勒伊岑·布劳威尔Luitzen Egbertus Jan Brouwer

勒伊岑·布劳威尔通常以这种全首字母的名字形式为人所知，但他的朋友们称他为Bertus，这是他三个名字中第二个的缩写。他在霍伦上高中，这是阿姆斯特丹以北须德海上的一个城镇。他在那里的表现非常出色，14岁就完成了学业。他在高中没有学过希腊语或拉丁语，但这两门都是进入大学所必需的，所以布劳威尔在接下来的两年里学习了这些科目。在此期间，他的家人搬到了阿姆斯特丹以西的哈勒姆，1897年，他就是在那里文理中学参加了阿姆斯特丹大学的入学考试。

布劳威尔开始学习时，科特韦格是阿姆斯特丹大学的数学教授，他很快意识到布劳威尔是一个杰出的学生。还在本科时，布劳威尔就证明了四维空间中连续运动的原创结果，科特韦格鼓励他提交发表。他这样做了，这成为他1904年由阿姆斯特丹皇家科学院发表的第一篇论文。布劳威尔感兴趣的其他主题是拓扑学和数学基础。他从大学的讲座中学到了一些这些主题的知识，但他也自己阅读了许多关于这些主题的著作。

他于1904年获得硕士学位，同年与Lize de Holl结婚，她比布劳威尔大11岁，并有一段前婚所生的女儿。婚后，这对夫妇搬到了阿姆斯特丹附近的布拉里库姆，他们没有孩子。三年后，Lize获得了药剂师资格，布劳威尔在许多方面帮助她，从记账到在药店服务。然而，布劳威尔并没有赢得继女的喜爱，他们之间的关系紧张。

从早期阶段起，布劳威尔就对数学哲学感兴趣，但他也着迷于神秘主义以及与人类社会相关的其他哲学问题。1905年，他在其论著Leven, Kunst, en Mystiek Ⓣ（《生命、艺术与神秘主义》）中发表了自己关于这一主题的想法。在这部作品中，他[1]：-

……将人类活动中从目标到手段的转变视为重要的推动原则之一，这种转变在经过若干次重复后可能导致与最初目标相反的活动。

布劳威尔的博士学位论文发表于1907年，对伯特兰·罗素与儒勒·昂利·庞加莱之间关于数学逻辑基础的持续争论做出了重大贡献。他的博士论文[13]：-

……揭示了支配他整个职业生涯的对数学的双重兴趣；他对批判性地评估数学基础的根本关注，这导致他创立了直觉主义，以及他对几何学的深厚兴趣，这导致了他在拓扑学中的开创性工作……

他很快发现，自己在数学基础方面的想法不会轻易被接受[13]：-

布劳威尔很快发现，他的哲学思想引发了争议。科特韦格，他的学位论文导师，对论文中较为哲学化的部分并不满意，甚至要求将初稿中的若干部分从最终答辩中删去。科特韦格敦促布劳威尔专注于更“体面”的数学，以便这位年轻人能够提升自己的数学声望，从而确保获得学术职位。布劳威尔极为独立，不追随任何人的脚步，但他显然接受了老师的建议……

布劳威尔在1908年发表的The Unreliability of the Logical Principles中继续发展其学位论文的思想。

布劳威尔此时开展的研究涉及两个领域。他继续研究数学的逻辑基础，同时也投入了极大的精力研究各种问题，他之所以研究这些问题，是因为它们出现在大卫·希尔伯特于1900年巴黎国际数学家大会上提出的问题清单上。特别是，布劳威尔研究了大卫·希尔伯特关于连续群理论的第五问题。他在1908年罗马国际数学家大会上就索菲斯·李群的拓扑基础作了报告。然而，在研究了阿图尔·舍恩弗利斯关于集合论的报告后，他写信给大卫·希尔伯特：-

