雷蒙·梅里儿·思木里安Raymond Smullyan

雷蒙·梅里儿·思木里安，被称为Ray，在纽约市的Far Rockaway长大。理查德·费曼也在思木里安出生前几个月出生在Far Rockaway。在[1]中，他讲述了自己六岁时接触逻辑谜题的经历：-

1925年4月1日，我卧病在床……早上，我的哥哥Emile（比我大十岁）走进我的卧室说：“好吧，思木里安，今天是愚人节，我要像你从未被愚弄过那样愚弄你！”我整天等着他愚弄我，但他没有。

Emile没有骗他，却把他骗了！思木里安写道[1]：——

我记得熄灯后很久还躺在床上，寻思自己究竟有没有真的被骗。

思木里安小时候既爱音乐又爱科学，而且在音乐上极有天赋。十二岁时，他在一场钢琴比赛中赢得金牌，看起来他会把音乐当作自己的事业。

思木里安十三岁时，全家搬到了曼哈顿。他在那里就读于布朗克斯区的Theodore Roosevelt High School，选择这所学校是因为它提供特别音乐课程，非常适合他的音乐目标。然而，这所学校最终并没有给思木里安他想要的东西。是的，他热切地热爱音乐，但他还有另一种热情，那就是数学。他想学习群、环和域、数学基础以及数理逻辑。Theodore Roosevelt High School没有给他这些，所以他离开学校自学。

几年的学习当然使他处于有利地位，可以参加大学董事会考试，他参加了考试并进入了俄勒冈州的Pacific College。不久思木里安转学到Reed College，然后他去了旧金山学习钢琴。看起来他在人生的这个阶段完全困惑于该学数学还是音乐，即使他已经在心里解决了这个问题，他似乎也没有发现学院和大学的传统教学方法合他的心意。

从旧金山回到纽约后，思木里安自学数学和逻辑，正是在这时他开始创作国际象棋排局。其实，他十六岁时就创作了第一道国际象棋排局，那是传统类型的排局，“白方走棋，两步将死”[3]：——

……那是一道传统的两步杀。我把它拿给几位年长的朋友看。其中一位说……“如果我来创作一道国际象棋题……那就要推断出棋局早先发生了什么”。我觉得这个想法很吸引人，立刻动手，创作了一道逆向分析题。

思木里安当时还没听说过逆向分析，而这类国际象棋题确实存在。它们是必须倒推求解的谜题。比如给出一个棋局局面，在其中一个格子上画个问号。题目要求找出那个格子上必然是什么缺失的棋子。听起来这种题似乎无法求解，而这恰恰是思木里安喜欢的那类题：有唯一解，却看上去完全不可能。在纽约的这段时间里，思木里安创作了许多逆向分析的国际象棋题，后来这些题被用在他关于这一主题的两本书[2]和[3]中。

此时他在纽约不仅研究数学和创作国际象棋排局，还学习了魔术表演，成为了一名非常出色的魔术师。1943年，他重返正规教育，进入威斯康星大学。在那里学习一年后，他搬到芝加哥，开始在大学听课，但仅一个学期后就放弃了。他继续自学，并通过在芝加哥罗斯福学院教授音乐来谋生。

随后他回到纽约，在那里度过了两年。在这几年里，他通过在格林威治村的夜总会表演魔术来赚钱。1949年，他回到芝加哥，在大学里修读各种课程，并在城里表演魔术以谋生。事实上，他的魔术表演非常受欢迎，思木里安虽然本质上是个害羞的人，但以“五张A”威妮弗雷德·埃杰顿·梅里尔的身份，配以令人捧腹的滑稽台词，呈现了一场精彩、有趣且富有娱乐性的表演。然而[1]：-

