埃米尔·博雷尔Émile Borel

埃米尔·博雷尔的父亲是奥诺雷博雷尔，他是一位新教牧师。奥诺雷博雷尔本人是来自蒙托邦的一位工匠的儿子，蒙托邦是南部-比利牛斯大区塔恩-加龙省的首府。埃米尔的母亲是埃米莉·泰西耶-索利耶，她是来自圣阿夫里克的一位富有的羊毛商人的女儿。埃米莉的父亲反对她嫁给一位牧师，想让她嫁给当地一位富有的地主。结果，他剥夺了她的继承权，埃米莉和父亲之间关系不好。博雷尔出生时，他的父母住在一栋漂亮的18世纪房子里，奥诺雷博雷尔在那里为附近新教家庭的孩子办了一所学校。奥诺雷和埃米莉有两个女儿，她们比博雷尔大得多，他出生时她们分别为十六岁和十四岁。奥诺雷博雷尔是个聪明人，在儿子生命的最初几年里，他在家里教育他。

当博雷尔十一岁时，他离开圣阿夫里克的家，去与他已婚的姐姐勒博夫人同住，她嫁给了蒙托邦的一位牧师。这意味着他能够就读于蒙托邦的公立中学。在那里他展现出非凡的才华，几年后，他作为奖学金生前往巴黎的圣巴布学院。在那里学习期间，他参加了路易大帝中学的课程，为参加进入巴黎综合理工学院和巴黎高等师范学院的考试做准备。他在这两项考试中都名列第一，可以选择这两所大学中的任何一所。他还在全国竞赛中获得了一等奖。人们可能会期望这一成就使这位才华横溢的学童受到高度赞扬。然而[5]：-

这一惊人的壮举引起了相当大的关注，并非所有关注都是同情的。因为有一位知名人士曾讥讽地提到学童成功的短暂性和其迅速褪色的桂冠，并向这位年轻的冠军发出挑战，要他在十年后证明自己。不到一半的时间过去，这位不宽容的批评者就可能显得愚蠢……

家里的朋友敦促博雷尔进入巴黎综合理工学院，这被认为是更有声望的学府，但他另有想法。

家里的朋友是商人和实业家，他们的建议是瞄准一个能赚大钱的职业。博雷尔被告知，巴黎综合理工学院的学位将为他提供在工业或商业领域就业的最佳机会。然而，他非常坚定地想要从事科学工作，尤其是数学，他确信在巴黎高等师范学院学习将最有可能获得这样的职位。他的父亲确实支持他的决定，他于1889年入学。我们应该提到他决定中的一个重要因素。在路易大帝中学时，他遇到了让·加斯东·达布的儿子，他是那里的学生。两人成为挚友，通过让·加斯东·达布家族，博雷尔结识了当时顶尖的数学家，特别是埃米尔·皮卡，他后来评论说此人对他在这一时期的成长有重大影响。

在大学学习期间，他曾在蒙彼利埃与工程师们一起服兵役一段时间，但他似乎能够做到这一点而不会对他的数学工作造成太大干扰。他在1890年发表了他的前两篇数学论文。这些是相当次要的作品，但对于一个刚刚开始学习的年轻本科生来说仍然很了不起。1892年，他被授予数学领域的教师资格会考（Agrégation），同年他的第六篇论文，也是他的第一篇重要论著，发表了。一年后，他在提交了学位论文Sur quelques points de la théorie des fonctions Ⓣ（关于函数论的某些问题）后获得了博士学位。他的学位论文导师是让·加斯东·达布。爱德华·科林伍德在[5]中关于这篇已发表的学位论文写道：-

……测度论、博雷尔的发散级数理论、他的非解析延拓理论以及拟解析函数理论，都源自在这篇论文中首次出现的思想。这篇论文还包含了对著名的覆盖定理的明确陈述和证明，而这个定理很不恰当地获得了爱德华·海涅-博雷尔定理的名称……

博雷尔谈到早年对他影响最大的数学家时，提到了卡米耶·若尔当、埃米尔·皮卡、保尔·阿佩尔、爱徳华·古尔萨、保罗·潘勒韦和马塞尔·布里渊等人。

几乎在他获得博士学位的同时，当时他还只有22岁，博雷尔被任命为里尔大学的讲师。他在里尔度过了三年，期间发表了22篇论文，离开时还有几篇正在印刷中。令人惊叹的不仅是数量，还有他所完成工作的卓越质量。他于1897年1月回到巴黎，被任命为巴黎高等师范学院的讲师。1897年晚些时候，他担任了8月9日至11日在苏黎世举行的第一届国际数学家大会的联合秘书。

