Émile Borel: A Life in Mathematics and Politics Across Two Centuries [1 ed.] 9783985470136, 9783985475131

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Pierre Guiraldenq

Émile Borel A Life in Mathematics and Politics Across Two Centuries

Pierre Guiraldenq

Émile Borel A Life in Mathematics and Politics Across Two Centuries Translated and edited by Arturo Sangalli

About the author and the translator Pierre Guiraldenq is professor emeritus at the École Centrale de Lyon, France. In addition to his scientific career as a metallurgist, his interest in the history of science has resulted in the publication of several biographical articles and books. Arturo Sangalli is the author of numerous scientific articles, books, and translations. He has a PhD in mathematics from the Université de Montréal.

This book is a translated and adapted version of Pierre Guiraldenq, Émile Borel, 1871–1956 : l’espace et le temps d’une vie sur deux siècles. Librairie Albert Blanchard, Paris, 1999. The translated quotes by Camille Marbo are taken from the book Camille Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967) © 1968 Éditions Grasset & Fasquelle and are used with permission. The cover drawing by André Aaron Bilis (1893–1971) comes from the Bibliothèque nationale de France, Réf. N2 – D 094 300.

ISBN 978-3-98547-013-6, eISBN 978-3-98547-513-1, DOI 10.4171/ST/17 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. Published by EMS Press, an imprint of the European Mathematical Society – EMS – Publishing House GmbH Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 136 10623 Berlin, Germany https://ems.press © 2022 European Mathematical Society Typesetting: Nicole Bloye, Cardiff, UK; Apostolos Damialis, Berlin, Germany Printed in Germany ♾ Printed on acid free paper

Foreword Émile Borel is one of those foremost scientists whose name rings so many bells to the ears of mathematicians while being almost unknown to the general public, save for those people who live in the French town of Saint-Affrique. There the hospital, in particular, is named after him. Not that he was a medical doctor nor was he involved in medicine. He was in fact the mayor of the town when the hospital was built, and it is far from being the only construction he can be credited with. Borel was indeed a builder, among many other facets of his personality. He also constructed, and ran as director until his death, the Institut Henri Poincaré in Paris. When I took over the direction of this famous and internationally renowned house of mathematics, the fact that it had been founded by Borel was quite impressive. So, I am honoured and delighted to have contributed a little bit to the publication of this volume, thanks to the links we have been developing with European Mathematical Society Press, which now publishes two of the four journals stemming from the original Annales de l’Institut Henri Poincaré. As far as I know, this is the first book ever written in English about Borel, even though one can have a glimpse of his life, for instance, in the book Mathematicians at war: Volterra and his French colleagues in World War I by Laurent Mazliak and Rossana Tazzioli. In fact, the few historians of mathematics interested in Borel have long refrained from writing down his biography. Hélène Gispert even wrote a paper (in French) in 2012 to explain their reasons. A biography of Borel would be too difficult a story telling, with too many choices to

be made within his numerous intellectual, mathematical, and political commitments, let alone the classical danger of instrumentalisation of his character. Nevertheless, Gispert accepted to curate an exhibition about him at the Institut Henri Poincaré library in 2021. Entitled Émile Borel, un mathématicien au pluriel, it was designed to avoid the pitfalls she had pointed out in her paper, and reflects her suggestion to focus on specific topics, and actually also on specific years in the life of the Borels, with information about his wife and writer Camille Marbo being showcased along with those about him. Thanks to our librarians and this group of historians, in particular Martha Cecilia Bustamante, the exhibition’s material was soon translated into Spanish and shown in Cali in Colombia, where Borel is a hero. In addition, we also published a booklet (in French) about the exhibition together with the Société Mathématique de France. Back in 1999, the biography written in French by the physicist and Borel enthusiast Pierre Guiraldencq filled a gap in a way that was not competed with until Michel Pinault’s monumental work entitled Émile Borel : une carrière intellectuelle sous la IIIe République. Today we are lucky enough to have a widely renewed version of Pierre Guiraldencq’s book in English, thanks to the joint work of the author and the translator Arturo Sangalli. I wish this book be enjoyed by a wide audience, ranging from professional mathematicians or historians to students and the general public. We will surely be proud to display it on the shelves of the Institut Henri Poincaré library. Paris

Sylvie Benzoni

vi

Foreword to the French edition Pierre Guiraldenq’s Émile Borel offers the portrait of a mathematician who had a considerable influence on the development of twentiethcentury mathematics, and, in addition, the discovery of his lifelong contribution to the awareness of this science’s cultural significance. Borel’s 1894 doctoral thesis contained the basis of a new theory of measure that he would expand in his Leçons sur la théorie des fonctions – an essential tool for the development of analysis – published in 1898; a theory that would completely revitalise the field and pave the way to the Lebesgue integral. This work also contained the first statement of the Borel–Lebesgue theorem, which would lead to one of the fundamental concepts of general topology, that of compactness. It is worth mentioning that in his proof of the theorem Borel used the notion of transfinite formulated by Cantor, demonstrating for the first time the practical application of Cantor’s hierarchy of infinities, which would later allow Baire to elaborate his famous classification of discontinuous functions. One of Borel’s defining traits was his desire, very early in his career, to devote his scientific knowledge, his prodigious intelligence, and his force of character to the service of his country. Ten years after his thesis, when he was at the summit of his scientific career and fame, he embarked in a new direction, favouring what seemed to him to possess a “practical value”, foremost the calculus of probability, until then considered a minor discipline, and for which he would develop the theory. He was the author of “the article” that in 1909 gave a new dimension to the field; but also, and especially, to all its applications, from statistical physics –

which had been reborn with the flourishing of molecular theories – to every human activity that involves statistics, including games of political and military strategies, of which he was a pioneer. It is impossible to understand this radical change of direction without considering Borel the man in his entirety and his deep attachment to his origins. In this sense, Pierre Guiraldenq’s book is unique. One cannot envision Borel without a land, a root, an anchoring point whose solidity rests on centuries of wisdom and human values. The author tells us a great deal about Borel, the man, the teacher, the scientist, the administrator, the politician. We discover, among other things, his foretelling vision of the necessity to build Europe, and his pledge for a greater representation of the “feminine element” in public administration. His involvement in the creation of the Institut Henri Poincaré in Paris, which he headed until his death, was decisive. Pierre Guiraldenq’s book will be of particular value to all those interested in the history of science and culture; the science and culture that Émile Borel so eminently represented. Paris

Bernard Bru and Pierre Dugac†

viii

Preface This book introduces the reader to the life of a scientist and man of action, Émile Borel, well known in mathematical circles but little known by the public at large. Borel was a member of the great French mathematical school of the beginning of the twentieth century, and an active participant in the remarkable scientific adventure of those pioneers of genius that revolutionised modern physics. But amid the profound upheavals that fell upon the continent, he also showed other talents: his energy and capacity to act decisively added a new, social and political dimension to his life. Borel’s theoretical and applied scientific works, so vast in number and scope, remain a permanent reference. However, apart from the short book La vie et l’œuvre d’Émile Borel written in 1965 by his former student Maurice Fréchet, little appears to have been written before the 1999 Colloque Émile Borel in Saint-Affrique to restore the memory of the man, his family life, his career, and his numerous and varied activities. Since then, a growing number of historians of mathematics, both in France and further afield, have manifested a real interest in his works: Hélène Gispert, Marie-France Bru, Bernard Bru, Roger Laurent, Gustave Choquet, André Warusfel, Laurent Mazliak, Jean-Michel Kantor, Jean-Baptiste Hiriart-Urruty, Henri Caussinus, Jean-Michel Guieu, Alexandre Klein, Jean-Paul Pier, Loren Graham, among others. Echoing the title of this book and underlying a renewed interest in Borel’s life and works, the study meeting “Borel : savant et politique” was held on November 17–18, 2021 at the Centre Alexandre-Koyré,

a research institute in the history of science and techniques located in a suburb of Paris. His wife, Marguerite (née Appell), a writer known by her pen name Camille Marbo, lived a dynamic and passionate existence at his side, but confessed to always having refused to write about the life of her husband of fifty years, as she writes in her last book À travers deux siècles : souvenirs et rencontres (1883–1967): Of such a complex man, what is to be said? One is simply struck by his extraordinary intelligence, his powerful will, his instinctive and rare kindness.

During a visit to the Princeton University Library in 1984, the author discovered on its shelves all the books written by Borel – in stark contrast with the handful of copies of his works held in the French National Library’s collection, in Paris. This prompted him to embark on a journey of historical research to restore the memory of his illustrious compatriot. Chance, a concept central to Borel’s work on probability, brought together (virtually) author and translator, during the long months of a global pandemic. The result of their collaboration is this English version of the author’s Émile Borel, 1871–1956 : l’espace et le temps d’une vie sur deux siècles, adapted for an international audience. The reader will find a chronological analysis of Borel’s “space in time”, with a focus on the most significant moments and the principal events in his rich life, across two centuries and two wars and against the backdrop of a complex contemporary history. As for his mathematical works, important as they are, they are only briefly discussed in the context of his scientific career. This book, which is also our way of marking the 150th anniversary of the birth of the great mathematician and statesman, would not have seen the light without the vision and enthusiasm of Apostolos Damialis, editorial director at European Mathematical Society Press, to whom we address a heartfelt “thank you”. Écully and Sherbrooke

Pierre Guiraldenq and Arturo Sangalli

x

Contents

Foreword Foreword to the French edition Preface 1

His youth: A string of accomplishments (1871–1900) Childhood, early training, and first achievements . . . . . . . Paris: Entrance exams, École Normale Supérieure, military service . . . . . . . . . . . . . . . . . . . . . . . . . . Beginnings of a university career . . . . . . . . . . . . . . .

v vii ix 1 1 2 8

The rise of a scientist (1900–1914) Scientific and pedagogical bloom . . . . . . . . . . . . . . . Married life begins: The Borel–Appell union . . . . . . . . New administrative duties and a foreign mission . . . . . . . A penchant for the concrete: Novel scientific and technical approaches . . . . . . . . . . . . . . . . . . . . . . .

11 11 17 20

3

The Great War and the return to peace (1914–1924) The draft: The front and the Ministry of Defense . . . . . . . The death of Fernand Lebeau . . . . . . . . . . . . . . . . . Postwar years: New scientific and international activities . .

31 31 34 34

4

From science to politics (1924–1940) The beginnings of a parliamentary career . . . . . . . . . . . The Ministry of the Navy and the Naval budget . . . . . . .

41 41 46

2

25

5

6

Life as a member of parliament: His three terms between the two wars . . . . . . . . . . . . . . . . . . . . . . . . . Mayor and councillor . . . . . . . . . . . . . . . . . . . . .

48 51

Scientific activity between the two wars (1925–1940) The early years of the Institut Henri Poincaré . . . . . . . . University life at the Sorbonne . . . . . . . . . . . . . . . Two presidencies: Académie des Sciences and the Institut . The scientific jubilee: Fifty years of research and teaching .

. . . .

55 55 57 61 64

. . . .

67 67 71 76 82

A long and final march towards the future (1940–1956) Dark years under occupation and new hopes . . . . . . . . Post-war period, c. 1950s: Promoting science outreach . . Reflections on current affairs: Secret gardens . . . . . . . The final years: Farewell from his hometown . . . . . . . .

Supplement to the English edition On the foundations of mathematical analysis from Borel to Bourbaki by Pierre Dugac . . . . . . . . . . . . . . . . Appendix Closing remarks and further reading . . . . . . . . Summary of Borel’s scientific career and public life Awards, honorary titles and honorary memberships Quotes and anecdotes . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

85 85

95 . 96 . 98 . 100 . 103 . 106

xii

Chapter 1

His youth: A string of accomplishments (1871–1900) Childhood, early training, and first achievements Félix Édouard Justin Émile Borel was born in Saint-Affrique, a commune in Southern France, on January 7, 1871, ten years after the death of the mathematician Pierre-Frédéric Sarrus,1 a native of the same town. His father, Honoré Borel, was a Protestant minister married to Émilie Tessié-Solier, who came from a very respectable family of the region. Much younger than his two sisters, Jeanne, sixteen, and Marthe, fourteen, little Émile was the centre of attention in an atmosphere of well-established family traditions. He received his primary education at the private Protestant school run by his father and located in his own house. In a difficult period when France, humiliated by her defeat in the 1870 war and with the French Republic still fragile, struggled to recover, it became important to instill in the young generation a sense of duty, effort, and responsibility. Of Émile’s early training few traces remain, except for some rare souvenirs recorded by his wife, who reported that Émile’s sharp mind didn’t go unnoticed, as his first academic successes confirmed.2 1 Also

known for the mnemonic rule of Sarrus for calculating determinants.

2 C. Marbo in F. Decuq, Retrospective de Saint-Affrique et de sa région, Maury, Millau

Borel’s formative years were also marked by a deep affection for the arduous and hardworking rural life of South Rouergue, his homeland, a trait that would later be reflected in his actions and decisions. In the property his father possessed at Saint-Paul-des-Fonts he discovered the demands of working the land, amid the damp heat, the yokes, and the herds. In 1882, young Émile was sent to live with his eldest sister and her husband, the pastor Élie Lebeau, in Montauban. He pursued his studies there at the Lycée Ingres and soon attracted his teachers’ attention for his outstanding abilities.3 In 1887, he obtained his baccalauréat ès sciences after winning nearly every first prize: in mathematics, physics and chemistry, history and geography, and German, besides, not surprisingly, the prize of excellence. In addition to his science degree he also successfully sat the baccalauréat ès lettres exams. Émile’s next goal was to prepare for the entrance examinations for the grandes écoles. These are elite education institutes, separate from French universities at the time, that employ highly selective admission policies, leading to a fiery competition among potential candidates. Following his teachers’ advice, Émile left his southern town and headed for Paris, home to such prestigious academic institutions.

Paris: Entrance exams, École Normale Supérieure, military service In October 1887, holding a higher education scholarship, Borel took a one-year course in advanced mathematics at the Collège Sainte-Barbe. He then continued his preparation for the exams at the famous Lycée Louis-le-Grand as a student of the reputed teacher Bolesias Niewenglowski.4 A string of achievements followed: in 1889, at age eighteen, young Borel obtained the first prize in mathematics in the general entrance (1971), pp. 143–147 3 J.-F. Delord, Le Lycée Ingres de Montauban, 1870–1914. Histoire, anecdotes, silhouettes et portraits, Archistra, (1993), no. 107, pp. 1–63 4 R. Deltheil, Émile Borel, Revue du Rouergue, tome X (1956), no. 1, janvier–mars, pp. 1–14

2

examinations, and also the top place in those of the École Normale Supérieure (ENS) and the École Polytechnique; a rare feat, reserved to distinguished alumni,5 such as Gaston Darboux, his future thesis advisor, in 1861, and Jacques Hadamard in 1884. The 1994 Fields medallist Jean-Christophe Yoccoz also swept all three top prizes in 1975. Borel was now at a crossroads, facing two alternative paths: either a life devoted to teaching and research, or a career in government – military or civil – by enrolling in the École Polytechnique, a first-class engineering school. In 1940, at the celebrations honouring his fifty years of public service, he would look back at those entrance examination days and the role played by the fortunes of life and the social environment around him: I was lucky to have as a friend Gaston Darboux’s son, thanks to whom I was introduced into the family of the eminent mathematician. The homely fifth-floor apartment on rue Gay-Lussac welcomed the young student from the province I was with warmth and good nature. I learned there many things, and in particular that the ENS was not only a pathway to a teaching career but also served as a preparation for scientific research. Gaston Darboux’s influence was thus crucial in determining the direction of my future training. . . 6

The dynamic spirit and technical achievements of the times could very well have lured him toward a successful future as an engineer. He already possessed the creativity and organisational talent that would be revealed later, during his Navy service and by his active role in the foundation of the Institut Henri Poincaré and the construction of a modern hospital in Saint-Affrique. But, in the end, his long-weighted decision to enroll at the ENS was driven by a need for abstraction and general knowledge, at least initially. He would spend three years at the prestigious school, from 1889 to 1892, and benefit from the lectures of some outstanding mathematicians: Charles Hermite (higher algebra), Paul Appell (rational mechanics), Gaston Darboux (higher geometry), Émile Picard (differential and integral 5 Maîtres et Élèves, célébrités et savants : l’École Normale Supérieure (1794–1994), Archives nationales, Paris (1994) 6 Idem, p. 71

3

calculus) and, at the Faculté des Sciences, Henri Poincaré (probability calculus and mathematical physics). During his time at the ENS Borel would also forge some lasting friendships, notably with Élie Cartan, Jean Perrin, and Paul Langevin. His friendship with Paul Valéry, whom he met in the Parisian intellectual circles, followed a different path, as it was anchored in their shared Southern origins, mutual passion for the sea, and months of military service in the same barracks in Montpellier.7 A true breeding ground of illustrious scholars,8,9 the ENS provided Borel with a worldview in the largest sense of the word (one that included culture and intellectual life), which perhaps explains and anticipates his future choices and trajectory. Even before obtaining his first university degree (licence ès sciences) in 1891, Borel submitted two articles – on theorems of Humbert and Fouret and on semi-convergent series – that were published in the reputable journals Nouvelles Annales de Mathématiques10 and Bulletin des Sciences Mathématiques,11 respectively. In 1892, he published a third article, on certain systems of differential equations, in the Annales de l’École Normale Supérieure.12 That same year, he sat his aggrégation examination to receive a teaching license and once again was top of his class; he was then only twenty-one. And then came the call for military service. The 1889 military law required all French citizens to serve for the same period of time, but ENS students underwent a special military training that allowed them to reduce their actual service period to one year,13 in order to prevent, in times of peace, too big a drop in the school’s enrolment. 7 É.

Borel in Paul Valéry, vivant, Cahiers du Sud, Marseille (1946) et Élèves, célébrités et savants : l’École Normale Supérieure (1794–1994),

8 Maîtres

op. cit. 9 A. Peyrefitte, Rue d’Ulm : chroniques de la vie normalienne. Édition du bicentenaire, Fayard, Paris (1994) 10 É. Borel, Note sur un théorème de M. Humbert et sur un théorème de M. Fouret, Nouv. Ann. Math. III. Sér., 9 (1890), pp. 123–129 11 É. Borel, Sur le changement de l’ordre des termes d’une série semi-convergente, Bull. Sci. Math. II. Sér., 14 (1890), pp. 97–102 12 É. Borel, Sur l’équation adjointe et sur certains systèmes d’équations différentielles, Ann. de l’Éc. Norm. III. Sér., 9 (1892), pp. 63–90 13 P. Jeannin, École Normale Supérieure : livre d’or, Office français de diffusion artistique et littéraire, Paris (1963)

4

The protestant church at Saint-Affrique and birthplace of Borel (From the author’s collection)

L’Avenue de Vabres where the Borel family settled in 1881–82 (From the author’s collection)

5

Émilie Tessié-Solier (1830–1911), mother of Émile Borel (J.-P. Cambon family collection)

Borel around the time of his military service

(J.-P. Cambon family collection)

Borel (third from left) with his classmates (1889)

(Collection of the Bilbiothèque de l’École Normale Superieure)

6

Entrance to the École Normale Superieure

(Collection of the Bibliothèque de l’École Normale Superieure)

7

From November 1892 to September 1893, Borel was enlisted as a second-class soldier in the 142nd Infantry Regiment, in Montpellier.14 During this period, he volunteered to give a course at the local university on higher and projective geometry for future mathematics teachers.15 With the permission of the local military authorities, he thus became one of the first “drafted scientists”.

Beginnings of a university career His ten-month “sabbatical” in Montpellier provided an opportunity for Borel to prepare his doctoral thesis under the supervision of Darboux. Back in Paris, and even before having defended his dissertation, he was appointed Maître de conférences at the Faculty of Science of the University of Lille on a temporary basis. Obliged to share his time between Paris and Lille, between his research and his teaching, Borel took the frequent trips in stride, as they were well-suited to his dynamic disposition and stimulating for his work. He successfully defended his thesis less than one year later, in June 1894, at the Faculté des Sciences de Paris, before a jury composed of some illustrious names: Darboux, his thesis advisor and president of the jury, Appell, his future father-in-law, and Poincaré, as a reviewer. The title of his dissertation’s first topic, Sur quelques points de la théorie des fonctions, was remarkably unassuming. His thesis also included a second topic: on third-order partial differential equations, characteristic16 of families in triple orthogonal systems of surfaces with plane lines of curvature – at the time, doctorats d’état required a second, imposed topic, consisting of an analysis of papers already published in order to assess the candidate’s teaching ability and summarizing skills. 14 É. Borel, État des Services Militaires, Archives du Service Historique de l’Armée de Terre, Château de Vincennes, Paris 15 Compte-rendu du Conseil de Faculté du 13 janvier 1893, sous la présidence du Doyen M. Sabatier, Archives de la Faculté des Sciences de Montpellier 16 Exact titles: (a) Équations aux dérivées partielles du 3e ordre caractéristiques des familles de surfaces pouvant appartenir à un système triple orthogonal; (b) Cas où les surfaces admettent des lignes de courbure planes.

