科恩Paul Cohen

科恩 的父母 马克斯·亚伯拉罕 和 Minnie 科恩 是从故国波兰移民到美国的犹太人。亚伯拉罕 科恩 基本上是个打零工的人，什么活都干；而他的妻子则靠做女装裁缝为家里挣来一些急需的钱。科恩 是他父母四个孩子中最小的一个，在纽约布鲁克林长大。由于父母在他九岁时分居，他由母亲抚养长大。他从小就对数学感兴趣，年纪不大便开始学习高等数学。他 [4]：-

……只有九岁，他的姐姐西尔维亚从纽约一家图书馆为他借了一本关于微积分的书。图书馆员不愿让她借这本书，更不用说给她弟弟了，理由是连一些大学教授都不懂微积分。

在整个青少年时期，他被视为数学神童，在数学竞赛中展现的能力令周围所有人惊叹。他就读于纽约市的斯图文森高中，1950年以十六岁的低龄毕业。这所学校以数学和科学的高声誉著称，只招收通过入学考试的最优秀学生。从斯图文森高中毕业后，科恩于1950年至1953年在布鲁克林学院学习，但在访问芝加哥讨论研究选择后被芝加哥大学研究生院录取，未取得学位便离开了。他在芝加哥攻读硕士学位，选修课程以符合他当时的目标，即从事数论研究。他在到达芝加哥之前对数论的了解来自他在学院期间自学阅读的若干经典著作。为了符合这一目标，他开始在安德烈·韦伊的指导下研究数论。他于1954年获得硕士学位，但他对某些数论结果不可判定这一事实比对数论本身更感兴趣。然而，数论在他整个职业生涯中始终是一个感兴趣的课题[14]：-

他养成了问教师和同学他们领域最重要的问题是什么的习惯，因为那些是他唯一想解决的问题。

他继续在芝加哥攻读博士学位，在Antoni Zygmund的指导下，于1958年因其博士论文Topics in the Theory of Uniqueness of Trigonometric Series获得博士学位。在这篇论文中，科恩表示他[6]：-

……希望向AZygmund教授表达最深切的感谢，感谢他在本学位论文准备期间不断的帮助和鼓励。

他在引言的开头把学位论文的主题置于背景之中[6]：-

三角级数的唯一性理论可以看作源于这样一个问题：在什么意义下，一个函数的Fourier series可以被视为该函数在无穷三角级数中的合法展开。当然，我们知道，如果该级数有界收敛于该函数，那么级数的系数确实必须由莱昂哈德·欧拉-约瑟夫·傅里叶公式给出。然而，在没有这种条件的情况下，我们可以问自己，两个三角级数是否可能处处收敛于同一个函数。这个问题的答案是否定的，本质上由波恩哈德·黎曼证明，证明由格奥尔格·康托尔完成。唯一性集理论所关注的，正是把处处收敛的条件替换为几乎处处收敛。

作为研究生的那些年对科恩来说是美好的岁月，他与同学结下了许多友谊，这些友谊将伴随他一生。约翰·格里格斯·汤普森就是芝加哥这样一位 fellow research student。通过这些友谊，科恩也开始对逻辑产生兴趣[14]：-

作为研究生，科恩与逻辑的联系来自他与一群后来成为逻辑学家的活跃学生的友谊：Michael Morley、Anil Nerode、Bill Howard、雷蒙·梅里儿·思木里安和Stanley Tennenbaum。有一段时间他住在Tennenbaum的房子里，通过潜移默化吸收了逻辑，因为芝加哥数学系没有逻辑课程。

1957年，在获得博士学位之前，科恩被任命为罗切斯特大学数学讲师，为期一年。随后他在1958-59学年在麻省理工学院度过，之后在1959-61年作为研究员在普林斯顿高等爱德华·斯图迪度过。这些年里，科恩取得了若干重大数学突破。在Factorization in group algebras（1959）中，他证明了局部紧群上的任何可积函数都是两个此类函数的卷积，解决了沃尔特·鲁丁提出的一个问题。在On a conjecture of Littlewood and idempotent measures（1960）中，科恩在解决利特尔伍德猜想方面取得了重大突破。他早些时候写信给哈罗德·达文波特告知他这个结果，哈罗德·达文波特回复[16]：-

