克林Stephen Cole Kleene

克林的父亲是古斯塔夫·Adolph Kleene，在儿子出生时是康涅狄格州哈特福德三一学院的经济学教授。他在那里度过了余下的职业生涯。他退休后成为荣休教授，并于1946年8月在缅因州尤尼恩的避暑别墅去世。这所房子位于缅因州的一个农场，该农场属于他的妻子，是从她的母亲德鲁西拉·克林那里继承的。古斯塔夫和他的儿子克林都认为这个农场是他们“真正的家”。古斯塔夫写了The problem of medical charity（1904年）和Profit and wages: A study in the distribution of income（1916年）等书。克林的母亲是爱丽丝·莉娜·克林，她是一位诗人和剧作家。结婚前，她发表了The lawsuit（发表于The Youth's Companion，1892年）、The dead bee（发表于The Century ，1899年）、The lost spell（发表于Atlantic Monthly，1900年）和Escape（发表于Atlantic Monthly，1900年）等诗作。结婚后，她以婚后姓名爱丽丝·克林发表了KIRSTIN; KIRSTIN. Play in Four Acts（1914年）。《纽约时报》的一位评论家将其描述为“一部歌词优雅的漂亮戏剧”。

克林在马萨诸塞州阿默斯特的阿默斯特学院攻读第一个学位，于1930年以最优等成绩获得学士学位。随后他前往普林斯顿大学，在那里他的博士研究由丘奇指导。正是奥斯瓦尔德·维布伦提出逻辑的发展需要数学家的仔细分析。丘奇，奥斯瓦尔德·维布伦的学生之一，于1929年被任命到普林斯顿，并在这个领域取得了显著进展。J Barkley Rosser也是丘奇在普林斯顿的博士生，于1933年到达，当时克林也在那里。这确实是一个令人兴奋的地方，可以进行将数学技术应用于逻辑的研究，来访者如库尔特·弗雷德里希·哥德尔——克林参加了他在高等爱德华·斯图迪研究所开设的课程。克林于1934年从普林斯顿获得博士学位，其学位论文题为A Theory of Positive Integers in Formal Logic。他在学位论文的引言中写道：-

……我们将主要关注基于丘奇提出的一组公理而建立的逻辑系统的发展。我们的目标是通过在该系统内构造该理论的一个重要部分，来经验性地证明该系统足以处理正整数理论。通过在丘奇的形式公理的某个子集的基础上进行构造，我们表明至少正整数理论的这一部分可以从逻辑中推导出来，而不使用否定、类和描述的概念。

获得博士学位后，克林在普林斯顿任教，直到1935年他加入威斯康星大学麦迪逊分校担任讲师。他于1937年在威斯康星晋升为助理教授，然后在1941年离开（因未能晋升而不满），回到他曾攻读第一个学位的阿默斯特学院担任助理教授。1942年，他与Nancy Elliot结婚；他们有四个孩子：Paul、Kenneth、Bruce和Pamela。Nancy Elliot是George Roy Elliott的女儿，George Roy Elliott是阿默斯特学院的英语教授，也是一位专门研究莎士比亚的文学评论家。同样在1942年，克林离开阿默斯特学院，以美国海军导航教官的身份在纽约海军预备役军官学校服役。后来他成为华盛顿特区海军研究实验室的项目主任。到第二次世界大战结束后离开海军时，克林已晋升为少校军衔。

1946年，他回到麦迪逊的威斯康星大学担任副教授，两年后晋升为正教授。1964年，他被任命为赛勒斯·C·达菲讲席教授，并一直担任该讲席直到1979年退休。他曾两届担任数学系主任，一届担任数值分析系（后更名为计算机科学系）主任。他还于1969-74年担任文理学院院长。在威斯康星大学期间，他是13名博士生的学位论文导师。

