Article from Volume 2, #1, of TopCom
Arthur Stone was born September 30, 1916 in London. He earned his B.A. from Cambridge University and his Ph.D. from Princeton University in 1938 and 1941, respectively. He married Dorothy Maharam in Pittsburgh, Pennsylvania (U.S.A.) on April 12, 1942. His positions in higher education include: Fellow of Trinity College, Cambridge, 1946-48; Lecturer and Senior Lecturer, Manchester University (England), 1948-61; Professor, University of Rochester (U.S.A.), 1961-87, emeritus 1987-present; Adjunct Professor, Northeastern University (Boston), 1988-present.
WWC. We are here at the suggestion of Melvin Henriksen, under the auspices of the Topology Atlas. I appreciate your willingness to be interviewed.
AHS. I am flattered to be asked.
WWC. Let's talk first about mathematics at the very personal level. Which of your many theorems do you believe is most admired by the topological community?
AHS. I don't think any of my theorems are actually admired by the topological community. At one time, my result implying the paracompactness of metrizable spaces was quite widely quoted, but now it seems to have receded to the usual obscurity.
WWC. Tell us something about the discovery and proof of that theorem. Had it already been conjectured by others? Did you know in your soul from the start that it was true, or did you oscillate between the search for a proof and the search for a counterexample?
ASH. It had been conjectured by Dieudonné, but at first I was fairly sure that it was false. I even had a candidate for a counterexample (some kind of linear function space, I think). It collapsed when I saw how to prove the theorem.
WWC. You must have been pleased and proud at the time. Do you have a recollection of the physical location or circumstances where you first knew for sure that the theorem was under control?
AHS. I wasn't sure until I had written it up, but I felt confident after brooding about it on a railway station platform while waiting for a train.
WWC. In general, concerning either that theorem or others of yours, do you have a preferred method of preparation for mathematical work? I mean, for example, do you demand total silence; a special favorite desk; a certain symphony in the background?
AHS. I often evolve ideas when walking, or occasionally when lying on a couch. Writing them up is another matter; I prefer silence, and I abominate background music!
WWC. On the basis of your own successes, do you have general advice for other younger workers - and for some of us not so young but still trying - as to how to approach mathematical research?
AHS. There are many different approaches to mathematical research, and I think each mathematician has to find his or her own. One hint I have sometimes given students: try simple cases first. But I don't know if this has helped them.
We all have our favorite stimuli. I recall being told at Princeton that the great J.W. Alexander liked to think about mathematics at the cinema. The moviehouse there in Palmer Square used to offer two programs every week and twice every week Alexander would be there.
There's a story about that moviehouse, by the way. The developer, preparing to plan the building, asked Princeton's professor of architecture to dinner and described his project. He especially wanted the acoustics to be good. The professor sketched a rough outline on the back of an envelope, and went away expecting to be asked to provide a complete architectural plan. But the cinema was built from the sketch! And the process worked: I can testify personally that the acoustics were really excellent.
WWC. Sometimes the theorems for which one is most admired and which other mathematicians find the most useful are not those from which one takes the greatest pride. Are there other of your results which bring you particular pleasure?
AHS. Actually I enjoy every result of mine, no matter how trivial.
WWC. You are one of the world's great topologists, married to one of the world's great measure theorists.
AHS. That is at any rate half true: Dorothy is a great measure theorist.
WWC. Obviously one wants to see one's spouse succeed in the chosen career. But I'm wondering whether there has ever been a competitive element in your relation with Dorothy.
AHS. Perhaps curiously, we have never felt any competitiveness between us. In fact, neither of us is very competitive; we enjoy our work, irrespective of what others are doing.
WWC. Are there special contributions over the years which you have made to her work, and she to yours? In general, would your own career have been as satisfactory, or more, or less, if Dorothy had been, say, a successful lawyer or chemist rather than a mathematician?
