Benoit Mandelbrot on Paul Lévy

We give two quotes by Benoit Mandelbrot about his teacher Paul Lévy. The first is a short extract from Mandelbrot's book The Fractalist. Memoir of a Scientific Maverick (Pantheon Books, New York, 2012), 177-178. In this Mandelbrot refers to the talk he was asked to give on 23 March 1973 in the amphitheatre Henri Poincaré of the École Polytechnique. Our second piece is an English translation of Mandelbrot's talk 'Paul Lévy, professor'.

1. Paul Lévy, by Benoit Mandelbrot.

Our pure mathematics teachers Gaston Julia and Paul Lévy differed in innumerable ways. When I was their student, the Paris mathematical world respected neither, and these two men and Szolem [Mandelbrot] had no love for one another. This did not matter to me, and they all influenced me profoundly.
Nearing sixty, Lévy was still viewed as a brilliant oddball of the first magnitude, but was "molting" into a great man in probability theory, arguably the greatest probabalist of all time. But Lévy's way of doing probability theory was too intuitive for some and too strange for others. As a result he was a loner, never to be an insider. His self-directed boldness and insight cost him much in his career and early recognition, but I found his independence admirable. I felt ready to pay the same price.
Getting to know Paul Lévy was one of my few academic accomplishments in 1954-55. He never had a formal disciple, I never had a formal teacher, and I never thought of becoming his clone or shadow. Yet much of probability theory has long consisted of filling logical gaps in his works, and in a real, though indirect, fashion, he was the teacher of several members of his family, and also mine.

He documented his life, thoughts, and opinions at length in a book well worth reading because of his lack of any attempt to appear better or worse than he was. The best passages are splendid. In particular, he describes in touching terms both his fear of being a "mere survivor of the last century," and his feeling of being a mathematician "unlike all the others." This feeling was widely shared. I recall John von Neumann saying in 1954, "I think I understand how every other mathematician operates, but Lévy is like a visitor from a strange planet. His own private methods of arriving at the truth leave me ill at ease."

When Lévy died in 1971, I lobbied for a memorial at Polytechnique, but very few people came. However, the centennial in 1986 was a different story. By then Lévy's mistakes and idiosyncrasies were forgotten and forgiven, and a large meeting was organised by pure mathematicians. (A Polytechnique building came to be called Lévy.) Late in the process, I was invited, discreetly informed of strong opposition to my participation, and advised to avoid the shrillest opponents. Sadly, I wondered whether Lévy himself would have been invited and - if so - would have felt comfortable. I did not.

Lévy was the least flashy person on earth, so how to explain the profound influence his work and manner had on me and on many other scientists? Herein lies a familiar and always surprising story concerning the very nature of probability theory.

One half of the story is part of the mystery the great mathematical physicist Eugene Wigner called the "unreasonable effectiveness of mathematics in the natural sciences." A symmetric mystery should never be forgotten: the unreasonable effectiveness of the sciences in mathematics. Together these mysteries acknowledge that human thinking is unified within itself (and even with feeling), not in a trendy New Age fashion but very fundamentally.

Georg Cantor claimed that "the essence of mathematics lies in its freedom." But mathematicians do not pick problems from thin air for the pleasure of solving them. To the contrary, a mark of greatness resides in the ability to identify the most interesting problem in the framework of what is already known. And the highest level of the label "interesting" is invariably accompanied by a restrictive label, such as "in mathematics" or "in physics." My admiration for Lévy's "mathematical taste" increases each time his mark is revealed on yet another tool I need when tackling a problem in science that he could not conceivably have had in mind.

What a contrast with the period around 1960! Then Lévy stability was viewed as a specialised and uninteresting concept. It received at most a page in textbooks, with the exception of one by Boris Gnedenko and Andrei Kolmogorov. The English translation expresses the hope that Lévy stable limits "will also receive diverse applications in time ... in, say, the field of statistical physics." But no actual application was either described or referenced - until my work.

Lévy's mini-courses - I attended several - have marked my whole life. Not a charismatic lecturer; he looked frail and withdrawn. The auditors were few, and I recall (wrongly, I hope) having often been alone. I also watched Lévy closely at the weekly seminar on probability. One speaker began by describing a problem on the blackboard, then faced Lévy squarely and invited him to guess the answer. The guess was correct. But how reliably could Lévy proceed beyond guesses? A book by Kiyosi Ito and Henry P McKean is pointedly dedicated to Lévy, "whose work has been our spur and admiration." It includes this comment: "The difficult point of this proof is the jump between [two equations on that page]; although the meaning is clear, the complete justification escapes us."

