勒内-路易·贝尔René-Louis Baire

勒内-路易·贝尔的父亲是一名裁缝，勒内是这个贫困工人阶级家庭三个孩子之一，他们不得不在艰难的经济状况下挣扎。勒内在巴黎长大，当时埃菲尔铁塔正在建造。1886年，当他十二岁时，勒内赢得了一项奖学金，使他尽管家境贫寒仍能接受良好的教育。他进入Lycée Lakanal，在那里寄宿，并成为一名出色的学生。他在Concours Général中获得两项荣誉提名，这是法国所有Lycée的顶尖学生之间的竞赛。

1890年，勒内完成了拉卡纳尔中学的高级课程，并进入亨利四世中学的特别数学班。在这所中学完成一年的预备学习后，他通过了巴黎综合理工学院和巴黎高等师范学院的入学考试。他选择后者作为学习的地方。科斯塔贝尔在[1]中写道：-

……在那里度过的三年里，[他]以其智力上的成熟引起了注意。他是一个安静的年轻人，独来独往，极其内省。在此期间，人们发现他身体娇弱。

在巴黎高等师范学校贝尔听了于乐·达奈希和爱徳华·古尔萨的课，此外，他还在索邦大学听了夏尔·埃尔米特、埃米尔·皮卡和儒勒·昂利·庞加莱的课。在学生时代，他协助编辑了儒勒·昂利·庞加莱1894年关于热传播的讲座，他听了这些讲座。获得学士学位后，贝尔朝着他的“教师资格会考”前进，但是，尽管他是这次考试笔试部分最好的学生，但在口试之后，他总排名仅为第三。

他在口试中的糟糕表现值得更详细地描述，因为这将对贝尔未来的研究方向产生重大影响。他被要求证明指数函数的连续性，但当他进行到证明中间时，他意识到[1]：-

……他对连续性的证明，是他在亨利四世中学学到的，纯粹是一种技巧，因为它没有充分涉及函数的定义。

考官们对贝尔很严厉，他对结果极其失望，但随后他决心在研究一般函数的连续性概念的同时，重新审视他的分析课程。然而，他通过教师资格会考的直接结果是，他获得了第一个职位，在一所中学担任教授。在巴勒迪克的任命给了他合理的经济保障，但他不高兴的是，住在巴勒迪克意味着他没有机会与大学生活密切接触。

在Bar-le-Duc的中学里，贝尔研究函数论和极限概念。大约在这个时候，他发现了函数是连续函数序列的极限的条件。此后不久，贝尔建立了他的函数分类。第1类函数是那些作为连续函数序列极限的函数。第2类函数是那些作为第1类函数序列极限的函数，而第3类函数是那些作为第2类函数序列极限的函数。

贝尔获得了一笔奖学金，使他能够继续在意大利学习，在那里他遇到了维多·沃尔泰拉，并与之建立了亲密的友谊。在中学工作期间，贝尔写了一篇关于不连续函数的博士论文。1899年3月24日，由让·加斯东·达布、保尔·阿佩尔和埃米尔·皮卡组成的委员会对他进行了考试，他们授予他博士学位。然而[1]：-

贝尔完全理解的少数反对意见证明，他已经走上了一条新路，并且不容易说服他的听众。

甚至在提交他的学位论文之前，贝尔的健康状况就一直不佳，而在获得博士学位之后，他只能断断续续地为数学做出贡献。他继续在中学任教（他曾在特鲁瓦、巴勒迪克和南锡任教），但并不满意在这样低的层次教学。1901年，贝尔被任命为蒙彼利埃大学的“会议讲师”。这个职位让他为学生准备“教师资格会考”，他比在中学任教更喜欢这个职位。在蒙彼利埃期间，他写了一篇关于无理数数和极限的论文。

1904年，他获得了Peccot基金会奖学金，该奖学金允许年轻学校教师在大学度过一个学期以发展他们的技能。贝尔在法兰西学院度过了这个学期，他就其论文主题进行了讲座，并于次年出版了这些讲座。贝尔回到蒙彼利埃，在那里他遭受了第一次严重的疾病发作，但一段时间后最严重的发作过去了，他能够再次工作。1905年，他被任命为大学职位，加入了第戎理学院。1907年，他被提升为第戎的分析学教授。

