让·迪厄多内Jean Dieudonné

让·迪厄多内 的父亲是 Ernest Dieudonné，一位实业家，母亲是 Léontine Labrun。迪厄多内 幼年时便不可抗拒地被词典、百科全书和世界通史所吸引。他在里尔中学就读，十四岁时开始学习代数，对数学的热爱由此蓬勃发展。完成中学学业后，他进入巴黎高等师范学校，在那里受到 埃米尔·皮卡、雅克·阿达马、埃利·嘉当、保罗·蒙泰尔、阿尔诺·当茹瓦 和 茹利亚 的启发。迪厄多内 在高等师范学校先后获得学士学位（1927年）和博士学位（1931年）。他的博士研究由 保罗·蒙泰尔 指导，学位论文属于经典分析领域。他从1933年起在雷恩理学院任讲师，1935年7月22日与 Odette Clavel 结婚；他们有两个孩子 迪厄多内-Pierre 和 Françoise。结婚时，他已经是 尼古拉·布尔巴基 的创始成员之一。这极大地改变了他的数学视野。他写道：——

若任由我自己发展，我无疑会一辈子固守在分析的一个狭窄领域里。

他还作为讲师在南锡理学院工作，从1937年到1946年。随后他被任命为巴西圣保罗的数学教授（1946—47年）。回到南锡后，他在理学院任教授，从1948年到1952年，之后接受密歇根大学数学教授的一年期任命。1953年至1959年在西北大学任教后，迪厄多内 回到法国，接受高等科学研究所数学教授的任命。1964年，在高等科学研究所工作五年后，他接受了尼斯理学院的讲席，担任此职直至1970年。

我们注意到，迪厄多内 的热爱在于研究而非教学。E Beckenstein 在评论 迪厄多内 的 Choix d'oeuvres mathématiques Ⓣ（数学杰作）时写道：——

他从未对教学有过丝毫兴趣。教学唯一吸引人之处在于，它为他提供了足够的时间从事自己的研究，而像恩斯特·爱德华·库默尔、卡尔·魏尔斯特拉斯、赫尔曼·格拉斯曼、威廉·基灵或保罗·蒙泰尔这样的人物，命运却没有给予他们这样的机会，他们职业生涯的大部分时间都花在了更耗时的中学教学上。即使教了四十年书，他在一张纸面前仍然比在听众面前更自在。讲课时他总是用讲稿，“pour éviter les catastrophes”。他并不为自己是典型的象牙塔型人物而道歉：只有通过不断的思考才能成就某些事情。但尽管身处象牙塔，他并非苦行僧，从未鄙视过生活的乐趣。

我们上面提到，迪厄多内是尼古拉·布尔巴基的创始成员。从该团体成立之时起，他就是尼古拉·布尔巴基系列文本的主要贡献者之一，并且在许多方面，他是这个团体的主要影响者，而这个团体的全部目标就是避免任何人承担这一角色。谈到他喜爱的尼古拉·布尔巴基大会，迪厄多内在[5]中写道：-

某些外国人应邀作为旁观者参加尼古拉·布尔巴基的会议，出来后总是留下这样的印象：这是一群疯子的聚会。他们无法想象这些人，大喊大叫——有时三四人同时喊——怎么可能想出什么聪明的东西……

谈到他自己参与尼古拉·布尔巴基及其对自己职业生涯的影响，迪厄多内写道[5]：-

根据我的个人经验，我相信，如果我没有被要求起草那些我完全不懂的问题，并且设法挺过来，我绝不会完成我所做的数学的四分之一，甚至十分之一。

他的数学生涯始于多项式分析。他在各种各样的数学领域工作，包括general topology、拓扑向量空间、代数几何、不变量理论和经典群。

他最著名的著作有La Géométrie des groupes classiques Ⓣ（《经典群的几何》）(1955)、Foundations of Modern Analysis (1960)、Algèbre linéaire et géométrie élémentaire Ⓣ（《线性代数与初等几何》）(1964)，以及九卷本的Éléments d'analyse Ⓣ（《分析原理》）(1960-1982)。C E Rickart在评论La Géométrie des groupes classiques Ⓣ（《经典群的几何》）时写道：-

