赫尔曼·外尔Hermann Weyl

赫尔曼·外尔的亲密朋友称他为彼得。他的父母是安娜·迪克和Ludwig Weyl，后者是一家银行的董事。还是个男孩时，雅各布·赫尔曼就已经显示出他在数学以及更广泛的科学方面具有非凡的才能。在完成中学毕业考试（Abiturarbeit）（见[17]）后，他准备进入大学学习。1904年，他进入慕尼黑大学，在那里学习了数学和物理课程，然后继续在哥廷根大学学习同样的科目。他完全被大卫·希尔伯特迷住了。他后来写道：-

我决心研究这个人所写的一切。在第一年结束时，我腋下夹着《数论报告》回家，在暑假期间，我努力钻研它——此前没有任何初等数论或埃瓦里斯特·伽罗瓦论的知识。这是我一生中最快乐的几个月，它的光辉，跨越了充满我们共同承担的怀疑和失败的岁月，仍然安慰着我的灵魂。

他的博士学位来自哥廷根，导师是大卫·希尔伯特。在提交他的博士学位论文Singuläre Integralgleichungen mit besonder Berücksichtigung des Fourierschen IntegraltheoremsⓉ（具有特殊应用于约瑟夫·傅里叶积分定理的奇异积分方程）后，他于1908年被授予学位。这篇论文研究了奇异积分方程，深入探讨了约瑟夫·傅里叶积分定理。正是在哥廷根，他担任了他的第一个教学职位，作为privatdozent，他一直担任这个职位直到1913年。他的教授资格论文（Habilitation）论文Über gewöhnliche Differentialgleicklungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher FunktionenⓉ（关于具有奇点的常微分方程及其对任意函数的展开）研究了奇异雅克·夏尔·弗朗索瓦·施图姆-约瑟夫·刘维尔问题的谱理论。在哥廷根期间，外尔为自己赢得了杰出数学家的声誉，他产生的作品对数学的进步产生了重大影响。他的特许任教资格论文就是这样的作品之一，但还有更多。他在1911-12学年讲授了一门关于波恩哈德·黎曼曲面的课程，从这门课程中诞生了他的第一本书Die Idee der Riemannschen FlächeⓉ（波恩哈德·黎曼曲面的概念），该书于1913年出版。它统一了分析、几何和拓扑学，使波恩哈德·黎曼发展的几何函数论变得严谨。这本书首次引入了[59]的概念：-

……二维可微流形、覆盖曲面，以及微分与1-闭链之间的对偶性。……外尔的空间概念还包括后来引入并普遍归功于费利克斯·豪斯多夫（1914年）的著名分离性质。

L Sario在1956年写道，外尔1913年的文本：-

……无疑对几何函数论的发展产生的影响比波恩哈德·黎曼的学位论文以来的任何其他出版物都大。

相当引人注目的是，这部1913年的著作于1997年重印。外尔本人推出了两个后来的版本，其中第三个（也是最后一个）版本于1955年问世，涵盖的主题与原书相同，但处理方式更为现代。然而，正是最初的1913年版于1997年重印，这或许比后来的版本更充分地表明，最初的1913年文本在数学发展中是何等重要。

作为哥廷根的编外讲师，外尔曾受到Edmund Husserl的影响，后者自1901年至1916年在该校担任哲学讲席。外尔于1913年与Helene Joseph结婚，她曾是Husserl的学生；他们有两个儿子。Helene出身犹太背景，是一位哲学家，当时从事西班牙语翻译工作。外尔与妻子不仅对哲学有共同兴趣，还共同拥有真正的语言天赋。语言对外尔具有特殊的重要性。他不仅德语写得优美，后来还写出了令人惊叹的英文散文，尽管用他自己在1939年一篇英文文本中的话来说：-

……诸神强加于我写作的，是一门外语的枷锁，这语言并非在我摇篮边吟唱。

从1913年到1930年，外尔担任苏黎世联邦理工学院数学讲席。在这个新职位上的第一个学年，他与阿尔伯特·爱因斯坦是同事，后者当时正在推敲广义相对论理论的细节。这一事件对外尔产生了很大影响，他很快便迷上了该理论背后的数学原理。