AHS. I have been enormously helped by Dorothy, not only in being able to discuss mathematical problems with her, but also with specific questions she has raised (and still raises). For instance, I have studied and used what we first called ``Dorothy's crazy topology,'' now renamed the ``topology of close approximation.'' I have contributed to her work, in a minor way, by solving some technical topological problems that arose in it, and in a major way by editing and typing her manuscripts. (She would probably interchange ``minor'' and ``major'' here.) I am also responsible, in a way, for two of her major papers. In one, I was unimpressed by a ``direct sum'' representation of measure spaces; as a result, Dorothy found a much better ``direct product'' normal form. In the other, I made a casual remark that topologists could ``smooth'' spaces by multiplying them by powers of the real line; this led her to smoothing measure spaces in a somewhat similar way. We have also written several joint papers.
WWC. As a very junior Assistant Professor at Rochester in the early 1960's, I recall sitting in one summer on a short course you taught on Uniform Spaces, this to an audience of select graduate students from around the country.
AHS. I enjoyed giving that course.
WWC. Your lectures on those occasions were really excellent.
AHS. Thank you!
WWC. It seems that some top-level mathematicians are blessed with, or manage to develop, lecturing talents to match; while others of equally creative research capacities are abysmal in the classroom. Would you attribute your own popularity and success in the classroom to hard work and careful preparation, or to some other serendipitous attribute?
AHS. I have had the advantage of lecturing (usually) to small classes of bright and well-motivated students. I don't think I did well with large classes in which the attitude was ``Are we responsible for this?''
WWC. Do you have advice for today's young scholars on how to increase their own effectiveness in this direction?
AHS. I found a hint from a former colleague, G.E.H. Reuter, to be helpful. It was to write up each lecture carefully after it was given. (Of course, one would prepare it before giving it, too.) That way one knew what additions and corrections were needed, for the next installment.
Another useful teaching tool, learned from my former mentor, A.S. Besicovitch, was to assign challenging homework problems, with the warning that some of them might be wrong; in that case, the student was to give a counterexample, guess what the correct problem was, and solve that. At first I would deliberately insert an occasional mistake, but I soon found this unnecessary; the mistakes would be there! Whether these devices would help other teachers, I can't say. Of course, it depends on the students, too.
I think that good teaching is a function of two variables - the teacher and the taught. My own experience as a student at Cambridge convinced me of this. The undergraduate course on Projective Geometry - yes, there was such a course, for undergraduates even! - was given by Mr. X, and that on Complex Variables by Mr. Y. (In American terminology they would have been Professors X and Y; in Britain, then and now, the title ``Professor'' is reserved for special appointments.) Mr. Y's lectures were beautifully clear and complete, easily transcribed and easily forgotten. Mr. X rarely got anything completely right, and I had to work hard to straighten out my notes; I really learned this subject! So ``bad'' lecturing can be better than ``good'' lecturing; it depends on the student. Dorothy had similar experiences as a graduate student.
WWC. I'd like to understand some historical specifics of your career. When did you come to Rochester?
AHS. In the fall of 1961.
WWC. From Manchester?
AHS. Yes.
WWC. Leonard Gillman had just taken over at Rochester as Chairman. If I recall correctly, he had a good bit of money from the Administration, with which to hire outstanding people at the senior level. Had you and Dorothy known Leonard personally, before coming to Rochester?
AHS. No. Our old friends Kay and George Whitehead suggested us to Leonard.
WWC. What convinced you to leave Manchester for Rochester?
AHS. Dorothy was anxious to move to the States to be near her family, and in particular her father, who was then nearly 80. Earlier (in 1946) we had moved to England to be near my family, especially my mother, who was then in poor health.
WWC. Leonard inherited a strong department, and made it stronger.
AHS1. He certainly made it stronger.
WWC. These days, at least in the United States, a senior professor contemplating a move will usually negotiate an extended leave of absence from the old university, in order to check into the situation at the new location for a year or two. Did you and Dorothy do that, or did you flat-out resign from Manchester to come to Rochester?
