Autobiography: My Higher Education

©1999 by Anil Nerode, Ithaca, NY

©2006 revised

read disclaimer first


When I was fourteen, I graduated from high school in Albuquerque, where we happened to be at the time. Early graduation then was rare. Now schools know how to handle it. But then... There was not much interest in Yoga by the public at that time. It seems that the West pursues Eastern thought for solace in periods when the materialistic culture is failing, such as the Great Depression. In boom times such as World War II and my graduation year 1947, the interest dies off. We were staying with my wife's next younger sister Cornelia in Albuquerque while my parents were trying to figure out how to make a living. We were broke. I selected the college I wanted to enter entirely on my own. My father was a graduate of the University of Calcutta, my mother did not graduate from UCLA. There was no such thing as a guidance counsellor. I had no advice whatsoever. I reasoned that I was more likely to get in if I did not apply for financial aid. I heard of Chicago because half the Los Alamos people migrated there. I found out that there would be a lot of students there my own age, a unique thing for a major university. I was accepted at Chicago, and went there with only a bus ticket and seventeen dollars in my pocket, without money for either tuition or board and room. Some would think this foolhardy, but I think I had acquired my father's attitude that Providence will provide. I have always felt that I live under a lucky star, that I am protected by unseen powers, that this is the best of all possible worlds, at least for me. This is surely derived from my father's entirely identical view of himself. I moved into the dormitories with some very fine roommates, and immediately went to the Bursar, Albert Cotton, to see what I could borrow. He suggested that perhaps I should get a job. I told him I was under the minimum age for employment in Illinois. He shrugged his shoulders and loaned me everything I needed, every year, till my Ph. D. I paid it all off after I finally finished. It shows how little I knew about financial aid. One of my friends had a full scholarship based on need; his parents were buying a bank and had no spare cash. The Bursar, I was told later, was the first to offer such long term loans to students, not only at Chicago, but in the United States. Providence was pretty effective.


The roommates I had the first year at Chicago, Robert L. Ross(deceased) and Daniel J. Robbins (deceased), remained life-long friends. But when I looked at a list of those going to the Chicago 50th reunion of my class of 49, I did not recognize a single name, and therefore did not go. Ross knew what he wanted to do, become a high school chemistry teacher, and that is exactly what he did in Peekskill, New York. Robbins was in process of becoming a painter, but switched, perhaps under the historical influence of the college, into art history. When writing his dissertation at the New York University Institute of Fine Arts, he concentrated on Albert Gleizes, a transitional figure in the development of cubismGleizes.. Gleizes' wife gave him the Gleizes' papers, including a shoeboxfull of photos of all his work. Robbins became THE Gleizes specialist. He was successively a modern art curator at the Metropolitan, at the Guggenheim (where he did a definitive exhibition and catalogue for Gleizes), Director of the Rhode Island Museum of the Rhode Island School of Design (a truly major museum most people have not heard of). I visited him there at the time of a Christmas art sale at the museum. He and I both bought Roy Lichtenstein works, op art from a very brief Lichtenstein period when he did op art using silvered paper and plastic sheetingLichtenstein Robbins was then offered a Professorship at Yale together with the directorship of their museum, alternately the Directorship of the Fogg at Harvard without a professorship. I strongly advised him that since he was introducing more 20th century into the museums, the old line Art Historians at Harvard would kill him if he were not a professor, while the Professorship in Art History at Yale would prevent that from happening there. Boston was closer to his farm in Vermont, so the Fogg is where he went. The Art History professors at Harvard were very unhappy with his direction. An opportunity came to boot him out, and they did. There was a robbery of gold coins at the Fogg. They awakened Robbins in the middle of the night, he said to leave it to the police, he would contact them in the morning. The fact that he did not immediately go to the scene of the crime was the pretext for getting rid of him. I am sure they knew he would have made a very poor Nero Wolfe.

Although he published a lot on Gleizes, he had not even turned his thesis in at that time, a common thing among museum curators and directors. The profssors attacked him as being an irresponsible graduate student in charge of the Fogg. He did eventually turn in his thesis at NYU. Then he took a Professorship in Art History at Union College, near his Vermont farm, and immediately irritated the President tthere by launching a campaign preventing him from tearing down a large building by an important 19th century architect. This time he was luckier. He held a tenured Chair. He died a few years ago of an incurable degenerative disease. He helped many people in their careers, artists, art historians, collectors, etc. I am told his funeral had attendence from all walks of life from all around the world. He was one of my favorite people.

