As an undergraduate at Tulane in New Orleans, 1922–1926, I was programmed to go to the University of Chicago and study celestial mechanics with Bliss, I turned to Bliss as an advisor in the calculus of variations.. My teacher, , had been a student of Moulton. That was an example of the great strength of the University of Chicago. Its PhD graduates made up a large part of the faculties of universities throughout the Mississippi Valley, Midwest and Southwest. So they sent their good students back to Chicago for graduate work. I went there first in the summer of 1926 and came to stay in 1928. In the interim I studied Moulton’s Celestial Mechanics and some of his papers in orbit theory. I met Moulton at a sectional meeting of the MAA where he was the invited speaker. He was a man of great charm and energy and was most encouraging to me. But by the time I got to Chicago in 1928 Moulton had resigned. I was told that he felt it was an ethical requirement, since he and his wife were getting a divorce. On the advice of , another former student of Buchanan, who was working with
It was a down cycle for mathematics at Chicago. All the great schools have their downs as well as ups, partly because great men retire, partly because their lines of investigation dry up. At Chicago at that time a young student could see the holdovers of the great period, 1892–1920, in Eliakim Hastings Moore, officially retired, Leonard E. Dickson, rounding out his work in algebra, Gilbert A. Bliss, busy with administration and planning for the projected Eckhart Hall. Also there was , teacher and doer, one of the original organizers of the Mathematical Association of America and its Monthly, even if he played only a supporting role in mathematics itself. He had an extrovert, friendly personality that reached out and got hold of you, whether he was organizing a department social or the Mathematical Association of America. He was the teacher of teachers and key figure in Chicago’s hold on education in the midwest and south. Every graduate department needs a man like Slaught if it is fortunate enough to find one. He was being succeeded by , a student of Bliss, an outstanding undergraduate teacher, though not the organizer Slaught was.
The University of Chicago was founded in 1892 with substantial financial support from John D. Rockefeller. William Rainey Harper, the first president, had bold educational ideas, one of which was that the United States was ready for a primarily graduate university, not just a college with graduate school attached. Harper brought E. H. Moore from Yale to establish his department of mathematics. Moore’s graduate teaching was done in a research laboratory setting. That is, students read and presented papers from journals, usually German, and tried to develop new theorems based on them. The general subject of these seminars was a pre-Banach form of geometric analysis that Moore called “general analysis.” It was itself not altogether successful. But even if general analysis did not succeed, Moore’s seminars on it generated a surprising number of new results in general topology, among them the Moore theorem on iterated limits and Moore–Smith convergence. Moore’s seminars also produced some outstanding mathematicians. His earlier students had included R. L. Moore, who took off in different mathematical directions. R. L. Moore developed the teaching method into an intensive research training regimen of his own, which was very successful in producing research mathematicians at the University of Texas., , and
I studied general analysis with other members of the faculty including, whom Moore had designated as his successor and whose notes record the second form of the theory, (Am. Philosophical Soc., Memoirs, v. 1, Philadelphia, 1935). Instead of taking the general analysis courses, my old friend , from New Orleans, worked in Moore’s small seminar in the foundations of mathematics. Although he was officially a student of Bliss, I think he was in a sense Moore’s last student.
Moore himself was meticulous in manners and dress. He would stop you in the hall, gently remove a pen from an outside pocket and suggest that you keep it in the inside pocket of your jacket. Nobody thought of not wearing a jacket. But Moore was less gentle if you used your left hand as an eraser, and he displayed towering anger at intellectual dishonesty. To understand him and his times one must read his retiring address as President of the Society [Science, March 1903]. In those days the Society accepted responsibility for teaching mathematics and Moore’s address was largely devoted to the organization of teaching, the curriculum, and the ideas of some of the great teachers of the time,, , , and, in this country, and John Dewey, whose ideas Moore supported by proposing a mathematics laboratory. This address was adopted as a sort of charter by the National Council of Teachers of Mathematics and republished in its first Yearbook (1925). By the time I got to Chicago the Association had been formed to relieve the Society of concern for college education, and NCTM to relieve it of responsibility for the school curriculum and training teachers. In the top universities only research brought prestige, even if a few, like Slaught, upheld the importance of teaching.
L. E. Dickson’s students tended to identify themselves strongly as number theorists or algebraists. I felt this particularly in Adrian A. Albert [sic],and . All his life Albert strongly identified himself, first as an algebraist, later with mathematics as an institution and certainly with the University of Chicago. I remember him as an advanced graduate student walking into Dickson’s class in number theory that he was visiting, smiling and self confident. He knew where he was going. Dickson was teaching from the galley sheets of his new Introduction to the Theory of Numbers (University of Chicago Press, 1929) with its novel emphasis on the representation of integers by quadratic forms. I think he requested Albert to sit in for his comments on this aspect. He was tremendously proud of Albert. I remember A\( ^3 \) too with his beautiful young wife, Frieda, at the perennial department bridge parties. He had superb mental powers; he could read a page at a glance. One could see even then that as heir apparent to Dickson he would do his own mathematics rather than a continuation of Dickson’s, however much he admired Dickson.
