冯·诺伊曼John von Neumann

冯·诺伊曼出生时名叫János 冯·诺伊曼。他小时候被称为Jancsi，这是János的昵称，后来在美国被称为Johnny。他的父亲Max Neumann是一位顶级银行家，他在一个大家庭中长大，住在布达佩斯，小时候从雇用的德语和法语女家庭教师那里学习语言。虽然这个家庭是犹太人，但Max Neumann并不遵守该宗教的严格习俗，家庭似乎混合了犹太教和基督教的传统。

还值得解释一下Max Neumann的儿子是如何获得“von”而成为János 冯·诺伊曼的。Max Neumann因为对当时成功的匈牙利经济做出了贡献，有资格申请世袭头衔，并在1913年支付费用获得了头衔，但他没有改姓。然而，他的儿子使用了德语形式冯·诺伊曼，其中“von”表示头衔。

童年时冯·诺伊曼就显示出惊人的记忆力。Poundstone在[8]中写道：-

六岁时，他已能用古典希腊语与父亲互开玩笑。冯·诺伊曼家有时会以演示Johnny记忆电话簿的能力来款待客人。客人会随机挑选电话簿的一页和一栏。年幼的Johnny把那栏读上几遍，然后把书交还给客人。他能回答向他提出的任何问题（谁的电话号码是多少多少？），或者按顺序背诵姓名、地址和号码。

1911年，冯·诺伊曼进入路德会文理中学。这所学校有着浓厚的学术传统，在冯·诺伊曼家和学校看来，这似乎比宗教归属更为重要。他的数学老师很快发现了冯·诺伊曼的天才，并为他安排了特别辅导。学校里还有另一位杰出的数学家，比冯·诺伊曼高一年级，即尤金·维格纳。

第一次世界大战对冯·诺伊曼的教育影响相对较小，但战争结束后，贝拉·库恩在1919年以共产主义政府控制了匈牙利五个月。冯·诺伊曼一家逃往奥地利，因为富裕阶层受到了攻击。然而，一个月后，他们返回面对布达佩斯的问题。当库恩的政府失败时，由于它主要由犹太人组成，犹太人受到了指责。这种情况毫无逻辑可言，而冯·诺伊曼一家反对库恩政府的事实并没有使他们免于迫害。

1921年，冯·诺伊曼在路德会中学完成了学业。他的第一篇数学论文是与迈克尔·费克特合作撰写的，后者是布达佩斯大学的助教，曾一直辅导他，该论文于1922年发表。然而，Max Neumann不希望儿子从事一个不能带来财富的学科。Max Neumann请西奥多·冯西奥多·冯·卡门与儿子谈话，说服他从事商业职业。也许冯西奥多·冯·卡门不是承担这项任务的合适人选，但最终大家都同意了一个折中方案，让冯·诺伊曼在大学学习化学。

由于多种原因，匈牙利对犹太裔来说并不是一个容易的国家，而且能够进入布达佩斯大学的犹太学生人数有严格限制。当然，即使有严格的配额，冯·诺伊曼的成绩也轻松为他在1921年赢得了一个学习数学的名额，但他没有去听课。相反，他也在1921年进入柏林大学学习化学。

冯·诺伊曼在柏林大学学习化学，直到1923年前往苏黎世。尽管没有上过任何课程，他却在布达佩斯大学的数学考试中取得了优异成绩。冯·诺伊曼于1926年在苏黎世的Technische Hochschule获得化学工程文凭。在苏黎世期间，尽管学习化学，他仍继续对数学感兴趣，并与当时都在苏黎世的赫尔曼·外尔和乔治·波利亚交流。当赫尔曼·外尔有一段时间不在苏黎世时，他甚至接替了赫尔曼·外尔的一门课程。乔治·波利亚说[18]：-

Johnny是我唯一害怕过的学生。如果在讲课过程中我提出一个未解决的问题，很可能讲座一结束他就会来找我，在纸条上潦草几笔给出完整的解答。

冯·诺伊曼也在1926年从布达佩斯大学获得数学博士学位，学位论文是关于集合论的。他20岁时发表了一个序数的定义，这个定义就是今天所使用的定义。

冯·诺伊曼 于1926年至1929年在柏林讲学，1929年至1930年在汉堡讲学。然而，他还获得了洛克菲勒奖学金，使他能够在哥廷根大学进行博士后研究。1926至1927年间，他在哥廷根的大卫·希尔伯特指导下学习。此时，冯·诺伊曼已经获得了名人地位[8]：-