AHS. We resigned and came. Len's offer was one we couldn't refuse, breaking new ground in at least two ways: First, by diluting the usual anti-nepotism rule to allow both of us to join the faculty. (That rule was diluted to forbidding either of us ever to ``boss'' the other by becoming chairman.) Second, by giving us equal rank and equal salaries. In those days a wife's salary and position were expected to be inferior to those of her husband, especially in academia. I'm not sure that things have changed much since.
WWC. Unquestionably, Rochester profited immensely. But was it a good move from your personal perspective? Were you and Dorothy generally happy in Rochester? I notice that you stayed there until retirement.
AHS. Yes, we were very lucky. Dorothy and I were happy with the Mathematics Department, the University and the town (except for the winter weather), and we stayed there for some 26 years. We retired in 1987, and stayed on as emeriti for another year before moving to the Boston area.
The move from England to Rochester was rather hard on our children, who hadn't finished high school; but they survived very well. Though rather young for it, they were able to start quite successfully as university freshmen because of the difference in pace of education in Britain and the U.S.
WWC. You and Dorothy have kept up - I should say, you never severed - your strong ties with England. For example, now in retirement you seem to prefer England's weather to that of Boston, at least in winter.
AHS. Yes, and in summer too!
WWC. Throughout the years you have both retained British citizenship?
AHS. Yes, but this perhaps requires clarification. Dorothy has dual citizenship, having acquired British citizenship by marrying me, without losing her American citizenship.
WWC. Let's go back to an earlier period. I recall that your Ph.D. is from Princeton. What year was that?
AHS. That was 1941. My supervisor was S. Lefschetz. His main contribution to my thesis was to listen encouragingly, which he did very well. As a lecturer he was stimulating - some might say exasperating. On one occasion he began a lecture by saying, ``Today I want to be totally rigorous. So, given \epsilon > 0, take \delta > 0 such that and so on and so on and so on.'' That was the last we heard of \epsilon and \delta, which in any case were not relevant to the lecture.
WWC. I would be interested to know of mathematicians or others who influenced your development.
AHS. As an undergraduate and beginning graduate student at Cambridge, I was particularly impressed by M.H.A. Newman and A.S. Besicovitch. And I was profoundly influenced by three fellow-students; the four of us collaborated in a paper which used the theory of electrical networks to settle an old problem - the dissection of a square into a finite number of unequal squares. One of the three, Leonard Brooks, after proving an important theorem in graph theory, now known as Brooks's Theorem, became an income tax inspector; another, Cedric Smith, is emeritus professor of genetics and statistics at London; and the third, Bill Tutte, is a distinguished graph theorist.
But the mathematician who most influenced my thesis was, curiously enough, Ralph Boas, who was visiting Cambridge in 1940. He had conjectured a curious result about translations of continua in the plane, and had proved it for convex sets. I observed that if it worked for a set it worked for the frontier of that set also.
WWC. Here's a napkin, and it's more or less planar. Can you show me?
AHS. The statement is quite simple. Call a plane set A ringable if there are 3 translates of A in the plane such that every two of them meet but no point is in all 3 of them. Conjecture: The only non-ringable plane continua are parallelograms, possibly degenerate ( i.e. reducing to a line segment or a single point). My thesis departed from my result about frontiers. His conjecture in full generality is still open, I believe.
I was also influenced by the early work of S. Eilenberg in Fundamenta Mathematicae , and of course by Paul Erdos.
WWC. What about Dorothy's degree? Where did you meet?
AHS. Dorothy's Ph.D. was from Bryn Mawr, under Anna Pell Wheeler. In 1941 we were both at Princeton; she was at the Institute, and I was at the University, later at the Institute too.
WWC. How were you occupied during the war? Were you engaged in intelligence work?
AHS. At first I taught at Purdue, where Ayres was rejuvenating the department. Then I moved to Washington D.C., doing what was called ``war work'' - aerodynamics, in this case.
