Born: 14 June 1903 in Washington, D.C., USA
Died: 11 Aug 1995 in Hudson, Ohio, USA

Alonzo Church's parents were Mildred Hannah Letterman Parker and Samuel Robbins Church. His father was a judge. He was a student at Princeton receiving his first degree, an A.B., in 1924, then his doctorate three years later. His doctoral work was supervised by Veblen , and he was awarded his doctorate in 1927 for his dissertation entitled Alternatives to Zermelo's Assumption. While he was still working for his doctorate he married Mary Julia Kuczinski at Princeton in 1926. They had three children, Alonzo Jr, Mary Ann and Mildred.

Church spent two years as a National Research Fellow, one year at Harvard University then a year at Göttingen and Amsterdam. He returned to the United States becoming Assistant Professor of Mathematics at Princeton in 1929. Enderton writes in [ 4 ]:-

Princeton in the 1930's was an exciting place for logic. There was Church together with his students Rosser and Kleene. There was John von Neumann. Alan Turing, who had been thinking about the notion of effective calculability, came as a visiting graduate student in 1936 and stayed to complete his Ph.D. under Church. And Kurt Gödel visited the Institute for Advanced Study in 1933 and 1935, before moving there permanently.

He was promoted to Associate Professor in 1939 and to Professor in 1947, a post he held until 1961 when he became Professor of Mathematics and Philosophy. In 1967 he retired from Princeton and went to the University of California at Los Angeles as Kent Professor of Philosophy and Professor of Mathematics. He continued teaching and undertaking research at Los Angeles until 1990 when he retired again, twenty-three years after he first retired! In 1992 he moved from Los Angeles to Hudson, Ohio, where he lived out his final three years.

His work is of major importance in mathematical logic, recursion theory, and in theoretical computer science. Early contributions included the papers On irredundant sets of postulates (1925), On the form of differential equations of a system of paths (1926), and Alternatives to Zermelo's assumption (1927). He created the -calculus in the 1930's which today is an invaluable tool for computer scientists. The article [ 10 ] is in three parts and in the last of these Manzano:-

... attempt[s] to show that Church's great discovery was lambda calculus and that his remaining contributions were mainly inspired afterthoughts in the sense that most of his contributions, as well as some of his pupils', derive from that initial achievement.

In 1941 he published the 77 page book The Calculi of Lambda-Conversion as a volume of the Princeton University Press Annals of Mathematics Studies. It is effectively a rewritten and polished version of lectures Church gave in Princeton in 1936 on the -calculus.

Church is probably best remembered for 'Church's Theorem' and 'Church's Thesis' both of which first appeared in print in 1936. Church's Theorem, showing the undecidability of first order logic, appeared in A note on the Entscheidungsproblem published in the first issue of the Journal of Symbolic Logic. This, of course, is in contrast with the propositional calculus which has a decision procedure based on truth tables. Church's Theorem extends the incompleteness proof given of Gödel in 1931.

Church's Thesis appears in An unsolvable problem in elementarynumber theory published in the American Journal of Mathematics58 (1936), 345-363. In the paper he defines the notion of effective calculability and identifies it with the notion of a recursive function. He used these notions in On the concept of a random sequence (1940) where he attempted to give a logically satisfactory definition of "random sequence". Folina [ 6 ] argues for the usually accepted view that Church's Thesis is probably true but not capable of rigorous proof. The background to Church's work on computability and undecidability, based on his correspondence with Bernays during the years 1934-1937, is examined by Sieg in [ 11 ].

Church was a founder of the Journal of Symbolic Logic in 1936 and was an editor of the reviews section from its beginning until 1979. In fact he published a paper A bibliography of symbolic logic in volume 4 of the Journal and he saw the reviews section as a continuation and expansion of this work. Its aim, he wrote, was to provide:-

...to provide a complete, suitably indexed, listing of all publications ... in symbolic logic, wherever and in whatever language published ... [giving] critical, analytical commentary.