In the conventional sense Dickson was not much of a teacher. I think his students learned from him by emulating him as a research mathematician more than being taught by him. Moreover, he took them to the frontier of research, for the subject matter of his courses was usually new mathematics in the making. As Antoinette Huston said, “He made you want to be with him intellectually. When you are young, reaching for the stars, that is what it is all about.” He was good to his students, kept his promises to them and backed them up. Yet he could be a terror. He would sometimes fly into a rage at the department bridge games, which he appeared to take seriously. And he was relentless when he smelled blood in the oral examination of some hapless, cringing victim. He was an indefatigable worker and in public a great showman, with the flair of a rough and ready Texan. An enduring bit in the legend is his blurt: “Thank God that number theory is unsullied by any application.” He liked to repeat it himself as well as his account of his and his wife’s honeymoon, which he said was a success, except that he got only two papers written.
The theme of beauty for its own sake was expressed more surprisingly by another Texan who worked in mechanics and potential theory,. According to the story he had come to Chicago as a mature man, without a college education, to sell his cattle. Having sold them, he went to Chicago’s Yerkes Observatory to see the Texas stars through the telescope. He was so fascinated that he stayed on to get his degrees in rapid succession, all summa cum laude. Then he remained as a member of the faculty. One day in his course on potential theory he wrote some important partial differential equations on the board with obvious pleasure, drawing the partial derivative signs with a flourish. Standing back to admire these equations, he said: “That is just beautiful. People who ask, ‘What’s it good for?’, they make me tired! Like when you show a man the Grand Canyon for the first time and you stand there as you do, saying nothing for a while.” And we could see that old Mac was really looking at the Grand Canyon. “Then he turns to you and asks, ‘What’s it good for?’ What would you do? Why, you would kick him off the cliff!” And old Mac kicked a chair halfway across the room. He was a prodigy, a good lecturer, an absolutely fascinating personality with a twinkling wit. Some of his work was outstanding, yet he had few doctoral students.
Celestial mechanics was being carried on by the young, who was, I think, Moulton’s last student. But celestial mechanics had gone into a barren period and Bartky with his superb talents turned to other applications of differential equations, to statistics and to administration.
was the principal hope of the department for carrying on the calculus of variations, which he did in the spirit of functional analysis. He was my favorite professor because he knew a lot of mathematics, knew it well, and in an unassuming way was glad to share it with you. Although he taught Moore’s general analysis, he pointed out the difficulties in it to me. His own brand of functional analysis was more oriented towards the use of the Frêchet differential in Banach space.
Research in geometry at Chicago was a continuation of Wylczinski’s projective differential geometry. There was no topology, though we heard that Veblen’s students studied something called analysis situs at Princeton. I knew so little about the subject that years later when I wanted to prepare for Morse theory I spent months studying Kuratowski’s point set topology before it dawned on me that what I wanted was algebraic topology.and his students carried on the study of projective differential geometry using rather crude analytical methods, that is, expansions in which one neglected higher order terms. We who were not Lane’s students tended to look on it with disdain as being non-rigorous. But the structure of the theory was beautiful, I thought. Lane was honest about the shortcomings of the methods, though he did not know how to overcome them.
Lane was a very fine man. I had come to Chicago in 1926 to run the high hurdles in the National Intercollegiate Track and Field Meet at Soldiers Field. I placed in the finals and some members of the U.S. Olympic Committee urged me to keep working for the 1928 Olympics. So I worked on the Stagg Field track until an accident set off a series of leg infections. I was very sick in Billings Hospital in the days before antibiotics and it was Lane who came to the hospital to see me and make sure that I got the best available care. The only way I was ever able to express my thanks to him was to do a similar service to some of my own students in later years. I guess that is the only way we ever thank our teachers.
Bliss was an outstanding master of the lecture-discussion. He could come into a class in calculus of variations obviously unprepared, because of the demands of his chairmanship, and still deliver an elegant lecture, drawing the students into each deduction or calculation, as he looked at us quizzically and waited for us to tell him what to write. His students learned their calculus of variations very thoroughly. Yet we did not work together, except in so far as we presented class assignments. Each research student reported to Bliss by appointment. The subject itself had come to be too narrowly defined as the study of local, interior minimum points for certain prescribed functionals given by integrals of a special form. Generalization came only at the cost of excessive notational and analytic complications. It was like defining the ordinary calculus to consist exclusively of the chapter on maxima and minima. A sure sign of the decadence of the subject was Bliss’s project to produce a history of it, like Dickson’s History of the Theory of Numbers. The history reached publication only in the form of certain theses imbedded in Contributions to the Calculus of Variations, 4 vols, 1930–1944, University of Chicago Press.
