Topology Atlas Document # topc-28.htm | Production Editor: Krzysztof Chris Ciesielski

AN INTERVIEW WITH W. WISTAR COMFORT

by

Neil Hindman

Article from volume 2, #3, of TopCom

LaTeX file is available for download.

This interview was conducted at the 1997 annual meeting of the American Mathematical Society in San Diego in January, 1997.

The format of presentation will be:

Q: ...

A: ...

where all questions are asked by the interviewer, Neil Hindman, and all answers are given by Wis Comfort.

Q: I understand you were born very young. What led you to become a mathematician?

A: It was a path of least resistance. Early on, by which I mean first grade, I was pretty good at arithmetic. I could add multi-digit numbers better than most, so I was branded by adoring relatives as a mathematician. As life went on, I found out slowly that what I was able to do is hardly what constitutes mathematics.

Actually, that early excellence served as an anchor to windward. I was not a very self assured young boy. It was a great convenience for my own psychological development to be able to tell anybody, and I guess that means tell myself: ``Yes, I'm a mathematician and that's all there is to it''. When I went to High School I knew that in at least one respect I would be near the top of the class, and when I went to college, I didn't have to go through the angst and sufferings that many people do trying to choose a major.

Q: Undoubtedly you had some help from others.

A: You bet. I must say a word or two about a wonderful man at Westtown School, Charlie Brown, or as we said, ``Master Charlie'', who fostered and encouraged my interest.

Q: ``Westtown School'', that was ...

A: That's a Quaker boarding school in the so-called ``Orthodox" tradition, outside of Philadelphia, where my parents sent me for the last three years of High School. It was a good school, a good social coed environment. Socially I learned a lot, or at least I became reasonably adept at hiding my insecurities.

Master Charlie was a good mathematician by the standards of the day, an excellent teacher who respected his subject but always had a quick sense of humor available to rescue any difficult situation. Like most other faculty at Westtown he had many non-academic duties: dorm master, athletic coach, chaperone.... It takes a while to realize that some people are honest to the core, that they will never twist the truth to ease an administrative problem. Charlie was a man of great integrity who inspired his students in that direction as well as with his evident love of mathematics. He was a graduate of Hamilton College and he just gave all of the support that one could want. As much as any man except my own father, he served as a role model for me. Of course that term hadn't been invented yet, back in 1950.

Q. What about college?

A. In those dark days, if you can believe it, calculus was viewed as too difficult for college Freshmen. When I got to Haverford College in 1950 we were given as Freshmen something which was a sort of fore-runner of what eventually became the ``new math''. It was an experimental course invented and pioneered by Carl Allendoerfer and Cletus Oakley, who were the top mathematicians teaching at Haverford. We were guinea pigs of sorts. There was a little statistics...

Q: Finite Math?

A: Yes. Finite Math and Statistics. Venn Diagrams and truth tables. It was not very difficult stuff but some of it was kind of fun and it was varied---that is, we never quite knew what was coming next. This was sort of a precursor of the Finite Math books that swept the country later. What got lost from the curriculum, I believe, to make room for the Oakley-Allendoerfer experiment, was classical plane and solid geometry in the style of Osgood and Graustein. To this day I couldn't tell you with any real understanding why a suitably inclined plane cuts a cone in a curve whose equation is x²/a²+y²/b²=1.

Q: What can you tell us about your biological family, before we get into your mathematical family?

A: It was always a wonder to my parents that I appeared to have some scientific ability or mathematical interests, because so far as is known nothing like this had surfaced elsewhere in our family. I enjoyed the great advantage of growing up in an intellectual environment, or shall we say, a professorial environment. My father was a leading classicist at Haverford College, where my grandfather was President for nearly a quarter-century. So, although an interest in mathematics came as a surprise to interested family spectators, in general terms my choice of a career in academia was almost programmed ab initio. So in making this choice I was not very imaginative. The course I followed was one of minimal resistance.

Q: When I first met you I recall seeing a rather crude drawing on your door labeled ``Martha''. You had young children at the time.