1970年，克林的妻子南希去世。八年后，他与珍妮·斯坦梅茨再婚。他因肺炎去世，享年85岁。

克林的研究涉及算法理论和递归函数理论，这是他创建并终生保持兴趣的领域。他与丘奇、库尔特·弗雷德里希·哥德尔、艾伦·图灵等人一起发展了递归理论领域。他为勒伊岑·布劳威尔创立的数学直觉主义做出了贡献。特别是，他于1950年在马萨诸塞州剑桥举行的国际数学家大会上就Recursive functions and intuitionistic mathematics进行了演讲。在这次演讲中，他谈到了他通过“实现”对直觉主义数论的解释如何可能扩展到直觉主义集合论。他在与Richard Vesley合著的The foundations of intuitionistic mathematics, especially in relation to recursive functions（1965年）一书中进一步探讨了这些想法。该书的第一、二、四章由克林撰写，而第三章由Vesley撰写。G 格奥尔格·克里泽尔在一篇评论中写道：-

第一章是目前为止对数学逻辑学家来说可用的关于直觉主义逻辑的最佳介绍。

迈克尔·达米特，一位权威人士，也对此印象深刻：-

……本书第一章首次系统地阐述了直觉主义分析的基础，将其表述为一个公理系统，处理勒伊岑·布劳威尔的扇定理、棒定理和连续性原则（称为勒伊岑·布劳威尔的原则）。在这些方面，这远远优于阿兰德·海廷早先著名的公理化。

克林在递归论方面的工作有助于为理论计算机科学奠定基础。通过提供确定哪些问题可解的方法，克林的工作引导了对哪些函数可计算的研究。他于1951年夏天在兰德公司度过，并发现了有限自动机的一个重要刻画。他关于那个夏天工作的兰德报告对理论计算机科学产生了非常大的影响。

1995年在芝加哥大学的一次演讲中，罗伯特·索尔以这些术语描述了克林的工作：-

克林通过六个模式对可计算函数的表述是最简洁和有用的之一，而他先前关于λ函数的工作在支持丘奇的论点——即这些类与直观可计算函数重合——方面发挥了重要作用。

从1930年代起，克林比任何其他数学家都更多地发展了可计算性和有效过程的概念，涵盖其所有形式，无论是抽象的与具体的，还是数学的与哲学的。他倾向于为一个领域奠定基础，然后转向下一个，因为每一个相继的领域都在他的引领下蓬勃发展成为一个主要的研究领域。

克林在可计算性中发展了多样化的主题：算术层级、可计算度、可计算序数和超算术理论、有限自动机和正则集（对计算机科学有巨大影响）、高阶类型的可计算性、直觉主义算术的递归可实现性（对哲学和计算机科学中的程序正确性有影响）。

克林最著名的著作是Introduction to Metamathematics（1952）和Mathematical Logic（1967）。克林在第一本书中写道：-

本书旨在对数学逻辑，特别是递归函数，以及一般较新的基础研究，提供一个连贯的导引。

J R Shoenfield在评论第二本书时写道：-

数学本科课程的扩展催生了许多教材，它们以比研究生教材更缓慢、更详细的方式处理严肃的数学成果。由于逻辑领域此类教材数量相当少，这本由该领域杰出权威撰写的书尤其受欢迎。……作者显然认为，对少数几个主题进行相当透彻的处理，胜过对每样东西都浅尝辄止。……困难的证明被分解为大量简单情形；其中一些情形通常留给读者。书中有许多富有启发性的例子；作者通常更感兴趣的是给出足够的例子来说明证明的要点，而不是给出证明的全部细节。清晰和简洁从不为了优雅而牺牲。历史注记和参考文献频繁出现，但不会喧宾夺主掩盖数学内容。

在克林因其杰出贡献而获得的奖项和荣誉中，我们提到他于1983年由美国数学会授予的Leroy P 凯瑟琳·斯蒂尔奖：-

……因三篇重要论文，它们为广义递归论和描述集合论的后续发展奠定了基础：“算术谓词与函数量词”、“论构造序数理论中谓词的形式（第二篇论文）”以及“数论谓词的层级”。