在接下来的几年里，博雷尔在事业和他所创造的杰出数学方面都取得了很大成就。从1899年到1902年，他在法兰西学院任教，并担任佩科课程的候补。1900年，他被任命为海军学校入学考试的考官，担任此职位十年。他于1898年获得科学院大奖，1901年获得让-维克托·彭赛列奖，1904年获得瓦扬奖，1905年获得佩蒂·多尔姆西奖。同样在1905年，他当选为法国数学会主席。

1901年10月，他与数学家保尔·阿佩尔的女儿玛格丽特·阿佩尔结婚。他们没有孩子，但收养了博雷尔的一个侄子费尔南·勒博，这是他大姐的儿子，他在蒙托邦中学读书时曾与大姐同住。他的姐姐和她的牧师丈夫都英年早逝，而悲剧的是，费尔南后来在第一次世界大战中阵亡。Marguerite Borel是一位杰出的作家，以笔名卡米耶·马尔博（马尔博取自玛格丽特的前三个字母和博雷尔的前两个字母）写作，并获得了1913年凭借La Statue voilée Ⓣ（《蒙面的雕像》）获得费米娜奖的殊荣。

1909年，博雷尔被任命为索邦大学专为他设立的函数论讲席，并一直担任该教授职位至1941年。1910年，在于乐·达奈希去世后，他获得了巴黎高等师范学院副校长的显赫职位，并担任该职十年。他后来说，他认为这次任命是他当时所获得的最大荣誉。第一次世界大战期间，他志愿参军，指挥一个炮兵连[5]：——

战争爆发时，由于他的学生都奔赴前线，他在巴黎和前线组织了一个关于声波测距的项目；但1915年，时任战争部长的保罗·潘勒韦任命他负责一个中央研究与发明部门，后来在担任总理时，又任命他为内阁办公室秘书长。1917年保罗·潘勒韦倒台后，博雷尔一度重返前线……

1918年，他因在第一次世界大战期间为祖国所做的努力而获得战争十字勋章。然而，战争结束后回到巴黎高等师范学院的职位，对他而言在情感上非常艰难。他的学生有一半阵亡，他自己的养子也在战争中牺牲。1920年，他辞去巴黎高等师范学院副校长一职，这一时期标志着他的职业生涯和研究兴趣都发生了方向性转变。1921年，他当选为Académie des Sciences院士，1933年成为其副主席，1934年成为其主席。同样在1921年，他被任命接替约瑟夫·布西内斯克担任概率论与数学物理讲席，次年，他创立了巴黎大学统计研究所。1928年，在洛克菲勒和罗斯柴尔德的资助下，他建立了儒勒·昂利·庞加莱研究所（博雷尔中心现在是该研究所的一部分），并管理该研究所三十年。

在[8]中，博雷尔的数学工作被分为以下主题：算术；数值级数；集合论；集合的测度；零测度集的稀疏化；实变量的实函数；复变量的复函数；微分方程；几何学；概率演算；以及数学物理。博雷尔创建了第一个有效的点集测度理论。这项工作，连同另外两位法国数学家勒内-路易·贝尔和昂利·勒贝格的工作，标志着实变量函数现代理论的开端。博雷尔虽然不是第一个定义发散级数之和的人，但却是第一个为发散级数发展出系统理论的人，他在1899年做到了这一点。1905年后，他对概率论产生了兴趣。他对这一领域的贡献在[9]中有详细描述，我们在此引用该论文摘要的一部分：——

除了许多教科书之外，博雷尔在1905年至1950年间发表了五十多篇关于概率演算的论文。它们主要受到儒勒·昂利·庞加莱、约瑟·伯特兰、汉斯·赖兴巴赫和约翰·梅纳德·凯恩斯的启发或影响。然而，由于他对数学的现实主义态度，他大多持相反的观点。他强调概率论的重要实用价值。他着重于在不同社会学、生物学、物理学和数学科学中的应用。他更倾向于阐明这些应用，而不是寻求概率论的公理化。对他来说，其本质特性是不可预测性、非决定性和不连续性。尽管如此，他对澄清概率概念仍感兴趣。

此外，在1921年至1927年间，博雷尔发表了一系列关于博弈论的论文，并成为第一个定义策略博弈的人。然而，在此期间，他已经开始了政治生涯。他在1924年写了一部重要的政治著作La politique républicaine Ⓣ（共和政策）[3]：-