外尔在苏黎世获得讲席后不久，第一次世界大战爆发。作为德国公民，他于1915年被征入德国军队，但瑞士政府特别请求允许他回到苏黎世的讲席，这一请求于1916年获批。1917年，外尔又开设了一门课程，通过微分几何对相对论提出了一种创新方法。这些讲座构成了外尔第二本书Raum-Zeit-MaterieⓉ（《空间-时间-物质》）的基础，该书于1918年首次出版，随后在1919年、1920年和1923年又出了更多版本，每一版都展示了他的思想如何发展。这些后来的思想包括一种度规（外尔度规），它导向了一种规范场论。然而阿尔伯特·爱因斯坦、沃尔夫冈·泡利、亚瑟·爱丁顿等人并未完全接受外尔的方法。同样在这一时期，外尔还在模1的均匀分布方面做出了贡献，这些贡献在解析数论中是基础性的。

1921年，埃尔温·薛定谔被任命到苏黎世，在那里他成为外尔的同事，并很快成为其最亲密的朋友。他们在数学、物理和哲学方面有许多共同兴趣。正如Moore在[5]中所述，他们的个人生活也变得纠缠在一起：-

那些在普林斯顿熟悉外尔严肃而魁梧形象的人，几乎认不出二十年代那个身材修长、相貌英俊、留着浪漫黑色小胡子的年轻人。他的妻子海伦·约瑟夫出身犹太背景，是一位哲学家和文人。她的朋友叫她赫拉，某种大胆和无忧无虑使她成为由科学家及其妻子组成的社交圈中无可争议的领袖。安妮[埃尔温·薛定谔的妻子]几乎与时尚而聪慧的赫拉截然相反，但也许正因如此，[外尔]觉得她有趣，不久她就疯狂地爱上了他。……他们在苏黎世所处的那个特殊圈子，比[美国]早一代人就享受了性革命。婚外情不仅被宽容，而且被期待，似乎很少引起焦虑。安妮在外尔身上找到了一个她全身心奉献的情人，而外尔的妻子赫拉则迷恋上了保罗·舍雷尔。

从1923年到1938年，外尔利用矩阵表示发展了连续群的概念。特别是他在1924年至1926年间发展的半单群的representations理论非常深刻，被外尔本人视为他最伟大的成就。这一理论背后的思想已由阿道夫·赫维兹和伊赛·舒尔引入，但正是外尔以其一般特征标公式将它们向前推进。然而，他并不是唯一发展这一理论的数学家，因为埃利·嘉当也在这一主题上做出了极其重要的工作。

从1930年到1933年，外尔在哥廷根担任数学讲席，他被任命填补大卫·希尔伯特退休后出现的空缺。如果政治环境不同，他很可能会在哥廷根度过余下的职业生涯。然而[8]：-

……纳粹的崛起使他在1933年接受普林斯顿新成立的爱德华·斯图迪高等研究院的职位，阿尔伯特·爱因斯坦也去了那里。在这里，外尔找到了一个非常适宜的工作环境，他能够指导和影响年轻一代的数学家，这项任务他极其适合。

人们还必须理解，外尔的妻子是犹太人，这在他们1933年离开德国的决定中必定起了主要作用。外尔留在普林斯顿的斯图迪高等研究院，直到1952年退休。他的妻子海伦于1948年去世，两年后他娶了来自苏黎世的雕塑家艾伦·洛恩斯坦·贝尔。

外尔在普林斯顿确实从事了极其重要的工作，但他最多产的时期无疑是在苏黎世度过的那些年。他试图将电磁学纳入广义相对论的几何形式体系。他提出了第一个统一场论，其中詹姆斯·克拉克·麦克斯韦电磁场和引力场表现为时空的几何性质。通过将群论应用于量子力学，他建立了现代学科。正是他在1927-28学年在苏黎世开设的群论与量子力学讲座课程，促成了他于1928年出版的第三部主要著作Gruppentheorie und Quantenmechanik Ⓣ（群论与量子力学）。John Wheeler写道[56]：-

