Celebratio Mathematica

Mary El­len Rud­in is now the first oc­cu­pant of a chair en­dowed in hon­or of the pi­on­eer­ing Eng­lish math­em­atician Grace Chisholm Young, and one of the most re­spec­ted work­ers in set the­or­et­ic to­po­logy, but for many years she had “no real job” in aca­demia — and like it that way, she says. She has a bib­li­o­graphy of some sev­enty re­search pa­pers and says she likes best to do math­em­at­ics sur­roun­ded by her fam­ily, in the liv­ing room of her “very open” Frank Lloyd Wright house. She be­lieves she might have fol­lowed any num­ber of oth­er in­terests if the le­gendary R. L. Moore had not signed her up for math­em­at­ics dur­ing fresh­man re­gis­tra­tion at the Uni­versity of Texas: “I am a math­em­atician be­cause R. L. Moore made me a math­em­atician.”

Math­em­at­ic­al People: Tell us something about Hills­boro, Texas, where you were born.

Rud­in: I don’t know any­thing about Hills­boro, Texas. I lived there only two weeks. My fath­er was a civil en­gin­eer. He worked for the State High­way De­part­ment so home would change as the pro­jects that he was work­ing on changed. He was build­ing something in Hills­boro. He was there for a brief peri­od of time and prob­ably stayed an ex­tra week or two, maybe an ex­tra six months, be­cause I was about to be born; but as soon as I was born, we moved on to the next pro­ject in a dif­fer­ent town.

MP: You men­tioned something to us earli­er about grow­ing up in prim­it­ive con­di­tions. Were you ac­tu­ally liv­ing where he was work­ing — in con­struc­tion camps?

Rud­in: No. We al­ways lived in towns near where he was work­ing. When I said prim­it­ive con­di­tions, I was re­fer­ring to the fact that when I was about six he was sent to a little town in south­w­est Texas called Leakey — spelled just like the name of the an­thro­po­lo­gist, and ac­tu­ally it is his fam­ily for whom the town is named. It is in the hills of south­w­est Texas. It is about 3000 feet high in the canyon formed by the Frio river with 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. It was the county seat, but to get to the oth­er town in the county, which was over a moun­tain, you had to go back out the fifty miles and take a dif­fer­ent road up a dif­fer­ent canyon. 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.

MP: So was your fath­er out of a job?

Rud­in: No. He bought land for the road, he sur­veyed for the road, he made plans for the road, but they nev­er got the money to build the road. He stayed there for a long time. I had al­most all my grow­ing up in this little town in south­w­est Texas. 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 that 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.

MP: What did your moth­er think about liv­ing there?

Rud­in: She was a little shocked at first, I think, but she learned to en­joy it.

MP: Where had she come from?

Rud­in: She had come from a town in the foot­hills of the Al­legheny moun­tains, which she thought of as the cen­ter of cul­ture and edu­ca­tion. Both my grand­moth­ers grew up in this same town, and they both went to col­lege there.

MP: Both grand­moth­ers?

Rud­in: Yes. The town was Winchester, Ten­ness­ee, and a col­lege there for wo­men had been foun­ded just at the close of the Civil War.

MP: What was the name of it?

Rud­in: It was, let me see — Mary Sharp Col­lege. It was a real col­lege for wo­men. They had art and mu­sic and things like that, but they also had philo­sophy and math­em­at­ics and so on. It was quite un­usu­al at that time and in that part of the world to have a col­lege for wo­men that wasn’t just a fin­ish­ing school. My pa­ternal grand­moth­er, who was born in 1852, at­ten­ded this col­lege and was very good at math­em­at­ics. She was proud of that. My oth­er grand­moth­er was not so par­tic­u­larly good at math­em­at­ics. At least she didn’t brag about it later in life.

MP: Did your grand­moth­ers teach after they gradu­ated?

Rud­in: No. They both mar­ried quite young and both had fam­il­ies — one had eight chil­dren and the oth­er had six. But they val­ued edu­ca­tion tre­mend­ously and al­ways talked about their own edu­ca­tions. They edu­cated their daugh­ters as well as their sons and saw to it that all their chil­dren had some kind of ad­vanced edu­ca­tion. By the time my moth­er came along, this same little town had a coedu­ca­tion­al school called Winchester Nor­mal, and she at­ten­ded that. Then she went to Pe­abody in Nashville, which was a teach­ers’ col­lege, and took some courses at Vander­bilt, too.

MP: What was your moth­er’s maid­en name?

Rud­in: Irene Shook. It’s Dutch, but I don’t think it was spelled that way when it was Dutch. My fath­er was Joe Jef­fer­son Es­till. The name is French, but it wasn’t spelled that way when it was French either. They were both des­cen­ded from people who came to the United States long be­fore it was the United States; that is, in the early 1700’s. They came over the moun­tains from Vir­gin­ia and settled in the val­leys of Ten­ness­ee. My fath­er’s fam­ily were mostly law­yers and doc­tors, but the story is that they made at least part of their liv­ing play­ing cards. My moth­er’s people were farm­ers primar­ily.

MP: Were you an only child?

Rud­in: Yes. No.

MP: Now wait a minute!

Rud­in: I was an only child of two. I am ten years older than my broth­er so I was raised in a way as an only child. When I left home, he was ready to enter first grade. I love him dearly — we are very close friends — but he’s more like a neph­ew or a son than a broth­er.

MP: Tell us about the school­ing you re­ceived in Leakey.