Hutchins Program

So I entered the Hutchins Great Books program at the University of Chicago, thinking of majoring later in physics. That was 1947, when ex-soldiers under the GI bill flooded Chicago and every other institution with dead serious intent to learn and make up for lost time. Their company was a quick maturing experience. I enjoyed the Hutchins program a lot. I read the complete great book whenever a snippet of one was assigned in one of the Hutchins courses. I had the noted southern poet Allan Tate as my preceptor in the metaphysical poets. I had seminars with T.S. Eliot. I remember speaking to and listerning to Nehru at a reception. I spent an evening with Dylan Thomas in a local tavern. Class attendance was not required in the Hutchins college, so I attended only lectures of enlightening teachers, usually not in the classes in which I was registered. Otherwise I read and read. This was all Hutchins' work. He really liked providing undergraduates with broad intellectual and cultural experiences and complete independence. Then there was the Hyde Park neighborhood, with the Compass group with Elaine May and Mike Nichols and crew, whom I and my friends went to see at virtually every opportunity during its heyday and before Second City. There was also Severn Darden and his fabled escapades at the university. Then there was Chicago. In that new postwar period, the Premier Grand Cru Bordeaux and Burgundy were two dollars a bottle; no one over here valued them and France needed currency. Many evenings were spent in intellectual discussions over a bottle of Margaux or Hermitage with cheese. There were cultural events at the Chicago Art Institute, at the Oriental Institute, at theatre groups on campus, there was acting as hatcheck for downtown legitimate theatres in return for free tickets, there was being a subject for autohypnosis experiments with Tony Eidson, who had studied with Andrew Salter. I had a wonderful time in a wonderful place. Most of the successful products of that milieu feel the same. Of course there were many others for whom either the standards or the freedom led them to disaster.

The memory that sticks most vividly from Hutchins College days was a little scary and unconnected with education. In my second year I commuted by bus and streetcar from a trailer where my family was living in the suburb of Oak Lawn. One day I felt very uncomfortable about the streetcar ride, and, for the only time- decided to stay overnight at the dormitory with a friend. I read the Chicago papers the next morning and discovered that the (hourly) streetcar I always took had been rammed by a gasoline truck. All passengers had been incinerated. Such are the ways of Providence.

Not everything was golden about the surrounding Hyde Park neighboorhood, either then or now. My parents settled in Hyde Park on Dorchester, below 55th, in 1951 in a small brick house built to house workers at the Chicago Exposition of 1893. They settled down because they had two younger children, Kiron and Rabindra (Robin), and travelling constantly with children had become a nightmare for my mother. My father hardly noticed. He did not leave on this earthly plane. My first night in that house, I was alone and heard screams on the street. I called the Hyde Park police and told them someone was being beaten up in front of the house. Their station was three blocks away, but no one came. I finally went to the door and a patrolman called to me that he had been in a fight with a suspect, and to call the police. I then called back and told them that a POLICEMAN was calling for help. Within one minute the entire street was filled with police cars and sirens, and they took them off. So much for getting help for the ordinary mortal from the Chicago police department. The next day I found the sidewalk drenched in blood, and heard that the policeman had beaten the suspect to death. Of course, it was not worth a mention in the Chicago papers, and nothing came of it. That was my introduction to Hyde Park outside the dormitories. The non-academic part of the university was a mixed blessing as well. To protect themselves from the black belt, the university administration was cooperating with the developers to get rid of the small home owners in the 55th street region and put in more expensive large apartment buildings as a buffer to the near south side black belt. We were called to a meeting at the local Buddhist temple down the block by Julian Levi (or was it his brother?) who told us that we the owners of the housing had champagne tastes and beer incomes, and would have to move out of the area to accomodate richer people. He said we had bought into a neighboorhood in which people of modest income should not live, and we should sell out to the developers on their terms. I got up and gave him hell. He tried to shut me up, but no one ever has, and we did own property. My mother was very capable. She formed a neighboorhood group, and the area is still into old remodelled housing 50 years later. I was very sorry when I heard that one of the Levi's had become president of my alma mater. I am sure they loved it, but I don't love them. I don't think Hyde Park has improved much since. My mother's house on Dorchester below 55th was broken into by burglers three times in the late 1990's, once possibly by alocksmith who had changed her locks! The last time my mother lambasted the burgler with her cane and drove him out of the house. Enough is enough, at 91 I moved her to a a 31st story apartment on Wells street on the near north side, with a doorman downstairs and cameras trained on all entrances. She died there at 94.

In the early 1960's, my mother proved her mettle again. There was a riparian war taking place near her childhood home in Mancos, Colorado, and her next younger brother Sharon was in jail accused of taking pot shots at the tractor of an upstream rancher trying to divert water. There appeared to be something fishy, and she flew to Denver to examine the materials on the case. All I know is that when she took information on collusion between prosecutor and the so-called victim back to Mancos, the prosecutor, who had a local reputation as a silver tongued orator, swallowed his tongue, had a heart attack, and dropped dead. The case was dropped.

