Dorothy L Bernstein: Miscellaneous writings

We present below four different pieces by Dorothy L Bernstein. The first is an autobiographical account given as part of the 1978 panel "Women Mathematicians before 1950". The second is the Foreword she wrote to her sister Yaffa Draznin's book "It Began with Zade Usher: The History and Record of the Families Bernstein-Loyev/Lewis-Mazur" published in 1972. The third is a piece of fiction written by Dorothy L Bernstein when she was a 16 year old schoolgirl. The fourth writing is a short extract from the interesting paper 'The Role of Applications in Pure Mathematics', The American Mathematical Monthly 86 (4) (1979), 245-253.

Click on a link below to go to that piece

Women Mathematicians Before 1950.

The history of the Bernstein family.

Sink or Swim.

The Role of Applications in Pure Mathematics.

1. Women Mathematicians Before 1950.

The Association for Women in Mathematics sponsored a panel on "Women Mathematicians before 1950" at the summer meeting in Providence, R.I., on 9 August 1978. A meeting was planned, moderated, and edited by Pat Kenschaft, Montclair State College. The speakers were Dorothy Bernstein of Goucher College, M Gweneth Humphreys of Randolph Macon Women's College, Anne F O'Neill of Wheaton College, and Mina Rees of City University of New York. We present below a version of the talk given by Dorothy L Bernstein.

Dorothy L Bernstein.

In looking over the article by Judy Green in the April, 1978, AWM Newsletter, "American Women Mathematicians - The First Ph.D.'s", I discovered that I had a direct connection with two of them - Florence Allen and Clara Bacon. Florence Allen, a 1907 University of Wisconsin Ph.D., was still there as an instructor when I arrived as a freshman in the early 1930's| I think she taught me analytic geometry. I know I shared an office with her in 1941 when I was back as an instructor myself. Clara Bacon, who received her Ph.D. from Johns Hopkins University in 1911 and was Hopkins' first woman Ph.D., I believe, and Florence P Lewis, a 1913 Hopkins Ph.D., were for many years the mainstay of the mathematics department at Goucher College, where I now teach. They started the tradition of an unusually strong mathematics curriculum for a woman's college, a tradition which has been steadily maintained to this day, under a succession of women chairmen.

But to return to Wisconsin - it was an exciting time to be a student, since Alexander Meiklejohn's experimental college was just finishing its short but brilliant career at the university, and one of its legacies was the system of advanced independent study, in which I spent my junior and senior years. For two full years, including summers, a student was allowed to study his major subject on his own, attending such lectures as he desired and talking with any member of the department he wished, but with no exams or grades. He was expected to write a thesis during his second year and take a comprehensive examination at the end of the period. On the basis of these two things (the thesis and the single examination), he was awarded either a B.A. degree or B.A. and M.A. degrees, together with the appropriate number of credits and a single grade that was determined by the department. Most students, of course, were unwilling to take the gamble, but the few of us who did thoroughly enjoyed it. In my case, I would study some subjects like advanced calculus or introductory abstract algebra for a month at a time, by myself, meeting weekly with my advisor, Mark Ingraham. For other things, such as complex variables or Galois theory, I attended the regular graduate course lectures. My thesis was on finding complex roots of polynomials by an extension of Newton's method. I received my B.A. and M.A. degrees at the end of four years and then returned to Wisconsin for an additional year of graduate work as a teaching fellow. Besides Professor Ingraham, I especially remember among my teachers Rudolf Langer and C C MacDuffee. All three encouraged me to continue graduate study in mathematics, but at another university.

So I came to Brown in 1935 and here became aware for the first time that some people made a distinction between men and women in mathematics. For example, I was assigned to teach a course in remedial algebra at Pembroke College, then a separate college for women at Brown, and Hugh Hamilton was assigned to teach the same course to the Brown boys. When we discovered that I had 3 girls in my class and he had 45 boys, it seemed natural to both of us, he from California and I from Wisconsin, to make two classes of 24 students each. But the chairman, C R Adams, would not hear of the idea, saying that the Brown boys would not stand being taught by a woman instructor. I pointed out that I had taught boys in Madison the previous year, but nothing was done, and we continued for a full semester - I with 3 girls and Hamilton with 45 boys.