我突然发现，关于平面拓扑的Schoenflies研究，我曾完全依赖它，但并非所有部分都正确，因此我的群论结果也变得可疑。

1909年，他被任命为阿姆斯特丹大学的privatdocent。1909年10月12日，他作了题为“几何的本质”的就职演讲，概述了他的研究计划。几个月后，大约在1909年圣诞节前后，他对巴黎进行了一次重要访问，并在那里会见了儒勒·昂利·庞加莱、雅克·阿达马和埃米尔·博雷尔。受巴黎讨论的启发，他开始研究维数不变性问题。

布劳威尔于1912年当选为皇家科学院院士，同年被任命为阿姆斯特丹大学集合论、函数论和公理学的特聘教授；他一直担任该职位，直到1951年退休。大卫·希尔伯特写了一封热情的推荐信，帮助布劳威尔在1912年获得讲席。尽管到此时他已对拓扑学作出了重大贡献，布劳威尔仍选择以直觉主义和形式主义作为其教授就职演讲的主题。次年，科特韦格辞去讲席，以便布劳威尔能被任命为正教授。

尽管他曾帮助布劳威尔在阿姆斯特丹获得讲席，1919年大卫·希尔伯特仍试图以哥廷根的一个讲席职位引诱他离开。同年，他还被提供了柏林的讲席。这些想必是诱人的提议，但尽管有其吸引力，布劳威尔还是拒绝了它们。或许，阿姆斯特丹对他的特殊待遇——巴特尔·伦德特·范德瓦尔登在下面引文中所提到的——帮助他做出了这些决定。

巴特尔·伦德特·范德瓦尔登，1919年至1923年在阿姆斯特丹学习，作为讲师撰写了关于布劳威尔的文章（例如见[14]）：-

布劳威尔来到[大学]授课，但住在拉伦。他每周只来一次。一般来说，这是不允许的——他应该住在阿姆斯特丹——但为他破例了。……有一次我在讲座中打断他提问。在下一周的课之前，他的助手来找我说，布劳威尔不希望有人在课堂上向他提问。他就是不想要这些问题，他总是看着黑板，从不面向学生。……尽管他最重要的研究贡献是在拓扑学，布劳威尔从未开设过拓扑学课程，而总是——并且只是——开设直觉主义基础课程。似乎他不再相信自己在拓扑学中的结果，因为从直觉主义的观点来看它们不正确，他根据他的哲学判断他以前所做的一切，他最伟大的成果，都是错误的。他是一个非常奇怪的人，疯狂地热爱他的哲学。

如这段引文所述，布劳威尔是拓扑学理论的主要贡献者，许多人认为他是其创始人。他开始研究时该学科的状况在[13]中有很好的描述：-

布劳威尔正开始他的数学生涯，集合论拓扑学处于原始状态。由于集合论悖论或矛盾，格奥尔格·康托尔的一般集合论备受争议。点集理论在分析中广泛应用，在几何中应用稍少，但它不具有统一理论的特征。有一些公认的基准。例如；普遍认为维数在一一连续映射下是不变的……

他在1909年至1913年的职业生涯早期几乎完成了所有拓扑学工作。他发现了笛卡尔平面的拓扑映射的特征以及若干不动点定理。他的第一个不动点定理，表明球面上保持定向的连续一一映射到自身总是至少固定一个点，源于他对大卫·希尔伯特第五问题的研究。最初对二维球面证明，布劳威尔后来将结果推广到$n$维球面。另一个特别重要的结果是证明维数的不变性。

除了在拓扑学中证明具有重大重要性的定理外，布劳威尔还发展了已成为该学科标准工具的方法。特别是他使用了单纯逼近，用分段线性映射逼近连续映射。他还引入了映射度的概念，将卡米耶·若尔当曲线定理推广到$n$维空间，并在1913年定义了拓扑空间。

在上述引文中，巴特尔·伦德特·范德瓦尔登说布劳威尔不会讲授他自己的拓扑学结果，因为它们不符合数学直觉主义。事实上，布劳威尔被许多数学家最为熟知的是作为数学直觉主义学说的创始人，该学说将数学视为受自明规律支配的心智构造的表述。他的学说与大卫·希尔伯特的形式主义和伯特兰·罗素的逻辑主义有很大不同。他1907年的博士论文攻击数学的逻辑基础，标志着直觉主义学派的开始。他的观点与儒勒·昂利·庞加莱的观点有更多共同之处，如果问他在伯特兰·罗素和儒勒·昂利·庞加莱之间的辩论中站在哪一边，那么他会站在后者一边。