Graduate School

My first physics teacher at Chicago (Lawson) told the class the first day that he would flunk at least half the students in order to cut the classes down in the subsequent courses. I took this as meaning that physicists there did not care much about the undergraduates. Anyone saying that today would be lynched. In addition his teaching assistant was incomprehensible, and on enquiry I found that the Physics Department (Voorhes) could not care less. I did not regard the physics department as hospitable. When I took mathematics and philosophy courses there, the teachers were friendly and I encountered no teaching assistants as teachers, teaching assistants just graded. I naturally inclined toward these disciplines. I was introduced to modern logic in the seminars of my much beloved philosophy professor, Rudolf Carnap, whose sixtieth birthday party in his apartment I remember well. I was thinking of working with him. He told me I was his best student, and suggested I go into mathematics because that was where the future of logic lay. At that time there were few logicians in mathematics departments (Church, Kleene, Rosser, Curry, Tarski); most were in philosophy departments ( Quine, Carnap, Reichenbach, etc. ). The obvious thing was to transfer to Berkeley (Tarski), Wisconsin (Kleene), or Princeton (Church), but it never occurred to me that he might be suggesting this. I therefore went into mathematics at Chicago, not realizing that I had stumbled into what was the beginning of Chicago's reign as arguably the most outstanding mathematics department in the world, what MacLane referred to in his autobiography as the "Stone Age" after its instigator Marshall Harvey Stone. I thought then that the math graduate students there were the run of the mill students you might find at any graduate school. I was simply very ill informed. This gave me unrealistically high expectations for graduate students when I finally became a professor. As is well known in the mathematical community, that student body of the Stone Age contained many of the most prominent mathematicians of the latter half of the twentieth century.

Chicago had the reputation of being a premier Bourbaki school with Andre Weil as its most famous personage, and Bourbaki did not believe logic was a branch of mathematics. Fot those not in mathematics, I should say that Bourbaki was a small French Mafia with Weil as the Godfather, writing a series of modern texts covering what they considered to be core mathematics. From the point of view of the rest of the mathematicians, they appeared to have taken upon themselves the role of arbiters of mathematical taste. Their mathematical horizons did not include Applied Mathematics, Probability, Statistics, or Logic. As to Logic, their foundations book showed that none of them had any understanding of the subject anyway. I looked over the Chicago faculty with great care for an advisor. Paul Halmos was creating his version of algebraic logic at the time, perhaps to show that logic was algebra, but he was not knowledgeable in recursive function theory, which I loved from Post's papers. I took the trouble to look up the background of all the faculty members, and found that MacLane had done his dissertation in logic, nominally under Hilbert, actually with Weyl and Bernays, and that his Category Theory was pretty close to logic anyway. No one then realized it was the typed lambda calculus in an arrowed disguise. There were no logic courses in the mathematics department, but I asked MacLane to sponsor a logic seminar, which he did.

The only students interested in logic when I entered the department were Ray Smullyan and Stanley Tennenbaum.. They were joined later by Bill Howard and Michael Morley.

Ray Smullyan

Ray was very smart and very slow. The phrase "slow" can be misinterpreted. He learned everything with an incredible attention to understanding every aspect of every definition and every construction, including alternate definitions and alternate constructions. This took a very long time, much longer than the courses ran. Much of his success has been due to finding elegant new ways of doing things which are the product of this kind of protracted analysis. So "slow" is not perjorative. He had been "discovered" by the algebraist McDuffee in a beginning course at Wisconsin, and sent to Chicago. He earned his living as a table magician at a popular downtown Chicago restuarant. Smullayn's special thing was to be able to do each trick 15 different ways, with the same effect, so you could not possibly figure out at all what he was doing as he switched between them. He did not really matriculate at Chicago. Finally Carnap, who had a very high opinion of him, got him a one year job at Dartmouth when John Kemeny became chairman of math and was short a person in September as he arrived back from a visit to Europe. The powers at Dartmouth discovered that Ray had no bachelor's degree, and gave him an ultimatum that he had to pick up one at the end of the year if he wanted to stay a second year. It is a tribute to Ed Spanier that when Smullyan came back to Chicago and told him this, he simply counted all the courses that Smullyan was teaching and all the ones he attended without grades at Chicago, and gave him his bachelor's degree. Try that in the modern Ivy League! Later, again due to a letter from Carnap, Smullyan became a student of Church at Princeton. Due to a tendency to want to overprepare , he was terrified at his preliminary exam. He was put at ease by Emil Artin, who knew he was an accomplished magician, and who told him at the exam to do some magic tricks instead. He spent his career at Yeshiva and CUNY, ever original and productive. His puzzle books are also a delight.