到二十多岁时，冯·诺伊曼的名声已在数学界传遍全球。在学术会议上，他常被指认为年轻天才。

奥斯瓦尔德·维布伦邀请冯·诺伊曼于1929年去普林斯顿讲授quantum theory。冯·诺伊曼回复奥斯瓦尔德·维布伦说处理完一些个人事务后就会来，随后前往布达佩斯，在那里与未婚妻玛丽埃塔·科韦西结婚，然后启程前往美国。1930年，冯·诺伊曼成为普林斯顿大学的访问讲师，1931年被任命为那里的教授。

1930年至1933年间，冯·诺伊曼在普林斯顿任教，但这并非他的强项[8]：-

他流畅的思路对于天赋较低的人来说难以跟上。他因在可用黑板上的一小部分飞快写出方程，并在学生抄写之前擦掉表达式而臭名昭著。

然而，相比之下，他有能力解释物理学中的复杂概念[3]：-

对于一位复杂的数学对他而言毫无困难的人来说，他能够以惊人的清晰度向门外汉解释他的结论。与他交谈之后，人们总是带着问题其实很简单明了的感觉离开。

1933年，他成为新成立的普林斯顿爱德华·斯图迪高等研究院最初的六位数学教授之一（詹姆斯·韦德尔·亚历山大、阿尔伯特·爱因斯坦、莫尔斯、奥斯瓦尔德·维布伦、J 冯·诺伊曼和赫尔曼·外尔），他终生担任此职位。

在美国的头几年，冯·诺伊曼在夏季继续返回欧洲。直到1933年，他仍在德国担任学术职位，但当纳粹上台后，他辞去了这些职位。与许多其他人不同，冯·诺伊曼不是政治难民，而是主要因为他认为那里的学术职位前景比德国更好而去了美国。

1933年，冯·诺伊曼成为Annals of Mathematics的联合编辑，两年后，他又成为Compositio Mathematica的联合编辑。他担任这两个编辑职务直到去世。

冯·诺伊曼和Marietta于1935年有了一个女儿Marina，但他们的婚姻于1937年以离婚告终。次年，他与同样来自布达佩斯的Klára Dán结婚，他是在一次欧洲之行中认识她的。结婚后，他们乘船前往美国，在普林斯顿安家。在那里，冯·诺伊曼过着对顶尖数学家来说相当不寻常的生活方式。他一直喜欢聚会[8]：-

聚会和夜生活对冯·诺伊曼有着特殊的吸引力。在德国教书时，冯·诺伊曼曾是卡巴莱时代的柏林夜生活圈的常客。

现在与Klára结婚后，聚会继续[18]：-

冯·诺伊曼家的聚会频繁、著名且漫长。

斯坦尼斯瓦夫·乌拉姆 概述了 冯·诺伊曼 在 [35] 中的工作。他写道：-

在他年轻时的著作中，他不仅关注数学逻辑和集合论的公理化，同时还关注集合论本身的实质内容，在 测度论 和实变量理论中获得了有趣的结果。也正是在这一时期，他开始了关于量子理论的经典工作、量子理论中测量理论的数学基础以及新的 统计力学。

他的著作 Mathematische Grundlagen der Quantenmechanik Ⓣ（量子力学的数学基础）（1932）为新量子力学建立了一个坚实的框架。Van Hove 在 [36] 中写道：-

量子力学确实非常幸运，在其1925年被发现后的最初几年里，就吸引了像 冯·诺伊曼 这样地位的数学天才的兴趣。结果，该理论的数学框架得以发展，其全新解释规则的形式方面由一个人在两年内（1927-1929）进行了分析。

在 大卫·希尔伯特 空间上的有界线性算子的自伴代数，在弱算子 拓扑学 中封闭，于1929年由 冯·诺伊曼 在 Mathematische Annalen 的一篇论文中引入。理查德·卡迪森 在 [22] 中解释道：-

他对 遍历理论、group representations 和量子力学的兴趣，极大地促进了 冯·诺伊曼 认识到算子代数理论是这一数学领域发展的下一个重要阶段。

这类算子代数被冯·诺伊曼称为“算子环”，后来又被其他一些数学家称为$W^{*}$-代数。1957年，雅克·迪克斯米耶在其专著Algebras of operators in 希尔伯特空间 (von Neumann algebras)中称它们为“冯·诺伊曼代数”。在20世纪30年代后半叶和40年代初，冯·诺伊曼与其合作者F J 詹姆斯·D·默里在一系列奠基性论文中为冯·诺伊曼代数的研究奠定了基础。