Rud­in: It was a very simple com­munity. Nobody had any money at all. We were rich be­cause we had a reg­u­lar salary com­ing in. But the prin­cip­al ob­ject that ex­is­ted in the town was the school, and it was a very good school. I was def­in­itely not the best stu­dent in my class. There were five of us who gradu­ated, and there was one girl who was much smarter than I. There was an­oth­er very bright kid. I was maybe third out of five. I went to the uni­versity think­ing that I would make C’s but I made A’s without any trouble.

MP: How do you ac­count for the fact that the school was so good in such a little town?

Rud­in: Many of the teach­ers were the chil­dren of ranch­ers who lived there, and they had come back home to teach. The num­ber of jobs avail­able in the world at that time was lim­ited. I sus­pect that they would have had oth­er op­por­tun­it­ies in an­oth­er time and place. But the com­munity val­ued edu­ca­tion.

MP: It would be in­ter­est­ing to track the chil­dren from that com­munity over time to see if the edu­ca­tion really has paid off.

Rud­in: Elton Lacey, who is chair­man of the math de­part­ment at Texas A. & M., also grew up in that same little com­munity. 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. We thought of wherever we were as some won­der­ful land. And this is my house and this tree is a castle and that’s whatever, de­pend­ing on wheth­er we were want­ing to be Hol­ly­wood stars or wheth­er we were want­ing to be in an­tiquity. The num­ber of books we had to read was fairly lim­ited. We had more books at home than there were in the school lib­rary. All my friends read the books in our house, too.

Al­though I came from a group of people who were edu­cated and who val­ued edu­ca­tion tre­mend­ously, I grew up in very prim­it­ive, simple sur­round­ings where I had lots of time just to think. But there was nev­er any ques­tion about the fact that I would go to the uni­versity and that I would do something with my edu­ca­tion. My moth­er had been a teach­er be­fore she mar­ried. She ex­pec­ted that I should earn my liv­ing and that what I did should be an in­ter­est­ing thing to do.

MP: What in­ter­ested you most in school at that time?

Rud­in: I en­joyed school, but it isn’t clear to me that I was in­ter­ested in one par­tic­u­lar thing or that one thing in­ter­ested me more than the oth­ers. I liked school as much for my friends and for talk­ing with my friends about the ideas we found there as for school it­self. I cer­tainly didn’t like school to the ex­clu­sion of play.

MP: You sound like a nor­mal kid.

Rud­in: Yes. I was a per­fectly nor­mal av­er­age kid.

MP: How did you hap­pen to go to the Uni­versity of Texas?

Rud­in: Well, my fath­er had gone there. He con­sidered it a first class place.

MP: When you set off for the Uni­versity, did you have in mind a spe­cial sub­ject that you wanted to study?

Rud­in: Oh, ab­so­lutely not! My fath­er went along with me be­cause he had an old girl­friend who taught Eng­lish there and he wanted to talk to her about what I should take. They both felt that I should take just a per­fectly gen­er­al lib­er­al arts course, partly to find out what I was in­ter­ested in and partly be­cause they both be­lieved very strongly — and my moth­er cer­tainly felt that way too — that one should be edu­cated in the sense of hav­ing a broad edu­ca­tion. So on the ap­pro­pri­ate morn­ing I went to the gym­nas­i­um to re­gister for the things they had de­cided on. There was a mass of people, but there were very few people at the math­em­at­ics table so I was sent over there. The man who was sit­ting at the table was an old white-haired gen­tle­man. He and I dis­cussed all kinds of things for a long time. I now know the kinds of things that he must have asked me. There would have been lots of sen­tences with if and then. I used if and then cor­rectly. I also used and and or cor­rectly from a math­em­atician’s stand­point. At the end of our con­ver­sa­tion he signed me up for the courses that I had writ­ten on my little slip of pa­per. When I went to my math class the next day, I found that the pro­fess­or was R. L. Moore — the same man who had talked to me at the re­gis­tra­tion table.

MP: You met him lit­er­ally on your first day on cam­pus?

Rud­in: On lit­er­ally my first day on cam­pus I met R. L. Moore. And lit­er­ally on my first day I was se­lec­ted by him.

MP: How large was the Uni­versity at that time?

Rud­in: El­ev­en thou­sand stu­dents.

MP: That’s still rather large for him to be sit­ting there in the gym and eval­u­at­ing in­di­vidu­al fresh­men.

Rud­in: He al­ways did that. He was look­ing for stu­dents.

MP: Well, he got you.

Rud­in: Yes. There was no one else in that first math class who was at all bright so far as I could tell.

MP: What kind of class was it?

Rud­in: It was a trig class. The next semester I took ana­lyt­ic geo­metry, and R. L. Moore taught it. The next semester I took cal­cu­lus, and R. L. Moore taught it. It nev­er oc­curred to me that that was really pe­cu­li­ar. I was not very smart! But I was fully aware that in some way he was teach­ing for my be­ne­fit. He would call on oth­er people first and let them fall on their faces. Then he would have me solve the prob­lem for them. I was al­ways the last per­son he would call on.

MP: Did this in­volve go­ing up to the board?

Rud­in: Yes. Everything was set up on the basis of prov­ing the­or­ems from ax­ioms. It nev­er oc­curred to me that there was any oth­er kind of math­em­at­ics. At the end of the cal­cu­lus course, for in­stance, I’m not sure I knew that the de­riv­at­ive of \( \sin x \) was \( \cos x \). But I could prove all sorts of the­or­ems about con­tinu­ity and dif­fer­en­ti­ab­il­ity and so on!

MP: In oth­er words, you were get­ting an in­tro­duct­ory ana­lys­is course.