It is perhaps surprising that this narrowly prescribed regimen turned out men who did important work in entirely different areas as, for example,did in biomathematics and numerical analysis, and did in computer theory. Among all of us has been most faithful to the spirit of Bliss’s teaching in carrying on research in the calculus of variations. Yet when Pontryagin’s optimal control papers revived interest in the subject many years later, students of Bliss were easily able to get into it. Optimal control theory really contained relatively little that was correct and not in the calculus of variations. In fact, optimal control was anticipated by the thesis of , loc. cit.
Quantum mechanics was breaking wide open in the twenties. Bliss himself got into it with his students by studying Max Born’s elegant canonical variable treatment of the Bohr theory. While that was going on, Sommerfeld’s Wellenmechanische Ergänzungsband to his Atombau und Spektrallinien (Vieweg, Braunschweig, 1929) came out. It was the first connected treatment of the new wave mechanics formulation of quantum mechanics due to de Broglie and Schrödinger. We dropped everything to study wave mechanics. Bliss was a remarkably knowledgeable mathematical physicist and quite expert in the boundary value problems of partial differential equations. That was not so remarkable in a mathematician of his generation. The narrowing of the definition of a mathematician and withdrawal into abstract specializations was just beginning. In fact Bliss had been chief of mathematical ballistics for the US Government in World War I, and later was commissioned to do a mathematical study of proportionate representation for purposes of reassigning Congressional districts. Bliss did not follow up his move into quantum mechanics but returned to the classical calculus of variations.
There were always more students in summers with all the teachers who came. Visiting professors like, , and came to teach. And there was the memorable visit of which was supposed to provide a uniting of Hardy’s analytic approach to Waring’s theorem with Dickson’s algebraic approach. Even with this infusion of talent, the offerings of the department were rather narrow. Besides having no topology as such, more surprisingly, there was little in complex function theory. And I do not recall being in a seminar, either a research or journal seminar. Essentially all teaching was done in lectures. Yet the only one of the abler students who I remember taking the initiative to go elsewhere was Saunders Mac Lane, when he did not find at Chicago what he was looking for.
I once asked, a famed universalist among mathematicians, how he came to switch from analysis to statistics at Yale. With a humorous twinkle he said: “An immutable law of academia is that the course must go on, no matter if all of the substance and spirit has gone out of it with the passing of the original teacher. So when (Josiah Willard) retired, his courses had to go on. And the department said: ‘Wilson, you are it’.” A graduate student at Chicago in the late twenties could see this immutable academic law in effect. In each line of study of the, then passing, old Chicago department, a younger Chicago PhD had been designated to carry on the work. If, in one’s immaturity, this was not apparent, the point was made loud and clear in a blast from Dickson during a colloquium with graduate students present. Dickson charged the chairman with permitting the department to slide into second rate status. It was true that the spirit of original investigation given way to diligent exposition in some of these fields. In some cases the fields themselves had gone sterile.
It was the lot of Bliss to preside over this ebb cycle of the department. He did an impressive best possible with what he had, with high mathematical standards, firmly, kindly and quietly. Most of the difficulties he had inherited. Bliss was able to appoint some outstanding young men but, if he had asked for the massive financial outlay to bring in established leading mathematicians to make a new start like the original one under President Harper, the support would not have been forthcoming, even with a mathematician,, as president and certainly not with the young Robert M. Hutchins, bent primarily on establishing his new college. It took the Manhattan Project, the first nuclear pile under the Stagg Field bleachers and Enrico Fermi to convince Hutchins of the importance of physical science and mathematics and to throw massive resources into the reorganization of the department near the end of World War II. Such reorganizations are necessary from time to time in every graduate department. They can be effective only when the time is right. It is the mark of a great university to recognize the necessity to break the immutable law of academia, and the opportunity, and to do it when the time is right. However, there were deep hurts, symbolized by Bliss’s refusal ever again to set foot in Eckhart Hall to his death. But this is really getting ahead of my story.