也许他最负盛名的奖项是1990年11月13日由布什总统在白宫东厅仪式上颁发的国家科学奖章：-

表彰他在递归论与有效可计算性理论方面的领导作用，以及将其发展为一个深广的数学研究领域。

其他荣誉包括当选国家科学院会士（1969年），当选符号逻辑协会主席（1956-58年），国际科学史与科学哲学联合会主席（1961年）以及该联合会逻辑、方法论与科学哲学分会主席（1960-62年）。他担任Journal of Symbolic Logic编辑长达十二年。

在[3]中，Keisler描述了克林在数学之外的兴趣：-

克林对自然和环境有着浓厚的兴趣，几乎每年夏天都会去他在缅因州的家族农场。他发现了一种蝴蝶Brenthis Bellona ab. Kleenei，并以他的名字命名。他是一位狂热的登山者，直到七十多岁，还带领每两年一次在麦迪逊举行的逻辑野餐会（现为克林纪念逻辑野餐会），徒步攀登魔鬼湖的悬崖。Steve Kleene对蘑菇的知识是传奇性的。

桑德斯·麦克兰恩在[5]中讲述了与克林一起攀登的一次经历：-

1949年，他和我即将参加美国数学会在达特茅斯举行的一次会议，我们凑在一起计划攀登总统山脉的所有山峰，还有第三位登山者加入我们，他是一位正在度假的旅馆侍者。然后，当我们三人终于站在最后一座山峰（麦迪逊山）顶上时，一场雷暴袭来。克林从山顶跳下来喊道：“下来；有闪电。”我往下走了一点，寻找我们的第三位同伴，却发现他平躺在地上，不省人事。史蒂夫（由于身高而动作更快）下到麦迪逊山口小屋寻求帮助。当我们把受伤的人带到那里时，我们发现闪电烧焦了他的头皮，并在他的脚上留下了一个水泡，对面是一个鞋钉。我们把他送到医院，我（裤子臀部完全烧焦）通知了他在优雅的东部斜坡旅馆的雇主。他后来康复了。史蒂夫和我继续攀登各种山峰，史蒂夫继续攀岩。

3的Keisler也描述了他的个性：-

尽管他是一个注重隐私的人，但他是一个技巧娴熟且热衷于讲轶事的人。他拥有强大的嗓音，总是让其他人无需看见他就能知道他在不在数学大楼里。

Stephen C Kleene's father was Gustav Adolph Kleene, a professor of economics at Trinity College, Hartford, Connecticut at the time of his son's birth. He remained there for the rest of his career. He retired, becoming professor emeritus, and died at his summer home in Union, Maine in August 1946. This home was on a farm in Maine which had belonged to his wife, inherited from her mother, Drucilla Cole. Both Gustav and his son Stephen considered this farm their 'real home.' Gustav wrote books such as The problem of medical charity (1904) and Profit and wages: A study in the distribution of income (1916). Stephen's mother was Alice Lena Cole who was a poet and writer of plays. Before marrying she published poems such as The lawsuit published in The Youth's Companion (1892), The dead bee published in The Century (1899), The lost spell published in Atlantic Monthly (1900) and Escape published in Atlantic Monthly (1900). After her marriage she published KIRSTIN; KIRSTIN. Play in Four Acts (1914) under her married name Alice Cole Kleene. It was described by a critic in the New York Times as a "pretty play with graceful lyrics."

Stephen Kleene studied for his first degree at Amherst College, Amherst, Massachusetts, being awarded his Bachelor's Degree summa cum laude in 1930. He then went to Princeton University where his doctoral studies were supervised by Alonso Church. It had been Oswald Veblen who had proposed that the development of logic required careful analysis by mathematicians. Church, one of Veblen's students, had been appointed to Princeton in 1929 and was making remarkable advances in this area. J Barkley Rosser was also a doctoral student of Church's at Princeton, arriving in 1933 while Kleene was there. It was certainly an exciting place to be undertaking research applying mathematical techniques to logic with visitors such as Kurt Gödel - Kleene attended a course he gave at the Institute for Advanced Study. Kleene received a doctorate from Princeton for his thesis entitled A Theory of Positive Integers in Formal Logic in 1934. He writes in the Introduction to his thesis:-