在他1908年的论文The Unreliability of the Logical Principles中，布劳威尔在数学证明中拒绝了排中律，该原理指出任何数学陈述要么真要么假。1918年，他发表了一种不使用排中律发展的集合论Founding Set Theory Independently of the Principle of the Excluded Middle. Part One, General Set Theory。他1920年的讲座Does Every Real Number Have a Decimal Expansion?在次年出版。布劳威尔对标题问题的回答是“不”。同样在1920年，他发表了Intuitionistic Set Theory，然后在1927年，他发展了不使用排中律的函数理论On the Domains of Definition of Functions。

他的构造性理论不容易建立，因为集合的概念不能作为基本概念，而必须使用更基本的概念来构建，在布劳威尔的情况下，这些是选择序列。粗略地说，集合的元素具有性质p，对布劳威尔来说意味着他有一个构造，允许他在有限步后决定集合的每个元素是否具有性质p。这些思想是当今理论计算机科学的基础。

布劳威尔职业生涯的后期包含一些有争议的事件。他于1914年被任命为Mathematische Annalen的编辑委员会成员，但在1928年，大卫·希尔伯特决定布劳威尔变得过于强大，特别是因为大卫·希尔伯特觉得自己活不长了（事实上他活到1943年）。他试图以一种与委员会设立方式不相容的方式将布劳威尔从委员会中移除。布劳威尔强烈反对这一举动，并得到其他委员会成员如阿尔伯特·爱因斯坦和康斯坦丁·卡拉西奥多里的大力支持。最终大卫·希尔伯特设法得逞，但这对布劳威尔来说是一个毁灭性的事件，他精神崩溃；详情见[26]。

1935年，布劳威尔进入地方政治，作为中立党候选人当选为布拉里库姆市议会议员。他继续在议会任职至1941年。他还积极创办新期刊，并成为Compositio Mathematica的创始编辑，该刊于1934年开始出版。

他在第二次世界大战中的行为引发了进一步的争议。布劳威尔积极帮助荷兰抵抗运动，特别是在这一困难时期支持犹太学生。然而，1943年德国人坚持要求学生签署效忠德国的声明，布劳威尔鼓励他的学生这样做。他后来说，他这样做是为了让他的学生有机会完成学业并为荷兰抵抗德国人的运动工作。然而，在阿姆斯特丹解放后，布劳威尔因他的行为被停职几个月。他再次深受伤害并考虑移民。

1951年退休后，布劳威尔于1952年在南非讲学，1953年在美国和加拿大讲学。他的妻子于1959年去世，享年89岁，而布劳威尔本人78岁，被提供在温哥华不列颠哥伦比亚大学的一年职位；他拒绝了。1962年，尽管他已年过八旬，仍被提供蒙大拿州的一个职位。他于1966年在布拉里库姆因交通事故去世。

Kneebone在[3]中关于布劳威尔对数学哲学的贡献写道：-

布劳威尔最为著名的是……他对数学哲学的贡献，以及他试图在直觉主义基础上重新构建数学，以应对他自己对迄今未经质疑的假设所作的深入批判。布劳威尔在能够跳出既定文化传统、以冷静客观的态度审视其最神圣预设方面，有些像尼采；而他对思维原理的质疑，使他在逻辑领域引发了一场尼采式的革命。事实上，他拒绝了由亚里士多德最初编纂、几乎原封不动地传至现代、并最近借助数学符号被扩展和推广到面目全非的、普遍接受的演绎推理逻辑。

Kneebone在[3]中也写到了布劳威尔关于数学基础的观点对其同行数学家的影响：-

布劳威尔对整个数学大厦的重建计划始终是一个梦想，但他的构造主义理想如今已融入我们整个数学思想的结构之中，并且已经激发、且仍在激发各种以构造主义精神进行的探索，这些探索带来了数学知识的重大进展。