The article [ 5 ] highlights Church's guiding role in defining the boundaries of the discipline of symbolic logic through this editorial work and testifies to his unflaggingindustry and conscientiousness and his high editorial standards. The aim of comprehensive coverage, which in 1936 had seemed quite practical, became less so as the years went by and by 1975 the rapid expansion in symbolic logic publications forced Church to give up this aspect and begin to provide only selective coverage. We mentioned above that Church retired from Princeton in 1967 and went to the University of California at Los Angeles. Perhaps this is the place where we should mention why he left Princeton after 38 years of service there. Enderton writes:-

Upon his retirement, Princeton was unwilling to continue accommodating the small staff working on the reviews for the Journal of Symbolic Logic.

Church wrote the classic book Introduction to Mathematical Logic in 1956. This was a revised and very much enlarged edition of Introduction to mathematical logic which Church published twelve years earlier in 1944. This first edition was, as he states in the Introduction:-

... the first half of an introductory course in mathematical logic given to graduate students in mathematics [at Princeton in 1943].

Haskell Curry in a review of the 1944 work writes:-

It is written with the meticulous precision which characterizes the author's work generally. ... The subject matter is more or less classical, namely, the propositional algebra and the functional calculus of first order, to which is added a chapter summarizing without proofs certain features of functional calculi of higher order. For the expert the chief interest in the tract is that it makes readily accessible careful detailed formulation and proofs of certain standard theorems, for example, the deduction theorem, the reduction to truth tables, the substitution rule for the functional calculus, Gödel 's completeness theorem, etc.

Manzano writes in [ 10 ] that the 1956 edition of the book:-

... defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed.

The book begins with an Introduction which discusses names, variables, constants and functions, and leads on to the logistic method, syntax and semantics. Chapters I and II are concerned with the propositional calculus, discussing tautologies and the decision problem, duality, consistency and completeness, and independence of the axioms and rules of inference. The first order functional calculus is studied in Chapters III and IV, while Chapter V deals mainly with second order functional calculi.

Another area of interest to Church was axiomatic set theory. He published A formulation of the simple theory of types in 1940 in which he attempted to give a system related to that of Whitehead and Russell 's Principia Mathematica which was designed to avoid the paradoxes of naive set theory. Church bases his form of the theory of types on his -calculus. Other work by Church in this area includes Set theory with a universal set published in 1971 which examines a variant of ZF-type axiomatic set theory and Comparison of Russell's resolution of the semantical antinomies with that of Tarski published in 1976. Another of Church's research interests was intensional semantics which is considered in detail in [ 3 ]. The idea developed here was similar to that of Frege , distinguishing between the extension of a term and the intension, or sense, of a term. Church considered this topic for about 40 years during the latter part of his career, beginning with his paper A formulation of the logic of sense and denotation in 1951.

Although most of Church's contributions are directed towards mathematical logic, he did write a few mathematical papers of other topics. For example he published Remarks on the elementary theory of differential equations as area of research in 1965 and A generalization of Laplace's transformation in 1966. The first examines ideas and results in the elementary theory of ordinary and partial differential equations which Church feels may encourage further investigation of the topic. The paper includes a discussion of a generalization the Laplace transform which he extends to non-linear partial differential equations. This generalization of the Laplace transform is the topic of study of the second paper, again using the method to obtain solutions of second-order partial differential equations.

Church had 31 doctoral students including Foster , Turing , Kleene , Kemeny , Boone , and Smullyan . He received many honours for his contributions including election to the National Academy of Sciences (United States) in 1978. He was also elected to the British Academy, and the American Academy of Arts and Sciences . Case Western Reserve (1969), Princeton (1985) and the State University of New York at Buffalo (1990) awarded him honorary degrees.