Stanley Tennenbaum

Stan came from high school in Cincinnati to the Hutchins Great Books Program and graduated with a Ph. B. a couple of years before I did. He was five years my senior, but my contemporary in entering mathematics.. After getting his Ph.B. Stan never got himself organized enough to finish courses in the graduate school, much less to get a degree beyond the Hutchins college Ph.B. But he did significant mathematical work then and later. Why did he not get an advanced degree? He was a true intellectual, a true product of the Hutchins' college as Hutchins had intended it. In that college, which I also went through without attending classes, there was no obligation to attend classes as an undergraduate, only an obligation to take a six hour exam in each course at the end of the year. Stan took advantage of this freedom and instead spent a lot of time reading the great books, in intellectual discussions with Chicago luminaries such as Robert Hutchins, Joe Schwab, Leo Strauss, Mortimer Adler, Bruno Bettleheim, and all others with an interest in the philosophy, psychology, and practice of education at all levels, an endless list. In graduate school he had none of the habits for success in a mathematics department such as attending classes, doing homework, or taking tests. I had the same habits he had, from the same source, for my first quarter in graduate school, but his habits were unchanged forever. He married Carol and had his first son early, and always supported them with a variety of jobs while a student.

There were many true anecdotes about him. One of his jobs was as a night cab driver. I got into his cab on a winter night, and discovered that the windows were completely covered with thick opaque snow, with only an eyehole cleared in front of the driver. This scared me, so I offered to clean the window. But Stanley drove off, saying it was much better to have the limited vision, because other cars would notice it and stay away. This was at night, with lots of traffic. He was fascinated by the idea that limiting information could improve performance. But in this instance...When Marcel Marceau visited Chicago, Stan took his small son to a performance. Stan wanted to meet Marceau, so he told the backstage manager that his son was a great fan and wanted to meet the great man. Marceau loved children, Stan loved children, and they spent the rest of the evening with him. I have not doubt that Stan used such strategies many times. He knew a lot of famous people outside the university that you don't run into on the street. One of the things that surprised me about him was a love of football, acquired in high school. My memory, which may be faulty, is that he played football in high school, but suffered an injury and had to stop. In any case he had the highest intellectual respect for certain football players, who were not very big, possibly not very fast, but out-psyched their opponents. Typical was his admiration for Charley Trippi of Georgia who could wiggle his hips deceptively, misleading pursuers, who then predicted incorrectly his twists and turns and tackled thin air. He was taken with the triumph of brain over brawn.. When Elvis became a sensation among the teenagers, and the rest of us were perplexed at the crowd hysteria we saw on tv, Stan expressed great admiration for Elvis for wiggling his pelvis while performing, saying that it was a great cultural advance that Elvis was a male permitted to express sexuality in the same way as women dancers in our overly inhibited society, a very thoughtful observation.

In 1959 he was the first to show that there were no recursive models of formal arithmetic, improving a corresponding result for set theory of Michael Rabin from 1958. In1963, while under support of one of my federal contracts, Stan proved the independence of Souslin's hypothesis. I was in Princeton at IAS and IDA. We made him an an appointment with Godel so Stan could show him the proof. Stan was understandably nervous, and arrived the day before to have me check the proof, since I had read Cohen's work the day Paul spoke at IAS earlier in the year. He was nervous and took a sleeping pill. I found minor errors. He then asked for dexedrine, which he knew I had. This made me extremely nervous, but I gave him a couple, and he corrected the errors that night before seeing Godel the next day. In his place I would not have taken a sedative and a stimulant at the same time, but that was Stan! Later, during and after a visit to Penn, he proved, with Solovay, the consistency of the Souslin hypothesis. These were the first concrete independence result in general mathematics after Paul Cohen's work. Cohen was a younger fellow student of ours at Chicago, who learned logic by osmosis while renting a room in Stan's house, where logicians and logic students congregated a decade earlier.

Stan always had a doubtful relation to MacLane, probably because Stan had never finished his courses at Chicago. When we were students, Stan said that whenever he had told Saunders anything, Saunders always checked with me afterwards to make sure it was right. It always was right, but it annoyed Stan greatly. Here is a Peter Freyd anecdote that illustrates this. "A story about the two of them: in 1964. I called Mac Lane in the middle of the summer to ask him why he was blocking a a one-year visitorship for Stanley in the Penn Philosophy department. (I had concluded that the Provost's friendship with Saunders must be obstructing the appointment. The Provost -- perhaps on orders from Saunders -- hadn't told me that there was even an obstruction, never mind that it came from Saunders.) In a somewhat heated conversation, Saunders said at one point that the Tennenbaum he knew could not possibly have proved the independence of the Suslin conjecture. I asked if I could quote him. Long pause. He then responded that he would call the Provost and undo the obstruction. Stanley's visit was actually a great success." Stan's proof, as I said earlier had been verified already by no less than Godel, but Saunders was not aware of this. He always underestimated Stan.