A: Both of our children were born in Boston Lying-In Hospital. Martha in 1959 and Howard in 1961. At the time I was enjoying a Peirce Instructorship at Harvard.

Q: Were you married as a graduate student?

A: Yes. Mary Connie and I got married in 1957. We had met at the University of Washington, where she was a drama major and I attended her many peformances and became a sort of stage-door Johnny. I finished my degree in 1958 and we moved to Harvard, where both children were born.

Q: You were speaking of the University of Washington.

A: The mathematician who sponsored me and made many things possible that would not otherwise have been possible was Edwin Hewitt. He had a way of sizing up graduate students rather quickly, and I think, to be frank, not always accurately. If you were taking a first-year graduate course with him, or even if you weren't---just through contacts in the halls or the lounge---he would decide on some basis that you were a good person or a bad person, that you were talented or not. Once you got on his good side things could go quite smoothly for you. And there were occasional stories of graduate students who did not get on his good side, and things did not go well for them. At any event, Ed picked me out, along with others, fairly early on.

Q: Weren't you in a group of six graduating at roughly the same time?

A: In 1958 Karl Stromberg and I were the two Hewitt students to receive a PhD. May I reminisce a while? As graduate students, Karl and I and many others would spend countless hours, I mean something like five or six nights a week throughout the academic year and through some of the summers, too, discussing mathematics at the Rainbow Tavern. The spirit was democratic and in principle we were all equals, but there was no question that Karl was de facto Dean of the group, the best source of information, the court of last resort when Hewitt's fiendish homework problems had stumped the rest of us. Karl was a patient teacher, who went on to do good work on the convolution of measures, and later on he and Saeki proved that every locally compact Abelian group is a Baire set, but never a Borel set except in trivial conditions, in its own Bohr compactification. You will be familiar with some of his later books. The faculty at Washington was terrific---Carl Allendoerfer had left Haverford in 1951 to become Chairman there and to build a strong department, and this he did quickly, with skill and enthusiasm---but my main professor if I can put it that way was Karl Stromberg. The debt I owe him is great indeed.

As graduate teaching assistants in those days we were paid if memory serves $125 a month, later raised to $150 after we passed the PhD Qualifying Examinations, an amount apparently shrewdly determined by some higher authority to keep us adequately in cigarettes and beer, but not sufficient to subsidize any serious trouble. Beer was available at the Rainbow Tavern at $.75 for a large pitcher, not a bad price, but cigarettes were expensive: 20 cents for a pack, although Karl kept a pipeline open down to Oregon which supplied him with a brand called Herbert Tarreytons at 18 cents a pack. Karl's devotion to Tarreytons was one of his few excursions into bad judgement and terrible taste but I think there was logic behind it: He was willing to bum a real cigarette off any of his buddies when occasion demanded, but you would think more than once before asking Karl to return the favor, knowing that a positive response would yield a Tarreyton.

Ken Ross was another Hewitt student, a couple of years behind me and Karl. He reminds me that at my graduation in 1958 the Department of Mathematics gave eight PhD degrees. I don't remember everyone, but some were pretty fine mathematicians: Robert Phelps, for example, who was later invited back as a faculty member at Washington, and the Canadian Colin Clark, who went on to do important work in Mathematical Biology.

Q. How did you happen to work for Hewitt?

A. Hewitt would pick you out as a graduate student, and sort of let it be known ``well now you're working for me, my boy''. That pretty effectively closed off any discussion as to whether you might possibly be working for someone else. At that stage I had strong topological interests, as I still do and always have, and left to my own devices I might very well have ended up working for Ernie Michael or Victor Klee, both of whom were on the faculty and were accepting graduate students. But I never approached either of them because it seemed pretty clear that I was working for Hewitt.

