My father, Joseph Moses Gillman, had come to the US from the Ukraine at the age of 18 and settled in Cleveland, where he had cousins. He spoke only Yiddish, knowing only few words of Russian. A thoughtful well-wisher advised him to enroll at Hiram College, a small, religious-oriented college outside Cleveland whose congenial atmosphere was conducive to learning English. After two years, he moved to Western Reserve University in Cleveland, where he completed a baccalaureate in philosophy. Joe's father settled in Cleveland at about the same time; Joe's mother had died when he was an infant. Etta was also a graduate of Reserve. Our family acquired grandpa's record collection when we moved to Pittsburgh in 1922; and I would listen by the hour. In my view, many of the performances were musically superior to those produced today.
At age 11/2 or so, when World War I was still in progress, I used to convulse the other passengers in the street car by singing Over There at the top of my lungs. (A year later, when my brother Bobby was an infant, a passenger asked me, "What is the name of your little brother?" whereupon I announced in a very serious voice, "His name is Baby Irresistible.) In Pittsburgh, a month or two before my seventh birthday, I started piano lessons, courtesy of a neighbor's piano. (These neighbors, the Meyerses, were our closest friends. Their son Fred was my age. The families have always kept in touch and in fact Reba and I visited Fred in the summer of 1997. (Our four parents died many years ago.)
I was interested in mathematics as a tot. I used to experiment with numbers in my head. When I was seven or eight, I noticed that 42-32=4+3, 52-42=5+4, and maybe a few more (though of course I did not use exponential terminology), and announced the general rule. Bobby remarked, "You're always figuring out stuff." Joe told me that when I got to algebra in school, I would learn that there was a formula, and what it was. I was crushed that my discovery was not new. I doubt that Joe saw right off what the formula was, but he was a highly educated man and I suspect had enough of a feeling about knowledge in general to realize that there surely was one. It was not long before I wondered about numbers that were two apart, and lo! 52-32 turned out to be 2 x (5+3), and so on.
My parents lived for their children. In Pittsburgh, I spent a year in Kindergarten in a public school. It is there that I first demonstrated my contempt for bureaucratic nonsense. I was standing in line to sharpen my pencil, when Miss McGinnis, the principal, who was reputed to know the name of every child in the school, walked into the classroom.
"Peter, what are you doing?" "I'm waiting to sharpen my pencil, Miss McGinnis."The next year I moved to first grade in a tiny private school that a group of parents, including mine, had organized. Fred Meyers was a classmate; he later became a professor of government at UT Austin (before I got here). Another was Philip Morrison, currently professor of physics at MIT, probably best known for having been a passenger in the Enola Gay when it dropped the atomic bomb on Hiroshima.
"Ann, what are you doing?" "I'm waiting to sharpen my pencil, Miss McGinnis."
"Leonard, what are you doing?" "You'll see."
But we were not getting any genuine mathematical guidance. When Walden remained in session on Lincoln's Birthday, which was a school holiday in New York, Ellis and I decided to play hooky. We met near the school and wondered what to do. Today's boys would have no hesitation: buy some drugs, slash a few tires, and molest some girls. First we got on the Broadway subway and rode uptown to the end of the line. That turned out to be unexciting, so we rode back all the way to 42nd Street, where we got out and walked to the main branch of the NY Public Library. There we went to the children's room and read the article on magic squares in the Encyclopedia Britannica, having no other ideas about finding anything mathematical.
The article included some oddball magic squares, including a 3 x 3 multiplication magic square using the numbers, 1, 2, 3, 4, 6, 9, 12, 18, 36. The next day when we arrived at school, I handed Ellis a piece of paper show ing the 3 x 3 magic square using the numbers 1,2,...,2 8, formed by placing 2 n in the same location as the number n+1 in the basic additive square - whereupon he handed me a sheet with the identical construction. The same morning during English class, we both did some high-level doodling, and at the end of class each of us handed the other the 5 x 5 square using the powers of 2 from 0 to 24, based on the same mapping formula. It is clear that we understood the basic idea of isomorphism.
From age nine through my high school years, I studied the piano with a truly wonderful man named Louis Kantorovsky, a sensitive and inspiring person and musician. (He had been one of the gifted children in the long-term study made by the psychologist Louis Terman.) He and his wife were much younger than my parents but the two couples became best friends. We were living in Queens, and the Kantorovskys lived in the Bronx; every Sunday afternoon my parents would take their two boys on the long subway ride to Manhattan, then transfer to the train to the Bronx, where Bobby and I had our piano lessons, after which the Kantorovskys served dinner.
At about age 10, I became an inveterate concert-goer. During the next half-dozen years, I heard pianists Bauer, Dohnanyi, Gabrilowitsch, Goldsand, Hess, Hofmann, Horowitz, Lhévinne, Levitzki, Paderewski, Rachmaninoff,and Rosenthal, and violinists Elman, Fran-cescatti, Heifetz, Menuhin, and Milstein.