There were five or six women graduate students in mathematics at Brown at the time, and though we felt a bit isolated, our fellow graduate students accepted us readily. I enjoyed my courses and began to think of writing a thesis in analysis under J D Tamarkin; but first I had to take my qualifying exams, or prelims. I did this after one year at Brown; they consisted of two full afternoons - about eight hours - of oral examination by the entire Brown mathematics department. It was quite an ordeal, but I was able to answer everything they asked. I found out later that some of my fellow graduate students had prelims which lasted only two or two and one-half hours. When I asked Professor Tamarkin about this, he admitted that my exam was extra long for two reasons: I was a woman and I had taken most of my courses at a Midwestern university. I am not sure which of the two was more prejudicial. (By the way, Brown's attitude toward women was typical of many other universities at that time - my experience is an illustration.)

I wrote my dissertation on multiple Laplace transformations. As in many analogous cases, the jump from one to two dimensions was by no means trivial and it turned into a very interesting problem with many ramifications into functions of several complex variables, functions of bounded variation in higher dimensions, and partial differential equations.

Meanwhile, I left Brown for my first full-time teaching job, as instructor in mathematics at Mount Holyoke College. I cannot resist one more Brown story in this connection. It concerns R D D Richardson, who was Dean of the Brown Graduate School and also Secretary of the American Mathematical Society for many years. In his latter capacity, he was consulted by many people about hiring personnel - I have heard stories, perhaps exaggerated, that he was a one-man employment bureau for mathematicians throughout the country. I do know, however, that when I came to see him, as we all did, about a college teaching job, he took out a map of the United States, covered the region west of the Mississippi and said: "You can't get a job there, because you are a woman." Then he covered the part south of the Ohio River and said: "And you can't get a job there, because you are Jewish." So that left the Northeast quadrant. Well, it happened that I heard of a job at Mt Holyoke and after a visit to South Hadley, I got the position. When I told the Dean, he expostulated "But I had that job all reserved for Hamilton!"

I was at Mt Holyoke from 1937 to 1940, meanwhile receiving my Ph.D. from Brown in 1939. Then I returned to Wisconsin and was an instructor at the university. During that time my mathematical interests expanded to include general integration and measure theory. I worked on this with Stan Ulam who was also at Wisconsin. Meanwhile the United States had entered World War II and in June, 1942, I went to the Statistical Laboratory of the University of California at Berkeley as research associate to Jerzy Neyman. They were doing a lot of secret computing for the Army, mostly with desk calculators, and I remember Evelyn Fix and Elizabeth Scott working steadily to all hours of the night in order to compile long lists of figures which they would mail every morning to Washington. I worked on theoretical problems in probability which Professor Neyman gave me, and also taught a graduate course in probability theory in the mathematics department. However, after eight months, I left - Neyman and I did not see eye-to-eye on what was the mathematical justification of a statistical procedure. Of course, since then I have learned
that this is not unusual - mathematicians and statisticians do have fundamentally different points of view, even though they may use the same techniques.

After a few months of unemployment, I came to the University of Rochester in 1943 and stayed there happily until 1959. I began teaching on the women's campus, but was soon transferred to the men's campus, or "river" campus, which had a Naval V-12 Unit. After the war I stayed on the river campus; by then all upper-division and science courses were taught there, with the girls commuting to their classes. Indeed, a few years later the University of Rochester became officially coeducational and the river campus had classes and dormitories for both men and women. I taught both undergraduates and graduate students and rose through the ranks to full professor.