Among his other talents, Hewitt had a masterful way of tailoring problems to the intellectual abilities of his students, as he perceived them. So under Hewitt you didn't struggle and waste a lot of time looking for a problem. In the 1950's he had in mind a major program, a multi-faceted project embracing at least and culminating later in his two-volume magnum opus with Ken Ross. At some moment he would simply call you in and say, ``well, here's your PhD question. This is a question which sounds as though the answer should be `yes' or `no'. Which is it?" My question, however, concerning the l₁-algebra of a communtative semigroup and its so-called Silov boundary, also had the advantage of being sufficiently open-ended that I suppose it was almost guaranteed that some results would appear given sufficient time.

Hewitt had another good facility, something many thesis advisers practice while many of us don't. It is very much worth while. Namely, I would have a weekly date with Ed, whether I liked it or not. It was scheduled, and it was earlier in the morning than I would have chosen if the decision had been mine alone. I forget the details, but ``early'' was probably nine o'clock. At any event, once a week, let's say Thursday morning at nine o'clock, I was going to spend an hour with Hewitt and in principle tell him about my progress in the last week. He was there, rain or shine, and so was I, whether I had anything to say or not. I think it's a good way to proceed. I tend to let things slide with my students, on the grounds that if they have anything to ask me or tell me, they'll probably find me.

Q: I recall a particular student who would probably see you four or five times a week, speaking of myself.

A: We have different ways of doing these things, and I kind of prefer to leave it extremely informal. Yes, it might very well be four or five times a week, and some of those sessions might last thirty seconds. Or they might last thirty minutes. I guess my philosophy is to try to be available, but not to force things. It seems to work with some students, but I think there are some for whom it doesn't work, and in those cases the advisor should be prepared to schedule regular appointments.

Q: You have mentioned being a student and you have mentioned your students. Let's go back in the easy direction of the genealogy first. Hewitt was a student of Marshall Stone. Do you know the genealogy further back?

A: Well, Stone was a student of G.D. Birkhoff at Harvard in the 1920's, and Birkhoff earned his degree from Eliakim Hastings Moore at Chicago. It's a matter of public record that Moore took his degree in 1885 at Yale from Hubert Anson Newton. It should be easy to push back at least one more generation, since apparently Newton was himself a Yale PhD. In the past year or so I have made countless inquiries of the folks at Yale to learn who was Newton's supervisor, and in what year. The Department of Mathematics at Yale, the Alumni Office, the Office of Yale History, the Department of Records and Archives, the Reference Desk and the Inter-Library Loan people---I have had no luck squeezing any quantum of interest in this matter from any of those folks. But the real fault here is mine, not Yale's. I have cursed myself many times for not having kept a careful record of a conversation that I had once with Hewitt. At one time, he was able to push things back many generations. I do remember the bottom line, or should I call it the top of the pyramid: Hewitt claimed we all go back to Gauss, and he had the data to prove it. I recall along the way the name of Gudermann, the teacher of Weierstrass. You can find reference to the complex-valued Gudermannian function G, which has the virtue that it transforms the hyperbolic functions of a complex variable z elementary trig functions of G(z). But a chain with a missing link is no chain at all and I am unable to furnish the data linking us to Gauss. ``Where did I come from?'' I guess is one of life's eternal questions. Someday one of our clan will find the time and have access to Yale's records, a privilege not given to Wesleyan faculty, and be able to pursue the matter.

Q: Going the other direction, I would think by most people's standards, you have been advisor to a large number of students. I know for sure you have great-grandchildren.

A: My first two students were at the University of Rochester, where I went after Harvard. I spent four years at the University of Rochester. First there was Negrepontis, who subsequently rose to great prominence back in Greece. He has served as Vice Rector of Athens University, a position of visibility and influence in Greek higher education, and of course he has turned out a tremendous number of very good students. Let me say Analysts, but Topological Analysts, doing tough problems on the fine combinatorial structure of Banach spaces.

Norman Noble was my second student, also from the University of Rochester. Very much a self-starter. Just point him vaguely in the right direction and was off and running. After those four years at Rochester I was as you know at the University of Massachusetts for two years. Then you and I and Victor Saks came down together from UMass to Wesleyan, and you were the first of my 12 Wesleyan PhD's.