8

The originality of his first topic prompted its swift publication in various journals.17 Years later, Borel would write a summary of his thesis,18,19 and elaborate on its genesis: It was mostly through Charles Hermite’s lectures that I developed an interest in the theory of functions. Thanks to him I came to appreciate and admire Cauchy, an admiration that only grew as I got to know better the work of he who was considered the chief founder of modern analysis. Those who came after Cauchy, in particular Weierstrass and Riemann, introduced new points of view of undeniable potential. Under their influence, analysis became more rigorous, more subtle, and more abstract, leading to the summit of abstraction that was reached with Georg Cantor’s theories. I must confess that in the beginning I was seduced by Cantor’s ideas, as were many other young mathematicians; I don’t regret it, because they help open up the mind. But I always believed that those abstract concepts should be a means to a goal, not an end in themselves; in my view, the study of sets was only justified by its usefulness in the study of functions.20

It is in this spirit that he defined the measure of a set of points on the unit interval21 and showed the interest of the notion of sets of measure zero in the theory of functions. Borel devoted the period from 1894 to 1897, during which he was teaching at Lille, to a generalisation of the theory of analytic functions. In particular, he contemplated regarding as multi-valued certain single-valued functions. His commitment to the generalisation of certain mathematical concepts was in phase with the explosion of science in general, and particularly modern physics, to which many of his friends and colleagues were attracted. 17 Comptes rendus de l’Académie des Sciences, Bulletin des Sciences Mathématiques, Nouvelles Annales de Mathématiques, Annales de l’École Normale Supérieure. 18 Notice sur les travaux scientifiques de M. É. Borel, Gauthier-Villars, Paris (1901), pp. 13–14 19 Notice sur les travaux scientifiques de M. É. Borel, Gauthier-Villars, Paris (1912) 20 M. Fréchet, La vie et l’œuvre d’Émile Borel; É. Borel, Œuvres, vol. I, CNRS, Paris (1972), p. 29 21 É. Borel, Leçons sur la théorie des fonctions, Gauthier-Villars, Paris (1898), p. 46

9

We give here a summary of Borel’s scientific production between 1895 and 1900: •

Twenty compte rendus to the Académie des Sciences;

•

Five articles published in Acta Mathematica, and as many in the Bulletin des Sciences Mathématiques;

•

The book Leçons sur la théorie des fonctions, published by GauthierVillars in 1898;

•

The lecture notes Introduction à l’étude de la théorie des nombres et de l’algèbre supérieure, co-written with fellow student Jules Drach and based on a course given at the ENS by their teacher, Jules Tannery, who was also Jacques Hadamard’s thesis advisor.

In 1898, at age twenty-seven, he was awarded the Grand Prix des Sciences Mathématiques by the Académie des Sciences for his article on divergent series,22 one year after his participation in the first International Congress of Mathematicians in Zurich, presided by Cantor himself. In the midst of so much success, what was to become of his private life? Borel’s hometown was not foremost in his mind and, despite the relatively long university holidays, his visits to Saint-Affrique were infrequent. But life went on – and so did death. Two family tragedies befell him just one year apart. In 1894, his brother-in-law died, leaving his sister Jeanne, then forty-nine, a widow with a four-year-old child. The following year his father suddenly passed away; an affective loss that might explain Borel’s need to so ardently serve the community to which his father had devoted almost fifty years of his life.

22 É. Borel, Mémoire sur les séries divergentes, Ann. de l’Éc. Norm. III. Sér., 16 (1899), pp. 9–131

10

Chapter 2

The rise of a scientist (1900–1914) Scientific and pedagogical bloom The unfolding of a university career In 1897, Borel left the University of Lille to become Maître de conférences at his alma mater, the École Normale Supérieure, a position he would occupy until 1904. Among his students we find René Baire, Paul Montel, and Henri Lebesgue. He would later describe this period of his life as one during which “you are not sure whether you’re already a professor or still a student”.1 From 1899 to 1902 he also gave a course at the Collège de France, introducing young scientists to new and potentially fruitful perspectives on various branches of pure and applied mathematics. This course, known as the Cours Peccot, was created in 1899. Borel was the first and unique recipient for two consecutive years (originally, the course was to be given only once). At the dawn of the twentieth century, the University of Paris, known as the Sorbonne, counted among its mathematics and physics faculty a number of distinguished names: Gaston Darboux, who was also the dean 1 Jubilé scientifique de M. Émile Borel (January 14, 1940), Gauthier-Villars, Paris (1940), p. 76

of the faculty, Paul Appell, Henri Poincaré, Paul Painlevé, and the future Nobel laureate in physics, Jean Perrin. Following the death of Pierre Curie in 1906, Marie Curie was appointed professor of general physics. As for Borel, who had assumed the role of professeur-adjoint in 1904, he had become the indisputable specialist of the theory of functions. In 1909 he was appointed to the chair Théorie des fonctions that had been created for him. Borel was then only thirty-eight years old. The doctoral theses presented in Paris during the years 1900–1910 demonstrate the diversity of research fields and the contributions of those French scholars who marked the contemporary history of science. Such an exceptional scientific community provided Borel with the occasion to imagine and develop new ways of communication addressed to a large audience. For example, the Revue du Mois was created by him in 1906 with the participation of several of his colleagues. However, the war would put a stop to his dynamic and promising university career. Later, in 1920, after peace had returned, Borel would occupy the chair of mathematical physics and probability calculus; he would remain in this post until his retirement in 1941.

The spirit of his teaching and research: Some firsthand accounts Here’s how some of Borel’s students remembered him: Robert Deltheil (ENS, 1909) – future professor and dean of the Faculty of Science of the University of Toulouse – observed that Borel’s teaching “was more like study supervision than formal lecture. There were impromptu discussions, with the master’s research work taking place before his students’ eyes, with his incisive mind and his constant concern for distinguishing what is essential from what is accessory”.2 Charles Maurain (ENS, 1890): “When I arrived at the École Normale, I found Borel’s presence everywhere. . . We were a group of students from the province, easily impressed by a world that was new to us. Intimidated by Borel’s reputation and prestige, we didn’t dare seek in2 R. Deltheil, Émile Borel, Revue du Rouergue, tome X (1956), no. 1, janvier–mars, pp. 1–14

12

formation or advice from him; it was he who would kindly come to us and, through informal conversations, helped us more than anyone else get a foothold on this new environment. It was for me the beginning of a long friendship, for which I always felt gratitude and respect”. Maurain still remembered Borel’s methods: “Thanks to his admirable insight he would come up with interesting ideas without taking the time to develop them himself, for his insatiable curiosity would soon take him in a new direction. Even though he left vast subjects for others to explore, he didn’t really abandon them, sparking their study and further development by his teaching and his numerous publications, such as the collection of monographs on the theory of functions – hence the large number of his students and the scope of his influence”.3 Gaston Julia (ENS, 1911) vividly recalled some traits of Borel’s personality and his behaviour during examinations: “A big man, dressed in black, sporting a black beard; with a brightness in the eye under the black eyebrow, he would walk back and forth along the rows, stopping from time to time to take a look over a student’s shoulder at the progress of his ‘research’. . . ” Julia adds: “The exam consisted of two parts: solving a problem in writing while the previous candidate was being assessed, and an oral examination at the blackboard. He would give us a small piece of paper with a question to be solved while the preceding candidate was plugging away at the board. Then, we picked up the chalk. . . and we had the impression of being dragged through every wet and muddy path of a wild forest. “I have a very precise memory of the ordeal I went through before admitting defeat. A single problem, at which we looked from every possible angle, and each time I succeeded in finding something new you would take me in a different direction. Your few but insidious questions; the occasional word to clarify some point; and then this final question over which I long struggled before hopelessly declaring, as I put down the chalk, that I didn’t understand anything anymore. But, to my astonishment, I didn’t come out completely defeated from this battle of Jacob with the Angel; your eyes lit up, and with a warm smile 3 C.

Maurain in Annuaire de l’École Normale Supérieure (1954–1957), January 13, 1957, pp. 29–32

13

and a convincing tone you told me that ‘I was not expected to know everything’ ”.4 Another former student of Borel’s, George Bruhat, whose books in general physics would for many years be essential reading for license exam candidates, recalled his teacher’s very personal style and pedagogical choices: “In your lectures as in your works, you focused on what is really fundamental; you knew how to make us understand the essential features of the theories you presented, and the importance of avoiding secondary demonstrations. It is thanks to your lectures that I came to appreciate how fascinating the teaching of mathematics could be; if I could resist its appeal, it was only because I was already initiated to the pleasures of building devices and performing experiments”.5 In a clever and efficient way, Borel gathered around him several of his colleagues to write a monograph on the theory of functions. He was at first the only author; later, a team of French assistants and foreign colleagues joined him in the project, resulting in the production of an impressive number of volumes, ten of which were written by Borel himself. The reason he was able to achieve such a feat was probably due, as his student Maurice Fréchet pointed out,6 to the fact that he would only develop the subjects orally during his lectures, leaving the task of writing them down to one of his students, just as Poincaré did. He also demonstrated his flair for choosing his assistants, since most of them would later become university professors. Here is a partial list of volumes based on the courses he taught, and which he published, like a metronome, over a period of about ten years: Lessons on: Theory of Functions (1898), Integral Functions (1900), Divergent Series (1901), Series with Positive Terms (1902), Meromorphic Functions (1903), Functions of Real Variables (1905), the Theory of Growth (1909), and Monogenic Functions (1917) – whose publica4 G. Julia in Jubilé scientifique de M. Émile Borel (January 14, 1940), Gauthier-Villars, Paris (1940), pp. 34–44 5 G. Bruhat in Jubilé scientifique de M. Émile Borel (January 14, 1940), GauthierVillars, Paris (1940), p. 53 6 M. Fréchet, La vie et l’œuvre d’Émile Borel; É. Borel, Œuvres, vol. I, CNRS, Paris (1972), p. 21

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tion was delayed by the war. The complete collection consists of some forty volumes.

Prizes and other teaching activities The years 1900–1905 saw Borel awarded a number of prizes: the Poncelet Prize (Geometry Section) awarded by the Académie des Sciences in 1901, and the 1904 Vaillant Prize for determining and studying all the displacements of an invariant figure of which the different points describe spherical curves. On this latter occasion he had entered the contest under a pseudonym.7 It was only after the results were announced that the jury learned the true identity of “Olinde Rodrigues” (the name of one of his wife’s relatives). In view of his repeated achievements, he was nominated several times to the Académie des Sciences, in 1900, 1901, and 1912, but each time he was passed over. Among his teaching activities, he served as a member of the École Navale entrance examination committee. A somewhat surprising undertaking, but one that showed Borel’s interest in, and passion for, marine sciences and, later, military navy. From 1901 to 1910 he performed his duties as examiner with distinction, as reports from the committee’s presidents testify. In 1901: “We couldn’t have made a better choice. The reputation of such an eminent mathematician could have intimidated the candidates, but he proved capable of asking even the hardest questions with extreme kindness and benevolence, thus allowing them to show their real competence”.8 In 1907: “An excellent examiner, absolutely impartial, with unequalled patience and insight. He would begin by asking simple questions to put the candidate at ease, and then, little by little, the examination often becomes extremely interesting”.9 7 Prix Vaillant, C. R. Acad. Sci. (1904), juillet–décembre, p. 1070; see also M. Husty, E. Borel’s and R. Bricard’s papers on displacements with spherical paths and their relevance to self-motions of parallel manipulators. In: International symposium on history of machines and mechanisms (Proceedings HMM 2000), pp. 163–171, Springer, Dordrecht (2000) 8 Service historique de la Marine, château de Vincennes, Bulletin individuel de Notes, Commission d’examen d’admission à l’École Navale (Année 1901), Paris, September 12, 1901, Dossier CC7 4o Moderne 186/4

15

Borel resigned from his duties as an examiner in December 1910 to become scientific director of the ENS, the position previously occupied by Jules Tannery.10

His interest in secondary education: Mathematics for all Besides his involvement in higher education, Borel set out to promote basic mathematical training, which he considered of primary importance. Between 1903 and 1910, he published a number of elementary mathematics texts in arithmetic, algebra, geometry, and trigonometry. These clearly written books, often illustrated with practical examples, were addressed not only to baccalauréat students but also to anyone seeking to learn without the pressure of having to take an exam. Years later, after the 1914–18 war, he would create the “Scientific Education Library”, a collection of volumes devoted to popular scientific culture. Here’s how he presented it: Our aim is to provide the reader with the basic tools for acquiring a true scientific education, without the burden and preoccupation of having to prepare for examinations. These days, we need scientific culture as much as we do literary and artistic cultures. If the former appears more difficult to acquire, it’s because most books, written for an audience with specific needs, often put off those seeking knowledge without the sanction of a diploma.11

Among the titles in the collection, we find Principles of Algebra and Analysis (1924), Geometry and Imaginary Numbers, co-authored by Robert Deltheil (1932), and Mechanics and Universal Gravitation (1942). Well aware of the importance of popularizing all branches of science without sacrificing rigour, he recruited other authors to contribute books in various fields, such as Energy, Matter, and Light, by the physicist Jean Perrin, and the zoologist Octave Duboscq’s The Cell in Primitive Organisms. 9 Idem,

(Année 1907), Paris, October 23, 1907 from É. Borel to the Minister of the Navy, Service historique de la Marine, château de Vincennes, December 7, 1910 11 É. Borel, Preface to the presentation of the series “Bibliothèque d’éducation scientifique” (Scientific Education Library). In: Principes d’algèbre et d’analyse, Albin Michel, Paris (1924) 10 Letter

16

Pedagogy for children and psychological research on creativity and imagination were also subjects that attracted Borel’s attention. To help develop different thought mechanisms, he did not hesitate to resort to very simple arithmetic questions, such as “How much is one-third plus half of one-third of 100?”12 “Someone who follows to the letter the statement of the question”, he wrote, “will, with more or less difficulty, first compute one-third of 100, that is, 33.333; then divide it by 2, and finally perform the addition. A person with some notions of arithmetic will observe that one-third plus half of one-third make 3/6, that is one-half, and therefore simply divide 100 by 2. Finally, someone used to employing geometric intuition will immediately visualise a segment OA divided in three equal parts by two points B and C, and intuitively see that the midpoint of BC is also the midpoint of OA. This person will arrive at the answer by taking a direct, immediate path, with no calculations”. O

B

C

A

M This example, almost trivial, conceals under its surface Borel’s subtler work on sets of points of the interval [0, 1] and their properties: measurable sets, open or closed, rarefaction – a notion he introduced to establish a classification and a scale among “zero” probabilities – , transfinite numbers, and geometric probability, among others.

Married life begins: The Borel–Appell union In her last book,13 Camille Marbo (née Marguerite Appell) charmingly described the cultural environment of her youth, the university life of her father, Paul Appell, and the first encounters with her future husband. At the time, she lived in Paris with her parents at 6, rue Le Verrié, near the Luxembourg Gardens, where her father would meet with friends 12 É. Borel, Documents sur la psychologie de l’invention dans le domaine des sciences, Organon (Warsaw), 1 (1935), pp. 33–42 13 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), pp. 53–63

17

and colleagues, in particular Darboux and Painlevé. She mentions a “scientific dinner” to which Borel was invited for the first time. But Émile forgot the date and did not show up. Made aware of his impoliteness by Darboux, he apologised to Mme Appell. This missed invitation would teach Borel a lesson: Punctuality became for him an imperative, and he made it a golden rule for family members, his assistants and, later, his drivers – an obsession with punctuality that became legendary. By 1898 Borel had become a frequent guest of the family. Marguerite was then fifteen. Borel continued his visits after the Appells left Paris for Saint-Germain-en-Laye, in the western suburbs. In a passage of her book, Marbo observes that “this mathematician likes to dance and is not averse to worldly pleasures. He is tall, dark-haired, with beautiful golden eyes and a thick beard”.14 He asked her hand in marriage on the day of her eighteenth birthday, April 11, 1901. The wedding took place at the City Hall six months later, on October 12, in the presence of their families. Among the witnesses of the wedding was Borel’s friend Paul Painlevé, who at thirty-seven was already a member of the Académie des Sciences. In the tradition of the “Belle Époque”, the newlywed couple left for a four-week honeymoon in Italy. They visited Bellagio, Venice and its palaces, and the island of Murano, before pursuing their tour of Italian cities with stops in Milan, Pisa, and Florence. The journey back to France was, just as their future would be, cosmopolitan and maritime, with a rough sea crossing from Genoa to Marseille in a cargo ship, punctuated by some picturesque stopovers. Back in Paris, they moved into the apartment Borel had occupied during his single years, at 30, boulevard Saint-Germain. An exhausted young Marguerite had a miscarriage that was followed by complications. To their great despair, they realised that they would never be able to have children. Their social life revolved around a group of ENS graduates, scientists, and intellectuals: Jean Perrin, Paul Langevin, the mathematicians Jules Drach and Paul Montel, the poet and essayist Charles Peguy, the politician and future prime minister Édouard Herriot, Pierre and Marie 14 Idem,

p. 20

18

Curie, and the poet and philosopher Paul Valéry. Such a rich intellectual environment provided a fertile ground for Marguerite’s literary aspirations. The sudden death of Émile’s eldest sister Jeanne Lebeau, in 1903, prompted the Borels to take under their care Jeanne’s son, Fernand, who was then only thirteen. Assuming the role of an adoptive father and now responsible for his nephew’s schooling, Borel enrolled Fernand at the Lycée Louis le Grand. In order to accommodate the larger family, the couple moved into a more spacious house, at 2, boulevard Arago.

The Revue du Mois In 1905, Borel won the Prix Petit d’Ormoy. The mathematical prize was accompanied by a monetary reward, and the Borels decided to use the money to create an imaginative periodical, La Revue du Mois. An editorial board headed by Borel and made up of friends and colleagues was appointed. Marguerite, who had adopted the pen name “Camille Marbo” – a combination of the first syllables of her given and married names – was in charge of proofreading and typesetting. The first issue of the magazine came out in January 1906, with articles by Vito Volterra, a professor at the University of Rome, on mathematics in biology and the social sciences, and by Gaston Darboux, on the life and works of Charles Hermite. Every month, the Revue would feature original perspectives on scientific culture, economic and social life, and political developments, while keeping abreast, at the approach of 1914, of German technical and military organisation. Among the articles contributed by Borel, and likely to capture a large readership, were “Graphology and the scientific method”, “The social role of amateur scientists”, and “Probability calculus and the individualist mentality”. The magazine, which at one point counted one-thousand subscribers, owed its success to the quality of the subjects treated on its pages, often connected to current events. But the war would gradually disrupt the operations of this distinctive publication. The 1915 issues were devoted to the dramatic events shaking the country, and the focus shifted to such topics as an inventory of the technical and military resources and the 19

analysis of public opinion on the war. Publication ceased following the general call-up and the increasing number of casualties. Among these, the Revue’s secretary, who was killed in action. Borel, for his part, joined the artillery regiment. La Revue du Mois would see light again in 1919 for a short period of time but remained faithful to its spirit. The December 1920 issue would be the last. In it, Borel insisted on the importance of a scientific vision of tomorrow’s world, based on international exchanges, and he expressed his hope of resuming publication in the future.15

New administrative duties and a foreign mission Scientific direction of the ENS Following the death of Jules Tannery in 1910, Borel was appointed scientific director and assistant director of the École Normale Supérieure, the institution he had entered as a student twenty-one years earlier. He considered this promotion as “an honour he didn’t deserve”, given the influence and reputation of the man he had admired at the beginning of his studies. He would later say that this was the most important change in his career since his admission to the ENS: The years between this date and the war of 1914 were among the happiest of my life. . . later darkened by the terrible slaughter of young men.16

The Borels moved into the apartment provided by the school, which had been occupied earlier by Louis Pasteur when he was administrator and director of scientific studies.17 From the beginning of his tenure Borel took a particular interest in student life, and organised weekly working lunches to facilitate the dialogue across disciplines. A series of foreign trips to represent the university then began; notably to Italy, where he met his friend Vito Volterra at the University 15 The magazine Vient de Paraître would feature a scientific section named La Revue du Mois, from 1923 to 1925. 16 Selecta : jubilé scientifique de M. Émile Borel, Gauthier-Villars, Paris (1940), p. 71 17 Maîtres et Élèves, célébrités et savants : l’École Normale Supérieure (1794–1994), Archives nationales, Paris (1994), p. 47

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of Rome, and Hungary, to attend the inauguration of Eötvös József Collegium, an institution modelled on the ENS. This administrative period weighed heavy on Borel.18 Shortly before the war, there were plenty of problems to worry about. Besides staff shortages, the recruitment of new candidates was made difficult by the appeal of other prestigious careers that attracted the French elites towards the business world. Borel sounded the alarm and warned the ministry that beyond a certain rank the only candidates left were those who didn’t gain admission to the École Polytechnique; action was needed to put things back on an even keel. In addition, in 1913 the ENS administration faced new challenges: The recently passed military law obliged students to participate in regular periods of military training, and for the 1913 academic year the school was compelled to double the number of admissions without receiving additional funding.

