奥斯瓦尔德·维布伦Oswald Veblen

奥斯瓦尔德·维布伦的父母是Andrew Anderson 维布伦和Kirsti Hougen。Andrew Veblen的父亲是挪威的木匠，但在1847年从挪威移民到美国后成为农民。Andrew Veblen的一个兄弟，Thorstein Bunde 维布伦，是一位著名的经济学家和社会批评家。Andrew Veblen是他父母十二个孩子中的长子，在维布伦出生时，他是爱荷华州迪科拉的Luther学院的教师。当他的儿子维布伦一岁时，他离开了Luther学院，成为爱荷华大学的数学和物理学教授。

维布伦 在进入爱荷华大学之前，在爱荷华城上学，1894 年入学，1898 年获得 A.B. 学位。在爱荷华大学担任实验室助理一年后，维布伦 在哈佛大学度过了一年，于 1900 年获得第二个 B.A. 学位，然后前往芝加哥大学进行研究。Archibald 在 [4] 中写道：-

他在芝加哥大学接受了大部分数学训练，师从那个鼓舞人心的三人组 Bolza、海因里希·马什克 和 伊莱基姆·黑斯廷斯·穆尔。在他们的指导下，他为他后来在几何基础、射影几何、拓扑学、微分不变量和旋量等领域取得的重要工作奠定了基础。他在 伊莱基姆·黑斯廷斯·穆尔 指导下经常被引用的学位论文，关于欧几里得几何的公理系统，遵循了 Pasch（1882）和 朱塞佩·皮亚诺（1889，1894）的发展趋势，而不是 大卫·希尔伯特（1899）和 玛利欧·派埃利（1899）的趋势。

事实上，维布伦在1901年参加伊莱基姆·黑斯廷斯·穆尔的讨论班时，第一次对伊莱基姆·黑斯廷斯·穆尔产生了深刻印象。次年，当Moore发表On the projective axioms of geometry时，他承认了维布伦的贡献，写道：-

……本课程成员的询问和评论，尤其是O先生维布伦的，一直是许多灵感的来源。

维布伦的博士学位论文，由伊莱基姆·黑斯廷斯·穆尔指导，题为A System of Axioms for Geometry，他于1903年在芝加哥大学获得博士学位。在论文中，维布伦给出了一个基于点和顺序而非传统的点、线和平面概念的公理系统。他提出了欧几里得几何的十二条公理，并证明这是一个完备的公理系统，他还证明了这些公理的独立性。获得博士学位后，他又在芝加哥待了两年，担任数学副教授。在此期间，他实际上指导了罗伯特·李·穆尔的博士研究，罗伯特·李·穆尔名义上是伊莱基姆·黑斯廷斯·穆尔的学生，于1905年因一篇题为Sets of Metrical Hypotheses for Geometry的学位论文而获得博士学位。在此期间，维布伦致力于将自己的论文整理成可发表的形式，A system of axioms for geometry于1904年作为一篇41页的论文发表在Transactions of the American Mathematical Society上。此时他已经开始进行拓扑学（或当时所称的分析位置）的研究，并于1905年发表了Theory on plane curves in non-metrical analysis situs。

维布伦从1905年到1932年在普林斯顿大学教授数学。他于1905年首次被任命为普林斯顿的导师，这是普林斯顿校长Woodrow Wilson为提高大学学术水平而引入的新职位。维布伦于1908年与Elizabeth Mary Dixon Richardson结婚。她是Owen Richardson的妹妹，Owen Richardson是普林斯顿的物理学教授，后来于1928年获得诺贝尔物理学奖。维布伦和他的妻子Elizabeth没有孩子。结婚两年后，即1910年，维布伦被提升为普林斯顿的数学教授。

在第一次世界大战期间，他先担任上尉，后来在军队中担任少校。他在1917年春天寻求在军械部获得委任，并被任命到马里兰州新的阿伯丁试验场的弹道研究办公室。在1917年夏天接受基础训练后，他于1918年1月4日抵达试验场。维布伦领导实验弹道学部，这是试验场的数学研究设施。他开始收集2.95英寸火炮射程的数据。然后他致力于将原始数据转换为结果表的问题，为此他有一批人员来进行冗长的手工计算。他雇用约瑟夫·里特来负责这些计算员。他的工作导致了新的弹道理论。关于他在此期间所从事工作的详细信息，见[7]。

