卡尔·门格尔Karl Menger

卡尔·门格尔的父亲Carl Menger（1840-1921）是一位著名经济学家，维也纳大学教授，奥地利经济学派的创始人。门格尔的母亲是Hermine Andermann（1869-1924），一位记者、作家和音乐家。Carl信奉罗马天主教，Hermine是犹太人，因此尽管他们同居，两人却不能结婚（当时奥地利的所有婚姻都是宗教仪式）。当他们唯一的孩子门格尔（本传记的主人公）于1902年出生时，维也纳的社会习俗认为这是不可接受的，Carl被迫退出公共生活。儿子一出生他就停止了在维也纳大学的教学，一年后他辞去了大学的讲席。Carl向奥地利皇帝Franz Joseph申请让他的儿子门格尔获得合法身份，他的请求最终获得批准。二十五年前，Carl Menger曾给Franz Joseph上过三个月经济学课程。

门格尔在维也纳的Döblinger 文理中学（Gymnasium）就读（1913-1920），他的两位同学是沃尔夫冈·泡利和Richard Kuhn（1900-1967）。值得注意的是，Kuhn于1938年获得诺贝尔化学奖，沃尔夫冈·泡利于1945年获得诺贝尔物理学奖。门格尔班上的一名学生是Heinrich Schnitzler（1902-1982），他后来成为演员和电影导演。Heinrich Schnitzler是著名作家Arthur Schnitzler（1862-1931）的儿子，在这个阶段，门格尔有写戏剧的想法。在维也纳成长并非易事，因为第一次世界大战爆发后，1914年9月Döblinger文理中学的建筑被改造成战地医院，教学转移到Krottenbachstrasse的一所中学建筑中。1916年2月战地医院关闭，学生们回到原来的建筑。1918年第一次世界大战结束后的几年对学校来说尤其艰难。君主制终结等政治事件迫使教育发生重大变革，文理中学成为公立学校。门格尔从Döblinger文理中学毕业后，于1920年进入维也纳大学学习物理学。他并没有放弃写戏剧的想法，因为此时他开始写一部关于伪经中的女教皇若安的戏剧。

在维也纳大学，门格尔听了理论物理学家Hans Thirring（1888-1976）的物理学讲座，Thirring对广义相对论理论做出了重要贡献。然而，汉斯·哈恩于1921年3月在维也纳成为讲师，门格尔参加了他讲授的What's new concerning the concept of a curve课程。Seymour Kass写道[21]：-

在第一讲中，汉斯·哈恩提出了精确表述曲线概念的问题，此前无人能够阐明这一问题，他提到了格奥尔格·康托尔、卡米耶·若尔当和朱塞佩·皮亚诺的不成功尝试。讲座中使用的拓扑学对门格尔来说是新的，但他“完全被迷住了，茫然地离开了讲堂”[1]。经过一周的完全沉浸，他给出了曲线的一个定义，并把它透露给同学Otto Schreier，后者找不出任何缺陷，但提醒门格尔注意费利克斯·豪斯多夫和路德维希·比贝尔巴赫最近关于该问题难以处理的评论，而汉斯·哈恩并未提及这些。在讨论班第二次会议之前，门格尔与汉斯·哈恩会面，后者不习惯认真听取一年级学生的意见，但仍然听了，经过一番思考后同意门格尔的尝试是对该问题一个有希望的攻击。