The Curie affair It was around this time that a campaign of slander against Marie Curie broke out in the press, prompting the Borels to stand up for her with tenacity and determination. Several books19,20,21 give an account of the facts: the events were brought to a head a few years after the death of Pierre Curie, during the Solvay Conference that took place in Brussels in October 1911. Among the participants we find Planck, de Broglie, Rutherford, Einstein, Lorentz, Poincaré, Perrin, Langevin, and Marie Curie. A smear campaign was launched to destroy Curie’s career. She was an easy prey for the tabloids, which denounced her origins, her work, and her friendship with Paul Langevin – a married man – misrepresenting her as a foreign Jewish home-wrecker.

18 P. Jeannin, École Normale Supérieure : livre d’or, Office français de diffusion artistique et littéraire, Paris (1963), pp. 123–124 19 È. Curie, Madame Curie, Gallimard, Paris (1938) 20 F. Giroud, Une femme honorable, Marie Curie, Fayard, Paris (1981) 21 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967)

21

Marguerite Appell (c.1901)

(G. Appell and Appell family collection)

Paul Appell (1855–1930)

(G. Appell and Appell family collection)

Marguerite at her office in the early days of the Revue du Mois (G. Appell and Appell family collection)

22

Marie Curie (1913)

(Archives of the Curie–Joliot-Curie Association, Paris (Ref. No. 149))

23

The Borels strongly supported the twice-Nobel Prize laureate. After an angry mob gathered in front of Curie’s home, they offered her shelter in their apartment, despite the opposition of Paul Appell, who asked his daughter and son-in-law to reconsider their decision. Borel was summoned by the minister of higher education and threatened with dismissal for allowing Curie to stay at government premises. With his future at the ENS in the balance, Borel stood his ground and replied: “Mme Curie shall stay at my place as long as she wishes. If I should be fired for having served a just cause, it will make me proud.”22 The minister did not insist. The story eventually went away, both in the press and among certain members of the university. It is important to emphasise the crucial role played by the Borels in this affair, at a time when Sweden was about to award Curie her second Nobel Prize for the discovery of radium in December 1911. A deep friendship developed between the Borels and Marie Curie that would last until her death in 1934.

Scientific mission to the USA A few months after the death of his mother, Borel, now aged forty, was chosen to replace Poincaré, whose health had deteriorated, as France’s representative to the inauguration of the recently created Rice University in Houston, Texas. It was an opportunity for the Borels to take some time off for themselves. They left in August 1912 on the transatlantic liner Espagne and would not be back in Paris until the beginning of winter. Rather than taking the most direct maritime route to New York (the tragedy of the Titanic had taken place only a few months earlier), they embarked on an original and colorful adventure that took them first to Cuba, then to Key West, Florida, and finally, after visiting New Orleans, by train to Houston. Borel gave two talks at the university’s inauguration: “Molecular theories and mathematics”, where he developed some novel concepts and showed the relevance of numerical analysis for the study of con22 Idem,

p. 113

24

tinuous, discontinuous and distribution functions, and “Aggregates of zero measure”, where he presented his latest results on zero-measure sets.23,24 Their journey of discovery of the New World would include other stops before reaching New York: the University of Illinois UrbanaChampaign, the University of Chicago, Cambridge, in Boston, and finally Princeton. They travelled back to Paris on the SS France, the first liner built at the French shipyards. Borel resumed his university duties, while Camille Marbo put the finishing touches on her novel La Statue Voilée, for which she would receive the 1913 Prix Femina.

A penchant for the concrete: Novel scientific and technical approaches At the dawn of the Great War, three books clearly indicated the new direction of Borel’s thought: In 1909 he published Éléments de la théorie des probabilités, in which probability and its applications become, in his own words, a tool for the experimental sciences. It was followed, in 1910, by l’Aviation, coauthored by Paul Painlevé, aimed at popularising and foreseeing the modern means of transportation. And in 1914, the release of Le Hasard provided a subtle and concrete vision of how the science of chance can be applied to numerous fields of activity. These three books seemed to mark a complete departure from the line of research he had pursued up until then. But, in fact, they were merely an expression of his personality: his penchant for concrete action, his interest in the applications of theory to real-world situations, and his search for practical solutions to complex problems in collective life. These character traits would later show up and contribute to his successful political career.

23 É. Borel, Molecular theories and mathematics, Annual Report of the Smithsonian Institution, Washington (1913); Rice Inst. Pamphlets, vol. I, May 1915, no. 2, pp. 163–193 24 É. Borel, Aggregates of zero measure, Rice Inst. Pamphlets, vol. IV, January 1917, no. 1, pp. 1–21

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Elements of the Theory of Probability This book was primarily written for “practical users”, given the number of applications to questions in physics, biology, and economics. As Borel put it in the preface, “those interested in applications do not always have the opportunity to study in depth the mathematical theories of probability; moreover, for them, these theories are of only secondary interest, since they are mostly looking for a theory of general methods”. Borel presents the basic notions of a probabilistic analysis and gives plenty of concrete examples (the Heads or Tails game, lottery drawings, and so forth). Besides these examples addressed to non-specialists, it is interesting to place Borel’s thinking in the context of theoretical physics, which in those years was making prodigious progress. Borel distances himself from Henri Poincaré’s views (for example, on the St. Petersburg’s paradox and the small planets problem) a topic so well discussed by Bernard Bru and Pierre Crépel at the 1999 Colloque in Saint-Affrique.25 To draw-in the reader, he chooses some simple cases, such as drawing balls from urns containing balls of different colours and the problem of ranking groups of students according to their level of achievement by random testing. He puts forward the idea that when the causes of an event are not well known, there is necessarily confusion in the analysis of the problem, but this analysis is essential in making decisions, as is evaluating the correlation between causes and events. Due to its huge success, the book was reprinted and augmented several times. Two other series on the same subject would follow: Traité du calcul des probabilités et de ses applications. Tome 1 à 4 (1925–1939) and Probabilités, Erreurs (1923; 1954). Finally, the well-known series “Que sais-je ?”, published by Presses Universitaires de France, would release two other titles by Borel: during the war years, Les statistiques et la vie (1943) and, only a few years before his death, Probabilités et certitudes (1950). It is also worth mentioning the series Monographies

25 P. Crépel and B. Bru, Émile Borel et le calcul des probabilités, Archives de la Maison de la Mémoire, Saint Affrique, 12400

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sur les Probabilités (Librairie Gauthier-Villars), of which he was the director, and featuring a number of authors besides Borel.

Chance Published five years later, the book is a natural sequel to the previous one, clearly addressed to a wide audience. Borel points out the relevance of the science of chance in facilitating reflection before action “by all reasonable beings”. He writes: “In the analysis of any question, common sense needs to be guided by the results of calculations; formulas do not create a shrewd mind, but they facilitate its use”. In this dawn of the twentieth century, when all of physics is still based on rational mechanics, he shows the need to embark on a true scientific revolution, since “the traditional mechanical explanations are insufficient, and must give way to statistical ones”. In those days of fundamental discoveries on radioactivity and its mechanisms, Borel presents the new concepts of quantum mechanics and irreversible thermodynamic processes. To capture the reader’s imagination, he chooses the right examples; the most famous is undoubtedly that of the monkeys typing at random. An example that is remarkable because it illustrates the notion of a practically impossible event, that is, one whose probability is extremely small, without any calculations: “Let’s imagine an army of one million monkeys, each sitting in front of a typewriter; they are trained to press the keys at random, and they work at this task ten hours per day under the surveillance of illiterate foremen. Each day, the foremen would collect the sheets typed by the monkeys and bind them into volumes. After one year, we can imagine that, by a miraculous coincidence, these volumes would contain the exact copy of all books of every kind in every language kept in libraries all over the world. . . and in the National Library”. The probability of such a “miracle”, even though extremely small, is not strictly speaking zero. Therein lies the subtlety of the argument. Other examples include the paradox of the “pile of grains of wheat”. How many grains constitute a pile? One grain doesn’t, neither do two. If we keep adding grains, we would eventually have a “pile”, although it 27

will be impossible to tell exactly at which point this happened – that is, the point at which a non-pile became a pile by the addition of a single grain. There is no logical way out of the impasse; the notion of “pile of grains” defies definition. Borel then transposes this idea to a subject he knows well: the expression of thought and the writing of a book. How many words? What should the number and the length of the sentences be? What choices are reasonable? He shows in this way that when we are confronted with making decisions, we should make them based on “statistical truths” as a way out of the logical deadlock resulting from “reasoning by continuity”. Finally, drawing on his earlier theoretical work and by an analysis of numbers in the context of set theory, he shows how to combine statistical analysis and the theory of functions, a synthesis he would later mention at his scientific jubilee in January 1940.

Aviation Borel also had a passion for aviation. The spirit of the times fostered such enthusiasm, with the exploits of those pioneers of the “mechanical genius”, in particular the French, who succeeded in building machines heavier than air that could fly ever faster and higher. Issy-les-Moulineaux, at the edge of Paris, witnessed the first flights. In 1908, the first Aeronautics Show was held in Paris. That same year, Louis Bleriot managed an eight-minute flight, and the American Wright brothers stunned the world at Avours. Eager to enlighten the general public with a better knowledge of the new technology, Painlevé and Borel published l’Aviation, as part of the “New Scientific Series”. It provides an analysis of flight mechanics based on the expertise and experience of the time, besides describing the various record-breaking airplanes that would soon be followed by the first fighter planes engaged in the Great War. A number of technical notes explain the role of each parameter, the future of certain peculiar aircrafts (helicopter, gyroplane), the importance of navigational instruments, and the ongoing research on decreasing the mass of those flying machines. There are also hints on what the future 28

of aviation will look like – non-stop and long courier flights, and air security and regulation problems. Besides treating a subject of current interest, the book fitted well with an initiative of the Faculté des Sciences in Paris, which had created an aero-technical research laboratory. In line with maintaining his commitment to the communication of knowledge before the onset of the war, Borel wrote Introduction géométrique à quelques idées physiques, a book that combines physics and mathematics. Released in 1914, it discusses the elements of relativity, statistical mechanics, and the new notions in atomic and molecular theories. He finished the book at the Saint-Paul-des-Fonts family home, where the Borels so much enjoyed receiving friends, notably the Perrins, Langevins, and Marie and Irène Curie.

29

Chapter 3

The Great War and the return to peace (1914–1924) The draft: The Front and the Ministry of Defense When war broke out in August 1914, Borel was not immediately put on active service for several reasons: First of all, his age (he was forty-three); also, the fact that since October 1911, he had been a Reservist1 and, finally, his important duties as director of the ENS, where his presence was deemed essential. But events were moving fast and the situation becoming so critical that new categories were drafted. Current and recently graduated students of the ENS left for training as Reserve sub-lieutenants, as did students from other grandes écoles. The ENS would no longer serve as a higher education institution, but became instead an auxiliary hospital as part of the effort to relieve the military hospitals in Paris. Élie Cartan was in charge of the administration. He could count on the help of several wives of university colleagues who acted as nurses; among these was Marguerite, who, in particular, cared for the morale of wounded newcomers. 1 P.

Jeannin, École Normale Supérieure : livre d’or, Office français de diffusion artistique et littéraire, Paris (1963)

During the first months of the conflict the loss among former ENS students was heavy: of the 265 enrolled at the École between 1910 and 1913, 109 were killed in action. Borel, for his part, was drafted on February 23, 1915. He initially reported to the High Command; then, one month later, to the Second Heavy Artillery Regiment. He was temporarily appointed Reserve sub-lieutenant and soon after commander of a sound localisation section on the Fourth Army front. His mission consisted of implementing the new electro-acoustic techniques to locate enemy batteries. These techniques had been developed at the ENS by the physicists Aimé Cotton and Pierre Weiss. In October 1915, Borel was awarded the Croix de Guerre avec palme for the successful accomplishment of the mission. His citation in the French Army’s “Order of the Day” reads: “He installed sound localisation stations all along the Army Corps’ front; on numerous occasions he carried out reconnaissance missions in heavily bombarded areas to set up new stations or to check the workings of the instruments”.2 That same year, Paul Painlevé entrusted Borel with the task of organising the recently created Service of Inventions of Interest to National Defense (SIDN). His new post would become an important support to the organisation of the defense. It was imperative to make up for Germany’s industrial lead – both in artillery and in combat materials in general – but first, he needed to recruit competent personnel and define the goals and objectives of the Service. Borel set up an applied mathematics and computation group headed by himself, with Charles Maurain and Henri Lebesgue as close assistants. Under difficult conditions, the varied characters and dispositions of these men were put to the test. Borel was true to his reputation as a quick and efficient organiser, while Lebesgue, a solitary man, worked in isolation and persisted in being a skeptical researcher. The letters Lebesgue addressed to Borel during those war years are most revealing; they show their often opposing reactions, and rekindle old rivalries about the paternity of certain mathematical concepts (definition of the measure of sets of points of the unit interval, zero-measure sets, and so forth).3 However, in the short 2 Ordre 3 P.

no. 408, IVème Armée, October 28, 1915 Dugac and B. Bru, Cahiers du séminaire d’Histoire des mathématiques, no. 12,

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term at least, Borel imposed his agenda so that artillery and adjustment of fire remained SIDN priorities. In 1917, as the war dragged on, Painlevé became prime minister, and Borel private secretary appointed to technical services, and later secretary general of the premier’s council. In a difficult political and military atmosphere that carried the weight of its recent past (the failure of the April 1917 offensive, the mutinies), Borel, now close to the circles of power, was confronted with the seriousness of the situation. In the autumn of 1917, the new government of Georges Clemenceau took the reins of the nation. Borel, promoted to lieutenant on October 1, 1917, and despite his age (he was now forty-six), requested to be redeployed to the Eighty-eighth Heavy Artillery Regiment. On January 1, 1918, he took the command of the Sixth Battery and once again would participate in war operations; first at Montdidier, Belgium, and later at Moreuil, France.4 He is cited in the Regiment’s “Order of the Day”: “Officer showing the highest military qualities and having the most extensive technical knowledge. His devotion and unrelenting energy were an inspiration to all his personnel. On August 28, ordered to occupy an advanced position, he carried out in the allotted time frame the particularly difficult mission despite the losses suffered under heavy bombardment”.5 He was promoted to reserve captain, and made Officer of the Legion of Honour by the minister of war in November 1918. During this period, he carried out a scientific mission for the Italian government, in particular in Rome, on behalf of the Office of Inventions. Italy had already honoured him with the rank of Commander of the Order of the Royal Crown in November 1917.6 After the armistice, Borel would retain his military status and serve in the Ministry of Industrial Reconstruction. He ceased active service on Université Paris VI, Laboratoire de mathématiques fondamentales (1991) 4 É. Borel, État des Services Militaires et feuillets individuels de campagne, dossier cote 6Ye 9935, Archives du Service Historique de l’Armée de Terre, Château de Vincennes, Paris 5 Ordre no. 393, September 9, 1918 6 For a detailed analysis of Borel’s participation in the conflict, see P. Guiraldenq, Émile Borel dans la grande guerre 1914–1918. Du scientifique au combattant, Histoire de la recherche contemporaine, CNRS Editions, tome V (2016), no. 1, pp. 21–30

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February 9, 1919, and went back to his former post as scientific director of the ENS. Although exempt from military duties, Captain Borel would remain a member of the army until 1929.

The death of Fernand Lebeau We recall Borel’s family situation. Borel’s nephew, Fernand Lebeau, became an orphan at thirteen and, in 1904, was taken in by his uncle to pursue his studies in Paris. Like his adoptive father, he was a brilliant student with a keen interest in the physical sciences,7 and in 1909 he obtained his engineering degree. After completing his second year of military service and soon after the war broke out, he was promoted to sub-lieutenant. As commander of a company, he was highly appreciated for his professionalism in a role he had not chosen. Fernand was killed in action in September 1915, at age twenty-five. News of his passing reached Marguerite Borel one month later, while her husband was at the Front. Fernand’s remains would join those of thousands of others in the graves of unknown soldiers.

Postwar years: New scientific and international activities On his return to civilian life, Borel’s grief was immense: The loss of his nephew was like a terrible open wound for him and Marguerite. The ENS, to which they were so attached, had paid a very heavy price during the war. In his post as scientific director Borel no longer felt the same commitment to his task; the spark that had animated him in the beginning had now gone: “The school became filled with shadows. Young faces that we will no longer see crop up at every corner. All of Fernand’s friends, except Deltheil (who was wounded twice) are dead”.8 Marguerite’s family had not been spared either – her brother Pierre 7 R. Deltheil in Annuaire de l’Association des anciens élèves de l’École Normale Supérieure (1917), pp. 130–133 8 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), p. 171

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was second-in-command of the submarine Monge, sunk by an Austrian cruiser along the Yugoslav coast in 1916.9 Borel could no longer bear the grieving atmosphere of the school and had lost interest in higher mathematics: “I’ll resign, even if we have to cut down on expenses”,10 he said to Marguerite. The Borels’ life then took on a new direction. They moved out of their residence at the school and rented the apartment his in-laws had vacated on rue du Bac. Three significant events took place between 1920 and 1924, in the aftermath of the war: an official trip to China with Painlevé, Borel’s appointment to the Académie des Sciences and, finally, on the scientific but also political front, Albert Einstein’s visit to Paris.

Mission to China Shortly after resigning from his post at the ENS, Borel was asked by Paul Painlevé to join him on a trip of several months to China. The former prime minister had been invited by the Chinese government as a technical advisor with the task of improving the organisation of the country’s railway and transportation systems. The invitation allowed for the possibility of adding a science education component: a series of lectures on new pedagogical methods to be given by Borel at various Chinese schools. The mission, sponsored by the ministry of foreign affairs,11 took place from June to September 1920, only a few months after the creation of the League of Nations. Their long journey was more like an expedition. They left from the port of Le Havre for New York, crossed the continent from east to west by train, and boarded a ship in Vancouver to finally arrive in China from the Pacific via Japan and Korea.12 During their stay, Borel and Painlevé shared with school authorities and teachers their rich common experience, insight, and vision on school curricula and 9 Pierre Appell (1887–1957), Toulon – Service Historique de la Marine, Musée national de la marine 10 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), p. 172 11 Paul Painlevé, Visa du Ministère des affaires étrangères, no. 494, May 4, 1920 12 P. Painlevé, Paroles et écrits, Rieder, Paris (1936)

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elementary training in mathematics. At the time, France was highly regarded as an authority on education. The province of Yunnan, in southwestern China, where Protestant churches were established by French missionaries coming from Indochina and teaching the French language,13 was most certainly one of the regions visited by the two mathematicians.14 As a token of appreciation and gratitude for his contribution to the mission’s success, the Kuomintang (Nationalist Party) government awarded Borel an official decoration, a medal stamped with old Chinese characters referring to culture and education. Following an ancient tradition, the symbols were coined in mirror image, to divert the attention of evil spirits.15

Membership to the Académie des Sciences 1921 was a year marked by the recognition of Borel’s scientific accomplishments. Besides his appointment to the mathematical physics and probability calculus chair – formerly occupied by Boussinesq and Poincaré – the fifty-year-old Borel finally became Member of the Académie des Sciences as the successor of the deceased Georges Humbert (Geometry section).16 He was elected after obtaining forty-eight of the fifty-four votes cast – Henri Lebesgue came a distant second with only four ballots. One year later, he would also serve as a member of the jury for several prestigious mathematical prizes, such as the Montyon and the Petit d’Ormoy.17 With a renewed interest in theoretical research, Borel aimed at linking probability to mathematical physics and unifying the various trends in modern physics. In a scientific note, he cautioned physicists against an overuse of a “mathematical framework that sometimes only serves to conceal the shortcomings of the hypotheses and the arguments”.18 13 P.