冯·诺伊曼以其在众多不同科学研究领域的广泛贡献而闻名。斯坦尼斯瓦夫·乌拉姆解释了[35]他是如何被引向博弈论的：-

冯·诺伊曼对其他数学家所获成果及其内在可能性的了解令人惊讶。在他工作的早期，埃米尔·博雷尔一篇关于极小极大性质的论文引导他发展了……这些想法后来在他最具原创性的创造之一——博弈论中达到了顶峰。

在博弈论中，冯·诺伊曼证明了极小极大定理。他逐步扩展了在博弈论方面的工作，并与合著者Oskar Morgenstern一起写出了经典著作Theory of Games and Economic Behaviour（1944）。

斯坦尼斯瓦夫·乌拉姆在[35]中继续写道：-

Koopman关于用函数空间上的算子处理经典力学问题的可能性的想法，激励他给出了遍历定理的第一个数学上严格的证明。阿尔弗雷德·哈尔在群上构造测度为他精彩地部分解决大卫·希尔伯特第五问题提供了灵感，在其中他证明了在紧群中引入解析参数的可能性。

1938年，American Mathematical Society将马克希莫·博谢奖授予冯·诺伊曼，以表彰其回忆录Almost periodic functions and groups。该文分两部分发表于Transactions of the American Mathematical Society，第一部分于1934年发表，第二部分于次年发表。大约此时，冯·诺伊曼转向应用数学[35]：-

30年代中期，Johnny对流体动力学湍流问题着迷。正是在那时，他意识到非线性偏微分方程这一主题背后的种种奥秘。从第二次世界大战开始，他的工作涉及流体动力学方程和激波理论的研究。这些非线性方程所描述的现象在分析上令人困惑，甚至用现有方法也无法获得定性认识。在他看来，数值工作似乎是获得对此类系统行为感觉的最有希望途径。这促使他研究在电子计算机上进行计算的新可能性……

冯·诺伊曼是计算机科学的先驱之一，对逻辑设计的发展做出了重要贡献。克劳德·香农在[29]中写道：-

冯·诺伊曼生命最后几年中有相当一部分时间用于[自动机理论]研究。对他而言，这代表了他早期对逻辑和证明论的兴趣与后来在第二次世界大战期间及之后从事大规模电子计算机工作的综合。自动机理论涉及纯数学与应用数学以及其他科学的混合，对于冯·诺伊曼广博的才智来说是一个理想领域。他为此带来了许多新见解，并至少开辟了两个新的研究方向。

他推进了元胞自动机理论，倡导采用比特作为计算机内存的度量单位，并解决了从不可靠的计算机组件中获得可靠答案的问题。

在第二次世界大战期间及之后，冯·诺伊曼担任武装部队的顾问。他的宝贵贡献包括提出了内爆方法以使核燃料爆炸，以及参与了氢弹的研制。从1940年起，他是马里兰州阿伯丁试验场弹道研究实验室科学咨询委员会的成员。1941年至1955年，他是海军军械局的成员，1943年至1955年，他是洛斯阿拉莫斯科学实验室的顾问。1950年至1955年，他是华盛顿特区武装部队特种武器项目的成员。1955年，艾森豪威尔总统任命他为原子能委员会委员，1956年，他获得了该委员会的恩里科·费米奖，当时他知道自己已患癌症，无法治愈。

尤金·维格纳在[18]中写到冯·诺伊曼的去世：-

冯·诺伊曼意识到自己已病入膏肓，他的逻辑迫使他认识到自己将不复存在，因而也将不再有思想……看着他的心智在一切希望都已破灭时，与在他看来不可避免却又无法接受的命运搏斗，那种挫败感令人心碎。

在[5]中，冯·诺伊曼的死亡是这样描述的：-

……他的心智，那枚他一直赖以依靠的护身符，正变得越来越不可靠。随后是彻底的心理崩溃；每晚都陷入恐慌，发出无法控制的恐惧尖叫。他的朋友Edward Teller说：“我认为冯·诺伊曼在心智不再运作时所遭受的痛苦，比我见过的任何人的痛苦都要深重。”

冯·诺伊曼那种刀枪不入的感觉，或者仅仅是求生的欲望，正在与不可改变的事实作斗争。他似乎直到最后都对死亡怀有极大的恐惧……如今，任何成就和影响力都无法像过去那样拯救他了。Johnny 冯·诺伊曼，那个懂得如何充分生活的人，却不懂得如何死去。

