# Laurent Schwartz on Paul Lévy

Laurent Schwartz married Paul Lévy's daughter, Marie-Hélène Jeanne Lévy, in 1938. Marie-Hélène was also a professional mathematician. After the death of Paul Lévy in 1972, a memorial meeting was held on 23 March 1973 in the amphitheatre Henri Poincaré of the École Polytechnique. At this meeting, Laurent Schwartz gave a talk entitled La pensée mathématique de P Lévy. We give an English version of this talk below.

The mathematical thought of Paul Lévy, by Laurent Schwartz, Professor at the Ecole Polytechnique.

Paul Lévy himself has given us many aspects of his mathematical and non-mathematical thinking, from his youth until the end of his life, in his book Some aspects of the thought of a mathematician, published by Blanchard in 1970. (He had a mathematical training relatively early; his grandfather was a scientist, and his father, Lucien Lévy, was an examiner at Ecole Polytechnique; they were both Polytechnicians as well as Paul Lévy, his son, and one of his sons-in-law.) Paul Lévy wrote this book relatively quickly; one has the impression that he felt his approaching end. Some of the stories he tells from his youth about his understanding of mathematics and science are truly amazing. At the age of seven, he gave his arm to a little girl a little older than himself; he noticed that he had to raise his arm to succeed, he was surprised at saying to himself: "She is taller but my arm is longer, things will compensate each other; but they do not compensate each other since I have to raise my arm." He naively believed that the difference was constant then that it was the ratio which remains constant. He declares in his book that he said to himself, from that moment on: "No I understood, everything is enlarged in the same proportion"; and he says that from that age of seven he had for ever the notion and the concept of similar figures. I would be astonished if he really pronounced the word proportion, which seems premature to me for a child of seven, but it is certain that from that age he had the idea and the understanding. At the age of eleven, when he was told the problem of Achilles and the tortoise, he immediately found the error of reasoning and said simply: "Time does not stand still to allow the philosopher to enumerate all the terms of a convergent series." Many other stories of this kind are told, and show that his thought as a mathematician was, as is very frequent, formed, at least in embryonic form, in his youth and in his childhood. He also says that he was astonished one day that adults speak of the mystery of death: "It is not death which is a mystery," he said, "it is life," and he stopped henceforth thinking about it.

Paul Lévy's family and mine have known each other for a very long time. I knew Paul Lévy's daughter while we were in high school together; we got engaged at the École Normale Supérieure in rue d'Ulm, where she was the only young girl in the class (and she too became a mathematician). While I was in the first and second year of preparatory classes for entry to the Grandes Écoles, it happened once or twice that I went to her place, and met Paul Lévy. From that moment on he particularly struck me with his tall and thin stature, and also with his extraordinary intelligence. From the point of view of mathematical intelligence, I mainly had contacts with my high school teachers and, on the other hand, with my great-uncle, Jacques Hadamard. It was obviously very quickly apparent to me that Jacques Hadamard and Paul Lévy were of a mathematical level considerably above all the teachers I knew. Paul Lévy played a very big role in my training as a mathematician, and particularly during my second year at the École Normale Supérieure at a time when French mathematics was relatively lagging behind; my contact with him was particularly fruitful for my training as an analyst and, to a certain extent, as a probabilist. He then introduced me to the most intuitive methods in analysis, transforming analysis, which was then for me a succession of theorems relatively difficult to demonstrate, into something intuitive where one could think about and prove things on one's own.

He particularly liked what he called asymptotic calculations: by that he meant the calculation of an integral or the sum of a series, of a rather complicated combinatorial expression which, when the values of the numbers involved are finite, appears as inextricable, but of which one can, by clever reasoning, have an asymptotic evaluation when the parameters tend towards 0 or towards infinity. This is where mathematical intuition could particularly make itself felt, both in analysis and in probability, and it was for me a real revelation and a tool that I have kept all my life. His intuition in these areas strikes me as the most fantastic I have ever known. When given a mathematical expression with at the same time the origin of the physical or probabilistic or geometrical problem, he proved almost immediately what it became when the parameters became infinite; you could tell he was swimming there like a fish in water.

