Paul Pierre Lévy

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15 September 1886
Paris, France
15 December 1971
Paris, France

Paul Lévy was a French mathematician who, after starting out as an expert on functional analysis, made important advances in probability theory.


Paul Lévy was born into a family containing several mathematicians. He was the son of Lucien Lévy (1853-1912) and Alice Flora Wolff (1856-1921). Lucien and Alice were married on 3 June 1878. Lucien Lévy had studied at the École Polytechnique from 1872 to 1877 and then had a career as a professor of mathematics, first at the lycée of Rennes and then in Paris, at the lycée Louis le Grand. As well as teaching in lycées, during most of his career he was an examiner at the École Polytechnique. He wrote several books: Arithmétique à l'usage de l'enseignement secondaire spécial  (1880), Éléments d'arithmétique (1883), Précis élémentaire de la théorie des fonctions elliptiques avec tables numériques et applications (1898), (with Eugène Rouché) Analyse infinitésimale à l'usage des ingénieurs Vol 1 (1900), (with Eugène Rouché) Analyse infinitésimale à l'usage des ingénieurs Vol 2 (1902). He was president of the French Mathematical Society in 1911. Lucien Lévy was the son of Edouard Lévy (1822-1900) and Hortense Breinlé Cahen (1832-1909). Edouard Lévy, Paul's paternal grandfather, had graduated from the École Normale Supérieure in 1843 and also had a career as a professor of mathematics. Paul's mother, Alice Flora Wolff, was the daughter of Samuel Wolff (1827-1913), Inspector General of the Ponts et Chaussées, and Fanny Goudchaux (1835-1888).

Paul attended three lycées in Paris, the lycée Montaigne from 1895 to 1898, the lycée Louis le Grand from 1898 to 1902 and the lycée Saint Louis from 1902 to 1904. At the lycée Montaigne, Lévy was taught literature and arts subjects and was certainly not the best student in his class, having three brilliant fellow-students, Jacques Massigli, Pierre Lachièze and Marc Bloch, who took the top three places. In early 1898 his teacher M de la Filolie told him he had won the Greek prize in the Concours général. Lévy wrote [2]:-
It was to be my only literary success.
Émile Mâle was the professor of rhetoric at the lycée Louis le Grand when Lévy began studying there later in 1898. His achievement in winning the Greek prize, however [2]:-
... was not enough to draw the attention of Émile Mâle to me, who, from a great height, let fall on the best in the class a small part of his knowledge of Latin and of archaeological science.
It was mathematics that fired Lévy's passion at the lycée Louis le Grand. He wrote [2]:-
It is especially in plane geometry that I excelled. I visualise poorly in space, and I do not have the memory to have been impassioned by algebra, which did not prevent me ... to be also, in this discipline, to be at the top of my class. I was passionate about plane geometry; certain theorems were revelations for me: the notion of the power of a point in relation to a circle, that of a radical axis, that of the transformation of a figure by inversion, delighted me. What admirable methods for finding new theorems! I always found the problems easy that were presented to me. I was hoping to have the mathematics prize at the Concours Général en rhetorique; I did have it, and have not had the opportunity to participate in this competition since. Yet I make a huge difference between what I was when I was in literary studies and what I was going to be the following year.
To prepare for entry to the Grands Écoles, Lévy's father moved him from the lycée Louis le Grand to the lycée Saint Louis for his two final years of schooling. He explained [2];-
On leaving rhetoric, after having learned the elements of trigonometry during the summer, I entered the Lycée Saint-Louis, in a class of elementary mathematics called higher .... My father thought that to prepare for the École Polytechnique, and possibly École Normale Supérieure, one should not waste time. I could prepare well for the baccalaureate in philosophy while doing mathematics in the lycée. In addition, as far as mathematics was concerned, I would have wasted my time if I had not skipped the preparatory class for the baccalaureate. But the upper class was of a higher standard in Louis Le Grand than in Saint Louis, and, in order not to require excessive effort from me, my father put me in Saint Louis. I have sometimes wondered if it would not have been better if I had stayed at Louis Le Grand. I'm sure I would have easily taken the mathematics class.
In his final year at the lycée Saint Louis, 1903-04, Lévy was taught by Émile Blutel (1862-1932). Blutel was a remarkable teacher who went on to publish a number of outstanding textbooks: Leçons de Mathématiques spéciales à l'École Normale Supérieure et des Étudiants des Facultés des Sciences (1914); Leçons de mathématiques spéciales à l'usage des candidats à l'École polytechnique: Géométrie analytique, courbes et surfaces (1914); and Leçons de mathématiques spéciales, candidats à l'École polytechnique et à l'École normale supérieure: Algébre, ligne droite et plan, trigonométrie, analyse, applications géométriques (1914).