Rud­in: Ex­actly. And it was a first-rate course in in­tro­duct­ory ana­lys­is. But it wasn’t un­til I taught cal­cu­lus my­self that I learned all those for­mu­las. Now I find it really in­cred­ible!

MP: So, after Moore las­soed you, you im­me­di­ately zer­oed in on math?

Rud­in: Oh no! I found the whole Uni­versity just fas­cin­at­ing. There was that rare books lib­rary that had everything in the world in it, and I loved to go there. I loved the his­tory courses. I loved the Eng­lish courses. I en­joyed phys­ics very much. I took a lot of philo­sophy. Even in my seni­or year I took courses all over the map — al­most as much philo­sophy as math­em­at­ics, al­most as much his­tory as math­em­at­ics, al­most as much Eng­lish as math­em­at­ics, al­most as much Span­ish as math­em­at­ics. I was equally good, as far as I could tell, in all these sub­jects. Even though I was hav­ing this spe­cial coach­ing, you might say, in math, it was still just one of many things I was tak­ing. When I was a seni­or, I went to the vo­ca­tion­al guid­ance of­fice and said, “I have just taken a broad gen­er­al course, and now I feel I should be spe­cial­iz­ing in something. I clearly need some ad­vice.” They gave me a whole bat­tery of tests. Af­ter­wards they said, “You’re kid­ding! You don’t know you’re good in math­em­at­ics?” I said, “I know I’m good in math­em­at­ics, but I can do oth­er things too. I don’t want to teach high school,” which was really the only thing that oc­curred to me that one could do with math­em­at­ics. They said, “Just re­lax. They’ll in­vite you to stay and go to gradu­ate school.” And in­deed with­in a week (I al­most sus­pec­ted that they had called the math de­part­ment) I was offered an in­struct­or­ship. So I stayed and went to gradu­ate school. I am sure Moore would have caught me any­way. He knew ex­actly what he was go­ing to do with me. He just hadn’t told me yet.

MP: Did you have any math­em­at­ics teach­ers be­sides Moore?

Rud­in: Well, yes, but not un­til my seni­or year. Then I had a course in al­gebra from some­body else and a course in dif­fer­en­tial equa­tions from some­body else.

MP: So you had a course from Moore every semester?

Rud­in: Every single semester dur­ing my en­tire ca­reer at the Uni­versity of Texas. 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.

MP: You are a child of Moore.

Rud­in: I’m a child of Moore. 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.

MP: You really feel that?

Rud­in: You give me the defin­i­tions, and I’ll solve the prob­lem. I’m a prob­lem solv­er, primar­ily a counter­example dis­cover­er. Part of that is a Moore thing, too. That is, he didn’t al­ways give us cor­rect the­or­ems, at least half of his state­ments were false. So we had to think about them as a re­search math­em­atician might. I still have this feel­ing that if a prob­lem can be stated in a simple form that I can really un­der­stand, then I should be able to solve it even if do­ing so in­volves build­ing some com­plic­ated struc­ture. Of course, I have had some fail­ures. You can guess how of­ten.

MP: But you’ve nev­er failed in con­fid­ence?

Rud­in: No, hav­ing failed 5,000 times doesn’t seem to make me any less con­fid­ent. At least I don’t feel bound by any ser­i­ous con­straints or doubts about my abil­ity.

MP: Tell us a little more about your gradu­ate school ex­per­i­ence.

Rud­in: I had entered the Uni­versity in 1941. That was the be­gin­ning of the war for the United States, and all Moore’s stu­dents went off to war ex­cept Bing, who had an old in­jury which didn’t al­low him to go. Be­cause of the speeded up war­time sched­ule, I had just three years as an un­der­gradu­ate, but gradu­ate school for me was fant­ast­ic. I star­ted in Decem­ber 1944, and by Septem­ber 1945 the war was over and the men were back. So I star­ted with R. D. An­der­son, R. H. Bing, Ed Moise, and Ed Bur­gess. There were five of us.

MP: That’s quite a col­lec­tion!

Rud­in: We were a fant­ast­ic class. Each of us could eat the oth­ers up. Moore did this to you. He some­how built up your ego and your com­pet­it­ive­ness. He was tre­mend­ously suc­cess­ful in that, partly be­cause he se­lec­ted people who nat­ur­ally had those qual­it­ies he val­ued. He im­me­di­ately sep­ar­ated Moise and Bing, who were fur­ther along than the rest of us. But still we were really to­geth­er, and 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. Ac­tu­ally in our group there was an­oth­er, a sixth, whom we killed off right away. He was a very smart guy — I think he went in­to com­puter sci­ence even­tu­ally — but he wasn’t strong enough to com­pete with the rest of us. Moore al­ways began with him and then let one of us show him how to solve the prob­lem cor­rectly. And, boy, did this work badly for him! It builds your ego to be able to do a prob­lem when someone else can’t, but it des­troys that per­son’s ego. I nev­er liked that fea­ture of Moore’s classes. Yet I par­ti­cip­ated in it.

MP: Did it hap­pen of­ten?

Rud­in: Very of­ten in the un­der­gradu­ate classes. I mean, I was the killer. He used me that way, and I was con­scious of be­ing used that way.

MP: You were Moore’s only wo­man stu­dent?

Rud­in: Moore had sev­er­al wo­men stu­dents after me, and he had had two wo­men stu­dents pre­vi­ously. The first, Anna Mul­likin, wrote a fant­ast­ic thes­is at Pennsylvania and then im­me­di­ately went off to China as a mis­sion­ary. The next was Har­lan Miller. She later taught at Texas Wo­men’s Col­lege and was very in­flu­en­tial as a teach­er and an ad­min­is­trat­or, but she nev­er did any re­search math­em­at­ics after her thes­is. He was tre­mend­ously dis­ap­poin­ted in both of these wo­men.