门格尔现在对这个主题着迷，并受到汉斯·哈恩的鼓励去研究它，他被诊断出患有肺结核，并前往奥地利南部施蒂里亚山区Aflenz的一家疗养院。门格尔在疗养院进行的数学研究使他独立于帕维尔·萨穆伊洛维奇·乌雷松得出了维数的定义。然而帕维尔·萨穆伊洛维奇·乌雷松在能够发表其工作之前死于溺水事故，门格尔并不知道此事。严重的肺病迫使门格尔在疗养院度过了一年多，但他带着在疗养院期间撰写的关于维数的重要论文回到维也纳，并在汉斯·哈恩的指导下，于1924年以他的学位论文Über die Dimensionalität von Punktmengen Ⓣ（《论点集的维数》）完成了博士学位。这并不是门格尔此时承担的唯一工作。他的父亲Carl Menger在门格尔住在疗养院期间去世，留下了他著作Grundsätze der Volkswirthschaftslehre Ⓣ（《经济学教学原理》）（第1版1871年）第二版未完成。门格尔完成了父亲的工作，并监督了第二版的出版，该版于1922年问世。到他完成这项工作时，门格尔已经在经济学方面获得了相当多的专业知识。

1925年3月，门格尔应勒伊岑·布劳威尔的邀请，利用他最近获得的洛克菲勒奖学金来到阿姆斯特丹大学，在那里他作为勒伊岑·布劳威尔的助手工作了两年。他从阿姆斯特丹写信给他在维也纳的女友Hilda Axamit，一位精算数学学生[22]：-

我正在讲授变分法。就我个人而言，我忙于各种几何学，此外还有认识论。我希望我能有精力把我关于真理问题的观点整理出来。在过去几周里，我有了如此多的想法，以至于我根本没有时间把它们写下来，每天晚上都跑出去分散自己的注意力……以免过度劳累。除此之外，我诅咒自己不在维也纳而是在这里。我无法习惯在这里生活，我会尽我最大的努力在六月份永远离开这里。

1927年，当Kurt Reidemeister前往柯尼斯堡时，门格尔受Hahn邀请，接受了维也纳大学几何学讲席。门格尔并不为离开阿姆斯特丹而感到遗憾，因为他已卷入与勒伊岑·布劳威尔的优先权之争，两人关系并不融洽。然而，他确实尊重勒伊岑·布劳威尔[21]：-

尽管他觉得勒伊岑·布劳威尔脾气急躁，但他对在阿姆斯特丹的逗留仍怀有温暖的回忆，并提到勒伊岑·布劳威尔对一些年轻数学家的帮助，以及“看着他聆听新发现报告的美好体验”。

门格尔成为维也纳学派的成员，该学派由哲学家、数学家和逻辑学家组成。他还在1928年于维也纳发起了数学讨论班，由顶尖数学家作报告，门格尔出版了会议录。Steve Abbott在评论1998年会议录重印本时写道[3]：-

《数学讨论班成果》于1929年至1937年间在维也纳出版。这八期包含了许多备受尊敬的数学家的贡献，但因发行范围有限而流传不广，完整套集所存无几。本书重印了全部文章（德文），并附有章节（英文），综述了《成果》中报道的经济学、逻辑学、拓扑学和几何学的重要进展。讨论班由门格尔在维也纳大学组织，最初是为了回应那里一些数学学生的请求。会议包括讲座、未解决问题讨论和近期工作综述。门格尔记录了会议情况，并通过Teubner Verlag或Deuticke出版笔记。……正是讨论班演讲者的明星阵容使得这些论文的重印具有价值。其中有来自爱德华·切赫、库尔特·弗雷德里希·哥德尔、门格尔、Popper、阿尔弗雷德·塔斯基、奥尔加·陶斯基-托德、冯·诺伊曼、亚伯拉罕·瓦尔德和诺伯特·维纳的贡献。

1928年，门格尔出版了DimensiontheorieⓉ（维数论）一书。Paul Althaus Smith（1900-1980）在评论中写道[37]：-

现代点集理论几何学的发展有一个重要阶段与维数概念密切相关——我们指的是试图用原始的非算术概念为我们直觉中的简单几何空间建立精确的数学意义。维数观念应当发挥作用，并本身得到研究和精确化，这确实是自然的，因为我们经验中的曲线、曲面和立体正是我们直觉维数观念的基础。……[维数的定义]由维也纳的门格尔教授引入，并在以优雅和普遍性为特征的一系列较短论文中得到发展。维数理论的要点，通过门格尔、维托尔德·胡列维茨、帕维尔·亚历山德罗夫等人近期的著作已达到相当完善的程度，在门格尔教授最近出版的一本书中以令人钦佩的清晰性和完整性得到阐述。

