Mandelbrot: Foreword to Abraham Robinson by Dauben

J W Dauben's book Abraham Robinson : The creation of nonstandard analysis. A personal and mathematical odyssey (Princeton University Press, Princeton, NJ, 1995) contains a Foreword written by Benoit B Mandelbrot. This Foreword tells us something of Robinson, but also something about Mandelbrot himself and his uncle Mandelbrojt.

Abraham Robinson's life was extraordinary in many ways, if only because his professional work spanned three significant fields: airplane design, symbolic logic, and mathematical analysis. His untimely death was a blow to many people worldwide, and I was one of them; in fact, Robinson and I had hoped to work together on a presentation of fractals in terms of nonstandard analysis (unfortunately, he was on sabbatical when I spent a term at Yale in 1970). This direct personal link, in addition to both my friendship with Renée Robinson and being the Robinson Professor at Yale, made me treasure the invitation to contribute this Foreword. But I did not accept until I had had the opportunity to read the book.

I was fascinated, and I hasten to congratulate Joseph Dauben for having produced a masterful and beautiful work. Very few mathematicians' stories justify a full biography or autobiography. In fact, most mathematicians would not want one, and I respect their feelings even when I disagree. A good example was my Uncle Szolem, who was delighted and proud that (excluding a few scary episodes that produced amusing anecdotes) his life ran according to a well-worn pattern and was not interesting. Thus, after I had taped hours of conversation with him, and eventually reduced this mass to an attractive text, he did not like it and was relieved when I offered not to publish it in his lifetime. To some extent his reaction was justified, because the "ghosted" autobiography that I had produced focused solely-as did the recent autobiography of one of my uncle's old friends - on superficial aspects of his life. I added glimpses of the daily life of Polish and French mathematicians in the 1920s and the 1930s but said nothing of the topics my uncle had worked on or of the ways his choice of topics had conformed to or contradicted his environment. The resulting contribution to history was amateurish and meagre, but it has helped me to appreciate the magnitude of the task that Joseph Dauben chose to face, and the quality of his achievement.

Of course, good history had best begin with a good story, and the case of Abraham Robinson, truly singular for a mathematician, allows Joseph Dauben to alternate two narratives. The first is the story of a life, with all its richness, its many picaresque and incongruous episodes. The second recounts the progression of Robinson's works, from airplane design to mathematical logic to nonstandard mathematical analysis.

What has emerged is a weighty book that covers many topics in great detail. Some readers will find themselves skimming over some details, but different readers will skim different parts. And I feel that in this instance there is great value in thoroughness. Let me elaborate. Many people in academia feel (as I do) that we have reached, not the end of a cycle, but the end of an era; to put it mildly, academic policy making will no longer be overwhelmed by headlong growth. If this is true, there is an urgent need, today, for documentation of the recent past. For the future historian, Robinson's tale has a marvellous virtue. As we follow in near-chronological order the life of a single individual of amazing brilliance, stamina, and versatility, we are guided back and forth without artificiality through at least three widely disparate academic cultures (pure and applied mathematics and logic-philosophy); six countries, representing six distinct flavours of Western culture; and, within the United States, two very different institutions. Indeed, Dauben depicts Robinson's long-term moves and his short-term moves during summers and sabbaticals against a rich background of general history, and he avails himself of this scope to discuss his subject's personal motivations carefully as well as delicately.

More specifically, in order to understand human creativity, I think we must investigate in detail how major creators have balanced the necessary but conflicting needs of deep-rootedness and personal boldness, how they have assessed the relative importance of their own private drumbeat, the drumbeat of the family and the professional community, and the drumbeat of society at large. For a reader craving variety in the backgrounds against which this question can be raised, Robinson's life has been a near-unique case. Thus, while it is not universally conceded that "God lives in the details," we should all be grateful to Joseph Dauben for having given future historians so much detail to think about. This tome's very size may insure its durability.

Throughout this book, Robinson is called a mathematician. I fully agree with this characterization, but I also hear the voice of the devil's advocate who would turn the same evidence around, interpret Abby's achievements backward, and assert that a person who spent much of his life outside of mathematics departments, working on airplane design and symbolic logic, was ipso facto not really a mathematician. Therefore, this book necessarily poses a larger question: What is mathematics? Opinions range the spectrum from a wide, liberal Open Mathematics to a small Fortress Mathematics. For proponents of the former school, which I favour and which I am sure Abraham Robinson also favoured, mathematics is a big rambling building permanently under construction, with many doors and many windows revealing beautiful and varied landscapes. For proponents of the latter, the highest ambition is to wall off the windows and preserve only one door. Fortress Mathematics is intolerant of individuals like Robinson, who, even in his specifically mathematical work, followed not the drumbeat of others but his own.

Incidentally, the influence of Leibniz on Robinson is well documented in this book. Today, other "hard" scientists (I am one of them) turn to Leibniz, but Robinson has preceded us, and I wonder how explicitly and to what extent he should be viewed as a forerunner on this account, as he is on so many others.

Once again, this book taught me a great deal and I recommend it heartily. My deep regret at not having known Robinson better has been to some extent mitigated, and the reasons for the existence of a Yale chair bearing his name have been clearly illuminated.

Benoit B Mandelbrot
Abraham Robinson Professor of Mathematical Sciences at Yale University
IBM Fellow Emeritus

JOC/EFR March 2006

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