I was born in Topeka, Kansas, on August 19, 1945, a couple of weeks after the bomb fell on Hiroshima. I grew up in Topeka, Dodge City, and Wichita, and graduated from Topeka High School in 1963. I spent four years at Caltech, where I studied both mathematics and physics, graduating in 1967. In my senior year I was exposed to mathematical logic, and decided that would be my subject. I remember Richard Feynman, from whom I took a lecture course on quantum electrodynamics, but who also would eat in the student cafeteria with us many a Friday afternoon, and who gave us advice on all subjects, for example: "Listen, son. You know those jokes in magazines about married couples? Those aren't jokes! They're all true."
I then went to graduate school at Stanford to study logic under Dana Scott. When I arrived, I went to see Dana and asked to do an independent study with him. "Fine," he said, "go work all the problems in Shoenfield's Mathematical Logic". They filled four notebooks. (However, after my first year there, Dana departed for Oxford, and I eventually got my Ph. D. under Friedman and Kreisel as joint advisors, with a lot of unofficial advice from Sol Feferman.) Stanford was a very stimulating environment. There I learned mathematical logic from Dana Scott, Sol Feferman, Georg Kreisel, and Harvey Friedman, artifical intelligence from John McCarthy, minimal surfaces from Bob Osserman and Bob Finn, number theory from Paul Cohen, and differential equations from Professors Schiffer and Gilbarg. I acquired a golden retriever, whom I named Socrates, because I was taking a course in Greek philosophy at the time. Socrates' mother used to ride a motorcycle to work with her graduate-student master. Eventually I wrote a thesis on the Metamathematics of constructive theories of effective operations, which marked the beginning of my research in logic.
In 1971 I passed my Ph. D. oral exams, at which Paul Cohen asked me to state the ergodic theorem and sketch the proof that a monotone function is continuous almost everywhere. I then taught at the University of California at Santa Cruz for two years, during which I worked with Michael Kahn, Bob Edgar, and Matt Sands to found Kresge College (UCSC is organized into Colleges and Boards of Studies.) During this period I wrote two papers containing my dissertation results, and a third on the same subject, solving a related problem left open in my thesis.
In 1974 I moved to Austin, Texas, to spend a year at the University of Texas. During this year I learned about Feferman's then-new formal systems for explicit mathematics and began to do new research in logic. In 1975-76 I came to Stanford for a year as visiting assistant professor. In the spring of 1976 I studied the sitar at the Ali Akbar Khan college, and in the fall I moved to Amsterdam for a year at the University of Amsterdam. I had never been east of the Mississippi before. In Amsterdam I had a fourth-floor apartment on the Beethovenstraat, within walking distance of the world-renowned Concertgebouw, where classical concerts take place almost every night. I studied sitar in Amsterdam with an Indian concert musician, Bhartiya. Every morning at 8 am I had to practice an hour in his living room, while he still slept. If I played out of tune, he would wake up and yell at me, so my ambition in those days was to play well enough that he would stay asleep until nine.
In the summer of 1977 I spent a month in Germany doing research on minimal surfaces. Bonn is a center of mathematics in the summer, and a very pleasant one, with boats on the Rhine, outdoor cafes, and conferences. I met my future wife on the train in Italy, but we didn't meet again for a couple of years. I then traveled to India, where I visited two gurus, gave a mathematics lecture at Kurukshetra University, and stayed with the director of the Bombay University music library, who had promised to give me access to tapes you could not hear in the West. I contracted dysentery to complete my Indian experience. I spent the fall of 1977 in Berkeley, California, in the home of my friend Marvin Jay Greenberg, whose hospitality I will always remember.
Spring of 1978 found me in Germany working on minimal surfaces at the University of Bonn. I stayed there fifteen months and then took a position at the University of Utrecht in the Netherlands. By this time I had connected with my future wife, and since she lived in Holland, the offer from Utrecht was attractive. She had been (and is) an elementary school teacher, but she had gone back to the University for a psychology degree. I managed to stay in the Netherlands, obtaining one temporary position after another, until she graduated. I also spent a month in Turin, Italy, at the University of Turin, which was delightful for its old books. I loved Italy, with its enthusiastic concert audiences, elaborate meals, late hours, siestas, and thick hot chocolate. When I left there was a large sending-off party in a restaurant and if my train hadn't been about to leave, they would have taken me down by the river to teach me Italian songs.