Lévy was placed first for entry to the École Normale Supérieur and second for entry to the École Polytechnique in the Concours d'entrée for the two institutions. He chose to attend the École Polytechnique, considered less prestigious at that time, probably because of his father. While still an undergraduate there, he published his first paper Sur les séries semi-convergentes in 1905 and his second Sur la densité des nombres premiers inférieurs à une grandeur donnée in the following year. Writing about his professors, he singled out Georges Humbert [2]:-
The most remarkable of our professors was certainly Georges Humbert, and the course of analysis he gave us was the one from which I benefited the most. His presentations were always remarkably clear. My fellow students even claimed that it was too clear, and that he avoided difficulties that we noticed when studying the course on the sheets that were distributed to us. I only saw once, about the fundamental Jacobian theorem, that this was true. Among our other professors, I will name only Henri Poincaré and Henri Becquerel. The first was, in theory, a professor of astronomy; but the course was given by a substitute, except for the beginning. Poincaré himself taught a lesson on spherical trigonometry and two lessons on error theory.
After graduating in first place, Lévy took a year doing military service before studying for an engineering degree at the École des Mines in 1907. While he studied at the École des Mines he also attended courses at the Sorbonne given by Gaston Darboux and Émile Picard. In addition he attended lectures at the Collège de France by Georges Humbert and Jacques Hadamard. Lévy wrote [31]:-
At the Ecole des Mines, I neglected many of my courses in order to attend mathematics courses at the Sorbonne and at the College of France. I attended Humbert's course for three years at the College of France. He is known for having been a marvellous calculator, and he contributed to the progress of classical mathematics. I was especially inclined towards the study of modern mathematics, and it was not Humbert's course that I profited from the most. It was also not Borel's, which I gave up on after 4 or 5 lectures, because he spent too much time on what was obvious. On the other hand, I profited a great deal from Emile Picard's course, which I attended for one year, and from Hadamard's, which I attended for three years. It was in Hadamard's course, in 1910, that I found my thesis topic.
As he explains, it was Hadamard who was the major influence in determining the topics on which Lévy would undertake research. Finishing his studies at the École des Mines in 1910 he began research in functional analysis. His thesis on this topic, Sur les equations intégro-differentielles définissant des fonctions de lignes , was examined by Émile Picard, Henri Poincaré and Jacques Hadamard in 1911 and he received his Docteur ès Sciences in 1912.

In 1913 Lévy married Suzanne Lévy (1892-1973), a daughter of the merchant Paul Lévy (1853-1903) and Berthe Weil (1862-1930). [Suzanne really did have a father and a husband with the same name!] Suzanne's maternal grandfather was Henri Weil (1817-1913), a philologist and member of the Institut de France. Paul and Suzanne Lévy had three children, Marie-Hélène (born November 1913), Denise (born 3 October 1916) and Jean Claude (born 10 April 1918). Marie-Hélène Lévy became a mathematician; she married Laurent Schwartz, taking the name Marie-Hélène Schwartz, and has a biography in this Archive. Denise became a professor of German at the lycée Molière in Paris. She married the engineer Robert Piron (mentioned below). Jean Claude studied at the École Polytechnique and became an engineer in the navy.