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

Born: 28 April 1906 in Brünn, Austria-Hungary (now Brno, Czech Republic)
Died: 14 Jan 1978 in Princeton, New Jersey, USA

Kurt Gödel 's father was Rudolf Gödel whose family were from Vienna. Rudolf did not take his academic studies far as a young man, but had done well for himself becoming managing director and part owner of a major textile firm in Brünn. Kurt's mother, Marianne Handschuh, was from the Rhineland and the daughter of Gustav Handschuh who was also involved with textiles in Brünn. Rudolf was 14 years older than Marianne who, unlike Rudolf, had a literary education and had undertaken part of her school studies in France. Rudolf and Marianne Gödel had two children, both boys. The elder they named Rudolf after his father, and the younger was Kurt.

Kurt had quite a happy childhood. He was very devoted to his mother but seemed rather timid and troubled when his mother was not in the home. He had rheumatic fever when he was six years old, but after he recovered life went on much as before. However, when he was eight years old be began to read medical books about the illness he had suffered from, and learnt that a weak heart was a possible complication. Although there is no evidence that he did have a weak heart, Kurt became convinced that he did, and concern for his health became an everyday worry for him.

Kurt attended school in Brünn, completing his school studies in 1923. His brother Rudolf said:-

Even in High School my brother was somewhat more one-sided than me and to the astonishment of his teachers and fellow pupils had mastered university mathematics by his final Gymnasium years. ... Mathematics and languages ranked well above literature and history. At the time it was rumoured that in the whole of his time at High School not only was his work in Latin always given the top marks but that he had made not a single grammatical error.

Gödel entered the University of Vienna in 1923 still without having made a definite decision whether he wanted to specialise in mathematics or theoretical physics. He was taught by Furtwängler, Hahn , Wirtinger , Menger , Helly and others. The lectures by Furtwängler made the most impact on Gödel and because of them he decided to take mathematics as his main subject. There were two reasons: Furtwängler was an outstanding mathematician and teacher, but in addition he was paralysed from the neck down so lectured from a wheel chair with an assistant who wrote on the board. This would make a big impact on any student, but on Gödel who was very conscious of his own health, it had a major influence. As an undergraduate Gödel took part in a seminar run by Schlick which studied Russell 's book Introduction to mathematical philosophy. Olga Taussky-Todd , a fellow student of Gödel's, wrote:-

It became slowly obvious that he would stick with logic, that he was to be Hahn 's student and not Schlick's, that he was incredibly talented. His help was much in demand.

He completed his doctoral dissertation under Hahn 's supervision in 1929 submitting a thesis proving the completeness of the first order functional calculus. He became a member of the faculty of the University of Vienna in 1930, where he belonged to the school of logical positivism until 1938. Gödel's father died in 1929 and, having had a successful business, the family were left financially secure. After the death of her husband, Gödel's mother purchased a large flat in Vienna and both her sons lived in it with her. By this time Gödel's older brother was a successful radiologist. We mentioned above that Gödel's mother had a literary education and she was now able to enjoy the culture of Vienna, particularly the theatre accompanied by Rudolf and Kurt.

Gödel is best known for his proof of "Gödel's Incompleteness Theorems". In 1931 he published these results in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. This ended a hundred years of attempts to establish axioms which would put the whole of mathematics on an axiomatic basis. One major attempt had been by Bertrand Russell with Principia Mathematica (1910-13). Another was Hilbert 's formalism which was dealt a severe blow by Gödel's results. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisaged by Hilbert . Gödel's results were a landmark in 20^th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions.

Gödel met Zermelo in Bad Elster in 1931. Olga Taussky-Todd , who was at the same meeting, wrote:-

The trouble with Zermelo was that he felt he had already achieved Gödel's most admired result himself. Scholz seemed to think that this was in fact the case, but he had not announced it and perhaps would never have done so. ... The peaceful meeting between Zermelo and Gödel at Bad Elster was not the start of a scientific friendship between two logicians.

Submitting his paper on incompleteness to the University of Vienna for his habilitation , this was accepted by Hahn on 1 December 1932. Gödel became a Privatdozent at the University of Vienna in March 1933.