那本书，我每读一次，都会有某种重要的新启示。

晚近，安娜·约翰逊·佩尔·惠勒曾尝试将电磁学纳入广义相对论。安娜·约翰逊·佩尔·惠勒的理论与外尔的理论一样，缺乏与量子现象的联系，而这种联系对于引力之外的相互作用是如此重要。惠勒在[56]中写到了第一次见到外尔的情形：-

1937年把我带到普林斯顿时，我第一次亲眼见到外尔：他身姿挺拔，目光炯炯，面带微笑。在那里，我听了他关于埃利·嘉当微分形式演算及其在电磁学中应用的讲座——雄辩、简洁、充满洞见。

我们在上文已经看到，外尔的伟大著作最初是如何作为讲座课程给出的。这是外尔有意为之的设计[56]：-

另有一次，外尔安排在普林斯顿大学讲授一门数学史课程。有一天他向我解释说，通过讲课来全面彻底地回顾他所关注的课题，对他而言是一种绝对的需要。他说，只有这样，他才能看到那些巨大的空白，那些需要更深入理解的地方，那些应当集中精力研究的地方。

外尔在普林斯顿的岁月里还出版了许多其他伟大著作。这些包括Elementary Theory of Invariants（1935）、The classical groups（1939）、Algebraic Theory of Numbers（1940）、Philosophy of Mathematics and Natural Science（1949）、Symmetry（1952）和The Concept of a Riemannian Surface（1955）。关于所有这些著作，可以说的太多了，但我们仅限于考察Symmetry的内容，因为这或许最能告诉我们外尔兴趣的完整范围。哈罗德·斯科特·麦克唐纳·考克斯特评论了这本书，他的评论出色地捕捉到了这本书的精神：-

这是1951年在普林斯顿大学所作的Louis Clark Vanuxem讲座的稍作修改的版本……第一讲首先展示双边对称的观念如何影响了绘画和雕塑，尤其是在古代。这自然地引向对“左与右的哲学”的讨论，包括如下问题。在自然界中，一种光学活性物质的两种对映形态之一出现，是否是生命物质的特征？在胚胎发育的哪个阶段，对称面被确定？第二讲包含了对变换群理论的简洁阐述，特别强调了相似群及其子群：全等变换群、运动群、平移群、旋转群，最后是任意给定图形的对称群。……循环群和二面体群通过雪花和花朵、被称为水母的动物以及对称建筑的平面图来举例说明。类似地，由螺旋相似生成的无限循环群通过鹦鹉螺壳和向日葵中小花的排列来举例说明。第三讲给出了二维晶体学十七个空间群枚举的基本步骤……[在第四讲中，他]展示了狭义相对论本质上是研究四维时空连续统的固有对称性，其中对称操作是亨德里克·洛伦兹变换；以及根据量子力学，原子的对称操作包括其外围电子的置换。从物理学转向数学，他给出了伽罗瓦理论的极其简洁的概要，引出了他的指导原则的陈述：“每当你处理一个具有结构的实体时，试着确定它的自同构群”。

1951年，外尔从普林斯顿高等斯图迪退休。事实上，他将Symmetry一书描述为他的‘天鹅之歌’。退休后，外尔和他的妻子Ellen部分时间在普林斯顿度过，部分时间在苏黎世度过。他在苏黎世时意外去世。他是在寄出感谢信给那些在他七十岁生日时祝愿他安好的人后步行回家时倒下并去世的。

我们必须稍微谈谈外尔工作的另一个方面，我们实际上没有提到，即他在数学哲学和数学基础方面的工作。有趣的是，我们引用的参考文献中有大量涉及他工作的这一方面，其重要性不仅在于工作本身，还在于外尔关于这些主题的思想在多大程度上构成了他其余数学和物理贡献的基础。外尔在他的观点上深受胡塞尔影响，也与勒伊岑·布劳威尔分享了许多思想。两人都认为直观连续统不能由格奥尔格·康托尔的集合论连续统准确表示。惠勒[56]写道：-

