儒勒·昂利·庞加莱Henri Poincaré

儒勒·昂利·庞加莱的父亲是Léon 庞加莱，母亲是Eugénie Launois。在庞加莱出生时，他们分别为26岁和24岁。庞加莱出生在Nancy，他的父亲是那里的大学教授。Léon 庞加莱的家族在庞加莱的一生中还产生了其他杰出人物。Raymond Poincaré曾多次担任法国总理，并在第一次世界大战期间担任法兰西共和国总统，他是Léon 庞加莱的兄弟Antoine Poincaré的长子。Antoine Poincaré的次子Lucien Poincaré在大学行政中获得了高位。

庞加莱是[2]：-

……双手俱利，且近视；童年时他肌肉协调性差，还曾一度因白喉而重病。他接受了他那有天赋的母亲的特别教导，还在小学时就在书面作文方面表现出色。

此处一段未译出，以下为英文原文 In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France.

庞加莱于1873年进入École Polytechnique，1875年毕业。他在数学上远远领先于所有其他学生，但也许并不奇怪，鉴于他协调能力差，他在体育和艺术方面表现并不比平均水平好。音乐是他的另一个兴趣，但尽管他喜欢听音乐，他在École Polytechnique期间学习钢琴的尝试并不成功。庞加莱广泛阅读，从科普作品开始，逐渐进展到更高级的文本。他的记忆力非凡，他从所读的所有文本中保留了很多内容，但不是死记硬背的方式，而是通过将他所吸收的思想联系起来，特别是以视觉方式。他能够将听到的内容视觉化的能力在他听讲座时特别有用，因为他的视力太差，无法正确看到讲师在黑板上写的符号。

从巴黎综合理工学院毕业后，庞加莱继续在巴黎矿业学院学习。他的[21]：-

……在那里学习期间所做的实地考察笔记细致入微，展现出对采矿业科学和商业方法的深刻了解；这是他一生都感兴趣的主题。

在巴黎矿业学院完成学业后，庞加莱在韦苏尔短暂担任采矿工程师，同时完成他的博士工作。作为夏尔·埃尔米特的学生，庞加莱于1879年从巴黎大学获得数学博士学位。他的学位论文是关于微分方程的，考官对这项工作有些批评。他们赞扬了论文开头部分的结果，但随后报告说（例如见[21]）：-

……论文的其余部分有些混乱，表明作者仍然无法清晰简单地表达他的思想。尽管如此，考虑到该主题的巨大难度和所展示的才能，学院建议授予庞加莱先生博士学位并享有所有特权。

获得博士学位后，庞加莱立即被任命在卡昂大学讲授数学分析。他在卡昂的教学报告并不完全是赞扬性的，提到他有时讲课风格缺乏条理。他在那里只待了两年，然后于1881年被任命为巴黎理学院的讲席。1886年，庞加莱被提名为索邦大学的数学物理和probability讲席。夏尔·埃尔米特的干预和支持确保了庞加莱被任命为该讲席，他还被任命为巴黎综合理工学院的一个讲席。他在巴黎给学生的讲座课程[2]：-

……他每年都更换讲授内容，会回顾光学、电学、流体团块的平衡、电学的数学、天文学、热力学、光和概率。

庞加莱在巴黎担任这些讲席直到他58岁早逝。

在简要审视庞加莱对数学和其他科学做出的许多贡献之前，我们应该稍微谈谈他的思维和工作方式。他被认为是所有时代最伟大的天才之一，有两个非常重要的来源研究他的思维过程。一个是庞加莱于1908年在巴黎l'Institute Général Psychologique所作的题为Mathematical invention的讲座，其中他审视了自己导致重大数学发现的思维过程。另一个是[30]一书，作者是Toulouse，他是巴黎l'École des Hautes Études心理学实验室的主任。尽管出版于1910年，这本书叙述了与庞加莱的对话以及Toulouse在1897年对他进行的测试。

在[30]中，Toulouse解释说庞加莱保持着非常精确的工作时间。他每天进行四小时的数学研究，从上午10点到中午，然后从下午5点到7点。他会在晚上较晚时阅读期刊上的文章。庞加莱工作的一个有趣方面是，他倾向于从基本原理出发发展他的结果。对许多数学家来说，存在一个构建过程，越来越多的东西建立在先前工作之上。这不是庞加莱的工作方式，不仅是他的研究，还有他的讲座和书籍，都是从基础开始精心发展的。也许最引人注目的是Toulouse在[30]中对庞加莱如何撰写论文的描述。庞加莱：-

