**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:-

On1April1925, 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 has one further hobby, namely astronony. He loves observing through his telescope, and he ground the six inch mirror himself.

**Article by:** *J J O'Connor* and *E F Robertson*