... we shall be concerned primarily with the development of the system of logic based on a set of postulates proposed by A Church. Our object is to demonstrate empirically that the system is adequate for the theory of positive integers, by exhibiting a construction of a significant portion of the theory within the system. By carrying out the construction on the basis of a certain subset of Church's formal axioms, we show that this portion at least of the theory of positive integers can be deduced from logic without the use of the notions of negation, class, and description.

After the award of his doctorate Kleene taught at Princeton until he joined the University of Wisconsin at Madison as an Instructor in 1935. He was promoted to Assistant Professor at Wisconsin in 1937 before leaving in 1941 (unhappy at his failure to be promoted) to become Assistant Professor back at Amherst College where he had studied for his first degree. In 1942 he married Nancy Elliot; they had four children, Paul, Kenneth, Bruce, and Pamela. Nancy Elliot was the daughter of George Roy Elliott, professor of English at Amherst College and a literary critic who specialized in Shakespeare. Also in 1942 Kleene left Amherst College to undertake war service with the US Navy as a navigation instructor at the Naval Reserve Midshipmen's School in New York. Later he was a project director at the Naval Research Laboratory in Washington DC. By the time he left the navy after the end of World War II, Kleene had risen to the rank of lieutenant commander.

He returned to the University of Wisconsin at Madison in 1946 as an associate professor being promoted to full professor two years later. In 1964 he was named Cyrus C Duffee Professor and continued to hold that chair until he retired in 1979. He served two terms as the Chair of the Department of Mathematics and one term as the Chair of the Department of Numerical Analysis (later renamed the Department of Computer Science). He also served as Dean of the College of Letters and Science in 1969-74. During his years at the University of Wisconsin he was thesis advisor to 13 Ph.D. students.

In 1970 Kleene's wife Nancy died. Eight years later he remarried Jeanne Steinmetz. He died of pneumonia aged 85.

Kleene's research was on the theory of algorithms and recursive function theory, an area which he created and retained an interest in throughout his life. He developed the field of recursion theory with Church, Gödel, Turing and others. He contributed to mathematical Intuitionism which had been founded by Brouwer. In particular he lectured on Recursive functions and intuitionistic mathematics at the International Congress of Mathematicians in Cambridge, Massachusetts, in 1950. In this lecture he spoke about how his interpretation of intuitionistic number theory by means of a "realization" might extend to intuitionistic set theory. He explored these ideas further in the book The foundations of intuitionistic mathematics, especially in relation to recursive functions (1965) written jointly with Richard Vesley. Chapters I, II, and IV of the book were written by Kleene while Chapter III is by Vesley. G Kreisel writes in a review:-

Chapter I is by far the best introduction to intuitionistic logic which is at present available for a mathematical logician.

Michael Dummett, a leading authority, was also greatly impressed:-

... chapter one of this book provides the first systematic exposition of the foundations of intuitionist analysis set out as an axiomatic system treating Brouwer's fan theorem, the bar theorem, and the continuity principle (called Brouwer's principle). In these respects, this was far superior to the earlier well-known axiomatization by Heyting.

Kleene's work on recursion theory helped to provide the foundations of theoretical computer science. By providing methods of determining which problems are soluble, Kleene's work led to the study of which functions can be computed. He spent the summer of 1951 at the RAND Corporation and discovered an important characterisation of finite automata. His RAND report on his work that summer has been very influential for theoretical computer science.

At a lecture in the University of Chicago in 1995, Robert Soare described Kleene's work in these terms:-

Kleene's formulation of computable function via six schemata is one of the most succinct and useful, and his previous work on lambda functions played a major role in supporting Church's Thesis that these classes coincide with the intuitively calculable functions.