本卷汇集了关于所谓经典群初等理论的现代成果的大部分内容。这里使用的“经典群”一词与作者专著《论经典群》(1948)中的用法相同，而“初等理论”大致指的是涉及子群和同态的成果，与例如涉及拓扑学、微分几何等的成果相对。其方法当然是代数的，但正如作者在这一领域工作的特点那样，深受几何观念的影響。尽管许多数学家对这一主题有所贡献，但这里呈现的大部分成果归功于作者，主要参考文献是上述专著及其论文《论经典群的自同构》(1951)。

迪厄多内在Foundations of Modern Analysis中写道，其意图是：-

……为涉及“分析”的现代数学所有分支提供必要的初等背景。

J L Kelly写道：-

该书最显著的特点是始终以几何方式表述结果。例如，微分学是以从斯特凡·巴拿赫空间的开子集到巴拿赫空间的函数的线性逼近来展开的。然而，若断言该书包含了对巴拿赫空间的研究，那就完全错了——书中没有证明关于此类空间的任何非平凡命题。研究的主题确实是初等分析，定理是以几何术语表述的分析定理。这种几何化颇像多年前发生的线性代数的几何化，而且正如线性代数的情况一样，在概念和技术上具有巨大的优势。在350页的正文中完成了相当多的内容。数学组织极为出色，表述清晰，有大量非常好的习题，每章都有极佳的阐述性导言（以作者惯有的谦逊风格写成）。简言之，这是一部优美的教材。

在撰写Algèbre linéaire et géométrie élémentaireⓉ（线性代数与初等几何）时，迪厄多内旨在为法国公立中学的教师提供足够的几何学背景，以便他们能适当地为学生准备大学学习。他以二维和三维的线性（或几何）代数来呈现这一主题。我们不对文本是否过于高深以致无法实现其预期目的作出评判，但我们注意到，在第一章中引入实数时，迪厄多内假定它们是一个有序域，其中中间值定理对三次多项式成立。

我们还应当考察迪厄多内作为数学史家的贡献。他出版了History of functional analysis（1981）、History of algebraic geometry（1985）、Pour l'honneur de l'esprit humain Ⓣ（《为了人类精神的荣誉》）（1987）、A history of algebraic and differential topology 1900—1960（1989），以及L'école mathématique française du XXe siècle Ⓣ（《二十世纪法国数学学派》）（2000）等著作。

History of functional analysis是：-

……一部详细而引人入胜的泛函分析历史与发展论述，从约瑟夫·拉格朗日和丹尼尔·伯努利开始，经过本世纪初埃里克·伊瓦尔·弗雷德霍姆、大卫·希尔伯特和里斯的工作，到大约1960年结束。

桑德斯·麦克兰恩在评论A history of algebraic and differential topology时写道：-

……是对代数拓扑学的问题与发展的一部见识广博且详尽的分析，从儒勒·昂利·庞加莱和勒伊岑·布劳威尔到让-皮埃尔·塞尔、约翰·柯西·亚当斯和勒内·托姆。作者考察了这条路径上的每一篇重要论文，并描述了其证明的步骤和策略及其与其他工作的关系。此前，20世纪数学众多技术发展的历史似乎给学术研究带来了不可逾越的障碍。本书以拓扑学为例表明这些障碍如何能够被克服，并带来了富有启发性的结果。

除历史文本外，迪厄多内还编辑了卡米耶·若尔当的著作。在第一卷中，迪厄多内贡献了一篇关于卡米耶·若尔当在有限群方面工作的文章，在第二卷中则是一篇有趣的116页导论，介绍卡米耶·若尔当在线性与多重线性代数以及数论方面的工作。迪厄多内还为埃瓦里斯特·伽罗瓦于1962年出版的数学著作与回忆录撰写了序言。

