哈斯凯尔·加里Haskell Curry

哈斯凯尔·加里的母亲是Anna Baright，父亲是皮埃尔·萨米埃尔 Silas 加里。萨米埃尔是波士顿表达学校的校长，Anna是该校的教务长。加里在高中时并未对数学表现出特别的兴趣，1916年毕业时他完全打算学医。他进入哈佛学院，即哈佛大学的本科部，在攻读医学学位的第一年学习中修了一门数学课程。

对他的研究方向产生重大影响的是美国在1917年春季参加第一次世界大战。加里想为国效力，并认为如果他接受数学训练，而不是继续他当时所学的医学预科课程，他更有可能参战。无论如何，他很喜欢自己所修的数学课程，并且在这门课程中表现得非常出色。他把主修科目改为数学，然后于1918年10月18日加入学生陆军训练团。然而，战争在此之后不久（11月）就结束了，1918年12月9日加里离开了军队。不过，他继续在哈佛学习数学课程，并于1920年毕业，获得A.B.学位。

加里现在决定要从事电气工程方面的职业，他在通用电气公司找了一份工作，这使他能够在麻省理工学院兼职学习电气工程。然而他很快发现，自己对课程的态度与其他人不同，因为他想知道一个结果为什么是正确的，而对其他人来说，结果正确与否才是唯一重要的。意识到自己更适合纯科学而非应用科学后，他于1922年转学物理。此外，哈佛似乎是学习纯科学的更好地方，因此他回到了那里，并已被任命为P W Bridgeman的研究助理半职，任期是1922-23学年。加里于1924年在哈佛大学获得物理学硕士学位，但此时他意识到适合自己的学科不是物理学，而是数学。他开始在哈佛大学攻读数学博士学位的研究。

在这段不断变换研究课题的时期，加里还有别的事情要忙。他的父亲于1921年去世，加里成了父亲遗产的受托人。当然，这份遗产的主要部分是波士顿的表达学校；在加里的母亲于1924年去世三年后，该校于1927年成为一家合法注册的公司。从表达公司成立之时起，加里就担任司库，但该公司于1928年被出售。

如果人们以为从1924年加里在哈佛开始攻读数学博士学位时，他终于找到了适合自己的课题，那就错了。他从乔治·戴维·伯克霍夫那里得到了一个微分方程理论方面的课题，但他开始阅读逻辑学方面的书籍，这些书在他看来比他的研究课题有趣得多。他询问了哈佛的多位教员，以及麻省理工学院的诺伯特·维纳，问他们是否认为他可以改做逻辑学研究。他们相当一致地劝他不要这样做。1926-27学年第一学期，他被哈佛聘为半时数学讲师，大约就在这个时候，他读了阿尔弗雷德·诺思·怀特海的第一卷和伯特兰·罗素的Principia Mathematica，后者出版于1910年。这对他的发展至关重要，因为正是在读了这部著作之后，他产生了用组合子来分析文本第一部分所特有的复杂替换规则的想法。他再次找到哈佛的多位教员，以及麻省理工学院的诺伯特·维纳，问他们是否认为他可以写一篇关于逻辑学的学位论文。这次他得到的回应与之前截然不同。诺伯特·维纳的回答很典型——除非你有话要说，否则避开逻辑学，但现在你当然有话要说了！

加里此时做出了他最后一次方向转变，决定放弃微分方程方面的博士研究，转而写一篇关于逻辑学的学位论文。在开始这个新课题的研究之前，他决定先教一年书，凭借乔治·戴维·伯克霍夫的强力推荐，他被聘为普林斯顿1927-28学年的数学讲师。在那里，他与奥斯瓦尔德·维布伦讨论了自己的研究计划，并且在普林斯顿图书馆翻阅Mathematische Annalen中的论文时，发现了M Schönfinkel 1924年的一篇论文Über die Bausteine der mathematischen Logik，该论文使用组合子的方式与他自己的一些想法相似。奥斯瓦尔德·维布伦向加里保证，这是一个积极的而非消极的发现，而且在詹姆斯·韦德尔·亚历山大告知他Schönfinkel正在精神病院、因此不会继续其研究路线之后，加里就谁最适合做博士导师征求了意见。奥斯瓦尔德·维布伦建议他，德国哥廷根的保罗·贝尔奈斯最合适。为了提高获得资助的机会，加里把自己的组合子想法写出来准备发表，这成了他的第一篇论文An analysis of logical substitution，于1929年发表在American Journal of Mathematics上。