MP: How did Moore con­duct a gradu­ate class?

MP: He must have had some male fail­ures, too.

Rud­in: Oh, he had plenty of male fail­ures. There’s no ques­tion.

MP: You were say­ing that you would talk about things oth­er than math­em­at­ics in class. What sort of things?

Rud­in: Moore would come in and stand at the board and sort of start the con­ver­sa­tion. “Miss Es­till, do you have any­thing to re­port today?” “Mr. An­der­son, what do you have to say?” There were maybe three people in the class, al­ways a small group.

MP: So if nobody had any­thing to re­port?

Rud­in: Then we would start dis­cuss­ing something — it could be polit­ics or any­thing. Ed Bur­gess, for in­stance, re­mem­bers the fol­low­ing in­cid­ent, which I don’t re­mem­ber at all. We were dis­cuss­ing lock­ing doors. I said that I would nev­er lock the door to my house un­less my hus­band in­sisted. Ed says that Moore lit­er­ally pounced on that, say­ing, “Hus­band! But, Miss Es­till, I thought that you were go­ing to be a math­em­atician.” Moore in­ten­ded to eli­cit a re­sponse from me; but al­though he may have had his doubts, I nev­er saw any con­tra­dic­tion in be­ing both a house­wife and a math­em­aticians — of the two I was more driv­en to be a house­wife. Now I don’t re­mem­ber this par­tic­u­lar in­cid­ent, but I do re­mem­ber that lock­ing doors was a sub­ject that we of­ten dis­cussed.

MP: Was there much in­ter­ac­tion among the Moore stu­dents out­side the classroom?

Rud­in: Very little. We went our own ways and did our own things.

MP: Were there any so­cial things with Moore and his wife?

Rud­in: Once in a blue moon there would be something so­cial, but Mrs. Moore didn’t do so­cial things very eas­ily and maybe he didn’t either. Ac­tu­ally there wer­en’t a lot of so­cial things in math­em­at­ics at the Uni­versity of Texas in those days.

MP: Tell us about your thes­is.

Rud­in: It was one of Moore’s many un­solved prob­lems. His tech­nique was to feed all kinds of prob­lems to us. He gave us lists of state­ments. Some were true, some were false, some he knew were true, some he knew were false, some were fairly easy to prove or dis­prove, oth­ers very hard. There would be all kinds of un­solved and solved prob­lems in the same batch, and there was no way of dis­tin­guish­ing between them. We worked on whatever we jolly well pleased. I solved one of the un­solved prob­lems. Ac­tu­ally I found a counter­example to a well-known con­jec­ture. The tech­nique I used is now called “Build­ing a Pix­ley–Roy Space.” Two math­em­aticians named Pix­ley and Roy tried to read my thes­is, which was writ­ten in Moore’s old-fash­ioned lan­guage and was not ter­ribly well writ­ten be­sides, and gave a beau­ti­ful sim­pli­fied de­scrip­tion of the tech­nique. At the time I wrote my thes­is, I had nev­er in my life seen a single math­em­at­ics pa­per!

MP: You had nev­er read any math­em­at­ics pa­pers?

Rud­in: I told you that I was pure and unadul­ter­ated. I only knew the math­em­at­ics that Moore fed me. The math­em­at­ic­al lan­guage that he used was his own. I didn’t know the stand­ard defin­i­tion of com­pact; I didn’t know the cor­rect defin­i­tion of lim­it point. I didn’t know how math­em­at­ic­al words were used at all. In­stead of open set, for in­stance, he used re­gion. His lan­guage was com­pletely dif­fer­ent from the lan­guage of the math­em­at­ic­al lit­er­at­ure. I didn’t know any oth­er lan­guage.

MP: He told his stu­dents — at least so I have heard — “I don’t want you go­ing to the lib­rary and read­ing pa­pers.”

Rud­in: I don’t re­mem­ber ever be­ing told that I shouldn’t read math­em­at­ic­al pa­pers, but I was nev­er temp­ted. It’s true, however, that he some­times en­cour­aged people to go out in­to the hall so that they would not hear a proof. I would not do such a thing. If some­body proved something first, he proved it first, and I would listen to it.

MP: Why do you think Moore did that?

Rud­in: He wanted to build your in­de­pend­ence.

MP: Wheth­er the oth­er per­son’s proof was right or wrong?

Rud­in: Wheth­er it was right or wrong. Of course if it was wrong, he’d be de­lighted to have you there be­cause then you could dis­cov­er that it was wrong.

MP: How would he know a proof was wrong in ad­vance?

Rud­in: First of all, he did have some in­ner sense. Second, he tried to get stu­dents to have the at­ti­tude that they didn’t want to listen to someone else’s proof. I re­belled against that, but there were people who did go out. I think that none of our group ever went out. That wasn’t our style. But when you read about Moore, you will read that he tried to get people to do that.

MP: He nev­er re­ferred in class to oth­er people’s work?

Rud­in: Nev­er, nev­er, nev­er.

MP: You grew up in a strange world.

Rud­in: In a strange, un­real world. Com­pletely. I still dis­like read­ing math­em­at­ic­al pa­pers, and I learn any oth­er way I pos­sibly can. My first two or three pa­pers were all writ­ten in Moore’s old-fash­ioned lan­guage.

MP: How did you feel about your math­em­at­ic­al edu­ca­tion later?

Rud­in: 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 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 so con­fid­ent as I was.