……在写论文时不会制定总体计划。他通常开始时不知道会在哪里结束。……开始通常很容易。然后工作似乎引导他前进，而他没有做出刻意的努力。在那个阶段很难分散他的注意力。当他搜寻时，他经常自动写下一个公式以唤起某种联想。如果开始是痛苦的，庞加莱不会坚持，而是放弃这项工作。

Toulouse接着描述庞加莱如何期望在他停止专注于问题时关键想法会来到他身边：-

庞加莱通过突然的冲击进行，拿起又放弃一个主题。在间隔期间，他假设……他的无意识继续反思工作。如果沒有足够强的分心，停止工作是困难的，特别是当他判断工作尚未完成时……因此，庞加莱从不在晚上做任何重要工作，以免打扰他的睡眠。

正如 Miller 在21中指出的：-

令人难以置信的是，他能够一页又一页地完成详细的计算，无论是最高深的抽象数学计算还是纯粹的数字计算，正如他在物理学中经常做的那样，几乎从不划掉任何东西。

让我们考察庞加莱用这种工作方法做出的一些发现。庞加莱是一位关注数学、物理学和哲学诸多方面的科学家，他常被描述为数学中最后一位通才。他对数学的众多分支、天体力学、流体力学、狭义相对论和科学哲学都做出了贡献。他的许多研究涉及不同数学主题之间的相互作用，而他对整个知识谱系的广泛理解使他能够从许多不同的角度来攻克问题。

在30岁之前，他发展了自守函数的概念，这是一类单复变函数，在一个变换群下保持不变，该群由线性项的比值代数地刻画。这一想法是以间接的方式来自他关于微分方程的学位论文工作。他的结果只适用于受限的函数类，庞加莱想要推广这些结果，但作为通向这一目标的一条途径，他寻找了一类解不存在的函数。这引导他得到了他以拉扎勒斯·福克斯命名、后来被称为自守函数的函数。关键的想法是在他正要上公共汽车时来到他脑海中的，正如他在Science and Method（1908）中所述：-

当我把脚踏上踏板的那一刻，一个想法来到我脑海中，此前的思想似乎没有任何东西为它铺平道路，那就是我曾用来定义 Fuchs 函数的变换与non-euclidean geometry的那些变换是相同的。

在菲利克斯·克莱因与庞加莱之间的通信中，许多深刻的思想得到了交流，自守函数理论的发展大大受益。然而，这两位伟大的数学家并未保持良好关系，菲利克斯·克莱因似乎因庞加莱对拉扎勒斯·福克斯工作的高度评价而变得不快。Rowe 在[149]中考察了这些通信。

庞加莱的Analysis situsⓉ（位置分析），于1895年出版，是对拓扑学的早期系统处理。可以说他是代数拓扑学的创始人，并且在1901年，他声称他在许多不同领域的研究，如微分方程和多重积分，都将他引向了拓扑学。在庞加莱于1894年发表他关于代数拓扑的六篇论文中的第一篇之后的40年里，该主题中基本上所有的想法和技术都基于他的工作。庞加莱猜想一直是代数拓扑中最令人困惑和最具挑战性的未解决问题之一，直到2002年被格里戈里·佩雷尔曼解决。

Homotopy theory通过将拓扑空间与各种群相关联，将拓扑问题简化为代数，这些群是代数不变量。庞加莱在他1894年的论文中引入了基本群（或第一同伦群），以区分不同类别的二维曲面。他能够证明任何具有与球面二维曲面相同基本群的二维曲面在拓扑上等价于球面。他猜想这个结果对三维流形成立，后来扩展到更高维度。令人惊讶的是，对于所有严格大于三维的维度，庞加莱猜想的等价物的证明是已知的。没有已知的三维流形的完整分类方案，因此没有可能的流形列表可以检查以验证它们都具有不同的同伦群。