当然，关于维度的更精细的思想仍在不断发展。然而，James Kesling 在1977年对另一部著作的评论中，写到 门格尔 的 Dimensiontheorie Ⓣ（维度论）：-

它同时揭示了早期研究者以现代标准来看的幼稚，以及他们对重要结果和理论未来方向的非凡洞察力。

门格尔 在1930-31学年在美国度过。他访问了哈佛大学和位于德克萨斯州休斯顿的莱斯研究所。在 门格尔 前往美国之前，库尔特·弗雷德里希·哥德尔 已加入了他的数学讨论班。在美国期间，门格尔 通过 Georg Nöbeling 与维也纳的讨论班保持联系，并与 库尔特·弗雷德里希·哥德尔 通信。例如，在1931年初他写道：-

我在行驶的火车上回复你迷人的来信，因此用打字机。Nöbeling 去年秋天已经写信告诉我你的伟大发现。我怀着极大的兴趣读了你的文章，并立即在此地作了一个报告。我将你的成就列为现代逻辑学最伟大的成就之一，并向你致以最衷心的祝贺。

回到维也纳后，门格尔 于1932年出版了 Kurventheorie Ⓣ（曲线论）。Wallace Alvin Wilson 写道[42]：-

所讨论的这本书将是一部题为“Mengentheoretische Geometrie”的著作的第二卷。除了是第一部关于曲线论及其拓扑性质的专著外，这本书还进一步将战后以来关于这一主题的大量研究工作有机地汇集在一起——这些工作，从其零散出现的性质来看，在许多人看来必定显得毫无意义。

本书包含门格尔的“$n$-弧定理”，Frank Harary将其描述为：-

……图论中最重要的结果之一……关于图的连通性的基本定理。

1932年9月，门格尔出席了在苏黎世举行的国际数学家大会，并作了关于Neuere Methoden und Probleme der Geometrie Ⓣ（几何学的新方法与新问题）的大会报告之一。他与女友Hilda Axamit订婚，1934年他们一起在拉姆绍和施特罗布尔度假。他们于1934年12月5日结婚，育有四个孩子：小门格尔（生于1936年7月9日）、双胞胎Rosemary和Fred（生于1937年12月13日）以及Eve（生于1942年）。1933年希特勒在德国掌权时，门格尔很快意识到奥地利将面临的问题。Steve Abbot在写到门格尔的讨论班时解释道[3]：-

《Ergebnisse》当时因“犹太撰稿人太多”而受到批评。随着纳粹在德国的兴起，奥地利的政治局势迫使讨论班会议结束只是时间问题。门格尔在德奥合并前一年离开了这个国家。

1936年6月，Moritz Schlick（维也纳哲学教授、维也纳学派创始人之一）被一名学生枪杀，这些问题对门格尔来说变得非常真实。一个月后，仍被这场悲剧震惊的门格尔在奥斯陆参加国际数学家大会。他向所有人解释了维也纳的局势如何迅速恶化。此后不久，他获得了印第安纳州圣母大学的讲席职位，并于1937年前往美国就任。在这个阶段，他保留着维也纳的讲席，但1938年3月，由于奥地利的政治局势，他辞去了维也纳的讲席。当然，在这个阶段他仍期望战后回到维也纳，但正如门格尔 Signund所写：-