关于迪厄多内的若干描述，尤其是那些在尼古拉·布尔巴基项目中与他共事的人的描述，颇为有趣。阿曼德·博雷尔在[3]中写道：-

大约二十五年来，他惯常以每天为布尔巴基写几页作为一天的开始（或许是在弹一小时钢琴之后）。特别是，但远非仅仅如此，他负责所有卷册（约三十卷）的最终定稿、习题以及付印准备，这些卷册是在他担任成员期间乃至稍晚一些时候问世的。这无疑在很大程度上解释了各卷风格的一致性，使得任何试图将某一贡献或另一贡献区分开来的努力都徒劳无功。但这其实并非迪厄多内的风格，而是他为布尔巴基所采用的风格。

皮埃尔·卡地亚在[10]中说：-

迪厄多内是一位相当不错的钢琴演奏者，属于业余水平，但相当不错，而且他有惊人的记忆力。他凭记忆熟知成百上千页的乐谱，能跟上每一个音符。我记得有几次和他一起去音乐厅。那真是引人入胜，他会看着手中的乐谱，如果乐团漏掉一个音符，他就会惊呼“哦！”他把自己生命中的最后六个月——当他决定自己的数学生涯已经结束，写完了最后一本书，退居家中——用来听录音并对照乐谱和音符。

迪厄多内是布尔巴基的抄写员，很多很多年里，每一个印出来的字都出自他的笔下。当然，有许多草稿和初步版本，但印刷版本总是出自迪厄多内的笔下。凭借他惊人的记忆力，他知道每一个字。我记得，这是个玩笑，你可以说：“迪厄多内，关于某某的结果是什么？”他就会走到书架前，取下那本书，翻到正确的那一页。

我们可以从若干来源了解Diedonné对数学的看法。首先我们引用迪厄多内在[5]中那个隐喻性的毛线球：-

这是我如今对数学的图景。它是一团毛线，一个纠缠的线团，其中所有数学以几乎不可预测的方式相互作用。而在这团毛线中，有一定数量的线头向四面八方伸出，并不与任何其他东西相连。那么布尔巴基方法非常简单——我们剪断这些线头。

在[9]中，Medvedev引用了迪厄多内在1976年一篇文章中写下的这些话：-

……数学发展的主要因素具有内在起源——对开放问题本质的反思，而不论其起源如何。

迪厄多内于1968年当选为科学院（巴黎）院士，1966年获得茹利亚奖，并被授予荣誉军团军官称号。

Jean Dieudonné's father was Ernest Dieudonné, who was an industrialist, and his mother was Léontine Labrun. As a young child Jean was irresistibly attracted to dictionaries, encyclopaedias, and universal histories. He studied at the Lycée in Lille where his love of mathematics flourished by age fourteen when he began to study algebra. After completing his school studies he entered the École Normale Supérieure in Paris where he was inspired by Émile Picard, Jacques Hadamard, Élie Cartan, Paul Montel, Arnaud Denjoy and Gaston Julia. Dieudonné received both his bachelor's degree (1927) and his doctorate (1931) from the École Normale. His doctoral studied were supervised by Montel and his thesis was in the area of classical analysis. He worked at the Faculty of Science at Rennes as Maître de Conférences from 1933, and on 22 July 1935 he married Odette Clavel; they had two children Jean-Pierre and Françoise. By the time of his marriage he had already become one of the founding members of Bourbaki. It greatly changed his mathematical outlook. He wrote:-

Left to myself, I would undoubtedly have remained billeted in a narrow sector of analysis my whole life.