I don't know for sure, but there must be more than great grandchildren by now. These things can move quickly. Stelios' first student, Grant Woods, would be my first grandson. And Grant has had many students.

Q: Including Alan Dow. Has Dow had any students finish?

A: As of this moment I think not, but I am told that at least one is nearing completion.

Alan is certainly someone for me to be---not that I have any right to be proud---but he's obviously very prominent, a genuinely significant mathematician.

In recent years Wesleyan has developed sort of a pipeline to Mexico which has fed some very good students to Wesleyan. Salvador Garcia was one, and Javier Trigos. I have a Mexican working for me now.

Q: Turning to mathematics itself, you have proved a large number of things. What of the things that you have proved would you classify as your favorite or favorites?

A: Well, I always find it fun to pin down something involving some specific infinite cardinal numbers. I guess that's a theme that has gone on through some of my work. That is, trying to evaluate specific cardinal numbers associated with specific spaces under hypotheses that are not too overbearing.

But any good work that I have done---I think there's no exception to this statement---has been done with someone else. The someone else varies. In other words, I've had a lot of co-authors. For some reason, just for fun, I counted them up not too long ago. The number came to 40. They are not only of diverse personalities and interests, but of diverse national origins. 15 countries are represented, including USA. It's strange, unexpected. When I got into this business I had no idea that one of the fringe benefits would be substantial international travel. But that has certainly been the way it has worked out.

It is a small fraternity. Whose work can I read perceptively and in detail? Very few people. Who would care to read any work of mine? Very few people. This involves finding each other. Sometimes I go where they are, and sometimes it works the other way around.

Q: I think I would like to dispute your assertion that all of your good work had been done with someone else. I think that there was a paper by the title ``A theorem of Stone-Cech type and a theorem of Tychonoff type, without the Axiom of Choice; and their realcompact analogues", which got a great deal of attention when it first came out. Didn't it?

A: Yes. That was kind of fun. And like most other things of some value, someone else will read it and show that it could have been done either much better or more briefly. In the case of that paper, both. But, yes, I enjoyed that and I guess the title as you just gave it more or less describes what goes on there.

Q: But that was an individual work, wasn't it?

A: Yes. It's surely the longest title I've thrust into the literature, and the only one bearing a semicolon.

The result that I have been most associated with---I don't know if it's the best theorem but it certainly has received considerable interest and follow up---goes back to the sixties with Ken Ross. This concerned pseudocompact groups. Namely, the product of pseudocompact groups is pseudocompact. The pseudocompact groups are exactly the totally bounded groups for which, when you take their group completion---of course that's compact---that group completion turns out to be the Stone-Cech compactification. Again, the theorems in that paper have been greatly generalized in ways I think it is fair to say neither Ken nor I had any remote thought of at the time. We showed that among totally bounded groups, the pseudocompact groups are those that are G_\delta-dense and subsequently C*-embedded in the group completion, which is compact. This has been worked over and improved and generalized in many ways by many people. A significant body of literature has grown out of that paper, which I find gratifying.

Of course one thing Ken and I did not do, though we certainly tried, was to prove a similar theorem for countably compact groups, namely that the product of countably compact groups is countably compact. It's a good thing we didn't prove that theorem since now at least consistent counterexamples are known. I don't believe that it is known whether there are models of ZFC where that theorem could hold.

You were asking for other theorems that I have enjoyed or taken some pride in. There's some recent work with Salvador Hernandez and Trigos concerning the Bohr topology on locally compact groups G, namely we showed that every closed subgroup of G is C*-embedded in G when G has its Bohr topology. I think there's some pretty good mathematics in there, at least something that we're all proud of. We answered a question asked by Eric Van Douwen, so the paper is hardly transparent. Will it turn out to be of any long-term significance or use? Who knows? In any case it was a lot of fun working with those people, and I take a lot of pride in that paper.