It was no ebb cycle for the University of Chicago as a whole in the twenties. There was intellectual excitement in many places in the university. I attended the physics colloquia where the great innovators of the day came to talk. With Mr. Bliss’s grudging consent, I took Arthur Compton’s course in X-rays. He already had the Nobel Prize for his work on the phenomena of X-rays colliding with electrons. Yet he seemed so naïvely simple minded to me, far less expert and mentally profound than other physicists in the department. Somewhere in here Einstein came for a brief visit. He permitted himself to be escorted by the physics graduate students for a tour of their experiments. To one he offered a suggestion. The brash young man explained immediately why it could not work. Einstein shook his head sadly. “My ideas are never good,” he said.
Michelson, another Noble Prizeman, was around, though retired. So was the great geologist, Chamberlin, with his planetesimal hypothesis in cosmology. In biology and biochemistry the great breakthroughs on the chemical nature of the steroid hormones and their effects on growth and development were excitingly unfolding. Young Sewall Wright was attracting students to his mathematical genetics. Economics promised a real breakthrough, though as it turned out, it was slow in coming. Linguistics was burgeoning. Anthropology and archeology were still actively following up the results of digs in Egypt, Turkey and Mesopotamia. The great debates over the truth of theories of relativity and quantum mechanics were raging. What was later to be planet Pluto had been observed as “Planet X” but heated arguments persisted on what it really was. On Sundays the University Chapel produced a succession of the leading Christian and Jewish spokesmen of the day. The textbook, The Nature of the World and of Man, H. H. Newman ed., University of Chicago Press, 1926, by illustrious Chicago faculty members was the best survey of physical and biological knowledge for college students that I have ever seen, though now dated, of course.
And outside the university the dangerous and ugly city of Chicago nevertheless had its charms, cultural and otherwise, that could take up all the time (and money) of a country boy. One could hear Mary Garden or Rosa Raisa at the Chicago Opera by getting a job as usher or super, or attend a fiesta in honor of the patron saint of some Halstead Street community that maintained its identity with the home village in the old country. One could drink wine at Alexander’s clandestine speakeasy. For recall that it was Prohibition and the height of the bootlegging days of Al Capone and rival gangs. The famous Valentine Day massacre was just one of the lurid stories in the Chicago Tribune. We students formed an informal protective association to promulgate rules to optimize safety for oneself and date. One old boy from Georgia, a graduate student in history, was so impressed by our admonition never to approach a car asking him to get in, that, when a police car challenged him with order to stop, he just took off in a blaze of speed. Caught later, out of breath, his one phone call brought some of us to police court to testify to his character. The officer who had made the arrest moved to dismiss the charges on the condition that “the defendant appear at Soldiers Field next Saturday and run for our company in the policemen’s track meet.” But it was grim business. Police, armed with machine guns, in such a car once arrested me on suspicion of rape on the Midway (not guilty!). Other students were mugged, raped, robbed and even killed.
Like today it was a time of inflation and most of us were poor. I had a full fellowship of \$410, of which \$210 had to be returned in tuition for three quarters. A dormitory room cost \$135 out of what was left. We could get cheap meals at the Commons, and on Sundays one could go to the Merit Cafeteria and splurge on a plate-sized slab of roast beef. It cost 28¢ but it was worth it. We all looked forward to a teaching job, I think. Those jobs required 15 hours of teaching for about \$2700. Soon the depression hit and, if we were lucky, we kept our jobs with salary cut to \$2400. Some beginning salaries for Chicago PhD’s were as low as \$1800 in the early thirties.
Before closing these recollections I must write something about women as graduate students in those times, not long after the victory of women’s suffrage. Only years later did I learn that it was considered unladylike to study mathematics. Many of the graduate students in mathematics were women. In fact there were 26 women PhD’s in mathematics at Chicago between 1920 and 1935. I shall mention only a few by name.(1921) was in the faculty of the department. (1931) was already showing the kind of ability that led her to a distinguished administrative career at Hunter College and CUNY. She did more than any other person to gain federal support for mathematics through her position as chief, Mathematics Branch ONR, when the National Science Foundation was established. Others included (1933), chairman at Vassar, and (1934) also of Vassar, (1933), chairman at Connecticut College, (1934), chairman, Mt. Holyoke, (1927) Northwestern, (1930) Penn State, and (1935) Randolph Macon. Graduate students married graduate students, though of necessity only after the man had his degree. In the department married . , student of Dickson, married and continued her publishing and teaching career at the University of Detroit. married . They both later taught at Rensselaer Polytech. married a non-Chicago mathematician, , and continued her teaching at Penn State. was in the department of Hunter College. These, and a number of others, were able to continue their professional work in spite of family obligations. Even intermarriage between departments was permitted! My wife to be, Mary Hardesty, was in zoology. We got our PhD degrees in the same commencement.
Looking back on those days, I wonder if the current women’s liberation has even yet succeeded in pushing the professional status of women to the level already reached in the twenties. Maybe this time women can hold their gains in universities.