外尔教导我们，连续统……是一种幻觉。它是一种理想化。它是一个梦。

外尔总结了他对数学的态度，写道：-

我自己的数学工作总是相当不系统，没有模式或联系。表达和形状对我来说几乎比知识本身更重要。但我相信，撇开我自己独特的本性不谈，数学本身与实验学科相比，具有一种更接近自由创造性艺术的特征。

你可以从外尔1946年在THIS LINK普林斯顿二百周年纪念会上的演讲中更多地了解他对数学的态度。

他经常被引用的一句话：-

我的工作总是试图将真与美结合起来，但当不得不选择其中之一时，我通常选择美……

虽然一半是玩笑，却概括了他的个性。

Hermann Weyl was known as Peter to his close friends. His parents were Anna Dieck and Ludwig Weyl who was the director of a bank. As a boy Hermann had already showed that he had a great talents for mathematics and for science more generally. After taking his Abiturarbeit (high school graduation exam) (see [17]) he was ready for his university studies. In 1904 he entered the University of Munich, where he took courses on both mathematics and physics, and then went on to study the same topics at the University of Göttingen. He was completely captivated by Hilbert. He later wrote:-

I resolved to study whatever this man had written. At the end of my first year I went home with the "Zahlbericht" under my arm, and during the summer vacation I worked my way through it - without any previous knowledge of elementary number theory or Galois theory. These were the happiest months of my life, whose shine, across years burdened with our common share of doubt and failure, still comforts my soul.

His doctorate was from Göttingen where his supervisor was Hilbert. After submitting his doctoral dissertation Singuläre Integralgleichungen mit besonder Berücksichtigung des Fourierschen Integraltheorems Ⓣ he was awarded the degree in 1908. This thesis investigated singular integral equations, looking in depth at Fourier integral theorems. It was at Göttingen that he held his first teaching post as a privatdozent, a post he held until 1913. His habilitation thesis Über gewöhnliche Differentialgleicklungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen Ⓣ investigated the spectral theory of singular Sturm-Liouville problems. During this period at Göttingen, Weyl made a reputation for himself as an outstanding mathematician who was producing work which was having a major impact on the progress of mathematics. His habilitation thesis was one such piece of work but there was much more. He gave a lecture course on Riemann surfaces in session 1911-12 and out of this course came his first book Die Idee der Riemannschen Fläche Ⓣ which was published in 1913. It united analysis, geometry and topology, making rigorous the geometric function theory developed by Riemann. The book introduced for the first time the notion of a [59]:-

... two-dimensional differentiable manifold, a covering surface, and the duality between differentials and 1-cycles. ...Weyl's idea of a space also included the famous separation property later introduced and popularly credited to Felix Hausdorff (1914).

L Sario wrote in 1956 that Weyl's 1913 text:-

... has undoubtedly had a greater influence on the development of geometric function theory than any other publication since Riemann's dissertation.

It is rather remarkable that this 1913 text was reprinted in 1997. Weyl himself produced two later editions, the third (and final) of these editions appearing in 1955 covering the same topics as the original text but with a more modern treatment. It was the original 1913 edition, however, which was reprinted in 1997 showing perhaps more fully than the later editions just how significant the original 1913 text was in the development of mathematics.

As a privatdozent at Göttingen, Weyl had been influenced by Edmund Husserl who held the chair of philosophy there from 1901 to 1916. Weyl married Helene Joseph, who had been a student of Husserl, in 1913; they had two sons. Helene, who came from a Jewish background, was a philosopher who was working as a translator of Spanish. Not only did Weyl and his wife share an interest in philosophy, but they shared a real talent for languages. Language for Weyl held a special importance. He not only wrote beautifully in German, but later he wrote stunning English prose despite the fact that, in his own words from a 1939 English text:-

... the gods have imposed upon my writing the yoke of a foreign language that was not sung at my cradle.

From 1913 to 1930 Weyl held the chair of mathematics at Zürich Technische Hochschule. In his first academic year in this new post he was a colleague of Einstein who was at this time working out the details of the theory of general relativity. It was an event which had a large influence on Weyl who quickly became fascinated by the mathematical principles lying behind the theory.