Lévy became professor of mathematics at the École des Mines in Paris and a tutor at the École Polytechnique in Paris both in 1913 [4]:-
A tutor's task was to help the students to understand the professors' lectures, and hence Lévy was well acquainted with the curriculum and the methods of instruction at the Polytechnique before his own appointment as professor.
World War I caused major disruption in France as it did in most of Europe. Lévy undertook military service and was asked to use his mathematical skills in the war effort. He was assigned practical problems of defence, especially those arising from the then new technology associated with aviation, in particular solving problems concerning defence against attacks from the air.

George Humbert, who had taught at the École Polytechnique from 1895, was one of the professors when Lévy was a tutor but he became seriously ill and he asked Lévy to substitute for him and take over part of his teaching commitments for session 1918-19. Humbert's poor health led him to resign and Lévy became a full professor of analysis at the École Polytechnique in 1920. Lévy remained at the École Polytechnique, with certain interruptions described below, until he retired in 1959.

A young mathematician René Gateaux was killed near the beginning of World War I and Hadamard asked Lévy to prepare Gateaux's work for publication. He did this but he did not stop at writing up Gateaux's results, rather he took Gateaux's ideas and developed them further publishing the material after the war had ended in 1919 as Publication des oeuvres posthumes de R Gateaux et notes personnelles complétant les résultats obtenus par cet auteur .

As we indicated above Lévy first worked on functional analysis [33]:-
... done in the spirit of Volterra. This involved extending the calculus of functions of a real variable to spaces where the points are curves, surfaces, sequences or functions.
In 1919, as part of his teaching duties, Lévy was asked to give three lectures at the École Polytechnique on (see [18]):-
... notions of calculus of probabilities and the role of Gaussian law in the theory of errors.
Taylor writes in [32]:-
At that time there was no mathematical theory of probability - only a collection of small computational problems. Now it is a fully-fledged branch of mathematics using techniques from all branches of modern analysis and making its own contribution of ideas, problems, results and useful machinery to be applied elsewhere. If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Lévy.
The contents of Lévy's lectures are described in [4], which the authors end with the following summary:-
Lévy's lecture notes at the École Polytchnique in 1919 nicely document his beginnings as a probabilist. Though they are evidently not developed in the detail needed (we must not forget that Lévy took the charge of the teaching at the drop of a hat), they indicate how Lévy perceived probability on the eve of his exceptional second mathematical career. They contain the seeds of the studies Lévy would pursue with virtuosity for 40 years: convergence in distribution, sums of random variables, the connection between mathematical probability and physics. They testify to what Lévy said about himself 50 years later in an interview to the French radio France Culture: "I have the feeling of being a mathematician, neither superior nor inferior to others, but different from them."
Loève, in [19], gives a very colourful description of Lévy's contributions:-
Paul Lévy was a painter in the probabilistic world. Like the very great painting geniuses, his palette was his own and his paintings transmuted forever our vision of reality. ... His three main, somewhat overlapping, periods were: the limit laws period, the great period of additive processes and of martingales painted in pathtime colours, and the Brownian pathfinder period.
Not only did Lévy contribute to probability and functional analysis but he also worked on partial differential equations and series. In 1926 he extended Laplace transforms to broader function classes. He undertook a large-scale work on generalised differential equations in functional derivatives. He also studied geometry.

World War II presented serious difficulties for Lévy. On 10 May 1940, Germany invaded Holland, Belgium and Luxemburg then quickly moved to attack France by early June. On 14 June German troops entered Paris and, on 22 June, French Marshal Pétain signed an armistice. This divided France into Vichy France, controlled by a government collaborating with Germany, and a part under direct German control. The École Polytechnique had a certain military status and Germany refused to allow it to remain in Paris. It was relocated to Lyon, in Vichy France, and Lévy moved to Lyon and began teaching. Up to this point we have not mentioned that Lévy was Jewish since it was not relevant. Now, of course, it became a life-threatening difficulty. Legislation of 3 October 1940 required all Jews to be dismissed from their teaching and, on 19 December 1940 Lévy received notice that he could no longer perform his duties. There were exemptions in the 3 October 1940 Act to allow those giving outstanding service to be exempted. An appeal by the director of the École Polytechnique succeeded and from 14 March 1941 he was allowed again to teach.