1926年，普林斯顿大学授予维布伦荣誉，任命他为Henry B Fine数学讲席教授。在1928-29学年，他作为与戈弗雷·哈罗德·哈代交换的一部分在牛津大学任教。1929年，普林斯顿为Fine Hall提供了资金，维布伦提供了其设计的大部分想法。他希望Fine Hall的数学家们能够：-

……组成团体，相互鼓励和支持。[它必须是一个]年轻新兵和老将[能够拥有]那些对每个人都如此重要的非正式和轻松的接触的地方。

然而，他也希望为教授们保留一个房间，因为：-

……那些试图拉近师生关系的人并不总能理解的是，在各种形式的社交往来中，为隐私所做的安排与为亲近所做的安排同样重要。

1932年，他在德国度过了一段时间，并在哥廷根、柏林和汉堡讲学。Zund写道，这段经历14：-

……让他 firsthand 瞥见了德国即将到来的动荡，随后他不懈努力，帮助安置来到美国的难民。

Leon Warren Cohen在1984年的一次采访中，回顾了维布伦对安置难民数学家的态度：-

那是大萧条时期。年轻的美国数学家很难找到职位，是否要引进外国数学家来占据那些本应留给美国数学家的位置，这个问题引发了争论。维布伦采取了我会称之为更宽广的视角。我犹豫是否要把观点归于维布伦，但似乎驱动他的考虑有两点：对数学本身福祉的关切，以及对某些有才华个人的出于人道的关切。维布伦是个了不起的人，他使之有可能来到美国的人为数学做出了巨大贡献。乔治·戴维·伯克霍夫在这件事上反对他。……[乔治·戴维·伯克霍夫说]“如果这些杰出人物来了并占据这些位置，年轻的美国数学家将沦为劈柴挑水的人。”

维布伦于1932年帮助在普林斯顿组织了高等爱德华·斯图迪，同年辞去Henry B Fine讲席教授职位，成为该研究院的首位教授。他让詹姆斯·韦德尔·亚历山大、阿尔伯特·爱因斯坦、冯·诺伊曼和赫尔曼·外尔——都是他挑选的数学家——成为研究院最初的成员。

维布伦对几何基础的兴趣促使他研究射影几何的公理系统。1906年，他与W H Bussey合作发表了Finite projective geometries，1907年发表了Collineations in a finite projective geometry，1908年发表了Non-Desarguesian and non-Pascalian geometries。他与弗瑞兹·约翰 Wesley Young合作发表了Projective geometry（1910-18）。这本书两卷中的第一卷由两位作者共同撰写，但第二卷则完全出自维布伦之手。这部著作的导言为研究基础提供了理由：

即使本卷中用于基础的有限篇幅，从教学法的角度来看，对某些数学家来说可能似乎是一个缺点。对此我们只能回答，在我们看来，如果不关注几何学的基础，就不可能获得对几何学的充分认识。我们还认为，抽象的处理方式在射影几何中尤为可取，因为正是通过射影几何，其他几何学科才最容易得到协调。由于从射影几何推导出与欧几里得、勒内·笛卡儿、罗巴切夫斯基等人名字相关的几何学科，比从其中某一学科推导出射影几何更为自然，因此将射影几何的基础作为所有几何学的基础是自然的。

我们上面提到，维布伦关于拓扑学的第一篇著作恰好在他抵达普林斯顿之前问世。他随后将普林斯顿确立为世界拓扑学研究的主要中心之一。Analysis Situs（1922）首次系统性地涵盖了拓扑学的基本思想，并促进了现代拓扑学的发展。

阿尔伯特·爱因斯坦的广义相对论理论出现后不久，维布伦将注意力转向了微分几何。这项工作导致了相对论理论中的重要应用，他的许多工作也在原子物理学中找到了应用。他的作品The invariants of quadratic differential forms（1927）是对波恩哈德·黎曼几何的系统处理，而他的作品，与他的学生Henry Whitehead共同撰写，The foundations of differential geometry（1933）给出了differentiable manifold的第一个定义。在Projective relativity theory（1933）中，他给出了旋量的新处理，用于表示电子自旋。