尽管未能使数学家们接受他的思想方式，布劳威尔仍因其杰出贡献而获得许多荣誉。我们在上文提到了他当选荷兰皇家科学院院士。其他荣誉包括当选伦敦皇家学会、Berlin Academy of Sciences和哥廷根科学院。他于1929年被奥斯陆大学授予荣誉博士学位，1954年被剑桥大学授予荣誉博士学位。1932年，他被授予荷兰狮骑士勋章。

L E J Brouwer is usually known by this form of his name with full initials, but he was known to his friends as Bertus, an abbreviation of the second of his three forenames. He attended high school in Hoorn, a town on the Zuiderzee north of Amsterdam. His performance there was outstanding and he completed his studies by the age of fourteen. He had not studied Greek or Latin at high school but both were required for entry into university, so Brouwer spent the next two years studying these topics. During this time his family moved to Haarlem, just west of Amsterdam, and it was in the Gymnasium there in 1897 that he sat the entrance examinations for the University of Amsterdam.

Korteweg was the professor of mathematics at the University of Amsterdam when Brouwer began his studies, and he quickly realised that in Brouwer he had an outstanding student. While still an undergraduate Brouwer proved original results on continuous motions in four dimensional space and Korteweg encouraged him to present them for publication. This he did, and it became his first paper published by the Royal Academy of Science in Amsterdam in 1904. Other topics which interested Brouwer were topology and the foundations of mathematics. He learnt something of these topics from lectures at the university but he also read many works on the topics on his own.

He obtained his master's degree in 1904 and in the same year married Lize de Holl who was eleven years older that Brouwer and had a daughter from a previous marriage. After the marriage, which would produce no children, the couple moved to Blaricum, near Amsterdam. Three years later Lize qualified as a pharmacist and Brouwer helped her in many ways from doing bookkeeping to serving in the chemists shop. However, Brouwer did not gain the affection of his step-daughter and relations between them was strained.

From an early stage Brouwer was interested in the philosophy of mathematics, but he was also fascinated by mysticism and other philosophical questions relating to human society. He published his own ideas on this topic in 1905 in his treatise Leven, Kunst, en Mystiek Ⓣ. In this work he [1]:-

... considers as one of the important moving principles in human activity the transition from goal to means, which after some repetitions may result in activities opposed to the original goal.

Brouwer's doctoral dissertation, published in 1907, made a major contribution to the ongoing debate between Russell and Poincaré on the logical foundations of mathematics. His doctoral thesis [13]:-

... revealed the twin interests in mathematics that dominated his entire career; his fundamental concern with critically assessing the foundations of mathematics, which led to his creation of intuitionism, and his deep interest in geometry, which led to his seminal work in topology ...

He quickly discovered that his ideas on the foundations of mathematics would not be readily accepted [13]:-

Brouwer quickly found that his philosophical ideas sparked controversy. Korteweg, his thesis advisor, had not been pleased with the more philosophical aspects of the thesis, and had even demanded that several parts of the original draft be cut from the final presentation. Korteweg urged Brouwer to concentrate on more "respectable" mathematics, so that the young man might enhance his mathematical reputation and thus secure an academic career. Brouwer was fiercely independent and did not follow in anybody's footsteps, but he apparently took his teacher's advice ...

Brouwer continued to develop the ideas of his thesis in The Unreliability of the Logical Principles published in 1908.

The research which Brouwer now undertook was in two areas. He continued his study of the logical foundations of mathematics and he also put a very large effort into studying various problems which he attacked because they appeared on Hilbert's list of problems proposed at the Paris International Congress of Mathematicians in 1900. In particular Brouwer attacked Hilbert's fifth problem concerning the theory of continuous groups. He addressed the International Congress of Mathematicians in Rome in 1908 on the topological foundations of Lie groups. However, after studying Schönflies's report on set theory, he wrote to Hilbert:-

I discovered all of a sudden that the Schoenfliesian investigations concerning topology of the plane, on which I had relied in the fullest way, could not be taken as correct in all parts, so that my group-theoretic results also became doubt.