Raibaud, personal communication (1997) remains to be confirmed. 15 R. Vollant, Professor of Chinese at École Centrale de Lyon, personal communication (1997) 16 Élections, C. R. Acad. Sci., April 11, 1921, p. 900 17 Table des auteurs, C. R. Acad. Sci., 192 (1931), janvier–juin, p. 1836 18 Supplément (1921) à la Notice (1912) sur les travaux scientifiques de M. É. Borel, Bibliothèque Nationale no. 48038 A, p. 387 14 This

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Fernand Lebeau (1890–1915), nephew of Émile Borel

Borel in academic dress

(G. Appell and Appell family collection)

(G. Appell and Appell family collection)

Élie Cartan with nurses at l’Hôpital 103. Marguerite Appell (second from right)

(Collection of the Bilbiothèque de l’École Normale Superieure)

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Borel and Painlevé visit a school during the mission to China (1920) (Centre Historique des Archives Nationales, Paris (AP 203))

Reception for Albert Einstein in Paris (April 1922). Front row: Paul Langevin, Albert Einstein, Anna de Noailles, Paul Painlevé. Back row: Thomas Barclay, Leo Strisower, Paul Appell, Émile Borel, Henri Lichtenberger (Wellcome Library, London (Ref. No. 14796i))

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His credo became “to facilitate the discussion of the theories formulated by physicists as long as they are not in conflict with experiments, which must always have the last word”.19 The study of radioactivity was a good example of this. Finally, in line with his early ideas on the theory of functions, he established the importance of probability in the study of sets of points of the real line.

Albert Einstein’s visit to Paris Borel’s position at the University of Paris and, in particular, his kinship with Paul Langevin, led him to support Einstein’s special theory of relativity – he was one of the first in France to do so – and later his general theory. In 1922, he published L’espace et le temps, a book popularizing the subject.20 With the painful memory of the war still vivid, news of the invitation that Langevin had extended to Einstein, then professor at the University of Berlin, caused a certain turmoil. The visit therefore required some special arrangements for the transportation and the stay of their illustrious colleague. The Collège de France organised a lecture to be given by Einstein on March 31, 1922 for a scientific audience. Another was given six days later at the French Philosophical Society. Among those present at the second talk were literary figures and philosophers such as Henri Bergson, Paul Valéry, and the Romanian-French writer and poet Anna de Noailles, who was said to be captivated by geometry and an admirer of abstract mathematical research.21 The visit to Paris resulted in the creation of close ties between Einstein and the French university community, aimed at promoting the goals of the League of Nations through the International Committee on Intellectual Cooperation. Contact between Borel and Einstein would continue until the latter’s departure for Princeton in 1933. In particular, at the Institut Henri 19 Ibid. 20 É.

Borel, L’espace et le temps, Librairie Félix Alcan, Paris (1922) Tome 4. Correspondances françaises. Lettres choisies et présentées par Michel Biezunski, Seuil, Paris (1989), p. 3 21 A. Einstein, Œuvres choisies.

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Poincaré, where Einstein gave two talks in 1929, and in connection with the German edition of Borel’s book L’espace et le temps. In a letter of July 10, 1930, Borel asked Einstein to write a few words in preface to the volume, but the famous physicist, at the time afflicted with health and family problems, never obliged. In the end, it was the book’s translator, H. Schutze, who wrote an introduction with the title “Der Leser und Einstein” (The Reader and Einstein).

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Chapter 4

From science to politics (1924–1940) The beginnings of a parliamentary career Borel’s political calling came late in his life – he was fifty-three – but its roots were deep. The war had changed his outlook, and what he now wanted above all things was to participate in the reconstruction of his country, victorious but weakened. He would face many challenges: Which methods to choose? By which means? Towards which goals? How to get organised? The victory, obtained at a high cost, had awakened in public opinion a desire for a lasting peace. Borel saw in the creation of the League of Nations a glimmer of hope, and in the International Committee on Intellectual Cooperation (ICIC), formed in Geneva in 1922, a powerful lever. The executive organ of the ICIC, the International Institute of Intellectual Cooperation, was established in Paris in 1924 with the aid of the French government. Its aim was to promote an exchange of knowledge and ideas essential to peace, by cooperation with regard to scientific, literary, and artistic progress. Paul Valéry called this “politics of the mind”, a vehicle for the large-scale development of culture as the foundation for sustainable peace.

A fervently committed Borel expressed his views forcefully in the press, and later collected and revised his articles in a book with the trenchant title Organiser.1 The urgency and importance of the problems naturally pushed him towards a new mission: political activism. In the work, he identified the four pillars of his action: (1) Peace, at a time when Europe was facing the critical problem of disarmament and France’s role in it, owing to her culture (French was widely used in official communications) and her ideals. “It is imperative”, he wrote, “that all over the world, peoples suffering injustice and yearning for freedom look up to France for protection and help. It is by pursuing an idealistic policy that we shall better serve our own interests; if we had not been known, in 1914, as defenders of human rights, the world would not have come to help us liberate our country from occupation”.2 (2) The citizen, whose future he couldn’t imagine without a larger worldview. “What is happening in the United States has a direct impact on the French farmer or worker, in regard to their taxes, benefits, and wages. The day citizens of free nations will be entirely convinced that the knowledge of foreign countries is now necessary to whoever has the interests of his country – that is, his own interests – at heart, that day world peace and prosperity would have taken a great leap forward”.3 (3) Europe. He promoted the idea that the problem of the organisation of Europe “requires the gradual construction of the United States of Europe, together with, for financial and economic questions, the transfer of some degree of national sovereignty to this super state that would be the League of European Nations [. . . ]. The country that would refuse to negotiate trade agreements with its neighbours and accept some of their demands would be condemned to isolation and ruin”. Troubled to see Europe so slow in recovering its moral equilibrium, he vigorously declared that “in order for Europe to heal, we must first heal the crisis of the mind” (an expression dear to Paul Valéry).4 1 É.

Borel, Organiser, Librairie Félix Alcan, Paris (1925) p. 33 3 Idem, p. 35 4 Idem, p. 218 2 Idem,

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(4) Education. He took up a subject familiar to him, that of the education system: the role to be played by science in education and culture in general (“science allows us to better understand the mechanisms of human reason”),5 a reorganisation of the disciplines to be taught in order to establish their complementarity,6 and, finally, technical training, long neglected and deserving an honourable place. He observed that “in the economic struggle, becoming more ruthless every day, it is the quality of their technical training that will determine which nations come out on top”. He concluded with an analysis of the fundamental question of the quality of entrance examinations (to the grandes écoles) and exams in general, pointing out the need for alternative ways of gaining access to professional training. It was in January 1924 that Borel declared his intention to run for deputy. He gave an account of the circumstances in a speech addressed to the minister of industry and commerce, Eugene Raynaldy, at a banquet in Paris in October of the same year: Last January, you called on me at the Académie des Sciences and asked me to join your party’s list. I initially responded that such honour belonged to my friend Étienne Fournol, whom you had not succeeded in recruiting as a candidate. You then told me that it was the duty of those from Aveyron who had left their homeland and came to Paris not to forget their fellow citizens that stayed behind. I must admit that your plea touched me deeply, and, after consultation with my friends from Saint-Affrique, a few weeks later I accepted your invitation.7

The election campaign got underway, with its rallies in support of the leftist cartel (Radical Socialist party), and confrontations with the traditional right, through the press, speeches, and posters.8 The results of the vote, although extremely close, handed Borel the success he had hoped for. The support of miners, railway workers and, to a lesser extent, glovemakers, proved decisive to his victory.9,10 5 Idem,

p. 238 p. 298 7 Speech given by É. Borel, Raynaldy Banquet, October 22, 1924, Imprimerie de Presles (S. et O.), Archives de l’Aveyron, Rodez 8 J. Vaizy, Quand nos grand-pères votaient, Le Midi-Libre, March 21, 1993 6 Idem,

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The Ministry of the Navy

(From the author’s collection)

Borel and Admiral Escadre on board (1925) (G. Appell and Appell family collection)

44

Émile Borel

(Collection of the Hôtel de Ville de Saint-Affrique)

45

Once elected deputy, Borel’s work routine changed. Marguerite became his personal secretary in charge of departmental affairs, while he travelled between Paris and Saint-Affrique, enduring the laborious railway journeys, to visit his constituency, lecture at the Sorbonne, and participate in parliamentary sittings and committees. His life, now divided between Paris and Rouergue, was to become even more full by his taking on new duties, far removed from the scientific realm that was his initial calling, but which would provide a new balance, answering to its own dynamics.

The Ministry of the Navy and the Naval budget Prime minister Paul Painlevé appointed Borel as minister of the navy in April 1925, a post he would occupy for only six months.11 But in this short period of time he and his team managed several noteworthy accomplishments: the organisation of naval schools, the statutes of the fleet crew and the mechanical engineer corps,12,13 and the development of rules for the definition of military materials (weaponry, tests, periodic ship maintenance).14 On the aviation front (the book written with Painlevé was proof of his knowledge of the field), Borel issued a directive for the organisation of naval air forces, including the creation of personnel training centres, the conversion of cruisers into aircraft carriers,15 and new promotion criteria for pilots that would take flight hours into consideration.16 Borel also initiated a program for the construction of seven submarines, whose names reflected his affinities: Pascal, Pasteur, Monge, Poincaré, Poncelet, Fresnel, and Archimède.17 9 La

Dépêche du Midi, May 13, 1924, pp. 1, 3; Archives de l’Aveyron, Rodez Catholique, no. 111, May 13, 1924; Archives de l’Aveyron, Rodez 11 Annuaire de la Marine 1925, Imprimerie Nationale, Paris (1925), p. 47 12 Bull. off. Mar., tome 152(2éme vol.) (1925), pp. 712–784 13 Idem, tome 151(1er vol.), p. 754 14 Idem, pp. 755–831 15 Idem, pp. 723–731 16 Bull. officiel des Ministères de la Guerre et des Pensions, (1925), pp. 2338–2341 17 Flottes de Combat, Éditions maritimes & d’outre-mer, Paris (1938), p. 38 10 l’Union

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By an irony of fate, all submarines in Borel’s program, except Archimède, would be lost during World War II, or scuttled at Toulon in November 1942, at the time of the German occupation.18 Following the defense agreements concluded between France and her protectorates, notably Tunisia, in 1925 Borel was promoted to Commander (second class) of the Order of Glory (Nichan Iftikhar, in Turkish). This medal, which he received from the Bey of Tunisia, Muhammad VI al-Habib, was a symbolic reference to his mathematical work. The anagram, carved in Arabic characters, signified that the “measures” taken with the Bey came from the book of Allah and the teachings of the Prophet Muhammad.19 Well aware of the role the navy could play in international missions, Borel invited his old friend Paul Valéry for a trip of several weeks aboard one of the ships of the Mediterranean fleet. To convince his high command of the relevance of the invitation, Borel alleged the interest surrounding the upcoming appointment of the “poet of the sea” to the Académie Française. Valéry was thus host of Admiral Dumesnil aboard the cruiser Provence in July 1925.20,21 Borel defended the interests of the navy not only during his short tenure as minister, but in multiple other occasions. As a member of the finance committee, he drew up the bill establishing the Navy’s 1934 budget and spelled out the resources to be engaged on the military level.22 In the text, after a critical analysis of the maritime situation and an assessment of the fire power capabilities of the major nations’ fleets, he proposed developing maritime aviation and requested a corresponding transfer of funds from the Air Force budget to that of the Navy. He had foreseen the tactical and strategic importance of aircraft carriers in future conflicts, and his analysis of the situation, which had little effect on France’s policy in the short term, would be amply vindicated and prove formidably efficient less than ten years later in the Pa18 Idem,

(1944–1945), p. 29 Krichen, doctoral student at École Centrale de Lyon, personal communication 20 É. Borel in Paul Valéry, vivant, Cahiers du Sud, Marseille (1946), pp. 111–115 21 G. Audisio in Paul Valéry, vivant, Cahiers du Sud, Marseille (1946), pp. 174–179 22 É. Borel, Rapport sur le budget général de l’exercice 1934, Marine Militaire, Chambre des Députés, no. 2728, Session extraordinaire de 1933, pp. 1–85 19 A.

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cific.23,24,25 Borel’s thoughts cannot be better expressed than by quoting him. In 1933 he wrote: “Given its important role in searching, fighting, reconnoitring and protecting ships, maritime aviation must be closely linked, by traditions, usages, combat methods, and commandment, to other naval forces, whose operations it amplifies and completes”.26 He emphasised the need to replace as soon as possible the outmoded fighter planes with more powerful and better adapted ones, and not to rely only on the single existing aircraft carrier, Bearn. He also insisted on the urgency to ensure that future navy officers were properly trained by completing the construction of the Naval School, the ministry of the budget having foolishly withdrawn its funding.

Life as a member of parliament: His three terms between the two wars The national issues of particular interest to Borel arose from his participation in various parliamentary committees: external affairs, military navy, mines, and education and fine arts. On each of these he worked incessantly, tirelessly, and enthusiastically. Camille Marbo quotes him: What I enjoy, is having lunch on Sunday with Rigaud (the mayor) in Broquiès (a commune in the Aveyron department) while we drink his favourite wine, and the day after going to the Élysée to have a chat with [Gaston] Doumergue (the president). In my opinion, a deputy must be the link in the chain that connects the humblest citizens to those in power and to the administrations. It is important to give proper consideration to the interests of common people, while at the same time work on the big issues, on which the fate of France, and even of Europe, depends.27 23 V. A. Le Pichon, l’Aviation embarquée et son histoire, l’Armement magazine, no. 51, March 1996, pp. 61–65 24 Amiral P. Barjot, Histoire de la Guerre Aéro-Navale, Flammarion, Paris (1961) 25 J. J. Antier, Histoire de l’Aviation Navale, De la Cité, Brest (1983) 26 É. Borel, Rapport sur le budget général de l’exercice 1934, Marine Militaire, Chambre des Députés, no. 2728, Session extraordinaire de 1933, pp. 1–85 27 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), p. 219

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First term (1924–1928)28 As a member of the National Council for Scientific and Industrial Research, one of Borel’s priorities was to develop scientific research. Funding for it would come from a portion of the taxes levied on production, the so-called “Laboratory Cent”. During this term, other topics affecting the lives of ordinary people were a source of concern: monetary stability, unemployment, and the high cost of living. He was also aware of the need to prepare the country should the lightning of war strike again; in particular, the recruitment for military service of students from the grandes écoles, a delicate situation of which he had first-hand experience.

Second term (1928–1932)29 Re-elected, but barely, in April 1928, he joined the group of independent leftists. He continued his efforts in favour of education and culture, and proved particularly innovative on the social domain by proposing a law to ensure effective representation of the “feminine element” in all municipal councils, a project that would only gradually see the light. His sense of precision and concern for rigour prompted him to suggest, as a way of improving and modernising voting procedures in the Chamber, the installation of electrical devices to guarantee the accuracy of vote counting and results.

Third term (1932–1936)30 Once again re-elected by a slim margin, after an epic campaign carried out before difficult and often tumultuous crowds,31 Borel joined the Republican Socialists group. Given the international context, in particular 28 Ann. Chamb. députés, Débats parlem., 13e Législature (1924–1928), Table nominative, Impr. Chambre des Députés, Paris, (1930), pp. 93–96 29 Ann. Chamb. députés, Débats parlem., 14e Législature (1928–1934), Table nominative, Impr. Chambre des Députés, Paris (1934), pp. 117–120 30 Ann. Chamb. députés, Débats parlem., 15e Législature (1932–1936), Table nominative, Impr. Chambre des Députés, Paris (1937), pp. 96–98 31 J. Vaizy, personal communication, arising from a local press article (Saint-Affrique)

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Hitler’s rise to power in Germany, he took part in shaping government policy, notably in national defense and the need for the creation of a national economic council. In 1934, when France was experiencing a very difficult period, Borel might have appeared to support extreme left views while in fact he was only a socialist radical, but devoted to the hierarchy of the grandes écoles despite being a humanist. The terrible February riots and their devastating consequences for the country’s social fabric were at the heart of a persistent economic crisis. Borel was appointed member of the commission of inquiry looking into the origin of those social movements. Having put so much hope on scientific progress as a vehicle for social justice and peace, he realised with dismay the pitiful state of Western Europe and the world.32 True to his beliefs, the events of 1936 – the victory of the Popular Front (an alliance of left-wing movements) and the Socialist leader Léon Blum becoming prime minister – pushed him to resign as deputy and to cut down on his activities. As Camille Marbo relates, 1936 was for Borel a year of reorientation and choices. He complained that “he was too old for taking up tasks imposed to conscientious deputies”.33 He therefore did not seek re-election and supported instead a little-known newcomer to the region,34 who was defeated by the conservative candidate. Borel was confronted with the same problem any scientist embarking on an administrative or political career must face: the difficulty for someone in a “hard” discipline such as mathematics to take on new responsibilities – for instance, director of the Institut Henri Poincaré, which we discuss in the next chapter – while at the same time fulfilling their duties as deputy, both in the field with his constituents and in the Chamber in Paris. For Borel, 1936 was nevertheless a year in which he was honoured by his country: He was appointed Commander of the Legion of Honour (External Affairs), in recognition for his international actions in favour 32 É. Borel, La crise économique et la science, La Revue de Paris, April 15, 1931, pp. 756–68 33 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), p. 285 34 J. Vaizy, personal communication, arising from a local press article (Saint-Affrique)

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of peace, in the framework of the League of Nations and as president of the French Committee for European Cooperation.35

Mayor and councillor Borel’s electoral success in 1924 and 1928, and the enthusiasm and encouragement of his friends from Saint-Affrique, undoubtedly influenced his (double) decision to run for mayor of his native town and for councillor of the commune of Cornus, in the Aveyron department. He scored a victory on both fronts, but, faced with new responsibilities combined with his already heavy agenda, the management of his personal time became a real challenge. Borel had to split his time between Paris (at the Sorbonne, the Chamber of Deputies, the Académie des Sciences) and Rouergue (at his deputy and mayoral offices); not to mention his numerous scientific missions abroad and his work for the League of Nations in Geneva. As in the days of the Revue du mois, Marguerite was of invaluable help with administrative and social tasks. Borel was elected mayor in May 1929. Among the main projects completed during his first term were the tarring of roads, the construction of a school for girls, a modern sewage system, and a one-hundred-bed hospital-hospice. The latter, built in an original style inspired by the colonial empire, was named after the mayor: Centre Hospitalier Émile Borel. For those who enjoyed playing boules, a well-established tradition among Saint-Affricans, he had a playing ground built on municipal land. Inaugurated in the summer of 1936, it would be known as Boulodrome Borel.36 In those difficult times of the Third Republic, Borel regularly wrote about national and international politics in the pages of the local newspaper, Le Progrès. In 1933, for instance, he didn’t hesitate, despite his profound desire for peace, to denounce Hitler’s rise to power as a frightening menace, and to warn his fellow countrymen: “In his propaganda, he calls himself national-socialist, seeking an equal balance between 35 J.

M. Guieu, Émile Borel et la coopération européenne, Bull. de l’Institut Pierre Renouvin, (1998), no. 5, pp. 13–32 36 F. Decuq, Rétrospective de Saint-Affrique et de sa région, Maury, Millau (1971)

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both adjectives characterizing his party. On the national front, he has made the most extreme demands, including the abolition of the Treaty of Versailles. On the socialist front, he has made fabulous promises to workers and lower-class employees. The big question will be to what extent he will try, once in power, to honour the promises he made to get there. If he wanted to make good on his national program of a Great Germany, he could only succeed by immediately resorting to war. That is a danger we cannot close our eyes to, and which will require our full vigilance if we are to foil it”.37 Borel’s second term as mayor, from 1935 to the 1939–1940 war years, coincided with one of the most painful political periods, due to the general deterioration of the situation, the so-called “funny war” followed by France’s surrender, and the beginning of the German occupation. Throughout those distressing years, Borel reassured and alerted his fellow citizens on subjects that had become crucial: the organisation and exploitation of statistics as a bookkeeping tool that would allow the country to operate like an enterprise;38 training and professional career counseling for the youth;39 the need to support a national agricultural industry and set up agricultural credit mechanisms;40 and the importance of applied research for national defense.41 In 1939, general mobilisation was called, and Borel, in his capacity as mayor, saw it as his obligation to support the civilian population’s morale: “Those who have been mobilised – he wrote – are ready to fulfill their duty to defend our soil and our freedom. Those staying behind will also do their duty, so that the life of the country can go on, and the necessary aid and protection to the children, the sick, the wounded, and the refugees is provided”.42 It was during those feverish times announcing the second world conflict that Borel lost his younger sister Marthe. 37 É.