要哪怕大致描述一下授予冯·诺伊曼的荣誉范围，几乎都是不可能的。他于1937年担任美国数学会的Colloquium Lecturer，并如上所述获得了该机构的马克希莫·博谢奖。他于1947年担任美国数学会的约西亚·威拉德·吉布斯 Lectureship，并于1951-53年担任该学会的主席。

他当选为许多科学院的院士，包括国家精确科学科学院（埃隆·拉格斯·利马，秘鲁）、Academia Nazionale dei Lincei（意大利罗马）、American Academy of Arts and Sciences（美国）、美国哲学学会（美国）、伦巴德科学与文学研究所（意大利米兰）、National Academy of Sciences（美国）和荷兰皇家科学院 and Letters（荷兰阿姆斯特丹）。

冯·诺伊曼获得了两次总统奖，1947年的功绩勋章和1956年的自由勋章。同样在1956年，他获得了阿尔伯特·爱因斯坦纪念奖和上述的恩里科·费米奖。

鲁道夫·佩尔斯写道[3]：-

他是“长发”数学教授的对立面。他总是衣着整洁，对国际政治和实际事务的看法，如同对数学问题一样生动。

John von Neumann was born János von Neumann. He was called Jancsi as a child, a diminutive form of János, then later he was called Johnny in the United States. His father, Max Neumann, was a top banker and he was brought up in a extended family, living in Budapest where as a child he learnt languages from the German and French governesses that were employed. Although the family were Jewish, Max Neumann did not observe the strict practices of that religion and the household seemed to mix Jewish and Christian traditions.

It is also worth explaining how Max Neumann's son acquired the "von" to become János von Neumann. Max Neumann was eligible to apply for a hereditary title because of his contribution to the then successful Hungarian economy and in 1913 he paid a fee to acquire a title, but he did not change his name. His son, however, used the German form von Neumann where the "von" indicated the title.

As a child von Neumann showed he had an incredible memory. Poundstone, in [8], writes:-

At the age of six, he was able to exchange jokes with his father in classical Greek. The Neumann family sometimes entertained guests with demonstrations of Johnny's ability to memorise phone books. A guest would select a page and column of the phone book at random. Young Johnny read the column over a few times, then handed the book back to the guest. He could answer any question put to him (who has number such and such?) or recite names, addresses, and numbers in order.

In 1911 von Neumann entered the Lutheran Gymnasium. The school had a strong academic tradition which seemed to count for more than the religious affiliation both in the Neumann's eyes and in those of the school. His mathematics teacher quickly recognised von Neumann's genius and special tuition was put on for him. The school had another outstanding mathematician one year ahead of von Neumann, namely Eugene Wigner.

World War I had relatively little effect on von Neumann's education but, after the war ended, Béla Kun controlled Hungary for five months in 1919 with a Communist government. The Neumann family fled to Austria as the affluent came under attack. However, after a month, they returned to face the problems of Budapest. When Kun's government failed, the fact that it had been largely composed of Jews meant that Jewish people were blamed. Such situations are devoid of logic and the fact that the Neumann's were opposed to Kun's government did not save them from persecution.

In 1921 von Neumann completed his education at the Lutheran Gymnasium. His first mathematics paper, written jointly with Fekete the assistant at the University of Budapest who had been tutoring him, was published in 1922. However Max Neumann did not want his son to take up a subject that would not bring him wealth. Max Neumann asked Theodore von Kármán to speak to his son and persuade him to follow a career in business. Perhaps von Kármán was the wrong person to ask to undertake such a task but in the end all agreed on the compromise subject of chemistry for von Neumann's university studies.

Hungary was not an easy country for those of Jewish descent for many reasons and there was a strict limit on the number of Jewish students who could enter the University of Budapest. Of course, even with a strict quota, von Neumann's record easily won him a place to study mathematics in 1921 but he did not attend lectures. Instead he also entered the University of Berlin in 1921 to study chemistry.

Von Neumann studied chemistry at the University of Berlin until 1923 when he went to Zürich. He achieved outstanding results in the mathematics examinations at the University of Budapest despite not attending any courses. Von Neumann received his diploma in chemical engineering from the Technische Hochschule in Zürich in 1926. While in Zürich he continued his interest in mathematics, despite studying chemistry, and interacted with Weyl and Pólya who were both at Zürich. He even took over one of Weyl's courses when he was absent from Zürich for a time. Pólya said [18]:-

Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper.