The situation became worse in the summer of 1942 and Lévy, anticipating events, left Lyon and went to live with his son-in-law Robert Piron, in Montbonnot, near Grenoble in the Italian controlled region of France. He had left Lyon by 4 November ahead of the German invasion of Vichy France on 11 November. He now lived in hiding, carrying false documents, and somehow survived. A letter he wrote to Maurice Fréchet on 29 November 1943 shows his extreme difficulties [25]:-
I just learned that I'm not a professor at the École Polytechnique anymore. In early November, a postal check issued for my October salary was returned to me as "not sufficient funds". I wrote to the administration of the École to ask for an explanation. On 24 November, I received a copy of a decision dated 30 June by which the École transferred me to the Corps des Mines on 1 October; it is also apparent from the text of the decision that on 29 April the Director of the École knew already that I was not to be "re-invested in my job" and asked the Minister how long he had to pay me!!! And all this is legal, if one considers as legal a 1941 regulation subjecting teachers to a decennial re-election, regardless of acquired rights. And now what will the Corps des Mines do with me, having left the place in 1913, and cannot give me work for which I have no longer any expertise and, in addition, with a delay of two months before taking up the post. Will it pay me? ... I deeply hope that 1944 will see the end of these miseries.
Already by the time Lévy wrote this letter, part of the École Polytechnique was operating again in Paris (it had returned in April 1943), but some teaching was still taking place in Lyon for escaped prisoners, Jewish students and others who could not, at that time, return to Paris. After the Allied invasion of France in June 1944 the German occupiers were driven back, and by August of that year the Vichy regime had fallen.

The war years were ones of great difficulties for Lévy, in Lyon, in Montbonnot, near Grenoble, and finally in the Loire, but he was still able to do mathematics [25]:-
In 1940, Lévy had become one of the world major specialists of stochastic processes. And then suddenly, the 55 years old mathematician who had produced a torrential flow of publications for more than 20 years was deprived of the possibility to send articles to journals. Moreover, as we have seen, his teaching tasks were also gradually reduced, and for some months in 1943, living in precarious silence in the Italian zone near Grenoble, Paul Lévy had nothing to do other than mathematics while waiting for better days. His luck was to remain in contact with Fréchet who played for him the role of a private registration journal and could also sometimes serve as a hub between Lévy and other mathematicians.
Back in Paris, he continued to teach at the École des Mines until 1951 and at the École Polytechnique until 1959. There were, however interruptions. He contracted tuberculosis and half his left lung was removed in 1951. Again in 1957-58 he was ill and could not teach, but recovered.

Lévy wrote ten books, the main ones being: Leçons d'analyse fonctionnelle (1922), Calcul des probabilités (1925), Théorie de l'addition des variables aléatoires (1937; 1954), and Processus stochastiques et mouvement brownien (1948). For information about all of Lévy's books, including contents of some of them and also extracts from some reviews, see THIS LINK.

For Lévy's views on the calculus of probabilities see the English version of his talk Les fondements du calcul des probabilités delivered to an audience at the École Polytechnique in the early 1950s, see THIS LINK.

For Levy's views on axiomatic set theory and the continuum hypothesis after the work of Kurt Gödel and Paul Cohen, see THIS LINK.

In 1963 Lévy was elected to honorary membership of the London Mathematical Society. In the following year he was elected to the Académie des Sciences.

Loève sums up his article [19] in these words:-
[Lévy] was a very modest man while believing fully in the power of rational thought. ... whenever I pass by the Luxembourg gardens, I still see us there strolling, sitting in the sun on a bench; I still hear him speaking carefully his thoughts. I have known a great man.
Lévy died in 1971 at the age of 85. A memorial meeting in his memory was held in the Henri Poincaré amphitheatre of the École Polytechnique on Friday 23 March 1973 when various talks were given. They were published in La Jaune et la Rouge in November 1973.