Stan was a bit of a conspiracy theorist. He was sure that that Freeman Dyson was part of a secret U S government sponsored group which controlled world weather and much of world affairs. He believed that this organization started in World War II with MacLane's deployment to an applied mathematics group at Columbia in New York City, also involving the man who brought me to Cornell, J. Barkely Rosser, who was very much involved with the military at a very high level. He viewed my work for many US military organizations over many years as cover for my participation in this project. My wife Sally's introduction to this was when she was walking on the Arts quad at Cornell and encountered Stan and said conversationally "It's a nice day". His reply was "and do you know why", proceeding to launch into this tale, hoping she would let something "slip" I could not disabuse him of these thoughts. He kept trying to trip me up into inadvertantly revealing the conspiracy. Since I did work for various agencies, whenever I said I was not at liberty to discuss some topic, he immediately interpreted this as protecting the "secret". Stan had a vivid imagination.

Stan was a truly exceptional teacher. In the early days he often taught at the night school of the downtown college of the University of Chicago. Later, at Cornell, I arranged for him to come during summers to teach prospective and established high school teachers in a program sponsored by Shell Oil. After the initial Souslin work, he got a tenure position at Rochester, without having an advanced degree, through the efforts of a brave chairman, Len Gillman, and references that I suggested from those few who then understood his work at that time. He stayed there for awhile, then got upset at a faculty meeting with then President Wallace, whom he had known at Chicago as a Statistics professor, walked to the front, spit on the President's shoes in contempt, left the meeting, and resigned from the University.

He left the academic community altogether to become a remarkably unsuccessful entrepreneur. He was a great teacher and felt he could find a mechanism for transferring this expertise to others in a way that would give him an income. But this did not happen. He never held another tenured position. He became entirely peripetetic, with visiting positions or no position for the rest of his life. He greatly valued his lifelong friendship with Hutchins. He told me a few years ago that the difference between him and me is that I grew up and he did not. His peregrinations were quite wild. After leaving Rochester, he came back and invited a large number of mathematicians to a meeting in a Rochester motel without informing the Rochester mathematics department. Afterwards, he sent the the department the bill. Instead of refusing to pay, as I would or you would, that ever kind Chairman Gail Young paid the bill. Finally, over his life he helped a great many people in their lives, giving freely of his time and energy whenever needed.

He had many excellent ideas for educational projects. People he knew occasionally gave him money to try to bring them to reality. It never happened. I don't think he had any idea how hard it is to make a concrete business plan for a venture which will convince backers that there would be a stream of income. He thought that an inspiring idea and a sketch of a plan would be enough to get interested parties to invest. It is not as easy as that. I would guess he acquired this excessive optimism from Hutchins, who had a motto that "ideas have consequences". This enthusiasm was an endearing quality to all who knew him. Another of his sterling qualities was that whenever anyone was in trouble, he would rush to their aid. If they needed money, he would borrow it from someone else and give it to them. If they required his time or professional help, he would offer it. This is one of the reasons he was so beloved by his friends.

Such was the fate of my fellow logic students in the year I entered.

My closest friend in the Hutchins college and in graduate school was Ed Nelson, who has spent his career at Princeton as a Professor of analysis and probability. He also made later contributions to nonstandard models of set theory and ultrafinitistic mathematics. I met him in 1948-9 when I was still in Hutchins' college. He had come in from a Roman Lycèe, taken the exams that are given when you entered Chicago, and placed out of all the courses. He was awarded his bachelor's degree without taking any of the required courses. Time magazine carried a picture of him next to a pile of all the required books he did not have to read, a pile nearly as tall as he was. He was best man at my first wedding. At that time he was not interested in logic. I had not seen him for a long time, we met again at the Tennenbaum memorial at CUNY in April of 2006. He said he was now a great-grandfather.

Michael Morley and Bill Howard

Michael Morley and Bill Howard joined me as part of the logic student contingent somewhat later. I think I can fairly claim to have induced both Michael Morley and Bill Howardright howard to do logic dissertations with MacLane. At least, Bill Howard has expressed gratitude for the suggestion ever since. I think he was too depressed at the time to think clearly. He had thought he solved a famous problem, told Weil, who telegraphed it to the elder Cartan. It turned out to be in error, Weil demolished him, and he became extremely depressed. He was also interested in proof theory from the beginning. Howard's most famous discovery was in the middle sixties, the Curry-Howard isomorphism for intuitionistic arithmetic. We now express this as the correspondence between natural deductions and typed lambda calculus terms, very important in computer science.

