The originality of P Lévy in mathematics

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, Paul-André Meyer gave a talk entitled L'originalité de P Lévy en mathématiques. We give an English version of this talk below.

The originality of P Lévy in mathematics, by Paul-André Meyer, Research director at the Centre national de la recherche scientifique.

Since the beginning of this century, mathematics has evolved towards an increasingly inflexible formal rigour. It was also at the beginning of this century that the modern calculus of probabilities was born, Emile Borel undoubtedly marking, in this respect, the link between two eras. All the big names of the new science: Wiener, Khinchin, Kolmogorov, Feller, Doob ..., are those of mathematicians who made it evolve, through painful controversies on the philosophical or axiomatic foundations of probabilities, towards the status of a "noble" discipline, as rigorous as the main traditional branches of mathematics.

In the midst of this ascent towards respectability, a bit like the birth of a painter into a family of bankers, Paul Lévy constitutes a unique and almost scandalous exception. He belonged to a French school formed, before the war of 1914-1918, by men who, by a certain mathematical aesthetic, excluded from all "excess of abstraction", who all refused to accept (Hadamard constituting a notable exception) the new forms of set theory; the French school, moreover, which was weakened by the war, and not to be reborn vigorously until much later, in a form on the contrary very inclined to abstraction. Everything seemed to lead Lévy to reject post-war mathematics, and to become a conservative mathematics teacher as we have seen so many times.

Now it is true that he was never interested in axiomatics: he seems to have formed very early on his own "system of probabilities", in which he could work conveniently, and never to have dealt with the foundations of probabilities. But he had such an extraordinary probabilistic intuition that some of his results are ahead, not only of the methods which were to allow them to be fully demonstrated, but even of the language necessary to state them with precision. I am thinking in particular of his work on the zeros of Brownian motion and his idea of the independence of the intervals between the zeros, which was only clarified last year by Ito. To take another example, there is his book of 1937 where he demonstrates the Khinchin-Lévy formula by counting the jumps of the processes with independent increments, while everyone was still working "in law", and there are still many more. A whole generation has worked to rigorously justify the results seen by Lévy, and there are undoubtedly some discoveries still to be made in his work. It is even more surprising, if one thinks that after all geometric or analytical intuition had had centuries to form, while no one had ever before encountered the beings that Lévy so described.

However, Lévy should not be reduced to an uncontrolled intuition: there are many magnificent, perfectly rigorous demonstrations from him. Even when he did not succeed in absolute rigour, he knew very well how to make himself understood. The best proof of this is the unanimous admiration that so many probabilists have shown him: Chung, Doob, Feller, Ito, McKean ...

There is, there too, something a little paradoxical, the rule being that revolutionaries are misunderstood. But Paul Lévy was not only a revolutionary, he was also a professor at the Ecole Polytechnique and, as such, he found himself sheltered from many quarrels and he had facilities to publish his work. Now imagine a mathematician like Paul Lévy, who would write as mathematicians talk to each other when there are no students in the room, and would launch a host of new ideas: I think that his work would be refused by most major mathematical journals. It's a bit worrying for our future.

Despite his title of professor, despite his election to the Institute (which he had very much desired, which is proof of his touching modesty), Paul Lévy was overlooked in France. His work was viewed with condescension, and it was frequently heard that "he was not a mathematician." He had very few students: since 1942, I have only seen one with him: Michel Loève. Yet he was extraordinarily welcoming, very easy to approach. I remember that one of my fellow students gave a seminar presentation, which Lévy came to attend. He was surprised to receive a letter from Lévy with compliments and a number of comments and suggestions of a mathematical nature. It is largely through Loève that Paul Lévy happens to have, in France itself, a direct scientific posterity. The hostility of the French probabilists of the time pushed Loève to settle in the United States immediately after the war, and he wrote there one of the first major treatises on probability. He returned to France to persuade French mathematicians of the importance of Lévy's scientific work in the last years of his life. Loève also welcomed several French probabilistic students to California, with great generosity. There is a kindness in Loève, a patience with regard to beginners which seem to me to be the reflection of these same qualities in Paul Lévy, in whom he had found them himself.

Paul Lévy was, despite his genius, universally loved among probabilists. I've never heard of any nastiness about him, and that does not mean probabilists are any less fierce than other mathematicians.

Last Updated September 2020