For an English version of Laurent Schwartz's talk, La pensée mathématique de P Lévy , see THIS LINK.

For an English version of Paul-André Meyer's talk L'originalité de P Lévy en mathématiques , see THIS LINK.

For an English version of Benoit Mandelbrot's talk Paul Lévy, professeur, and another description of Lévy by Mandelbrot, see THIS LINK.

We end this biography of Paul Lévy by quoting an extract from the talk Paul Lévy, delivered by Jean Ullmo (1906-1980) at the meeting on 3 March 1973 [35]:-
Paul Lévy was one of the greatest mathematicians of his time and it is an honour for the Corps des Mines to have facilitated the full realization of his exceptional gifts. As a scholar, he was undoubtedly one of the last examples of absolute individualism: he was a solitary researcher who was only concerned with posing problems that interested him and pursuing their solution through only work of interior reflection. He read very little of the work of others, he did not participate in international congresses, except exceptionally at the end of his life. His working methods, artisanal one might say, had their drawbacks: he often found, without knowing it, already known results; more often still, he discovered important results without giving them the necessary publicity, sometimes because he believed them already known. It cannot be denied that there has been wasted effort, but the counterpart has often been admirable: it is the profound originality of a thought indifferent to fashions and schools which did not hesitate to embark on absolutely new paths because he had no fear of solitude. Thus, after having been one of the main precursors of functional analysis, Paul Lévy was the great creator of the theory of probability. It can be said that most of the essential concepts of this theory derive from him.

His teaching at the École Polytechnique, which was of exceptional duration, left a deep mark on his innumerable students: they saw in it a model of conciseness, a requirement vis-à-vis the reader who must recognize all the difficulties hidden in a laconic text; in short, an admirable piece of intellectual gymnastics. Let us add for the record that this very classic course which was little modified until 1957 did not mean that Paul Lévy was losing interest in modern developments in mathematics since he had the courage and the elegance to undertake, in the last two years of his teaching, a redesign of his course to introduce the most recent language and methods.

Finally, we would like to evoke a personal memory: Paul Lévy spoke to me one day about how he chose the questions on which he wanted to focus his research: "I am asking myself a not too difficult problem," he said to me, "so as not to break my teeth in front of an excess of difficulties, but all the same to have to make a big effort which occupies me and gives me the satisfaction of finding something." We will admire this modesty mixed with legitimate pride. These "not too difficult problems" were generally new avenues open to the human mind.

References (show)