Much of his probabilistic work has followed the threads of this intuition that anyone who knew him immediately noticed. From that moment on, I noticed that he read particularly little, even less than I if I may say so: he found everything himself. When he needed a theorem, he would prove it again, often even not taking note of it, then forgetting the proof and finding it again. I remember once, when I asked him if he knew a simple proof of Lebesgue's density theorem, he gave me, after half an hour, a very nifty personal demonstration. About six months later, it was he who asked me if I had a good proof of Lebesgue's density theorem; I brought out his that I had carefully noted down, he replied: "Oh! that is really very clever, I would never have found it on my own!" All his life, he remained someone who read very little, largely ignoring the work of others, finding some work already proven and sometimes feeling a certain bitterness.

It was during this year 1935, when I saw him practically all the time, that he introduced me to the marvels of the beginning of the calculus of probabilities and to the marvels of these results which almost all of them he had himself proved. Instead of probabilities having a density, he systematically introduced the distribution function, which is no longer considered today and which is replaced by the notion of measure. He introduced the characteristic function by putting the exponential $e^{itx}$ in place of Poincaré's $e^{-tx}$, which required particular convergence conditions. He gave his inversion formula, which at the time was relatively very complicated. Thus showing the one-to-one correspondence between convolution of the laws of probability, which served for the addition of the independent random variables, and produces characteristic functions, between convergence of the laws of probability (which we now call close convergence and which was defined at this period in a very "convoluted" way) and uniform convergence of the characteristic functions on any compact space, he founded an essential tool which is still essential today.

He thus gave a new proof of the central limit theorem, which led him to study the fundamental role of Gauss's law. He studied the sums of independent random variables, and wondered in particular when series were almost surely convergent and when sums of independent random variables were approximately Gaussian. We find here two of his favourite themes, which he has developed enormously. In studying the sums of a large number of independent random variables and in studying the problem of the Gaussian nature of this sum, he found a theorem whose statement is in fact relatively complicated to specify, difficult to prove, but that completely solves the problem. When we have such a sum, the law of probability that it follows is approximately Gaussian if, and only if (in terms that I am deliberately simplifying and which are therefore partially wrong), the largest of these variables are Gaussian or have only an order of magnitude negligible compared to the total sum. This is what makes it possible to enunciate the so important role of Gauss's law for all physical measurements.

If we admit that the error of a physical measurement follows the law of the Gaussian curve, this is because this error is the sum of a large number of very small errors and that the largest of them is itself negligible in the face of the total error. As this circumstance occurs very frequently, one understands why a large part of the measurement errors are indeed Gaussian. The proof of this theorem required the proof of a conjecture that he omitted at that time: "The sum of two independent random variables can be Gaussian only if each of them is Gaussian." He would have been perfectly capable of demonstrating this conjecture, but it turns out that it was not demonstrated until some time later by Kramer, and Paul Lévy regretted all his life not having found it himself. He also studied, at that time, the indefinitely divisible laws, the stable laws, which are still called the stable laws of Gauss-Lévy, and which generalise the usual Gaussian law, and we can say that the general form of the central limit theorem which makes it possible to have the appearance of the sum of a large number of small independent random variables remained at that time a fundamental problem for him.

The study of series of independent random terms and their almost certain convergence is also a firework display of wonders; he first noted that these series are: either almost surely convergent, or else almost surely divergent. By dint of finding thus only zero or one probabilities, he finally enunciated a law that is still called the zero or one probability theorem, which is in constant use, and for which he proved the underlying reasons. All this was only the beginning of his work. I followed it less then, having turned more to analysis than to probabilities, only to take up probabilities in recent years; but precisely, in recent years, in my growing interest in probability, I have rediscovered the influence of Paul Lévy on the training of my youth.

He introduced into martingales and Markov processes some of the deepest ideas; he studied the curve of Brownian motion down to the smallest detail, proving for example, with Paley and Zygmund, that the trajectory is almost surely non-derivable. He also conceived the addition of independent random variables as leading directly to the study of the trajectories of functions with independent random increases (additive process). His work in probability is therefore immense; we can safely say that he was, if not the founder, at least one of the main founders of the calculus of probabilities.