在动身前往哥廷根之前，加里与Mary Virginia Wheatley结婚，他是在表达学校认识她的，当时她是那里的学生。他们于1928年7月3日结婚，并一起前往德国。差不多整整一年之后（1929年7月24日），他参加了题为Grundlagen der kombinatorischen Logik的学位论文答辩。形式上由大卫·希尔伯特指导他，但实际上是为他的工作提供日常支持的是保罗·贝尔奈斯。他的学位论文于1930年发表在American Journal of Mathematics上。

回到美国后，加里于1929年9月被聘到宾夕法尼亚州立学院（现宾夕法尼亚州立大学）。加里和Virginia此时开始了他们的家庭生活，Anne Wright 加里生于1930年7月27日，Robert Wheatley 加里生于1934年7月6日。大萧条始于1929年，所以加里能在那个时候获得职位是幸运的。大萧条的那些年，一个数理逻辑学家几乎不可能获得职位。尽管他一直留在宾夕法尼亚州立大学任教，直到1966年退休，但他确实在其他机构度过了一些时间，特别是在芝加哥大学，他在1931-32年间是国家研究委员会研究员，以及1938-39年间在普林斯顿的高等爱德华·斯图迪研究所。他早期研究生涯中发表的一些论文包括The universal quantifier in combinatory logic（1931）、Some additions to the theory of combinators（1932）、Apparent variables from the standpoint of combinatory logic（1933）和Some properties of equality and implication in combinatory logic（1934）。

符号逻辑协会于1936年成立，加里是创始人之一。他在1936-37年间任副主席，然后在1938-40年间任协会主席。他的卸任主席演讲The combinatory foundations of mathematical logic于1942年发表在Journal of Symbolic Logic上。在非常清晰地阐述了组合逻辑的基础、展示了它与丘奇发展的λ演算的密切关系之后，加里接着描述了他最近的工作。他考察了在不一致逻辑系统中推导悖论（如Richard和伯特兰·罗素的悖论）的简化方法，并且还发展了一种方法，将概括性或形式蕴涵等未定义的普遍性概念引入组合逻辑，使得像丘奇和Rosser那样的相容性定理仍然可以推导出来。

20世纪40年代初，加里已成为世界领先的数理逻辑学家之一。他被邀请向数学家们作一次阐述性演讲，解释形式主义的基本概念并提出新的建议。1941年发表在Bulletin of the American Mathematical Society上的论文Some aspects of the problem of mathematical rigor就是这次演讲的文本。他在演讲中提出了：对非形式理论的批判；形式系统的概念（以伦纳德·尤金·迪克森的群公理为例说明）；演算的概念；对元理论的讨论；数学的定义；以及形式系统的可接受性，并讨论了直觉主义者和形式主义者的批评。20世纪40年代期间，加里还重新与波士顿的表达学校建立了联系，此时该校已更名为加里学院。他于1940年加入该学院的董事会，并在董事会任职十余年。

加里从事应用数学研究。他于1943年发表了The Heaviside operational calculus。在该文中，他提出了一种非常简单的代数方法，但也意识到其局限性，写道：

……这一优势当然意味着对处理范围的限制，因为它仅限于常系数常线性微分方程所产生的有理方面。对于偏微分方程、分数阶算子等更一般的情形，积分变换理论无疑是不可避免的。