Now 1933 was the year that Hitler came to power. At first this had no effect on Gödel's life in Vienna; he had little interest in politics. In 1934 Gödel gave a series of lectures at Princeton entitled On undecidable propositions of formal mathematical systems. At Veblen 's suggestion Kleene , who had just completed his Ph.D. thesis at Princeton, took notes of these lectures which have been subsequently published. However, Gödel suffered a nervous breakdown as he arrived back in Europe and telephoned his brother Rudolf from Paris to say he was ill. He was treated by a psychiatrist and spent several months in a sanatorium recovering from depression.

Despite the health problems, Gödel's research was progressing well and he proved important results on the consistency of the axiom of choice with the other axioms of set theory in 1935. However after Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by a NationalSocialist student in 1936, Gödel was much affected and had another breakdown. His brother Rudolf wrote:-

This event was surely the reason why my brother went through a severe nervous crisis for some time, which was of course of great concern, above all for my mother. Soon after his recovery he received the first call to a Guest Professorship in the USA.

He visited Göttingen in the summer of 1938, lecturing there on his set theory research. He returned to Vienna and married Adele Porkert in the autumn of 1938. In fact he had met her in 1927 in Der Nachtfalter night club in Vienna. She was six years older than Gödel and had been married before and both his parents, but particularly his father, objected to the idea that they marry. She was not the first girl that Gödel's parents had objected to, the first he had met around the time he went to university was ten years older than him.

In March 1938 Austria had became part of Germany but Gödel was not much interested and carried on his life much as normal. He visited Princeton for the second time, spending the first term of session 1938-39 at the Institute for Advanced Study. The second term of that academic year he gave a beautiful lecture course at Notre Dame. Most who held the title of privatdozent in Austria became paid lecturers after the country became part of Germany but Gödel did not and his application made on 25 September 1939 was given an unenthusiastic response. It seems that he was thought to be Jewish, but in fact this was entirely wrong, although he did have many Jewish friends. Others also mistook him for a Jew, and he was once attacked by a gang of youths, believing him to be a Jew, while out walking with his wife in Vienna.

When the war started Gödel feared that he might be conscripted into the German army. Of course he was also convinced that he was in far too poor health to serve in the army, but if he could be mistaken for a Jew he might be mistaken for a healthy man. He was not prepared to risk this, and after lengthy negotiation to obtain a U.S. visa he was fortunate to be able to return to the United States, although he had to travel via Russia and Japan to do so. His wife accompanied him.

In 1940 Gödel arrived in the United States, becoming a U.S. citizen in 1948 (in fact he believed he had found an inconsistency in the United States Constitution, but the judge had more sense than to listen during his interview!). He was an ordinary member of the Institute for Advanced Study from 1940 to 1946 (holding year long appointments which were renewed every year), then he was a permanent member until 1953. He held a chair at Princeton from 1953 until his death, holding a contract which explicitly stated that he had no lecturing duties. One of Gödel's closest friends at Princeton was Einstein . They each had a high regard for the other and they spoke frequently. It is unclear how much Einstein influenced Gödel to work on relativity, but he did indeed contribute to that subject.

He received the Einstein Award in 1951, and National Medal of Science in 1974. He was a member of the National Academy of Sciences of the United States, a fellow of the Royal Society , a member of the Institute of France, a fellow of the Royal Academy and an Honorary Member of the London Mathematical Society . However, it says much about his feelings towards Austria that he refused membership of the Academy of Sciences in Vienna, then later when he was elected to honorary membership he again refused the honour. He also refused to accept the highest National Medal for scientific and artistic achievement that Austria offered him. He certainly felt bitter at his own treatment but equally so about that of his family.