……我的魔术生意曾短暂低迷，我不得不想办法补充收入。我决定尝试找一份推销员的工作。我向一家吸尘器公司提出了申请……

到1954年，他仍在芝加哥，从事研究生水平的研究，但仍未积累到获得第一个学位所需的足够学分。

思木里安在芝加哥大学的老师之一是鲁道夫·卡尔纳普，著名的逻辑学家和逻辑实证主义哲学家。他现在推荐思木里安去新罕布什尔州汉诺威的文理学院达特茅斯学院担任数学职位。思木里安此时没有正式资格，但已经在为未来的出版物进行数学研究。他从1954年到1956年在达特茅斯学院任教，并于1955年获得芝加哥大学的学士学位。他从未完成足够的课程以获得该学位，但为了凑够数量，芝加哥大学将他从未上过但正在教授的微积分课程计入了他的学分。

思木里安于1957年在Journal of Symbolic Logic上发表了Languages in which self reference is possible。次年，Undecidability and recursive inseparability问世，证明了算术中两个不可判定性的结果，其中一个是由保罗·贝尔奈斯提出的。当这些文章中的第二篇发表时，思木里安正在普林斯顿大学在丘奇指导下攻读博士学位。他于1957年入学，1959年获得博士学位。1958年被任命为普林斯顿的职位，他在那里工作到1961年。

在此期间，他发表了多篇数学文章。与希拉里·普特南合著的Exact separation of recursively enumerable sets within theories于1960年发表，而思木里安也在同年发表了Theories with effectively inseparable nuclei，然后在1961年发表了三篇论文Extended canonical systems; Elementary formal systems和Monadic elementary formal systems。

1961年，他还出版了由普林斯顿大学出版社出版的专著Theory of formal systems。格奥尔格·克里泽尔在评论该书时说，它给出了：-

……现存对递归可枚举（r.e.）集理论最优雅的阐述。……所有关于r.e.集的著名结果都给出了，包括变体和改进……[有]对先前阐述的显著改进……

1957年，当思木里安还是普林斯顿大学的研究生时，他向一位研究生同学展示了他的一些国际象棋谜题，这位同学[2]：-

……提供了许多有用的建议。

事实上，这名研究生把其中一个谜题寄给了他在英格兰的父亲，父亲又把它寄给了Manchester Guardian，报纸将其刊登。Guardian此前不知道作者是谁，当思木里安联系他们时，他们很乐意承认他的作者身份，并发表更多他的国际象棋问题。

1961年，思木里安被任命到纽约的犹太叶史瓦大学，在那里任教至1968年，之后他转到雷曼学院，该学院前身是亨特学院的布朗克斯校区，同年加入了纽约市立大学。从1982年起，他成为纽约市立大学雷曼学院和研究生中心的荣休教授。随后，他被任命为印第安纳大学的奥斯卡·尤因哲学讲席教授。

思木里安的出版物相当卓越，包括两本关于逆向分析棋题的杰出著作[2]和[3]，一系列精彩的普及谜题书如[1]和[4]，以及一些关于数学基础和数理逻辑的书籍，这些书在许多方面都独树一帜。

这些谜题书向公众提供了一种愉快的入门，引导他们了解数学基础中一些最深刻的思想。例如，[1]一书在封面上被描述如下：——

从充满乐趣的猴子把戏和带有刁钻新转折的经典脑筋急转弯开始，思木里安教授编织了一个逻辑迷宫，其中包含更加复杂和具有挑战性的问题，他深入探讨了逻辑和集合论中一些最深刻的悖论，包括库尔特·弗雷德里希·哥德尔的革命性不可判定性定理。

马丁·加德纳在Scientific American中这样描述这本书：——

有史以来写得最原创、最深刻、最幽默的消遣逻辑与数学问题合集。

在他的谜题书4中，思木里安写道，他给出了：-

……一次对无穷的导览，解释了伟大数学家格奥尔格·康托尔的开创性发现，他是第一个将这一主题置于逻辑上可靠基础之上的人。……令人惊叹的是，整个迷人的无穷主题对公众竟如此鲜为人知！为什么中学不教它呢？它并不比代数或几何更难理解，而且收获如此之大！