Among the co-authors I've enjoyed working with are long-term Wesleyan colleagues Tony Hager and Lew Robertson. Tony and I in the 1970's showed that every infinite countably complete Boolean algebra B satisfies |B|=|B|^\omega, and in other work we found a number of reasonable upper bounds for cardinal numbers of the form |C(X)|, in terms of other cardinals associated with X. Working with Tony---if you want to keep up with him---can be pretty intense; during the period of serious progress you need to suspend or at least reschedule your normal daily routine---eating, sleeping, athletic activity and the like. Working with Lew is a bit more relaxed but the rewards are equally tangible. He and I wrote several papers in the 1980's in and around the general question whether every pseudocompact topological group of uncountable weight has a proper dense pseudocompact subgroup. I guess our best result in this direction is to the effect that for zero-dimensional Abelian groups the answer is ``Yes". Later in a sequence of papers Jan van Mill and I found a number of groups with various properties P which do and do not have proper dense subgroups with property Q. Working with Jan brings special pleasure and a jolt of humility. He can be amazingly quick and incisive. From a vast store of tricks and techniques he seems able to recall just the right tool at the right time. I expect that a look at Jan's curriculum vitae would confirm the thought that he achieves a great deal, both in his own research and as an editor, by moving through life with little wasted moton and little need for back-tracking. Get it right the first time, and move on!

Q: Who have you been working with most, lately?

A: In addition to the work with Trigos and Hernandez in and around the Bohr topology, I have enjoyed a long and productive association with a young German mathematician, Dieter Remus. We became acquainted nearly 15 years ago in the classic awkward context---ignorant of each other's work in progress we had each devoted a lot of time to the same circle of problems, and sure enough we had achieved very similar theorems.

Q: What happened then?

A: For my part I recalled a comment Hewitt had made once discussing a similar instance in his own life: Upon inspection, such contretemps are usually less serious than at first glance. Around the periphery, he said, each team will probably know some useful things that the other one missed. That was certainly the case here. Eventually Dieter and I pooled our talents and we have gone on to write several papers suggested by what is known as Markov's Fourth problem: Does every infinite group admit a non-discrete Hausdorff group topology? The brief answer there is ``No" but for Abelian groups it is ``Yes", as has been known for decades. Refinements of that question---for example, How many?, and What is the shape of the poset of such topologies on a given group?---have been intriguing and profitable. Working with Dieter in what are presumably my latter productive years has been great fun, an invigorating challenge.

Q: Let's turn to more general things. What would you think are the most significant developments in your kind of mathematics during your career?

A: I'd have to think about that a while, because the theorems that I have found absolutely devastating or absolutely intriguing---there haven't been that many of them. There haven't been that many surprises.

Q: I remember something you wrote in one of your surveys that everybody agreed that the P-point independence proof was the most significant result in that short span.

A: There are certainly special theorems that have come up and are extremely interesting. Yes, the independence of the existence of P-points in \betaN\setminusN was a surprise and maybe a shock. To go to really blockbuster theorems, I'd probably have to go back to the Godel incompleteness theorem. I've never been real happy with independence proofs or consistent solutions. I always like to see something settled, and consistency results are unsettling.

Q: What are some of your favorites?

A: There's this question known as ``Blumberg's Question'', namely: Does every compact Hausdorff space have the property that every real-valued function on it is continuous when restricted to a suitably chosen dense subset? Of course the real line is not compact, but it was Blumberg's theorem on the real line that generated the general question. He proved, in the Transactions back in the early 1920's, that every real-valued function on the reals is continuous when restricted to a suitably chosen dense subset. But Bill Weiss' example for the compact Hausdorff case, if I recall the situation correctly, has great logical appeal. Weiss he built two spaces. These spaces are absolutely defined. You don't need any special axioms beyond those of ZFC to build either of the spaces. They are simply there. You define them. Then, for one of the spaces he proves that if the continuum hypothesis fails, then this particular compact space, X let's call it, answers Blumberg's question in the negative. And, if the continuum hypothesis holds, then this other space, Y, answers it in the negative. Then one can take the disjoint union of these two spaces, and that exists in any model of ZFC---there it is---and just by elementary logic it is clear that this space is an example which answers the question in the negative. In each model there is a real-valued function on that space which is not continuous on any dense subset. In that model either CH holds or it doesn't, so we focus attention either on Y or on X. I've always enjoyed the argument immensely.