Von Neumann received his doctorate in mathematics from the University of Budapest, also in 1926, with a thesis on set theory. He published a definition of ordinal numbers when he was 20, the definition is the one used today.

Von Neumann lectured at Berlin from 1926 to 1929 and at Hamburg from 1929 to 1930. However he also held a Rockefeller Fellowship to enable him to undertake postdoctoral studies at the University of Göttingen. He studied under Hilbert at Göttingen during 1926-27. By this time von Neumann had achieved celebrity status [8]:-

By his mid-twenties, von Neumann's fame had spread worldwide in the mathematical community. At academic conferences, he would find himself pointed out as a young genius.

Veblen invited von Neumann to Princeton to lecture on quantum theory in 1929. Replying to Veblen that he would come after attending to some personal matters, von Neumann went to Budapest where he married his fiancée Marietta Kovesi before setting out for the United States. In 1930 von Neumann became a visiting lecturer at Princeton University, being appointed professor there in 1931.

Between 1930 and 1933 von Neumann taught at Princeton but this was not one of his strong points [8]:-

His fluid line of thought was difficult for those less gifted to follow. He was notorious for dashing out equations on a small portion of the available blackboard and erasing expressions before students could copy them.

In contrast, however, he had an ability to explain complicated ideas in physics [3]:-

For a man to whom complicated mathematics presented no difficulty, he could explain his conclusions to the uninitiated with amazing lucidity. After a talk with him one always came away with a feeling that the problem was really simple and transparent.

He became one of the original six mathematics professors (J W Alexander, A Einstein, M Morse, O Veblen, J von Neumann and H Weyl) in 1933 at the newly founded Institute for Advanced Study in Princeton, a position he kept for the remainder of his life.

During the first years that he was in the United States, von Neumann continued to return to Europe during the summers. Until 1933 he still held academic posts in Germany but resigned these when the Nazis came to power. Unlike many others, von Neumann was not a political refugee but rather he went to the United States mainly because he thought that the prospect of academic positions there was better than in Germany.

In 1933 von Neumann became co-editor of the Annals of Mathematics and, two years later, he became co-editor of Compositio Mathematica. He held both these editorships until his death.

Von Neumann and Marietta had a daughter Marina in 1935 but their marriage ended in divorce in 1937. The following year he married Klára Dán, also from Budapest, whom he met on one of his European visits. After marrying, they sailed to the United States and made their home in Princeton. There von Neumann lived a rather unusual lifestyle for a top mathematician. He had always enjoyed parties [8]:-

Parties and nightlife held a special appeal for von Neumann. While teaching in Germany, von Neumann had been a denizen of the Cabaret-era Berlin nightlife circuit.

Now married to Klára the parties continued [18]:-

The parties at the von Neumann's house were frequent, and famous, and long.

Ulam summarises von Neumann's work in [35]. He writes:-

In his youthful work, he was concerned not only with mathematical logic and the axiomatics of set theory, but, simultaneously, with the substance of set theory itself, obtaining interesting results in measure theory and the theory of real variables. It was in this period also that he began his classical work on quantum theory, the mathematical foundation of the theory of measurement in quantum theory and the new statistical mechanics.

His text Mathematische Grundlagen der Quantenmechanik Ⓣ (1932) built a solid framework for the new quantum mechanics. Van Hove writes in [36]:-

Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann's stature. As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analysed by one single man in two years (1927-1929).

Self-adjoint algebras of bounded linear operators on a Hilbert space, closed in the weak operator topology, were introduced in 1929 by von Neumann in a paper in Mathematische Annalen . Kadison explains in [22]:-

His interest in ergodic theory, group representations and quantum mechanics contributed significantly to von Neumann's realisation that a theory of operator algebras was the next important stage in the development of this area of mathematics.

Such operator algebras were called "rings of operators" by von Neumann and later they were called $W^{*}$-algebras by some other mathematicians. J Dixmier, in 1957, called them "von Neumann algebras" in his monograph Algebras of operators in Hilbert space (von Neumann algebras). In the second half of the 1930's and the early 1940s von Neumann, working with his collaborator F J Murray, laid the foundations for the study of von Neumann algebras in a fundamental series of papers.

However von Neumann is known for the wide variety of different scientific studies. Ulam explains [35] how he was led towards game theory:-

Von Neumann's awareness of results obtained by other mathematicians and the inherent possibilities which they offer is astonishing. Early in his work, a paper by Borel on the minimax property led him to develop ... ideas which culminated later in one of his most original creations, the theory of games.