贝尔从小身体就不好，但从他在Bar-le-Duc的中学时起，他的健康开始恶化到妨碍他工作的程度。糟糕的时期变得更加频繁，使他长时间无法行动。除了自青年时期就困扰他的食道问题外，他还患上了一种心理障碍，用他自己的描述，这种障碍偶尔会“使他虚弱”。显然，他最终无法从事需要集中注意力的工作，在这些时候数学研究变得不可能。1909年至1914年间，他继续尝试履行教学职责，但这变得越来越困难。然后在1914年初，他请求休假，以便尝试恢复健康。

贝尔先去了阿莱西亚，然后去了洛桑。正是在洛桑期间，第一次世界大战爆发，他无法返回法国。从1914年到1918年的战争岁月，他都在洛桑度过，经济状况相当困难。

考察他的同代人提出的关于他病因的各种说法是很有意思的。有人提出，他问题的根源在于学生时代的智力过度消耗。他的近亲以及其他与他亲近的人，则将他的病归咎于他因学术权威未认可其成就而产生的深深挫败感。贝尔觉得自己理应获得巴黎的教授职位，据认为，未能如愿使他抑郁，进而导致健康不佳。

贝尔觉得像昂利·勒贝格这样比贝尔年轻的人，却受到不公平的优先对待。1904年，他在法兰西学院授课时，首次与昂利·勒贝格发生争执，争论谁最有资格教授这门课程。他们的竞争在贝尔晚年演变成更严重的争吵。贝尔也与夏尔-让·德拉瓦莱·普桑闹翻了，这对于那些知道贝尔的思想是通过夏尔-让·德拉瓦莱·普桑的著名论著进入数学主流的人来说，可能会感到惊讶。夏尔-让·德拉瓦莱·普桑写给贝尔的信件，收录在[7]中，让人了解到他们争吵的原因，这些原因似乎集中在夏尔-让·德拉瓦莱·普桑按重要性顺序对昂利·勒贝格和贝尔的数学发现进行了分类这一事实上。

说到信件，我们应当指出，[4]收录了贝尔写给埃米尔·博雷尔的五十封信。前五封写于1898年，始于贝尔在意大利期间。从第五封信（日期为1898年5月22日）到第六封信（日期为1902年2月4日）之间有一段空白。这段空白可以用贝尔在巴勒迪克任教期间第一次重病来解释。在[4]收录的信件中，贝尔非常详细地写了他的研究想法，包括函数的贝尔分类、第一范畴与第二范畴的集合以及半连续性。在信中他还讨论了格奥尔格·康托尔的集合论和数学基础。

看来不仅是贝尔的家人觉得他受到了不公正对待，因为1918年之后，数学界许多人似乎都在试图弥补他未获认可的状况。1918年，有人建议，法兰西学院的一个讲席——他无疑配得上这个职位——会缓解贝尔的抑郁，帮助他恢复智力活力，但显然这些建议从未实现。由于无法恢复职务，贝尔住在日内瓦湖畔，他就是在那里获得了荣誉军团骑士勋章，并于1922年当选为Académie des Sciences院士。他于1925年退休，晚年独自在莱芒湖畔的旅馆房间里度过。尽管他领取的养老金还算合理，但这些年的通货膨胀很快意味着他临终时的经济困难与他年轻时相似。

尽管无法长时间工作，贝尔还是写了许多重要的分析著作，包括Théorie des nombres irrationels, des limites et de la continuitéⓉ（无理数、极限与连续性理论）（1905年）和Leçons sur les théories générales de l'analyseⓉ（分析一般理论教程），2卷（1907-8年）。贝尔在摆脱函数与连续性的直观观念方面迈出了决定性的一步，他清楚地看到，无穷集合的理论是严格实数分析的基础。他在他的学位论文中写道：-

一般来说，在此时我们关注的思想框架中，函数理论中的每一个问题都会导致集合论中的某些问题，而正是随着这些后一问题得到解决，才有可能或多或少地完全解决给定的问题。

当他健康状况良好时，他的讲座质量在学生中得到的评价相当不同。有些人形容他的讲座非常清晰，但另一些人声称他所教的内容太难，超出了人类理解能力。贝尔意识到这些评论，写道：-