MP: Read­ing oth­er people’s work is a great way to des­troy one’s con­fid­ence.

Rud­in: Maybe it is. At least that was Moore’s opin­ion.

MP: Wer­en’t there any de­part­ment­al re­quire­ments at Texas? Any qual­i­fy­ing ex­ams for the doc­tor­ate?

Rud­in: He was the de­part­ment. I took ex­ams in philo­sophy, which was my gradu­ate minor, but I nev­er took an ex­am in math­em­at­ics in my life. Moore stu­dents were good in dir­ect pro­por­tion to how fast they learned after they got out. I still feel ser­i­ously de­prived by the short­age of the things I learned. I re­sent that, I guess, but at the same time I’m con­scious of how much — well, I wouldn’t have been a math­em­atician at all if it hadn’t been for R. L. Moore.

MP: How about find­ing a po­s­i­tion after you got your Ph.D.?

Rud­in: Get­ting a po­s­i­tion was just like go­ing to gradu­ate school. I nev­er ap­plied for one. 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.”

MP: What feel­ings to­ward Moore, as a per­son, did you de­vel­op over time?

Rud­in: Oh, I had very warm, en­thu­si­ast­ic feel­ings for him, al­though I also had very neg­at­ive feel­ings. I was con­scious of both levels.

MP: And the neg­at­ive ones?

Rud­in: I was aware that he was big­oted — he was — but I also was aware that he played the role of a big­ot some­times in or­der to get our re­ac­tions, maybe even to keep us from be­ing big­ots. I’m nev­er sure to what ex­tent that was true. Moise, for in­stance, was a Jew. Moore al­ways claimed that Jews were in­feri­or. I was a wo­man. He al­ways poin­ted out that his wo­men stu­dents were in­feri­or. Moise and I both loved him dearly, and we knew that he sup­por­ted us fant­ast­ic­ally and did not think that we were in­feri­or — in fact, he thought that we were su­per spe­cial. On the oth­er hand, he wanted us to be very con­fid­ent of ourselves in what he un­doubtedly viewed as a some­what dis­ad­vant­aged po­s­i­tion. Now then, did he play the role of a big­ot to eli­cit a re­sponse? I have no idea.

MP: His talk­ing about his former wo­men fail­ures prob­ably made you say to your­self, I’m not go­ing to be one of those!”

Rud­in: I can’t say that I really ever iden­ti­fied with them. Something else Moore built in­to all of us was our re­spons­ib­il­ity to be part of the math­em­at­ic­al com­munity — to take part in the Amer­ic­an Math­em­at­ic­al So­ci­ety, very strongly, and to take part in The Math­em­at­ic­al As­so­ci­ation of Amer­ica even though it was not a re­search or­gan­iz­a­tion. He be­lieved in go­ing to meet­ings of pro­fes­sion­al or­gan­iz­a­tions and par­ti­cip­at­ing in the meet­ings. That’s something that all of us have done more than our share of. Moore was pres­id­ent of the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence; and Moore, Why­burn, Wilder and Bing were all pres­id­ents of the AMS as well as col­loqui­um lec­tur­ers for the AMS and mem­bers of the Na­tion­al Academy of Sci­ences. Wilder, Bing and Moise were MAA pres­id­ents. All of us have served on end­less com­mit­tees for these or­gan­iz­a­tion.

MP: What about edit­ing?

Rud­in: He be­lieved very strongly in do­ing that, too.

MP: Even though he nev­er read?

Rud­in: Oh yes, he con­sidered pub­lic­a­tion ab­so­lutely vi­tal. We should pub­lish and be very in­volved, even if we shouldn’t read too much about what oth­er people were do­ing.

MP: But he sent you off to Duke nev­er hav­ing read any­thing?

Rud­in: Right.

MP: At Duke was where you met Wal­ter Rud­in. Will you tell us a little about Wal­ter?

Rud­in: Wal­ter had grown up in Vi­enna. He was part of an old Aus­tri­an Jew­ish fam­ily. His great grand­fath­er, who was a phil­an­throp­ist, had been knighted by Franz Josef. However, when the Ger­mans came in­to Aus­tria in 1938, the Aus­tri­ans re­jec­ted their Jews, and Wal­ter now totally re­jects his Aus­tri­an back­ground. His par­ents man­aged to send Wal­ter and his sis­ter to school in Switzer­land. It was maybe six months be­fore they them­selves could leave Aus­tria. They left without a thing ex­cept the clothes on their backs. They showed up in Switzer­land think­ing that they would live there forever, but it very soon be­came clear that that was not a pos­sib­il­ity. So they moved on to France where they again ex­pec­ted to live forever. But then the French be­came very frightened about hav­ing so many Ger­man-speak­ing refugees in the coun­try so they put Wal­ter and his fath­er in in­tern­ment camps in dif­fer­ent parts of France. Wal­ter vo­lun­teered for the work­ing corps in the French army. He was 19, I guess, and he was in the French army for about two weeks be­fore the Ger­mans came in. He was told by the French that he could just walk away. He got to Eng­land and spent the war years there. After the war he came to vis­it in this coun­try, where his par­ents had man­aged to come. He went down to Duke to see his sis­ter, who was in gradu­ate school — she’s a chem­ist — and while he was there he talked to the people in the math de­part­ment. He had nev­er gone to col­lege, but he per­suaded them that maybe he could be a ju­ni­or. Four years later he had a Ph.D. in math­em­at­ics. So we were both fresh young Ph.D.’s. We went to­geth­er, but we didn’t — I think there were times dur­ing that year when I was in­ter­ested in mar­ry­ing Wal­ter and there were times when he was in­ter­ested in mar­ry­ing me, but it was nev­er at the same time. We were both just start­ing out in life.