他于1942年5月至1944年1月在弗兰克福德兵工厂工作，随后在约翰斯威廉·霍普金斯大学应用物理实验室工作至1945年3月。之后他前往阿伯丁试验场，这是位于马里兰州东北部哈福德县的一个军事武器试验场。在那里，他参与了ENIAC计算机的工作，并于1946年发表了A study of inverse interpolation on the ENIAC和A study of fourth order interpolation on the ENIAC。他于1946年9月回到宾夕法尼亚州立大学，试图说服大学当局购置一台计算机，但未能成功。

他的主要著作包括Combinatory Logic（1958年，与Robert Feys合著）和Foundations of Mathematical Logic（1963年）。加里于1950年开始撰写Combinatory Logic，当时他获得了富布赖特资助，得以在鲁汶与Robert Feys合作。加里回到美国后，他们继续合作撰写此书，并于1956年完成。E J Cogan在评论此书时，对组合逻辑作了很好的描述：

组合逻辑关注的是数学基础中某些基本概念，这些概念通常以直觉的、未经分析的方式使用。这类概念包括替换——通常通过使用变量引入——以及将系统中的实体分类为类型——通常由辅助于系统但并非系统一部分的规则来提供。组合逻辑中关注诸如替换这类涉及变量的根本性质问题的部分，称为组合子理论。

在Foundations of Mathematical Logic中，加里从代数基础出发，运用格哈德·根岑的方法发展了这一主题。J Tucker写道：-

这种方法最显著的效果是，首先处理蕴涵、合取和析取的有限肯定运算，而否定和量化则在后面的单独章节中引入。每个基本联结词的意义不像经典方法那样在开头就规定好，而是通过推理规则推导出来。

1966年，他接受了阿姆斯特丹的逻辑学、逻辑史和科学哲学教授职位。他担任该职位四年，之后回到宾夕法尼亚州州学院居住。

[3]的作者们对加里和他的妻子作了一些很好的评论：-

认识Curry夫妇的人都知道他们总是多么友好和乐于助人。加里对同事和学生所做的，一直远不止是重要思想的来源。他总是愿意倾听每一个想和他交谈的人，讨论他们的想法，并给予他所能给予的鼓励。……他的办公室门总是敞开的。这无疑对我们在组合逻辑领域工作的许多人的热情做出了重要贡献。Curry夫妇无论住在哪里，他们所表现出的好客也是众所周知的。总是有许多聚会和其他不那么正式的集会，我们猜想Virginia的烹饪也对组合逻辑兴趣的增长起了一定作用。

Haskell Curry's mother was Anna Baright and his father was Samuel Silas Curry. Samuel was the president of the School of Expression in Boston and Anna was the Dean of the School. Haskell did not show particular interest in mathematics when at high school and when he graduated in 1916 he fully intended to study medicine. He entered Harvard College, the undergraduate school of Harvard University, and took a mathematics course in his first year of study as part of his studies towards a degree in medicine.

A major influence on the direction that his studies took was the entry of the United States into World War I in the spring of 1917. Curry wanted to serve his country, and decided that he would be more likely to see action if he had a mathematics training rather than if he continued the pre-medical course he was on. Anyhow he had enjoyed the mathematics course he had taken and had done very well in the course. He changed his major subject to mathematics and then enlisted in the Student Army Training Corps on 18 October 1918. The war, however, ended shortly after this (in November) and on 9 December 1918 Curry left the army. He continued on the mathematics course at Harvard, however, and graduated in 1920 with an A.B. degree.

Curry now decided that he would look for a career in electrical engineering and he took a job with the General Electric Company which allowed him to study electrical engineering part-time at the Massachusetts Institute of Technology. However he soon discovered that he had a different attitude to the others taking the courses, for he wanted to know why a result was correct when for everyone else it only mattered that it was correct. Realising that he was more suited to pure science than to applied science, he changed course to study physics in 1922. Also Harvard seemed a better place to study pure science so he returned there, having been appointed to a half-post as a research assistant of P W Bridgeman for the session 1922-23. Curry graduated with a Master's Degree in physics from Harvard in 1924 but by now he realised that the subject for him was not physics but it was mathematics. He began to undertake research for his doctorate in mathematics at Harvard.