From 1930's on Kleene more than any other mathematician developed the notions of computability and effective process in all their forms both abstract and concrete, both mathematical and philosophical. He tended to lay the foundations for an area and then move on to the next, as each successive one blossomed into a major research area in his wake.

Kleene developed a diverse array of topics in computability: the arithmetical hierarchy, degrees of computability, computable ordinals and hyperarithmetic theory, finite automata and regular sets with enormous consequences for computer science, computability on higher types, recursive realizability for intuitionistic arithmetic with consequences for philosphy and for program correctness in computer science.

Kleene's best known books are Introduction to Metamathematics (1952) and Mathematical Logic (1967). Kleene writes in the first of these:-

The aim of this book is to provide a connected introduction to the subjects of mathematical logic and recursive functions in particular, and to the newer foundational investigations in general.

J R Shoenfield, reviewing the second, writes:-

The growth of the undergraduate curriculum in mathematics has given rise to a number of texts treating serious mathematical results in a slower and more detailed manner than is customary in graduate texts. Since the number of such texts in the field of logic is quite small, this book by an outstanding authority in the field is especially welcome. ... The author clearly feels that a fairly thorough treatment of a few topics is preferable to a little bit of everything. ... Difficult proofs are broken down into a large number of simple cases; some of these cases are usually left to the reader. There are many illuminating examples; the author is usually more interested in giving enough examples to illustrate the important points of the proof than in giving complete details of the proof. Clarity and simplicity are never sacrificed for elegance. Historical notes and bibliographical references are frequent, but are not allowed to overshadow the mathematics.

Among the awards and honours that Kleene received for his outstanding contributions we mention the Leroy P Steele Prize which he was awarded by the American Mathematical Society in 1983:-

.. for three important papers which formed the basis for later developments in generalized recursion theory and descriptive set theory "Arithmetical predicates and function quantifiers", "On the forms of the predicates in the theory of constructive ordinals (second paper)", and "Hierarchies of number-theoretic predicates".

Perhaps his most prestigious award was the National Medal of Science presented by President Bush at a White House East Room Ceremony on 13 November 1990:-

For his leadership in the theory of recursion and effective computability and for developing it into a deep and broad field of mathematical research.

Other honours included election to the National Academy of Sciences (1969), election as President of the Association for Symbolic Logic (1956-58), president of the International Union of the History and the Philosophy of Science (1961) and of the Union's Division of Logic, Methodology and Philosophy of Science (1960-62). He was editor of the Journal of Symbolic Logic for twelve years.

In [3], Keisler describes Kleene's interests outside mathematics:-

Kleene had a strong interest in nature and the environment and visited his family farm in Maine almost every summer. He discovered a variety of butterfly Brenthis Bellona ab. Kleenei which was named after him. He was an avid climber and, until well into his seventies, led the biannual logic picnic at Madison (now the Kleene Memorial Logic Picnic) on hikes up the cliffs at Devil's Lake. Steve Kleene's knowledge of mushrooms was legendary.

Mac Lane. in [5], recounts an experience climbing with Kleene:-

In 1949 he and I, about to attend a meeting at Dartmouth of the American Mathematical Society, got together on a project to climb all the peaks in the Presidential range, and we were joined by a third climber, a vacationing bellhop. Then, as we three stood finally on top of the last peak (Mount Madison), a thunderstorm struck. Kleene bounded down from the peak shouting, "Get down; it's lightning." I stepped down a bit, searched for our third companion only to find him flat on the ground, unconscious. Steve (quicker by way of his height) went down to the Madison Pass hut for help. When we got the injured man there we found that the lightning had singed his scalp and left a blister on his foot opposite a hobnail. We ferried him to a hospital, and I (complete with a burned-out seat of the pants) notified his employer at the elegant Eastern Slopes Inn. He later recovered. Steve and I continued to climb assorted mountains and Steve kept on rock climbing.

Keisler [3] also describes his personality:-

Although a private man, he was a skilful and enthusiastic teller of anecdotes. He possessed a powerful voice that always made it possible for others to know without seeing him whether he was in the maths building.