Gödel's mother had left Vienna before he did, for in 1937 she returned to her villa in Brno where she was openly critical of the National Socialist regime. Gödel's brother Rudolf had remained in Vienna but by 1944 both expected German defeat, and Rudolf's mother joined him in Vienna. In terms of the treaty negotiated after the war between the Austrians and the Czechs, she received one tenth of the value for her villa in Brno. It was an injustice which infuriated Gödel; in fact he always took such injustices as personal even although large numbers suffered in the same way.

After settling in the United States, Gödel again produced work of the greatest importance. His masterpieceConsistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940) is a classic of modern mathematics. In this he proved that if an axiomatic system of set theory of the type proposed by Russell and Whitehead in Principia Mathematica is consistent, then it will remain so when the axiom of choice and the generalized continuum-hypothesis are added to the system. This did not prove that these axioms were independent of the other axioms of set theory, but when this was finally established by Cohen in 1963 he built on these ideas of Gödel.

Concerns with his health became increasingly worrying to Gödel as the years went by. Rudolf, Gödel's brother, was a medical doctor so the medical details given by him in the following will be accurate. He wrote:-

My brother had a very individual and fixed opinion about everything and could hardly be convinced otherwise. Unfortunately he believed all his life that he was always right not only in mathematics but also in medicine, so he was a very difficult patient for doctors. After severe bleeding from a duodenal ulcer ... for the rest of his life he kept to an extremely strict (over strict?) diet which caused him slowly to lose weight.

Adele, Gödel's wife, was a great support to him and she did much to ease the tensions which troubled him. However she herself began to suffer health problems, having two strokes and a major operation. Towards the end of his life Gödel became convinced that he was being poisoned and, refusing to eat to avoid being poisoned, essentially starved himself to death [ 3 ]:-

A slight person and very fastidious, Gödel was generally worried about his health and did not travel or lecture widely in later years. He had no doctoral students, but through correspondence and personal contact with the constant succession of visitors to Princeton, many people benefited from his extremely quick and incisive mind. Friend to Einstein , von Neumann and Morgenstern, he particularly enjoyed philosophical discussion.

He died [ 18 ]:-

... sitting in a chair in his hospital room at Princeton, in the afternoon of 14 January 1978.

It would be fair to say that Gödel's ideas have changed the course of mathematics [ 3 ]:-

... it seems clear that the fruitfulness of his ideas will continue to stimulate new work. Few mathematicians are granted this kind of immortality.

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

Alan Turing was born at Paddington, London. His father, Julius Mathison Turing, was a British member of the Indian Civil Service and he was often abroad. Alan's mother, Ethel Sara Stoney, was the daughter of the chief engineer of the Madras railways and Alan's parents had met and married in India. When Alan was about one year old his mother rejoined her husband in India, leaving Alan in England with friends of the family. Alan was sent to school but did not seem to be obtaining any benefit so he was removed from the school after a few months.

Next he was sent to Hazlehurst Preparatory School where he seemed to be an 'average to good' pupil in most subjects but was greatly taken up with following his own ideas. He became interested in chess while at this school and he also joined the debating society. He completed his Common Entrance Examination in 1926 and then went to Sherborne School. Now 1926 was the year of the general strike and when the strike was in progress Turing cycled 60 miles to the school from his home, not too demanding a task for Turing who later was to become a fine athlete of almost Olympic standard. He found it very difficult to fit into what was expected at this public school, yet his mother had been so determined that he should have a public school education. Many of the most original thinkers have found conventional schooling an almost incomprehensible process and this seems to have been the case for Turing. His genius drove him in his own directions rather than those required by his teachers.

He was criticised for his handwriting, struggled at English, and even in mathematics he was too interested with his own ideas to produce solutions to problems using the methods taught by his teachers. Despite producing unconventional answers, Turing did win almost every possible mathematics prize while at Sherborne. In chemistry, a subject which had interested him from a very early age, he carried out experiments following his own agenda which did not please his teacher. Turing's headmaster wrote (see for example [ 6 ]):-

If he is to stay at Public School, he must aim at becoming educated. If he is to be solely a Scientific Specialist, he is wasting his time at a Public School.

