保罗·莱维Paul Pierre Lévy

保罗·莱维出生于一个包含多位数学家的家庭。他是Lucien Lévy（1853-1912）和Alice Flora Wolff（1856-1921）的儿子。Lucien和Alice于1878年6月3日结婚。Lucien Lévy于1872年至1877年在巴黎综合理工学院学习，之后开始了数学教授生涯，先是在雷恩的中学，然后在巴黎的路易大帝中学。除了在中学任教外，在他职业生涯的大部分时间里，他还是巴黎综合理工学院的考官。他写了几本书：Arithmétique à l'usage de l'enseignement secondaire spécialⓉ（供特殊中等教育使用的算术）（1880年），Éléments d'arithmétiqueⓉ（算术基础）（1883年），Précis élémentaire de la théorie des fonctions elliptiques avec tables numériques et applicationsⓉ（椭圆函数理论初等精要及数值表和应用程序）（1898年），（与欧仁·鲁谢合著）Analyse infinitésimale à l'usage des ingénieurs第1卷Ⓣ（工程师用无穷小分析第1卷）（1900年），（与欧仁·鲁谢合著）Analyse infinitésimale à l'usage des ingénieurs第2卷Ⓣ（工程师用无穷小分析第2卷）（1902年）。他于1911年担任法国数学会主席。Lucien Lévy是Edouard Lévy（1822-1900）和Hortense Breinlé Cahen（1832-1909）的儿子。Edouard Lévy，即保罗·莱维的祖父，于1843年毕业于巴黎高等师范学校，也从事数学教授职业。保罗·莱维的母亲Alice Flora Wolff是皮埃尔·萨米埃尔 Wolff（1827-1913，桥梁公路工程局总监）和Fanny Goudchaux（1835-1888）的女儿。

保罗·莱维在巴黎上了三所中学：蒙田中学（1895年至1898年）、路易大帝中学（1898年至1902年）和圣路易中学（1902年至1904年）。在蒙田中学，保罗·莱维学习文学和艺术科目，肯定不是班上最好的学生，有三位才华横溢的同学Jacques Massigli、保罗·莱维 Lachièze和Marc Bloch占据了前三名。1898年初，他的老师M de la Filolie告诉他，他在 Concours général 中获得了希腊语奖。保罗·莱维写道[2]：-

这将是我唯一的文学成就。

当保罗·莱维于1898年晚些时候开始在路易大帝中学学习时，Émile Mâle是该校的修辞学教授。然而，他获得希腊语奖的成就[2]：-

……不足以引起Émile Mâle对我的注意，他从高处将他对拉丁语和考古科学知识的一小部分传授给班上最好的学生。

正是数学点燃了保罗·莱维在路易大帝中学的热情。他写道[2]：-

我尤其擅长平面几何。我在空间想象方面很差，也没有记忆力去对代数产生热情，但这并不妨碍我……在这门学科上也名列前茅。我热衷于平面几何；某些定理对我来说是启示：点关于圆的幂的概念、根轴的概念、通过反演变换图形的概念，都让我欣喜。这些寻找新定理的方法多么令人赞叹！别人给我的问题我总是觉得很容易。我曾希望在修辞学年级的普通竞赛中获得数学奖；我确实得到了，此后却没有机会再参加这项竞赛。然而，我对自己在文学学习阶段的状态和第二年将要成为的状态之间有着巨大的区别。

为了准备进入大学校，保罗·莱维的父亲把他从路易大帝中学转到圣路易中学，完成最后两年的学业。他解释说[2]：-

修辞学年级结束后，在暑假期间学习了三角学的基础知识，我进入了圣路易中学，进入一个称为高等的初等数学班……父亲认为，为了准备巴黎综合理工学院，可能还有巴黎高等师范学校，不应该浪费时间。我可以在中学学习数学的同时，很好地准备哲学业士学位考试。此外，就数学而言，如果我没有跳过业士学位预备班，我就会浪费时间。但路易大帝中学的高年级班比圣路易中学的水平更高，为了不让我付出过多的努力，父亲把我放在了圣路易中学。我有时会想，如果我留在路易大帝中学是否会更好。我相信我会轻松地进入数学班。

