Overture Up till about five years ago, I was a normal mathematician. I didn’t do risky and unorthodox things, like writing a book such as this. I had my “field”—partial differential equations—and I stayed in it, or at most wandered across its borders into an adjacent field. My serious thinking, my real intellectual life, used categories and evaluative modes that I had absorbed years before, in my training as a graduate student. Because I did not stray far from these modes and categories, I was only dimly conscious of them. They were part of the way I saw the world, not part of the world I was looking at. My advancement was dependent on my research and publication in my field. That is to say, there were important rewards for mastering the outlook and ways of thought shared by those whose training was similar to mine, the other workers in the field. Their judgment would decide the value of what I did. No one else would be qualified to do so and it is very doubtful that anyone else would have been interested in doing so. To liberate myself from this outlook—that is, to recognize it, to become aware that it was only one of many possible ways of looking at the world, to be able to put it on or off by choice, to compare it and evaluate it with other ways of looking at the world—none of this was required by my job or my career. On the contrary, such unorthodox and dubious adventures would have seemed at best a foolish waste of precious time—at worst, a disreputable dabbling with shady and suspect ventures such as psychology, sociology, or philosophy. The fact is, though, that I have come to a point where my wonderment and fascination with the meaning and purpose, if any, of this strange activity we call mathematics is equal to, sometimes even stronger than, my fascination with actually doing mathematics. I find mathematics an infinitely complex and mysterious world exploring it is an addiction from which I hope never to be cured. In this, I am a mathematician like all others. But in addition, I have developed a second half, an Other, who watches this mathematician with amazement, and is even more fascinated that such a strange creature and such a strange activity have come into the world, and persisted for thousands of years. I trace its beginnings to the day when I came at last to teach a course called Foundations of Mathematics. This is a course intended primarily for mathematics majors, at the upper division (junior or senior) level. My purpose in teaching this course, as in the others I had taught over the years, was to learn the material myself. At that time I knew that there was a history of controversy about the foundations. I knew that there had been three major “schools” the logicists associated with Bertrand Russell, the formalists led by David Hilbert, and the constructivist school of L. E. J. Brouwer. I had a general idea of the teaching of each of these three schools. But I had no idea which one I agreed with, if any, and I had only a vague 1 http://dx.doi.org/10.1090/mbk/083/01

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