I appreciate very much the opportunity to participate in this conference honoring
Donald J. Newman, and to say a few words about him and his influence on me.
I have always regarded Donald as my teacher. Even if my thesis adviser officially was Norman Levinson, to whom I was attracted by his book "Gap and Density Theorems", I never got a problem, nor any help from him. When I told him I was reading that book he simply replied "Oh, don't read that." Levinson was then no longer interested in harmonic analysis, but in ordinary differential equations, which (at that time) I had no interest in. But, I did get a problem - from Donald! It was a typical Newman problem: an inequality to prove something was bounded by Sqrt[n], when the conventional wisdom only sufficed to give the bound Sqrt[n+1]. I became fascinated with this problem, which was the leitmotif for my Master's dissertation in 1951 (with the so-called Rudin-Shapiro polynomials as spinoff) and also stimulated my interest in function theoretic extremal problems, which was the subject of my doctoral dissertation and has remained a lifelong interest. So, to all intents and purposes Donald functioned as my thesis adviser.
But, he was my teacher in a deeper sense than that! To explain what I mean, I would like to quote a story I enjoyed from Martin Buber's book "Tales of the Hasidim". Rabbi Leib, son of Sarah (he was a famous mystic) attached himself for some time to the house of study of a renowned holy man (Dov Baer, the Maggid of Mezritch, also known as the Great Maggid). "Ah", said a friend later, "you went to hear Torah from the Great Maggid". "No", replied Rabbi Leib, "I did not go to the Maggid to hear Torah, but only to watch how he laced and unlaced his felt boots."
The meaning of the story (and there are many like it in the lore of mysticism) is, of course, that the Maggid did not have to say Torah ... he was Torah! Also, that the most important things one learnes from a teacher are not this or that fact, but are more in the realm of the intangible: attitudes, a certain style. One of my students expressed this very well in the phrase "the body language of mahematicians". The beginner becomes wiser as he ponders his teacher's remark "In proving this inequality there is no loss of generality in assuming the right hand side equals 1". Later in life, he cannot remember when he learned this idea.
Donald's "house of study" was, for me, the lunchroom at City College in the years 1947-48. I remember the scene well: he would be seated at the table reading Landau's "Vorlesungen über Zahlentheorie". He didn't know much German, but could understand Landau's crisp prose, carried along by the formulas. From time to time some "disciple" would come by and say something like "I see how to get that estimate", and Donald would reply something like "OK, now try to get it without the log term. To change the metaphor, he was like a chessplayer playing twenty games simultaneously. He knew perfectly the position on each board, and could tell you all the previous moves. And I was taking all this in, understanding almost nothing, and yet learning. He didn't have to speak mathematics to me - he was mathematics.
So, what did I learn as I watched him opening and closing his copy of Landau? I suppose the most important thing was the obvious joy he derived from all things mathematical. The elation at seeing a new theorem (whether one's own or another's, it didn't matter) or an elegant solution to a problem struck a responsive chord in me. This was far removed from the dispassionate "professionalism" I was later to encounter in graduate school. I came from a background (on my father's side) where Torah was studied in this spirit, but my disposition was not suited to this kind of scholarship. Mathematics was a new world, without bounds...it was already clear that it was so rich that the smallest of its problems could engage one for a lifetime. Mathematics was an art form. To Donald mathematics and music were inseparable, and through him I learned to love classical music.
The fundamental unit of mathematics, and the measure of all things, was the problem. Most conversations began with "Here's a problem". A problem could be something whose answer was unknown to the poser, or an old chestnut making the rounds; or, a tantalizing special case of a general theorem, it didn't matter. As long as it was a good problem, which meant: it was easy to state, and had some "point", like possessing an elegant or surprising solution, or an air of paradox. Most of what I have learned in mathematics to this day is by way of problems in this sense.
I would like to mention two very specific features of Donald as a mathematician. One was what I would call his democratic attitude. No one was too humble to be taken seriously (and especially, if he came with a good problem, or a good solution). Pedigree didn't matter, if Donald had any bias at all it was against "establishment" representatives. This also applied to the mathematics itself. To illustrate how precious this was to me, I can contrast it with what was said to me by an established mathematician who came to give a Colloquium talk at M.I.T. when I was a graduate student. In the Common Room, at coffee before his talk, he asked me what I was working on. Flattered that such a person should take note of my existence, I told him my thesis problem. "Oh", he said, "I had no idea anyone is still interested in that sort of thing. That's 1870 mathematics." Those were tough words, but I had my "network" and could withstand them. To Donald 1870 mathematics wasn't bad, although 1770 (Eulerian) mathematics was better.
A second characteristic of Donald was his concrete approach to mathematics. It was (and still is) fashionable amond the graduate students to generalize everything. So if, say, you had found a cute theorem on polygons and showed it to someone, that person would say something like "Can you generalize this to polyhedra in R^n? I guess what you really are looking for is a theorem on convex sets in topological vector spaces, etc., etc.". Donald would take note of the theorem and say "What does this say in case the polygon is a triangle? What if it's an equilateral triangle?" This, I learned from him and I feel it has stood me in good stead.
Well, those are some reminiscences of Donald in the years 1947-52. I restricted myself to that period, because afterwards we met infrequently, and also I don't know if I could do justice to his more recent ideas, like probabilistic truth. But thanks, Donald, for all you gave me in those formative years!
One can distinguish two types of scholar which I may
call: Guardian of the conventional wisdom, and Dreamer of the impossible dream.
I would say roughly that I am the former type and Ambjörn
the latter. I am paid to teach the basics, like integration theory according to
Rudin's book, and do that as best I can.
The other type is much more interesting. These individuals turn up
throughout history, often obsessed with a strange idea
such as turning base metals into gold, constructing machines that fly, devices that
enable us to see distant stars, etc.
Actually, what is the conventional wisdom, other than the impossible dreams of 300 years ago? Who could have foretold 300 years ago that we would be sitting in a room tonight illuminated by the force of water falling in the north of Sweden, that some people arrived here in flying machines and others in machines rolling in tunnels blasted out of granite by a chemical mixture one can hold in one's hand?
I remember my first meetings with Ambjörn, it was culture shock to learn his interest in magic, alchemy and like things. Now I don't regard these things as so strange. The alchemists didn't make gold from base metals but ultimately they (and their followers) achieved something far more amazing...modern chemistry. And even some apparent magic is explainable by long chains of logical causal steps.
Ambjörn long ago developed an interest bordering on obsession for ray congruences and focal figures. Strangely, my own mathematical interests have been focussing there despite a totally different starting point. His dissertation is partly borne along on the thought that this branch of geometry, perhaps just because of its sheer magnificence, has an important role to play in the computational aspects of surfaces, perception, etc. How this is to happen is as yet not obvious, but one could, not long ago, say the same of rooms illuminated by the force of falling water, refrigerators driven by the sun's rays, etc. His faith is great, and it is hard not to share his enthusiasm. Maybe his "worst fears" will be realized and his ideas will succeed so well that they are transformed, in his own time, into conventional wisdom. Then we may see the spectacle of Sing Sing packed to capacity with teknologer taking their exams on focal surfaces. Qui vivra verra!
I propose a toast, not to Ambjörn alone, but to all those past, present and future dreamers of the impossible dream. And for personal reasons I would include not only those who succeeded, but the others...those whose dream really was impossible, those that fell by the wayside, and those whose names we don't know because their good ideas were stolen or ursurped by those richer, more powerful or more unscrupulous.
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