战后，民主的新奥地利将重建被炸毁的国家歌剧院列为最高优先事项。然而，像……门格尔这样的人却被礼貌地告知，维也纳大学没有他们的位置。

门格尔是埃米尔·阿廷的同事，后者当时正逃离纳粹，并在那里度过了1937-38年。同样在圣母大学任职的还有Paul Milton Pepper（1909-2010），他刚刚在辛辛那提大学完成博士学位。门格尔与这两位同事以及其他几人一起，在圣母大学建立了一个博士项目。门格尔还组织了一个数学讨论班，以他在维也纳建立的那个为基础，并于1938年开始出版Reports of a Mathematical Colloquium。他安排了库尔特·弗雷德里希·哥德尔访问圣母大学，但未能说服他接受那里的职位。然而，当战争在1941年开始影响美国后，学术生活被打乱，门格尔的数学讨论班未能像维也纳学派那样具有影响力。Reports于1946年停止出版。Rudolf Carnap是门格尔在维也纳学派的同事之一，他也移民到了美国，并建立了芝加哥学派。即使门格尔在圣母大学主持自己的讨论班时，他仍尽可能努力参加Carnap的芝加哥学派。

大约在这个时候，门格尔对数学的兴趣变得更加广泛，他开始研究双曲几何、概率几何和函数代数。门格尔在几何方面的工作未能产生他在维数理论方面的工作那样的影响。这几乎肯定是因为几何在当时是一个相当被忽视的数学领域，尤其是在美国。同样在战争年代，门格尔对战争的贡献是作为1942年至1944年运行的V-12海军学院培训计划的一部分，向海军学员教授微积分。这导致了他对数学教育的兴趣，并在20世纪50年代和60年代，他撰写了关于数学教育的文章，并出版了关于教授微积分、几何和其他数学分支的新思想的书籍。

门格尔的一篇关于教学的文章可在THIS LINK获取
并且
门格尔书籍的评论在THIS LINK。

门格尔对Carnap的芝加哥学派的访问使他感到芝加哥对他工作和他孩子受教育都是更好的地方。芝加哥伊利诺伊理工学院的数学系主任是莱斯特·佛特，门格尔从1931年访问德克萨斯州休斯顿的莱斯学院时就认识他。门格尔与莱斯特·佛特谈论了想搬到芝加哥的愿望，莱斯特·佛特很快就能向他发出邀请。1948年，门格尔去了伊利诺伊理工学院，并在芝加哥度过了余生。Seymour Kass是门格尔在20世纪60年代在芝加哥的同事，他写道[21]：-

门格尔被描述为一个性格火爆的人。作为20世纪60年代伊利诺伊理工学院的初级教员，我发现他亲切、迷人且活泼。门格尔对学生很关心。从他在维也纳的早期开始，他就邀请学生和教员到他家。在芝加哥，这包括参观他装饰瓷砖的收藏，这些瓷砖排列在他客厅的墙上。他有时还邀请博士生清晨沿着密歇根湖进行数学散步。他的办公室是一片混乱的展示场，桌面上覆盖着一片汹涌的纸海。他知道每一片纸的确切位置。在电话里，他能准确地指示秘书如何找到他需要的东西。有一次，在他不在时，一位新秘书着手“整理”，在他的桌子上堆成小堆。当他回来发现这场灾难时，他几乎要哭了，因为“现在我不知道东西都在哪里了。”

门格尔在芝加哥教书直到1971年退休。

我们已经看到门格尔的兴趣如何超越数学，延伸到哲学和经济学。Donald Gillies在他那篇关于‘门格尔作为哲学家’的引人入胜的论文中写道[13]：-

门格尔作为哲学家并不总是得到应有的认可。事实上……维也纳学派的许多重要思想都源自门格尔——尽管它们常常被归功于他人。此外，门格尔自己的数学哲学（我为此创造了名称：‘自由放任形式主义’）是许多在职数学家的隐含哲学。总的来说，门格尔可以被描述为他们中最逻辑实证主义的人。