在圣路易中学的最后一年，1903-04年，保罗·莱维由埃米尔·布吕泰尔（1862-1932）授课。布吕泰尔是一位杰出的教师，后来出版了许多优秀的教科书：Leçons de Mathématiques spéciales à l'École Normale Supérieure et des Étudiants des Facultés des Sciences Ⓣ（巴黎高等师范学校及理学院学生的特殊数学课程）（1914）；Leçons de mathématiques spéciales à l'usage des candidats à l'École Polytechnique: Géométrie analytique, courbes et surfaces Ⓣ（供巴黎综合理工学院考生使用的特殊数学课程：解析几何、曲线与曲面）（1914）；以及Leçons de mathématiques spéciales, candidats à l'École Polytechnique et à l'École Normale Supérieure: Algébre, ligne droite et plan, trigonométrie, analyse, applications géométriques Ⓣ（特殊数学课程，巴黎综合理工学院和巴黎高等师范学校考生：代数、直线与平面、三角学、分析、几何应用）（1914）。

在两所院校的入学考试中，保罗·莱维被巴黎高等师范学校录取为第一名，被巴黎综合理工学院录取为第二名。他选择进入当时被认为声望较低的巴黎综合理工学院，可能是由于他父亲的原因。还在那里读本科时，他于1905年发表了他的第一篇论文Sur les séries semi-convergentes Ⓣ（论半收敛级数），次年发表第二篇Sur la densité des nombres premiers inférieurs à une grandeur donnée Ⓣ（论小于给定数的素数的密度）。在谈到他的教授时，他特别提到了Georges Humbert [2]：-

我们最杰出的教授无疑是Georges Humbert，他给我们讲授的分析课程是我受益最多的。他的讲解总是非常清晰。我的同学们甚至声称这太清晰了，说他回避了我们在学习发给我们的讲义时注意到的困难。我只见过一次，关于基本的雅可比定理，确实如此。在我们的其他教授中，我只提儒勒·昂利·庞加莱和亨利·贝克勒尔。前者在理论上是一位天文学教授；但除了开头之外，课程是由一位代课教师讲授的。儒勒·昂利·庞加莱本人讲授了一节球面三角学课和两节误差理论课。

以第一名毕业后，保罗·莱维服了一年兵役，然后于1907年在矿业学校攻读工程学位。在矿业学校学习期间，他还参加了索邦大学由让·加斯东·达布和埃米尔·皮卡讲授的课程。此外，他还参加了法兰西学院由Georges Humbert和雅克·阿达马讲授的讲座。保罗·莱维写道[31]：-

在矿业学校，我忽视了许多课程，以便参加索邦大学和法兰西学院的数学课程。我在法兰西学院听了Humbert'三年的课程。他以出色的计算能力而闻名，并为经典数学的进步做出了贡献。我特别倾向于现代数学的研究，而Humbert'的课程并不是我受益最多的。也不是埃米尔·博雷尔的课程，我在听了四五节课后就放弃了，因为他在显而易见的内容上花费了太多时间。另一方面，我从埃米尔·皮卡的课程中受益匪浅，我听了一年，还有雅克·阿达马的课程，我听了三年。正是在1910年雅克·阿达马的课程中，我找到了我的学位论文题目。

正如他所解释的，正是雅克·阿达马对确定保罗·莱维将从事研究的主题产生了主要影响。1910年完成矿业学校的学业后，他开始研究functional analysis。他关于这个主题的学位论文，Sur les equations intégro-differentielles définissant des fonctions de lignes Ⓣ（论定义线函数的积分微分方程），于1911年由埃米尔·皮卡、儒勒·昂利·庞加莱和雅克·阿达马审查，并于1912年获得理学博士学位。

1913年，保罗·莱维与苏珊娜保罗·莱维（1892-1973）结婚，她是商人保罗·莱维（1853-1903）和贝尔特安德烈·韦伊（1862-1930）的女儿。[苏珊娜的父亲和丈夫确实同名！]苏珊娜的外祖父是Henri Weil（1817-1913），一位语文学家，法兰西学会成员。保罗·莱维和苏珊娜保罗·莱维育有三个孩子：玛丽-埃莱娜（1913年11月出生）、德尼丝（1916年10月3日出生）和让·克洛德（1918年4月10日出生）。Marie-Hélène Lévy成为了一名数学家；她嫁给了洛朗·施瓦茨，改姓Marie-Hélène Schwartz，在本档案中有传记。德尼丝成为巴黎莫里哀中学的德语教授。她嫁给了工程师罗贝尔·皮龙（下文提及）。让·克洛德就读于巴黎综合理工学院，成为海军工程师。