I decided to concentrate on Juilliard full time for the first semester, but in Spring (1934) I enrolled in the Columbia Extension division (evening school), signing up for French and Analytic Geometry. I considered music my major field at Columbia but wished to learn more mathematics. I did not consult Juilliard about Columbia for fear they would deny permission, though I probably leaked it out during the semester. At any rate, my Juilliard "report card" at the end of the school year consisted of one item: "Should not let work at Columbia interfere with Fellowship."
The piano faculty consisted of eight concert artists: Carl Friedberg, a pupil of Brahms; James Friskin, a Scottish scholar and Bach specialist with a prodigious memory, who was my teacher; Ernest Hutcheson, the Dean; Olga Samaroff, ex-wife of Leopold Stokowski, the famous conductor of the Philadelphia Orchestra (remember Fantasia?); Alexander Siloti, a cousin of Rachmaninoff and pupil of Liszt; Oscar Wagner, a Hutcheson student; and Josef and Rosina Lhévinne, husband and wife who had met as students at the Moscow Conservatory. Josef was a renowned virtuoso whose recordings appear these days in collections with names like "Piano Giants of the Past". By universal agreement among the students, playing for these eight at the final exam was the most dreaded and uncomfortable 15 minutes of the year.
The school was not an academic institution - there were no degrees or even certificates - but was dedicated to producing concert performers. Still, the folks in charge, well educated themselves, were aware that the students' training should not be narrow. So there were offerings outside music. One I remember was called Literary Philosophy. The teacher was Rhoda Erskine, sister of John Erskine, who was a Professor of English at Columbia and an amateur pianist and student of Hutcheson; he also happened to be the President of the Juilliard School of Music. The School comprised the Institute of Musical Art, which was a conservatory open to the general public, and the Graduate School. (Thus I grew up regarding "The Institute" as a pejorative term.)
Rhoda died mid-semester and was replaced by Harold Hutcheson, son of Ernest. (How cozy can you get?) But he was wonderful. Courses were also offered in French, German, and Italian, but I think they were limited to the voice majors, for whom they were required.
There was extensive broadening within music. Every nonpianist was required to take "secondary piano" - standard practice in music schools. Their teachers were piano fellows. I never got to do any of this teaching, possibly because the plum was reserved for older chaps or those who really needed the money. I had courses in harmony, counterpoint, theory, fugue, composition, score reading, orchestration, conducting, chamber music (a performance class, not book-larnin), and a specialized class in music for the layman, organized by Madame Samaroff, and Bernard Wagenaar's course called General Musicianship, in which he analyzed works of Bach, Beethoven, Schumann, Wagner, Prokofieff, Gershwin, and others.
This course was the most interesting of all. Wagenaar was a learned musician, former violinist with the NY Philharmonic, and a composer whose first symphony had been performed by Toscanini. He was a fabulous person and everyone's favorite teacher. He also liked and told good jokes, especially dirty jokes, both in class and at lunch. He was my teacher in harmony, counterpoint, theory, fugue, and composition.
A typical setting for these courses was a small studio containing one or two grand pianos, with the teacher on a piano bench, and seven students on folding chairs in a semicircle. Classes met once a week for an hour. Generally, we would hand in an assignment on arrival, and the teacher would look at and comment on it then and there. The students were happy goof-offs like students everywhere, but everyone was relaxed about it. In this free spirit, I once handed in a piece in marching-band style that I had marked "Sousando"; and a chap from Brooklyn handed in a composition at a 9:00 a.m. Saturday class with the subtitle "Subway Special". Wagenaar was a master raconteur, and we quickly perfected the ploy, when we had arrived unprepared (the usual case), of getting him to tell stories.
One morning, at home, I noted that my composition was three weeks overdue and I jolly well had to have it in by 9 the next morning. But my heart wasn't in it, and I decided to leave the chore to a crash job in the evening. Unfortunately, Charlie Chaplin in Modern Times was opening that evening all over town, including at our neighborhood theater. But I stuck to my resolve and sat down and glumly dribbled out a miserable specimen of a composition, which I dutifully showed up with the next morning. As the class was about to get underway, Wagenaar remarked that he had seen Modern Times the night before. Then some subhuman student announced that he too had seen it. The ultimate in insult added to injury than befell. The two of them started talking to each other about the movie. "Remember this?" (Joint laughter) "And that?" (ditto) "And the feeding machine!" (Ditto) "And the boss spying on him in the men's room!" (Ditto) - for the entire goddam hour.
The school offered an annual series of concerts by the Juilliard Orchestra featuring concertos with faculty soloists. I first learned of César Franck's it Symphonic Variations and of Beethoven's G major piano concerto (#4), when Friskin performed them in these series. One program of unusual interest included Vivaldi's concerto for four violins (and orchestra), followed by Bach's magnificent transcription of the work to a concerto for four pianos (and orchestra), in which the soloists were Friskin, Hutcheson, Rosina Lhévinne, and Wagner, with the program duly noting that the men were playing Steinways but Madame Lhévinne a Baldwin).