在Austrian Marginalism and Mathematical Economics（1973年）中，门格尔写道：-

作为《Grundsätze》作者的儿子以及一名数学家，“我胸中居住着两个灵魂”。我希望这一事实能帮助我客观地讨论争议中的各个要点。

Karl Sigmund在评论门格尔的论文时写道：-

通过将经济学家和数学家聚集在一起，门格尔发挥了催化作用，而他的成长和教育背景使他特别适合扮演这一角色。

在[1]中，他的兴趣被描述如下：-

他非常热爱音乐。……他收集了来自世界各地的装饰瓷砖，藏品颇为可观。……他吃肉很少，尤其是在晚年。但他总是乐于品尝对他来说是新奇的菜肴，从古巴菜到埃塞俄比亚菜。他喜欢烤苹果。

Karl Menger's father, Carl Menger (1840-1921) was a famous economist, a professor at the University of Vienna, and founder the Austrian School of Economics. Karl's mother was Hermine Andermann (1869-1924), a journalist, author and musician. Carl was Roman Catholic and Hermine was Jewish so, although they lived together, the two could not marry (all marriages in Austria at this time were religious ceremonies). When their only child Karl (the subject of this biography) was born in 1902 this was seen as unacceptable by the Viennese social conventions and Carl was forced to withdraw from public life. He stopped teaching at the University of Vienna as soon as his son was born and he resigned his chair at the University a year later. Carl applied to Emperor Franz Joseph of Austria to have his son Karl made legitimate and his request was eventually granted. Twenty-five years earlier, Carl Menger had given Franz Joseph a three month course on economics.

Karl Menger attended the Döblinger Gymnasium in Vienna (1913-1920) where two of his fellow students were Wolfgang Pauli and Richard Kuhn (1900-1967). It is worth noting that Kuhn was awarded the Nobel Prize for Chemistry in 1938 and Pauli was awarded the Nobel Prize for Physics in 1945. One of the students in Menger's class was Heinrich Schnitzler (1902-1982) who went on to become an actor and film director. Heinrich Schnitzler was the son of the famous author Arthur Schnitzler (1862-1931) and, at this stage, Menger had ideas of writing dramas. This was not an easy time to be growing up in Vienna for, after the outbreak of World War I, in September 1914 the Döblinger Gymnasium building was converted into a war hospital, and teaching was transferred to a secondary school building in the Krottenbachstrasse. In February 1916 the war hospital closed and pupils returned to the original building. The years following the end of World War I in 1918 were particularly hard at the school. Political events such as the end of the monarchy forced major educational changes and the Gymnasium became a state school. After Menger graduated from the Döblinger Gymnasium, he entered the University of Vienna in 1920 to study physics. He had not given up his idea of writing dramas, for at this time he began to write a drama about the apocryphal Pope Joan.

At the University of Vienna, Menger attended physics lectures by the theoretical physicist Hans Thirring (1888-1976) who had made significant contributions to the theory of general relativity. However Hans Hahn became a lecturer in Vienna in March 1921 and Menger attended a course he gave on What's new concerning the concept of a curve. Seymour Kass writes [21]:-

In the first lecture Hahn formulated the problem of making precise the idea of a curve, which no one had been able to articulate, mentioning the unsuccessful attempts of Cantor, Jordan, and Peano. The topology used in the lecture was new to Menger, but he "was completely enthralled and left the lecture room in a daze" [1]. After a week of complete engrossment, he produced a definition of a curve and confided it to fellow student Otto Schreier, who could find no flaw but alerted Menger to recent commentary by Hausdorff and Bieberbach as to the problem's intractability, which Hahn hadn't mentioned. Before the seminar's second meeting Menger met with Hahn, who, unaccustomed to giving first-year students a serious hearing, nevertheless listened and after some thought agreed that Menger's was a promising attack on the problem.