我们在上文提到了他的一本关于数理逻辑的书。他在1968年出版了另一部教材First-order logic：-

本书主要讨论一阶逻辑完备性定理各种表述形式的证明及其相互联系。……本书兼具优雅与清晰、详细的阐述；一个好学生几乎不需要老师就能读懂它。

1992年，他出版了Gödel's incompleteness theorems。思木里安在序言中解释说，他写这本书是为了：-

……面向一般数学家、哲学家、计算机科学家以及任何其他好奇的读者，只要他们对一阶逻辑的符号体系略知一二，并能识别几个基本公式的逻辑有效性。一门标准的为期一学期的数理逻辑课程，对于理解本卷来说绰绰有余。

这本书是迅速接连问世的一系列教材中的第一本。1993年，他出版了Recursion theory for metamathematics，它是上文所述他1992年教材的续篇。该系列的第三卷Diagonalization and self-reference于1994年出版，它把一个非常困难的专题讲述得既易于理解又引人入胜。

1996年，思木里安与Melvin汉斯·费汀合著了Set theory and the continuum problem。Plotkin在评论这本书时写道：-

一致性证明与独立性证明就其本质而言是琐细、形式化且高度技术性的。两位作者写得极为清晰，令人钦佩。书中关于可数与不可数以及数学归纳法的一些段落确实非常迷人……读者能感受到作者力求表述优雅与内容完整的意志。

思木里安这本书的合著者描述了思木里安的工作方式[5]：-

有些人会同时做几件事。[思木里安]不是这样。Ray的工作向来是一段一段的。他会对某件事产生兴趣，然后或多或少放弃其他一切。他曾在某个时候写了一篇文章，然后在接下来几年里，出现了大量文章。……文章阶段之后，他开始做智力题。接下来两三年里，一切都是智力题。它们逐渐进入了他的所有工作。现在他又回到了数学，但智力题的元素仍然在那里。

作为一名教师，思木里安的风格与大多数讲课者不同。Mothner[5]写道：-

在课堂上，思木里安 绝非悠闲或安静。……我观看他教授一门研究生层次的逻辑课程，他摇摇晃晃地走向黑板（在那里他用可辨认的笔迹和完整的句子书写），在讲桌旁踱步，坐立不安，轻声笑着。对于似乎让学生们更加困惑而非觉得有趣的问题，他会突然发出一阵轻微的咝咝笑声。上课前，他试图活跃课堂气氛，抛出一些简单的谜题……

最后让我们提一下，他还有一个进一步的爱好，即天文学。他喜欢用望远镜观察，并且自己磨制了六英寸的镜片。

Raymond Smullyan, known as Ray, was brought up in Far Rockaway in New York City. Richard Feynman had also been born in Far Rockaway just months before Smullyan. In [1] he recounts his introduction to logical puzzles when he was six years old:-

On 1 April 1925, I was sick in bed ... In the morning my brother Emile (ten years my senior) came into my bedroom and said: "Well, Raymond, today is April Fool's Day, and I will fool you as you have never been fooled before!" I waited all day for him to fool me, but he didn't.

Emile had fooled him by not fooling him! Smullyan writes [1]:-

I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled.

As a young boy Smullyan loved both music and science and he was extremely talented musically. When he was twelve years old he won a gold medal in a piano competition and it looked as if he would make music his career.

When Smullyan was thirteen years old his family moved to Manhattan. There he attended the Theodore Roosevelt High School in the Bronx, this school being chosen because it offered special music courses which would be well suited to his musical aims. However, the school did not in the end give Smullyan what he wanted. Yes, he was passionately interested in music but he had another passion and that was mathematics. He wanted to learn about groups, rings and fields, the foundations of mathematics and mathematical logic. This the Theodore Roosevelt High School did not give him so he left the school to study on his own.

A few years of study certainly put him in a good position to sit the College Board examinations, which he did and entered Pacific College in Oregon. Soon Smullyan moved to Reed College, and then he went to San Francisco where he studied the piano. It looks as if he was totally confused at this stage of his life whether to study mathematics or music and even if he had sorted out this problem in his mind, he does not seem to have found that the conventional teaching methods in colleges and universities were to his liking.