He worked, also as Maître de Conférences, at the Faculty of Science at Nancy from 1937 to 1946. He was then appointed professor of mathematics at São Paulo in Brazil (1946-47). Returning to Nancy, he was professor there in the Faculty of Science from 1948 to 1952 when he accepted a one year appointment as professor of mathematics at the University of Michigan. After teaching at Northwestern University from 1953 to 1959 Dieudonné returned to France to take up an appointment as professor of mathematics at the Institut des Hautes Études Scientifiques. In 1964, after five years at the Institut des Hautes Études Scientifiques, he accepted a chair at the faculty of Science at Nice, a post which he held until 1970.

We note that Dieudonné's love was for research and not for teaching. E Beckenstein, reviewing Dieudonné's Choix d'oeuvres mathématiques Ⓣ, writes:-

He never had the least inclination to teaching. Its only attractive feature was that it provided him with enough time to pursue his own research, an opportunity that fate did not grant to men of the stature of Kummer, Weierstrass, Grassmann, Killing or Montel, who spent most of their careers in the far more time-consuming endeavour of secondary teaching. Even after forty years of teaching, he was still more at ease in front of a sheet of paper than a listener. He always used notes when he lectured "pour éviter les catastrophes". He makes no apologies for being a typical ivory tower type: only by incessant thought can some things be achieved. But ivory tower notwithstanding, he is not an ascetic, never having disdained the pleasures of existence.

We mentioned above that Dieudonné was a founder member of Bourbaki. He was one of the main contributors to the Bourbaki series of texts from the time that the group came into existence and in many ways he was the leading influence in a group whose whole object was to avoid anyone taking on this role. Speaking of Bourbaki congresses, which he loved, Dieudonné writes in [5]:-

Certain foreigners, invited as spectators to Bourbaki meetings, always come out with the impression that it is a gathering of madmen. They could not imagine how these people, shouting -- some times three or four at the same time -- could ever come up with something intelligent ...

Speaking of his own involvement in Bourbaki and its influence on his own career, Dieudonné writes [5]:-

In my personal experience, I believe that if I had not been submitted to this obligation to draft questions I did not know a thing about, and to manage to pull through, I should never have done a quarter or even a tenth of the mathematics I have done.

He began his mathematical career working on the analysis of polynomials. He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.

His best known books are La Géométrie des groupes classiques Ⓣ (1955), Foundations of Modern Analysis (1960), Algèbre linéaire et géométrie élémentaire Ⓣ (1964) and nine volumes of Éléments d'analyse Ⓣ (1960-1982). C E Rickart, reviewing La Géométrie des groupes classiques Ⓣ writes:-

This volume brings together most of the modern results concerning the so-called elementary theory of the classical groups. Here the term "classical group" is used as in the author's monograph, Sur les groupes classiques (1948) and the "elementary theory" refers roughly to results which involve subgroup and homomorphisms as opposed to results concerned for example with topology, differential geometry, etc. The approach is, of course, algebraic but, as is characteristic of the author's work in this field, is strongly influenced by geometrical notions. Although many mathematicians have contributed to the subject, the bulk of the results presented here are due to the author, the main references being the monograph cited above and his paper, On the automorphisms of the classical group (1951).

Dieudonné writes in Foundations of Modern Analysis that it is intended:-

...to provide the necessary elementary background for all branches of modern mathematics involving 'analysis'.

J L Kelly writes:-

The most remarkable feature of the text is the consistently geometrical formulation of the results. For example, the differential calculus is developed in terms of linear approximation to functions on an open subset of a Banach space to a Banach space. Yet it would be completely false to assert that the book contained a study of Banach spaces - no non-trivial proposition on such spaces is proved. The subject of study is indeed elementary analysis, and the theorems are theorems of analysis stated in geometrical terms. This geometrization is rather like the geometrization of linear algebra which occurred some years ago, and, as in the linear algebra case, there are enormous conceptual and technical advantages. A good deal is accomplished in the 350 pages of the text. The mathematical organization is superb, the presentation lucid, there are a large number of very good problems, and there are excellent expository introductions to each chapter (couched in the author's customary diffident style). In brief, it is a beautiful text.