Although Menger was now fascinated in the topic and was encouraged by Hahn to work on it, he was diagnosed with tuberculosis and went to a sanatorium in Aflenz in the mountains of Styria in southern Austria. Menger's mathematical investigations, carried out in the sanatorium, led him to a definition of dimension independently of Pavel Urysohn. However Urysohn had died in a drowning accident before he could publish his work and Menger was not aware of it. The severe lung disease forced Menger to spend more than a year in the sanatorium, but he returned to Vienna with important papers he had written on dimension while in the sanatorium and, advised by Hahn, completed his doctorate in 1924 with his thesis Über die Dimensionalität von Punktmengen Ⓣ. This was not the only work Menger undertook at this time. His father, Carl Menger, had died while Menger was in the sanatorium and had left the second edition of his book Grundsätze der Volkswirthschaftslehre Ⓣ (1st edition 1871) unfinished. Menger completed his father's work and supervised the publication of the second edition which appeared in 1922. By the time he had completed this work, Menger had gained considerable expertise in economics.

In March 1925 Menger was invited by L E J Brouwer to use his recently won Rockefeller Fellowship to come to the University of Amsterdam where he spent two years working as Brouwer's assistant. He wrote from Amsterdam to his girlfriend Hilda Axamit, an actuarial mathematics student, in Vienna [22]:-

I am lecturing on the calculus of variations. Personally, I am occupied by geometry of all kinds, furthermore by epistemology. I hope I'll get the energy to put together my views about the problem of truth. In the last weeks, I have had so many ideas that I don't have any time at all to write them down, and run away every evening to distract myself ... in order not to overwork. Apart from that, I curse the fact that I am not in Vienna but rather here. I can't get used to living here, and I will try my best to leave here forever in the month of June.

In 1927 Menger was invited by Hahn to accept the chair of geometry at the University of Vienna when Kurt Reidemeister left for Königsberg. Menger was not sorry to leave Amsterdam since he had become involved in a priority dispute with Brouwer and they were not on the best of terms. However, he certainly respected Brouwer [21]:-

Though he found Brouwer testy, he retained warm feelings about his stay in Amsterdam and cited the good Brouwer did for some young mathematicians and "the beautiful experience of watching him listen to reports of new discoveries".

In Vienna, Menger became a member of the Vienna Circle which comprised philosophers, mathematicians and logicians. He also started up the Mathematical Colloquium in Vienna in 1928 which was addressed by leading mathematicians and Menger published the Proceedings. Steve Abbott, reviewing a 1998 reprinting of the Proceedings, writes [3]:-

The 'Ergebnisse eines Mathematischen Kolloquiums' were published in Vienna between 1929 and 1937. The eight issues included many contributions by highly respected mathematicians but suffered from a limited distribution, and few complete sets remain. This book reprints all the articles (in German) along with chapters (in English) surveying the important developments in economics, logic, topology and geometry that were reported in the 'Ergebnisse'. The 'Kolloquiums' were organised by Karl Menger at the University of Vienna, initially in response to a request from some of the mathematical students there. The meetings included a mixture of lectures, discussions of unsolved problems and reviews of recent work. Menger kept a record of the meetings and published the notes through Teubner Verlag or Deuticke. ... It is the stellar collection of 'Kolloquium' speakers that justifies the reprinting of the papers. There are contributions from Čech, Gödel, Menger, Popper, Tarski, Taussky, von Neumann, Wald and Wiener.

In 1928 Menger published the book Dimensiontheorie Ⓣ.. Paul Althaus Smith (1900-1980) writes in a review [37]:-

There is an important phase in the development of modern point set theoretical geometry which has been closely associated with the concept of dimensionality, - we refer to the attempt to create precise mathematical meaning for the simple geometric spaces of our intuition in terms of primitive non-arithmetical concepts. That the idea of dimensionality should have come into play and itself have been studied and made precise is indeed natural, since the curves, surfaces, and solids of our experience furnish the very basis for our intuitive ideas of dimensionality. ... [A definition of dimension] was introduced by Professor Karl Menger of Vienna and developed in a sequence of shorter papers characterized by their elegance and generality. The essentials of the dimensionality theory, which has by now attained a considerable perfection through the recent writings of Menger, Hurewicz, P S Aleksandrov and others, have been developed with admirable clarity and completeness in a recently published book by Professor Menger.