Soon after I came to Rochester, C B Tompkins who was working at Engineering Research Associates on an ONR contract under Mina Rees asked me to undertake a study of what was known in existence theorems in partial differential equations, since some of the proofs could be used as basis for the computational solutions of non-linear problems that were just being tackled by high-speed digital computers. This resulted in a book: Existence Theorems in Partial Differential Equations, published by Princeton Press, which appeared in 1950, while I was spending a sabbatical year at the Institute of Advanced Study in Princeton.

This is the end of the period we were supposed to discuss, according to my understanding of the assignment, so I will pass over the next 28 years by saying that I was at Rochester until 1959, except for a sabbatical year at the Institute for Numerical Analysis at UCLA. Since 1959, I have been at Goucher College, again with the exception of 2 sabbatical years, one as Visiting Professor of Applied Mathematics here at Brown, and the other spent half at Brown and half at the University of Tennessee. I have continued research and teaching and have become particularly interested in how to combine pure and applied mathematics in the undergraduate curriculum, but that is another story.

I would like to speak briefly about the contrast between attitudes and opportunities for women with bachelor's degrees in mathematics in the late 1930's and at the end of the period we are discussing, 1950. Before World War II, the only opportunities open to them were high school teaching, a few civil service jobs doing mostly routine statistical calculations, a few low-level actuarial jobs, and graduate school with the remote possibility of a college teaching job at the end. This changed drastically after the war, for two reasons. During the war, women had taken jobs formerly held by men and had shown they could handle them, and also the computer explosion opened up many new areas to mathematics applications with many new jobs resulting from this.

On the other hand the attitude of women also changed. In 1950, most of the women graduating as mathematics majors, like most of their sisters, were interested only in getting married and raising a family. The numbers going to graduate school and entering professions decreased markedly. In contrast, before WW II, there was a slow but steady number of women going seriously into mathematics. For example all six of the women who were graduate students at Brown when I was there continued in mathematics. Some married and some did not; but they are all, or were for many years, college teachers, industrial mathematicians, editors of mathematical journals, and so on.

One final observation: There always has been a very nice, strong tradition of older women in mathematics helping younger ones as much as they could. Julia Bower, who has a Chicago Ph.D., 1932, reminded me yesterday of Anna Pell-Wheeler of Bryn Mawr, who was very helpful in this way. I remember Marie Weiss and Marie Litzinger, both of whom died relatively young, as being very supportive. But then, I can also say that, with a few glaring exceptions, the men I have known in mathematics have been equally kind and generous in their support.

2. The history of the Bernstein family.

Yaffa Draznin, Dorothy Bernstein's sister also known as Clarice, published the book "It Began with Zade Usher: The History and Record of the Families Bernstein-Loyev/Lewis-Mazur" in 1972. Dorothy Bernstein wrote a Foreword for the book and we give a version of this below.

Foreword by Dorothy L Bernstein.

One of my earliest recollections of life on the farm in Jackson is that of meeting a stranger at the supper table, being told that he was a cousin from Milwaukee (or Chicago or elsewhere), and then running to my mother with the question: " But how is he related to me?" Fifty years later, I attended the bas mitzva of my niece Joan in Los Angeles. There I saw her being introduced to a stranger, being told that he was a cousin, and then seeing her running to her mother with the question: "But how is he related to me?" We have all had to face this question many times, especially at weddings, funerals, and other special occasions; as the family has grown and dispersed, it has become more and more difficult to answer. Several of us have felt the need of some kind of family history, and a few desultory starts were made; but not until my sister Yaffa undertook it was the project finally brought to fruition.

The result is this book, and what a magnificent achievement it is! It presents a pageant of 200 years in the life of a family, clearly and beautifully told, with complete intellectual honesty and with all the thousands of details of names and places and dates carefully authenticated. Some of us will be most interested in the early story about the beginnings in Russia; some in the chronology of the various branches in this country; some in the specific details about certain persons and their descendants; and some in finding out about the new branches that are beginning to flourish. I found the whole story fascinating.