We must be aware of the particularly difficult mathematical situation in which these theories developed. At that time, general topology and linear algebra were hardly developed. He himself timidly used compact spaces, introducing them cautiously as "the notion of a compact set". But it was above all on the topic of random variables that no precise axioms yet existed. I never really understood back then, from a mathematical point of view, what random variables were and he could never explain it to me. He explained it to me like a physicist; the random variables are independent if they correspond to random draws of which each one ignores the result of the others, this is a physical definition; fairly quickly we managed to draw a certain number of deductions which can be written very clearly in mathematical form, for example the fact that the probability of occurrence of two independent simultaneous events is the product of the probabilities of these events; we could also write that with the notion of conditional probabilities; but all this remained extremely vague and very difficult to understand in depth in terms of mathematics, as we see it today.

I do not believe that Paul Lévy could have stopped at a rigorous axiomatic approach to the calculus of probabilities. Everything still had to be done in this area; he practically should have spent years of his life there, and that obviously was not what interested him the most. He thought that the physical notion of independence or conditional dependence, or correlation, was enough for him; he was striding forward and demonstrating a host of theorems, most of which could not be expressed with the ideas of his time. In this sense he has been considerably ahead and it is only in the last one or two decades or even just in the last years that some of these theorems have become understandable in modern axiomatic language.

Paul Lévy was for 39 years a professor at the École Polytechnique, in charge of the analysis course. As Mrs Paul Lévy has often told me, he did not have as much to do as I have to do today at École Polytechnique! The professor came to teach his course, the role of small classes and of assistant lecturers for a very long time remained non-existent or little, and as a result he had a lot of free time. He used this constantly for his research. I have seen him work and research for several decades and now I know his method well. At certain times, there were no problems. At others, he was focused on a problem and worked for hours on end, all alone in his office, sometimes with very little correspondence with foreign mathematicians; when he had results worthy of publication, he wrote them by hand with his quill, in that calligraphic handwriting that many of his friends have known, almost even more readable than a current typed text.

As probability did not play a large role at the École Polytechnique and he was cut off from the University, he had very few students in France. He trained the young German mathematician Wolfgang Doeblin, but he died in the war. His real pupil was Michel Loève. This student, of Egyptian nationality, worked for several years before the war with Paul Lévy, was trained by him, even more than myself since he remained a probabilist, following very closely Lévy's work and his thought. But the French laws on naturalisation are, as we know, a considerable administrative mess; Loève was unable to obtain the naturalisation he requested (which is perhaps ultimately fortunate since he was an Egyptian Jew and the war came soon afterwards ...); Loève therefore went into exile in the United States, at the University of California, at Berkeley.

It was Loève who, in turn, was, with Jerzy Neyman, at the origin of the formation of this magnificent Californian school of probabilities, of this department of statistics of the University of California, which then made the scientific thought of Paul Lévy shine in the world of probabilities. It is also through him that other American probabilists, such as Joseph Doob, for example, learned of the work of Paul Lévy. Thus occurred this paradox that the work of Paul Lévy was underestimated, or not appreciated, or unknown in France, whereas it was an essential subject of the mathematical work of the Americans.

When Paul Lévy was invited to the first international colloquium on probability and statistics at the University of California, which was moreover one of his first and only major trips abroad, he was there with great modesty, wondering a little what he was going to do there, and was completely amazed to be greeted by Jerzey Neyman who said to him: "This conference will be that of the works of Paul Lévy."

Loève was, a number of years ago, invited as an associate professor at the University of Paris, he gave a course and trained the young student at the Ecole Normale Supérieure,Paul-André Meyer who, at that time, was looking for his direction and which very quickly became that of probability. It is through this intermediary that the probability of Paul Lévy returned to France. Probability, which was hardly accepted as a mathematical science at the time of Paul Lévy, and kept a little apart, is today in France one of the leading branches of mathematics.

To conclude, I could only point out the exceptionally gentle and modest character of Paul Lévy; I never saw him get angry at anyone, he always had the best relationship with everyone. When he had given his opinion, if the contrary opinion was expressed before him, he did not continue the conversation, possibly keeping his opinion to himself, and not attaching too much importance to it. He remained the same as a man and a mathematician until the end of his life. In his mathematical production he found, after 75 years, some very important theorems. From the point of view of his character, he remained practically the same until the end, until the disease carried him off in a few months.

He was treated by his wife; they had the happiness of living happily together for 59 years during which Mrs Paul Lévy has always worked to give him in his family an atmosphere of serenity perfectly favourable to his research.

Paul Lévy will surely remain one of the very great mathematical figures of the 20th century.

Last Updated September 2020