Each year a half dozen students were selected to appear as soloist with the Juilliard Orchestra (conducted by a conducting major). In May of 1938, I had the honor of appearing as soloist in the Liszt A major concerto. The conductor was Charles Kent, a pleasant young man who ended up as director of the renowned Peabody Conservatory in Baltimore. That summer I played the same work with the WPA Orchestra. (The Works Progress Administration was one of FDR's programs to help bolster the economy of the country.) The conductor this time was Edgar Schenkman, who taught conducting to those who like me were not majoring in the subject. Edgar's younger brother Eugene became a fairly well-known group theorist, who ended up at Purdue after I had left.
In May of 1938, after five years at Juilliard, I was told I had now graduated. In June, Reba graduated from Bennington College as a music major (voice). We married in December. (Our loving fathers continued their financial support.) By that time Reba and I had been classmates and friends for nine years, so no one can complain that we rushed into it. An elementary computation reveals that we recently celebrated our H. J. Heinz anniversary. At first we lived with my parents. (Once when Reba answered the phone and the caller asked for Mrs. Gillman, she responded, "Do you mean the young Mrs. Gillman or the it old Mrs. Gillman?") After a couple of months we got our own apartment near Columbia - the first time I had ever lived away from "home". (One day, when we were having a minor tiff, she admonished me: "Don't talk to me that way or I'll go home to it your mother."
Two or three years later, a Juilliard diploma arrived out of the blue. The school had apparently decided diplomas were not a bad thing after all and gallantly set about playing catch-up by awarding a batch of them retroactively.
My career as a musician was feeble, consisting of giving a handful of piano lessons, some privately and some at a couple of schools, tutoring a student in theory, and playing in the pit for a show that folded after three days.
One day I naively walked into a music instructor's office at Columbia, told him about my Juilliard background, and suggested my Juilliard transcript be evaluated for transfer credit. After a week I was informed that I had been awarded 12 credit hours. I protested that they had merely counted hours, pointing out that the students in the Juilliard class were highly selected and could assimilate and understand in 10 seconds a musical point that a student in an ordinary college class could spend several hours on and still might not get. This did not fly.
Later I enrolled for two advanced music courses. One was Literature of the Quartet with Daniel Gregory Mason, a distinguished elderly gent from a venerable musical family (e.g., Mason & Hamlin pianos) and a well-known composer, whom I hit it off with to the extent that he would invite me to play two-piano works him for demonstrations in his other classes; I also performed one of his piano compositions over radio station WNYC.
The other class was Literature of the Symphony with Douglas Moore, a composer of better-known works ( The Devil and Daniel Webster, The Ballad of Baby Doe) and quite a friendly guy. Sometimes he would come to class unprepared and try to bluff his way through, looking at the score for the first time and then attempting to explain the fine points to us. I found I could read a score better than he. At any rate, after he had got to know me fairly well, I asked whether he would reevaluate my Juilliard record. He was glad to, and this time I was awarded 34 credit hours, which were not only worth at least a full year's time and $408 (34 hours @ $12) but provided me with the requisite credits to graduate with a major in music.
Then one day in 1940, just before the start of the fall semester, my mother pointed out, gently, that there was no point, given my Juilliard background, to adding a degree in music from Columbia: instead why not major in mathematics? I agreed at once. So far I had never taken more than two college courses at a time, as I was still practicing the piano diligently. But then I noticed from the catalogue that if I took five math courses all at once I would have my degree at the end of the semester and moreover as a mathematics major.
I signed up for Calculus III (multivariable), an undergraduate course; Probability, and Advanced Calculus (dual level); and Number Theory, and Real Variable (graduate). All met for three hours per week, but undergraduate courses counted 3 credits, dual level counted 3 for undergraduates and 41/2 for graduate students, and graduate courses counted 41/2 for everybody. Calculus was taught by Professor L.P.Siceloff, who had done some good work in finite groups many years earlier but had long since stopped doing research and was now the departmental undergraduate advisor and man-of-all-work.
Probability was taught by Assistant Professor Bernard O.Koopman, and Advanced Calculus by W.B.Fite, the department head and author of the text; Number Theory by Koopman, and Real Variable by Assistant Professor Andrew C. Berry, who shortly afterward moved to Lawrence College (now University) in Appleton, Wisconsin. It was the most exhilarating semester of my life: I was walking on air. I used to come home at the end of the day and tell poor Reba everything I had learned, exclaiming, "It's a whole new language!"
One may wonder what it was like taking Calculus III (undergraduate), Advanced Calculus (dual level), and Real Variable (graduate) all at the same time. The two closest were the two advanced courses. But Berry in Real Variable spent time on sets (including Russell's paradox) and cardinal numbers, which were not even mentioned in the other courses. When Berry defined completeness, he did it five different ways, including Cauchy sequences, nested intervals, and Dedekind cuts. Then he showed that all five were equivalent, by proving five implications in a circle. Finally, he picked one of the conditions and showed it to be true for R. (I forget how he defined R.) It was the first time I had encountered a proof by cyclic implication.
