quickly as no longer relevant. It is an example of a woman who was able to successfully combine marriage, motherhood, and mathematics. She was able to stay active and productive as a researcher. She had four children and was actively involved in raising them, and she did not feel that she had to deny the domestic side of her life. Perhaps most important, she was very happy.

Is it possible, therefore, to modify her model to meet the needs of contemporary women, and increasingly of men as well? Is it not possible, for example, for mathematicians to be part-time for certain segments of their career, and still be defined as professionals? Is it necessary to maintain such a degree of separation between work and family in order to be taken seriously as a mathematician? Is it necessary to relegate family to a lesser priority in order to be defined as a real mathematician? Certainly Mary Ellen Rudin suggests a different way of looking at these questions. She would argue that maintaining some connection to research at all times, even when children are young, is very important, but as her life illustrates, there are many ways to do this.