Dorothy also did some teaching at Purdue. We later found that she had acquired a tremendous reputation there for having flunked an entire class in trigonometry; this, however, was an exaggeration - it was only about half the class that flunked.
WWC. Your children must have been born during World War II or shortly thereafter. They both went on to successful careers in Mathematics.
AHS. It was clear from an early age that they would both take up mathematics. We didn't mean to push it on them, but circumstances had led us to give David some coaching quite early, at age six or so, and then Ellen naturally wanted it, too. They lapped it up. David went on to do undergraduate work at Harvard; his PhD is from Princeton. Ellen began her undergraduate career as Rochester's youngest freshman; she took her PhD from Cornell in due course, continuing after some university teaching to a career in industrial computer science. Their theses being in piecewise linear topology and differential topology, respectively, they could (and did) talk freely to each other, and, with some difficulty, to me.
WWC. I recall from visiting your home in Rochester that you and Dorothy kept a religiously observant Jewish household. Has that tradition been helpful to you in your work? Have you experienced resentment or anti-semitism at different moments in your careers?
AHS. Thank you for raising the subject, but I don't have much to say about my Jewish religion. In childhood I was sometimes jeered by other children, just for being different, but since then I have experienced hardly any anti-semitism, and Dorothy not much. Our timing was fortunate. Judaism is, for most Jews, more a matter of behavior than belief. It was important, I think, for our children (and also for our relatives) that we maintained the traditions. In fact, even as we prepare this Interview, we are getting ready for Chanukah.
WWC. Let's go back for a moment to an even earlier period, your own childhood and youth. Like David and Ellen a generation later, were you mathematically precocious?
AHS. I suppose I was mildly precocious. At any rate I enjoyed playing mentally with numbers and geometrical figures. In elementary school, aged around 10, I noticed on my own that n^2 - 1 = (n-1)(n+1), and a bit later I set myself and solved the problem: given a+b and ab, find a and b - thus in effect, though I didn't realize it at the time, solving quadratic equations. I was always good at written tests and exams in arithmetic, my usual method being to answer the questions as quickly as possible, then to return to make the (often necessary) corrections. This attitude has persisted, not always helpfully.
WWC. Can you tell me something about your own upbringing and training?
AHS. In those days the London County Council elementary school system, like those of other County Councils, led up to a Scholarship Examination; this later became the ``11+''. I was lucky enough to win a scholarship to a sort of public school, that is, to a private school in U.S. terminology, namely Christ's Hospital. I say ``sort of'' because it didn't have quite the social prestige or snob appeal of such schools as Eton. Like those, it awarded full scholarships to a number of able examination-passers, but unlike them it didn't charge large fees to the non-scholarship majority. It gave an excellent all-round education. We were ``streamed'' initially for different subjects, independently; a student might be in the top stream for mathematics, the third stream for French and the fifth for Latin, English and History. Science was taught in functioning laboratories; we measured densities, produced oxygen, counted bacterial plaques on Petri dishes. The higher forms did more highbrow things, of course. Year by year we specialized more and more, choosing our subjects with advice from the staff.
It was then, like most public schools, a unisex boarding school, situated in open country. School life had its austerities. Windows stayed wide open year-round, providing healthy fresh air. The heating system did what little it could, with warm-water central heating, to mitigate this in winter. We were compelled to exercise daily - ``physical jerks'' mid-morning, football (``rugger'') in winter, cricket in summer, and running all year round. And school meals were undelicious. Being forced to accommodate to such austerities has proved extremely valuable in later life.
The school also featured a good choir and orchestra, ``manual training'' in woodwork and metalwork, and some ``art.''
From the age of 14 on, which was then the legal limit, many of the pupils would leave for jobs. The intellectually successful would stay on. On arrival at the top form we would spend most of the school time on our specialities, being coached for the University scholarship examinations - University meaning Oxford or Cambridge, of course. It was here that our having taken Latin helped, since Latin was then an entrance requirement. Not surprisingly we nearly all received scholarship awards of one kind or another. These would be supplemented by grants from the appropriate County Council, so that usually, as in my case, both tuition and living expenses were covered at University.