This says far more about the school system that Turing was being subjected to than it does about Turing himself. However, Turing learnt deep mathematics while at school, although his teachers were probably not aware of the studies he was making on his own. He read Einstein 's papers on relativity and he also read about quantum mechanics in Eddington 's The nature of the physical world.

An event which was to greatly affect Turing throughout his life took place in 1928. He formed a close friendship with Christopher Morcom, a pupil in the year above him at school, and the two worked together on scientific ideas. Perhaps for the first time Turing was able to find someone with whom he could share his thoughts and ideas. However Morcom died in February 1930 and the experience was a shattering one to Turing. He had a premonition of Morcom's death at the very instant that he was taken ill and felt that this was something beyond what science could explain. He wrote later (see for example [ 6 ]):-

It is not difficult to explain these things away - but, I wonder!

Despite the difficult school years, Turing entered King's College, Cambridge, in 1931 to study mathematics. This was not achieved without difficulty. Turing sat the scholarship examinations in 1929 and won an exhibition, but not a scholarship. Not satisfied with this performance, he took the examinations again in the following year, this time winning a scholarship. In many ways Cambridge was a much easier place for unconventional people like Turing than school had been. He was now much more able to explore his own ideas and he read Russell 's Introduction to mathematical philosophy in 1933. At about the same time he read von Neumann 's 1932 text on quantum mechanics, a subject he returned to a number of times throughout his life.

The year 1933 saw the beginnings of Turing's interest in mathematical logic. He read a paper to the Moral Science Club at Cambridge in December of that year of which the following minute was recorded (see for example [ 6 ]):-

A M Turing read a paper on "Mathematics and logic". He suggested that a purely logistic view of mathematics was inadequate; and that mathematical propositionspossessed a variety of interpretations of which the logistic was merely one.

Of course 1933 was also the year of Hitler's rise in Germany and of an anti-war movement in Britain. Turing joined the anti-war movement but he did not drift towards Marxism, nor pacifism, as happened to many.

Turing graduated in 1934 then, in the spring of 1935, he attended Max Newman 's advanced course on the foundations of mathematics. This course studied Gödel 's incompleteness results and Hilbert 's question on decidability. In one sense 'decidability' was a simple question, namely given a mathematical proposition could one find an algorithm which would decide if the proposition was true of false. For many propositions it was easy to find such an algorithm. The real difficulty arose in proving that for certain propositions no such algorithm existed. When given an algorithm to solve a problem it was clear that it was indeed an algorithm, yet there was no definition of an algorithm which was rigorous enough to allow one to prove that none existed. Turing began to work on these ideas.

Turing was elected a fellow of King's College, Cambridge, in 1935 for a dissertation On the Gaussian error function which proved fundamental results on probability theory , namely the central limit theorem. Although the central limit theorem had recently been discovered, Turing was not aware of this and discovered it independently. In 1936 Turing was a Smith's Prizeman.

Turing's achievements at Cambridge had been on account of his work in probability theory. However, he had been working on the decidability questions since attending Newman 's course. In 1936 he published On Computable Numbers, with an application to the Entscheidungs problem. It is in this paper that Turing introduced an abstract machine, now called a "Turing machine", which moved from one state to another using a precise finite set of rules (given by a finite table) and depending on a single symbol it read from a tape.

The Turing machine could write a symbol on the tape, or delete a symbol from the tape. Turing wrote [ 13 ]:-

Some of the symbols written down will form the sequences of figures which is the decimal of the real number which is being computed. The others are just rough notes to "assist the memory". It will only be these rough notes which will be liable to erasure.

He defined a computable number as real number whose decimal expansion could be produced by a Turing machine starting with a blank tape. He showed that π was computable, but since only countably many real numbers are computable, most real numbers are not computable. He then described a number which is not computable and remarks that this seems to be a paradox since he appears to have described in finite terms, a number which cannot be described in finite terms. However, Turing understood the source of the apparent paradox. It is impossible to decide (using another Turing machine) whether a Turing machine with a given table of instructions will output an infinite sequence of numbers.