2. Paul Lévy, professor, by Benoît Mandelbrot, Scientific advisor to the Director of Research, IBM.

I was asked to talk about Paul Lévy as a teacher, as I saw and heard him for the first time in this same amphitheatre, and also about the influence that his work as a probabilist had outside pure mathematics. Not only do I admire Lévy deeply, but I consider myself to be his disciple. More precisely, since Lévy had no disciple in the usual sense, I count myself among those whom he influenced in a particularly direct way. I would therefore like to say how the same man was able to have such an influence, not only on several mathematicians - who have already called themselves "gentrified probabilists" - but also in other circles.

Let's start with the course he delivered at the École Polytechnique. Chance having given me a place at the back of the amphitheatre, and Lévy's voice being rather weak and not amplified, this spoken lesson left me with a simply blurred image. The most vivid memory is that of a resemblance that a few of us were to see between his long, neat and grey silhouette, and the rather special way he had of tracing the symbol of integration on the board!

But the written course, that was unusual. Lévy, in his autobiography, says he has "the very clear feeling of being a mathematician like no other", and this was already visible in the sheets he gave us. First of all, they were extremely concise. Above all, they did not leave me the memory of the traditional, tidy ordering, starting with a host of definitions and lemmas, followed by theorems all the assumptions of which are clearly repeated, interrupted by a few results not demonstrated but clearly underlined as such. Rather, I remembered a tumultuous stream of remarks and observations. In his autobiography, which I will continue to quote, Lévy repeatedly describes himself as an explorer who - having reached the top of a mountain - describes the most salient features of a new landscape, without focusing on the obvious or the uninteresting. (This is how I saw him proceed, orally, in a few series of lectures, to a derisory audience, which he persisted around 1950 in offering to a recalcitrant Faculty of Sciences.)

Elsewhere, Lévy suggests that: "To get children interested in geometry, it would be necessary to arrive as quickly as possible at theorems that they are not tempted to consider as obvious." In his course at the École Polytechnique, intermediate between the junior high school and research, Lévy's method was not that different. In short, he avoided the description of mountain climbing, a process that others who are more keen on rigour, on the contrary, want to describe by following the most "direct" variant possible, which, alas! is rarely the easiest: they let the reader know exactly where he is at all times, but without telling him, where he is going, or why he has to go to so much trouble. What does it matter if fans of beautiful landscapes are not all mountaineering enthusiasts, and that many mountaineers are too tired on arrival to have retained the desire to look at the landscape!

Needless to say, Lévy's sheets were not universally popular. For many excellent students in their final year preparing for the Grandes Écoles, they were - pending the general examination - a source of concern. In the final revision, which I experienced in 1957, being its Senior Lecturer, all these traits were further accentuated; for example the presentation of the theory of integration was frankly approximate. "You don't do a good job trying to force students' talent," he wrote. It seemed that in his last lesson his talent had been forced; but of the course given to the class of 1944, I have kept an extraordinarily positive memory. If intuition cannot be taught, it is all too easy to thwart it. I think that is what Lévy was trying to avoid above all, and I think he succeeded.

While still at school, I had heard many allusions to Lévy's creative work. It was, people would say, very important, but they added that the most urgent thing was to make it rigorous. This has been done, and Lévy's intellectual grandchildren are delighted to be accepted as mathematicians in their own right. I fear that this acceptance was paid at a price "too dear". There seems to me to be, in any branch of knowledge, insufficient levels of precision and generality, unsuitable for tackling anything other than very simple problems. In almost every branch of knowledge there are also, increasingly, excessive levels of precision and generality. For example, one may need a hundred additional pages of preliminaries to be able to prove a single known theorem in a form whose only merit is to be a little more general. Finally, in some branches of knowledge there are levels of precision and generalities, say, "classic". The almost unique greatness of Paul Lévy is that in his field he was both the precursor and the classic.

Blow after blow, I found in the applications which occupied me - of which I will only allow myself to say that they have a certain variety - that what I needed is either Lévy's original idea, or a generalisation in his spirit. So, one of two things. Perhaps my education simply stopped after I got to know him, and I simply knew how to accompany what little I did know with particularly varied sauces. But I rather believe that the inner world, of which Lévy made himself the geographer, had with the world around us this precise degree of agreement which quite simply is the mark of genius.

Last Updated September 2020