In the case of Morley, he had discovered some extension theorems for 1-1 endomorphisms of groups to automorphisms of larger groups. I suggested he should do the same for first order logic. Overnight, he read up on what first order logic was, and extended his theorems. He had no previous experience with logic. This was the beginning of the saturated models work. He also developed recursive analysis. He presented his work to MacLane, who said it was not enough for a thesis. Morley pointed out Abraham Robinson's recent book on the Metamathematics of Algebra on MacLane's bookshelf, and said his work was certainly as good as that. MacLane agreed, took the book, and told Morley he could keep it. More about how he got his degree later. Michael has always been a down to earth fellow, stemming I am sure from his upbringing in a Youngstown blue collar environment. When he and fellow student Vivienne Brenner decided to marry, they took a streetcar downtown, were married by a judge, and returned to attend an afternoon math tea. No one is less pretentious.

As a student I realized that I should try to make a good impression on the rest of a department dominated by that ultimate sceptic about logic, Andre Weil. In those days for the Ph.D. qualifying exam the student chose three areas for examination, areas in which one might write a thesis. I choose logic as one, and algebraic topology for MacLane. I noticed that Weil was giving an elliptic and modular functions year course, so I decided to take it and choose that subject as my third topic. The course of Weil was most memorable. The audience at the beginning contained Armand Borel (visitor), Serge Lang (instructor), Ed Nelson (student), and me, as I remember it. Weil almost always turned up with some large red volumes and gave the lectures while looking at these volumes and then going to the board and working things out in Weierstrass p-function terms. At the end of the course, I located the books; they were Kronecker's collected works, and at sight he was translating proofs from the Kronecker notation to the Weierstrass notation, no mean feat! This was part of his preparation for writing his "Basic Number Theory"; he always consulted the classics. This reinforced my interest in reading 19th century mathematics, which I have done ever since. I have occasionally given courses in the history of nineteenth century mathematics as a result. I remember the last week of his course, when almost everyone had stopped attending. I came late, the chairs were empty, and Weil had already started lecturing and had filled up three blackboards!

I took a lot of courses from almost everyone there including all visitors, since I like every branch of mathematics. This has stood me in good stead for over fifty years. When I need to use things from subjects very remote from logic, they are usually among the topics I learned as " mother's milk" at Chicago. Without the Chicago background from the early 1950's, I certainly don't think I would have been able to recognize Finsler metrics arising from Lagrangians and use them to compute connections to be used as control policies for physical devices. I learned "modern" differential geometry from Chern that long ago! He had had one of my graduate student friends, Louis Auslander (now deceased) write a Finsler Space paper which I read at that time.

A significant logical influence on me was provided by three visitors in about 1952-3. They were John Myhill in Philosophy, Jim Dekker in mathematics, and Burt Dreben in Philosophy. They became my lifelong friends, but all three are now deceased. Dekker got his degree under Paul Rosenbloom at Syracuse. Rosenbloom was an analyst who wrote a really nice short book on logic using Post productions. Dreben and Myhill were students of Quine.

Dana ScottScott turned up as an instructor at Chcago for a year, which was a delight. I believe he met his wife Irene Schreier there. Irene was the stepdaughter of a well-known music teacher in Chicago (Jonas) who knew Smullyan, also a pianist. She is the daughter of the mathematician Schreier. Everyone in the mathematics world passed through Chicago. For a logician, especially memorable were lectures and teas with Von Neumann and Brouwer. Brouwer was really hard to communuicate with, even though I knew his work quite completely.