维布伦是美国数学会的活跃成员，1915年担任该学会的副主席，1923-24年担任主席。1916年，他作为该学会的学术讨论会讲师，做了一系列关于拓扑学的讲座。作为美国数学会的主席，他是美国数学界的领袖，他为增加研究经费所做的努力在[6]中有解释，Feffer写道：

在两次世界大战之间，美国的所有科学专业都经历了巨大的发展。科学领袖们在战时的经历激发了他们对继续开展某种集中、资金充足的研究项目的渴望。物理学家和化学家不是求助于政府，而是转向慈善事业，努力将他们战后的声望转化为对不受限制的研究的更大支持。美国数学界的领袖们也想扩大他们的支持基础，但发现自己面临着独特的障碍。为支持数学研究而筹集资金的尝试结果好坏参半。[最重要的是]维布伦在20世纪20年代领导的努力，在物理科学扩张的背景下，通过普林斯顿大学和美国数学会为支持数学筹集资金。

维布伦当选为美国哲学会和国家科学院（美国）的成员，并获得了世界各地许多其他学会的荣誉会员资格。例如，他是伦敦数学会的成员，1928年他在牛津接替戈弗雷·哈罗德·哈代时担任理事会成员。牛津大学还在1929年授予他荣誉理学博士学位，同年，在尼尔斯·阿贝尔百年庆典之际，奥斯陆大学也授予他荣誉。他当选为丹麦科学院、Academy of Sciences（巴黎）、Polish Academy of Sciences、Royal Society of Edinburgh以及爱尔兰、意大利和秘鲁等国的许多其他国家科学院的成员。除了牛津大学的荣誉学位外，他还获得了爱丁堡大学、格拉斯哥大学、汉堡大学和奥斯陆大学的类似荣誉。他被授予挪威皇家圣奥拉夫勋章的骑士称号，这一荣誉也曾授予他的父亲。

关于他的个人特点，他是14：-

……一个讨人喜欢的人，非常谦逊，富有个人魅力。

L W Cohen 说：-

维布伦有许多极好的品质。他是个善良的人。

维布伦热爱户外，正是他的倡议促使普林斯顿为高等斯图迪购买了一片林地，以便成员们可以去乡间散步。他自己的家位于八十英亩的林地中，他将其命名为赫伦敦森林。他和妻子把赫伦敦森林留给了默瑟县，作为一个可以……的地方：-

……远离汽车，只是散步和坐着。

在他生命的最后几年，维布伦部分失明，他设计了一些盲人辅助工具，其中一件由美国盲人基金会制造并分发。

Oswald Veblen's parents were Andrew Anderson Veblen and Kirsti Hougen. Andrew Veblen's father was a Norwegian cabinet maker but became a farmer after emigrating from Norway to the United States in 1847. One of Andrew Veblen's brothers, Thorstein Bunde Veblen, was a well-known economist and social critic. Andrew Veblen, was the eldest of his parents twelve children and he was a teacher at Luther College in Decorah, Iowa, at the time Oswald was born. He left Luther College when his son Oswald was one year old and he became professor of mathematics and physics at the University of Iowa.

Veblen attended school in Iowa City before entering the University of Iowa in 1894, receiving his A.B. in 1898. After a year spent as a laboratory assistant at the University of Iowa, Veblen spent a year at Harvard University where he was awarded a second B.A. in 1900, before going to the University of Chicago to undertake research. Archibald writes in [4]:-

He received the major part of his mathematical training at the University of Chicago from that inspiring trio Bolza, Maschke, and Eliakim Moore. Under their direction he laid the basis for the important work he was later to achieve in the fields of foundations of geometry, projective geometry, topology, differential invariants and spinors. His often quoted dissertation under Moore, on a system of axioms of Euclidean geometry, followed the trend of development of Pasch (1882) and Peano (1889, 1894) rather than that of Hilbert (1899) and Pieri (1899).