庞加莱 也被认为是多复变解析函数理论的创始人。他在1883年的一篇论文中开始了对这一主题的贡献，其中他使用 约翰·彼得·古斯塔夫·勒热纳·狄利克雷 原理证明两个复变量的 亚纯函数 是两个 整函数 的商。他还在 代数几何 中工作，在1910-11年撰写的论文中做出了基础性贡献。他研究了代数曲面 $F(x, y, z) = 0$ 上的代数曲线，并发展了方法，使他能够轻松证明由 埃米尔·皮卡 和 弗朗切斯科·塞维里 得出的深刻结果。他首次正确证明了由 Castelnuovo、费代里戈·恩里克斯 和 弗朗切斯科·塞维里 陈述的结果，这些作者曾提出错误的证明方法。

他对数论的第一个重大贡献是在1901年对[1]的研究中作出的：-

... Diophantine问题，即在曲线$f (x, y) = 0$上寻找有理坐标点，其中f的系数为有理数。

在应用数学方面，他研究了光学、电学、电报学、毛细现象、弹性、热力学、potential theory、quantum theory、相对论和宇宙论。在天体力学领域，他研究了三体问题以及光和电磁波理论。他与阿尔伯特·爱因斯坦和亨德里克·洛伦兹一起被公认为狭义相对论的共同发现者。我们应当更详细地描述庞加莱在3-body problem方面的重要工作。

瑞典和挪威国王奥斯卡二世于1887年发起了一场数学竞赛，以庆祝他1889年的六十岁生日。庞加莱 因提交了一篇关于天体力学中三体问题的论文而获奖。在这篇论文中，庞加莱 首次描述了同宿点，首次对混沌运动进行了数学描述，并首次大量使用了不变积分的概念。然而，当这篇论文即将在 Acta Mathematica 上发表时，负责编辑出版该论文的 Phragmen 发现了一个错误。庞加莱 意识到他确实犯了一个错误，约斯塔·米塔格-莱弗勒 竭尽全力阻止发表该论文的错误版本。在1887年3月至1890年7月之间，庞加莱 和 约斯塔·米塔格-莱弗勒 交换了五十封信，主要涉及生日竞赛，其中第一封由 庞加莱 写给 约斯塔·米塔格-莱弗勒，告诉他打算提交参赛作品，当然，这50封信中的后期信件讨论了有关错误的问题。有趣的是，这个错误现在被认为标志着混沌理论的诞生。庞加莱 的论文修订版于1890年出现。

庞加莱 关于天体力学其他主要著作包括 Les Méthodes nouvelles de la mécanique céleste Ⓣ（天体力学新方法），共三卷，于1892年至1899年间出版，以及 Leçons de mecanique céleste Ⓣ（天体力学教程）（1905年）。在第一部著作中，他旨在完全刻画机械系统的所有运动，并援引流体流动的类比。他还表明，以前用于研究三体问题的级数展开是收敛的，但并非一致收敛，因此对 约瑟夫·拉格朗日 和 皮埃尔·西蒙·拉普拉斯 的稳定性证明提出了质疑。

在法国公众对科学尚不热衷的时期，他还撰写了大量科普文章。正如Whitrow在[2]中所写：-

庞加莱作为数学家崭露头角后，便将自己卓越的文学天赋投入到向公众描述科学和数学的意义与重要性这一挑战中。

庞加莱的科普著作包括Science and Hypothesis（1901年）、The Value of Science（1905年）和Science and Method（1908年）。这些著作中的一段引文与本数学史档案尤为相关。1908年他写道：-

预见数学未来的真正方法是研究它的历史和它的实际状态。

最后我们来看庞加莱对数学哲学和科学哲学的贡献。首先要指出的是，庞加莱认为逻辑和直觉在数学发现中都发挥作用。他在Mathematical definitions in education（1904）中写道：-

我们靠逻辑来证明，靠直觉来发明。

在后来的一篇文章中，庞加莱再次强调了这一点，方式如下：-

因此，逻辑若不经直觉的滋养，就仍然是贫瘠的。

McLarty [119] 举出例子表明，庞加莱并没有费心去做到严格。他在数学上取得成功的途径在于他充满激情的直觉。然而对庞加莱来说，直觉并不是他在找不到逻辑证明时才使用的东西。相反，他相信形式论证可以揭示直觉的错误，而逻辑论证是确认洞见的唯一手段。庞加莱相信，仅靠形式证明无法通向知识。知识只能来自包含内容的数学推理，而不只是形式论证。