……对科恩说，如果科恩的证明成立，他就胜过了整整一代在这个问题上辛勤工作的英国分析学家。科恩的证明确实成立；事实上，哈罗德·达文波特是第一个改进科恩结果的人。

1961年，科恩被任命为斯坦福大学数学系助理教授。次年，他晋升为数学副教授，同样在1962年，他获得了阿尔弗雷德·P·斯隆研究奖学金。1962年8月，科恩参加了在斯德哥尔摩举行的国际数学家大会。他作为受邀演讲者作了题为Idempotent measures and homomorphisms of group algebras的报告。大会结束后，在从斯德哥尔摩到列宁格勒的游轮上，科恩遇到了来自瑞典马隆的克里斯蒂娜·卡尔斯。他们于1963年10月10日结婚，育有三个儿子：双胞胎埃里克和史蒂文，以及查尔斯。

1964年，他晋升为斯坦福大学正教授，此时他已解决了数学中最具挑战性的未解决问题之一。科恩使用了一种称为“力迫”的技术，证明了选择公理和广义连续统假设在集合论中的独立性。安格斯·麦金太尔写道[13]：-

连续统假设研究的一个戏剧性方面是，科恩在逻辑学领域是一位自学成才的局外人。他在集合论和p进域方面的工作具有非常独特的风格，组合式的，且相当不受一般理论束缚。

在[7]中，科恩解释了他如何通过阅读库尔特·弗雷德里希·哥德尔的The Consistency of the Continuum Hypothesis而产生力迫的想法，这本书由1938-39年在斯图迪高等研究院所授课程的笔记组成。连续统假设问题是大卫·希尔伯特在1900年巴黎第二届国际数学家大会上提出的著名23个问题中的第一个。大卫·希尔伯特的著名演讲The Problems of Mathematics挑战（今天仍然挑战）数学家们解决这些基本问题，而科恩有解决第1个问题的殊荣。

他在1962年底开始研究连续统假设的独立性。到1963年4月，他感觉事情豁然开朗[7]：-

在任何数学发现中，都有某些时刻，问题的解决发生在如此潜意识的层面，以至于回想起来，似乎无法剖析并解释其起源。相反，整个想法一下子呈现出来，往往也许是模糊的形式，但逐渐变得更加精确。

在阅读了科恩在1963年5月9日的一封信中寄来的证明后，库尔特·弗雷德里希·哥德尔回复他说：-

让我再说一遍，读到你关于连续统假设独立性的证明真是一件乐事。我认为在所有本质方面你都给出了尽可能最好的证明，而这种情况并不常见。读你的证明给我带来的愉快感受，就像看一出真正的好戏一样。

科恩在1963年7月4日于乔治·伯克利举行的“模型论”国际研讨会上所作的讲座Independence results in set theory中，谈到了他关于选择公理以及连续统假设相对于恩斯特·策梅洛-亚伯拉罕·弗兰克尔集合论公理的独立性的工作。他的证明发表在The independence of the continuum hypothesis（1963）和The independence of the continuum hypothesis. II（1964）两篇论文中。安杰伊·莫斯托夫斯基在评论其中第一篇时写道：-

这些结果给出了公理集合论中最突出的未决问题期待已久的解答，应当被评价为自库尔特·弗雷德里希·哥德尔1940年专著《连续统假设的一致性》（1940）出版以来公理集合论研究中最重要的进展。……在这位评论者看来，几乎可以肯定，科恩的发现对元数学的影响至少会像对数学的一般哲学（或许不只是数学的哲学）的影响一样深远。

安格斯·麦金太尔（Angus MacIntyre）于1964年至1967年在斯坦福大学读研究生，他写道[4]：-

在我还是个年轻数学家时，他激励了我。我从未听过他讲授集合论，而是听他讲授代数几何和p进域。他有非常独特的风格，充满热情，非常“亲自动手”。他尽可能少用一般理论，并且总是传达出一种他触及了事物核心的感觉。他的技巧，即使在像集合论这样抽象的东西中，也非常具有构造性。他聪明得令人望而生畏，在60年代，要把自己的“最难的问题”提给我所认识的那个科恩，一个人非得天真或者格外无私才行。

参见 科恩 在 THIS LINK 上关于数学与教学的一篇文章

1966年，科恩 出版了一部专著 Set theory and the continuum hypothesis，该书基于他1965年春季在哈佛大学讲授的一门课程。Azriel 保罗·莱维（他最初是在 伯克利 模型论会议上听到 科恩 的结果的）写道：-