In fact Veblen first impressed Moore when he attended his seminar in 1901. When Moore published On the projective axioms of geometry in the following year he acknowledged Veblen's contribution, writing:-

... queries and remarks of members of this course, in particular Mr O Veblen, have been a source of much stimulus.

Veblen's doctoral dissertation, supervised by Moore, was entitled A System of Axioms for Geometry and he was awarded his doctorate from the University of Chicago in 1903. In it Veblen gave an axiom system based on point and order rather than on the traditional notions of point, line and plane. He presented twelve axioms for Euclidean geometry which he proved to be an complete system of axioms and he also proved the independence of the axioms. He spent another two years at Chicago after the award of his doctorate as an associate in mathematics. During this time he effectively supervised the doctoral studies of Robert Moore, officially a student of Eliakim Moore's, who was awarded a doctorate in 1905 for a dissertation entitled Sets of Metrical Hypotheses for Geometry. During this period Veblen worked on putting his own thesis into a form for publication and A system of axioms for geometry appeared as a 41 page paper in the Transactions of the American Mathematical Society in 1904. Already at this time he had begun to undertake research in topology (or analysis situs as it was then called) and he published Theory on plane curves in non-metrical analysis situs in 1905.

Leaving Chicago, Veblen taught mathematics at Princeton University from 1905 to 1932. His initial appointment to Princeton in 1905 was as a preceptor, a new position which Woodrow Wilson, the president of Princeton, introduced in an attempt to raise the level of scholarship at the university. Veblen married Elizabeth Mary Dixon Richardson in 1908. She was the sister of Owen Richardson who was the professor of physics at Princeton, and later went on to receive the Nobel prize for physics in 1928. Veblen and his wife Elizabeth had no children. Two years after his marriage, in 1910, Veblen was promoted to professor of mathematics at Princeton.

During the First World War he served first as a captain, later as a major in the army. He sought a commission in the Ordnance Department in the spring of 1917 and was appointed to the Office of Ballistics Research at the new Aberdeen Proving Ground in Maryland. After basic training during the summer of 1917, he arrived at the Proving Ground on 4 January 1918. Veblen headed the Division of Experimental Ballistics which was the mathematical research facility at the Proving Ground. He began collecting data on the range of a 2.95 inch gun. He then worked on the problem of transferring raw data into tables of results and for this he had a staff of men to undertake the lengthy hand calculations. He employed Joseph Ritt to take charge of these computers. His work led to new ballistics theory. See [7] for details of the work he undertook during this period.

Veblen was honoured by Princeton in 1926 when they named him Henry B Fine Professor of Mathematics. In the academic year 1928-29 he taught at Oxford as part of an exchange with G H Hardy. In 1929 funds were provided for Fine Hall at Princeton and Veblen provided most of the ideas that went into its design. He wanted the mathematicians in Fine Hall to be able to:-

... group themselves for mutual encouragement and support. [It had to be a place where] the young recruit and the old campaigner [could have] those informal and easy contacts that are so important to each of them.

However he also wanted a room reserved for professors since:-

... not always understood by those who try to bring about closer relations between faculty and students [is] that in all forms of social intercourse the provisions for privacy are as important as those for proximity.

In 1932 he spent time in Germany and lectured at Göttingen, Berlin and Hamburg. Zund writes that this experience [14]:-

... gave him a first hand glimpse of the approaching turbulence in Germany, and he subsequently worked tirelessly to help place refugees who came to the United States.

Leon Warren Cohen, in an interview in 1984, recalled Veblen's attitude towards placing refugee mathematicians:-

It was the Depression. Young American mathematicians were finding it hard to get appointments, and the question of whether to bring in foreign mathematicians to occupy positions which would then not be available to American mathematicians was debated. Veblen took what I would call the broader view. I hesitate to attribute views to Veblen, but the considerations that seem to have actuated him were two: a concern for the welfare of mathematics itself, and a humane concern for certain individuals who had talent. Veblen was a grand man, and the people for whom he made it possible to come to the United States made a great contribution to mathematics. G D Birkhoff opposed him on this. ... [Birkhoff said] "If these distinguished people come and take the positions, the young American mathematicians will become hewers of wood and drawers of water."