合理的问题是，庞加莱所说的“直觉”是什么意思。这并不简单，因为他在物理学工作和数学工作中对它的看法相当不同。在物理学中，他把直觉视为用数学方式概括他的感官所告诉他的世界。但要解释数学中的“直觉”是什么，庞加莱只能退而说它是逻辑无法推出的部分：-

……要使几何……成为某种非纯逻辑的东西是必要的。要描述这个“某种东西”，我们除了直觉没有别的词。

庞加莱在评论大卫·希尔伯特的Foundations of geometry（1902）时再次提出了同样的观点：-

似乎只有逻辑观点让[大卫·希尔伯特]感兴趣。给定一串命题，他发现所有命题都从第一个命题逻辑地推出。对于这第一个命题的基础，对于它的心理起源，他并不关心。

然而，我们不应给人留下这篇评论是负面的印象，因为庞加莱对大卫·希尔伯特的这项工作非常肯定。在[181]中，Stump探讨了庞加莱的直觉的含义，以及它在数学上可接受和不可接受的形式之间的区别。

庞加莱相信人们可以选择欧几里得几何或非欧几里得几何作为物理空间的几何。他相信，由于这两种几何在拓扑上等价，人们可以将一种几何的性质翻译成另一种，因此两者都不正确也不错误。为此，他论证说物理学家总是会偏好欧几里得几何。然而，这并未被证明是正确的，实验证据现在清楚地表明物理空间不是欧几里得的。

然而，庞加莱在批评像伯特兰·罗素那样希望将数学公理化的人时是绝对正确的；他们注定要失败。庞加莱声称，数学归纳法原理不能逻辑地推导出来。他还声称，如果像大卫·希尔伯特那样用公理系统来定义算术，那么算术永远不能被证明是一致的。庞加莱的这些主张最终被证明是正确的。

我们应该注意到，尽管庞加莱对他那个时代的数学产生了巨大影响，但他从未建立自己的学派，因为他没有任何学生。虽然他的同时代人使用了他的结果，但他们很少使用他的技巧。

庞加莱因其真正天才的贡献而获得了最高荣誉。他于1887年当选为Académie des Sciences院士，并于1906年当选为科学院院长。他研究的广度使他成为唯一一位当选为科学院所有五个学部（即几何学、力学、物理学、地理学和航海学学部）成员的院士。1908年，他当选为法兰西学院院士，并在他去世那年当选为院长。他还被授予荣誉军团骑士称号，并受到世界各地大量学术团体的表彰。他赢得了无数奖项、奖章和荣誉。

庞加莱去世时年仅58岁[3]：-

庞加莱，尽管他的大多数朋友都不知道，最近在一家疗养院接受了一次手术。他似乎恢复得很好，今天早上正准备第一次开车外出。他在穿衣时突然去世。

他的葬礼有许多科学界和政治界的重要人物参加[3]：-

参议院议长和大多数部会成员出席了葬礼，还有来自法国科学院、Académie des Sciences、索邦大学和许多其他公共机构的代表团。摩纳哥亲王出席了葬礼，突尼斯的贝伊由他的两个儿子代表，罗兰·波拿巴亲王以巴黎地理学会会长的身份出席。Royal Society由其秘书约瑟夫·拉莫尔和皇家天文学家F W 弗里曼·戴森先生代表。

让我们以葬礼上的一段讲话作为结束：-

[M 庞加莱是]一位数学家、几何学家、哲学家和文人，他是某种无穷的诗人，某种科学的吟游诗人。

Henri Poincaré's father was Léon Poincaré and his mother was Eugénie Launois. They were 26 and 24 years of age, respectively, at the time of Henri's birth. Henri was born in Nancy where his father was Professor of Medicine at the University. Léon Poincaré's family produced other men of great distinction during Henri's lifetime. Raymond Poincaré, who was prime minister of France several times and president of the French Republic during World War I, was the elder son of Léon Poincaré's brother Antoine Poincaré. The second of Antoine Poincaré's sons, Lucien Poincaré, achieved high rank in university administration.

Henri was [2]:-

... ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school.