In 1909 he was appointed as a privatdocent at the University of Amsterdam. He gave his inaugural lecture on 12 October 1909 on 'The nature of geometry' in which he outlined his research programme. A couple of months later he made an important visit to Paris, around Christmas 1909, and there met Poincaré, Hadamard and Borel. Prompted by discussions in Paris, he began working on the problem of the invariance of dimension.

Brouwer was elected to the Royal Academy of Sciences in 1912 and, in the same year, was appointed extraordinary professor of set theory, function theory and axiomatics at the University of Amsterdam; he would hold the post until he retired in 1951. Hilbert wrote a warm letter of recommendation which helped Brouwer to gain his chair in 1912. Despite the substantial contributions he had made to topology by this time, Brouwer chose to give his inaugural professorial lecture on intuitionism and formalism. In the following year Korteweg resigned his chair so that Brouwer could be appointed as ordinary professor.

Although he had helped Brouwer to obtain his chair in Amsterdam, in 1919 Hilbert tried to tempt him away with an offer of a chair in Göttingen. He was also offered the chair at Berlin in the same year. These must have been tempting offers, but despite their attractions Brouwer turned them down. Perhaps the exceptional way he was treated by Amsterdam, mentioned in the following quote by Van der Waerden, helped him make these decisions.

Van der Waerden, who studied at Amsterdam from 1919 to 1923, wrote about Brouwer as a lecturer (see for example [14]):-

Brouwer came [to the university] to give his courses but lived in Laren. He came only once a week. In general that would have not been permitted - he should have lived in Amsterdam - but for him an exception was made. ... I once interrupted him during a lecture to ask a question. Before the next week's lesson, his assistant came to me to say that Brouwer did not want questions put to him in class. He just did not want them, he was always looking at the blackboard, never towards the students. ... Even though his most important research contributions were in topology, Brouwer never gave courses on topology, but always on -- and only on -- the foundations of intuitionism. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy. He was a very strange person, crazy in love with his philosophy.

As is mentioned in this quotation, Brouwer was a major contributor to the theory of topology and he is considered by many to be its founder. The status of the subject when he began his research is well described in [13]:-

When Brouwer was beginning his career as a mathematician, set-theoretic topology was in a primitive state. Controversy surrounded Cantor's general set theory because of the set-theoretic paradoxes or contradictions. Point set theory was widely applied in analysis and somewhat less widely applied in geometry, but it did not have the character of a unified theory. There were some perceived benchmarks. For example; the generally held view that dimension was invariant under one-to-one continuous mappings ...

He did almost all his work in topology early in his career between 1909 and 1913. He discovered characterisations of topological mappings of the Cartesian plane and a number of fixed point theorems. His first fixed point theorem, which showed that an orientation preserving continuous one-one mapping of the sphere to itself always fixes at least one point, came out of his researches on Hilbert's fifth problem. Originally proved for a 2-dimensional sphere, Brouwer later generalised the result to spheres in $n$ dimensions. Another result of exceptional importance was proving the invariance of dimension.

As well as proving theorems of major importance in topology, Brouwer also developed methods which have become standard tools in the subject. In particular he used simplicial approximation, which approximated continuous mappings by piecewise linear ones. He also introduced the idea of the degree of a mapping, generalised the Jordan curve theorem to $n$-dimensional space, and defined topological spaces in 1913.

Van der Waerden, in the above quote, said that Brouwer would not lecture on his own topological results since they did not fit with mathematical intuitionism. In fact Brouwer is best known to many mathematicians as the founder of the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are governed by self-evident laws. His doctrine differed substantially from the formalism of Hilbert and the logicism of Russell. His doctoral thesis in 1907 attacked the logical foundations of mathematics and marks the beginning of the Intuitionist School. His views had more in common with those of Poincaré and if one asks which side of the debate between Russell and Poincaré he came down on then it would have with the latter.