Veblen helped organise the Institute for Advanced Study in Princeton in 1932, resigning the Henry B Fine Professorship to become the first professor at the Institute in the same year. He had Alexander, Einstein, von Neumann and Weyl, all mathematicians he had chosen, as original Institute members.

Veblen's interest in the foundations of geometry led to his work on the axiom systems of projective geometry. He published Finite projective geometries with W H Bussey in 1906, Collineations in a finite projective geometry (1907), and Non-Desarguesian and non-Pascalian geometries (1908). Together with John Wesley Young he published Projective geometry (1910-18). The first of the two volumes in this book were jointly written by the two authors but the second volume was due the Veblen alone. The introduction to this work justifies the study of foundations:-

Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view to some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to its foundations. We believe, moreover, that the abstract treatment is particularly desirable in projective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Since it is most natural to derive the geometrical disciplines associated with the names of Euclid, Descartes, Lobachevsky etc. from projective geometry than to derive projective geometry from one of them, it is natural to take the foundations of projective geometry as the foundations of all geometry.

We mentioned above that Veblen's first work on topology appeared just before he arrived in Princeton. He went on to establish Princeton as one of the leading centres in the world for topology research. Analysis Situs (1922) provided the first systematic coverage of the basic ideas of topology and contributed to the development of modern topology.

Soon after Einstein's theory of general relativity appeared Veblen turned his attention to differential geometry. This work led to important applications in relativity theory, and much of his work also found application in atomic physics. His work The invariants of quadratic differential forms (1927) is a systematic treatment of Riemann geometry while his work, written jointly with his student Henry Whitehead, The foundations of differential geometry (1933) gives the first definition of a differentiable manifold. In Projective relativity theory (1933) he gave a new treatment of spinors, used to represent electron spin.

Veblen was an active member of the American Mathematical Society, serving the Society as vice-president in 1915 and president in 1923-24. He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology. As president of the American Mathematical Society he was the leader of American mathematics and his efforts to increase the funding for research is explained in [6] where Feffer writes:-

Between the world wars, all the scientific professions in the United States underwent tremendous growth. The wartime experiences of scientific leaders whetted their appetites for the continuation of some kind of concentrated, well-funded research programs. Turning not to government but instead to philanthropy, physicists and chemists worked to parlay their postwar prestige into greater support for unfettered research. Leaders of the American mathematical community also wanted to expand their base of support but found themselves facing unique obstacles. Attempts to raise money to support mathematical research had mixed results. [Most important were] the efforts led by Oswald Veblen in the 1920s to collect funds to support mathematics through Princeton University and the American Mathematical Society in the context of this climate of expansion for the physical sciences.

Veblen was elected to the American Philosophical Society and the National Academy of Sciences (United States) and was honoured with memberships of many other societies around the world. For example he was a member of the London Mathematical Society, serving on the council in 1928 when he was replacing Hardy at Oxford. Oxford further honoured him with an honorary D.Sc. in 1929, while in the same year he was honoured by the University of Oslo on the occasion of the centenary celebrations for Abel. He was elected to the Danish Academy of Sciences, the Academy of Sciences (Paris), the Polish Academy of Sciences, the Royal Society of Edinburgh and a number of other national academies in, for example, Ireland, Italy, and Peru. In addition to an honorary degree from Oxford, he received similar honours from the universities of Edinburgh, Glasgow, Hamburg, and Oslo. He was made a Knight of Norway's Royal Order of St Olaf, an honour which had also been bestowed on his father.

As to his personal characteristics he was [14]:-

... An engaging person of great modesty and personal charm.

L W Cohen said:-

Veblen had many wonderful characteristics. He was a kind man.

Veblen loved the outdoors and it was his initiative which led to Princeton purchasing a woodland site for the Institute for Advanced Study so that members could go for countryside walks. His own home was in eighty acres of woodland which he named Herrontown Wood. He and his wife left Herrontown Wood to Mercer County as a place where:-

... you can get away from cars and just walk and sit.

In his last years Veblen became partially blind and he devised a number of aids for blind people, one of which was manufactured and distributed by the American Foundation for the Blind.