保罗·莱维于1913年成为巴黎矿业学校的数学教授和巴黎综合理工学校的辅导教师[4]：-

辅导教师的任务是帮助学生理解教授的讲课，因此保罗·莱维在被任命为教授之前就非常熟悉综合理工学校的课程和教学方法。

第一次世界大战给法国带来了重大破坏，就像给欧洲大部分地区带来的破坏一样。保罗·莱维服了兵役，并被要求将他的数学技能用于战争工作。他被分配了实际的防御问题，尤其是那些由当时与航空相关的新技术引起的问题，特别是解决有关防御来自空中攻击的问题。

George Humbert自1895年起在巴黎综合理工学院任教，当保罗·莱维担任辅导教师时，他是教授之一，但他病重，请求保罗·莱维代他上课，并在1918-19学年接管他的部分教学任务。Humbert健康状况不佳，使他辞职，保罗·莱维于1920年成为巴黎综合理工学院的分析学正式教授。保罗·莱维留在巴黎综合理工学院，其间有下文所述的某些中断，直到1959年退休。

年轻数学家René Gateaux在第一次世界大战初期阵亡，雅克·阿达马请保罗·莱维整理Gateaux的著作以便出版。他这样做了，但他并没有止步于整理Gateaux的成果，而是采纳了Gateaux的思想并进一步发展，在战争结束后的1919年出版了这些材料，题为Publication des oeuvres posthumes de R Gateaux et notes personnelles complétant les résultats obtenus par cet auteurⓉ（《R Gateaux遗著出版及补充该作者所获成果的个人笔记》）。

正如我们上面所指出的，保罗·莱维首先研究的是分析[33]：-

... 本着维多·沃尔泰拉的精神完成的。这涉及将实变量函数的微积分扩展到点可以是曲线、曲面、序列或函数的空间。

1919年，作为其教学职责的一部分，保罗·莱维被要求在巴黎综合理工学院就以下内容作三次讲座（见[18]）：-

... 概率微积分的概念以及高斯定律在误差理论中的作用。

Taylor在[32]中写道：-

当时还没有关于probability的数学理论——只有一堆小的计算问题。如今它已是数学的一个成熟分支，运用现代分析各个分支的技巧，并在思想、问题、结果和可用于其他地方的有用工具方面作出自己的贡献。如果有一个人对概率论的建立和发展产生的影响超过其他任何人，那个人必定是保罗·莱维。

保罗·莱维的讲座内容在[4]中有描述，作者在结尾给出如下总结：-

保罗·莱维于1919年在巴黎综合理工学院的讲义很好地记录了他作为概率论学者的起步。尽管这些讲义显然没有发展到所需的详细程度（我们不应忘记保罗·莱维是临时仓促接手教学的），但它们表明了保罗·莱维在其非凡的第二次数学职业生涯前夕是如何看待概率的。它们包含了保罗·莱维将以精湛技艺追求40年的研究的种子：依分布收敛、随机变量之和、数学概率与物理学之间的联系。它们印证了保罗·莱维50年后在接受法国电台France Culture采访时对自己的评价：“我感觉自己是一个数学家，既不比别人优越也不比别人逊色，但与他们不同。”

Loève在[19]中对保罗·莱维的贡献作了非常生动的描述：-

保罗·莱维是概率世界中的画家。如同那些极其伟大的绘画天才一样，他的调色板是他自己的，他的画作永远改变了我们对现实的看法。……他的三个主要时期（略有重叠）是：极限律时期，以路径时间色彩绘制的可加过程与鞅的伟大时期，以及布朗运动探路者时期。

保罗·莱维不仅对概率论和泛函分析作出贡献，还研究偏微分方程和级数。1926年，他将皮埃尔·西蒙·拉普拉斯变换推广到更广泛的函数类。他开展了关于泛函导数中广义微分方程的大规模工作。他还研究几何学。