Although this paper contains ideas which have proved of fundamental importance to mathematics and to computer science ever since it appeared, publishing it in the Proceedings of the London Mathematical Society did not prove easy. The reason was that Alonzo Church published An unsolvable problem in elementary number theory in the American Journal of Mathematics in 1936 which also proves that there is no decision procedure for arithmetic. Turing's approach is very different from that of Church but Newman had to argue the case for publication of Turing's paper before the London Mathematical Society would publish it. Turing's revised paper contains a reference to Church 's results and the paper, first completed in April 1936, was revised in this way in August 1936 and it appeared in print in 1937.

A good feature of the resulting discussions with Church was that Turing became a graduate student at Princeton University in 1936. At Princeton, Turing undertook research under Church 's supervision and he returned to England in 1938, having been back in England for the summer vacation in 1937 when he first met Wittgenstein . The major publication which came out of his work at Princeton was Systems of Logic Based on Ordinals which was published in 1939. Newman writes in [ 13 ]:-

This paper is full of interesting suggestions and ideas. ... [It] throws much light on Turing's views on the place of intuition in mathematical proof.

Before this paper appeared, Turing published two other papers on rather more conventional mathematical topics. One of these papers discussed methods of approximating Lie groups by finite groups . The other paper proves results on extensions of groups, which were first proved by Reinhold Baer , giving a simpler and more unified approach.

Perhaps the most remarkable feature of Turing's work on Turing machines was that he was describing a modern computer before technology had reached the point where construction was a realistic proposition. He had proved in his 1936 paper that a universal Turing machine existed [ 13 ]:-

... which can be made to do the work of any special-purpose machine, that is to say to carry out any piece of computing, if a tape bearing suitable "instructions" is inserted into it.

Although to Turing a "computer" was a person who carried out a computation, we must see in his description of a universal Turing machine what we today think of as a computer with the tape as the program.

While at Princeton Turing had played with the idea of constructing a computer. Once back at Cambridge in 1938 he starting to build an analogue mechanical device to investigate the Riemann hypothesis , which many consider today the biggest unsolved problem in mathematics. However, his work would soon take on a new aspect for he was contacted, soon after his return, by the Government Code and Cypher School who asked him to help them in their work on breaking the German Enigma codes.

When war was declared in 1939 Turing immediately moved to work full-time at the Government Code and Cypher School at Bletchley Park. Although the work carried out at Bletchley Park was covered by the Official Secrets Act, much has recently become public knowledge. Turing's brilliant ideas in solving codes, and developing computers to assist break them, may have saved more lives of military personnel in the course of the war than any other. It was also a happy time for him [ 13 ]:-

... perhaps the happiest of his life, with full scope for his inventiveness, a mild routine to shape the day, and a congenial set of fellow-workers.

Together with another mathematician W G Welchman, Turing developed the Bombe, a machine based on earlier work by Polish mathematicians, which from late 1940 was decoding all messages sent by the Enigma machines of the Luftwaffe. The Enigma machines of the German navy were much harder to break but this was the type of challenge which Turing enjoyed. By the middle of 1941 Turing's statistical approach, together with captured information, had led to the German navy signals being decoded at Bletchley.

From November 1942 until March 1943 Turing was in the United States liaising over decoding issues and also on a speech secrecy system. Changes in the way the Germans encoded their messages had meant that Bletchley lost the ability to decode the messages. Turing was not directly involved with the successful breaking of these more complex codes, but his ideas proved of the greatest importance in this work. Turing was awarded the O.B.E. in 1945 for his vital contribution to the war effort.

At the end of the war Turing was invited by the National Physical Laboratory in London to design a computer. His report proposing the Automatic Computing Engine (ACE) was submitted in March 1946. Turing's design was at that point an original detailed design and prospectus for a computer in the modern sense. The size of storage he planned for the ACE was regarded by most who considered the report as hopelessly over-ambitious and there were delays in the project being approved.