……并开始了为统一欧洲而奋斗，并以不屈不挠的乐观精神继续余生。……与此同时，他加入了共和-社会党，即白里安和保罗·潘勒韦所属的团体，并于1924年当选为阿韦龙省的议员，1928年当选为圣阿弗里克的议员。

多年来，他出版了许多杰出著作，包括Leçons sur la théorie des fonctions Ⓣ（函数论讲义）（1898年）、Leçons sur les séries entières Ⓣ（整级数讲义）（1900年）、Leçons sur les fonctions divergentes Ⓣ（发散函数教程）（1901年）、Leçons sur les fonctions de variables réelles et les développements en séries de polynômes Ⓣ（实变函数与多项式级数展开教程）（1905年）、Le Hasard Ⓣ（机遇）（1913年）、Leçons sur les fonctions monogènes uniformes d'une variable complexe Ⓣ（复变单演函数教程）（1917年）、L'Espace et le temps Ⓣ（空间与时间）（1921年）、Méthodes et problèmes de la théorie des fonctions Ⓣ（函数论的方法与问题）（1922年）、Traité du calcul des probabilités et ses applications Ⓣ（概率论及其应用）（1924-1934年）以及Principes et formules classiques du calcul des probabilités Ⓣ（概率的原理与经典公式）（1925年）。

1924年后，博雷尔活跃于法国政府，在法国众议院任职（1924-1936年），并担任海军部长（1925年4月至11月）。第二次世界大战期间，他继续创作极具原创性的数学著作，如Valeur pratique et philosophique des probabilités Ⓣ（概率的实践与哲学价值）（1939年）、（与安德烈·谢龙合著）Théorie mathématique du bridge à la portée de tous Ⓣ（面向大众的桥牌数学理论）（1940年）、Le jeu, la chance et les théories scientifiques contemporaines Ⓣ（机遇与当代科学理论）（1941年）、Les probabilités et la vie Ⓣ（概率与生命）（1943年）以及Evolution de la mécanique Ⓣ（力学的演化）（1943年）。然而，他也参与了抵抗运动，并于1941年被捕，在弗雷斯内监狱关押了一个月，但很快获释。在维希政权下被监禁后，他继续为抵抗运动工作。战后，他被授予带花结的抵抗运动勋章和荣誉军团大十字勋章（1950年）。1946年，当他75岁时，博雷尔出版了引人入胜的著作Les paradoxes de l'infini Ⓣ（无限的悖论）。但更多作品出自他的笔下，如Eléments de la théorie des ensembles Ⓣ（集合论基础）（1949年）、Les nombres inaccessibles Ⓣ（不可达数）（1952年）以及L'imaginaire et le réel en mathématiques et en physique Ⓣ（数学与物理中的虚与实）（1952年）。他还做出了[3]：-

……对阿尔伯特·爱因斯坦相对论知识的若干宝贵贡献。

这位杰出人物还担任了其他职务，例如1946年当选为经度局成员，1948年被任命为联合国教科文组织科学委员会主席。由于他的科学工作，他于1955年获得了国家科学研究中心的首枚金质奖章。

Émile Borel's father was Honoré Borel who was a Protestant minister. Honoré Borel himself was the son of a craftsman from Montauben, the capital of Tarn-et-Garonne in the Midi-Pyrénées region. Émil's mother was Émilie Teissié-Solier who was the daughter of a rich wool merchant from Saint Affrique. Émilie's father was opposed to her marrying a minister, wanting her to marry a rich landowner from the district. As a result he disinherited her and bad feeling existed between Emilie and her father. At the time Émile was born, his parents were living in a fine eighteenth century house in which Honoré Borel had a school for the children of Protestant families in the neighbourhood. Honoré and Émilie had two daughters who were much older than Émile, being sixteen and fourteen years old when he was born. Honoré Borel was an intelligent man and for the first years of his son's life he educated him at home.

When Émile was eleven years old he left home in Saint Affrique and went to live with his eldest sister, Madame Lebeau, who was married to a minister from Montauben. This meant that he was able to attend the Lycée at Montauben. There he showed extraordinary talents and, several years later, went to Paris as a bursar to the Collège Sainte-Barbe. While studying there, he attended courses at the Lycée Louis-le-Grand in preparation for taking the examinations to enter the École Polytechnique and the École Normale. He had the distinction of being ranked first in both these examinations and could choose either of the two universities. He also took first prize in the Concours Général. One might have expected this achievement to cause high praise to he heaped on the brilliant schoolboy. However [5]:-

The spectacular feat attracted a good deal of publicity, not all of it sympathetic. For there was a sneering reference by a prominent man to the ephemeral nature of schoolboy success and its quickly fading laurels with a challenge to the young champion to show himself in ten years time. Before half that time had passed the ungenerous critic could have been made to look foolish ...