这部专著主要是对作者著名结果的阐述，即连续统假设和选择公理的独立性。此外，它还介绍了逻辑和集合论中的主要经典结果。……这本书呈现了一种新鲜而直观的方法，并让人窥见引导作者走向其发现的心智过程。读者会在这本书中发现，对于一部数学专著而言恰到好处的哲学评论。

同年，科恩 因其在集合论基础方面的奠基性工作而被授予 菲尔兹奖。该奖由 苏联科学院 主席 姆斯季斯拉夫·克尔德什 在1966年于莫斯科举行的国际数学家大会上颁发给他。只有一位 约翰·查尔斯·菲尔兹 奖章得主（拉尔斯·阿尔福斯）在更年轻的年龄获得了 约翰·查尔斯·菲尔兹 奖章。丘奇 在大会上就 Paul J Cohen and the continuum problem 作了演讲，描述了 科恩 的卓越成就。然而，约翰·查尔斯·菲尔兹 奖章并不是 科恩 获得的第一个奖项。1964年，他获得了 美国数学会 颁发的 马克希莫·博谢 纪念奖：-

……因其论文《论Littlewood的一个猜想与幂等测度》，American Journal of Mathematics 82 (1960), 191-212。

三年后，即1967年，科恩 获得了国家科学奖章：-

因在数理逻辑方面取得划时代成果，这些成果使数学基础研究活跃起来并得到拓展。

1968年2月13日，他在白宫的一个仪式上从总统罗杰·林登 B Johnson手中接受了该奖项。他还当选为国家科学院、American Academy of Arts and Sciences的成员，以及伦敦数学会的名誉外籍成员。

除了集合论方面的工作外，科恩还研究微分方程和调和分析。Dawn Levy在[12]中报道了Peter Sarnak（普林斯顿大学数学教授，曾是科恩的博士生，学位论文为Prime Geodesic Theorems（1980））对科恩的评论：-

科恩是20世纪最杰出的数学家之一。像许多伟大的数学家一样，他的数学兴趣和贡献非常广泛，从数学分析和微分方程到数理逻辑和数论。去年9月在斯坦福举行的一次会议庆祝了科恩的工作和他的72岁生日，这次会议凸显了这种广度。与会者包括不同领域的顶尖专家，他们通常不会听同一组讲座。……科恩是一位充满活力和热情的讲师和教师。他使数学看起来简单而统一。他总是渴望分享他在不同领域的许多想法和见解。他对数学的热情从未减退。

Macintyre撰文论述了科恩在连续统假设[13]上取得杰出成果之后发表的重要论文：-

1969年，科恩发表了一篇关于p进胞腔分解的极具原创性的论文，给出了Ax-Kochen-Ersov著名结果的一个构造性版本。它现在是动机积分逻辑分析的基础。从1969年起，科恩投身于一些最具挑战性和最棘手的问题，例如波恩哈德·黎曼假设。他是一位充满热情和鼓舞人心的数学家。

Kathy Owen，20世纪70年代曾在斯坦福度过一段时间，她写下了当时关于科恩的回忆[16]：-

科恩是一个令人惊叹的人。急躁、不安分、好胜、挑衅且才华横溢。他是研究生和教员咖啡时间的常客。他喜欢就任何话题进行辩论和争论的激烈交锋，一旦发现对方观点中的逻辑弱点，就毫不留情。简直无处可藏！他以其极其敏锐的才智、对重大问题的着迷、对“绝对音感”的奇怪兴趣（他带着音叉来咖啡时间测试每个人）以及对少数确实有绝对音感的人的轻微恼怒而脱颖而出。他是一个非凡的人，一位对我生活产生重大影响的挚友，一道拥有全色谱色彩的光芒。

科恩于1972年被任命为斯坦福大学定量科学Marjorie Mhoon Fair讲席教授，是该讲席的第一位持有者。他于2004年正式退休，但继续在斯坦福任教，直到去世前不久。他在帕洛阿尔托的斯坦福医院因一种罕见的肺部疾病去世。

至于科恩在数学之外的兴趣，他既弹钢琴又拉小提琴，在斯坦福合唱团唱歌，并且是一个瑞典民间团体的成员。他是一位有成就的语言学家，会说瑞典语、法语、西班牙语、德语和意第绪语。他和妻子经常为学生、同事和朋友举办晚宴。他喜欢带游客参观旧金山及周边地区。