In lieu of a final exam, we were asked to write up any part of the course. I took the five conditions and supplied proofs for all 20 implications, managing in all but a few cases to avoid using any of the others as part of the argument. I handed it in early enough for Berry to read it and say that while he was not taking it lightly, the fact was that once one has caught on to the general flavor, there is not any great difference among the ideas, and to ask me to tack on some other small project. So I wrote up the "Cantor intersection theorem", stating that the intersection of a nested sequence of bounded closed sets is nonempty. This was the only course of the five in which no homework was assigned.
Getting back to the relations among the three courses: the Mean-Value Theorem arose in all three. We got to it first in Calculus III. When I saw the proof I thought, hell, why construct that complicated function? - just rotate the axes so that the chord lies along the new x-axis and apply Rolle's Theorem. But I was afraid to ask for fear it was a dumb question. A little while later it came up in Advanced Calculus, and this time I had to fight myself hard not to ask. But when it appeared in Real Variable, I could no longer hold out, and asked. Berry said, right off: "The function might not be single-valued" (today we would say the graph might not represent a function). I thought he had misunderstood the question, and repeated it. He waited patiently for me to finish, smiled, and gave the same answer as before, this time drawing a picture. This incident taught me some mathematics as well as something about teaching.
A graduation requirement for a major in mathematics was to pass a "comprehensive" exam in elementary calculus and geometry. I graduated as a math major.
At the close of the Spring semester, Fite took me aside at a colloquium to offer me the assistantship for the Fall term. There was just one assistantship, which had traditionally been a full-year appointment, but for the preceding several years had been divided into two; apparently no one thought of asking the administration to double the funding. The assistantship paid $500 per semester plus free tuition. It was formally a faculty appointment: I was invited to join the Men's [sic] Faculty Club, attend the weekly math department lunch, and march with the faculty at Commencement. The duties had traditionally been to sit in the department head's office, probably to ward off angle trisectors, and grade papers for selected classes. But with the wartime increase in enrollments, I was asked if I would mind teaching a class. What a question! My dream come true! I was given a trig class of 40+ students, which for those days was transfinite. The first day, Siceloff walked in one minute before the bell, apparently to check that I would remember to dismiss the class. That was the only visit he or anyone else ever made.
Richard Cohn, my counterpart for the spring semester, was drafted into the Army and had to bow out, and they gave me his half of the assistantship as well. I was delighted to have it but at the same time felt bad about getting it at his expense as it were. (I had a deferment as a married man.) Cohn ended up at Rutgers, from which he is now retired.
Koopman, who knew me best mathematically, became my benefactor. He told me that Harold Hotelling, the statistics bigshot, controlled some fellowship money, and arranged for me to visit him. The catch was that I had to register for courses in statistics. Well, what the hell. It was a Carnegie Corporation fellowship paying $1800 for the year (but not covering tuition). In the absence of a Statistics department, Hotelling was a member of the Economics department. His sidekicks in statistics were Abraham Wald and Henry Mann. I took just about every course offered by Hotelling or Wald. Mann was a research associate and did not teach. He was a group theorist, who wrote an engaging little book on the design of statistical experiments. During that period he won the Cole Prize in Number Theory for his solution of the \alpha\beta-problem on the density of the sum of two sets of integers.
Academic life was different in those days from now. Lorch was an instructor for 12 years, and Koopman an assistant professor for 13 - although once they were promoted they moved up to full professor rapidly.
Hotelling had his pompous streak - one chap characterized him as the only person who could strut sitting down - but he was a kind and friendly person. He and his wife hosted a monthly get-together for faculty and graduate students at their suburban home in New Jersey. That's where I met Saunders Mac Lane, who at the time was head of the Applied Mathematics group at Columbia, which spent its time solving problems for the Army. In real life, he was Associate Professor of Mathematics at Harvard. We also met Hotelling's daughter, Muriel, whom we have kept in contact with over the years. (She and her husband visited us in Austin a few years ago, and we visited then in Schenectady shortly afterward on our way to the summer meeting in Burlington.) Hotelling was also quick to permit me to teach Columbia classes while holding the fellowship, recognizing that $1800 was really not enough to live on. That was timely, as our first child, Jonathan was born in the fall of 1942.
Other courses I took were Automorphic Functions, using Lester Ford's book, from J. F. Ritt, the departmentUs biggest-shot mathematician and a superb teacher; a topology course from George Adam Pfeiffer; and Linear Operators with Francis J. Murray (of Murray and von Neumann) based on his book of the same title.