In the few months between the examinations and graduation, the mathematics staff put the time to good use by giving the examinees short courses based on the staff's own mathematical training (they all had Oxford or Cambridge degrees in mathematics), so we were kept stimulated and educated. This was, I believe, very unusual, even in those days.
I was duly awarded a scholarship to Trinity College, Cambridge, ``reading'' mathematics. This meant studying mathematics exclusively, but that wasn't as narrow as it might seem since it included both pure and applied mathematics. (As we use the word, ``Applied'' mathematics would be covered in most American universities by the Engineering and Physics departments). Life at Cambridge was refreshingly non-austere. The lectures were organized university-wide, but each college also arranged for individual, or nearly individual, tutoring in one's speciality. One was thus encouraged to keep working. But there were also very many student societies - debating, musical, dramatic, to say nothing of sports - to occupy one's spare time. In particular, each of the larger colleges had its own mathematical society, with several smaller colleges combining jointly, and the University as a whole had (and has) its mathematical society, the ``Archimedeans.''
After three years, with an examination at the end of each, one received one's B.A. Like many graduates I stayed on for a further year of research, and for the degree of M.A.; but it should be explained that an Oxford or Cambridge M.A. is not a testimonial to academic merit, but merely an entitlement to vote in University affairs.
WWC. The standards of rigor and the expectations made of English school children in those days certainly differ from those most Americans grow up with these days. Does Britain continue to hold the line?
AHS. Every few years the Department of Education investigates the school programs, consults educationists, and issues revised curricula. Then, to judge by what the newspapers and friends say, there is not much tangible gain. What is constant is the desire to minimize spending on education. (The situation is perhaps not dissimilar to that on this side of the Atlantic.) But in the long run there have been changes, not all bad. In my youth the system provided an excellent education for the privileged few - the well-off and the academically able - and perhaps an adequate one for the rest. Things have been slowly leveling off. When my children went to school in Manchester in the 1950's, secondary education was determined by the results of the ``11+'', the examination taken at that age. Those who did well in it went on to ``grammar schools'' for a traditional education. The others went to ``comprehensive'' schools, which of course were looked down on and which were, partly for that reason, in fact inferior. More ``redbrick'' universities became available, and many grammar-school students (perhaps most) were able to get a university education. It was not absolutely unknown for a ``comprehensive'' scholar to switch over to a grammar school, or even to reach a university, but this was rare.
Since then, the reputation of the comprehensive schools has been rising, and to judge from anecdotal evidence the standard of the grammar schools has been falling, so the two systems today are probably not markedly different. The schools are now being reclassified on a quite different principle, according to whether or not they ``opt out'' of local governmental funding. I don't understand the details, but it appears once more to be a question of saving governmental money. In any case, all the British university colleagues with whom I have discussed the matter agree that students come to the universities knowing less and less. Again, I have heard similar comments on the U.S. side.
WWC. Could you compare the British and American training in mathematics?
AHS. The British school system starts at the age of five, as against the American start at six. That puts Britain a year ahead, and my impression is that this lead increases up through the undergraduate years, a British B.A. or B.Sc. being roughly the equivalent of an American Masters degree. But in graduate school the Americans catch up; the Ph.D. program typically takes three years in Britain, as against four or more in the U.S. And the American graduate students work harder than their British counterparts. Thus the Ph.D. levels are about the same. However, the British system does ensure that its Ph.D.'s in pure mathematics have some acquaintance with applied mathematics, a very valuable feature.
There is another major difference in the two systems. In America, college students are expected to work during the academic year to help pay for their education. In Britain, at least in my time, college students, being financed by scholarships, had leisure for study and recreation. Both systems have merit; the British system encourages self-education, the American system encourages self-reliance. But the British system promotes more mathematical training, I think.