MP: When did you and Wal­ter get mar­ried?

Rud­in: We got mar­ried in ’53. Wal­ter was go­ing to Rochester that year, and jobs were very scarce. Dur­ing the time at Duke I had worked on prob­lems re­lated to Souslin’s con­jec­ture and had con­struc­ted a lot of weird con­nec­ted sets in the plane, one of which dis­ap­proved a con­jec­ture of Wilder’s, so I was be­gin­ning to be reas­on­ably well known. Wilder ar­ranged for me to go to the Uni­versity of Michigan. You see, he had ap­plied for a grant for me. I hadn’t ap­plied, he had ap­plied.

MP: So it was an ex­ten­sion of the Moore meth­od by a Moore stu­dent?

Rud­in Well, when Wilder wired that he had the money for me, I wired back, “I’m sorry but I’m get­ting mar­ried and go­ing to Rochester.” And Wilder ar­ranged for the grant to be trans­ferred from Michigan to Rochester!

MP: The Uni­versity of Texas took care of its own.

Rud­in: I al­ways had this built-in fam­ily that really took care of me. For in­stance, I gave col­loquia at Vir­gin­ia, where Why­burn was. It nev­er oc­curred to me that it was slightly dif­fi­cult to be a wo­man math­em­atician.

MP: But when you mar­ried, there was ab­so­lutely no guar­an­tee that you would have any­thing to do at Rochester.

Rud­in: No, but it ab­so­lutely nev­er oc­curred to me to worry. I was mar­ried.

MP: Did you and Wal­ter ever dis­cuss the ques­tion of your ca­reer?

Rud­in: No. Of course not. I didn’t know any couple that did. I didn’t even think of math­em­at­ics as a ca­reer. The Uni­versity of Rochester hadn’t known I was com­ing with Wal­ter, but they im­me­di­ately gave me a cal­cu­lus class to teach. So I taught. I had a private of­fice. I didn’t really have a po­s­i­tion, but — oh well, I was a tem­por­ary part-time something. And that’s the kind of job I have had al­most all my life un­til 1971, when I be­came a full pro­fess­or. I have had non-jobs wherever we happened to be, I loved my non-jobs. I al­ways had all the good­ies that go with be­ing a math­em­atician. I had gradu­ate stu­dents, I had sem­inars, I had col­leagues who loved me. I nev­er had com­mit­tees. I nev­er had a tri­go­no­metry class. I was al­ways just asked, “Is there something you’d like to teach this semester?” And I would say, “I think I’ll teach a gradu­ate course in — ” or maybe, “I think I’ll take this semester off.” I of­ten had grant money. It didn’t make that much dif­fer­ence in my life though, since Wal­ter al­ways had a good job. But I was a math­em­atician, and I al­ways thought of my­self as a math­em­atician.

MP: And a moth­er.

Rud­in: Oh yes. We had two ba­bies in the first two years. So 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.

MP: Did you have ex­tra help with your chil­dren?

Rud­in: At first, at Rochester and later at Yale, I had great re­spons­ib­il­it­ies at home be­cause I had friends rather than hired people help­ing me with my chil­dren. When we moved to Madis­on, however, I happened on an ab­so­lutely fant­ast­ic wo­man named Lila Hil­gen­dorf. She had been the wife of a farm­er and had had six chil­dren. She was won­der­ful with chil­dren.

MP: By this time you had your four?

Rud­in: There were still just two; but from the be­gin­ning, after we moved to Madis­on, she came reg­u­larly to my house. 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 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, un­pres­sured ca­reer as mine.

MP: You were, nev­er­the­less, crank­ing out pa­pers at quite a rate.

Rud­in: Oh yes. Ter­rif­ic. But 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.

MP: A great reas­on.

Rud­in: Then, when the time came that I was re­l­at­ively free, it all of a sud­den be­came pos­sible for me to be­come a pro­fess­or. I was in­stantly made a pro­fess­or. I mean, the guilt feel­ings in the math de­part­ment were such that nobody even asked me if I wanted to be a pro­fess­or. I was simply presen­ted with this full pro­fess­or­ship. My first re­ac­tion was to say no, but Wal­ter per­suaded me that I should say yes. He said, “It’s in­sur­ance I can’t buy you. Why don’t you just ask for half­time?” So I had a full pro­fess­or­ship half­time un­til I dis­covered that I was build­ing no re­tire­ment. They mul­tiply by half three times: half the num­ber of hours, half the amount of salary, and half the years of ser­vice.

MP: An eighth.

Rud­in: Pre­cisely. The chil­dren were grown by then so I de­cided to teach full time, but I’d really like to go back to half­time. Teach­ing takes a lot of time; and even though I en­joy it and have had some won­der­ful gradu­ate stu­dents, I think a half­time ca­reer is just great.

MP: Es­pe­cially when you have a hus­band.

Rud­in: It’s a func­tion of hav­ing some­body to earn a liv­ing for you.

MP: You say that you en­joy teach­ing and you have had some great stu­dents. Do you use the Moore meth­od of teach­ing?

Rud­in: No. Not at all.

MP: Why not?

Rud­in: I guess I don’t be­lieve in it.

MP: What is your way of teach­ing?

Rud­in: Oh, I’m a per­fectly straight­for­ward, en­thu­si­ast­ic lec­turer.

MP: You bubble.