We are particularly fortunate in having Yaffa as the author. For the past seventeen years, she has worked full-time, first as a research associate at the University of Chicago, then as the editor of various internal and technical publications (for which she has won several national and international awards), and now as a freelance writer whose articles are regularly published. This background has given her the skill and competence needed for such a project; but, from my experience as a mathematician for thirty-five years, I appreciate that this is not enough. She also brought patience, enthusiasm, imagination, and tireless effort to the task; and the result is, in my opinion, a real work of creative scholarship.

Although this book gives the bare bones of the family chronicle, I think that everyone who participated in the story recorded here can be proud. Except for the early generations, there are few details given about private lives, careers, and special events. (Some of this will appear in a forthcoming volume.) But one does get a sense of a family whose members are always active in the life of their times. Whatever movement was afoot - pioneering in Palestine, coming to America and settling in many parts of the country, homesteading in Utah, raising large families, sending children to college when this was a rare thing, having women work in business and professions, being actively concerned with Socialism and Zionism, fighting in wars, successfully battling depression - members of this family were in the thick of it. The zest for living, which shows through all the generations, is to me the most outstanding characteristic of this mischpucha.

My parents, to whom this book is dedicated, would have been very pleased to see it appear. From my earliest childhood, I know that the family was important to them, and they showed this in their actions. To this day, I still hear tales from this or that individual, about the quiet but crucial support they received at some point from Jake and Tillie. To us, their children, they gave a very great gift: the courage to lead our lives. Sometimes this was in the direction different from the patterns to which they were accustomed; but their love and help was always there. For this we shall always remember and honour them.

Baltimore, Md. November, 1971

3. Sink or Swim.

Dorothy Bernstein graduated from the North Division High School, Milwaukee, in 1930, at age sixteen, as valedictorian. She contributed "Sink or Swim" to The Commencement Tattler 1930, The North Division High School, Milwaukee, Wisconsin. We present a version below.

Sink or Swim by Dorothy Bernstein.

"The manager wants to see you, Burt."

Surprised and a little frightened, I jumped up to obey the summons, for an invitation to Mr Kane's office was unusual. I paused before the door marked "Parker and Larson, Advertising Agency, Private," and then entered.

As I closed the door, Mr Kane, our genial grey-haired manager, who knew advertising and men better than any one else I have ever known, rose.

"Good morning, Burt. This gentleman wants to see you."

He motioned toward a short, red-faced man who had been standing beside the table, but who now came forward.

"Good morning. I'm the vice-president of the Carter Swimming Suit Company, and I suppose you're the young man who has been writing our ads these last four weeks." This was stated, rather than asked. "Well, the ads don't go over, understand? They don't click; people don't buy. All of this," waving his hand vaguely to include the whole room, "is expensive, but we get nothing out of it." He said this rapidly and emphatically.

Then he turned abruptly and picked up his hat. His voice calmed down. "That was all I wanted to say. As Kane will tell you, we are allowing you two weeks more, and if nothing happens by then - " The sentence was left unfinished: he had left the room.

"Sit down, Burt."

Dazed as well as astonished, I obeyed. "Does he mean my ads are no good?" I demanded.

"No, not quite that," answered Mr Kane, soothingly. "The style is wonderful. Its the best you've done yet. And that idea of yours, of running a series of lessons on how to swim, is great."

"Well, what's the matter, then?"

Mr Kane shook his head. "I don't know. It's just that the ads haven't - well, they don't ring true. That's what makes them so stiff; as if they were written by someone who didn't know how to swim." Suddenly he turned to me. "You can swim, of course, can't you?"

I shook my head and blushed. "I - I - never cared very much for the water."

He thought for a while and then spoke. "Burt, I want you to take your vacation now, instead of a month later. Go to some lake and learn how to swim. Learn to swim like a fish."

I must have turned pale, for he added. "Remember, it's not only for yourself, but for the company. Carter's ads are just the thing we need to put us over, big."