World War I broke out not long after Weyl took up the chair in Zürich. Being a German citizen he was conscripted into the German army in 1915 but the Swiss government made a special request that he be allowed to return to his chair in Zürich which was granted in 1916. In 1917 Weyl gave another course presenting an innovative approach to relativity through differential geometry. The lectures formed the basis of Weyl's second book Raum-Zeit-Materie Ⓣ which first appeared in 1918 with further editions, each showing how his ideas were developing, in 1919, 1920, and 1923. These later ideas included a gauge metric (the Weyl metric) which led to a gauge field theory. However Einstein, Pauli, Eddington, and others, did not fully accept Weyl's approach. Also over this period Weyl also made contributions on the uniform distribution of numbers modulo 1 which are fundamental in analytic number theory.

In 1921 Schrödinger was appointed to Zürich where he became a colleague, and soon closest friend, of Weyl. They shared many interests in mathematics, physics, and philosophy. Their personal lives also became entangled as Moore relates in [5]:-

Those familiar with the serious and portly figure of Weyl at Princeton would have hardly recognised the slim, handsome young man of the twenties, with his romantic black moustache. His wife, Helene Joseph, from a Jewish background, was a philosopher and literateuse. Her friends called her Hella, and a certain daring and insouciance made her the unquestioned leader of the social set comprising the scientists and their wives. Anny [Schrödinger's wife] was almost an exact opposite of the stylish and intellectual Hella, but perhaps for that reason [Weyl] found her interesting and before long she was madly in love with him. ... The special circle in which they lived in Zürich had enjoyed the sexual revolution a generation before [the United States]. Extramarital affairs were not only condoned, they were expected, and they seemed to occasion little anxiety. Anny would find in Hermann Weyl a lover to whom she was devoted body and soul, while Weyl's wife Hella was infatuated with Paul Scherrer.

From 1923-38 Weyl evolved the concept of continuous groups using matrix representations. In particular his theory of representations of semisimple groups, developed during 1924-26, was very deep and considered by Weyl himself to be his greatest achievement. The ideas behind this theory had already been introduced by Hurwitz and Schur, but it was Weyl with his general character formula which took them forward. He was not the only mathematician developing this theory, however, for Cartan also produced work on this topic of outstanding importance.

From 1930 to 1933 Weyl held the chair of mathematics at Göttingen where he was appointed to fill the vacancy which arose on Hilbert's retirement. Given different political circumstances it is likely that he would have remained in Göttingen for the rest of his career. However [8]:-

... the rise of the Nazis persuaded him in 1933 to accept a position at the newly formed Institute for Advanced Study in Princeton, where Einstein also went. Here Weyl found a very congenial working environment where he was able to guide and influence the younger generation of mathematicians, a task for which he was admirably suited.

One also has to understand that Weyl's wife was Jewish, and this must have played a major role in their decision to leave Germany in 1933. Weyl remained at the Institute for Advanced Study at Princeton until he retired in 1952. His wife Helene died in 1948, and two years later he married the sculptor Ellen Lohnstein Bär from Zürich.

Weyl certainly undertook work of major importance at Princeton, but his most productive period was without doubt the years he spent at Zürich. He attempted to incorporate electromagnetism into the geometric formalism of general relativity. He produced the first unified field theory for which the Maxwell electromagnetic field and the gravitational field appear as geometrical properties of space-time. With his application of group theory to quantum mechanics he set up the modern subject. It was his lecture course on group theory and quantum mechanics in Zürich in session 1927-28 which led to his third major text Gruppentheorie und Quantenmechanik Ⓣ published in 1928. John Wheeler writes [56]:-

That book has, each time I read it, some great new message.

More recently attempts to incorporate electromagnetism into general relativity have been made by Wheeler. Wheeler's theory, like Weyl's, lacks the connection with quantum phenomena that is so important for interactions other than gravitation. Wheeler writes about meeting Weyl for the first time in [56]:-

Erect, bright-eyed, smiling Hermann Weyl I first saw in the flesh when 1937 brought me to Princeton. There I attended his lectures on the Élie Cartan calculus of differential forms and their application to electromagnetism - eloquent, simple, full of insights.