Throughout this period of changing topics Curry had other things to keep him busy. His father had died in 1921 and Curry became a trustee of his father's estate. Of course the main part of this estate was the School of Expression in Boston and, three years after the death of Curry's mother in 1924, it became a legal corporation in 1927. Curry acted as treasurer to the Expression Company from the time it was formed, but it was sold in 1928.

If one imagines that from 1924 when Curry embarked on his doctorate in mathematics at Harvard he at last found the topic for him, then one would be mistaken. He was given a topic in the theory of differential equations by George Birkhoff but he began reading books on logic which seemed to him far more interesting that his research topic. He asked various faculty members at Harvard, and Norbert Wiener at MIT, if they thought he might change to undertake research in logic. They were quite unanimous, advising him against it. He was employed as a half-time instructor in mathematics by Harvard during the first semester of 1926-27 and it was around this time that he read the first volume of Whitehead and Russell's Principia Mathematica which had been published in 1910. This was fundamental in his development, for it was after reading this work that he had the idea to use combinators to analyse the complicated rules of substitution which characterise the first part of the text. He again approached various faculty members at Harvard, and Norbert Wiener at MIT, asking whether they thought that he might write his doctoral dissertation on logic. He now got a very different response from what he had received earlier. Wiener's reply was typical - avoid logic unless you have something to say, but now you certainly have something to say!

Curry now made his final change in direction and decided to give up his doctoral studies on differential equations and to write a doctoral dissertation on logic. Before beginning research on this new topic he decided to teach for a year and, with a strong recommendation from Birkhoff, was appointed as an instructor in mathematics at Princeton for session 1927-28. There he discussed his research plans with Veblen and, while looking at papers in Mathematische Annalen in the Princeton library, discovered a 1924 paper by M Schönfinkel Über die Bausteine der mathematischen Logik which used combinators in a similar way to his own ideas. Veblen assured Curry that this was a positive, rather than negative, discovery and, after Alexander had informed him that Schönfinkel was in a mental hospital and therefore not continuing his line of research, Curry sought advice on who would be the best Ph.D. supervisor. Veblen advised him that Bernays at Göttingen, in Germany, would be best. In order to improve his chances of financial support, Curry wrote up his ideas on combinators for publication and this became his first paper An analysis of logical substitution which appeared in the American Journal of Mathematics in 1929.

Before setting off for Göttingen, Curry married Mary Virginia Wheatley whom he had met at the School of Expression when she was a student there. They married on 3 July 1928 and travelled together to Germany. After almost exactly a year (on 24 July 1929) he was examined on his thesis entitled Grundlagen der kombinatorischen Logik. Formally he was supervised by Hilbert, but in fact it was Bernays who provided day to day support for his work. His dissertation was published in the American Journal of Mathematics in 1930.

Returning to the United States, Curry was appointed to State College, Pennsylvania (now Pennsylvania State University) in September 1929. Haskell and Virginia began their family at this time with Anne Wright Curry born on 27 July 1930 and Robert Wheatley Curry born on 6 July 1934. The Great Depression began in 1929 so it was fortunate that Curry obtained his position when he did. Certainly the years of the Great Depression would have been ones when a mathematical logician would hardly have been likely to obtain a post. Although he remained on the faculty at Pennsylvania State until he retired in 1966, he did spend time at other institutions, particularly at the University of Chicago, where he was a National Research Council Fellow during 1931-32, and the Institute for Advanced Study at Princeton during 1938-39. Some papers published during the early years of his research include The universal quantifier in combinatory logic (1931), Some additions to the theory of combinators (1932), Apparent variables from the standpoint of combinatory logic (1933), and Some properties of equality and implication in combinatory logic (1934).