John Myhill

I remember first meeting Myhill in an apartment in Hyde Park in about 1952. He had tried to paint it blue, and blue had been transferred to floors and tables. He ended up having all objects in the room bright blue with no contrast. It was a blinding sight when I met him in that room in bright sunlight. I bumped into everything in the room, they all were invisible. There are many stories about Myhill. One he was hardly able to live down. He had gone to a local tavern with friends, and a women there insisted that Myhill was a child and needed a women's care. When they left the tavern, Myhill passed by a baby carriage parked in front of an apartment building. He climbed into it, and she began to roll him along the sidewalk. The owner looked out the window, saw his diappearing baby carriage, and called the Hyde Park Police. They took him in, and one of the party happened upon Bill Howard, who rushed down to the station to rescue Myhill. He told them that Myhill was a professor of philosophy. The Irish cop, with an Irish Brogue, asked Bill what HIS profession was. Bill replied that he was a mathematician. The cop replied skeptically "Oh, a mathematician, eh, lets see if you can extract the cube root of 7". Bill desperately tried to remember high school algebra, and painfully started to work it out, with the cop saying "that's right, you're getting it." This was probably not a typical Chicago cop, even among the Irish ones. Unfortunately a stringer for the Chicago Tribune was in the station. The next morning the headline on the front page of the Tribune read LOGIC GOES BUGGY, and started out roughly saying "a man, claiming th distinction of a Harvard Ph. D. and Professorships at Yale and Chicago, was apprehended in a ..." The Philosophy Department Chair at Chicago claimed he was not a faculty member, which may be technically true since he was merely a visiting assistant professor from Yale. This tale followed him for quite a while. At the end of the fifties he had an unpleasant divorce and became a bit erratic. He took too many leaves, and lost his tenured position. He was regarded as unstable. I put a huge effort into convincing my friends at Buffalo to take him on. He was a great positive influence there for upgrading the department, since he was very social nationally with the best mathematicians. He built a terrific constructive mathematics group, married one of them, Miss Kino. His life ended in tragedy. He told me that Ms. Kino committed suicide without warning, and that her family blamed it on him. I have heard differing accounts on her death from others. Then he succumbed to lung cancer from a lifetime of smoking. One of the last things he did was to invite me over to meet his latest student, Andre Scedrov, who would have no one looking after him after Myhill's imminent death. He asked me to, and I did. They also left a teenage daughter, of whom I have lost track.

Walter Bartky's Institute

In 1954 I met Moe Schreiber (deceased) , also a graduate student in mathematics, and noticed that he could afford decent clothes and apartment etc., unlike the TA's and RA's. I ask him how, and he mention Prof. Walter Bartky's Institute for Air Weapons Research, located in the upper reaches of the Museum of Science and Industry, which did classified work for the US Air Force. Bartky was Dean of the Physical Sciences Division of Chicago at the time. Moe said the work was consistent with writing a dissertation and probably comparable in time consumption to being a TA or RA, but paid two or three times as much. Bartky (deceased) was the only applied mathematician in the Mathematics Department, a celestial mechanist. Schreiber offered to get me an interview. I was interviewed by Bernie Howard, an MIT Ph.D. who was running a new Institute for Systems Research under Bartky, sponsored by the Air Force. He hired me in 1954, and I spent the first three months reading the MIT radiation series from cover to cover, till a secret clearance came through. Then I went into the business of simulating rival designs for airborne weapon systems. At that point, Michael Morley was stuck so far as getting a dissertation going, and I suggested he apply too. Eventually he and I and his wife Vivienne were all there.

James Corbett

There was a very impressive engineer, James Corbett (deceased), highly mathematical, who was Bartky's advisor. From him and on the job I learned many branches of applied mathematics and mechanical and electrical engineering and control theory, and participated in weapons system design and simulation. At the end of Corbett's career he introduced the use of the Euler characteristic to check that when maps are fitted together, from aerial observation, the roads are connected properly. It is in use world wide. My job also meant that I was deferred from the Korean War. My Chicago draft board was in a working class neighborhood, and resented mightily my not serving. But Bartky and his Air Force program manager told them I was doing essential military work, and that was that.

It is because of the work with Corbett that I have been a consultant for many military and inductrial projects over fifty years. Bernie Howard went on to found the computer science department at the University of Miami at Coral Gables. Bartky died a premature death. Corbett lasted till 1998. Morley often visited him.

First Marriage

I met my first wife, Sondra Raines, in 1952 when she was the girlfriend of one of my closest friends, Robert L. Ross. I more or less grabbed her away, which irritated Ross, but he found a very fine wife a few years later, and we were all friends again later.We started dating when she was 18, I was 20. She was my very first girlfriend, and by everybody's accounts a considerable beauty. I actually didn't notice, I just liked her company. I think most mathematicians choose wives without a lot of regard for looks. She was my first female good friend. Sondra was an undergraduate in the Hutchins College, I was in mathematics graduate school. In 1955 the income from the Bartky Institute job meant that we could afford to marry. The wedding ,My First Wedding was in Barnes Chapel at the Chicago Theological Seminary next to the Chicago campus February 12, 1955. She went to graduate school in art history, but this did not work out. Much later, after divorcing me and remarrying in 1968, she went to Law School in Buffalo and became a lawyer. Our oldest son Christopher, was born in 1957,our middle son Gregory in 1965.