让我们以引用科恩关于他在连续统假设方面工作的回忆来结束这本传记[11]：-

……有点奇怪的是，在某种意义上，连续统假设和选择公理并不是真正困难的问题——它们不涉及技术上的复杂性；然而，在当时它们被认为是困难的。可以幽默地说，对我的证明的态度是这样的。当它首次提出时，有些人认为它是错的。然后它被认为极其复杂。然后它被认为很容易。但当然，它之所以容易，是因为有一个清晰的哲学思想。你知道，有一些技术要点困扰着我，但基本上它并不是一个真正极其复杂的组合问题；它是一个哲学思想。

Paul Cohen's parents, Abraham and Minnie Cohen, were Jewish immigrants to the United States from their native land of Poland. Abraham Cohen was basically an odd job man, turning his hand to a variety of different jobs, while his wife brought in some much needed money to the family from dressmaking. Paul was the youngest of his parents' four children and he was brought up in Brooklyn, New York. He was brought up by his mother from the age of nine since at that time his parents separated. Interested in mathematics from childhood, he began to study advanced mathematics from a young age. He [4]:-

... was only nine years old when his sister Sylvia checked out a book about calculus from a New York library for him. Librarians were reluctant to let her have the book, much less for her younger brother, arguing that even some college professors didn't understand calculus.

Throughout his teenage years he was regarded as a mathematical prodigy, amazing all around him with the abilities he displayed in mathematics competitions. He attended Stuyvesant High School in New York City, graduating in 1950 at the young age of sixteen years. This school, with a high reputation for mathematics and science, accepted only the best students after taking an entrance examination. After graduating from Stuyvesant High School, Cohen was a student at Brooklyn College from 1950 until 1953 but left without taking a degree having been admitted to graduate studies at the University of Chicago after making a visit to discuss his research options at Chicago. He studied for his master's degree at Chicago, taking courses to fit in with his aim at the time which was to undertake research in number theory. His knowledge of number theory before arriving in Chicago was from a number of classic texts that he had read on his own while at College. To fit in with this aim he began to work on number theory supervised by André Weil. He was awarded his Master's degree in 1954 but he came to be more interested in the fact that certain results in number theory were undecidable than in number theory itself. Number theory, however, remained a topic of interest to him throughout his career [14]:-

He made a habit of asking the faculty and fellow students what the most important problems were in their fields because those were the only problems he wanted to solve.

Continuing to study at Chicago for his doctorate under the supervision of Antoni Zygmund he was awarded his PhD in 1958 for his doctoral thesis Topics in the Theory of Uniqueness of Trigonometric Series. In this thesis, Cohen states that he [6]:-

... wishes to express his deepest gratitude to Professor A Zygmund for his constant aid and encouragement during the preparation of this dissertation.

He begins the Introduction by putting the topic of the thesis into context [6]:-

The theory of uniqueness of trigonometrical series can be regarded as arsing from the question of deciding in what sense the Fourier series of a function may be considered as the legitimate expansion of the function in an infinite trigonometrical series. We know, of course, that if the series converges boundedly to the function, then indeed the coefficients of the series must be given by the Euler-Fourier Formulas. However, in the absence of such a condition, we may ask ourselves whether two trigonometrical series may converge to the same function everywhere. The answer to this question is in the negative and was essentially proved so by Riemann, the proof being completed by Cantor. It is with the replacement of the condition of convergence everywhere with that of convergence almost everywhere, that the theory of sets of uniqueness is concerned.

The years as a research student were good ones for Cohen and he made many friendships with fellow students, friendships that would last throughout his life. John Thompson was one such fellow research student at Chicago. Cohen, through these friendships, had also begun to take an interest in logic [14]:-

As a graduate student Cohen's connection with logic were his friendships with a lively group of students who became logicians; Michael Morley, Anil Nerode, Bill Howard, Ray Smullyan, and Stanley Tennenbaum. For a while he lived in Tennenbaum's house and absorbed logic by osmosis, for there were no courses in logic in the Chicago mathematics department.

In 1957, before the award of his doctorate, Cohen was appointed as an Instructor in Mathematics at the University of Rochester for a year. He then spent the academic year 1958-59 at the Massachusetts Institute of Technology before spending 1959-61 as a fellow at the Institute for Advanced Study at Princeton. These were years in which Cohen made a number of significant mathematical breakthroughs. In Factorization in group algebras (1959) he showed that any integrable function on a locally compact group is the convolution of two such functions, solving a problem posed by Walter Rudin. In On a conjecture of Littlewood and idempotent measures (1960) Cohen made a significant breakthrough in solving the Littlewood Conjecture. He had earlier written to Harold Davenport telling him about this result and Davenport replied [16]:-

... to Paul saying that if Paul's proof held up, he would have bettered a generation of British analysts who had worked hard on this problem. Paul's proof did hold up; in fact, Davenport was the first to improve on Paul's result.