Q: I remember as a student at the University of Massachusetts, first signing up for a course called ``Rings of Continuous Functions", which I did on the recommendation of some fellow students. And the recommendation was ``Gee, you've got to take Comfort. He's a great lecturer." You mentioned when talking about your early life, that one gets into mathematics for reasons which aren't really related to what mathematicians actually do. And I, at least, was led to you for reasons totally independent of what kind of mathematics that you did. I suspect that that's not uncommon. I wonder if you have any thoughts on the relationship between mathematics and pedagogy.

A: We are a more human discipline than I guess is often recognized or spoken about. Everybody needs encouragement, or stroking, or whatever the word is. For some people this is very much a necessity, for others not so much. I try to give encouragement when I think it may be needed or may do some good. As a student you personally didn't need any mathematical help. You would have written a strong creative thesis in any area, extreme points or Poincare stuff or even PDE. But I bet that eventually you would have found your way into the combinatorics of \beta N\setminus N and Ramsey theory, with or without a gentle push from me.

As to being a good or bad teacher, I guess the techniques I have tried to use at the graduate level would be about the same as the techniques that I use in an elementary course or in any other lecture. Essentially, it's clarity. Put yourself in the place of the naive, ignorant student. I try to adhere to two rules, which is not always easy because they have elements of potential conflict. First, remember how little they know; second, don't talk down to them. It helps to be prepared. It helps to know what you plan to cover that day. Don't try and fake them out. Don't pretend that you understand what you are talking about if you don't. You'd better withdraw and regroup and find out what you are talking about and come back a little later and get it right instead of trying to bluff your way through. That's one lesson I more or less taught myself. And don't try to give them a snow job. At least at the graduate level there's undoubtedly someone out there who is smarter than you are and will be real happy to come back and nail you later.

I consider mathematics to be a very human profession. My personal contacts with students, my administrative obligations with the American Mathematical Society---these contacts on a personal level meant a great deal to me. They are as important as the mathematics itself.

Q: What do you think are the most significant issues confronting the mathematical community today?

A: One can't help being concerned with this phenomenon: Why are we not understood? Why are we not appreciated? How does it happen that the level of mathematical sophistication and understanding that we find in American society appears to be deteriorating? If one may judge on the basis of the high school students that we see entering college, it's worse now than it was twenty years ago. Why should that be? Why should it be that people still sort of laugh at mathematics? They say with pride and without shame, ``Well, I never got beyond ...'' wherever it is. That's a pity, but it's worse than a pity; it's a loss for society. They do not have understanding. They're not getting it at an early age. They're not getting it at an intermediate age. Our vice president for academic affairs at Wesleyan is in many ways an intelligent, sophisticated individual, but he has absolutely no concept of what is involved in mathematics or its teaching. Our President got off a good line in a recent faculty meeting, where the Mathematics department was enjoying a mild going-over from some sort of faculty committee charged with oversight of the curriculum: ``I never got to Calculus myself, but I understand it was simply thrilling." His little joke helped to defuse an unpleasant confrontation, and that I appreciate, but it makes nevertheless the point that even high-achievers can be quite innocent of anything approaching scientific appreciation. It's not simply that he doesn't know any Calculus; my guess is that he has never spent a moment pondering the issues and problems associated with encouraging mathematical understandings and insights. Our Trustees are successful men and women, deeply interested in fostering the continued excellence of Wesleyan University. At some silly social level, they admire mathematicians, even stand in awe. But when educational decisions are made, the same catch phrases that determine policy for Romance Languages and Sociology are applied to the sciences and Mathematics, too.