He arose and walked with me to the door. "Be ready to start tomorrow. Of course, it's at the company's expense. You can send your ads from the lake. Good luck."

I was too bewildered to say anything, and I was still slightly perturbed when I found myself on a train, the next day, speeding toward Lake Mulidin, with only a small suitcase, in which the most prominent article was a new, red and black bathing suit.

Lake Mulidin turned out to be much better than I had expected, and the cottages looked quiet and restful. That night I met some college chums, and they gave me a fitting reception. The evening would have been perfect, if it had not been for Ted's parting remark, "I'm giving a swimming party out at the cottage next week, and I'm going to count you in."

I didn't sleep much that night, you can be sure. Next morning, the sun found me on the edge of the water, sitting on the sand in my bathing suit and shivering. It was very early, but I had decided I might as well get it over with, soon. Then there was the pride that had kept me from asking one of my friends to teach me to swim and which now urged me to go down before the beach was crowded. I looked at myself and admired the fit of the suit. I began composing my next ad, but a noise interrupted me.

A little boy of ten ran past me and jumped into the water. He splashed gleefully there, and then waded out into the lake. When he got near the end of the pier, that extended for some distance out into the lake, he began to swim. I had watched him, enviously; but now, when he struck out, I felt ashamed.

Moved by a sudden resolution, I walked out on the pier and along it to the end, intending to jump off into the water and be done with it quickly. At the end I hesitated. The small boy was off to one side, watching me curiously, so I walked quickly over to the other side.

My heart sank as I looked down on the clear, cold water, but I told myself sternly that a few feet of water was nothing to be afraid of. So taking one last look around, and closing my eyes, I jumped.

I felt the cold water on my face, and then I realised that my feet had not yet touched bottom. I opened my eyes and struck out blindly. I came to the surface, gasped and went down again. Suddenly a phrase from one of the "Carter" swimming lessons came to me. "If you are suddenly dumped into the water, never struggle. Lie still and you'll come to the surface."

I obeyed automatically, and so was once more on the surface, face down. I recalled sentence after sentence on the lessons as though in a dream. I struck out with my arms and legs, and began working toward the shore.

Curiously enough, I was not the least bit afraid, but I was angry, very angry, for being tricked into believing that the water was only four or five feet deep. I did not know at the time that there was an abrupt drop of ten feet on my side of the pier.

I was about half-way to shore, when I realised that I was really swimming. I stopped for a moment, but then went on again. Why, it was fun! Especially since I was getting proficient as I neared shore.

An idea entered my head. Why not write up my experiences for Carters'? Make it seem to happen to a stranger, and change it around to emphasise the suit I was wearing? As soon as my feet touched bottom, I dashed to my room, sat down at my typewriter, and wrote us fast as I could. Hastily correcting the sheet, I ran to the post-office and mailed it, all the while in my wet bathing suit. Three days later I received a telegram from Kane. "Ad was wonderful! Keep it up and take the rest of month off!"

* * * *

The lifeguard in blue turned over on his back and gazed at the other, admiringly. "To look at you, one would never dream you've been swimming only a year, Burt."

The guard in black and red yawned and admiringly looked down at his bronzed skin. "Well, do you know, it's a funny thing about swimming. Once you start you must keep it up. And now that the office let's me have a month's vacation, I'm here all the time."

4. The Role of Applications in Pure Mathematics.

We give a short extract from Dorothy L Bernstein's interesting paper 'The Role of Applications in Pure Mathematics', The American Mathematical Monthly 86 (4) (1979), 245-253. She gives ten examples in the paper, but we give only the first example.

Applications in Pure Mathematics.