Returning to New York from San Francisco, Smullyan studied mathematics and logic on his own and it was at this time that he began to compose chess puzzles. Well in fact he had composed his first chess puzzle at age sixteen and it was the conventional type of chess puzzle "white to play and mate in two moves" [3]:-

... it was a conventional two-mover. I showed it to several of my older friends. One of them said ... "if I were to compose a chess problem ... it would be to deduce what happened earlier in the game". This struck me as a fascinating idea, and I straightway set to work and composed a problem in retrograde analysis.

Although Smullyan had not heard of retrograde analysis at this time, such a field of chess problems did exist. They were puzzles where one has to work backwards. For example a chess position would be given and a question mark would be on one of the squares. The problem would be to find what the missing piece was that has to be on that square. It sounds as if such a problem could not be solved, and this is exactly the type of problem that Smullyan liked. Problems which had a unique solution, yet looked quite impossible. During this spell in New York, Smullyan composed many chess problems in retrograde analysis and they later were used in his two books on the topic [2] and [3].

Studying mathematics and composing chess problems were not the only things he did in New York at this time, for he also learnt to do magic tricks, becoming a very good magician. In 1943 he returned to formal education entering the University of Wisconsin. After studying there for a year he moved to Chicago where he began to take courses at the university but gave up after only one semester. He continued to study on his own and earned his living teaching music in Roosevelt College in Chicago.

He then returned to New York where he spent two years. During these years he earned money performing magic acts in nightclubs in Greenwich Village. In 1949 he returned to Chicago, took various courses at the university and performed his magic act around the town to earn his living. In fact his magic act was very popular and Smullyan, although basically a shy man, gave a wonderful amusing and entertaining act as Five Ace Merrill, with hilariously funny patter. However [1]:-

... my magic business was slow for a brief period and I had to supplement my income somehow. I decided to try getting a job as a salesman. I applied to a vacuum cleaner company ...

By 1954 he was still in Chicago, undertaking graduate level research but still not having amassed the right number of credits for the award of a first degree.

One of Smullyan's teachers at the University of Chicago had been Rudolf Carnap, the famous logician and philosopher of Logical Positivism. He now recommended Smullyan for a mathematics post at Dartmouth College, the liberal arts college in Hanover, New Hampshire. Smullyan had no formal qualifications at this time but was already working on mathematical research for future publications. He taught at Dartmouth College from 1954 until 1956, being awarded his B.S. from the University of Chicago in 1955. He had never completed sufficient courses to merit the award, but to make up the number Chicago credited him with a calculus course which he had never taken but was teaching.

Smullyan published Languages in which self reference is possible in the Journal of Symbolic Logic in 1957. In the following year Undecidability and recursive inseparability appeared which proves two results on undecidability in arithmetic, one of which had been suggested by Bernays. By the time the second of these articles appeared, Smullyan was at Princeton University working under Alonzo Church for his doctorate. He entered in 1957 and was awarded his Ph.D. in 1959. Appointed to a post at Princeton in 1958, he worked there until 1961.

He published several mathematical articles during this period. Exact separation of recursively enumerable sets within theories written jointly with Hilary Putnam was published in 1960 while Smullyan also published Theories with effectively inseparable nuclei in that year and then in 1961 the three papers Extended canonical systems; Elementary formal systems; and Monadic elementary formal systems.

In 1961 he also published the monograph Theory of formal systems published by Princeton University Press. Kreisel, reviewing the book says that it gives:-

... the most elegant exposition of the theory of recursively enumerable (r.e.) sets in existence. ... All the well-known results on r.e. sets are given, including variants and refinements ... [there is] striking improvement over previous expositions ...

In 1957, when Smullyan was a graduate student at Princeton he showed some of his chess puzzles to a fellow graduate student who [2]:-

... provided a host of helpful suggestions.

In fact this graduate student sent one of the puzzles to his father in England who in turn sent it to the Manchester Guardian and the newspaper published it. The Guardian had not known who the author was and, when Smullyan contacted them, they were pleased to acknowledge his authorship and to publish more of his chess problems.