In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France.

Poincaré entered the École Polytechnique in 1873, graduating in 1875. He was well ahead of all the other students in mathematics but, perhaps not surprisingly given his poor coordination, performed no better than average in physical exercise and in art. Music was another of his interests but, although he enjoyed listening to it, his attempts to learn the piano while he was at the École Polytechnique were not successful. Poincaré read widely, beginning with popular science writings and progressing to more advanced texts. His memory was remarkable and he retained much from all the texts he read but not in the manner of learning by rote, rather by linking the ideas he was assimilating particularly in a visual way. His ability to visualise what he heard proved particularly useful when he attended lectures since his eyesight was so poor that he could not see the symbols properly that his lecturers were writing on the blackboard.

After graduating from the École Polytechnique, Poincaré continued his studies at the École des Mines. His [21]:-

... meticulous notes taken on field trips while a student there exhibit a deep knowledge of the scientific and commercial methods of the mining industry; a subject that interested him throughout his life.

After completing his studies at the École des Mines Poincaré spent a short while as a mining engineer at Vesoul while completing his doctoral work. As a student of Charles Hermite, Poincaré received his doctorate in mathematics from the University of Paris in 1879. His thesis was on differential equations and the examiners were somewhat critical of the work. They praised the results near the beginning of the work but then reported that the (see for example [21]):-

... remainder of the thesis is a little confused and shows that the author was still unable to express his ideas in a clear and simple manner. Nevertheless, considering the great difficulty of the subject and the talent demonstrated, the faculty recommends that M Poincaré be granted the degree of Doctor with all privileges.

Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Caen. Reports of his teaching at Caen were not wholly complimentary, referring to his sometimes disorganised lecturing style. He was to remain there for only two years before being appointed to a chair in the Faculty of Science in Paris in 1881. In 1886 Poincaré was nominated for the chair of mathematical physics and probability at the Sorbonne. The intervention and the support of Hermite was to ensure that Poincaré was appointed to the chair and he also was appointed to a chair at the École Polytechnique. In his lecture courses to students in Paris [2]:-

... changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability.

Poincaré held these chairs in Paris until his death at the early age of 58.

Before looking briefly at the many contributions that Poincaré made to mathematics and to other sciences, we should say a little about his way of thinking and working. He is considered as one of the great geniuses of all time and there are two very significant sources which study his thought processes. One is a lecture which Poincaré gave to l'Institute Général Psychologique in Paris in 1908 entitled Mathematical invention in which he looked at his own thought processes which led to his major mathematical discoveries. The other is the book [30] by Toulouse who was the director of the Psychology Laboratory of l'École des Hautes Études in Paris. Although published in 1910 the book recounts conversations with Poincaré and tests on him which Toulouse carried out in 1897.

In [30] Toulouse explains that Poincaré kept very precise working hours. He undertook mathematical research for four hours a day, between 10 am and noon then again from 5 pm to 7 pm. He would read articles in journals later in the evening. An interesting aspect of Poincaré's work is that he tended to develop his results from first principles. For many mathematicians there is a building process with more and more being built on top of the previous work. This was not the way that Poincaré worked and not only his research, but also his lectures and books, were all developed carefully from basics. Perhaps most remarkable of all is the description by Toulouse in [30] of how Poincaré went about writing a paper. Poincaré:-

... does not make an overall plan when he writes a paper. He will normally start without knowing where it will end. ... Starting is usually easy. Then the work seems to lead him on without him making a wilful effort. At that stage it is difficult to distract him. When he searches, he often writes a formula automatically to awaken some association of ideas. If beginning is painful, Poincaré does not persist but abandons the work.

Toulouse then goes on to describe how Poincaré expected the crucial ideas to come to him when he stopped concentrating on the problem:-

Poincaré proceeds by sudden blows, taking up and abandoning a subject. During intervals he assumes ... that his unconscious continues the work of reflection. Stopping the work is difficult if there is not a sufficiently strong distraction, especially when he judges that it is not complete ... For this reason Poincaré never does any important work in the evening in order not to trouble his sleep.

As Miller notes in [21]:-

Incredibly, he could work through page after page of detailed calculations, be it of the most abstract mathematical sort or pure number calculations, as he often did in physics, hardly ever crossing anything out.