We have seen above how Weyl's great works were first given as lecture courses. This was a deliberate design by Weyl [56]:-

At another time Weyl arranged to give a course at Princeton University on the history of mathematics. He explained to me one day that it was for him an absolute necessity to review, by lecturing, his subject of concern in all its length and breadth. Only so, he remarked, could he see the great lacunae, the places where deeper understanding is needed, where work should focus.

Many other great books by Weyl appeared during his years at Princeton. These include Elementary Theory of Invariants (1935), The classical groups (1939), Algebraic Theory of Numbers (1940), Philosophy of Mathematics and Natural Science (1949), Symmetry (1952), and The Concept of a Riemannian Surface (1955). There is so much that could be said about all these works, but we restrict ourselves to looking at the contents of Symmetry for this perhaps tells us most about the full range of Weyl's interests. Coxeter reviewed the book and his review beautifully captures the spirit of the book:-

This is slightly modified version of the Louis Clark Vanuxem Lectures given at Princeton University in 1951 ... The first lecture begins by showing how the idea of bilateral symmetry has influenced painting and sculpture, especially in ancient times. This leads naturally to a discussion of "the philosophy of left and right", including such questions as the following. Is the occurrence in nature of one of the two enantiomorphous forms of an optically active substance characteristic of living matter? At what stage in the development of an embryo is the plane of symmetry determined? The second lecture contains a neat exposition of the theory of groups of transformations, with special emphasis on the group of similarities and its subgroups: the groups of congruent transformations, of motions, of translations, of rotations, and finally the symmetry group of any given figure. ... the cyclic and dihedral groups are illustrated by snowflakes and flowers, by the animals called Medusae, and by the plans of symmetrical buildings. Similarly, the infinite cyclic group generated by a spiral similarity is illustrated by the Nautilus shell and by the arrangement of florets in a sunflower. The third lecture gives the essential steps in the enumeration of the seventeen space-groups of two-dimensional crystallography ... [In the fourth lecture he] shows how the special theory of relativity is essentially the study of the inherent symmetry of the four-dimensional space-time continuum, where the symmetry operations are the Lorentz transformations; and how the symmetry operations of an atom, according to quantum mechanics, include the permutations of its peripheral electrons. Turning from physics to mathematics, he gives an extraordinarily concise epitome of Galois theory, leading up to the statement of his guiding principle: "Whenever you have to do with a structure-endowed entity, try to determine its group of automorphisms".

In 1951 Weyl retired from the Institute for Advanced Study at Princeton. In fact he described the Symmetry book as his 'swan song'. After his retirement Weyl and his wife Ellen spent part of their time at Princeton and part at Zürich. He died unexpectedly while in Zürich. He was walking home after posting letters of thanks to those who had wished him well on his seventieth birthday when he collapsed and died.

We must say a little about another aspect of Weyl's work which we have not really mentioned, namely his work on mathematical philosophy and the foundations of mathematics. It is interesting to note what a large number of the references we quote deal with this aspect of his work and its importance is not only in the work itself but also in the extent to which Weyl's ideas on these topics underlies the rest of his mathematical and physical contributions. Weyl was much influenced by Husserl in his outlook and also shared many ideas with Brouwer. Both shared the view that the intuitive continuum is not accurately represented by Cantor's set-theoretic continuum. Wheeler [56] writes:-

The continuum ..., Weyl taught us, is an illusion. It is an idealization. It is a dream.

Weyl summed up his attitude to mathematics, writing:-

My own mathematical works are always quite unsystematic, without mode or connection. Expression and shape are almost more to me than knowledge itself. But I believe that, leaving aside my own peculiar nature, there is in mathematics itself, in contrast to the experimental disciplines, a character which is nearer to that of free creative art.

You can discover more of Weyl's attitude to mathematics from his address to the Princeton Bicentennial in 1946 at THIS LINK.

His often quoted comment:-

My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful ...

although half a joke, sums up his personality.