第二次世界大战给保罗·莱维带来了严重的困难。1940年5月10日，德国入侵荷兰、比利时和卢森堡，随后于6月初迅速转向进攻法国。6月14日德军进入巴黎，6月22日法国元帅贝当签署停战协定。这使法国分裂为由与德国合作的政府控制的维希法国和直接受德国控制的部分。巴黎综合理工学院具有一定的军事地位，德国拒绝允许它留在巴黎。它迁至维希法国的里昂，保罗·莱维搬到里昂并开始教学。至此我们尚未提及保罗·莱维是犹太人，因为这与主题无关。现在，这当然成了一个危及生命的困难。1940年10月3日的立法要求所有犹太人被解除教职，1940年12月19日保罗·莱维收到通知，他不能再履行职责。1940年10月3日的法案中有豁免条款，允许那些做出杰出服务的人获得豁免。巴黎综合理工学院院长的上诉成功，从1941年3月14日起他再次被允许教学。

1942年夏天情况变得更糟，保罗·莱维预见到事态发展，离开里昂，去与他的女婿Robert Piron同住，地点在格勒诺布尔附近的Montbonnot，位于法国意大利控制区。他在11月11日德国入侵维希法国之前，已于11月4日前离开里昂。他现在过着藏匿的生活，携带假证件，不知怎么地活了下来。他于1943年11月29日写给莫里斯·弗雷歇的一封信显示了他极度的困难[25]：-

我刚得知我不再是巴黎综合理工学院的教授了。11月初，一张为我10月工资开出的邮政支票被退回给我，理由是“资金不足”。我写信给学院的行政部门要求解释。11月24日，我收到一份日期为6月30日的决定副本，学院据此于10月1日将我转到矿业团；从决定文本中还可以看出，4月29日学院院长已经知道我不会被“重新聘用”，并询问部长他还要付我多久工资！！！而这一切都是合法的，如果人们认为1941年的一项规定——使教师每十年重新选举一次，不论既得权利——是合法的话。现在矿业团会怎么处理我，我1913年就离开了那个地方，无法给我一份我已不再具备专业知识的工作，而且还要等两个月才能上任。它会付我工资吗？……我深切希望1944年能结束这些苦难。

到保罗·莱维写这封信时，巴黎综合理工学院的一部分已在巴黎重新运作（它已于1943年4月返回），但一些教学仍在里昂进行，面向逃出的战俘、犹太学生和其他当时无法返回巴黎的人。1944年6月盟军入侵法国后，德国占领者被击退，到那年8月维希政权已经垮台。

战争年代对保罗·莱维来说是极其困难的时期，在里昂、格勒诺布尔附近的Montbonnot，最后在卢瓦尔，但他仍然能够做数学[25]：-

1940年，保罗·莱维已成为世界主要的随机过程专家之一。然后突然间，这位55岁的数学家，在20多年里发表了大量出版物，却被剥夺了向期刊投稿的可能性。此外，正如我们所见，他的教学任务也逐渐减少，1943年的几个月里，保罗·莱维在格勒诺布尔附近的意大利区过着不稳定的沉默生活，除了数学无事可做，等待更好的日子。他的幸运是保持与莫里斯·弗雷歇的联系，后者为他扮演了私人登记期刊的角色，有时也可以作为保罗·莱维和其他数学家之间的枢纽。

回到巴黎后，他继续在矿业学校任教至1951年，在巴黎综合理工学院任教至1959年。然而，其间有过中断。他感染了肺结核，1951年左肺被切除了一半。1957-58年他又生病无法授课，但后来康复了。

保罗·莱维写了十本书，主要的是：Leçons d'analyse fonctionnelle Ⓣ（泛函分析教程）（1922），Calcul des probabilités Ⓣ（概率计算）（1925），Théorie de l'addition des variables aléatoires Ⓣ（随机变量的加法理论）（1937；1954），以及Processus stochastiques et mouvement brownien Ⓣ（随机过程和布朗运动）（1948）。关于保罗·莱维所有书籍的信息，包括其中一些书的内容以及一些评论的摘录，见THIS LINK。

保罗·莱维关于概率计算的看法，见他的演讲Les fondements du calcul des probabilités Ⓣ（概率计算基础）的英文版，该演讲于1950年代初在巴黎综合理工学院向听众发表，见THIS LINK。