  1. Biography in Encyclopaedia Britannica.
  2. Paul Lévy, Quelque aspects de la pensée d'un mathématicien (Paris, 1970).
  3. M Barbut, B Locker and L Mazliak (eds.), Paul Lévy and Maurice Fréchet. 50 Years of Correspondence in 107 Letters (Springer, London, Heidelberg, New York, 2014).
  4. M Barbut and L Mazliak, Commentary on the notes for Paul Lévy's 1919 lectures on the probability calculus at the École Polytechnique, Electronic Journal for History of Probability and Statistics 4 (1) (2008).
  5. R Bard, Sur les travaux de Paul Lévy, La Jaune et la Rouge (September 1964).
  6. B C Brookes, Review: Théorie de l'Addition de Variables Aléatoires, by Paul Lévy, The Mathematical Gazette 39 (330) (1955), 344.
  7. W D Cairns, Review: Calcul des Probabilités, by Paul Lévy, Amer. Math. Monthly 33 (6) (1926), 328-330.
  8. L Le Cam, Paul Lévy, 1886-1971, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability III: Probability theory (Berkeley, Calif., 1972), xiv-xx.
  9. K L Chung, Reminiscences of some of Paul Lévy's ideas in Brownian motion and in Markov chains, Colloque Paul Lévy sur les Processus Stochastiques, Astérisque 157-158 (1988), 29-36.
  10. K L Chung, Reminiscences of some of Paul Lévy's ideas in Brownian motion and in Markov chains, Colloque Paul Lévy sur les Processus Stochastiques, Astérisque 157-158 (1988), 99-107.
  11. A Church, Review: Axiome de Zermelo et Nombres Transfinis, by Paul Lévy, J. Symbolic Logic 15 (3) (1950), 201-202.
  12. R Dautray, Paul Lévy, vu par un de ses élèves, La Jaune et la Rouge (November 1973).
  13. J L Doob, Paul Lévy, Annals of Probability 1 (1971), 1-18.
  14. J L Doob, Obituary: Paul Lévy, J. Appl. Probability 9 (1972), 870-872.
  15. D Dugué, Souvenirs sur P Lévy, La Jaune et la Rouge (November 1973).
  16. T H Hildebrandt, Review: Leçons d'Analyse Founctionelle, by Paul Lévy and J Hadamard, Amer. Math. Monthly 32 (6) (1925), 309-311.
  17. D G Kendall, Obituary: Paul Lévy, J. Roy. Statist. Soc. Ser. A 137 (1974), 259-260.
  18. D G Kendall, Review: Processus Stochastiques et Mouvement Brownien, by Paul Levy, Biometrika 53 (1/2) (1966), 293-294.
  19. M Loève, Paul Lévy, 1886-1971, Ann. Probability 1 (1) (1971), 1-18.
  20. B B Mandelbrot, The Paul Lévy I knew, Lévy flights and related topics in physics, Lecture Notes in Phys. 450 (Berlin, 1995), ix-xii.
  21. B Mandelbrot, P Lévy, professeur, La Jaune et la Rouge (November 1973).
  22. J Marshall, Review: Calcul des Probabilités, by Paul Lévy, The Mathematical Gazette 13 (184) (1926), 214.
  23. K O May, Review: Quelque aspects de la pensée d'un mathématicien, by Paul Lévy, Isis 62 (3) (1971), 415-416.
  24. L Mazliak, How Paul Lévy saw Jean Ville and Martingales, Electronic Journal for History of Probability and Statistics 5 (1) (2009).
  25. L Mazliak, The difficulties of scientific life in occupied France: The examples of Emile Borel, Paul Lévy and others ... (2016).
  26. P-A Meyer, L'originalité de P Lévy en mathématiques, La Jaune et la Rouge (November 1973).
  27. J Neveu, Notes sur les travaux de P Lévy en probabilités, La Jaune et la Rouge (November 1973).
  28. Publications of Paul Lévy, The Annals of Probability 1 (1) (1973), 9-18.
  29. W V Quine, Review: Les Paradoxes de la Théorie des Ensembles Infinis, by Paul Lévy, J. Symbolic Logic 4 (2) (1939), 102.
  30. L Schwartz, La pensée mathématique de P Lévy, La Jaune et la Rouge (November 1973).
  31. G Shafer and L Mazliak (eds.), An autobiographical note by Paul Lévy, written for Takeyuki Hida in 1969, Electronic Journal for History of Probability and Statistics 5 (1) (2009).
  32. L Svarc, Paul Lévy (1885-1971) (Bulgarian), Fiz.-Mat. Spis. Bulgar. Akad. Nauk. 15 (48) (1972), 171.
  33. S J Taylor, Paul Lévy, Bull. London Math. Soc. 7 (3)(1975), 300-320.
  34. J Ullmo, Biographie de Paul Lévy, Annales des Mines (February 1972).
  35. J Ullmo, Paul Lévy, La Jaune et la Rouge (November 1973).
  36. F P W, Review: Calcul des Probabilités, by Paul Lévy, Science Progress in the Twentieth Century (1919-1933) 20 (80) (1926), 712-713.
  37. D B White, Review: Sampling of Populations: Methods and Applications, by Paul Lévy and Stanley Lemeshow, SIAM Review 34 (2) (1992), 347-349.

Additional Resources (show)

Honours (show)

Honours awarded to Paul Lévy

  1. LMS Honorary Member 1963

Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update September 2020