It's difficult to complain without appearing to whine. Forty million dollars, which is roughly the National Science Foundation's total subsidy of research in ``pure" mathematics, still seems like a lot of money to me, until I recognize that the total federal budget is around 1.6 trillion dollars. So despite the lip-service given to what is sometimes called curiosity-driven mathematical research, it comes in at about one part in forty thousand, or one four-hundredth of one percent, of the total---not even third-order round-off error on what we devote to so-called defense and weapons of destruction. Of course, it was science that brought us the bomb. But history does teach us that today's subsidy of pure science brings huge benefits tomorrow. Anyone should be able to read and absorb that message, from the legislators in Washington to the trustees of Wesleyan University. It is certainly true that we are losing our place as world leaders in mathematical education. You can live for a long time on your reputation, but not forever. My impression is that many countries which have far fewer assets than we do spend much more on pure research in mathematics. For example, in the past few months I have attended topological conferences in Mexico and Spain. Their mathematicians are well-subsidized to act as hosts, bringing in visitors at local expense from around the world, and they are themselves generously subsidized to travel out of the country, too. Look at the demographics of our graduate programs. Essentially the programs would be dead, even the best ones, if it were not for the availablity of masses of foreign graduate students. Over half the PhD's given last year in the mathematical sciences, by all US graduate schools, were to non-citizens. Short-term, it's terrific. We at Wesleyan have profited from some really excellent students, from Ethiopia, Mexico, Russia, China, Chile and Peru. Everyone says the supply of talented Chinese applicants is infinite. But what an artificial situation this is! For the long haul, we are educating none of our own. And eventually, the center of gravity will shift closer to Peking, and when we need a huge pool of mathematical talent to carry the torch, it won't be here.

Q: I got a disturbing little report, talking to Bruce Rothschild about the job prospects, and he said ``well, when we retire and aren't replaced" which he sort of took as a given. There are a large number of mathematicians who are my age and older, that is within twelve years of sixty five, and you would normally think ``OK, that's good for the community because there will be a lot of vacancies''. But then, these folks don't understand mathematics, and so they don't appreciate it.

A: One of the things we are finding, even at a good school, a prestigious school like Wesleyan, is that we are being asked, either overtly or sotto voce, to restructure our classes and their content in such a way as to please the students. That's not quite the way it is stated of course, but the idea is to give the students more of what they want, which is to say, less. It appears to be an invitation to water things down. So, the standards of rigor that one would employ not so long ago in the classroom, not only do the students not adapt to those very readily, but it appears that the administration itself takes no pride in having things taught up to standards. At least, they offer no tangible encouragement. Even at Wesleyan, I'm afraid that we are not teaching as much or as rigorously as we did some years ago. And I don't think there's anything unusual about us.

Q: Well, I have the experience of having taught students from a much wider variety of backgrounds, and my observation has always been that a calculus student is a calculus student is a calculus student. That the students at Wesleyan have the same troubles at the same places that the students at Howard do. This is not true of the lower courses. When I was there, Wesleyan didn't even have anything below calculus. I don't know whether they do now. Binghamton for instance did have classes below calculus and their students in those classes were much better than Cal State students or Howard students. But the calculus students---they select themselves out. What are your lower courses now?

A: We've introduced a high school level precalculus course. The way it came about tells something of the story. As you indicated, for many years there was no need at Wesleyan for such a course. Then the need became apparent. Initially the department toed the line. We said: OK, we'll teach a course like this, but it will be a non-credit course because this is not collegiate material. And that seemed quite logical. And some people took it, and a few of them even went on successfully, to Calculus. But after all, there are only so many hours in a student's day. Many people who needed the course didn't take it. So, after a period of three to four years on that basis, essentially we sold out under pressure and said ``fine, we're already teaching this course using college faculty and college facilities, let's call it a college course and give them credit for it''. And that's what we're doing now. You can take a course at Wesleyan University in High School algebra and trigonometry, and you get college credit for it.

Q: Thank you very much, Wis. Do you have any parting words of wisdom?

A: You mean advice: Keep your sense of humor.