Postdoctoral Years

I wrote my dissertation in 1955-6 on a topic of my choice without any external advice, while MacLane was on leave in France. He went through it by mail, and I received my Ph.D. in Mathematics in 1956.graduation. The thesis was on an algebraic abstract formulation of substitution in many-sorted free algebras and its relation to equational definitions of the partial recursive functions. I unfortunately acceded to the referee's request that the one-sorted case was enough to publish, the many-sorted subject by now has been revived several times in computer science under the name instantiation theory and also abstract theory of data types. I was awarded an NSF postdoctoral fellowship, but decided to spend one more year (1956-7) at Bartky's Institute completing projects no one else would have completed. It was there that I was able to support some of Myhill's early work on Automata from Air Force grants and occasionally Tennenbaum. This led to my work in 1958 on linear automata, and the Nerode's theorem that is in many beginning Computer Science textbooks. I am not going to include my mathematical autobiography for years before 1992, because there already is a very complete mathematical biography eighty five pages long by my former student J. B. Remmel in the 1992 Festschrift volume "Logical Methods".

For me a signal event was the Cornell NSF Summer 1957 Institute in Logic, arranged by J. Barkley Rosser. I was told to go by Paul Halmos, bless his heart, and I did so. Except for Gödel, every significant logician in the world was there, a group of no more than 60 people. Gödel thought he was was not healthy enough to attend, but he sent Kreisel with a paper for the conference. It was the FIRST time many major figures had met, and everyone had a wonderful time. It was a society of equals; everyone was asked to say what they wanted to present, and to give an abstract. I gave two. One was on my thesis, the other had two theorems: the continuous functions on the partial functions are precisely the relative recursive functionals; the everywhere defined reduction procedures are precisely the truth table reductions. The published version of this paper is referenced by Scott as one of the origins of his continuous functionals interpretation of computer programs. I thought Cornell and Ithaca were the prettiest places I had ever been. The other signal event of 1957 was the birth of my eldest son Christopher.

In 1958-9 I went to the Institute for Advanced Study with Deane Montgomery as sponsor, since MacLane regarded him as more likely to complete administrative matters than Gödel. My officemate there was C. C. Chang. Gödel was the person I worked with. There were so many rumors about him that a Polish friend in Chicago gave me a formal European letter of introduction to a rich Princeton friend of Gödel who had followed Gödel from Europe. I met the friend first, and he gave me a second letter of introduction to Gödel, which I used. All of this was entirely unnecessary; Gödel was happy to talk once a week, and was a pretty fair administrator. There are many anecdotes about Gödel, a few of which are true. What I remember that year about him is the two of us taking an Institute car every week of the winter into town. He wore a European heavy black overcoat and fedora. As it got to be a warm spring and a hot summer, I wondered when he would leave these at home. One day in June in blazing heat he had. He had changed over to a heavy European brown overcoat and brown fedora, which he wore the rest of the year. He was very helpful . He took seriously questions about automata and the question whether there were nonstandard models within the Dekker isols. I proved a few years later there were. Dekker was around Princeton in 1958-9, as was Myhill. I got extremely interested in Myhill's new combinatorial functions, and spent a number of years working out the metatheory of isols using them. I was also priveleged to meet Heyting as he passed through. I spent a couple of hours talking to him. Paul Halmos was in the room, and afterwards said, "ah, you have been conversing with the AntiChrist". This reflected the general opinion about intuitionism in those days. Of course, those of us in recursive function theory understood it through its recursive function theory interpretations (realizability) established a decade earlier by Kleene.

I wrote and asked Tarski, whom I had met at the 1957 Cornell conference, if he could arrange a visit, and he arranged a visiting assistant professorship at Berkeley for 1958-9. I was the officemate of Leon Henkin. He and his wife were most hospitable, as were Julia and Raphael Robinson and the Tarskis. Julia cooked us some excellent dinners. I learned there from Tarski how to run a real logic seminar in which papers in all logic subjects were covered and the students wrote up things that had previously been written badly. I copied this when I got to Cornell. At Berkeley my first semsester teaching assignments were for courses in differential geometry, existence theorems in differential equations, and foundations of mathematics. I apparently had succeeded in creating a persona of being a genuine mathematician, rather than a specialized logician, since I did not ask for these courses. That same reputation meant that I have had to referee papers in a vast variety of subjects only remotely related to logic, which was probably good for me. At a regional meeting I heard Bob Vaught of Berkeley give a talk, half the theorems of which were those of Morley's incomplete thesis of earlier years. I told Vaught about Morley, and Vaught got the manuscript from Morley. They then wrote the Morley-Vaught paper on saturated models. Vaught also told Michael to come out and try to find something interesting to do for a thesis. One of the first things Morley heard at Berkeley was Los's conjecture, which he solved that fall. That satisfied MacLane, and he got a Chicago Ph.D. in about 1961. He cast the thesis in category theory form in honor of his advisor. He never felt MacLane should have accepted the previous manuscripts, even though their content included many theses accepted elsewhere. Morley believed in high standards too.

My Childhood among the Yogis

My Life at Cornell

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