Murray taught by reading from the book. He was one of those readers who sound as though they don't understand what they are reading, though in this case at least, he knew it backwards, forwards, and sideways. Moreover, the book has an exasperating way of referring to results as Corollary 3 of Theorem 2 of Section 4 of Chapter 5; it is bad enough when you are reading it alone, but try to picture following a mathematical argument when it is read to you at a brisk pace. Incidentally, the Schwarz inequality was invariably referred to as B-11, its position in the list of axioms for a Hilbert space. But Murray was totally conscientious. He told us that the final would include choices. When we entered the room, we found the board already half-full, with Murray busily writing more and more. He continued until he had stated 17 theorems, then asked us to prove any two of them. To show off, I did four. Of course there were a dozen I couldn't touch. Murray eventually got more interested in numerical analysis and moved to Duke, from which he is now retired. As far as I know, he and Berry are the only teachers of mine who are still living - at any rate, they are still listed in the latest Combined Membership List (CML).
Fellow graduate students at Columbia included Ernst Straus, who became Einstein's assistant at the Institute and who wrote two joint papers with him, so that Einstein's Erdös number eventually became 2; Louise Miller, who became Straus's wife; Bernard Gelbaum, at Minnesota for many years, where he wrote a book of counterexamples with John Olmsted, and at SUNY Buffalo after that; Alan Hofmann, retired IBM fellow; Abe Hillman, New Mexico, retired, who for many years was high up in Putnam exam circles and may still be; Martin Klein, for many years a distinguished historian of science at Yale; and Fritz Steinhardt.
Fritz was an Austrian refugee, a magnificent human being with a wonderful sense of humor, who became our very close friend. During a period that he and I were working with the closure and negation operations, he suggested that our young son Jonathan need not bother to learn the word "cold" but could instead use "hot-bar". Fritz had an outstanding command of English, including the ability to make puns; he taught at CCNY for ages, and died two or three years ago.
According to my grapevine at the time, a student once walked into Ritt's office with an enthusiastic account of some research he had done on his own. Without saying a word, Ritt pulled out a manuscript from a desk drawer and held it up for the student to look at. It contained the results that the poor fellow had just been describing, and the chap left the room crest-fallen. A contrasting story has a student come to tell his results to Sammy Eilenberg, who listened attentively and offered suggestions for further progress; when the elated student left the room, Sammy pulled out a manuscript from a drawer, tore it up, and threw it in the wastebasket. And there is the one about Paul Erdös, who on one of his visits to Budapest found that he was a member of a doctoral committee scheduled to convene in a few days for the final thesis defense. In the interim, while browsing in the mathematics library, he found the student's main result in a journal article of many years earlier - whereupon he checked out the journal and secreted it at home, ensuring that no one else on the committee would find it.
The topology course with Pfeiffer was based on Max Newman's Topology of plane sets. Once during a discussion of continuity, one of the students (a very good one) became momentarily confused and asked a question that Pfeiffer did not understand. After they had talked at cross purposes for a while, the student finally blurted out, "But suppose I choose epsilon large?" Ah," Pfeiffer replied with a twinkle, "but you don't choose epsilon. I choose epsilon."
By the end of the school year (1942-43), at the age of 26, I had amassed the 60 credit hours required for the PhD; remaining were a dissertation, a departmental oral, two comprehensives (algebra and analysis). and the dissertation defense (a formality).
The project was attached to the Special Devices Section of the Training Division of the Bureau of Aeronautics. Special Devices occupied a vacated Chevrolet dealership within easy walking distance of Union Station in Washington. Rulon was the Project Director; he was a professor at the Harvard Graduate School of Education, but had a strong background in engineering mathematics and was smart. He of course stayed in Cambridge, but he was on the phone with us almost daily. Each week, one or both of us would visit Special Devices in Washington. The work was interesting and often exciting, and we worked our asses off, putting in full days working side by side planning, organizing, and churning out results, producing 100+ weekly memos in our two years on the job. They were short, usually two or three typed pages, but nevertheless represented work accomplished. For the second year, our salaries were increased to $3900.
It is interesting to see how very elementary mathematics will often solve a problem that appears difficult. One such was the problem of the "spot-light animator", a device designed for anti-aircraft training. A beam of light was reflected by a irregularly rotating mirror onto a screen; the trainee, sitting at a mock gun-mount, tries to track the spot of light as it moves erratically on the screen. Eventually, the Naval officers in charge realized that a better way to train gunners would be to have the light follow a realistic path, such as would result from an attacker closing in. The problem for us, then, was to instruct them how to cut the cam governing the mirror's rotation so as to create a desired path on the screen. They laughed when they presented it, feeling that it was impossibly difficult. We took the device back to New York, where we made careful measurements of its construction. We derived seven equations in seven variables. But they were all linear or quadratic, and none of them contained all the variables. Solving them was therefore very easy. We sent in our report with instructions for a cam to create a straight-line path and one for a U-shaped path. When it worked, our standing, already high, reached Olympus.