Of course, more sophisticated ideas on dimension continue to be developed. However, James Kesling, in a 1977 review of another work, wrote of Menger's Dimensiontheorie Ⓣ:-

It reveals at one and the same time the naiveté of the early investigators by modern standards and yet their remarkable perception of what the important results were and the future direction of the theory.

Menger spent the academic year 1930-31 in the United States. He visited Harvard University and the Rice Institute in Houston, Texas. Before Menger went to the United States, Kurt Gödel had joined his Mathematical Colloquium. While in the United States, Menger kept in touch with the Colloquium in Vienna through Georg Nöbeling and also corresponded with Gödel. For example, in early 1931 he wrote:-

I am replying to your charming letter in a moving train and therefore by typewriter. Nöbeling had already written to me last autumn about your great discovery. I read your article with the utmost interest and immediately delivered a report on it here. I rank your achievement among the greatest of modern logic and send you my heartiest congratulations.

Back in Vienna, Menger published Kurventheorie Ⓣ in 1932. Wallace Alvin Wilson writes [42]:-

The book under discussion is to be the second volume of a work entitled 'Mengentheoretische Geometrie'. In addition to being the first treatise on the theory of curves and their topological properties, the book performs the further service of gathering together in an organic whole much of the research work on this subject done since the war - work which, from the nature of its piecemeal appearance, must have seemed to many to be of a pointless character.

This book contains Menger's '$n$-Arc Theorem', described by Frank Harary as:-

... one of the most important results in graph theory ... the fundamental theorem on connectivity in graphs.

Menger attended the International Congress of Mathematicians in Zürich in September 1932 when he gave one of the plenary addresses on Neuere Methoden und Probleme der Geometrie Ⓣ. He became engaged to his girlfriend Hilda Axamit and in 1934 they spent a holiday together in Ramsau and Strobl. They married on 5 December 1934 and had four children; Karl Jr (born 9 July 1936), twins Rosemary and Fred (born 13 December 1937), and Eve (born 1942). When Hitler came to power in Germany in 1933, Menger soon realised the problems that lay ahead for Austria. Steve Abbot, writing about Menger's Colloquium, explains [3]:-

The Ergebnisse was criticised at the time for having 'too many Jewish contributors'. With the rise of the Nazis in Germany, it was only a matter of time before the political situation in Austria forced the end of the Kolloquium meetings. Menger left the country a year before the Anschluss.

The problems had become very real to Menger when, in June 1936, Moritz Schlick (professor of philosophy in Vienna and one of the founders of the Vienna Circle) was shot dead by a student. A month later Menger, still stunned by the tragedy, was in Oslo at the International Congress of Mathematicians. He explained to everyone how the situation in Vienna was deteriorating fast. Soon after this he was offered a chair at the University of Notre Dame, Indiana and he went to the United States in 1937 to take up the post. At this stage he kept open his chair in Vienna but, in March 1938, as a result of the political situation in Austria, he resigned his chair in Vienna. Certainly at this stage he was still expecting to return to Vienna after the war but, as Karl Signund writes:-

After the war, the reconstruction of the bombed-out State Opera was accorded highest priority by democratic new Austria. Men like ... Menger, however, were politely told that the University of Vienna had no place for them.