In recent years, the Mathematical Association of America and other mathematical groups have paid considerable attention to applied mathematics and the use of mathematics in natural and social sciences and in government and industry. A glance at the programs of the national and sectional meetings of the past few years will show a generous sprinkling of hour-long talks and panels dealing with applied topics. The usual diagram that is supposed to represent the role of mathematics in applications goes something like this:

Real-world problem → Mathematical formulation → Mathematical solution → Real-world solution

Of course, one might proceed by experiment or observation directly from the real-world problem to the real-world solution; but this may be impossible or prohibitively expensive, and so the fruitful detour through mathematical models is taken. I have no quarrel with this general description, although I point out that it says nothing about who formulates the mathematical problem, who solves it, and who interprets the solution in real-world terms. I am concerned with the impression it leaves, either expressed or implied, that this is all irrelevant to the development of pure mathematics. Mathematicians stand over here, inventing axiom systems and abstract spaces and proving theorems about them, which are available for use when needed in the above scheme. Physicists and biologists and economists and all the others are over there, proposing their problems and then gratefully receiving the solutions.

This point of view is expressed in the following quotation, translated from a letter from Hermite to Stieltjes, 28 November 1882: "I am only an algebraist and have never gone outside the domain of pure mathematics. I am nevertheless completely convinced that the most abstract speculations of Analysis are evidence of realities that exist outside ourselves and will eventually come to our attention. I even think that the work of the pure geometers is directed, without their being aware of it, toward such an end, and the history of Science seems to me to prove that a mathematical discovery comes about at the precise time when it is needed for each new advance in the study of those phenomena of the real world that are amenable to calculation."

Real-world problems and their mathematical formulation often are the source of interesting problems in pure mathematics, and mathematical solutions of these models are sometimes the stimulation for generalisations which yield important concepts and theorems in mathematics. I propose that we examine some specific areas of mathematics in which applications have played an important part and then try to draw some general conclusions about the role of applications in mathematics (let's drop the "pure" - I'm never quite sure what it means in a given discussion).

Example 1. Our first example is geometry. As its name indicates, it began as an empirical science, that of land measurement by the Egyptians. Because of the flooding of the Nile, the size of a man's farm was never the same from one year to the next; and since the amount of taxes he paid was determined by how much land he owned, it was extremely important to have procedures for accurate measurement of land. It was the Greeks, culminating in Euclid, who made geometry into a postulational system with theorems and proofs that was a model of mathematical thinking for the next 2,000 years. However, it was still considered a description of the real world, and much of its authority during this long period came from this fact. In the eighteenth and nineteenth centuries, mathematicians failed in attempts to prove Euclid's fifth postulate by showing that its denial led to a contradiction. Instead, there came a gradual realisation, probably first by Gauss, that the hyperbolic geometry of Saccheri, Bolyai, and Lobachevsky did not lead to a contradiction. In fact, as Klein, Beltrami, and Poincaré showed, both hyperbolic geometry and the elliptic geometry of Riemann and Schlafli were as logically consistent as Euclidean geometry. All were abstract mathematical systems in the modern sense, each with its own power and beauty, but none with a special claim as a description of the physical space we inhabit. Cayley was able to show that elliptic geometry was closely related to projective geometry, which had also been developed during the eighteenth and nineteenth centuries.

In the first part of the twentieth century, however, precisely at the time mathematicians were completing the abstraction of geometry, the flow was reversed. Einstein, looking for a basis for his general theory of relativity, found it in the geometry of Riemann. The idea that physical space is finite but unbounded is a cliché of modern-day physics; but, to quote the physicist Freeman Dyson, "Einstein took the revolutionary step of identifying our physical space-time with a curved non-Euclidean space ... on the basis of very general arguments and aesthetic judgments. The observational tests of the theory were made only after it was essentially complete, and they did not play any part in the creative process. Einstein himself seems to have trusted his mathematical intuition so firmly that he had no feeling of nervousness about the outcome of the observations. The positive results of the observations were, of course, decisive in convincing other physicists that he was right". That it was possible for the same man to hold both points of view is illustrated by Hilbert, who made important contributions to both axiomatic geometry and to general relativity.