On July 22, 1980, I married Henny Nijland, who has been my faithful companion ever since, and in May 1981, our daughter Amanda was born. When she was two months old, we flew to the United States. She napped on the airline dinner tray. We landed in Los Angeles, bought a car from my aunt, and drove to San Jose with all our belongings in or hanging on the outside of that Toyota. In August, 1981, I began teaching computer science at San Jose State University, without knowing where to find the on switch of a personal computer (IBM introduced the PC in that year.) We settled in Felton, where I struggled to finish writing my book, Foundations of Constructive Mathematics , which I had tried unsuccessfully to finish before leaving Holland. Now that I had a baby and more than a full-time job (it was necessary to teach part-time at UCSC to make ends meet on my meager salary), it took a lot of 2 a.m. nights to finish that book, and it was 1985 before it was finally published.
In 1984, we purchased our present home near Santa Cruz. One week after we moved in, our second daughter Helen was born. She lived in that same house all her life until she went to college. Living in the same place for over twenty years was a pleasant change for me, since I lived in at least 35 different places since graduating from college. Since that time I have quietly taught classes, pursued my research in theoretical computer science and in automated deduction, developed my software MathXpert, played the piano, and watched my children grow.
MathXpert turned into a 165,000 line C program, and consumed about seven man-years over the twelve-year period 1985-1997. MathXpert was published in 1997, and after that I turned to automated deduction. I have always been interested in the connections between logic and computation--MathXpert approached that question from the computational side, doing logically correct computations. I wanted to approach it from the logical side, writing programs that could find proofs using computations. In 2002 the NSF agreed to fund my research, and for four years I worked on a new algorithm for "lambda-unification", the related theory of Lambda Logic, and an implementation of the algorithm in the existing theorem-prover Otter. You can read about that project at the Otter-lambda home page.
In 2000 I was on leave from teaching for the entire calendar year, and during that time I wrote several papers. In the summer of 2001 I attended a conference in Italy and a three-week conference on minimal surfaces at the Mathematical Sciences Research Institute in Berkeley, California, and wrote a paper on minimal surfaces, completing a problem I had been unable to solve completely twenty years before. I've been checking this paper, off and on, and filling in more details, for four years now. The proof is 67 pages long. It's posted on my website if you'd like to help me check it. I think it's right and I'm about to submit it for publication.
In 2004 I re-acquired the rights to MathXpert, which had been owned by others who paid me a royalty. Now it's sold through my website at www.HelpWithMath.com.
My wife, Henny, taught in the Santa Cruz Montessori School while our children were students there, but since then she has been teaching in the public elementary schools. She has taught fourth and third grades, and then served as the "mathematics facilitator" for a year, then the "language arts facilitator", and in 2006-2007 she is the "science and art facilitator". These positions means that she has a classroom of her own, but not a class of her own. She teaches science and art to students from all the classes, and assists the other teachers in preparation and assessment. She enjoys playing her flute and guitar (not at the same time, of course), keeps our flower and vegetable gardens thriving, reads books, and watches a lot of nature videos.
My daughter Amanda entered MIT as a freshman in the fall of 1999. We visited her in Stuttgart, Germany, in the summer of 2001, where she had a summer job at the fluid dynamics lab of Bosch. She graduated from MIT (with a B.S. in Mathematics with Computer Science) in June 2003, with her proud parents in attendance. In fall 2003 she moved to La Jolla and became a graduate student in mathematics at the University of California, San Diego. Her interest is in number theory; she is now about to write a thesis under Dr. Harold Stark. In her spare time she surfs and takes photographs.
My daughter Helen graduated from high school in May, 2002. She is in the class of 2007 at the University of California, Berkeley, where she is majoring in biology. She spent summer 2004 working in a biology lab at Berkeley and worked there part-time during fall semester 2004. She spent spring semester 2005 studying Spanish at the University of Guanajuato, Mexico. She returned to UC Berkeley in the fall, where she worked in an entomology lab. She spent the summer of 2006 at the Rocky Mountain Biological Lab, 10000 feet high in the Rocky Mountains, and in the fall of 2006 embarked on her senior year as a UC biology student. In her spare time she paints pictures, takes photographs, throws pots, surfs, goes camping, and jumps off of tall cliffs into lakes, rivers, and oceans.