Family friends urged Borel to enter the École Polytechnique, which was considered the more prestigious establishment, but he had other ideas.

The family friends were merchants and industrialists, and their advice was to aim at an occupation where he would make a lot of money. Borel was advised that a degree from the École Polytechnique would give him the best opportunities for a job in industry or business. He, however, had very strong views that he wanted a job in science, particularly in mathematics, and he was sure that the best opportunities for such a position would be achieved by studying at the École Normale. He did have the support of his father for his decision, entering in 1889. We should mention a significant factor in his decision. While at the Lycée Louis-le-Grand he had met Gaston Darboux's son who was a pupil there. The two became firm friends and through the Darboux family Borel came to know leading mathematicians of the day, in particular Émile Picard whom he later remarked was a major influence on him at this time.

While studying at university he undertook military service with the engineers at Montpellier for a while but he seemed to be able to do this without too much disruption to his mathematical work. He published his first two mathematical papers in 1890. These were fairly minor works but still remarkable for a young undergraduate student just starting his studies. In 1892 he was awarded the agrégation in mathematics and in the same year his sixth paper, which is his first major memoir, was published. One year later he was awarded his doctorate after submitting his thesis Sur quelques points de la théorie des fonctions Ⓣ. His thesis advisor was Gaston Darboux. Collingwood writes [5] about this published thesis:-

... the theory of measure, Borel's theory of divergent series, his theory of non-analytic continuation and the theory of quasi-analytic functions all derive from ideas which make their first appearance in this paper. And it contained the explicit statement and proof of the famous covering theorem which, quite inappropriately, acquired the name of the Heine-Borel theorem ...

Later Borel spoke about the mathematicians who had influenced him most in his early years mentioning, among others, Camille Jordan, Émile Picard, Paul Appell, Édouard Goursat, Paul Painlevé and Marcel Brillouin.

At almost exactly the same time that he was receiving his doctorate, when still only 22 years of age, Borel was appointed Maître de Conférence at the University of Lille. He spent three years at Lille during which time he published 22 papers, with several others in the hands of the printer when he left. It was not only the quantity that is amazing but also the remarkable quality of the work that he produced. He returned to Paris in January 1897 when appointed Maître de Conférence at the École Normale Supérieure. Later in 1897 he was joint secretary at the first International Congress of Mathematicians held in Zürich from 9 August to 11 August.

Borel achieved much over the next years, both in his career and in the outstanding mathematics which he produced. From 1899 to 1902 he taught at the Collège de France and was reserve for the Cours Peccot. He was appointed examiner for entry to the École Navale in 1900, holding this position for ten years. He was awarded the Grand Prix of the Academy of Sciences in 1898, he was awarded the Poncelet Prize in 1901, he received the Vaillant Prize in 1904, and the Petit d'Ormcy Prize in 1905. Also in 1905 he was elected president of the French Mathematical Society.

In October 1901 he had married Marguerite Appell, daughter of the mathematician Paul Appell. They had no children but adopted one of Borel's nephews, Fernand Lebeau, the son of his eldest sister whom he had lived with while a pupil at the Lycée at Montauben. Both his sister and her pastor husband had died young and, tragically, Fernand was later killed during World War I. Marguerite Borel was an outstanding author, writing under the pen name Camille Marbo (Marbo being the first three letters of Marguerite and the first two of Borel), and received the distinction of being awarded the Prix Femina in 1913 for La Statue voilée Ⓣ.

In 1909 Borel was appointed to a chair of Theory of Functions created specially for him at the Sorbonne and he went on to hold this professorship until 1941. The year 1910 saw him attain the prestigious position of Deputy Director of the École Normale Supérieure, on the death of Jules Tannery, which he held for ten years. He later said that he regarded this appointment as the greatest honour he had received up to that time. During World War I he volunteered for military service and commanded an artillery battery [5]:-

On the outbreak of war, as his students left for the front, he had organised a project partly in Paris and partly at the front on sound-ranging; but in 1915 Painléve, then Minister of War, placed him in charge of a central department of research and inventions and later, when Prime Minister, appointed him Secretary-General in the Cambinet Office. On the fall of Painléve in 1917, Borel returned for a time to the front ...