关于Levy在库尔特·弗雷德里希·哥德尔和科恩的工作之后对公理集合论和连续统假设的看法，见THIS LINK。

1963年，保罗·莱维被选为伦敦数学会的名誉会员。次年，他被选为Académie des Sciences。

Loève用以下的话总结了他的文章[19]：-

[保罗·莱维]是一个非常谦逊的人，同时完全相信理性思考的力量。……每当我经过卢森堡公园，我仍能看到我们在那里散步，坐在长椅上晒太阳；我仍能听到他仔细地说出他的想法。我认识了一位伟人。

保罗·莱维于1971年去世，享年85岁。1973年3月23日星期五，在巴黎综合理工学院的儒勒·昂利·庞加莱阶梯教室举行了纪念他的追思会，会上进行了各种演讲。这些演讲于1973年11月发表在La Jaune et la Rouge Ⓣ（黄色与红色）上。

关于洛朗·施瓦茨演讲的英文版，La pensée mathématique de P LévyⓉ（P 保罗·莱维的数学思想），见THIS LINK。

关于Paul-André Meyer演讲的英文版L'originalité de P Lévy en mathématiquesⓉ（P 保罗·莱维在数学中的独创性），见THIS LINK。

关于曼德尔布罗演讲的英文版Paul Lévy, professeur，以及曼德尔布罗对保罗·莱维的另一描述，见THIS LINK。

我们以Jean Ullmo（1906-1980）在1973年3月3日会议上所作的演讲Paul Lévy的摘录来结束这篇保罗·莱维的传记[35]：-

保罗·莱维是他那个时代最伟大的数学家之一，矿业 corps 能够促进他非凡天赋的充分实现，这是一种荣誉。作为一名学者，他无疑是绝对个人主义的最后典范之一：他是一位孤独的研究者，只关心提出自己感兴趣的问题，并通过纯粹的内在反思工作来寻求解答。他很少阅读他人的著作，不参加国际会议，除了在生命末期例外。他的工作方法，可以说是手工式的，有其缺点：他常常在不知情的情况下发现已知的结果；更常见的是，他发现重要结果却没有给予必要的宣传，有时是因为他认为它们已经为人所知。不可否认，有过浪费的努力，但对应的往往是令人钦佩的：那是一种漠视时尚和学派的思想的深刻原创性，这种思想毫不犹豫地踏上全新的道路，因为他不惧怕孤独。因此，在成为泛函分析的主要先驱之一后，保罗·莱维成为了概率论的伟大创造者。可以说，这一理论的大部分基本概念都源自于他。

他在巴黎综合理工学院的教学，持续时间异常长，给他无数的学生留下了深刻的印记：他们从中看到了简洁的典范，对读者的要求，读者必须认识到简洁文本中隐藏的所有困难；简而言之，这是一项令人钦佩的智力体操。让我们补充记录，这门非常经典的课程直到1957年几乎没有修改，并不意味着保罗·莱维对数学的现代发展失去了兴趣，因为在他教学的最后两年，他有勇气和优雅地重新设计他的课程，引入最新的语言和方法。

最后，我们想唤起一段个人回忆：保罗·莱维有一天和我谈起他是如何选择自己想要集中研究的问题的：“我问自己一个不太难的问题，”他对我说，“以免在过多的困难面前碰得头破血流，但同时又必须付出巨大的努力，让我投入其中，并给我发现某种东西的满足感。”我们会钦佩这种谦逊与正当自豪的混合。这些“不太难的问题”通常是向人类心智敞开的新途径。

Paul Lévy was born into a family containing several mathematicians. He was the son of Lucien Lévy (1853-1912) and Alice Flora Wolff (1856-1921). Lucien and Alice were married on 3 June 1878. Lucien Lévy had studied at the École Polytechnique from 1872 to 1877 and then had a career as a professor of mathematics, first at the lycée of Rennes and then in Paris, at the lycée Louis le Grand. As well as teaching in lycées, during most of his career he was an examiner at the École Polytechnique. He wrote several books: Arithmétique à l'usage de l'enseignement secondaire spécial Ⓣ (1880), Éléments d'arithmétique Ⓣ (1883), Précis élémentaire de la théorie des fonctions elliptiques avec tables numériques et applications Ⓣ (1898), (with Eugène Rouché) Analyse infinitésimale à l'usage des ingénieurs Vol 1 Ⓣ (1900), (with Eugène Rouché) Analyse infinitésimale à l'usage des ingénieurs Vol 2 Ⓣ(1902). He was president of the French Mathematical Society in 1911. Lucien Lévy was the son of Edouard Lévy (1822-1900) and Hortense Breinlé Cahen (1832-1909). Edouard Lévy, Paul's paternal grandfather, had graduated from the École Normale Supérieure in 1843 and also had a career as a professor of mathematics. Paul's mother, Alice Flora Wolff, was the daughter of Samuel Wolff (1827-1913), Inspector General of the Ponts et Chaussées, and Fanny Goudchaux (1835-1888).