At Notre Dame, Menger was a colleague of Emil Artin who was escaping from the Nazis and spent the year 1937-38 there. Also on the staff at Notre Dame was Paul Milton Pepper (1909-2010) who had just completed his doctorate at the University of Cincinnati. Menger, with these two colleagues and a couple of others, set up a Ph.D. programme at Notre Dame. Menger also organised a Mathematical Colloquium based on the one he had set up at Vienna and began publishing Reports of a Mathematical Colloquium in 1938. He arranged a visit by Gödel to Notre Dame but failed to persuade him to accept a post there. However after the war began to affect the United States in 1941, academic life was disrupted and Menger's Mathematical Colloquium failed to become as influential as the Vienna Circle had been. The Reports stopped publication in 1946. Rudolf Carnap, one of Menger's colleagues in the Vienna Circle, had also emigrated to the United States and had set up the Chicago Circle. Even while Menger was running his own Colloquium at Notre Dame, he still made the effort to attend Carnap's Chicago Circle whenever possible.

Around this time Menger's interests in mathematics broadened and he began to work on hyperbolic geometry, probabilistic geometry and the algebra of functions. Menger's work on geometry failed to have the impact that his work on dimension theory had. This is almost certainly because geometry, at this time, was a rather neglected area of mathematics, particularly in the United States. Also during the war years Menger's contribution to the war effort was teaching calculus to Naval cadets as part of the V-12 Navy College Training Program which ran from 1942 to 1944. This led to his interest in mathematical education and, during the 1950s and 1960s, he wrote articles on mathematical education and published books with new ideas on teaching calculus, geometry and other branches of mathematics.

An article by Menger on teaching is available at THIS LINK
and
reviews of Karl Menger's books are at THIS LINK.

The visits that Menger made to Carnap's Chicago Circle led to him feeling that Chicago would be both a better place for him to work and a better place for his children to be educated. The chairman of the mathematics department at the Illinois Institute of Technology in Chicago was Lester R Ford whom Menger had known from the time of his 1931 visit to the Rice Institute in Houston, Texas. Menger talked to Ford about wanting to move to Chicago and Ford was soon in a position to make him an offer. In 1948 Menger went to the Illinois Institute of Technology and he was to remain in Chicago for the rest of his life. Seymour Kass, who was a colleague of Menger's at Chicago during the 1960s, writes [21]:-

Menger has been described as a fiery personality. As a junior faculty member at IIT in the 1960s, I found him gracious, charming, and vivacious. Menger was solicitous of students. From his early days in Vienna onward he invited students and faculty to his home. In Chicago it included a tour of his decorative tile collection, which lined the walls of his living room. And he sometimes invited doctoral students for early morning mathematical walks along Lake Michigan. His office was a showplace of chaos, the desktop covered with a turbulent sea of papers. He knew the exact position of each scrap. On the telephone he could instruct a secretary exactly how to locate what he needed. Once, in his absence, a new secretary undertook to "make order", making little stacks on his desk. Upon his return, discovering the disaster, he nearly wept, because "Now I don't know where anything is."

Menger taught in Chicago until he retired in 1971.

We have seen how Menger's interests extended beyond mathematics to philosophy and economics. Donald Gillies, in his fascinating paper on 'Karl Menger as a Philosopher', writes [13]:-

Menger is not always given his due recognition as a philosopher. In fact ..., many important ideas of the Vienna Circle originated with Menger - though they are often attributed to others. Moreover, Menger's own philosophy of mathematics (for which I have coined the name: 'laissez-faire formalism') is the implicit philosophy of many working mathematicians. Generally, Menger could be described as the most logical positivist of them all.

In Austrian Marginalism and Mathematical Economics (1973), Menger writes:-

Since I am the son of the author of the 'Grundsätze' as well as a mathematician, 'two souls reside within my breast '. I hope that this fact will help me in discussing various points of the controversy objectively.

Karl Sigmund, reviewing Menger's paper, writes:-

By bringing together economists and mathematicians, Menger had played a catalytic role for which he was uniquely predisposed by his upbringing and education.

In [1] his interests are described as follows:-

He had a great love of music. ... he built up a notable collection of decorative tiles from all over the world. ... he ate meat sparingly, particularly in his last years. But he was always glad to sample cuisines, from Cuban to Ethiopian, that were new to him. He liked baked apples.