restricted access Acknowledgments
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ACKNOWLEDGMENTS If the history of science is a secret history, then the history of mathematics is doubly secret, a secret within a secret. —George Sarton ANY BIOGRAPHY written over the course of fifteen years will have incurred many debts. It is with pleasure and gratitude that I devote the next few pages to thanking the many individuals and institutions whose resources, assistance, and support have proven invaluable to the reconstruction of the life and work of Abraham Robinson. The idea for this book began with a telephone call I received late in 1979. My first biography, a study of the nineteenth-century German mathematician Georg Cantor, was devoted to the origins of transfinite set theory, a revolutionary new conceptualization of the infinite. The book had just appeared, and I had given a lecture on the subject at the California Institute of Technology in Pasadena that fall. Shortly thereafter , W.A.J. Luxemburg called from Caltech to ask if I might be interested in writing a biography of the mathematical logician, Abraham Robinson. What I knew of Robinson at that point was limited to nonstandard analysis—Robinson's own extension of the real numbers which was as controversial and compelling in its own way as Cantor's introduction of transfinite numbers. Using the power of model theory and mathematical logic to define infinitesimals (as well as infinitely large elements), Robinson's work complemented what Cantor had achieved a century earlier. In Robinson's case, however, it was the rigorous treatment of infinitesimals that was especially ingenious. The infinite has always been regarded as troublesome in the history of mathematics, and infinitesimals especially so. Following their prominent introduction by Newton and Leibniz as key elements of the calculus in the seventeenth century, debates over their legitimacy (and apparently self-contradictory qualities) were often intense. By the nineteenth century Cauchy and, later, Weierstrass and their followers found an effective way to bar infinitesimals from mathematics (through the well-known method of deltas and epsilons), while retaining the basic idea of arbitrarily small quantities. In a similar but even more dogmatic spirit (despite his own success in creating transfinite set theory), Cantor condemned infinitesimals as "the cholera bacillus of mathematics." Thus xiv — Acknowledgments the fact that Abraham Robinson succeeded a century later in providing a rigorous foundation for infinitesimals, although others before him had largely failed, is all the more remarkable. What Luxemburg did not know when he first contacted me was that I was already familiar with the details of nonstandard analysis, specifically in the form Luxemburg himself had made popular and widely accessible to mathematicians. As an undergraduate mathematics major at Claremont-McKenna College (one of the Claremont Colleges in California ), I had written a senior honors thesis on nonstandard analysis in 1966, taking Luxemburg's approach via ultrafilters to the subject, rather than Robinson's own path to discovery which had relied specifically on logic and model theory. With respect to my earlier work on Cantorian set theory, Luxemburg emphasized how, in a very direct way, a biography of Robinson would be a natural sequel. Since Luxemburg was planning to visit Yale University that spring (1980), he suggested we meet there. He arranged for us to have lunch with Renee Robinson, and for me to meet members of the Yale Mathematics Department who had been Abby's colleagues before his untimely death in 1974.1 wanted to be certain that Mrs. Robinson and the Department at Yale were amenable to the idea of my writing Robinson's biography, for their cooperation was clearly essential. It was decided that we would meet for lunch at the Park Plaza Hotel; Mrs. Robinson had just returned from a trip to Morocco and much of our conversation was devoted to the many trips she and Abby Robinson had taken in the course of their life together. They had been inveterate travelers, and in this we had an immediate common interest, as it turned out we did in archaeology as well. Agreed that I could rely upon Mrs. Robinson's cooperation in writing the biographical portions of Robinson's life, I was subsequently assured of the support of the Yale Department of Mathematics. This much settled, I arranged to spend a week working in the Yale University Archives, where I made a preliminary survey of what was available for study in seventeen boxes of Robinson's collected writings, correspondence, papers and various miscellanea. Two years later, the Department of Mathematics invited me to give the Harvard...