Paul attended three lycées in Paris, the lycée Montaigne from 1895 to 1898, the lycée Louis le Grand from 1898 to 1902 and the lycée Saint Louis from 1902 to 1904. At the lycée Montaigne, Lévy was taught literature and arts subjects and was certainly not the best student in his class, having three brilliant fellow-students, Jacques Massigli, Pierre Lachièze and Marc Bloch, who took the top three places. In early 1898 his teacher M de la Filolie told him he had won the Greek prize in the Concours général. Lévy wrote [2]:-

It was to be my only literary success.

Émile Mâle was the professor of rhetoric at the lycée Louis le Grand when Lévy began studying there later in 1898. His achievement in winning the Greek prize, however [2]:-

... was not enough to draw the attention of Émile Mâle to me, who, from a great height, let fall on the best in the class a small part of his knowledge of Latin and of archaeological science.

It was mathematics that fired Lévy's passion at the lycée Louis le Grand. He wrote [2]:-

It is especially in plane geometry that I excelled. I visualise poorly in space, and I do not have the memory to have been impassioned by algebra, which did not prevent me ... to be also, in this discipline, to be at the top of my class. I was passionate about plane geometry; certain theorems were revelations for me: the notion of the power of a point in relation to a circle, that of a radical axis, that of the transformation of a figure by inversion, delighted me. What admirable methods for finding new theorems! I always found the problems easy that were presented to me. I was hoping to have the mathematics prize at the Concours Général en rhetorique; I did have it, and have not had the opportunity to participate in this competition since. Yet I make a huge difference between what I was when I was in literary studies and what I was going to be the following year.

To prepare for entry to the Grands Écoles, Lévy's father moved him from the lycée Louis le Grand to the lycée Saint Louis for his two final years of schooling. He explained [2]:-

On leaving rhetoric, after having learned the elements of trigonometry during the summer, I entered the Lycée Saint-Louis, in a class of elementary mathematics called higher .... My father thought that to prepare for the École Polytechnique, and possibly École Normale Supérieure, one should not waste time. I could prepare well for the baccalaureate in philosophy while doing mathematics in the lycée. In addition, as far as mathematics was concerned, I would have wasted my time if I had not skipped the preparatory class for the baccalaureate. But the upper class was of a higher standard in Louis Le Grand than in Saint Louis, and, in order not to require excessive effort from me, my father put me in Saint Louis. I have sometimes wondered if it would not have been better if I had stayed at Louis Le Grand. I'm sure I would have easily taken the mathematics class.

In his final year at the lycée Saint Louis, 1903-04, Lévy was taught by Émile Blutel (1862-1932). Blutel was a remarkable teacher who went on to publish a number of outstanding textbooks: Leçons de Mathématiques spéciales à l'École Normale Supérieure et des Étudiants des Facultés des Sciences Ⓣ (1914); Leçons de mathématiques spéciales à l'usage des candidats à l'École Polytechnique: Géométrie analytique, courbes et surfaces Ⓣ (1914); and Leçons de mathématiques spéciales, candidats à l'École Polytechnique et à l'École Normale Supérieure: Algébre, ligne droite et plan, trigonométrie, analyse, applications géométriques Ⓣ (1914).