In 1961 Cohen was appointed to the faculty at Stanford University as an assistant professor of mathematics. He was promoted to associate professor in mathematics in the following year and, also in 1962, was awarded an Alfred P Sloan research fellowship. In August 1962 Cohen participated in the International Congress of Mathematicians in Stockholm. He was an invited speaker giving the address Idempotent measures and homomorphisms of group algebras. On a cruise from Stockholm to Leningrad, following the Congress, Cohen met Christina Karls from Malung, Sweden. They married on 10 October 1963 and had three sons, twins Eric and Steven, and Charles.

He was promoted to full professor at Stanford University in 1964 having, by this time, solved one of the most challenging open problems in mathematics. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalised continuum hypothesis. Angus MacIntyre writes [13]:-

A dramatic aspect of the continuum hypothesis work is that Cohen was a self-taught outsider in logic. His work on set theory and p-adic fields has a very characteristic style, combinatorial and rather free of general theory.

In [7] Cohen explains how he came to the idea of forcing from reading Kurt Gödel's The Consistency of the Continuum Hypothesis, a book consisting of notes of a course given at the Institute for Advanced Study in 1938-39. The continuum hypothesis problem was the first of David Hilbert's famous 23 problems delivered to the Second International Congress of Mathematicians in Paris in 1900. Hilbert's famous speech The Problems of Mathematics challenged (and today still challenges) mathematicians to solve these fundamental questions and Cohen has the distinction of solving Problem 1.

He had begun working on the independence of the continuum hypothesis towards the end of 1962. By April 1963 he felt things click into place [7]:-

There are certain moments in any mathematical discovery when the resolution of a problem takes place at such a subconscious level that, in retrospect, it seems impossible to dissect it and explain its origin. Rather, the entire idea presents itself at once, often perhaps in a vague form, but gradually becomes more precise.

After reading Cohen's proof which he sent in a letter of 9 May 1963, Kurt Gödel replied to him:-

Let me repeat that it is really a delight to read your proof of the independence of the continuum hypothesis. I think that in all essential respects you have given the best possible proof and this does not happen frequently. Reading your proof had a similarly pleasant effect on me as seeing a really good play.

Cohen spoke about his work on the independence of the axiom of choice and the continuum hypothesis from the axioms of Zermelo-Fraenkel set theory in a lecture Independence results in set theory delivered at the international symposium on the 'Theory of Models' at Berkeley on 4 July 1963. His proof appeared in the two papers The independence of the continuum hypothesis (1963) and The independence of the continuum hypothesis. II (1964). Andrzej Mostowski, reviewing the first of these, writes:-

These results present the long-awaited solutions of the most outstanding open problems of axiomatic set theory and should be rated as the most important advance in the study of axiomatic set theory since the publication of Gödel's 1940 monograph 'The consistency of the continuum hypothesis' (1940). ... to this reviewer it seems more than probable that the influence of Cohen's discovery will be at least as deep in metamathematics as in the general philosophy of mathematics (and perhaps not only of mathematics).

Angus MacIntyre, who was a graduate student at Stanford from 1964 to 1967, writes [4]:-

He inspired me when I was a young mathematician. I never heard him lecture on set theory, but rather on algebraic geometry and p-adic fields. He had a very special style, full of enthusiasm and very 'hands on.' He used as little general theory as possible and always conveyed a sense that he got to the heart of things. His techniques, even in something as abstract as set theory, were very constructive. He was dauntingly clever, and one would have had to be naive or exceptionally altruistic to put one's 'hardest problem' to the Paul I knew in the '60s.

See an article by Paul Cohen on mathematics and teaching at THIS LINK

In 1966 Cohen published the monograph Set theory and the continuum hypothesis based on a course he gave at Harvard in spring 1965. Azriel Lévy (who first heard Cohen's results at the Berkeley model theory conference) writes:-

This monograph is mostly an exposition of the celebrated results of the author, namely the independence of the continuum hypothesis and the axiom of choice. In addition it presents also the main classical results in logic and set theory. ... This book presents a fresh and intuitive approach and it gives some glimpses into the mental process that led the author to his discoveries. The reader will find in this book just the right amount of philosophical remarks for a mathematical monograph.