In his 1908 paper The Unreliability of the Logical Principles Brouwer rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false. In 1918 he published a set theory developed without using the Principle of the Excluded Middle Founding Set Theory Independently of the Principle of the Excluded Middle. Part One, General Set Theory. His 1920 lecture Does Every Real Number Have a Decimal Expansion? was published in the following year. The answer to the question of the title which Brouwer gives is "no". Also in 1920 he published Intuitionistic Set Theory, then in 1927 he developed a theory of functions On the Domains of Definition of Functions without the use of the Principle of the Excluded Middle.

His constructive theories were not easy to set up since the notion of a set could not be taken as a basic concept but had to be built up using more basic notions which, in Brouwer's case, were choice sequences. Loosely speaking, that the elements of a set had property p, meant to Brouwer that he had a construction which allowed him to decide after a finite number of steps whether each element of the set had property p. Such ideas are fundamental to theoretical computer science today.

The later part of Brouwer's career contains some controversial episodes. He had been appointed to the editorial board of Mathematische Annalen in 1914 but in 1928 Hilbert decided that Brouwer was becoming too powerful, particularly since Hilbert felt that he himself did not have long to live (in fact he lived until 1943). He tried to remove Brouwer from the board in a way which was not compatible with the way the board was set up. Brouwer vigorously opposed the move and he was strongly supported by other board members such as Einstein and Carathéodory. In the end Hilbert managed to get his own way but it was a devastating episode for Brouwer who was left mentally broken; see [26] for details.

In 1935 Brouwer entered local politics when he was elected as Neutral Party candidate for the municipal council of Blaricum. He continued to serve on the council until 1941. He was also active setting up a new journal and he became a founding editor of Compositio Mathematica which began publication in 1934.

Further controversy arose due to his actions in World War II. Brouwer was active in helping the Dutch resistance, and in particular he supported Jewish students during this difficult period. However, in 1943 the Germans insisted that the students sign a declaration of loyalty to Germany and Brouwer encouraged his students to do so. He afterwards said that he did so in order that his students might have a chance to complete their studies and to work for the Dutch resistance against the Germans. However, after Amsterdam was liberated, Brouwer was suspended from his post for a few months because of his actions. Again he was deeply hurt and considered emigration.

After retiring in 1951, Brouwer lectured in South Africa in 1952, and the United States and Canada in 1953. His wife died in 1959 at the age of 89 and Brouwer, who himself was 78, was offered a one year post in the University of British Columbia in Vancouver; he declined. In 1962, despite being well into his 80s, he was offered a post in Montana. He died in 1966 in Blaricum as the result of a traffic accident.

Kneebone writes in [3] about Brouwer's contributions to the philosophy of mathematics:-

Brouwer is most famous ... for his contribution to the philosophy of mathematics and his attempt to build up mathematics anew on an Intuitionist foundation, in order to meet his own searching criticism of hitherto unquestioned assumptions. Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic. He in fact rejected the universally accepted logic of deductive reasoning which had been codified initially by Aristotle, handed down with very little change into modern times, and very recently extended and generalised out of all recognition with the aid of mathematical symbolism.

Kneebone also writes in [3] about the influence that Brouwer's views on the foundations of mathematics had on his fellow mathematicians:-

Brouwer's projected reconstruction of the whole edifice of mathematics remained a dream, but his ideal of constructivism is now woven into our whole fabric of mathematical thought, and it has inspired, as it still continues to inspire, a wide variety of inquiries in the constructivist spirit which have led to major advances in mathematical knowledge.

Despite failing to convert mathematicians to his way of thinking, Brouwer received many honours for his outstanding contributions. We mentioned his election to the Royal Dutch Academy of Sciences above. Other honours included election to the Royal Society of London, the Berlin Academy of Sciences, and the Göttingen Academy of Sciences. He was awarded honorary doctorates the University of Oslo in 1929, and the University of Cambridge in 1954. He was made Knight in the Order of the Dutch Lion in 1932.