Lévy was placed first for entry to the École Normale Supérieur and second for entry to the École Polytechnique in the Concours d'entrée for the two institutions. He chose to attend the École Polytechnique, considered less prestigious at that time, probably because of his father. While still an undergraduate there, he published his first paper Sur les séries semi-convergentes Ⓣ in 1905 and his second Sur la densité des nombres premiers inférieurs à une grandeur donnée Ⓣ in the following year. Writing about his professors, he singled out Georges Humbert [2]:-

The most remarkable of our professors was certainly Georges Humbert, and the course of analysis he gave us was the one from which I benefited the most. His presentations were always remarkably clear. My fellow students even claimed that it was too clear, and that he avoided difficulties that we noticed when studying the course on the sheets that were distributed to us. I only saw once, about the fundamental Jacobian theorem, that this was true. Among our other professors, I will name only Henri Poincaré and Henri Becquerel. The first was, in theory, a professor of astronomy; but the course was given by a substitute, except for the beginning. Poincaré himself taught a lesson on spherical trigonometry and two lessons on error theory.

After graduating in first place, Lévy took a year doing military service before studying for an engineering degree at the École des Mines in 1907. While he studied at the École des Mines he also attended courses at the Sorbonne given by Gaston Darboux and Émile Picard. In addition he attended lectures at the Collège de France by Georges Humbert and Jacques Hadamard. Lévy wrote [31]:-

At the École des Mines, I neglected many of my courses in order to attend mathematics courses at the Sorbonne and at the College of France. I attended Humbert's course for three years at the College of France. He is known for having been a marvellous calculator, and he contributed to the progress of classical mathematics. I was especially inclined towards the study of modern mathematics, and it was not Humbert's course that I profited from the most. It was also not Borel's, which I gave up on after 4 or 5 lectures, because he spent too much time on what was obvious. On the other hand, I profited a great deal from Emile Picard's course, which I attended for one year, and from Hadamard's, which I attended for three years. It was in Hadamard's course, in 1910, that I found my thesis topic.

As he explains, it was Hadamard who was the major influence in determining the topics on which Lévy would undertake research. Finishing his studies at the École des Mines in 1910 he began research in functional analysis. His thesis on this topic, Sur les equations intégro-differentielles définissant des fonctions de lignes Ⓣ, was examined by Émile Picard, Henri Poincaré and Jacques Hadamard in 1911 and he received his Docteur ès Sciences in 1912.

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

Lévy became professor of mathematics at the École des Mines in Paris and a tutor at the École Polytechnique in Paris both in 1913 [4]:-

A tutor's task was to help the students to understand the professors' lectures, and hence Lévy was well acquainted with the curriculum and the methods of instruction at the Polytechnique before his own appointment as professor.

World War I caused major disruption in France as it did in most of Europe. Lévy undertook military service and was asked to use his mathematical skills in the war effort. He was assigned practical problems of defence, especially those arising from the then new technology associated with aviation, in particular solving problems concerning defence against attacks from the air.

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

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

As we indicated above Lévy first worked on functional analysis [33]:-

... done in the spirit of Volterra. This involved extending the calculus of functions of a real variable to spaces where the points are curves, surfaces, sequences or functions.

In 1919, as part of his teaching duties, Lévy was asked to give three lectures at the École Polytechnique on (see [18]):-

... notions of calculus of probabilities and the role of Gaussian law in the theory of errors.

Taylor writes in [32]:-

At that time there was no mathematical theory of probability - only a collection of small computational problems. Now it is a fully-fledged branch of mathematics using techniques from all branches of modern analysis and making its own contribution of ideas, problems, results and useful machinery to be applied elsewhere. If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Lévy.

The contents of Lévy's lectures are described in [4], which the authors end with the following summary:-

Lévy's lecture notes at the École Polytchnique in 1919 nicely document his beginnings as a probabilist. Though they are evidently not developed in the detail needed (we must not forget that Lévy took the charge of the teaching at the drop of a hat), they indicate how Lévy perceived probability on the eve of his exceptional second mathematical career. They contain the seeds of the studies Lévy would pursue with virtuosity for 40 years: convergence in distribution, sums of random variables, the connection between mathematical probability and physics. They testify to what Lévy said about himself 50 years later in an interview to the French radio France Culture: "I have the feeling of being a mathematician, neither superior nor inferior to others, but different from them."

Loève, in [19], gives a very colourful description of Lévy's contributions:-

Paul Lévy was a painter in the probabilistic world. Like the very great painting geniuses, his palette was his own and his paintings transmuted forever our vision of reality. ... His three main, somewhat overlapping, periods were: the limit laws period, the great period of additive processes and of martingales painted in pathtime colours, and the Brownian pathfinder period.