In writing Algèbre linéaire et géométrie élémentaire Ⓣ Dieudonné aims to provide teachers in the lycées of France with sufficient background in geometry so that they can prepare their pupils properly for entry to university study. He presents the topic in terms of linear (or geometric) algebra of two and three dimensions. Let us not pass judgement on whether the text is too sophisticated to fulfil its intended purpose but we do note that in introducing the real numbers in the first chapter Dieudonné assumes they are an ordered field in which the intermediate value theorem is valid for polynomials of degree 3.

We should also examine Dieudonné's contributions as a historian of mathematics. He published texts such as History of functional analysis (1981), History of algebraic geometry (1985), Pour l'honneur de l'esprit humain Ⓣ (1987), A history of algebraic and differential topology 1900-1960 (1989), and L'école mathématique française du XXe siècle Ⓣ (2000).

The History of functional analysis is:-

... a detailed and absorbing account of the history and development of functional analysis, beginning with Lagrange and Daniel Bernoulli, through the work of Fredholm, Hilbert, and Frigyes Riesz at the turn of this century, and ending about 1960.

Mac Lane, in a review of A history of algebraic and differential topology, writes:-

... is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of its proofs and its relation to other work. Previously, the history of the many technical developments of 20th -century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.

In addition to the historical texts, Dieudonné edited the works of Camille Jordan. In the first volume Dieudonné has contributed an article on Jordan's work on finite groups and in the second volume an interesting 116-page introduction to Jordan's work on linear and multilinear algebra and on the theory of numbers. Dieudonné also wrote a preface to the mathematical writings and memoirs of Galois which were published in 1962.

Several descriptions of Dieudonné, particularly of those involved with him in the Bourbaki project, are interesting. Armand Borel writes in [3]:-

For about twenty-five years he would routinely start his day (maybe after an hour of piano playing) by writing a few pages for Bourbaki. In particular, but by far not exclusively, he took care of the final drafts, exercises, and preparation for the printer of all the volumes (about thirty) which appeared while he was a member and even slightly beyond. This no doubt accounts to a large extent for the uniformity of style of the volumes, frustrating any effort to try to individualize one contribution or the other. But this was not really Dieudonné's style, rather the one he had adopted for Bourbaki.

Pierre Cartier said in [10]:-

Dieudonné was quite a good piano player, at an amateur level, but quite good, and he had a fantastic memory. He knew hundreds and hundreds of pages of score by heart and could follow every single note. I remember I had a few occasions to go to the concert hall with him. It was fascinating, he would look at the score in his hand and exclaim "OH!" if a note was missing from the orchestra! He devoted the last six months of his life - when he decided that his mathematical life was finished, he had written his last book, and he retreated to his home - to listening to recordings and following the scores and the notes.

When Dieudonné was the scribe of Bourbaki, for many many years, every printed word came from his pen. Of course there had been many drafts and preliminary versions, but the printed version was always from the pen of Dieudonné. And with his fantastic memory, he knew every single word. I remember, it was a joke, you could say, "Dieudonné, what is this result about so and so?" and he would go to the shelf and take down the book and open it to the right page.

We can obtain an appreciation of Diedonné's views of mathematics from a number of sources. First we quote Dieudonné's metaphorical ball of yarn from [5]:-

Here is my picture of mathematics now. It is a ball of wool, a tangled hank where all mathematics react upon another in an almost unpredictable way. And then in this ball of wool, there are a certain number of threads coming out in all directions and not connecting with anything else. Well the Bourbaki method is very simple-we cut the threads.

In [9] Medvedev quotes these words of Dieudonné written in a 1976 article:-

... the principal factor in the development of mathematics has an internal origin - reflection on the nature of the open problems, independently of their origin.

Dieudonné was elected to the Academy of Sciences (Paris) in 1968, received the Gaston Julia prize in 1966, and he was made an Officer of the Légion d'Honneur.