Borel, Hitler au pouvoir, Le Progrès Saint-Affricain, March 25, 1933 Progrès Saint-Affricain, December 31, 1938 39 Idem, May 21, 1937 40 Idem, June 4, 1938 41 Idem, September 17, 1938 42 Idem, September 3, 1939 38 Le

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Saint-Affrique became a refuge town. Court and hospital rooms were transformed into dormitories and all efforts were deployed to accommodate the large number of refugees fleeing the regions taken over or threatened by the invaders.43 After the rout and the capitulation, the Vichy regime – a nominal government of a large part of France that sought to preserve sovereignty by collaborating with Hitler – removed Borel from his position as mayor.44 He would be reinstated after the war for a period of two years. Borel addressed the Aveyron General Council for the last time on September 29, 1950. He stressed the importance of having a strong representation of France in European international councils, but he also insisted on the “moral and material” interests of his beloved Rouergue.45 Under the weight of his 80 years, he would be slowly forgotten by this illustrious assembly. The following year marked the end of his long and devoted service to the people of his native community.

43 Idem,

September 16, 1939 Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), p. 298 45 Archives Départementales de l’Aveyron, Conseil Général de l’Aveyron (1950), September 29–30, 1950, pp. 1–3 44 C.

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Chapter 5

Scientific activity between the two wars (1925–1940) The early years of the Institut Henri Poincaré In a new era for modern physics, the fact that Borel held the chair of probability and mathematical physics at the Faculté des Sciences in Paris may explain his interest in the creation of an institute that would bring together mathematicians and physicists. At the beginning of 1926, the American mathematician George Birkhoff came to Paris as a representative of the International Educational Board, an organisation incorporated in 1923 with funding from John D. Rockefeller and dedicated to the promotion and advancement of education throughout the world. Birkhoff’s mandate was to explore possible ways of promoting mathematical physics in the country, in consultation with his French colleagues. Louis de Broglie and Émile Borel were among these. It was Borel who came up with the idea to create a research centre in mathematics and theoretical physics. His initiative was approved by the Board and a 100,000-dollar grant was awarded to the project, together with 25,000 dollars donated by Edward de Rothschild. Borel proposed to name the new centre Institut Henri Poincaré (IHP). He became its natural director in 1928 and would remain at the helm until his death in 1956.

Construction began in 1926 on the university’s grounds bordering rue Pierre et Marie Curie. Under Borel’s watchful eye, the building was completed in 1928 with a substantial budget surplus. This sum was allocated to the Faculté des Sciences as a contribution towards its operating costs.1 The institute, inaugurated on November 17, 1928, had a dual mission: teaching the calculus of probability and carrying out research in theoretical physics at an international level, with the possibility of inviting each year well-known scientists to present their work.2,3 In a letter of November 2, 1928, Borel invited Einstein to the inauguration, and he added: “It goes without saying that we would be particularly delighted to count you among the institute’s first guest lecturers. We are pleased to offer you a remuneration of one-thousand francs per talk, their number and topics being entirely left to your discretion. I hope it will be possible for you to give at least a dozen lectures, in the first months of 1929”.4 Einstein eventually accepted the invitation and gave two talks at the IHP, on November 8 and 12, 1929. In the 1928–1929 academic year, Borel delivered a series of lectures on the theory and applications of probability, while Maurice Fréchet, then a senior lecturer, taught modern results on the principles of probability theory. Léon Brillouin, holder of the physical theories chair, lectured on quantum theory, and Louis de Broglie on wave mechanics. Very soon, famous names were invited to teach and give talks at the institute. Among these, Vito Volterra from Rome (biological dynamics) and Enrico Fermi (quantum mechanics). Borel also invited colleagues from other French universities and abroad, such as George Darmois from Nancy (statistical laws, correlation), and Vladimir A. Kostitzin from Moscow (applications of integral equations).5

1 Courrier

du Recteur de l’Académie de Paris au Doyen de la Faculté des Sciences, Archives de l’Insitut Henri Poincaré, November 20, 1930 2 Nominations, C. R. Acad. Sci., November 12, 1928, p. 867 3 M. Fréchet, The inauguration of the Institut Henri Poincaré in Paris, Bull. Am. Math. Soc., 35 (1929), pp. 198–200 4 Albert Einstein Archives, Hebrew University of Jerusalem, Israel 5 Cours et conférences, Archives de l’Insitut Henri Poincaré, 1928–1929

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Other guest lecturers included Paul Dirac from Cambridge (statistical bases of quantum mechanics, relativistic electron theory); Chandrasekhara Raman from Calcutta (molecule structures); Max Born from Göttingen (problems in quantum mechanics); and Pierre Debye from Leipzig (strong electrolytes). Later would also come Charles-Jean de la Vallée Poussin from Louvain (potential theory, Dirichlet’s problem); Arnold Sommerfeld from Munich (photoelectricity, X-ray spectrum); and Erwin Schrödinger from Berlin (electron theory),6 among others. At Borel’s initiative, some lectures he considered particularly remarkable were published in the Annales de l’Institut Henri Poincaré. Over the course of time and under Borel’s efficient stewardship the Institut Henri Poincaré would become an international reference. Thanks to some extension work that increased its capacity to receive students, the IHP became a school associated to the University of Paris VI, while remaining the “home” of mathematics and theoretical physics, and a showcase for several learned societies. The Centre Émile Borel, one of IHP’s historical services, organises thematic programs and the “Research in Paris” program. The institute also housed an important archive and documentation centre. Borel jealously watched over this treasure trove of archival material, to ensure its future preservation.7,8 The creation and subsequent development of the IHP marked a significant stage in Borel’s professional life. Unfortunately, Paul Appell, who had always been a source of encouragement for him, passed away in October 1930,9 too soon to be able to witness the full blossoming of his son-in-law’s career.

University life at the Sorbonne We have seen that over the years, and by his teaching style, Borel had influenced and made an impression on many students, first at the ENS, 6 Idem,

1930–1931 de l’Ing. Général Wilmet à É. Borel, Archives de l’Académie des Sciences, September 10, 1953 8 Courrier réponse de É. Borel, Archives de l’Académie des Sciences, September 26, 1953 9 Le Progrès Saint-Affricain, November 1, 1930 7 Courrier

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and later at the Faculté des Sciences and the IHP. His teaching method was not the mere transmission of the course content, but rather a series of analyses leading the audience towards an approach favouring research. However, it does not appear that Borel “had particularly advised” the rare ENS student specializing in mathematics to choose a subject in probability for the thesis, corresponding to what he had been teaching since 1914, and later taught from 1920 on.10 The war had in effect changed Borel’s perspective, and he was now instinctively oriented towards scientific applications and team work. His strong preference for directing journals and other publications with the help of large teams of assistants and collaborators paints a better image of him than that of a typical boss working with a handful of subordinates. Engaged, as always, in the pursuit of new challenges, he participated in a project of the Faculté de droit initiated by Fernand Faure: the creation of the Institut de Statistiques de l’Université de Paris (ISUP),11 while being actively involved in thesis supervision and jury duty. Preparing and defending a thèse d’État (more or less equivalent to a PhD) was a drawn-out task, and often a lifelong one, since there was no time limit for its completion. From 1904 to 1940, Borel took part in some forty PhD juries, as examiner, reviewer – the member of the jury responsible for writing a report on the candidate’s dissertation – or president.12 The long list of candidates, generally in the field of the theory of functions, includes, before World War I: Maurice Fréchet (“On certain aspects of functional calculus”, 1906), Paul Montel (“On infinite sequences of functions”, 1907), and Arnaud Denjoy (“On canonical products of infinite order”, 1909). Among the members of the juries, we find, besides Borel, Appell, Painlevé, Picard, Drach, and Hadamard. After the war, the theses defended were notably those of Robert Deltheil (“On the theory of geometric probability”, 1920), Yves-André Rocard – a physicist who helped develop the atomic bomb for France – (“Hydrodynamics and the kinetic theory of gasses”, 1927), Henri Cartan 10 B. Bru and P. Crepel, Colloque Émile Borel, Archives de la Maison de la Mémoire, Saint-Affrique, July 16–17, 1999 11 Ibid. 12 Service des Thèses, Archives de l’Université Paris VI

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(“On systems of holomorphic functions and their applications”, 1928), and Francis Perrin (“A mathematical study of rotating Brownian motion”, 1928). The authors gratefully acknowledged Borel’s help and guidance on various topics in pure mathematics and theoretical physics. There were also theses prepared outside France but defended at the Sorbonne that confirmed the role played by Borel at IHP and his international influence. Among those defended in the 1930s, let us mention in particular Wolfgang Doeblin (Berlin, 1932) and Djuro Kurepa (Belgrade, 1935). It was in the early 1930s that the famous Nicolas Bourbaki group was born, founded by André Weil, Henri Cartan, Jean Dieudonné, Jean Delsarte, and Claude Chevalley.13 This peculiar team of young mathematicians intended to carry out a vast synthesis and unification of modern mathematics in a collective and anonymous fashion. The rules governing the group were strict; in particular, they required that members must be less than fifty years old – at the time of the group’s founding Borel was sixty-three. As their project progressed, the Bourbakists would make reference to Borel’s seminal work (the notion of Borel tribes, and so forth); meanwhile, even if he favoured basic research, Borel was involved in a program of application-oriented books and older popular editions: L’application du calcul des probabilités aux jeux du hasard (1938), and Valeur pratique et philosophique des probabilités (1939), among others. An original work, Théorie mathématique du bridge à la portée de tous, published in 1940 and co-authored by André Cheron – a French chess player and endgame theorist – was further proof of Borel’s strategist’s mind. He introduced here the notion of “subjective probability” (which is not the same for all players), as opposed to objective probability, associated with random distributions – Bruno de Finetti gave a talk on this topic at the IHP. Borel’s Traité du calcul des probabilités, co-authored by Jean Ville and published in 1939, was an important contribution to the further development of game theory by John von Neumann.14 It was during his final years of university activity that 13 M. 14 J.

Chouchan, Nicolas Bourbaki : faits et légendes, Du Choix, Argenteuil (1995) von Neumann and O. Morgenstern, Theory of games and economic behavior,

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Borel was president of the jury for the theses of Gustave Malécot and Jean Ville (1939), and Raphaël Salem (1940). The latter was exceptionally excused from the oral examination before the jury, probably due to the disorganisation provoked by the chaotic situation the country experienced in June 1940. After the 1939–1945 war, Borel produced a remarkable synthesis of set theory – a concept at the heart of the Bourbakist project – expressed in a simple and accessible language that would be the reason for its success. The International Exhibition of Art and Technology in Modern Life was held in Paris from May to November 1937. For the occasion, Jean Perrin, then state secretary for scientific research, commissioned the building of a science museum known as Palais de la Decouverte (Discovery Palace). Wishing to help Perrin and in the interest of science Borel delivered, to the delight of large audiences, a number of talks in mathematics, because “abstract thought cannot be displayed in a showcase”, as he put it. With the use of some anecdotes and numerical calculations, he presented “mathematical recreations”, similar in spirit to the popular “Science Days” held by the Centre National de la Recherche Scientifique (CNRS).15 With the aim of establishing research as a career, and convinced of the need for a permanent structure at the national level even if “returns do not cover expenses”,16 Borel helped those in charge, and Perrin in particular, to create the CNRS. The new organisation would see the light six weeks after the beginning of World War II in October 1939. Its goal was to merge into a single body several existing organisations, such as the National Fund for Scientific Research (offering student bursaries and created ten years earlier, when Borel was a deputy), the council managing research grants, and the National Bureau of Scientific Research and Inventions. Resorting to concrete images to get his message across, Borel insisted on numerous occasions on the need to promote science among the Princeton University Press (1944); third ed. (1967) 15 É. Borel, Œuvres, vol. 4, CNRS, Paris (1972), pp. 2345–2350, 2357–2361 16 Ibid.

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younger generation, in the hope that one of them may one day become a Louis Pasteur: A single prestigious success, whose benefit would be immense, would be worth the large number of almost useless, but necessary, expenses.

To that end, he demanded that the recruitment of young mathematicians be maintained, because, he continued: Progress is only possible if the tree of science keeps on drawing its roots from the mathematical sap, essential to the life of all of science.17

Two presidencies: Académie des Sciences and the Institut A highlight in Borel’s career was about to follow when he was elected vice-president of the Académie des Sciences in 1933, since the holder of that position automatically becomes president the following year. In 1934 it was sciences’ turn to preside over the Institut de France, a learned society comprising five academies. Borel therefore became president of the Institut for one year. In his opening address,18 he emphasised two points: The progress of science and the organisation of science should proceed simultaneously without interfering with each other. He mentioned to the audience Jean Perrin’s initiative in favour of the creation of a research council, with the Academy playing a major role in it – those efforts would bear fruit in 1939 with the founding of the CNRS. The realisation that despite the benefits of science there was a crisis, and that some turbulent and dissatisfied spirits tended to blame scientists, inventors, and laboratories for unemployment and misery. He strongly reacted against those claims and the mistake of shifting responsibility, while acknowledging the difficulty for scientific progress to contribute to improving material existence through a better organisation of production and better management of working time. 17 Ibid. 18 C.

R. Acad. Sci., January 3, 1934, pp. 18–20

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Émile Borel

(Archives de l’Académie des Sciences, Paris; from the author’s collection)

Entrance to the Institut Henri Poincaré (From the author’s collection)

Émile Borel and Vito Volterra

(G. Appell and Appell family collection)

Jean Perrin, 1926

(Collection of the Bibliothèque nationale de France)

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He also insisted on the need to bring about a greater organisation, circulation, and distribution of wealth. But he was aware that those economic questions, unfortunately too often blended with political ones, were beyond the Academy’s jurisdiction. And he concluded with a wish, today more or less forgotten: “To serve science means first and foremost the unselfish quest for truth, which must also be the fruitful source of material and moral progress. . . ” As a recently elected deputy in 1925, Borel had proposed the socalled “Laboratory Cent” bill – later voted into law – to fund scientific research and laboratories with a tax of five cents per one-hundred francs on all industrial and commercial wages.19 In the years 1933 and 1934, Borel’s life also had its share of sorrow: the death of his old and dear friend Paul Painlevé, and that of Marie Curie eight months later. Given his official position, it fell upon him to deliver the eulogies of his two colleagues. Painlevé, exhausted by his work and still at the head of the ministry of aviation, passed away on October 29, 1933. The next day, Borel paid him an eloquent tribute before the Académie des Sciences and summarised his career: the strength of the man, a man of action whose life was devoted to the service of his country; and on the scientific front, his approach, which was a model for Borel, of constantly bringing together theory and applications.20 Marie Curie died on July 3, 1934. It was again Borel who informed the Academy and six days later paid homage to her exceptional career: her two Nobel Prizes, in 1903 and 1911, and the countless titles and honours she received from all over the world.21 He reminded members that in 1911 her candidacy had encountered a strong opposition on the very principle of the admission of women to the institution, and that she had made it a point of honour never to apply again. He did not fail to mention her role during the war – she not only invented the radiological car but also drove one to the front herself – nor her work in the Committee on Intellectual Cooperation of the League of 19 C.

R. Acad. Sci., March 2, 1925, p. 629 R. Acad. Sci., October 30, 1933, pp. 953–955 21 È. Curie, Madame Curie, Gallimard, Paris (1938) 20 C.

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Nations, where she passionately pushed in favour of education and the international organisation of research, alongside Painlevé, Lorentz, and Einstein.22 In an almost symbolic gesture, and as a way of making their friendship last, Borel sold part of his Saint-Paul-des-Fonts property in order to purchase Marie Curie’s house, “La Crête”, which she had built in Cavalaire. The picturesque villa, at the top of a hill overlooking the town and the Hyères islands, would become the place where the Borels often went to rest and to enjoy the charms of Provence. Besides the scientific notes he published in the Comptes rendus,23 Borel was keen to participate in award juries, often for prizes far removed from his original field of interest – such as the Plumey Prize for navigation, and the Binoux Prize for the history and philosophy of science – as if he hadn’t forgotten that he had been a young prizewinner himself, and the importance of rewarding talent and merit.

The scientific jubilee: Fifty years of research and teaching The ceremony, which should have taken place in 1939 to celebrate Borel’s fifty years of scientific activity (he had entered the ENS in 1889), was postponed to July 14, 1940. The delay, due to the imminent threat of war, resulted in a number of foreign guests being unable to travel to Paris to participate in the festivities (the “family reunion” wished for by Borel) in the great halls of the Sorbonne. A number of eminent personalities and former students of Borel’s addressed the audience.24 Among these, Louis de Broglie, Nobel laureate in physics, Michèle Vergne, president of the Mathematical Society, Gaston Julia, Henri Cartan, and Torsten Carleman, director of the MittagLeffler Institute, on behalf of foreign mathematicians. 22 C.

R. Acad. Sc., July 9, 1934, pp. 105–107 Borel, Œuvres, vol. 1, CNRS, Paris (1972) 24 Jubilé scientifique de M. Émile Borel (January 14, 1940), Gauthier-Villars, Paris (1940) 23 É.

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A number of French political, military, literary, and mathematical celebrities participated as members of the organising committee: Prime Minister Édouard Herriot, Irène and Fréderic Joliot-Curie – joint winners of the 1935 Nobel Prize in chemistry – poets Paul Valéry and André Breton, Paul Langevin, Henri Lebesgue, Jacques Hadamard, and André Lichnerowicz; and, of course, distinguished foreign colleagues: John von Neumann from Princeton, John Littlewood from Cambridge, Wacław Sierpiński from Warsaw, Ivan Vinogradov from Moscow, and Harald Cramér from Stockholm. Borel was presented with a book25 containing a collection of long excerpts from his most significant works, in the theory of functions, the calculus of probability, and mathematical physics, as well as articles on his ideas about research and invention. In his address, after giving thanks for the gift, Borel reflected upon the development of his career, shaped from the beginning by the ENS. He also expressed his profound friendship for the foreign colleagues that could not be present, Vito Volterra, Wacław Sierpiński, and Ernst Lindelöf among these. His thoughts then turned to those French soldiers sent to the front, conveying his hope that the new German aggression would not resemble that of twenty years earlier. The bitter memories of the years 1914–1918 were still vivid, and he feared the consequences of an untenable situation, as his career entered its twilight.

25 Selecta :

jubilé scientifique de M. Émile Borel, Gauthier-Villars, Paris (1940)

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Chapter 6

A long and final march towards the future (1940–1956) Dark years under occupation and new hopes Efforts amid the collapse As the war broke out, laboratories housed in the IHP building were requisitioned in order to provide scientific assistance to the French army. A similar initiative was implemented at the laboratories of Jean Perrin, Irène and Fréderic Joliot-Curie, and Paul Langevin. In January 1940, Louis de Broglie noted that “the house” (that is, the IHP) had somewhat changed: a new type of personnel is around; one often sees uniforms, and conversations are almost as much about canons as about electrons. A most surprising transformation, but no doubt necessary in the circumstances.1

Alas, a few months later the country collapsed: the enemy broke through the Maginot line – fortifications built to deter invasion by Germany – and the retreat degenerated into a rout, followed by a mass exodus of the population in confusion and fear. 1 L.

de Broglie in Jubilé scientifique de M. Émile Borel (January 14, 1940), GauthierVillars, Paris (1940), pp. 16–23

On June 17, Marshal Pétain announced on the radio that the fighting must stop. General de Gaulle’s appeal the next day restored the confidence of those who had chosen to resist. Borel explained the French Academy’s and the other academies’ reaction on this issue, in particular by a courageous speech from Paul Valéry. The poet, who after the Montoire accord between Pétain and Hitler – marking the beginning of the collaboration with Nazi Germany – had categorically refused any kind of traitorous cooperation with the enemy.2 The Borels retreated to the family home in Saint-Affrique, where they provided shelter to family, friends, and refugees. Among these, Geneviève Appell, their niece, and Bernard Langevin, who had put his life at risk through actions against the occupying forces. On May 12, 1941, the government of Vichy removed Borel as mayor of his home town.