Rud­in: I bubble, and I get stu­dents en­thu­si­ast­ic. I’m able to ex­plain things. I’m a good lec­turer, I think. I’ve nev­er tried to use the Moore meth­od — I guess be­cause I got burned by hav­ing been a stu­dent un­der it. Bing, however, al­ways used the Moore meth­od. And there are oth­er Moore stu­dents and stu­dents of Moore stu­dents, and so on, who have used it and used it very ef­fect­ively — com­bin­ing it with teach­ing stu­dents to read. Nobody ever does it Moore’s way. For­tu­nately. I do try, however, to give stu­dents in­ter­est­ing prob­lems to work on.

MP: How do you find in­ter­est­ing prob­lems?

Rud­in: Oh it’s easy. There are a lot of in­ter­est­ing prob­lems around.

MP: When you give a stu­dent a prob­lem, have you more or less worked it out in your head?

Rud­in: Some­times.

MP: Pólya us that he nev­er gave a stu­dent a dis­ser­ta­tion prob­lem un­less he knew that it could be done. We pressed him. Did that mean that he had done it him­self. He said, well, yes.

Rud­in: Ef­fect­ively. Yes. Of course, I have had stu­dents who don’t need any as­sist­ance of that kind. That’s the kind of stu­dent that a teach­er loves to have. They teach you all kinds of things. You don’t teach them. But you also have stu­dents who need a par­tic­u­lar prob­lem, and you try to give it to them. My tech­nique is not so much to have solved the prob­lem com­pletely as to have in mind two or three things that I think can be done and to have done enough think­ing about them to see that they are not too dif­fi­cult. I try to give my stu­dents two or three things to work on. I be­lieve in work­ing on sev­er­al things at once, even though I find it dif­fi­cult to do that my­self. I really like to con­cen­trate on one thing. That’s one of my worst flaws as a math­em­atician.

MP: In­cid­ent­ally, how do you ac­tu­ally do math­em­at­ics?

Rud­in: I lie on the sofa in the liv­ing room with my pen­cil and pa­per and think and draw little pic­tures and try this thing and that thing. I’m in­ter­ested in how ideas fit to­geth­er. Ac­tu­ally I’m very geo­met­ric in my think­ing. I’m not good at num­bers at all. Al­though I do like com­bin­at­or­ics, I’m not really in­ter­ested in num­bers. Wal­ter says that I think one num­ber is just like any oth­er num­ber.

MP: So you do math­em­at­ics ly­ing on the couch in the liv­ing room of your Frank Lloyd Wright house?

Rud­in: Yes. It’s a very easy house to work in. It has a liv­ing room two stor­ies high, and everything else sort of opens onto 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.

MP: What about Wal­ter and his math­em­at­ics?

Rud­in: He likes more pri­vacy. We have a study that is quite private, and that is Wal­ter’s study. My study is in the middle of the liv­ing room — in the middle of our one-room house.

MP: Then in ef­fect you have no study?

Rud­in: No. Nev­er have had a study. Nev­er have had any in­terest in hav­ing one.

MP: A num­ber of cre­at­ive wo­men seem to be able to op­er­ate that way. I re­mem­ber read­ing that Har­riet Beech­er Stowe wrote her books on her knee in her kit­chen with all her chil­dren around her.

Rud­in: I like that situ­ation. 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.

MP: It’s in­ter­est­ing that you now oc­cupy a pro­fess­or­i­al chair hon­or­ing the Eng­lish math­em­atician Grace Chisholm Young. She was a wo­man with six chil­dren, I be­lieve. Do you hap­pen to know if she op­er­ated the same way?

Rud­in: I think she must have, ba­sic­ally. Let me get you some things about her from her son, L. C. Young, who is in Madis­on. You will love them. There’s a let­ter writ­ten to her by her hus­band, who was the math­em­atician W. H. Young, say­ing, “Mine the glory now and the know­ledge. Yours the know­ledge only. But later, when all the loaves and fishes are in, you will get your glory.” Something like that.

MP: A wo­man math­em­atician needs a very spe­cial man for a hus­band.

Rud­in: Right. Right. Right.

MP: You men­tioned earli­er that dur­ing your col­lege years your so­cial friends were not math­em­aticians. Now who are your friends?

Rud­in: They’re more math­em­aticians now, and the reas­on is that math­em­at­ics in Madis­on is a very large thing. It in­cludes all kinds of people so it’s easi­er for Wal­ter and me to know math­em­aticians than to know any­one else. We have very close math­em­at­ic­al friends.

MP: Your math­em­at­ic­al fam­ily.

Rud­in: Yes. We have both had lots of gradu­ate stu­dents so now we have stu­dents and stu­dents of stu­dents. We’re as close to our grand­chil­dren — math­em­at­ic­al grand­chil­dren — as we are to our stu­dents. Then we’ve both had very strong math­em­at­ic­al fam­il­ies be­hind us. I’ve de­scribed the Moore fam­ily to you. And Wal­ter also has worked with a spe­cial group of math­em­aticians in France, Sweden, and the U.S. It in­cludes a whole group of people from The Uni­versity of Chica­go who were stu­dents of Zyg­mund’s. This group has been a very strong math­em­at­ic­al fam­ily for Wal­ter. We re­cently went to Yugoslavia for a month, and every place we went we were tre­mend­ously wel­comed and very well taken care of by mem­bers of our math­em­at­ic­al fam­il­ies who are there.

MP: What vo­ca­tions have your chil­dren fol­lowed?

