I started learning mathematics in high school, out of books. I was especially drawn to the editions put out in paperback by Dover Books, because they were cheap. It was many years later that I finally realized that the reason Dover could publish books so cheaply was that those books were dreadfully out of date. For me, though, they were very exciting books.

There was a book on vector analysis, which seemed like quite useful stuff and not too difficult, though I did have a hard time getting my mind around the idea of differentiating and integrating vectors.

There were Knopp's three volumes on The Theory of Functions. I knew what a function was, but I couldn't imagine how one could create a theory about them, and I was desperately curious to find out. (I didn't actually buy these until after my first year of college.)

Likewise I was extremely curious about a book by someone named Kamke, as I recall, on the Theory of Sets. I couldn't imagine how one could find much of anything interesting to say about sets. However, I think I never actually got that one at all. (I couldn't afford to buy very many, even at Dover prices.)

I didn't completely understand these books, but I worked very hard at them. My desire to understand as much as possible was passionate.

One book I devoted an immense amount of effort to was Hermann Weyl's Space, Time, and Matter. The title seemed to promise that if I could just master that one book, I'd understand the whole universe. Besides, it was largely about the theory of relativity, which I very much wanted to understand, because at that time it had the reputation of being the most difficult subject in the world. (Later on, as a graduate student and mathematician, I would make a valiant attempt, extending over several years, to understand algebraic geometry, for much the same reason.)

Between 1957, when I graduated from high school, and 1966, when I enrolled in graduate school, the mathematical world, along with the rest of the academic world, changed in other ways too. Before 1960, being an academic was a calling, something involving a major sacrifice, which one did because one cared passionately about one's subject. And, as far as I could tell as an undergraduate at the University of Arizona, the number one concern of academics was with their teaching. Grants were essentially unknown. Colloquia were so unusual at Arizona that they were announced in the undergraduate mathematics classes, and I and some of my friends went to some of them, assuming that there would be something of interest to us.

By the time I got to graduate school in 1966, being an academic had become simply a career choice. If one liked going to school, and didn't like the idea of a forty-hour per week job, then one went to graduate school. (Another major consideration was the Vietnam War. Having a wife and daughter had always kept me safe from the draft, and by the time I enrolled in graduate school I was already over the age limit, but many of my male friends were quite frank about the fact that they had no intention of getting their degree until they were 26 and no longer needed the student deferment. A few years later, the lottery would be introduced and the draft would become a little more sane.)

By 1966, professors' salaries had become quite good -- it was taken for granted that getting a Ph.D. would pay off in financial terms. (Those Ph.D.s who weren't good enough to get a job at a major university could always get one in industry, where the money was even better.)

When I went to the University of Arizona after my one year at Hopkins, I wanted to take only mathematics that was useful -- vector analysis, partial differential equations, matrix algebra. (Somehow I had missed out on ordinary differential equations, which hadn't been part of the calculus sequence at Hopkins, so I taught it to myself out of Schaum's Outline.)

The Math Department had recently added a requirement that all majors had to take the first semester of real analysis, and that was actually the first math course I took at Arizona. The class was mostly full of graduate students, who had an extremely hard time with all the epsilon-delta proofs, but for me it was mostly just a review of familiar material I'd learned out of Knopp. We were all, though, very puzzled by the question of just what all that stuff might be good for.

Matrix Algebra, which I took (along with two other Math courses) my second semester at Arizona, was taught by Evar Nering. He and Harvey Cohn had the reputation of being the top mathematical minds in the department, and the most demanding teachers. Nering anounced to us that he would not be using the text by Franz Hohn from the bookstore, but instead would be teaching us a subject called Linear Algebra from a dittoed set of notes he would hand out. I found Nering's talk about vector spaces and linear transformations to be quite fascinating, and I retained a fascination with linear algebra for years later. However I didn't actually learn the material very well, because for the first time in a mathematics course I was having a hard time forcing myself to actually read the book. Fortunately, Nering was so nervous about the fact that he was deviating from the prescribed curriculum and using this new abstract approach that he gave us essentially trivial tests, and I wound up with an A in the course. (The following summer, I did read through his notes and discovered the material to be for the most part fairly easy. I couldn't understand why I hadn't been more conscientious about reading the notes during the semester.)

By my senior year, I was aware that there was a sea change going on in the mathematics program at Arizona. Some new faculty had been hired who seemed to have a different, very modern attitude toward mathematics. The graduate program was being expanded and a number of new courses were added to the curriculum, including one which I was especially excited by: topology.

I had heard that topology could be considered to be the foundation of all mathematics, but I hadn't been able to learn very much about it. Some books described it as geometry done on rubber sheets, which seemed to me mildly interesting but hardly of fundamental importance. For the most part, though, I couldn't make heads nor tails out of the books on topology in the library.

I enrolled in topology the first semester of my senior year. Up till then, I had never taken notes in a mathematics course. I would just sit in class and follow what the professor was saying in the textbook, making occasional notes in the margin. But Louise Lim (a logician) came into class the first day, didn't tell us anything of what the subject was about but just started putting theorems and proofs on the board. I started leafing wildly through my book, trying to figure out where she was, but what she was doing didn't seem to be in the book at all, so I pulled out a notebook and frantically tried to catch up. She started talking about open sets and closed sets, which should have been familiar material from real analysis, but what she was doing didn't seem to relate to the concepts of openness and closedness I knew about at all. And what she was saying didn't seem to have anything to do with geometry, on rubber sheets or otherwise.