Muriel Hotelling worked in our office for a time, as did Louise Miller and Fritz Steinhardt. Once when briefing Fritz on progress since his last visit, I mentioned that what we had been calling k we were now calling \psi. His instant response: "O \Psi.''
Our prize contribution was an analysis of ordinary and generalized pursuit curves and of countermeasures that a target ship can take to defend itself from an incoming homing torpedo. Our results appeared first as memos, but then the idea got through that we should organize the collection into a book, which we did. It came to 250 pages, including a half-dozen pages of tables in which one could look up either one of the polar coordinates in terms of the other - each table for a different ratio of torpedo speed to target speed. I typed the entire report, "Mathematical Analysis of Ordinary and Deviated Pursuit Paths," as well as every one of our memos, on my portable Smith-Corona. Many years later, I typed Rings of Continuous Functions on the same machine, 500 manuscript pages on Ditto masters in 10 days. In 1969, when Reba and I were leaving Rochester for Texas, I sold the typewriter at our garage sale for $7. It still hurts to think of it, although of course I would never use it today.
Both Harry and I had long since acquired the necessary 30 credit hours for the masters degree. For theses, I wrote up our theoretical results about pursuit curves, while he did the same with some associated numerical results. We submitted them to Professor Murray, who accepted them.
One day, Rulon, who was also a consultant for the Army Air Force, gave us a problem concerning admission to pilot training school. The AAF administered an admissions test to candidates. The grading in the school was strictly pass-fail. An important question is how well the admissions test predicts school achievement The setup has two exasperating features: we don't know how well those who were not admitted to training would have fared, and the only scores we have for the trainees are 0 or 1.
One assumes that performance on the admissions test is a normally distributed continuous variable X and that performance in the training program is a normally distributed continuous variable Y. Problem: Determine the correlation coefficient between X and Y. Harry and I worked out a (not very subtle) solution, calling our statistic, naturally enough, G. It was published in the Harvard Educational Review, January 1946, pp. 52-55, under the title An estimate of the correlation coefficient of a bivariate normal population when X is truncated and Y is dichotomized. This was the first of my published papers, and the title remains the longest of all. It certainly made a hit with our friends at Special Devices.
After the war, Harry stayed with the Navy as an administrator and this led to his appointment as director of the Willow Run Research Laboratory and Professor of Engineering at the University of Michigan. He died in the fall of 1960, just before Kennedy was elected President His death affected me deeply. Lisa, his older daughter is currently a professor of harpsichord at Oberlin College, and Erica, the younger, is a senior editor at U. S. News & World Reports. (Elsie, their mother, lives in Oberlin near Lisa).
Finding an apartment was extremely difficult, but my father had some acquaintances who got us one. It was in Southeast Washington in a section no one would call hoity toity. OEGUs headquarters were in the Navy Building, a World War I relic at 17th and Constitution. Getting to town in the absence of expressways or direct buses was difficult. The Navy's parking lot was located at the end of Constitution Avenue, by the river; since I was a late sleeper, the parking spaces nearest our building were all taken by the time I got there. The long walk was then even longer, and in bad weather I sometimes took a cab from the lot to the building. I soon solved this problem by buying a used motorcycle from a colleague; then the gendarmes at the Navy Building would simply wave me through into the alleys separating the wings of the building, where admirals parked their cars. I had bought some surplus Navy foul-weather gear, including a pair of magnificent elbow-length fur mittens, and red, green, and silver reflective tape ("Scotchlite") for night driving. (Two cops once drew alongside on their bikes saying, "You look like a Christmas tree!")
The work at OEG was interesting, and I often brought problems home to work on in the evening. I also had one wife, two children, and a piano. So much for the dissertation! For some time I worked on antisubmarine problems, notably evasive action by target ships, such as following zigzag paths. But after a while, the theory of games captured my attention. Recall that von Neumann and Morgenstern's 1944 book had just appeared People at the Rand Corporation, particularly, were greatly interested in its applicability to military strategy.
One of the first problems, solved at Rand and independently at the Operations Research Group, was to find the best strategy for a target submarine to follow when faced with traversing a hostile channel. It went something like this: Enemy searchers are lined along the two shores; from each point, the probability of detecting the submarine if on the surface is assumed known (e.g., it is low where the channel is wide, high when narrow); a submerged submarine is assumed undetectable, but the submarine's batteries can maintain it under water for only a limited total distance (also assumed known). What is the distribution of points at which the sub should choose to be under water? I discussed this problem in a talk at Purdue about eight years later and got a good response by remarking that while I had met several submarine skippers, I had yet to hear one give an order to submerge when x is rational and surface when x is irrational. David Blackwell, then at Howard University (but long since at UC Berkeley), had been a consultant at Rand and solved the same problem, and in fact published his solution in the Bulletin of the AMS (before the Proceedings had been invented), using the same joke though in a more subdued way. He visited OEG a number of times and he and I became good friends.