……但看看阿尔诺·当茹瓦——他理解了它，因此它一定不会那么难……

阿尔诺·当茹瓦，作为贝尔最著名的学生，当然理解了贝尔的思想，并在自己的工作中发展了它们。他写道，贝尔是：-

……不是一个讨人喜欢的性格……[而且]不是一个有深厚文化修养的人……[但]由于大脑的疲劳而不断受到折磨。

另一方面，阿尔诺·当茹瓦这样描述他：-

……一个优秀的人。

他的另一位学生Reault写得要积极得多，他描述贝尔具有：-

……父亲般的关怀……[他]有很高的智力素质……一个敏锐的数学头脑[具有]他知识的广度和深度……[以及]他品格的高尚。

René Baire's father was a tailor and René was one of three children from the poor working class family who had to struggle under difficult financial circumstances. René grew up in Paris at the time when the Eiffel tower was being constructed. In 1886, when he was twelve years old, René won a scholarship to enable him to have a good education despite his family's poverty. He entered the Lycée Lakanal where he boarded and he became an outstanding student. He won two honourable mentions in the Concours Général, a competition between the top pupils from all the Lycées across France.

In 1890 René completed the advanced classes at the Lycée Lakanal and entered the special mathematics section of the Lycée Henri IV. After completing one year of preparation at this Lycée, he passed the entrance examinations for both the École Polytechnique and the École Normale Supérieure. He chose the latter as the place to study. Costabel writes in [1]:-

... during his three years there [he] attracted attention by his intellectual maturity. He was a quiet young man who kept to himself and was profoundly introspective. During this period he was found to be in delicate health.

At the École Normale Supérieure Baire attended lectures by Jules Tannery and Goursat and, in addition, he attended lectures by Hermite, Émile Picard and Poincaré at the Sorbonne. While he was a student he assisted with the editing of Poincaré's lectures, which he attended in 1894, on the propagation of heat. Having received his licentiate Baire proceeded toward his "agregation" but, although he was the best student in the written parts of this examination, he was only third overall after the oral examination.

His poor performance in the oral is worth describing in more detail since it was to have a great affect on the future direction of Baire's research. He was asked to prove the continuity of the exponential function but when he was in the middle of the proof he realised that [1]:-

... his demonstration of continuity, which he had learnt at the Lycée Henri IV, was purely an artifice, since it did not refer sufficiently to the definition of the function.

The examiners were hard on Baire and he was extremely disappointed with the outcome, but he then determined to examine again his analysis course while researching into the concept of continuity of a general function. However, the immediate result of his passing his agregation was that he obtained his first post as a professor at a lycée. An appointment in Bar-le-Duc gave him reasonable financial security but he was unhappy that living in Bar-le-Duc meant that he had no opportunity for close contacts with university life.

At the Lycée in Bar-le-Duc Baire worked on the theory of functions and the concept of a limit. Around this time he discovered conditions under which a function is a limit of a sequence of continuous functions. Shortly after this Baire set up his classification of functions. Class 1 functions were those functions which were the limit of a sequence of continuous functions. Class 2 functions were those functions which were the limit of a sequence of Class 1 functions, while Class 3 functions were those functions which were the limit of a sequence of Class 2 functions.

Baire was awarded a scholarship to allow him to continue his studies in Italy and there he met and established a close friendship with Volterra. While he worked in the lycée, Baire wrote a doctoral thesis on discontinuous functions. He was examined on 24 March 1899 by a board consisting of Darboux, Appell and Émile Picard, and they awarded him the doctorate. However [1]:-

The few objections, which Baire fully appreciated, proved that he had embarked on a new road and would not find it easy to convince his listeners.

Even before presenting his thesis Baire had suffered from poor health and, after the award of his doctorate, he was only able to contribute to mathematics for a few short spells. He continued to teach in lycées (he taught in Troyes, Bar-le-duc and Nancy) but was not happy teaching at this low level. In 1901 Baire was appointed to the University of Montpellier as a "Maitre des conferences". This post saw him preparing students for the "agregation" examination, a position he enjoyed much more than teaching in lycées. While at Montpellier he wrote a paper on irrational numbers and limits.

In 1904 he was awarded a Peccot Foundation Fellowship which was to allow young school teachers to spend a semester in a university developing their skills. Baire spent the semester at the Collège de France where he lectured on the subject of his thesis and had the lectures published the next year. Baire returned to Montpellier where he suffered the first severe attack of illness but after a while the worst of the attack passed and he was able to work again. He was appointed to a university post in 1905 when he joined the Faculty of Science at Dijon. In 1907 he was promoted to Professor of Analysis at Dijon.