Let us examine some of the discoveries that Poincaré made with this method of working. Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics. He made contributions to numerous branches of mathematics, celestial mechanics, fluid mechanics, the special theory of relativity and the philosophy of science. Much of his research involved interactions between different mathematical topics and his broad understanding of the whole spectrum of knowledge allowed him to attack problems from many different angles.

Before the age of 30 he developed the concept of automorphic functions which are functions of one complex variable invariant under a group of transformations characterised algebraically by ratios of linear terms. The idea was to come in an indirect way from the work of his doctoral thesis on differential equations. His results applied only to restricted classes of functions and Poincaré wanted to generalise these results but, as a route towards this, he looked for a class functions where solutions did not exist. This led him to functions he named Fuchsian functions after Lazarus Fuchs but were later named automorphic functions. The crucial idea came to him as he was about to get onto a bus, as he relates in Science and Method (1908):-

At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformation that I had used to define the Fuchsian functions were identical with those of non-euclidean geometry.

In a correspondence between Klein and Poincaré many deep ideas were exchanged and the development of the theory of automorphic functions greatly benefited. However, the two great mathematicians did not remain on good terms, Klein seeming to become upset by Poincaré's high opinions of Fuchs's work. Rowe examines this correspondence in [149].

Poincaré's Analysis situs Ⓣ, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work. The Poincaré conjecture remained as one of the most baffling and challenging unsolved problems in algebraic topology until it was settled by Grisha Perelman in 2002.

Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants. Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces. He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere. He conjectured that this result held for 3-dimensional manifolds and this was later extended to higher dimensions. Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than three. No complete classification scheme for 3-manifolds is known so there is no list of possible manifolds that can be checked to verify that they all have different homotopy groups.

Poincaré is also considered the originator of the theory of analytic functions of several complex variables. He began his contributions to this topic in 1883 with a paper in which he used the Dirichlet principle to prove that a meromorphic function of two complex variables is a quotient of two entire functions. He also worked in algebraic geometry making fundamental contributions in papers written in 1910-11. He examined algebraic curves on an algebraic surface $F(x, y, z) = 0$ and developed methods which enabled him to give easy proofs of deep results due to Émile Picard and Severi. He gave the first correct proof of a result stated by Castelnuovo, Enriques and Severi, these authors having suggested a false method of proof.

His first major contribution to number theory was made in 1901 with work on [1]:-

... the Diophantine problem of finding the points with rational coordinates on a curve $f (x, y) = 0$, where the coefficients of f are rational numbers.

In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of celestial mechanics he studied the three-body-problem, and the theories of light and of electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity. We should describe in a little more detail Poincaré's important work on the 3-body problem.

Oscar II, King of Sweden and Norway, initiated a mathematical competition in 1887 to celebrate his sixtieth birthday in 1889. Poincaré was awarded the prize for a memoir he submitted on the 3-body problem in celestial mechanics. In this memoir Poincaré gave the first description of homoclinic points, gave the first mathematical description of chaotic motion, and was the first to make major use of the idea of invariant integrals. However, when the memoir was about to be published in Acta Mathematica, Phragmen, who was editing the memoir for publication, found an error. Poincaré realised that indeed he had made an error and Mittag-Leffler made strenuous efforts to prevent the publication of the incorrect version of the memoir. Between March 1887 and July 1890 Poincaré and Mittag-Leffler exchanged fifty letters mainly relating to the Birthday Competition, the first of these by Poincaré telling Mittag-Leffler that he intended to submit an entry, and of course the later of the 50 letters discuss the problem concerning the error. It is interesting that this error is now regarded as marking the birth of chaos theory. A revised version of Poincaré's memoir appeared in 1890.

Poincaré's other major works on celestial mechanics include Les Méthodes nouvelles de la mécanique céleste Ⓣ in three volumes published between 1892 and 1899 and Leçons de mecanique céleste Ⓣ (1905). In the first of these he aimed to completely characterise all motions of mechanical systems, invoking an analogy with fluid flow. He also showed that series expansions previously used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace.

He also wrote many popular scientific articles at a time when science was not a popular topic with the general public in France. As Whitrow writes in [2]:-

After Poincaré achieved prominence as a mathematician, he turned his superb literary gifts to the challenge of describing for the general public the meaning and importance of science and mathematics.