The problem I tackled and solved, as did Blackwell, was a different one: the fighter-bomber duel. The goal of the bomber is to get to his target and drop his bombs. The goal of the defending fighter plane is to shoot down the bomber so as to prevent that. The bomber is interested in shooting down the fighter only insofar as it permits him to reach his target. Thus we have a zero-sum game. It is assumed that the bomber can fire one salvo at the fighter, and he has to decide when. The fighter on the other hand is assumed to have a supply of ammunition that he can distribute over time, and he has to decide on the distribution. The game-theoretic solution was interesting in that it featured a jump discontinuity at the last possible instant of the duel.
I dressed up the problem for public consumption as finding advertising strategies for two business people, Mr. Big, who has a steady clientele and is prosperous, and Mr. Little, whose situation is precarious. The two are competing for a new customer who has appeared on the scene at time 0. Mr. Little's goal is to make the sale in order to stay in business. Mr. Big's goal is that Mr. Little be forced out of business: he wins even if the customer decides not to buy from either one. Mr. Little will choose an instant t in [0,1] at which to advertise, while Mr. Big will choose a function S(t), his rate of spending over the interval. Mr. Little's optimal strategy turns out to be such that the probability that he will advertise at some time later than t is the function with the jump at t=1. I presented this model at a meeting in New York of the American Statistical Association, under the title Operations analysis and the theory of games: an advertising example and wrote it up for publication in the Journal of the A.S.A., vol.45, December, 1940, pp. 541-545, with the same title.
A year later, OEG-er John Danskin and I solved the more difficult two-fighter duel and published the result in Revista Mat. Univ. Parma 4 (1953), pp. 1-12. Danskin had spent a couple of years at Rand, where he became immersed in the culture of functional analysis, and our paper, titled A game over function space, reflected this fact, rendering it unintelligible to a great many people who would have profited from a less sophisticated approach. By this time, no one felt the need to couch the results in nonmilitary terms.
In 1948, I visited Professor Art Copeland at the University of Michigan, who was directing a Navy project of his own and had read some of my work and apparently was impressed by it. He and his wife Dorothy put me up in their home for the three days I was in town. During the visit I met Arthur Jr., a mathematics student, and his wife Lynda, who were both working on Art Sr.'s project. And I met mathematicians Wilfred Kincaid and Max Woodbury, who also were members of the project. Art Sr. and Dorothy have passed on. Art Jr. was a colleague of mine at Purdue for a while, then moved to Northwestern, and is now at New Hampshire. Kincaid is listed in the CML as a retired professor at Michigan and Woodbury is listed as a retired professor at Duke, at its Center for Demographic Studies. Woodbury came within an inch of immortality during the Dewey-Truman contest of 1948. Every poll on earth and every study were predicting a win for Dewey. Woodbury had developed his own program, but noted with consternation that it was predicting a win for Truman. As this was obviously wrong, he kept the result to himself. I kick myself every time I think of it.
The Copelands had a Steinway grand, and I played some pieces for them, including Liszt's big B minor sonata. One evening we attended a party at the home of Ted Hildebrandt, the longtime department chairman. He had two Steinway grands. (No, the attempted induction ends there: the dean did not have three Steinway grands.) I picked up the idea that Art had suggested me to Ted as a possible faculty appointment. Supporting evidence materialized when they asked me when I got my PhD and their faces fell when I told them that I did not have it yet. I was disappointed that the possibility fell through but encouraged by the idea that they had considered it.
At about that time, I spent six weeks at the submarine base in New London to learn more about the boats first-hand. I had experience in such things as interpreting the display in the Torpedo Data Computer and hunting for a destroyer (the "enemy") through a periscope (of a submerged submarine). It was clear to me, and undoubtedly to the naval officers, that I was better suited to sitting at a desk.
I used to read mathematics at home. One book I was interested in was Landau's Grundlagen der Analysis. In the preface, he tells about Grandjot's objection to Peano's method of definition by induction (recursion). Let x' denote the successor of x. Peano's definition of addition was that x+1= x' and x+y' = (x+y)'. Grandjot objected that, well and good, but x+y has not been defined. I could not figure this out, and one evening walked over to consult Leonidas Alaoglu (of Alaoglu's Theorem), who lived around the corner, who clarified the issue. While I was there he showed me Sierpinski's book Leçons sur les nombres transfinis , and encouraged me to borrow it. It is a small book -- 240 pages -- and before the evening was over, I had read it from cover to cover in one sitting, the first and last time a book had mesmerized me so.
My colleagues at OEG expected me to base my dissertation on the air duels I had worked on, but, thanks to Alaoglu, I was hooked on set theory. To be practical, I set myself one month to get something started in set theory. If nothing materialized, I would revert to the game theory. When I told Harry Goode about this, he hit the roof: " First write the dissertation on game theory. Then, if there is time left, try the set theory'' - but I bull-headedly decided to stick with my announced plan.