Baire's health had never been good since he was young but from the time he was at the Lycée at Bar-le-Duc it began to deteriorate to the stage that it prevented him from working. The bad spells became more frequent, immobilising him for long periods. Apart from problems with his oesophagus that had plagued him since his youth, he developed a kind of psychological disorder which, using his own description, "debilitated" him occasionally. Apparently he eventually became unable to undertake work which required him to concentrate, and research in mathematics became impossible at these times. Between 1909 and 1914 he continued trying to undertake his teaching duties, but this became more and more difficult. Then near the beginning of 1914 he requested leave so that he might try to recover his health.

Baire went first to Alésia, then he went to Lausanne. It was while he was in Lausanne that World War I began and he was not able to return to France. He spent the war years from 1914 until 1918 in Lausanne in quite difficult financial circumstances.

It is interesting to consider the various causes suggested by his contemporaries to account for his illness. Some suggested that the cause for his problems lay in intellectual overexertion in his student days. His close family, and others close to him, blamed his illness on his deep feelings of frustration that his achievements were not being recognised by the academic authorities. Baire felt that he deserved a professorship in Paris and failing to achieve this, it was suggested, caused him depression and hence his ill health.

Certainly Baire felt that men such as Lebesgue, who was younger than Baire, had been unfairly preferred to him. He first fell out with Lebesgue in 1904, when he taught his course at the Collège de France, over who had the most right to teach such a course. Their rivalry turned into a more serious argument later in Baire's life. Baire also fell out with de la Vallée Poussin which may be surprising to those who know that Baire's ideas entered the mainstream of mathematics through de la Vallée Poussin's well-known treatise. The letters written to Baire by de la Vallée Poussin, and reproduced in [7], give an idea of the reasons for their arguments which seem to centre round the fact that de la Vallée Poussin had classified by order of importance mathematical discoveries of Lebesgue and Baire.

While on the topic of letters, we should remark that [4] contains fifty letters written by Baire to Émile Borel. The first five are written during 1898 beginning during the time that Baire was in Italy. There is a gap from the fifth letter, dated 22 May 1898 to the sixth dated 4 February 4 1902. The gap is explained by Baire's first serious illness over the period he taught at Bar-le-Duc. In the letters reproduced in [4], Baire writes in great detail about his research ideas, including the Baire classification of functions, sets of first and second category, and semicontinuity. In the letters he also discusses Cantor's set theory and the foundations of mathematics.

It appears that it was not only Baire's family who felt he had been hard done by, for after 1918 many in the mathematical community seemed to be trying to make amends for his lack of recognition. In 1918 some suggested that a chair at the Collège de France, which he undoubtedly deserved, would lift Baire's depression, helping him to regain his intellectual vigour, but apparently these suggestions never materialised. Unable to resume his duties, Baire lived on the shores of Lake Geneva and he was there when he received the Chevalier de la Legion d'Honneur and in 1922 when he was elected to the Académie des Sciences. He retired in 1925 and spent his last years in the solitude of hotel rooms on the shores of the Lake of Leman. Although he received a reasonable pension, inflation over these years soon meant that he ended his life with financial difficulties similar to those of his youth.

Despite being unable to work for long periods, Baire wrote a number of important analysis books including Théorie des nombres irrationels, des limites et de la continuité Ⓣ (1905) and Leçons sur les théories générales de l'analyse Ⓣ, 2 Vols. (1907-8). Baire made a decisive step in moving away from the intuitive idea of functions and continuity and he saw clearly that a theory of infinite sets was fundamental for rigorous real analysis. He wrote in his doctoral thesis:-

Generally speaking, in the framework of ideas that here concern us, every problem in the theory of functions leads to certain questions in the theory of sets, and it is to the degree that these latter questions are resolved, that it is possible to solve the given problem more or less completely.

When his health was good, the quality of his lectures received rather differing opinions from his students. Some described his lectures as very clear, but others claimed that what he taught was so difficult that it was beyond human ability to understand. Baire, aware of these comments, wrote:-

... but look at Denjoy - he understood it, hence it must not be so difficult ...

Denjoy, who was Baire's most famous student, certainly understood Baire's ideas and developed them in his own work. He wrote that Baire was:-

.. not an agreeable character ... [and] not a person of enormous culture ...[but] constantly tormented due to the fatigue of his brain.

On the other hand Denjoy described him as:-

... an excellent person.

Another of his students, Reault, wrote much more positively describing Baire as having:-

... paternal concern ... [He had] high intellectual qualities ... a penetrating mathematical mind [with] the extent and depth of his knowledge ... [and] the greatness of his character.