In 1918 he received the Croix de Guerre for his efforts for his country during World War I. However returning to his position at the École Normale after the war ended proved emotionally very difficult for him. Half of his students had been killed, and his own adopted son had also been killed in the war. He resigned as Deputy Director of the École Normale in 1920 and this period marks a change in direction in both his career and in his research interests. In 1921 he was elected to the Académie des Sciences, becoming its vice-president in 1933 and its president in 1934. Also in 1921 he was appointed to succeed Joseph Boussinesq in the chair of Probability and Mathematical Physics and, in the following year, he founded the Institut de Statistique de l'Université de Paris (Institute of Statistics at the University of Paris). In 1928, with financial support from Rockefeller and Rothschild, he set up the Institut Henri Poincaré (the Centre Émile Borel is now part of the Institute) and he ran the Institute for thirty years.

In [8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics. Borel created the first effective theory of the measure of sets of points. This work, along with that of two other French mathematicians, René Baire and Henri Lebesgue, marked the beginning of the modern theory of functions of a real variable. Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899. After 1905 he became interested in the theory of probability. His contributions to this area are described in detail in [9] and we quote here part of the abstract of that paper:-

In addition to many textbooks, Borel published more than fifty papers between 1905 and 1950 on the calculus of probability. They were mainly motivated or influenced by Poincaré, Bertrand, Reichenbach, and Keynes. However, he took for the most part an opposed view because of his realistic attitude toward mathematics. He stressed the important and practical value of probability theory. He emphasized the applications to the different sociological, biological, physical, and mathematical sciences. He preferred to elucidate these applications instead of looking for an axiomatization of probability theory. Its essential peculiarities were for him unpredictability, indeterminism, and discontinuity. Nevertheless, he was interested in a clarification of the probability concept.

In addition, between 1921 and 1927, Borel published a series of papers on game theory and became the first to define games of strategy. During this period, however, he had already embarked on a political career. He wrote an important political work in 1924 La politique républicaine Ⓣ [3]:-

... and began the struggle for a united Europe, and with unbending optimism continued it for the rest of his life. ... In the meantime he joined the Republican-Socialist Party, the group to which Briand and Painlevé belonged, and was elected deputy for Aveyron in 1924, for St Affrique in 1928.

He published many outstanding works over the years including Leçons sur la théorie des fonctions Ⓣ (1898), Leçons sur les séries entières Ⓣ (1900), Leçons sur les fonctions divergentes Ⓣ (1901), Leçons sur les fonctions de variables réelles et les développements en séries de polynômes Ⓣ (1905), Le Hasard Ⓣ (1913), Leçons sur les fonctions monogènes uniformes d'une variable complexe Ⓣ (1917), L'Espace et le temps Ⓣ (1921), Méthodes et problèmes de la théorie des fonctions Ⓣ (1922), Traité du calcul des probabilités et ses applications Ⓣ (1924-1934), and Principes et formules classiques du calcul des probabilités Ⓣ (1925).

After 1924, Borel became active in the French government serving in the French Chamber of Deputies (1924-36) and as Minister of the Navy (April to November 1925). During World War II he continued to produce mathematical works of great originality such as Valeur pratique et philosophique des probabilités Ⓣ (1939), (with André Chéron) Théorie mathématique du bridge à la portée de tous Ⓣ (1940), Le jeu, la chance et les théories scientifiques contemporaines Ⓣ (1941), Les probabilités et la vie Ⓣ (1943), and Evolution de la mécanique Ⓣ (1943). However he was also involved in fighting with the Resistance and he was arrested and imprisoned at Fresnes for a month in 1941, but soon released. After his imprisonment under the Vichy regime he continued working for the Resistance. After the war, he was awarded the Médaille de la Résistance with rosette, and the Grand Croix Légion d'Honneur (1950). In 1946, when he was 75 years old, Borel published the fascinating book Les paradoxes de l'infini Ⓣ. But further works were to come from his pen such as Eléments de la théorie des ensembles Ⓣ (1949), Les nombres inaccessibles Ⓣ (1952), and L'imaginaire et le réel en mathématiques et en physique Ⓣ (1952). He also made [3]:-

... a number of valuable contributions to the knowledge of Einstein's theory of relativity.

Further roles were given to this distinguished man such as being elected as a member of the Bureau des Longitudes in 1946 and being made president of the Science Committee of UNESCO in 1948. For his scientific work he received the first gold medal of the Centre National de la Recherche Scientifique in 1955.