In the same year Cohen was awarded a Fields Medal for his fundamental work on the foundations of set theory. It was presented to him by Mstislav Vsevolodovich Keldysh, President of the USSR Academy of Sciences, at the 1966 International Congress of Mathematicians in Moscow. Only one Fields Medalist (Lars Ahlfors) has been awarded the Fields Medal at a younger age. Alonzo Church gave an address to the Congress on Paul J Cohen and the continuum problem describing Cohen's remarkable achievements. The Fields Medal, however, was not the first award that Cohen received. In 1964 he was awarded the Bôcher Memorial Prize from the American Mathematical Society:-

...for his paper, On a conjecture of Littlewood and idempotent measures, American Journal of Mathematics 82 (1960), 191-212.

Three years later, in 1967, Cohen received the National Medal of Science:-

For epoch-making results in mathematical logic which have enlivened and broadened investigations in the foundation of mathematics.

He received the award from President Lyndon B Johnson in a ceremony in the White House on 13 February 1968. He has also been elected to the National Academy of Sciences, the American Academy of Arts and Sciences, and as an honorary foreign member of the London Mathematical Society.

In addition to his work on set theory, Cohen worked on differential equation and harmonic analysis. Dawn Levy reports in [12] comments made about Cohen by Peter Sarnak (professor of mathematics at Princeton and a former doctoral student of Cohen's with the thesis Prime Geodesic Theorems (1980)):-

Paul Cohen was one of the most brilliant mathematicians of the 20th century. Like many great mathematicians, his mathematical interests and contributions were very broad, ranging from mathematical analysis and differential equations to mathematical logic and number theory. This breadth was highlighted in a conference held at Stanford last September celebrating Cohen's work and his 72nd birthday. The gathering consisted of leading experts in different fields who normally would not find themselves listening to the same set of lectures. ... Cohen was a dynamic and enthusiastic lecturer and teacher. He made mathematics look simple and unified. He was always eager to share his many ideas and insights in diverse fields. His passion for mathematics never waned.

Macintyre writes about the important papers Cohen produced after his outstanding results on the continuum hypothesis [13]:-

In 1969 Cohen published a highly original paper on p-adic cell decomposition, giving a constructive version of the famous results of Ax-Kochen-Ersov. It is now fundamental for logical analysis of motivic integration. From 1969 on Cohen devoted himself to some of the most challenging and unyielding problems, such as the Riemann Hypothesis. He was a passionate and inspiring mathematician.

Kathy Owen, who spent time at Stanford in the 1970s, wrote about Cohen at that time [16]:-

Paul was an astonishing man. Impatient, restless, competitive, provocative and brilliant. He was a regular at coffee hour for the graduate students and the faculty. He loved the cut-and-thrust of debate and argument on any topic and was relentless if he found a logical weakness in an opposing point of view. There was simply nowhere to hide! He stood out for his razor-sharp intellect, his fascination for the big questions, his strange interest in "perfect pitch" (he brought a tuning fork to coffee hour and tested everyone) and his mild irritation with the few who do have perfect pitch. He was a remarkable man, a dear friend who had a big impact on my life, a light with the full spectrum of colours.

Cohen was named Marjorie Mhoon Fair Professor in Quantitative Science at Stanford in 1972, being the first holder of this chair. He formally retired in 2004, but continued teaching at Stanford until shortly before his death. He died of a rare lung disease at Stanford Hospital in Palo Alto.

As to Cohen's interests outside mathematics, he played both the piano and violin, sang in a Stanford chorus, and was a member of a Swedish folk group. He was an accomplished linguist speaking Swedish, French, Spanish, German and Yiddish. He and his wife hosted frequent dinner parties for students, colleagues and friends. He loved showing visitors round San Francisco and the surrounding area.

Let us end this biography by quoting Cohen's reminiscences about his work on the continuum hypothesis [11]:-

... it's somewhat curious that in a certain sense the continuum hypothesis and the axiom of choice are not really difficult problems - they don't involve technical complexity; nevertheless, at the time they were considered difficult. One might say in a humorous way that the attitude toward my proof was as follows. When it was first presented, some people thought it was wrong. Then it was thought to be extremely complicated. Then it was thought to be easy. But of course it is easy in the sense that there is a clear philosophical idea. There were technical points, you know, which bothered me, but basically it was not really an enormously involved combinatorial problem; it was a philosophical idea.