Poincaré's popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908). A quote from these writings is particularly relevant to this archive on the history of mathematics. In 1908 he wrote:-

The true method of foreseeing the future of mathematics is to study its history and its actual state.

Finally we look at Poincaré's contributions to the philosophy of mathematics and science. The first point to make is the way that Poincaré saw logic and intuition as playing a part in mathematical discovery. He wrote in Mathematical definitions in education (1904):-

It is by logic we prove, it is by intuition that we invent.

In a later article Poincaré emphasised the point again in the following way:-

Logic, therefore, remains barren unless fertilised by intuition.

McLarty [119] gives examples to show that Poincaré did not take the trouble to be rigorous. The success of his approach to mathematics lay in his passionate intuition. However intuition for Poincaré was not something he used when he could not find a logical proof. Rather he believed that formal arguments may reveal the mistakes of intuition and logical argument is the only means to confirm insights. Poincaré believed that formal proof alone cannot lead to knowledge. This will only follow from mathematical reasoning containing content and not just formal argument.

It is reasonable to ask what Poincaré meant by "intuition". This is not straightforward, since he saw it as something rather different in his work in physics to his work in mathematics. In physics he saw intuition as encapsulating mathematically what his senses told him of the world. But to explain what "intuition" was in mathematics, Poincaré fell back on saying it was the part which did not follow by logic:-

... to make geometry ... something other than pure logic is necessary. To describe this "something" we have no word other than intuition.

The same point is made again by Poincaré when he wrote a review of Hilbert's Foundations of geometry (1902):-

The logical point of view alone appears to interest [Hilbert]. Being given a sequence of propositions, he finds that all follow logically from the first. With the foundations of this first proposition, with its psychological origin, he does not concern himself.

We should not give the impression that the review was negative, however, for Poincaré was very positive about this work by Hilbert. In [181] Stump explores the meaning of intuition for Poincaré and the difference between its mathematically acceptable and unacceptable forms.

Poincaré believed that one could choose either euclidean or non-euclidean geometry as the geometry of physical space. He believed that because the two geometries were topologically equivalent then one could translate properties of one to the other, so neither is correct or false. For this reason he argued that euclidean geometry would always be preferred by physicists. This, however, has not proved to be correct and experimental evidence now shows clearly that physical space is not euclidean.

Poincaré was absolutely correct, however, in his criticism that those like Russell who wished to axiomatise mathematics; they were doomed to failure. The principle of mathematical induction, claimed Poincaré, cannot be logically deduced. He also claimed that arithmetic could never be proved consistent if one defined arithmetic by a system of axioms as Hilbert had done. These claims of Poincaré were eventually shown to be correct.

We should note that, despite his great influence on the mathematics of his time, Poincaré never founded his own school since he did not have any students. Although his contemporaries used his results they seldom used his techniques.

Poincaré achieved the highest honours for his contributions of true genius. He was elected to the Académie des Sciences in 1887 and in 1906 was elected President of the Academy. The breadth of his research led to him being the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections. In 1908 he was elected to the Académie Francaise and was elected director in the year of his death. He was also made chevalier of the Légion d'Honneur and was honoured by a large number of learned societies around the world. He won numerous prizes, medals and awards.

Poincaré was only 58 years of age when he died [3]:-

M Henri Poincaré, although the majority of his friends were unaware of it, recently underwent an operation in a nursing home. He seemed to have made a good recovery, and was about to drive out for the first time this morning. He died suddenly while dressing.

His funeral was attended by many important people in science and politics [3]:-

The President of the Senate and most of the members of the Ministry were present, and there were delegations from the French Academy, the Académie des Sciences, the Sorbonne, and many other public institutions. The Prince of Monaco was present, the Bey of Tunis was represented by his two sons, and Prince Roland Bonaparte attended as President of the Paris Geographical Society. The Royal Society was represented by its secretary, Sir Joseph Larmor, and by the Astronomer Royal, Mr F W Dyson.

Let us end with a quotation from an address at the funeral:-

[M Poincaré was] a mathematician, geometer, philosopher, and man of letters, who was a kind of poet of the infinite, a kind of bard of science.