Just before leaving Washington, I vowed independently of mathematics, to give up smoking for the year. I had been a light-to-moderate smoker - less than a pack a day. The thought lurked deep down that in new surroundings, the task would be easier. In order to help make it easy, I decided to give up coffee as well - after all, how can you drink a cup of coffee without a cigarette? Well, we got settled in, and I signed up for two courses: general topology with Witold Hurewicz and something closer to continuous groups with Warren Ambrose. And I started thinking about set theory, notably about Souslin's problem (whether a totally ordered set in which every family of mutually disjoint intervals is countable has a countable dense subset), which had been bugging me ever since I had learned of it from Sierpinski's book. (A few years later, with the help of a running program, I quite smoking for good; and these days I rarely drink coffee.)
After 27 days of nothing, my wife stepped in. She pointed out that I was there to write a dissertation, which was something difficult under the best of circumstances. But here I was making it harder for myself by tacking on another difficult, and totally irrelevant problem at the same time: namely, quitting coffee and cigarettes. At this, I got up from my desk and went out and bought a carton of Camels, a pound of coffee, and a pint of half-and-half cream. The next day I had a theorem: A necessary and sufficient condition that every totally ordered set of cardinal \aleph_\omega admit a family of mutually disjoint intervals is that 2^{\aleph_n} <\aleph_\omega for every finite n.
I wondered whether the result was known and asked Hurewicz. He didn't know but promised to write to Sierpinski to find out. I thought that would take too long, so I sat down and wrote to Tarski (at Berkeley), whom I had never met, though I had heard him lecture. He replied promptly, saying he believed it was not known; he also supplied a proof. This seemed strange to me, as though we were in competition, as I had told him I was just a graduate student - though I was familiar with his paper with Lindenbaum that states over 100 theorems, each one designated either L or T or LT.
Where my proof had used simple induction, Tarski invoked a very general theorem that he and Erdös had published in the Annals a few years earlier. I could see that this was just what I needed to extend my result to arbitrarily large cardinals; I wrote him this and he confirmed it. His proof also used a result of Hausdorff for all infinite cardinals that even Urysohn and the editors of Fundamenta Mathematica had not known about, having published a paper by Urysohn solving the special case \aleph_0; that too was immensely useful. Finally, he suggested additional reading, which led me to some additional results. Still, he lived up to his reputation by detailing just who had sent which result to whom at which time. My quip was that I was in residence at MIT, on leave from a job in Washington, writing a dissertation for Columbia, with a thesis advisor in Berkeley.
When I wrote to Koopman (now back at Columbia) to tell him that I was on my way to a thesis in set theory, he replied that the Columbia faculty were not up on the topic but that Ray Lorch had offered to read the work and serve as my official thesis advisor. Tarski told me some time later that he refers to me as his PhD by mail.
We made many new friends at MIT: Irvin Sol Cohen; Witold Hurewicz; Phil Haas, a student of his in differential equations, who became provost at Purdue, and his wife Violet; Norman Levinson, who later won the Bôcher Prize, and his wife Fagie; math chairman Ted Martin and his wife Lucy; George and Kay Whitehead; John Moore, a student of Whitehead's, who ended up as a professor at Princeton; Allen Shields, a fellow student in Hurewicz's class, who became a distinguished analyst at Michigan; and George Thomas, of later textbook fame; and others - not to mention the four C. L. E. Moore instructors: Walter Rudin, Izzy Singer, George Springer, and Earl Coddington (who used to seduce every girl within a half-mile radius); also Dick Kadison, a close friend of Izzy's, whom we met when he was in town on a social visit. And we met Felix and Eva Browder, Felix having been a Moore instructor the year before. Rudin was still a bachelor, and our having him over for dinner apparently made a big impression on him. At any rate he was very kind to me later on.
At the end of the year, we returned to the Washington area, to a small house in Falls Church, VA. I resumed work at OEG; but after my heady year at MIT my heart wasn't in it, and I resigned after a couple of weeks to concentrate on studying for the comprehensives at Columbia. These exams were finals rather than qualifiers. I passed the analysis but failed the algebra, though I made it the next time around.
In the meantime, Glen Camp, a physicist friend from OEG, had left the Group to make big bucks (22k) at Melpar, a small company that lived on Navy contracts. Their contracts operated on a cost plus 7% basis. They consistently turned in a final report three months after starting a project. Glen quoted their president: "That's a good way to live. Where else can you get a steady 28% on your investment?"
Glen wanted me to sign up permanently with Melpar, mentioning a salary of $10,000. At the same time some Army outfit mentioned the same salary, as did Convair, a military aircraft company in Texas. And so did my buddy Harry Goode at Willow Run. My quip for the year was that this was reassuring: if worse came to worst and I failed to land a $4000 instructorship somewhere, I could always fall back on a $10,000 job in military work. When I explained this to Harry and a couple of his associates, Harry turned to them and said, "I bet you think he's kidding. He means that."