The Fresnes Prison Back in Paris, Borel resumed his duties at the Académie des Sciences and participated in the selection of candidates for various prizes to be awarded in 1941. But events rushed on, and a search of his house on rue Froidevaux by the Gestapo resulted in his arrest and detention in the Fresnes prison in November 1941. The same fate befell the physicist Aimé Cotton,3 just as Paul Langevin before them, who after being detained for several months was sent on house arrest.4 Borel’s internment at the beginning of the winter, in isolation and under unsanitary conditions, was extremely painful. Set free after over one month in prison and weakened by his ordeal, Borel’s recovery was slow – he was now seventy. Camille Marbo recounts how those days and nights in prison took a toll on his morale; unable to read and write freely, he could only surmount them thanks to his capacity for concentration and abstraction.5 2 É.

Borel in Paul Valéry, vivant, Cahiers du Sud, Marseille (1946), pp. 114–115 Cotton, Œuvres scientifiques, CNRS, Paris (1956), p. 9 4 L. de Broglie, Notice sur la vie et l’œuvre de Paul Langevin, Institut de France, Académie des Sciences, Gauthier-Villars, Paris (1947), p. 33 5 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), pp. 299–301 3 A.

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Retreat to Saint-Affrique A new arrest of Borel in Paris during the first semester of 1942, followed by his release the same day, pushed the Borels to leave the capital’s poisonous atmosphere in October 1942 and retreat to Saint-Affrique, this time for good. The town was located, but not for long, in the free zone – the unoccupied southern part of France. It was during this inglorious period in France’s history that several of his friends and colleagues passed away. Among these, Henri Lebesgue and Émile Picard, both in 1941, and Jean Perrin, who died in New York in May 1942. From his retreat, Borel kept in touch with and offered his help to young students who had still been spared compulsory work service – the enlistment and deportation of French workers to Nazi Germany to work as forced labour for the German war effort. One of these was the young André Viales, whom he advised to enrol at Clermont-Ferrand, where the best Bourbakists of the Faculty of Sciences of Strasbourg – Henri Cartan, André Weil, and Laurent Schwartz – had found refuge. After successfully sitting the exams, Viales sent the questions to Borel, who dissected them in record time despite his age.6 This episode illustrates the old man’s power of abstraction and mental agility, and his capacity to analyse the most complex examination questions. It was also in the shadows of the occupation that Borel pursued his reflections on probability and wrote short popularisation books: Le jeu, la chance et les théories scientifiques modernes in 1941; Les probabilités et la vie in 1942, in which he shows his perfect knowledge of the demographic situation confronting medicine; also, in 1942, La mécanique et la gravitation universelle, another subject he was passionate about; and, finally, l’Évolution de la mécanique in 1943. Borel wrote ceaselessly, but he also remained concerned with the situation in the country. Not only was the enemy still there, but German military presence had increased. The Resistance and its networks must be helped. Clandestine cells were organised throughout the region, ready to act on orders received through coded messages. Whoever lived in 6 A.

Viales, former inspector general for mathematics, personal communication

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France in those days cannot have forgotten the voice of hope coming on the radio airwaves from across the English Channel and reaching the depths of a basement or an old kitchen: “Ici Londres”, sending secret messages for the free French forces, or the radio program “Les Français parlent aux Français”. All those efforts paid off: on June 6, 1944, the allied forces landed in Normandy, then in Provence on August 15. Rouergue was liberated, but at a heavy price; the white crosses at La Pezade cemetery remind us even today of the sacrifice of those young men.

The return of peace A local Liberation Committee, to which Borel belonged, was charged with managing current affairs and planning Saint-Affrique’s future. The local newspaper’s printing presses were operating again, and Borel’s timely articles regularly featured in its pages. In the winter of 1944, with fighting still going on in a France not completely liberated, Borel set out on a mission to London to prepare for peace, but without losing sight of his main goal: “We must count on ourselves to restore our own ruins; we must also put an end to the war.”7 Reinstated as mayor in May 1945, Borel was once again honoured: he was made Grand Officer of the Legion of Honour by the President, Félix Gouin, on March 4, 1946;8,9 he also received the Médaille de la Résistance, avec rosette (Medal of the Resistance, with rosette) for his exemplary behaviour and his actions during the occupation.10 In Paris, Borel had to deal with an increasingly demanding workload. He had resumed the direction of the IHP, and regularly participated in the meetings of the Académie des Sciences. He was also a member of the Central Committee of the Human Rights League and of the national council of the CNRS. New appointments were added to this already heavily-charged agenda: to the Longitudes Bureau, and, as a technical 7 Le

Progrès Saint-Affricain, December 23, 1944 officiel de la République française, March 4, 1946 9 Le Progrès Saint-Affricain, March 30, 1946 10 Ordre de la Libération, Commission Nat. de la Médaille de la Résistance, courrier, December 16, 1994 8 Journal

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expert, to the recently created Atomic Energy Commission headed by Fréderic Joliot-Curie.11

Post-war period, c. 1950s: Promoting science outreach The Longitudes Bureau In 1946, Borel was appointed as permanent member of the Longitudes Bureau.12 This research organisation, associated to the Académie des Sciences, dated back to 1793 and the National Convention – the first government of the French Revolution. Originally, its aim was to improve the accuracy of maps used by French ships navigating the seas and oceans of the world. It was also the place where nautical ephemerides were constructed. Over time, the organisation’s mandate was expanded to astronomy, and therefore to works in celestial mechanics and the analysis of the motion of bodies in space. Universal gravitation theories and the complex calculations involved resulted in new and unexpected discoveries about the cosmic world. Working at the Longitudes Bureau in the 1950s was for Borel the source of stimulating reflections and presented him with new opportunities to write. Besides his practical interest in meteorology and climatology, he published Les paradoxes de l’infini in 1946, Probabilités et certitudes in 1950, and L’imaginaire et le réel and Les nombres inaccessibles in 1952. These four books are the final expression of his worldview. Taking a historical path, they show the magic of numbers and their combinations in probabilistic terms. Borel seeks to demonstrate, by a scaling effect, how the power of the human mind can develop in the abstract and the imaginary, beyond the “terrestrial measurable quantities”.

11 N.

Loriot, Irène Joliot-Curie, Presses de la Renaissance, Paris (1991) du Bureau des longitudes, communication du 17-2-1997 de Y. de Kergrohen, Secrétaire administratif : arrêté de nomination par le ministre de l’Éducation de M. Émile Borel (11 juin 1946) 12 Archives

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Cavalaire harbour

(From the author’s collection)

Émile Borel at his home office in Paris (1950s) (G. Appell and Appell family collection)

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Émile Borel (c.1945–1950)

(G. Appell and Appell family collection)

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The return of Jean Perrin In 1941, after France’s surrender to Germany, Jean Perrin saw the Borels for the last time, before leaving for the United States with the purpose of creating a French free university in New York. However, he would not be able to see this original and courageous project through: He died one year later, on April 17, 1942, away from his homeland. After the end of the war, France sought the repatriation of his remains to deposit them in the Pantheon, the Temple of the French nation, and so honour his memory. Perrin’s ashes arrived in Paris in June 1948. For the occasion, an official ceremony was organised at the Sorbonne, where Borel delivered an address on behalf of both the Académie des Sciences and the University of Paris.13 He spoke of Perrin’s scientific works and the reasons for his departure to the United States seven years earlier. He explained how Perrin, eager to serve his country in the best possible way, hesitated between remaining in France, where the Resistance was beginning to get organised, and emigrating to the States. The choice of New York imposed itself on him “in order to enlighten American opinion on the true feelings of the French people towards the German occupation and the Vichy regime”. Perrin expected in this way to encourage the United States to take up arms and once again come to the rescue of the country of Lafayette – a point Borel was keen to emphasise.

Borel’s final years at the Académie des Sciences From 1945, and up to his death in 1956, Borel regularly put pen to paper, either to write articles on subjects he had developed in the past (set theory, probability), or as a reviewer for papers submitted to various journals, often in relation to concrete problems in operations research (waiting lines, stocking problems, logistic organisation). He also participated in an impressive number of committees for prizes awarded by the Académie des Sciences. 13 Transfer des cendres de Jean Perrin. Allocution de M. Émile Borel à la Sorbonne, Académie des Sciences, Institut de France, Paris, June 18, 1948

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Considering that France’s presence abroad was essential to forging links between nations through scientific and technical cooperation, he was tirelessly committed to representing the Academy at numerous international events (see Table 1 for a partial list). Year

Place

Event

194714 Paris 195115 Paris 195116 New Delhi17 195118 195219

195320 195421 195522

Member of the National Commission for UNESCO Member of the National Commission for UNESCO President of the Mathematical Statistics Section of the International Statistics Institute London Represents the Academy at the Institut de France House Amsterdam Vice-president of the International Council of Scientific Unions Rome General Assembly of the International Mathematical Union Rome 28th Session of the International Statistics Institute Paris 100th anniversary of the birth of Henri Poincaré Rio de Janeiro Represents the Academy, with M. Fréchet, G. Darmois and D. Dugué, at the International Statistics Institute 29th Meeting

Table 1. Some of Borel’s international missions as a representative of the Académie des Sciences. 14 C.

R. Acad. Sci., (1947), p. 142 C. R. Acad. Sci., May 21, 1951, p. 1891 16 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), pp. 329–348; see also Désignations, C. R. Acad. Sci., June 4, 1951, p. 2069 17 On this occasion, in a letter to his colleagues of the Academy, he tells them about having listened to “several interesting talks”, and also of the “particularly warm welcome my wife and I received from the Indian government. We are guests of the President of the Republic at Government House, the former Palace of the Viceroy”. 18 C. R. Acad. Sci., (1951), p. 1014 19 Comité secret, C. R. Acad. Sci., March 10, 1952, p. 1328 20 Désignations, C. R. Acad. Sci., April 27, 1953, p. 1626 21 Désignations, C. R. Acad. Sci., May 3, 1954, p. 1762 22 Comité secret, C. R. Acad. Sci., March 7, 1955, p. 1159 15 Désignations,

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Reflections on current affairs: Secret gardens His chronicles in Le Progrès After having been a political propaganda organ for the elections between 1924 and 1939, Le Progrès Saint-Affricain remained for Borel an important means of communication. Even before the conclusion of the 1939–1945 war and until the end of his life, he regularly wrote on a wide range of current affairs topics to inform, enlighten, and advise his readers in a spirit of national reconciliation. Long excerpts of his writings deserve quoting, to show their relevance even today. Shortly after the Liberation and in the wake of several foreign missions (London, Moscow), he emphasised the need to continue the work of the League of Nations, but with a military force to enforce its decisions and maintain the peace. . . ;

and he added: we must count on ourselves to restore our own ruins. . . A major effort is required from everybody if we don’t want the reconstruction of our country to go on forever.23

Already in 1945 he saw the potential of atomic energy, which he envisaged as a new source of exploitable energy. But he also recognised the need for an accord to supervise the fabrication of atomic bombs, accord that should not be difficult to find, since all nations would be equally interested in this surveillance.24

Years later, on the questions of confidentiality and espionage, he would admit that in a free nation, it is in the long term difficult to keep secrets, since the curiosity of politicians and journalists regarding the difficult problems of modern physics results in their not being aware of the significance of revealing certain facts, as they repeat confidential information without understanding it. 23 Le

Progrès Saint-Affricain, December 23, 1944 September 8, 1945

24 Idem,

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Regarding the use of nuclear weapons, he was not optimistic: I do not see the time approaching when men will be wise enough so as not to devote a considerable part of their activities to finding the means of destroying themselves [. . . ] Given the speed of the ‘progress’ made in the improvement of engines of annihilation, I fear that we are not far from the day when an excess of delusions of grandeur could result in the destruction of vast regions of the planet. Let us hope that wisdom will prevail.25

He wrote this in the midst of the cold war. During the preparation of the referendum on constitutional reform, in 1946, he sought to show the pertinence of the Radical Party such as he conceived it: Above all, we should understand that in parliament, the power of an important radical group is crucial to a balanced republic, and therefore we must prevent at all costs the creation of two antagonistic blocs: the far right and the far left. . . Not to mention the grave danger of this incoherence for our economic and financial recovery, and the even more disastrous consequences for our foreign policy. . . .

He went on: It is necessary to have in parliament men who advocate the boldest reforms; it is also necessary to have the presence of conservatives, in the most honourable sense of the term, because in the traditions and institutions we inherited from our fathers not all is to be discarded. It is essential to have sensible men, willing to act as arbiters and carry out the necessary reforms without changing too much that which deserves to be preserved. Such will be the role of the Radical Party. In this way, the social reforms and the nationalisations supported by the great majority of the electorate will be carried out in an atmosphere of order, justice, and freedom. France will maintain all the friendships and alliances she requires to raise from her ruins. This restoration will be swift if we quickly get rid of the useless and often harmful organisations hindering agriculture, commerce, industry, and work.26 25 Idem, 26 Idem,

November 3, 1951 June 1, 1946

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In 1948, he tackled the delicate question of stabilizing the currency and the problem of the exchange market and the free evolution of prices, which he saw as getting dangerously out of hand27,28,29 “at a time when our national economy must make the transition from widespread agricultural and industrial restrictions to a gradual return to a liberalisation that ought to be extended to international trade”. Beginning in 1950, a pivotal year that “marks the end of the restrictions”,30 he proposed some ideas on the priorities of the times: the arms race, the future of Europe, and science and education. Given the suffering provoked by the war and the still-present threat due to the power of atomic weapons, he again raised “the question of the good and pacific intentions of each nation and the constant denunciation of their eventual rivals’ spirit of aggression. Such an attitude soon leads countries to a policy that deserves to be called policy of fear, and which unfortunately becomes a danger to peace”. Using as an example certain recent events, he came to the conclusion that “the mechanism by which fear generates the arms race and, as a result, increases fear, may appear as unavoidable”. He then went on to denounce the well-known method of the media: “The press too often seeks, through sensationalist headlines, to unsettle public opinion and increase fear”.31 To counter that fear he relied on the mission and role of the United Nations: To remove the spectre of war, we must try to ease fear by acting on the psychology of frightened governments and peoples. This does not necessitate costly armament or millions of soldiers: it is enough to have a forum where one can speak with a reasonable chance of being heard by the entire world. Such a forum already exists, it is that of the United Nations, but it doesn’t seem that pacifist nations are using it as much as they should for the specific purpose of putting an end to the politics of fear.32 27 Idem,

April 24, 1948 November 11, 1948 29 Idem, August 6, 1949 30 Idem, August 13, 1949 31 Idem, March 4, 1950 32 Ibid. 28 Idem,

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Regarding Europe and her future, Borel exposed his views on the problem beginning in 1950. He explained how the Council of Europe was created and the role of Robert Schuman in setting up a common authority to oversee steel and coal production in France, Germany, and other European countries: This economic unit could later be extended to other domains; it is the only way of unifying the European economies and so allowing them to efficiently compete with the American industry.33

He went on: But the most delicate point will be that of the relationship between this body and the governments, as the latter will not accept to relinquish their sovereign rights, which permit them to intervene in a variety of ways in the economic sphere. This leads to the idea of a political organisation to control this economic body. [. . . ] In my view, it will be necessary to resort to the Council of Europe, whose members are independent from governments and represent the various political parties of each country. The increasing importance of economic factors in the life of peoples should not detract us from realizing the vital importance of political institutions. It is thanks to the Constitution of the United States that that great country achieved its economic supremacy. It is only by creating and improving its political organisation that Europe will succeed in surmounting its present difficulties.34

He was receptive to the idea of a European flag. A typically French initiative by Daniel Villey, a law professor at Poitiers, regarding the creation of such a symbol, was thought by Borel to be a worthwhile idea, as long as French citizens, their municipal and departmental representatives, and parliament itself, be sufficiently interested in the construction of Europe as to be willing to seize every occasion for flying the European flag.

And he added: 33 Idem, 34 Ibid.

July 8, 1950

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But couldn’t we go a step further and have a law voted by the National Assembly and the Council of the Republic to the effect that the European flag would be flown alongside the French banner in our public buildings, our ministries, and the Élysée Palace?35

The problems of education, a major concern of Borel’s, were also present in articles addressed to his local friends – and not directly to the authorities in the capital. As always, schooling remained a crucial issue, one that conditioned the future in those years when an increasing number of births was coupled with an important decline in infant mortality. The construction of new schools became an urgent necessity, and he made a pressing appeal for the implementation of “specific programs, strictly limited to the essentials, but easily adaptable, to prevent the arrival of new students from becoming an unexpected cataclysm.”36 The solution he proposed consisted in designing a certain number of building types “among which it would be possible to choose according to local needs”. He wished to show the importance of “rapidly ending the present crisis and the overcrowding of classes,” but “without wasting public funds in extravagant constructions”.37 He wrote this at eightyfour, less than five months before his death. Aware of the problem of education reform, he warned his constituency and Parliament against decisions influenced by too much modernism and resulting in “a completely new system, for then an entire generation of students would risk paying the price for such an experiment”. Even if he recognised the positive role played by the grandes écoles, he thought that a reform might be necessary, “and could go so far as dismantling them, in favour of the universities”.38 As an example, he mentioned the case of the ENS, which in 1903 became part of the University of Paris while retaining a certain autonomy, notably its very selective entrance examinations. On the role of scientists, the place of science in society, and the need to make scientific concepts and discoveries accessible to the general 35 Idem,

April 15, 1950 February 10, 1951 37 Idem, October 1, 1955 38 Idem, April 22, 1950 36 Idem,

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public, Borel had very strong views, which he communicated through his numerous writings and at public lectures, philosophical as well as popular. He advocated for funding all fields of knowledge, rather than favour those that appear profitable in the short run, in terms of scientific results or financial gain. In his view, a bureaucratically controlled system risked clipping the wings of every original thought, of every innovative idea.

Farewell to Parisian life: Secret gardens A final official distinction: In August 1950 Borel received the Grand Croix de la Légion d’Honneur (Great Cross of the Legion of Honour), the highest tribute awarded by the French government. One year later, on November 10, 1951, Émile and Marguerite celebrated their golden wedding anniversary in the hall of the FranceAmerica Committee, an organisation presided by Borel. It was of course an exceptional opportunity to gather with their countless Parisian friends and acquaintances, and to mark that long journey of life together, so dynamic and multifaceted – intellectual, administrative, artistic, literary, diplomatic. But many good friends were no longer among them; as Marguerite put it, that gathering was “a way to say farewell to society life”.39 There were still a number of foreign trips to be made, but it was also the time to look back into the past: the absence of children and grandchildren and the loss of their nephew Fernand Lebeau represented the backdrop of two passionately lived lives. But the couple’s journey went on, along a path where memories, beloved places, and habits crossed. On the occasions when they got away to their house in Saint-Affrique, “Borel”, writes Camille Marbo, “would spend long periods in the garden, near the beds of his beloved pink and red polyanthas, and the old serpentine wisteria. [. . . ] To me, this garden has a special significance – he would say – here, I reconnect with my past”.40 39 C. Marbo, À travers deux siècles : souvenirs et rencontres (1883–1967), Bernard Grasset, Paris (1967), p. 328 40 Idem, p. 324

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There was another secret garden at the villa so dear to Marie Curie, “La Crête”, in Cavalaire, which the Borels had purchased following her death.41 “My husband would sit facing the sea, on the patio he had had built”, writes Marbo. “He would spend long hours watching the wrinkles in the water and the light spots that appeared sometimes against the dark blue seafloor and listening to the sound of the waves against the rocks”. After Borel’s death, Marguerite could not afford to keep this villa, heavy with history,42 and was forced to sell it in October 1956.