Rud­in: Our old­est daugh­ter is a lin­guist. She is an ex­pert on syn­tax, Bul­gari­an syn­tax in par­tic­u­lar, but she knows many oth­er lan­guages — south Slavic lan­guages are her spe­cialty. She’s mar­ried to an an­thro­po­lo­gist. Our second daugh­ter is an en­gin­eer. She got an un­der­gradu­ate de­gree in phys­ics. She works for 3M in Min­neapol­is and is mar­ried to a com­puter sci­ent­ist and en­gin­eer. Our young­est son is go­ing to be a bio­chem­ist and an M.D. He’s still in school. And our re­tarded son is the jan­it­or for the loc­al pizza par­lor. Wal­ter says he’s our greatest suc­cess. He’s liv­ing way bey­ond his in­tel­li­gence while the oth­ers are just liv­ing up to theirs! I claim that all these kids have in­her­ited the fam­ily tal­ent. They’re all math­em­aticians of a sort.

MP: Wait a minute now.

Rud­in: How does math­em­at­ic­al tal­ent show it­self? It’s in pat­tern re­cog­ni­tion. And the lin­guist daugh­ter, the en­gin­eer­ing daugh­ter, the ge­net­i­cist — it’s ob­vi­ous with them. But even our re­tarded son has a tre­mend­ous amount of this abil­ity. He doesn’t have very good judg­ment, but he has cer­tain spe­cial­ized tal­ents which seem to me to be very much of the pat­tern re­cog­ni­tion type. He loves his­tory. He can tell you what happened on cer­tain dates. He doesn’t know what the facts mean, but he likes to fit them to­geth­er. He also knows the bus sys­tem in Madis­on ab­so­lutely per­fectly. If you want to go from any­where to any­where at a cer­tain time, he can tell you when the bus will come and where it will go.

MP: When you are at a party with a lot of people who are not math­em­aticians, and people ask what you do, what do you say?

Rud­in: I say, “I’m a math­em­atician.” And then they tell me that math­em­at­ics was their worst sub­ject and they need me to help them bal­ance their check­books. The same thing they would tell a male math­em­atician.

MP: Why do you think that there are not more wo­men in math­em­at­ics?

Rud­in: Well, it isn’t clear to me. I re­cently talked to some high school stu­dents on “What is math­em­at­ics?” and, among oth­er things, I talked about that very ques­tion. Math­em­at­ics is ob­vi­ously something that wo­men should be able to do very well. It’s very in­tu­it­ive. You don’t need a lot of ma­chinery, and you don’t need a lot of phys­ic­al strength. You just need stam­ina, and wo­men of­ten have a great deal of stam­ina. So why do not more wo­men be­come math­em­aticians? These were high school kids, and I gave them the stat­ist­ics. When they start out as first graders, the girls are ac­tu­ally a little ahead of the boys be­cause they read bet­ter. We know that this is ge­net­ic, and we don’t worry about it. At the end of ele­ment­ary school, the girls say that they don’t do math­em­at­ics very well and the boys say that they do, even though there’s evid­ence to the con­trary. At that point girls and boys are ap­prox­im­ately the same ex­cept for their at­ti­tude to­ward the sub­ject. But then when they get to high school, things change. At Wis­con­sin we send out to the high school stu­dents in the state some really won­der­ful prob­lems that the math de­part­ment has worked very hard to de­vel­op. These are prob­lems that don’t re­quire spe­cial know­ledge — they just re­quire tal­ent and hard work. We send them around to the high schools in the state and tell the stu­dents if they are will­ing to look at these prob­lems and send us their solu­tions, we’ll grade them and we’ll start com­mu­nic­at­ing dir­ectly with them, giv­ing them more prob­lems.

MP: So you’re de­vel­op­ing a tal­ent bank?

Rud­in: That’s right. It’s called the Tal­ent Search. It was star­ted by Grace Chisholm Young’s son. But the point of my story is that we al­most nev­er get any high school girls to re­spond. They won’t do the hard prob­lems. They’re per­form­ing bet­ter than the boys at that point. But why won’t they look at the hard prob­lems?

MP: Do you have any ideas?

Rud­in: I really don’t. It must be something. There are people at Johns Hop­kins who say that it’s ge­net­ic. Maybe it is, but I really don’t be­lieve it. I think that for some reas­on, prob­ably so­ci­olo­gic­al, girls are re­fus­ing to look — they just simply won’t try something that they view as a hard prob­lem in math­em­at­ics. But boys for some reas­on are will­ing and eager to look at the hard prob­lems. We’ll get forty kids from the state to come to the uni­versity at the end of the year, and we’re very lucky if there are two girls among them. Now at the uni­versity some girls do be­gin to get really in­ter­ested in math­em­at­ics, and by the time they get to gradu­ate school some are very good — ac­tu­ally a fairly sub­stan­tial group of them. They seem to re­cov­er later, but it’s very weird at the high school level.

MP: You’ve said that do­ing math­em­at­ics is dif­fer­ent for wo­men today than it was for you. Prob­ably not as good?

Rud­in: Oh, I don’t know that it’s not as good. In fact, I think that maybe it’s bet­ter for the fol­low­ing reas­on. 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 were en­thu­si­ast­ic am­a­teurs, but we were am­a­teurs. I’m of the house­wives’ gen­er­a­tion. 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.

MP: But you didn’t quit.

Rud­in: No. I had the en­thu­si­asm to do it. Ju­lia Robin­son cer­tainly didn’t quit. But there were those who did. I think we’re get­ting a lot bet­ter young wo­men math­em­aticians now than in my day.

MP: Cer­tainly a lot more.

Rud­in: I think that if you get a lot more you’ll get more that are bet­ter. I think that’s hap­pen­ing. And that’s really very sat­is­fy­ing to me. I see it com­ing.

Ju­ly 1986 in Oak­land, Cali­for­nia.