In 1961 Smullyan was appointed to the Jewish Yeshiva University in New York where he taught until 1968 when he moved to Lehman College, formerly Hunter College's Bronx campus, which joined the City University of New York in that year. From 1982 he became Professor Emeritus of the City University of New York - Lehman College and Graduate Center. He was then appointed Oscar Ewing Professor of Philosophy at Indiana University.

Smullyan's publications have been quite remarkable with the two outstanding books on retrograde analysis chess problems [2] and [3], a whole series of marvellous popular puzzle books such as [1] and [4], and some books on the foundations of mathematics and mathematical logic which are in many ways in a class of their own.

The puzzle books present to the general public an enjoyable introduction to some of the deepest ideas in the foundations of mathematics. For example the book [1] is described on the cover as follows:-

Beginning with fun-filled monkey tricks and classic brain-teasers with devilish new twists, Professor Smullyan spins a logical labyrinth of even more complex and challenging problems as he delves into some of the deepest paradoxes of logic and set theory, including Gödel's revolutionary theorem of undecidability.

Martin Gardner described this book in Scientific American as:-

The most original, most profound and most humorous collection of recreational logic and mathematics problems ever written.

In his puzzle book [4] Smullyan writes that he gives:-

... a guided tour of Infinity, explaining the pioneering discoveries of the great mathematician Georg Cantor, who was the first to put the subject on a logically sound basis. ... it must be wondered at that the whole fascinating subject of Infinity is so little known to the general public! Why isn't it taught in high schools? It is no harder to understand than algebra or geometry, and it is so rewarding!

We have mentioned one of his books on mathematical logic above. He published another text First-order logic in 1968:-

This book deals primarily with the proofs of, and the interconnections between, various formulations of the completeness theorem for first-order logic. ... This book combines elegance with clear, detailed exposition; a good student should be able to read it almost without a teacher.

In 1992 he published Gödel's incompleteness theorems. Smullyan explains in the Preface that he has written the book:-

... for the general mathematician, philosopher, computer scientist and any other curious reader who has at least a nodding acquaintance with the symbolism of first-order logic, and who can recognize the logical validity of a few elementary formulas. A standard one-semester course in mathematical logic is more than enough for the understanding of this volume.

This book was the first of a series of texts which appeared in quick succession. In 1993 he published Recursion theory for metamathematics which is a sequel to his 1992 text described above. A third volume in the series Diagonalization and self-reference was published in 1994 and presents a very difficult topic in such a way as to make it both understandable and enjoyable.

In 1996 Smullyan co-authored with Melvin Fitting Set theory and the continuum problem. Plotkin, reviewing this book writes:-

Consistency and independence proofs are by their very nature picky, formal, and highly technical. The authors write with admirable lucidity. There are some truly charming set pieces on countability and uncountability and on mathematical induction ... The reader can feel the authorial will striving for elegance of presentation and completeness.

Melvin Fitting, Smullyan's co-author for this text, has described the way that Smullyan works [5]:-

Some people do things simultaneously. [Smullyan] doesn't. Ray has always been episodic in his work. He will get interested in something and more or less abandon everything else. He wrote an essay at one point and then, for the next couple of years, there was this enormous stream of essays. ... After the essay stage he started doing puzzles. For the next two or three years everything was puzzles. They were finding their way into all of his work. Now he's gone back to maths, but the puzzle element is still there.

As a teacher Smullyan's style is different from most lecturers. Mothner [5] writes:-

In the classroom, Smullyan is anything but leisurely or quiet. ... I watched him teach a graduate level logic course, as he lurched to the blackboard (where he writes in a serviceable hand and in complete sentences) and paced about his desk, fidgeting and chuckling. He would break into a small sibilant laugh at problems that seemed to leave his students more confused than amused. Before the class began, he tried to warm up the group, tossing out some simple puzzles ...

Finally let us mention that he had one further hobby, namely astronomy. He loved observing through his telescope, and he ground the six inch mirror himself.