The final years: Farewell from his hometown Relentlessly, faithful to his goals and despite his age, Borel pursued his mission as a scientific ambassador. At eighty-four, he agreed to represent the Académie des Sciences at the twenty-ninth meeting of the International Statistical Institute that took place in Rio de Janeiro43 from June 24 to July 2, 1955. He was most certainly encouraged by the marks of respect from the scientific community: In 1954, he was the recipient of the first médaille d’or awarded by the Centre national de la recherche scientifique (CNRS),44 for his crucial role and decisive actions in favour of scientific research. That same year, the Institut honoured him with the Osiris Prize,45 which rewards the most remarkable scientific discovery, and which had previously been awarded to Marie Curie.46 In spite of a difficult winter due to some lung complications, his health recovered after some weeks of rest in Saint-Affrique and Cavalaire. But, shortly before his departure for Brazil, Borel was the victim of a traffic accident on his way to attend his last board meeting at the FranceAmerica Committee, on avenue Franklin Roosevelt in Paris. Visibly 41 Idem,

p. 289 de Germond, Madame Marie Curie – Villa la Crète à la Vigie, Histoire et histoires de Cavalaire, éditions Sainte-Maxime (1992), pp. 525–34 43 Borel’s first trip to Rio took place in 1922. It is described by Camille Marbo in Souvenirs et rencontres, pp. 199–204, and by A. Micali, Émile Borel et le Brésil. In: Les mathématiques en France au début de XX siécle. Colloque Émile Borel (Saint-Affrique, July 1999). Ière session, Saint-Affrique, Aveyron (1999). 44 Le Progrès Saint-Affricain, April 9, 1954 45 Prix Osiris, C. R. Acad. Sci., December 13, 1954, p. 1739 46 È. Curie, Madame Curie, Gallimard, Paris (1938), p. 307 42 J.-D.

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shaken by the incident, he nevertheless decided to go ahead with the trip, accompanied by his wife and his colleagues Fréchet, Darmois, and Dugué. Another fall in his cabin during the crossing worsened his condition and further complicated his stay. It was an exhausted Borel that returned to Paris. He spent the end of the summer in Saint-Affrique and Cavalaire, under the attentive care of his physician, and the last, slow days of the year, with difficulty, in Paris. Regarding his final wishes, Borel gave some strict instructions to his wife: no ceremonies in Paris; for his final rest, he wished to be buried near his parents, in the Protestant section of the cemetery in Saint-Affrique. Émile Borel let out his last breath at his house on rue Froidveaux on the evening of February 3, 1956. His funeral took place on February 9 in Saint-Affrique. It was a freezing cold day, but a large crowd came to pay him a last tribute of respect, a final homage, at the ceremony attended by the president of the general council of Aveyron and many other official personalities.47 Emotional speeches were delivered on the steps of City Hall – by the mayor, who praised his predecessor’s “republican faith and courage in the most difficult circumstances”, and a former student of Borel’s – followed by a religious service at the Temple. Despite not being a religious man, he asked for this ceremony because he believed he owed as much to the memory of his parents. In Paris, at the Académie des Sciences, Jacques Hadamard expressed his condolences at the February 6 meeting,48 Paul Montel recounted the highlights of the distinguished member’s career;49 the Faculté des Sciences paid a last homage to their former colleague with a solemn speech by Joseph Pérès on March 22.50 On December 9, 1957, on behalf of the Institut de France, Louis de Broglie, secretary of the Académie des Sciences, read a long memoir on Borel’s life and works.51 47 Revue

du Rouergue, tome X, (1956), no. 1, janvier–mars, pp. 83–91 et communications, C. R. Acad. Sci., February 6, 1956, p. 701 49 Mémoires et communications, C. R. Acad. Sci., February 13, 1956, pp. 845–50 50 J. Pérès, Étude des marchés par la méthode des sondages, Archives de l’Académie des Sciences, Doxométrie, Brussels, September 1956, pp. 3–7 51 L. de Broglie, Notice sur la vie et l’œuvre de Émile Borel, Institut de France, 48 Mémoires

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Finally, the ENS, the institution to which Borel had remained so devoted, would express, through the pen of the geophysicist Charles Maurain, the man’s principles and the traits of his character: native of a land with a harsh climate that left its mark on him, possessing a sharp intelligence, seizing at once the important points of every problem, at times annoyed at the slowness and superficialities of a conversation or discussion and letting it be known – an attitude that concealed, in fact, under a lucid criticism, a profound humanism.52

This strong presence of Borel, imprinted in the memory of those who knew him, was perfectly expressed by Paul Ramadier: he knew, with a noble regard for human personality, how to be entirely himself and still respect the other’s convictions. The sentiment of this simple greatness, which wished to remain hidden and would only reveal itself by deeds, commanded respect to all those who had the rare privilege of penetrating his intimate self.53

And so, in that beginning of 1956, a rare star went out. A thought lingers: This twenty-first century, of uncertain times and uncertain future, could have benefited from his talents, his worldview, his unwavering dynamism, his rigour, his sense of duty, his advice.

Académie des Sciences, Séance Annuelle des Prix, December 9, 1957 52 C. Maurain in Annuaire de l’École Normale Supérieure (1954–1957), January 13, 1957, pp. 29–32 53 P. Ramadier, La Dépêche du Midi, February 10, 1956

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Supplement to the English edition On the foundations of mathematical analysis from Borel to Bourbaki by Pierre Dugac This is a transcript of a talk that Pierre Dugac delivered at the Colloque Émile Borel: “Mathematics in France at the Beginning of the Twentieth Century”, held in Saint-Affrique, on July 16 and 17, 1999. This was the first colloquium of the World Mathematical Year 2000.

Émile Borel: The “practical usefulness” of set theory In his thesis, On Certain Aspects of the Theory of Functions, where he follows up on the works of Weierstrass, Poincaré, and Hadamard on analytic functions, Borel rejects, in a note, “the proofs of certain propositions that are linked to set theory”1 and proves, using Cantor’s transfinite numbers, the following theorem:2 If we have an infinite number of partial intervals of a line, such that every point of the line lies inside at least one of these intervals, we can effectively determine a limited number of intervals chosen among the given ones and having the same property (every point of the line lies inside at least one of them). 1 É. Borel, Sur quelques points de la théorie des fonctions, Ann. de l’Éc. Norm. III. Sér., 12 (1895), pp. 9–55; Œuvres, vol. 1, CNRS, Paris (1972), p. 241 2 Idem, pp. 281–82

We shall discuss this theorem only briefly, since later in the colloquium André Warusfel will talk about the history of the Borel–Lebesgue theorem. Even if the theorem has already a long history behind it,3 it is interesting for our purpose for two reasons: on the one hand, it brings to light more explicitly than its predecessors the idea of a cover – which will eventually lead to a good definition of compact set – and, on the other, in its proof Borel uses for the first time Cantor’s concept of transfinite number, which over several decades will play a non-negligible role in mathematical analysis. In the spring of 1897, Borel gave a series of lectures at the École Normale Supérieure that he would later publish with the title: Leçons sur la théorie des fonctions (1898). Among the audience were Baire and Lebesgue. Borel’s aim4 was “to present, at an elementary level” certain “relatively recent” results and concepts “whose relevance is increasing day by day”. Set theory is among these: “It is to that theory that this work is devoted. I decided however to give it the title Leçons sur la théorie des fonctions because in my use of sets, I tried to never lose sight of the applications.” This is a constant of Borel’s thought, as he will later forcefully put it: I must confess that in the beginning I was seduced by Cantor’s ideas, as did many other young mathematicians; I don’t regret it, because they help open up the mind. But I always believed that those abstract concepts should be a mean to a goal, not an end in themselves.5

Leçons became the first volume of his famous series “Monographs on the Theory of Functions”, which had a decisive influence on the development of analysis at the beginning of the twentieth century, despite Laurent Schwartz’s slightly ironic tone in his autobiography, Un mathématicien aux prises avec le siècle, published in 1997: I cannot fail to mention the volumes in the Borel Series, on a wide variety of subjects. They read like novels. Their style is simple, intuitive, 3 P. Dugac, Sur la correspondance de Borel et le théorème de Dirichlet–Heine– Weierstrass–Borel–Schoenflies–Lebesgue, Arch. Internat. Hist. Sci., 39 (1989), no. 122, pp. 69–110 4 É. Borel, Leçons sur la théorie des fonctions, Gauthier-Villars, Paris (1898), VII–VIII 5 M. Fréchet, La vie et l’œuvre d’Émile Borel; É. Borel, Œuvres, vol. I, CNRS, Paris (1972), p. 29

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easy, more sentimental than rigorous, barely divided into chapters. In them we can find intriguing “we have earlier proved”, but where? One had to be clever enough to dig out the proof in question – sometimes, admittedly, in vain. We also find: “It’s true that the theorem previously proved was not exactly the same we are using here, but the reader would easily set things right.” Not so sure!6

René Bair: “Every problem in the theory of functions leads to certain questions in set theory” The torch of the new analysis initiated by Borel will be immediately picked up by Baire, who introduces7 around November 25 the notions of lower and upper semi-continuity, and towards January 25, 1897, proves that functions possessing those properties are point-wise discontinuous in an interval 𝐼, that is, that the set of points of 𝐼 where they are continuous is dense in 𝐼. Throughout his research Baire makes use for the first time, in a systematic way, of all the notions introduced by Georg Cantor, those in general topology as well as the ones regarding transfinite numbers. In March 1898, he proves the following theorem: For a real function of a real variable to be the limit of a sequence of continuous functions it is necessary and sufficient that it be pointwise discontinuous relative to every perfect set, that is, a closed subset without isolated points of a topological space. This theorem will be the starting point of a new theory of real functions of a real variable and will mark an important stage in the history of analysis.

Baire also develops a classification of functions that will appear in his Note to the Académie des Sciences of June 6, 1898 8 containing all the discontinuous functions known at the end of the nineteenth century, and 6 L.

p. 75

Schwartz, Un mathématicien aux prises avec le siècle, Odile Jacob, Paris (1997),

7 P. Dugac, Notes et documents sur la vie et l’œuvre de René Baire, Arch. History Exact Sci., 15 (1975/76), no. 4, pp. 297–383 8 R. Baire, Œuvres scientifiques. Published under the direction of P. Lelong in collaboration with P. Dugac, Gauthier-Villars, Paris (1990), p. 41

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which will influence the works of the greatest analysts of the beginning of the next century. In his papers, Baire introduces the concept of a set of first category, what Bourbaki calls a meager set – that is, a countable union of nowhere dense sets – and that will pervade the whole of analysis. In his 1899 thesis, Baire shows that the transformation that took place in the tools of mathematical analysis at the end of the nineteenth century consists in the fact that “every problem in the theory of functions leads to certain questions in set theory; and it is to the extent that those questions are or can be resolved that it is possible to solve more or less completely the given problem”.9 In this way, the most difficult questions in analysis are transferred to what will later be called general topology. In his report on the development of the theory of sets of points released only one year after Baire’s thesis, Arthur Schoenflies had observed that in his “hugely promising” work, Baire had travelled “totally new” paths, and that it was “novel ideas” that had led him to his results.10

For David Hilbert, questions on the foundations of analysis do not belong to the “future problems of mathematics” On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert gave a memorable talk “On the future problems of mathematics”. He considers that the most suggestive and important event in analysis in the nineteenth century is “the arithmetical point of view of the concept of continuum that one finds in the works of Cauchy, Bolzano, and Cantor”.11 This is why the first future problem is the one formulated by Cantor in relation with the continuum question, and Hilbert considers as highly likely the correctness of the following theorem: As regards the equivalence of sets, there would only be “two sets of numbers, the countable set and the continuum”.12 9 Idem,

p. 169 Schoenflies, Die Entwicklung der Lehre von Punktmannigfaltigkeiten, Deutsche Math. Ver., 8 (1900), no. 2, p. 220 11 D. Hilbert, Mathematische Probleme, vol. IV, Gesammelte Abhandlungen, Springer, Berlin (1935), pp. 290–329; French translation: Comptes rendus du deuxième Congrès international des mathématiciens, Gauthier-Villars, Paris (1900), p. 69 12 Idem, p. 70 10 A.

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A question closely related to the previous one is to determine whether a totally ordered set can be well ordered, that is, whether there exists “not only in the set itself, but in every partial set, a number that precedes all the others”.13 The possibility of a “well order” quickly became a question involving the axiom of choice. It could be noted in this regard that Cantor’s focus on metamathematics since 1895 may explain his lack of interest in the works of Borel, Baire, and Lebesgue, even though they were using in a decisive way the concepts he had introduced. On April 29, 1901, Lebesgue presented to the Académie des Sciences his note On a generalisation of the definite integral, which, in particular, allows him to integrate all functions in the Baire classification.14 In a 1905 paper, he shows the equivalence between the functions in the Baire classification and the set of Borel functions, and then constructs a function that does not belong to the Baire classification – and which therefore is not analytically representable – and it is not a Borel function.15

Maurice Fréchet and Felix Hausdorff: the fathers of general topology In 1904 Maurice Fréchet used for the first time the word “compact”, but the set in question is in fact precompact, that is, its completion is compact.16 In his 1906 thesis, with the encouragement of Jacques Hadamard, Fréchet adopts “a completely general point of view”. It is “a first attempt to systematically establish certain fundamental principles of functional calculus”.17 13 Idem,

p. 71

14 H. Lebesgue,

Sur une généralisation de l’intégrale, Comptes Rendus Acad. Sci., 132 (1901), pp. 1025–28; Œuvres scientifiques, vol. I, Institut de Mathématiques de l’Université de Genève, Geneva (1972), p. 199 15 H. Lebesgue, Sur les fonctions représentables analytiquement, Journal Math. pures appl., (6), 1 (1905), pp. 139–216; Œuvres scientifiques, vol. III, Institut de Mathématiques de l’Université de Genève, Geneva (1972), pp. 103–80 16 M. Fréchet, Généralisation d’un théorème de Weierstrass, C. R. Acad. Sci., 139 (1904), juillet–décembre, pp. 848–50 17 M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), pp. 1–2

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The most important concept introduced by Fréchet in his thesis18 is that of metric space, which paves the way to the introduction of spaces more general than finite-dimensional Euclidean spaces. He also defines the notion of a complete space,19 a space that admits a generalisation of Cauchy’s theorem – every sequence of points of the space that “satisfies Cauchy’s conditions” has a unique limit. Hausdorff’s 1914 book Foundations of the theory of sets, played a crucial role in the development of analysis and mathematics in general. His approach, based on the notion of neighborhood20 remained a model of an axiomatic theory. He defines the notion of limit21 with the help of that of neighborhoods, thus achieving what Euclid had sketched almost twenty-five centuries earlier. He gives the present-day definition of a connected set:22 We shall call 𝐴 connected if 𝐴 is non-empty and “if it cannot be decomposed into two disjoint closed subsets different from the empty set”. He also shows that every metric space can be extended to a complete metric space.23 In June 1920, Stefan Banach defended his thesis On operations on abstract sets and their application to integral equations. In the introduction, he describes his method, initiating a new style in the writing of mathematical treatises: The aim of the present work is to establish certain theorems that are valid in a number of functional domains to be specified later. However, in order to avoid having to prove them separately for each particular case, which would be rather painful, I have chosen the following different path: I consider sets of elements that I assume possess certain properties; I prove theorems about them, and finally I show that each particular functional domain satisfies the assumed properties.24 18 Idem,

p. 30 p. 16 20 F. Hausdorff, Grundzüge der Mengenlehre, Veit, Leipzig (1914) and Chelsea, New York (1949), p. 211 21 Idem, p. 233 22 Idem, p. 244 23 Idem, p. 315 24 S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Math., 3 (1922), pp. 133–81; Œuvres, vol. II, PWN Editions Scientifiques de Pologne, Warsaw (1979), p. 305 19 Idem,

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Banach introduces in his thesis the concept of a complete normed vector space now named after him.25 In 1924, Pavel Aleksandrov and Pavel Urysohn introduce the notion of compact space, a part of a topological space satisfying the following property: If the space is contained in the union of a set (of any power) of domains, then it is already contained in the union of a finite number of domains in the set.26

That same year, Aleksandrov defines locally compact spaces and rightly states that it is a concept of “great topological significance”.27

Stefan Banach’s Theory of Linear Operations Back in 1963, Marcel Brelot told us that Banach’s Theory of Linear Operations was one of the most important books in the mathematical literature.28 According to Nicolas Bourbaki this treatise marked “the coming of age of the theory of normed spaces”, and one of the “immediate effects of its publication was the almost-universal adoption of Banach’s terminology and notation”.29 In 1935, John von Neumann gives the definition of a locally convex topological vector space.30 Jean Dieudonné devoted a significant part of his mathematical activity to the study of topological vector spaces.31 In this regard, Gottfried Köthe wrote in 1969: 25 Idem,

pp. 306–7 S. Aleksandrov and P. S. Urysohn, Zur Theorie der topologischen Räumen, Math. Ann., 92 (1924), p. 259 27 P. S. Aleksandrov, Über die Metrisation der im Kleine kompakten topologischen Räumen, Math. Ann., 92 (1924), pp. 294–95 28 P. Dugac, Autour de la notion de limite et de ses voisinages : les mathématiques dans la vie de leur temps, in preparation at the time of the talk. This 439-page manuscript was never published, but it is mentioned in P. Dugac, L’histoire de l’analyse, Vuibert, Paris (2003), p. 383 29 N. Bourbaki, Eléments d’histoire des mathématiques, Masson, Paris (1984), p. 273 30 J. von Neumann, On complete topological spaces, Trans. Amer. Math. Soc., 37 (1935), pp. 1–20; Collected Papers, vol. V, Pergamon Press, Oxford (1961), p. 511 31 P. Dugac, Jean Dieudonné mathématicien complet, Jacques Gabay, Paris (1995), pp. 28–30 26 P.

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Let us consider the situation of the theory of topological vector spaces before the war. On the one hand, we had the theory of Banach and his school, highly developed and with its applications to analysis. On the other, there was a parallel theory of spaces of sequences named after Toeplitz and myself where duality played a certain role. Of the more general theory of locally convex spaces there were only a few preliminary results due to von Neumann. It was Dieudonné who developed the fundamental ideas of a powerful theory of locally convex spaces that contained the two aforementioned theories as special cases.32

Nicolas Bourbaki and the structures of analysis According to Jean Dieudonné, one of the founders of the Bourbaki group, “in the beginning, Bourbaki’s model was certainly van der Waerden’s excellent algebra treatise”.33 In 1939, Bourbaki published the first book of a treatise that will dominate mathematical style from 1950 on: The theory of sets: Fascicle of results. The treatise is titled Elements of Mathematics, and it is built around the notion of structure. The first part is devoted to “the structures of analysis”. It is interesting to note that Jean-Pierre Serre writes in his report on André Weil that he “always had a very clear awareness of the unity of mathematics” (and it is certainly under his influence that Bourbaki speaks of mathematics using the singular: “la” mathématique).34 In 1940 begins the publication of General Topology, where Bourbaki, following in particular Hausdorff, gives a precise mathematical meaning to the concepts of neighborhood, limit, and continuity. 32 P.

p. 29

Dugac, Jean Dieudonné mathématicien complet, Jacques Gabay, Paris (1995),

33 J. Dieudonné, Regards sur Bourbaki, Analele Univ. Bucaresti, mat.-mec., 18 (1969), no. 2, p. 16 34 P. Dugac, Autour de la notion de limite et de ses voisinages : les mathématiques dans la vie de leur temps, in preparation at the time of the talk. See footnote 28 of this Supplement.

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Today’s renewed interest in the applications of mathematics: where we rediscover Borel’s ideas At a time when mathematics strongly takes over the other sciences and is itself taken over by them, we find it interesting to quote Borel: The general trend of my research work and my textbooks is the following: I try to show that mathematics is not a purely abstract game of the mind, but, on the contrary, that it is closely connected to concrete reality.35

To illustrate this fact, let’s revisit the dispute between Borel and Lebesgue at the Council meeting of the Faculté des Sciences in Paris on January 24, 1920.36 Borel, holder since 1909 of the theory of functions chair that had been created for him, and devoting more and more time to the calculus of probability and its applications to theoretical physics – then undergoing a significant boom – requested, in December 1919, to be transferred to the calculus of probability and mathematical physics chair left vacant after Joseph Boussinesq’s retirement on November 1. The question then came up about whether to keep or transform Borel’s theory of functions chair, and it is the reason for the Council meeting. At Borel’s suggestion, the committee appointed to examine the question proposed to transform Borel’s chair into a “theoretical physics and celestial physics” one. The only person strongly opposed to this transformation was Lebesgue. The following excerpt from Lebesgue’s speech is telling of the gap between him and Borel, who was then devoted entirely to “useful mathematics” and its “practical value”: Mr. Borel thinks that the teaching of mathematics for the licence and the doctorate must be treated differently. For the first one, because of its obvious usefulness, he requests its growth. The second, he believes, can 35 M. Fréchet, La vie et l’œuvre d’Émile Borel; É. Borel, Œuvres, vol. I, CNRS, Paris (1972), p. 29 36 B. Bru and P. Dugac, Borel et Lebesgue : la fin d’une amitié, Enseign. Math., to appear (at the time of the talk). This article in progress was replaced, after the death of Pierre Dugac, by B. Bru and P. Dugac (eds.), Les lendemains de l’intégrale : lettres de Henri Lebesgue à Émile Borel, Vuibert, Paris (2004)

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be diminished, and because mathematics is useful it doesn’t have the right to be studied by itself. Now, while in the other sciences the latest discoveries have an impact on the teaching of the basics, it is not the same in mathematics because of its logical structure, and the teaching for the licence rests on knowledge acquired a long time ago. Therefore, there are only two or three chairs devoted to living mathematics, where learning how to work takes place, where the subjects with a potential to lead to theses are taught. And we are now asked to eliminate one of those chairs! If Mr. Borel succeeds, and he will, his teaching will attract physicists, and if it also appeals to mathematicians, the latter will be lost for mathematics.