Not only did Lévy contribute to probability and functional analysis but he also worked on partial differential equations and series. In 1926 he extended Laplace transforms to broader function classes. He undertook a large-scale work on generalised differential equations in functional derivatives. He also studied geometry.

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

The situation became worse in the summer of 1942 and Lévy, anticipating events, left Lyon and went to live with his son-in-law Robert Piron, in Montbonnot, near Grenoble in the Italian controlled region of France. He had left Lyon by 4 November ahead of the German invasion of Vichy France on 11 November. He now lived in hiding, carrying false documents, and somehow survived. A letter he wrote to Maurice Fréchet on 29 November 1943 shows his extreme difficulties [25]:-

I just learned that I'm not a professor at the École Polytechnique anymore. In early November, a postal check issued for my October salary was returned to me as "not sufficient funds". I wrote to the administration of the École to ask for an explanation. On 24 November, I received a copy of a decision dated 30 June by which the École transferred me to the Corps des Mines on 1 October; it is also apparent from the text of the decision that on 29 April the Director of the École knew already that I was not to be "re-invested in my job" and asked the Minister how long he had to pay me!!! And all this is legal, if one considers as legal a 1941 regulation subjecting teachers to a decennial re-election, regardless of acquired rights. And now what will the Corps des Mines do with me, having left the place in 1913, and cannot give me work for which I have no longer any expertise and, in addition, with a delay of two months before taking up the post. Will it pay me? ... I deeply hope that 1944 will see the end of these miseries.

Already by the time Lévy wrote this letter, part of the École Polytechnique was operating again in Paris (it had returned in April 1943), but some teaching was still taking place in Lyon for escaped prisoners, Jewish students and others who could not, at that time, return to Paris. After the Allied invasion of France in June 1944 the German occupiers were driven back, and by August of that year the Vichy regime had fallen.

The war years were ones of great difficulties for Lévy, in Lyon, in Montbonnot, near Grenoble, and finally in the Loire, but he was still able to do mathematics [25]:-

In 1940, Lévy had become one of the world major specialists of stochastic processes. And then suddenly, the 55 years old mathematician who had produced a torrential flow of publications for more than 20 years was deprived of the possibility to send articles to journals. Moreover, as we have seen, his teaching tasks were also gradually reduced, and for some months in 1943, living in precarious silence in the Italian zone near Grenoble, Paul Lévy had nothing to do other than mathematics while waiting for better days. His luck was to remain in contact with Fréchet who played for him the role of a private registration journal and could also sometimes serve as a hub between Lévy and other mathematicians.

Back in Paris, he continued to teach at the École des Mines until 1951 and at the École Polytechnique until 1959. There were, however interruptions. He contracted tuberculosis and half his left lung was removed in 1951. Again in 1957-58 he was ill and could not teach, but recovered.

Lévy wrote ten books, the main ones being: Leçons d'analyse fonctionnelle Ⓣ (1922), Calcul des probabilités Ⓣ (1925), Théorie de l'addition des variables aléatoires Ⓣ (1937; 1954), and Processus stochastiques et mouvement brownien Ⓣ (1948). For information about all of Lévy's books, including contents of some of them and also extracts from some reviews, see THIS LINK.

For Lévy's views on the calculus of probabilities see the English version of his talk Les fondements du calcul des probabilités Ⓣ delivered to an audience at the École Polytechnique in the early 1950s, see THIS LINK.

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

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

Loève sums up his article [19] in these words:-

[Lévy] was a very modest man while believing fully in the power of rational thought. ... whenever I pass by the Luxembourg gardens, I still see us there strolling, sitting in the sun on a bench; I still hear him speaking carefully his thoughts. I have known a great man.

Lévy died in 1971 at the age of 85. A memorial meeting in his memory was held in the Henri Poincaré amphitheatre of the École Polytechnique on Friday 23 March 1973 when various talks were given. They were published in La Jaune et la Rouge Ⓣ in November 1973.

For an English version of Laurent Schwartz's talk, La pensée mathématique de P Lévy Ⓣ, see THIS LINK.

For an English version of Paul-André Meyer's talk L'originalité de P Lévy en mathématiques Ⓣ, see THIS LINK.

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

We end this biography of Paul Lévy by quoting an extract from the talk Paul Lévy, delivered by Jean Ullmo (1906-1980) at the meeting on 3 March 1973 [35]:-

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

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

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