The Association for Symbolic Logic was founded in 1936 with Curry as one of the founders. He was vice-president during 1936-37 and then president of the Association in 1938-40. His retiring presidential address The combinatory foundations of mathematical logic was published in the Journal of Symbolic Logic in 1942. After giving a very clear exposition of the fundamentals of combinatory logic, showing its close relationship to the λ-calculus developed by Church, Curry went on to describe his recent work. He had examined simplified methods of deriving the paradoxes (such as those of Richard and Russell) in systems of logic which are inconsistent, and had also developed a method of introducing into combinatory logic undefined notions of generality, such as quantification or formal implication, in such a way that a consistency theorem like that of Church and Rosser could still be derived.

As the 1940s began, Curry had reached the position of being one of the leading mathematical logicians in the world. He was asked to give an expository address to mathematicians to explain the fundamental concepts of formalism and to add new suggestions. The paper Some aspects of the problem of mathematical rigor published in the Bulletin of the American Mathematical Society in 1941 is the text of this address. He presented in his address: a critique of non-formal theories; the notion of a formal system (illustrated by Dickson's postulates for a group); the notion of a calculus; discussion of a metatheory; the definition of mathematics; and acceptability of a formal system, discussing criticisms from intuitionists and formalists. During the 1940s Curry also renewed his association with the School of Expression in Boston, which by this time had been renamed Curry College. He joined the Board of Trustees of the College in 1940 and remained on the Board for over a decade.

During World War II Curry undertook research in applied mathematics. He published The Heaviside operational calculus in 1943. In it he presented a very simple algebraic approach but was aware of its limitations writing:-

... this advantage, of course, implies a restriction on the scope of the treatment, because it is limited to the rational aspects such as arise from ordinary linear differential equations with constant coefficients. For the more general cases of partial differential equations, fractional operators, etc., the theory of integral transforms is doubtless unavoidable.

He worked at the Frankford Arsenal from May 1942 to January 1944, then at the Applied Physics Laboratories at Johns Hopkins University until March 1945. He then went to the Aberdeen Proving Ground, a military weapons testing site in Harford county northeastern Maryland. There he became involved with the ENIAC computer publishing A study of inverse interpolation on the ENIAC and A study of fourth order interpolation on the ENIAC both in 1946. He returned to Pennsylvania State University in September 1946 and tried to persuade the University authorities to acquire a computer, but he failed.

His major texts include Combinatory Logic (1958) (with Robert Feys), and Foundations of Mathematical Logic (1963). Curry began working on Combinatory Logic in 1950 when he was awarded a Fulbright Grant that enabled him to work with Robert Feys at Louvain. They continued collaborating on the book after Curry returned to the United States and completed the text in 1956. E J Cogan, reviewing the book, gives a nice description of combinatory logic:-

Combinatory logic is concerned with certain basic notions of the foundations of mathematics which are usually used in an intuitive and unanalysed way. Such notions include substitution, usually introduced by the use of variables, and classification of the entities of a system into types, which is usually provided for by rules which are auxiliary to, but not part of, the system. The part of combinatory logic which is concerned with questions of a fundamental nature which, like substitution, involve variables, is called the theory of combinators.

In Foundations of Mathematical Logic Curry develops the topic from an algebraic basis using Gentzen's methods. J Tucker writes:-

The most striking effect of this approach is that the finite positive operations of implication, conjunction and alternation are dealt with first, while negation and quantification are brought in later and in separate chapters. The meaning of each fundamental connective is not laid down at the beginning as it is in the classical approach, but is derived by inferential rules.

In 1966 he accepted the position of Professor of Logic, History of Logic, and Philosophy of Science at Amsterdam. He held this position for four years after which he returned to live in State College, Pennsylvania.

The authors of [3] make some nice comments about Curry and his wife:-

Everyone who knows the Currys is aware of how friendly and helpful they always are. Haskell has always done more for colleagues and students than be a source of important ideas. He has always been willing to listen to everyone who wanted to talk to him, to discuss their ideas, and to give whatever encouragement that he could. ... His office door has always been open. And this has undoubtedly been an important contribution to the enthusiasm of many of those of us working in combinatory logic. Also well known wherever the Currys have lived has been the hospitality they have shown. There are always many parties and other, less formal gatherings, and we conjecture that Virginia's cooking has also played a role in the growth of interest in combinatory logic.