And now there is a final swinging of the pendulum. Says Roger Penrose: "The debt to pure geometry that relativity had owed has now been amply repaid. For many of the ideas of the modern subject of differential geometry received their initial stimulus from concepts arising from Einstein's general relativity." These include manifolds, tangent spaces, and parts of complex geometry.

... let us try to formulate some general conclusions:

  1. There has been a continuous and fruitful interplay between science and mathematics from the very beginning. Von Neumann, in his essay "The Mathematician", classifies science into three groups: descriptive, experimental, theoretical. Or rather, these are three stages through which a science passes. The descriptive ones call most on mathematics and have the least to give. The experimental sciences use mathematics freely and begin to return the investment. In the theoretical sciences, there is a free interchange of ideas with mathematics. Notice, I said interchange, not identification. A mathematician's motives are internal: his intellectual curiosity, his sense of form and pattern, his taste. A scientist, even a theoretical one, finds his motives elsewhere, as von Neumann points out.

  2. No one can predict what mathematics will become useful for applications or when this will happen.

  3. Mathematical ideas may originate as abstract concepts and then have useful applications or they can originate in the context of applications and be generalised to abstract concepts. Well, what about the future? The danger of overspecialisation that many people have pointed out is not one that bothers me. At present, most of us are willing to sit and listen to what others have to say about advances in their fields. But that is not the same as becoming creatively interested. I was reading the book by F D Thompson in which he describes the remarkable breakthrough that occurred in numerical weather prediction during the period 1951-55. In 1949, von Neumann had organised a special group at the Institute of Advanced Study to go over the partial differential equations of weather prediction, which had been formulated by Richardson in 1922. Computers had been introduced, but they could only handle data. By the time the programmers had written the instructions necessary to handle the data collected from weather stations, it was too late to predict the weather for a given day. It was under the pressure of this problem that he conceived the revolutionary idea of storing programs as well as data-that is, essentially letting the computer decide how to handle data. With this established, the project was under way and, according to Thompson, "It was a small dedicated group of people who knew something about meteorology, physics, mathematics, numerical analysis, and computer science who were able, in the short period of four years, to completely solve the problem of mathematical weather prediction." Incidentally, now it is a new ballgame, since one considers relativistic hydrodynamics of probabilistic solutions.
I have said nothing about new problems facing our society in conserving energy and handling the environment, nor about the whole range of biomedical problems. All of these need new mathematical ideas, and I am convinced that it is only by a combined attack like that just described that they can be handled. How should this be accomplished? I am not a believer in giant national programs; I think interested individuals can accomplish more. To the young Ph.D.'s I say: Talk to your friends - the young men and women of your own age in physics, biology, economics, anthropology, psychology. Find out what they are interested in-at some point, you will find a common bond with someone, and from then on, with both of you talking furiously and each learning from the other, you may very well find something of real mathematical interest to you as well as something helpful to them. Remember, they probably know the cut-and-dried mathematics; it's the dreamy, kooky ideas I'm talking about. Next, to the graduate students-if you have the time, or even if you haven't, try to take a graduate course in some field outside of mathematics in which you have some interest and knowledge. I do not mean an introductory survey course-I mean an advanced graduate course where you really find out what the new thinking is in the subject. You may find yourself over your head sometimes, but it will open your eyes to new possibilities; and as you talk to other graduate students, you may give them an idea of what modern mathematics is all about. The undergraduates - well, arrange some joint meetings of your mathematics clubs with clubs in other fields. And, again, talk to your friends. You note there is one group I have left out, the one to which I belong, the senior faculty. We know something of what is going on elsewhere, and some of us, as my examples have shown, have been able to bridge the gap between mathematics and applications. But our methods of thought are set, which makes it much harder for us to see new connections and new relations. I believe that young people, starting in the ways I have indicated, can make real progress in established areas by solving new problems, and in new fields by defining mathematical problems and beginning to solve them.

Last Updated December 2021