It was Émile Picard who replied to Lebesgue that in the committee’s opinion there was a need for a chair in theoretical physics treated from the physics perspective – which was Borel’s highly novel point of view in mathematical physics – and that it was more important than the theory of functions. The committee would rule in favour of this recommendation. Today, at a time when mathematics and physics go hand in hand, a mathematician working in physics is certainly not one “lost for mathematics”!

Final note We consider it interesting to include another view on the Borel Series, that of Polish-born Szolem Mandelbrojt: I didn’t like Polish mathematics, after all. Because there is such a thing we call, or then used to call, Polish mathematics. It was something very important, for I’m certain there would be no Bourbaki, not even abstract mathematics, without this Polish school where set theory and functional calculus played an extremely significant role. We practically did nothing but that. Only, as it happened, I read at the time certain works of Lebesgue in the Émile Borel Series. The theory of functions really pleased me a lot. I liked, from the start, things that I could touch, if not with my hand, at least with my mind. The mind has hands too, doesn’t it? With the mind, there are things that we can touch, others that we cannot touch. For example, I have never understood, I still don’t understand, why it is so important to study structures from the outside,

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why only care about the form of things when every thing has its soul, its core, on which one can put a finger. That’s what I liked about the theory of functions. Yes. When I read the books in the Borel Series and Hadamard’s works, I was moved.37

This story is all the more interesting given the fact that Mandelbrojt was a member of the Bourbaki group when it was first created.

37 S. Mandelbrojt, Souvenirs à bâtons rompus de Szolem Mandelbrojt, recueillis en 1970 et préparés par Benoît Mandelbrot. With notes by Pierre Dugac, Cah. Sémin. Hist. Math., 6 (1985), pp. 4–5

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Closing remarks and further reading The original French version of this biography of Émile Borel, deliberately chronological, was completed shortly before the Colloque Émile Borel in Saint-Affrique, more than twenty years ago. Despite its age, we believe that it can be useful to historians of mathematics and in other domains wishing to carry out new research on the life and works of such a rich personage. Let us begin by mentioning the following work, originally published as an article and later as a short book: M. Fréchet, La vie et l’œuvre d’Émile Borel, Monographies de l’Enseignement Mathématique, 14. Université de Genève, Geneva (1965). Reprinted from Enseign. Math. II. Sér, 11 (1965), pp. 1–98; see also É. Borel, Œuvres, vol. I, CNRS, Paris (1972), pp. 5–98. Of interest is also the book: J.-P. Pier, Histoire de l’intégration, Masson, Paris (1996) which was released a few years before the Colloque. After the Colloque, the following titles appeared: P. Dugac, L’histoire de l’analyse, Vuibert, Paris, (2003). B. Bru and P. Dugac (eds.), Les lendemains de l’intégrale : lettres de Henri Lebesgue à Émile Borel, Vuibert, Paris (2004). R. Siegmund-Schultze, The Institute Henri Poincaré and mathematics in France between the wars, Rev. Hist. Sci., 62 (2009), no. 1, pp. 247–283. L. Graham and J.-M. Kantor, Naming infinity: A true story of religious mysticism and mathematical creativity, Belknap Press, Cambridge, MA (2009); French edition, Au nom de l’infini : une histoire vraie de mysticisme religieux et de création mathématique, Belin, Paris (2010). M.-C. Bustamante, M. Cléry and L. Mazliak, Le Traité du calcul des probabilités et de ses applications : étendue et limites d’un projet borélien de grande envergure (1921–1939), North-West. Eur. J. Math., 1 (2015), pp. 85–123. 96

M. Pinault, Émile Borel : une carrière intellectuelle sous la IIIe République, L’Harmattan, Paris (2017). C. Ehrhardt and H. Gispert, La création de la Revue du mois : fabrique d’un projet éditorial à la Belle Époque, Philosophia Scientiae, 22 (2018), pp. 99–118. A. Bernard, Émile Borel (1908) : le calcul des probabilités et la mentalité individualiste, Cahiers philosophiques, 4 (2018), pp. 81–95. M. Cléry, La théorie des probabilités et l’Institut Henri Poincaré (1918– 1939), PhD thesis, Université Paris Saclay (2020). Finally, we would like to note the exhibition that took place at the Institut Henri Poincaré from September 2021 through February 2022, honouring Borel on the occasion of the one-hundredth anniversary of his appointment to the Academy of Sciences – a timely initiative bound to encourage further research on the multifaceted mathematician and statesman.

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Summary of Borel’s scientific career and public life Age

Years

Major events

16 18

1871 1887 1889

21

1889–92 1892

22

1892–93 1893–97

23 26 30

1894 1897–1904 1901

33

1904–09

38

1909–41

39

1910–20

44 50

1915–19 1921–56

53 54 56

1924–36 1925 1927–56

Born in Saint-Affrique (January 7) Baccalaureate (Sciences and Letters) First Prize in the Concours Général de Mathématiques First place in the École Normale Supérieure entrance examinations First place in the École Polytechnique entrance examinations Student at the ENS First place in the Concours d’Agrégation de Mathématiques Military service (Montpellier) Maître de Conférences at the Faculty of Sciences, University of Lille (4 years) Docteur ès Sciences Mathématiques (Sorbonne) Maître de Conférences at the ENS (7 years) Married to Marguerite Appell in St.-Germain-en-Laye (October 12) Adjunct Professor at the Faculty of Science of Paris (5 years) Chair Professor at the Faculty of Science of Paris (32 years) Assistant Director, Scientific Director of the ENS (10 years) Drafted to fight against Germany (3 years) Elected member of the Academy of Sciences (35 years) Elected Député de l’Aveyron (12 years) Minister of the Navy (6 months) Founder and Director of the Institut Henri Poincaré (29 years)

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Age

Years

Major events

58

1929–41

63

1934

65

1936

67

1938

69 70 74

1940 1941 1945–47

Elected Mayor of the Town of Saint-Affrique (12 years) Elected Conseiller Général du Canton de Cornus (12 years) President of the Academy of Sciences and the Institut de France (1 year) Declined seeking a fourth mandate as Député de l’Aveyron President of the International Union of Associations for the League of Nations Jubilé Scientifique (50 years of university activity) Imprisoned by the Gestapo at Fresnes (1 month) Reelected Mayor of Saint-Affrique for 2 years (Total: 14 years) Reelected Conseiller Général de l’Aveyron for 6 years (Total: 18 years) Member of the Bureau des Longitudes (10 years) Died in Paris (February 3)

1945–51 75 85

1946–56 1956

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Partial list of awards, honorary titles and honorary memberships France Order of the Legion of Honour – – – – –

Chevalier (1909) Officier (1920) Commandeur (1936) Grand officier (1946) Grand Croix (1950)

Croix de Guerre 1914–1918 – Mention in the Order of the Army (with palm) (1915) – Mention in the Order of the Regiment (Bronze Star) (1918) Médaille de la Résistance – Rosette (1947) Academic Palms – Officier d’académie (1898) – Officier de l’Instruction Publique (1909) Academy of Sciences Prizes – – – – –

Grand Prize of Mathematical Sciences (1898) Poncelet Prize (1901) Vaillant Prize (1904) Petit d’Ormoy Prize (1905) Osiris Prize (1954)

Centre National de la Recherche Scientifique – CNRS Gold Medal (awarded for the first time, 1954) Abroad Order of the Crown of Italy – Commander (1917) 100

Order of the British Empire – Commander (1919) Order of the Nichan Iftikhar (Tunisia) – Second Class Commander (1925) Order of Mérite (Haiti) (1925) Order of Saint-Sava (Serbia) – Grand Cross (1925) From the Chinese Government (Kuo-Ming-Tang Nationalist Party) – Medal of Recognition (1922) Foreign memberships and doctorates honoris causa Belgium – Société Royale des Sciences de Liège: Membre correspondant (1927) – Université de Liège: Doctorate honoris causa (1946) Bulgaria – University of Sofia: Doctorate honoris causa (1947) Denmark – University of Copenhagen: Doctorate honoris causa (1924) Ireland – Trinity College, University of Dublin: Doctorate honoris causa (1921) Italy – Accademia Reale dei Lincei: Foreign associate member (1918) – University of Rome: Doctorate honoris causa (1919) Poland – University of Warsaw: Doctorate honoris causa (1930) Sweden – Royal Swedish Academy of Sciences: Foreign member (1941) 101

Ukraine – Kharkiv Mathematical Society: Corresponding member (1908)

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Quotes and anecdotes Let us not forget that if it is alright to take advice from numbers, we should never become their slaves. Le Hasard, Félix Alcan, Paris (1914), p. 185 The science of chance can only, such being science’s role, facilitate the reflection that precedes the action of every reasonable person. Le Hasard, Félix Alcan, Paris (1914), p. 11 If chance is nothing more than the name we give to ignorance, the aim of the theory of probability is therefore to create science out of ignorance, that is, something out of nothing. There is here a misunderstanding about the meaning of the word ignorance. Le Hasard, Félix Alcan, Paris (1914), p. 13 I would like to warn physicists against certain theorists of physics’ abuse of a mathematical framework whose relevance only serves to mask the shortcomings of their hypotheses and arguments. Notice Individuelle d’Émile Borel, Titres et travaux (1921), Archives de l’Académie des Sciences, p. 6 Facilitate the discussion of the physicists’ theories but never impinge on experience, which must always have the last word. Notice Individuelle d’Émile Borel, Titres et travaux (1921), Archives de l’Académie des Sciences, p. 7 I believe that the unity of the human mind is more important than its diversity. . . and the invention, imagination, and discovery methods much more similar than it’s generally believed in the various domains in which the mind is put to work. Organon (Warsaw), 1 (1935), pp. 33–42 A great country must operate like a large industrial company that does not hesitate in spending whatever is necessary to accurately ascertain its costs: this is achieved through the use of statistics. Le Progrès Saint-Affricain, December 31, 1938

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Every advancement of modern science and technology, every invention that changed the face of the world and profoundly modified the very conditions of everyday life, are due, in the final analysis, to geometric works that during two thousand years appeared to have no practical applications. It is the most remarkable example of the mathematical miracle. La mécanique et la gravitation universelle, Albin Michel, Paris (1942), pp. 303–04 When faced with taking a risk, especially if our health or life are at stake, no one is so rich as to afford neglecting probabilities, except in case when higher considerations, on questions of morality or honour, compel us to risk our life whatever the cost. In this case, it is preferable to ignore the probability of the danger. Les probabilités et la vie, Presses Universitaires de France, Paris (1943), p. 45 Ignoring probabilities may result in taking greater risks while wishing to avoid smaller ones. Les probabilités et la vie, Presses Universitaires de France, Paris (1943), p. 46 What we wish to try to give them [the readers] it’s above all the aesthetic pleasure derived, and experienced by all those who enjoy science, from the beautiful harmony of the mathematician’s logical constructions. Les paradoxes de l’infini, Gallimard, Paris (1946), p. 10 Any false equality must be considered as totally false, regardless of the error, because from it one can deduce every other false equality. Probabilités et certitude, Presses Universitaires de France, Paris (1950), p. 68 Ignorance we are aware of is not error. Probabilités et certitude, Presses Universitaires de France, Paris (1950), p. 117

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There are no degrees of error, just as there are no large or small numbers, for all depends on the unit chosen: one billion is for us a large number if we are talking years or tons of gold; it is a small number in the case of hydrogen molecules or even drops of water in the ocean. Probabilités et certitude, Presses Universitaires de France, Paris (1950), p. 68 It was always through the contact with Nature that mathematical analysis renewed itself, and it was only thanks to that permanent contact that it could avoid the danger of becoming pure symbolism, circling around itself. Talk given at the Rice Institute, Houston, USA, 1912. In: M. Fréchet, La vie et l’œuvre d’Émile Borel; É. Borel, Œuvres, vol. I, CNRS, Paris (1972), p. 93 It is impossible to learn a language possessing several spellings – this word hates the plural form. At the International Congress of Mathematics (Zurich, August 1897); Œuvres, vol. IV, CNRS, Paris (1972), p. 2327 Life is too short to be making useless efforts in order to please people not interested in you and in whom you’re not interested either. In: M. Fréchet, La vie et l’œuvre d’Émile Borel; É. Borel, Œuvres, vol. I, CNRS, Paris (1972), p. 11 When he was minister of the navy and received the journalists, who were surprised and curious at the appointment of a mathematician to that office, in order to cut short any discussion he would tell them: “But I know why navy cannons are rifled to the left and not to the right, like in the army”, and of course he would not tell them the reason. Private communication from Jean Coulomb (1992), honorary professor at the Sorbonne and member of the Institut de France Lack of respect for the other is the worst form of egotism. Private communication from Geneviève Appell (c. 1990s)

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Acknowledgments The writing of the French original of this book would not have been possible without the assistance, encouragement, and advice of numerous witnesses and friends in bringing back, as faithfully as possible, memories, major traits, and defining moments of the life of an exceptional compatriot. We hereby present our warmest thanks to all of them. For the English edition in particular, we would like to thank Nicole Bloye and Apostolos Damialis for their help with editing the text. Relatives and friends First of all, the Borel–Appell–Tessié-Solier families, relatives and longtime friends of Émile and Marguerite Borel. Geneviève Appell and her entire family. Geneviève Appell has given us permission to reproduce her family’s photographs and encouraged the publication of this English version. Jean-Pierre Cambon, Rose Tessié-Solier. Henri Cartan, Bernard Langevin. Institutions of which Borel was a member in various capacities Académie des Sciences: Jean Delcourt, permanent secretary. Jean Coulomb, honorary director of the CNRS, the Longitudes Bureau, and the National Centre for Space Studies. Pierre Dugac, member for the history of science, who did us the honour of prefacing the original French edition of this book together with our colleague Bernard Bru. Archives (Service des archives): Christiane de Meuleneare-Douyere, Florence Greffe, and the photographer Jean-Loup Chernet. Institut Henri Poincaré (IHP): Joseph Oesterlé, director (1999). Later, Cédric Villani, and the current director, Sylvie Benzoni. IHP Documentation and Archives Department: Hélène Nocton. École Normale Supérieure and Bibliothèque des Lettres: Pierre Petitmengin, director, and Mme Dauphragne. National Assembly: Jacques Godfrain, former deputy minister-mayor of the commune of Millau (Aveyron), and Michel Mopin, library director.

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Higher education and research institutions Université René Descartes (Paris V): Bernard Bru. Université Pierre-et-Marie-Curie – Jussieu (Paris VI) and Theses Department: Mme Grais, curator. Université Paul Sabatier (Toulouse III): Pierre Coulomb. Université Sciences et Techniques du Languedoc (Montpellier II): JeanClaude Pieri. École Centrale de Lyon: Anne-Marie Mougenot and Michèle Weiss, Documentation Department. Norbert Philippe, Michel Serres Library. Raymond Volant, International Relations Department, and former PhD graduates Abdelkader Krichen and Mahmoud Daoud. École Nationale Supérieure de Mécanique et d’Aéronautique (Poitiers): François Armanet, director, former scientific attaché at the French embassy in Washington, DC, USA. History Archives National Library (Prints Department): Zoubida Zerkane. National Archives (Centre d’Accueil et de Recherches). Army Archives (château de Vincennes). Navy Archives (château de Vincennes and base navale Toulon). National Navy Museum (Paris): Jérome Legrand, documents clerk. Curie Museum and Curie–Joliot-Curie Association: Ginette Gablot and Lenka Brochard. Tarn-et-Garonne Departmental Archives (Montauban) and Suzanne Moussié, from the “Friends of the Archives” group. Writers Society of France: Marie-France Briselance. Éditions Grasset (Paris): Marie-Hélène d’Ovidio and Anna Nikitina for having granted us permission to reprint and translate excerpts from Camille Marbo’s works. Art and Archaeology Library (Paris) and Elsa Thulin, for her memories of Jean Perrin.

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Regional and local press La Dépêche du Midi (Toulouse), Midi-Libre (Montpellier). Le Progrès Saint-Affricain and Imprimerie du Progrès: Jean-Claude and Annick Aufrère and their children, Eric and Bruno, thanks to whom the original French edition of this book could see the light, and also to former journalists André Calmels and Lucien Ramond. La Revue de Rouergue (Rodez). Saint-Affrique and South Rouergue Fernand Sambucy, mayor of the town of Saint-Affrique (1999). André Vigouroux, former mayor, and present mayor Sébastien David, their assistants and municipal councillors. La Maison de la Mémoire and Bernadette Gervais-Damoiseau, its president. Le Service de l’État-Civil and the Town Archives. Édouard Peyre, a.k.a. Jacques Vaizy, who carefully revised our French manuscript and whom we emphatically thank for his preface to the French edition. Girard Photo Studio. Numerous Saint-Affricans, the list of whom would be too long to enumerate. In Millau, Jacques Cros-Saussol, president of the Popular University of South Rouergue. In Saint-Paul-des-Fonds, the late Renée Roucoules, and Pierre Marechal and his family. Elsewhere Marc and Elizabeth Chaix, Cavalaire (Var). Jean-François Delord, professor at the Ingres Lycée (Montauban), and the family of André Viales (Toulouse). Rudolf Kundera, painter from Cassis, and his Association Rudolf Kundera, with whom we had some excellent discussions regarding Paul Valéry. 108

Abroad Mittag-Leffler Institute of the Royal Swedish Academy of Sciences and its director, Kiell-Ove Widman. Hebrew University of Jerusalem, for the Albert Einstein archives, and Ze’ev Rosenkranz, curator. General documentary sources France National Archives: Centre d’Accueil et de Recherche des Archives Nationales (Paris). National Library and Print Room (Paris). Archives of the Académie des Sciences (Paris). Archives of the Bureau des Longitudes (Paris). Library of the National Assembly (Paris). Les Services Historiques de l’Armée de Terre (Château de Vincennes). Les Services Historiques de la Marine (Château de Vincennes, Paris and Toulon). Document Centre and Library, Institut Henri Poincaré (Paris). UNESCO Archives (Paris). Sorbonne Archives (Faculté des Sciences, Paris IV). Bibliothèque des Lettres (École Normale Supérieure, Paris). Médiathèque (École Centrale de Lyon). Curie Institute Museum and Curie and Joliot-Curie Association (Paris). Société des Gens de Lettres de France (Hotel de Massa, Paris). Aveyron Department Archives (Rodez). Municipal Archives (Saint-Affrique). Archives of the Imprimerie du Progrès (Saint-Affrique).

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Abroad Manuscripts Department and Archives (Hebrew University of Jerusalem, Israel). United Nations Archives (Geneva). Mittag-Leffler Institute of the Royal Swedish Academy of Sciences (Stockholm). French Embassy (Washington, DC).

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Émile Borel, one of the early developers of measure theory and probability, was among the first to show the importance of the calculus of probability as a tool for the experimental sciences. A prolific and gifted researcher, his scientific works, so vast in number and scope, earned him international recognition. In addition, at the origin of the foundation of the Institut Henri Poincaré in Paris and longtime its director, he also served as member of the French Parliament, minister of the Navy, president of the League of Nations Union, and president of the French Academy of Sciences. The book follows Borel, one of France’s leading scientific and political figures of the first half of the twentieth century, through the various stages and the most significant events of his life, across two centuries and two wars. Originally published in French, this new English edition of the book will appeal primarily to mathematicians and those with an interest in the history of science, but it should not disappoint anyone wishing to explore, through the life of an exceptional scientist and man, a chapter of history from the Franco-Prussian War of 1870 to the beginnings of contemporary Europe.

https://ems.press ISBN 978-3-98547-013-6