In game theory von Neumann proved the minimax theorem. He gradually expanded his work in game theory, and with co-author Oskar Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944).

Ulam continues in [35]:-

An idea of Koopman on the possibilities of treating problems of classical mechanics by means of operators on a function space stimulated him to give the first mathematically rigorous proof of an ergodic theorem. Haar's construction of measure in groups provided the inspiration for his wonderful partial solution of Hilbert's fifth problem, in which he proved the possibility of introducing analytical parameters in compact groups.

In 1938 the American Mathematical Society awarded the Bôcher Prize to John von Neumann for his memoir Almost periodic functions and groups. This was published in two parts in the Transactions of the American Mathematical Society, the first part in 1934 and the second part in the following year. Around this time von Neumann turned to applied mathematics [35]:-

In the middle 30's, Johnny was fascinated by the problem of hydrodynamical turbulence. It was then that he became aware of the mysteries underlying the subject of non-linear partial differential equations. His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks. The phenomena described by these non-linear equations are baffling analytically and defy even qualitative insight by present methods. Numerical work seemed to him the most promising way to obtain a feeling for the behaviour of such systems. This impelled him to study new possibilities of computation on electronic machines ...

Von Neumann was one of the pioneers of computer science making significant contributions to the development of logical design. Shannon writes in [29]:-

Von Neumann spent a considerable part of the last few years of his life working in [automata theory]. It represented for him a synthesis of his early interest in logic and proof theory and his later work, during World War II and after, on large scale electronic computers. Involving a mixture of pure and applied mathematics as well as other sciences, automata theory was an ideal field for von Neumann's wide-ranging intellect. He brought to it many new insights and opened up at least two new directions of research.

He advanced the theory of cellular automata, advocated the adoption of the bit as a measurement of computer memory, and solved problems in obtaining reliable answers from unreliable computer components.

During and after World War II, von Neumann served as a consultant to the armed forces. His valuable contributions included a proposal of the implosion method for bringing nuclear fuel to explosion and his participation in the development of the hydrogen bomb. From 1940 he was a member of the Scientific Advisory Committee at the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland. He was a member of the Navy Bureau of Ordnance from 1941 to 1955, and a consultant to the Los Alamos Scientific Laboratory from 1943 to 1955. From 1950 to 1955 he was a member of the Armed Forces Special Weapons Project in Washington, D.C. In 1955 President Eisenhower appointed him to the Atomic Energy Commission, and in 1956 he received its Enrico Fermi Award, knowing that he was incurably ill with cancer.

Eugene Wigner wrote of von Neumann's death [18]:-

When von Neumann realised he was incurably ill, his logic forced him to realise that he would cease to exist, and hence cease to have thoughts ... It was heartbreaking to watch the frustration of his mind, when all hope was gone, in its struggle with the fate which appeared to him unavoidable but unacceptable.

In [5] von Neumann's death is described in these terms:-

... his mind, the amulet on which he had always been able to rely, was becoming less dependable. Then came complete psychological breakdown; panic, screams of uncontrollable terror every night. His friend Edward Teller said, "I think that von Neumann suffered more when his mind would no longer function, than I have ever seen any human being suffer."

Von Neumann's sense of invulnerability, or simply the desire to live, was struggling with unalterable facts. He seemed to have a great fear of death until the last... No achievements and no amount of influence could save him now, as they always had in the past. Johnny von Neumann, who knew how to live so fully, did not know how to die.

It would be almost impossible to give even an idea of the range of honours which were given to von Neumann. He was Colloquium Lecturer of the American Mathematical Society in 1937 and received the its Bôcher Prize as mentioned above. He held the Gibbs Lectureship of the American Mathematical Society in 1947 and was President of the Society in 1951-53.

He was elected to many academies including the Academia Nacional de Ciencias Exactas (Lima, Peru), Academia Nazionale dei Lincei (Rome, Italy), American Academy of Arts and Sciences (USA), American Philosophical Society (USA), Instituto Lombardo di Scienze e Lettere (Milan, Italy), National Academy of Sciences (USA) and Royal Netherlands Academy of Sciences and Letters (Amsterdam, The Netherlands).

Von Neumann received two Presidential Awards, the Medal for Merit in 1947 and the Medal for Freedom in 1956. Also in 1956 he received the Albert Einstein Commemorative Award and the Enrico Fermi Award mentioned above.

Peierls writes [3]:-

He was the antithesis of the "long-haired" mathematics don. Always well groomed, he had as lively views on international politics and practical affairs as on mathematical problems.