WWC. The American press nowadays, sometimes conveniently ignoring our own oddities and failures, seems to take pleasure describing a sort of Fall from Grace of some of England's most praised institutions, including the education system. Do you think the British School system is as strong today as it was, say, 70 years ago?
AHS. In moving to a more egalitarian system, the British schools have perhaps retained the same total strength, but have redistributed it. There are still more new universities, some of these being technical schools renamed. One important and relatively recent development is the encouragement of women. Many schools and colleges are now co-educational, my own former school and college among them. Incidentally, Oxbridge no longer requires even a slight knowledge of Latin. In Britain, foreign languages are taught insufficiently and badly. In America the situation is even worse.
The British system seems to have retained its admirable attitude towards school games. In the U.S., each school or college will usually have at most one team for each sport; the other pupils are not involved. (Cheer leading is a separate sport, of course.) In the better British schools, all students are encouraged to take some form of physical exercise. Thankfully, British universities do not have quasi-professional athletic departments and teams. But now, under pressure from Government cut-backs, they are trying various fund-raising expedients: appeals for donations, catalog sales, and the like. So far, such popular events as the annual Boat Race between Oxford and Cambridge have not been commercialized, but you never know. Maybe professional games could yet be on the way!
WWC. Let's return now, if you will, from the wide arena to the specifics of topology. Ever since I found myself topologically inclined, back in graduate school, I've been hearing that the subject was dead or dying. Has this finally happened?
AHS. I hope not! It is true that many old problems have been solved, and their solutions have been generalized sometimes to the point of aridity. But there are always new questions.
WWC. From the texture of your own work over the decades, it's pretty clear that you see set-theoretic topology as a subject of interest and worth in its own right, rather than solely as a subservient tool.
AHS. Certainly!
WWC. Others might have it otherwise.
AHS. The attitude of many mathematicians to general topology is exactly the attitude of many physicists to Mathematics. In each case there's a belief, often mistaken, that whatever they need from the despised subject they can do just as well by themselves, and whatever they don't need is superfluous.
WWC. You have plenty of theorems which depart from a ``general infinite cardinal m,'' and you know the axioms of ZFC better than I do, but I'll ask anyway: Do you believe in cardinals greater than continuum c?
AHS. I don't really believe in any infinite cardinals. Even \aleph_0 is only a figment of our imagination. If you accept \aleph_0 you may as well accept c, measurable cardinals and the rest. Of course a contradiction may eventually arise and demolish some or all of them.
WWC. What has been your relation to arguments based on forcing, to independence results? Are you pleased when some axiom outside of ZFC settles a particular question one way, and another axiom settles it the other way? Have you personally found a ``consistent solution'' or a ``consistent counterexample'' to a problem you were working on?
AHS. I have occasionally used CH and GCH, mostly for positive results, but at least once for a counterexample. But I feel happier when a result can be settled in ZFC.
WWC. Which way do you see general topology (or more generally, pure Mathematics broadly construed) heading these days?
AHS. A glance at (for instance) a recent Topology Proceedings [of the 1994 conference at Auburn University] suggests that general topology is heading in all directions! Pure mathematics as a whole seems to me to oscillate between (roughly) periods of mainly self-absorption and periods of responding to outside stimuli. Both tendencies are vital.
WWC. Given the state of the job market and the diminution of support for curiosity-driven abstract research, two phenomena which appear pervasive and ubiquitous throughout the world, would you advise a bright scholar today to pursue ``pure mathematics'' as a career?
AHS. A career in pure mathematics should be attempted only by someone who will ignore all advice to the contrary.
WWC. If you had an opportunity to give a bit of advice to people in authority - elected leaders in government or education, say, or the President or faculty Dean at a major Institution - what would that be?
AHS. I can be free with advice here, since certainly my advice would in any case be totally ignored. Perhaps the main thing is to educate the general public to the importance of education.
WWC. Arthur, this has been valuable, informative and good fun. Thank you very much.
Middletown, Connecticut
December 4, 1996