Turing returned to Cambridge for the academic year 1947-48 where his interests ranged over many topics far removed from computers or mathematics; in particular he studied neurology and physiology. He did not forget about computers during this period, however, and he wrote code for programming computers. He had interests outside the academic world too, having taken up athletics seriously after the end of the war. He was a member of Walton Athletic Club winning their 3 mile and 10 mile championship in record time. He ran in the A.A.A. Marathon in 1947 and was placed fifth.

By 1948 Newman was the professor of mathematics at the University of Manchester and he offered Turing a readership there. Turing resigned from the National Physical Laboratory to take up the post in Manchester. Newman writes in [ 13 ] that in Manchester:-

... work was beginning on the construction of a computing machine by F C Williams and T Kilburn. The expectation was that Turing would lead the mathematical side of the work, and for a few years he continued to work, first on the design of the subroutines out of which the larger programs for such a machine are built, and then, as this kind of work became standardised, on more general problems of numerical analysis.

In 1950 Turing published Computing machinery and intelligence in Mind. It is another remarkable work from his brilliantly inventive mind which seemed to foresee the questions which would arise as computers developed. He studied problems which today lie at the heart of artificial intelligence. It was in this 1950 paper that he proposed the Turing Test which is still today the test people apply in attempting to answer whether a computer can be intelligent [ 1 ]:-

... he became involved in discussions on the contrasts and similarities between machines and brains. Turing's view, expressed with great force and wit, was that it was for those who saw an unbridgeable gap between the two to say just where the difference lay.

Turing did not forget about questions of decidability which had been the starting point for his brilliant mathematical publications. One of the main problems in the theory of group presentations was the question: given any word in a finitely presented groups is there an algorithm to decide if the word is equal to the identity. Post had proved that for semigroups no such algorithm exist. Turing thought at first that he had proved the same result for groups but, just before giving a seminar on his proof, he discovered an error. He was able to rescue from his faulty proof the fact that there was a cancellative semigroup with insoluble word problem and he published this result in 1950. Boone used the ideas from this paper by Turing to prove the existence of a group with insoluble word problem in 1957.

Turing was elected a Fellow of the Royal Society of London in 1951, mainly for his work on Turing machines in 1936. By 1951 he was working on the application of mathematical theory to biological forms. In 1952 he published the first part of his theoretical study of morphogenesis, the development of pattern and form in living organisms.

Turing was arrested for violation of British homosexualitystatutes in 1952 when he reported to the police details of a homosexual affair. He had gone to the police because he had been threatened with blackmail. He was tried as a homosexual on 31 March 1952, offering no defence other than that he saw nothing wrong in his actions. Found guilty he was given the alternatives of prison or oestrogen injections for a year. He accepted the latter and returned to a wide range of academic pursuits.

Not only did he press forward with further study of morphogenesis, but he also worked on new ideas in quantum theory, on the representation of elementary particles by spinors, and on relativity theory. Although he was completely open about his sexuality, he had a further unhappiness which he was forbidden to talk about due to the Official Secrets Act.

The decoding operation at Bletchley Park became the basis for the new decoding and intelligence work at GCHQ. With the cold war this became an important operation and Turing continued to work for GCHQ, although his Manchester colleagues were totally unaware of this. After his conviction, his security clearance was withdrawn. Worse than that, security officers were now extremely worried that someone with complete knowledge of the work going on at GCHQ was now labelled a security risk. He had many foreign colleagues, as any academic would, but the police began to investigate his foreign visitors. A holiday which Turing took in Greece in 1953 caused consternation among the security officers.

Turing died of potassium cyanide poisoning while conducting electrolysis experiments. The cyanide was found on a half eaten apple beside him. An inquest concluded that it was self-administered but his mother always maintained that it was an accident.

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