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Alan Turing in America Alan Turing in America Author: David E. Zitarelli (Temple University) Overview Alan Turing visited the United States during 1936-38 and 1942-43. This article describes how two of Turing's greatest accomplishments, in logic and computer design, were influenced by the first of his two visits to the U.S. Contents . Alan Turing in America – Introduction . Alan Turing in America – Logic . Alan Turing in America – Computers . Alan Turing in America – Cryptography . Alan Turing in America – 1942–1943 . Alan Turing in America – Conclusion . Alan Turing in America – References and About the Author Alan Turing in America – Introduction Figure 1. Alan Turing (1912-1954) (Source: MacTutor History of Mathematics Archive) The British mathematician Alan Turing has perhaps achieved greater fame today than during his lifetime. Thirty-two years after his death in 1954 a play based on part of his work, Breaking the Code, opened in London’s West End theater district; it played in New York the next year. The Broadway production was nominated for three Tony Awards, a rare accomplishment for a theatrical piece about a mathematician. But Turing was no ordinary mathematician. The play deplored his sad demise apparently caused by his homosexuality. More recently, the movie The Imitation Game depicted Turing’s early education, role in deciphering Enigma in World War II, and the tragedy of his death at age 41. Here I describe how two of his greatest accomplishments were influenced by one of his two visits to the U.S. The authoritative account of the life and career of Alan Turing is a book by Andrew Hodges [10]. Two more recent works from different viewpoints are [3] and [11]. In addition, online files from a Princeton conference held to commemorate Turing’s 100th birthday can be accessed at: http://www.princeton.edu/turing/ Alan Turing in America – Logic It does not seem to be well-known that Alan Mathison Turing (1912-1954) spent two academic years at Princeton University, from the summer of 1936 to the summer of 1938. Before then Turing had entered Cambridge University in 1931 and while there in the spring of 1935 took a course given by the topologist M.H.A. Hodges called “Foundations of Mathematics” that would dramatically alter the direction of his life. One of the primary topics in the course was Hilbert’s Entscheidungs problem, which asked if it was possible to find an algorithm that would determine whether an inference in a formal logic system was valid. Turing soon showed that no such algorithm exists. (See [8] for a modern analysis of this work.) This was such a critical breakthrough that Hodges encouraged him to put it in print at once. But the painfully shy Turing hesitated and did not write up his results until the spring of 1936. Within days of handing the first draft to Hodges for review, his mentor received the most recent issue of the American Journal of Mathematics, which contained a devastating article by Princeton’s Alonzo Church (1903-1995) establishing the same result [2]. Figure 2. This photo of a young Alan Turing is believed to have been taken during the years he was at Princeton University. (Source: Convergence Portrait Gallery) Turing could have wilted under such bad news but instead he flashed resilience. Events then moved quickly. That very day Hodges wrote directly to Princeton logician Alonzo Church seeking support for his star student to visit the university. Church was totally supportive and so that fall Turing crossed the Atlantic to study with Church himself. By the time Turing arrived he realized that although he had reached the same negative solution to the Entscheidungs problem as Church, his approach was sufficiently different to be worthy of separate publication. Therefore Turing submitted the paper, “On computable numbers with an application to the Entscheidungsproblem,” to the Proceedings of the London Mathematical Society. However, mathematics journals generally do not publish papers covering results that have already appeared in print. By then Hodges was convinced of Turing’s talents and “canvassed vigorously to gain support for [his] paper, writing to [the editor] F.P. White … to argue for its publication” [4, p. 178]. By the time the paper appeared in 1937, Turing had added an appendix “Computability and effective calculability” outlining a proof that his concept of computability was equivalent to Church’s \(\lambda\)-definability. Turing listed his address as “The Graduate College, Princeton University, New Jersey, U.S.A.” Moreover, the appendix began, “Added August 28, 1936.” These two facts misled some people to conclude erroneously that his work had been completed under Church’s direction. Figure 3. Alonzo Church in October of 1943, Princeton University, New Jersey (Source: Wikimedia Commons) Alan Turing did study under Alonzo Church over the next two years, however, culminating in his Ph.D. in 1938. Princeton was so impressed with the quality of this work during his first year as a graduate student that the university awarded him a prestigious Proctor Fellowship for the second year. His doctoral dissertation, “Systems of logic based on ordinals,” dealt with the undecidability of transfinite sequences of certain formal systems, a study that had been inaugurated by Kurt Gödel. Thus Turing was deeply influenced by two logicians in the U.S.— Church at Princeton University and Gödel at the Institute for Advanced Study in Princeton. Before Alan Turing came to America in the summer of 1936 he had resolved one of Hilbert’s famous problems in the negative. This result alone would have earned him immediate recognition but it was the approach he adopted that has gained him an international reputation that is as strong today as ever primarily because it transcended mathematical logic and became fundamental to the study of modern computers. Alan Turing in America – Computers Figure 4. Alan Turing on a 2000 "millennium" stamp commemorating his 1937 theory of digital computing. Turing was based at Princeton University throughout 1937. (Source: MacTutor History of Mathematics Archive) Alan Turing’s viewpoint can be derived from one telling statement in his initial paper. He wrote, “The behaviour of the computer at any moment is determined by the symbols which he is observing, and his ‘state of mind’ at that moment” [12, p. 250]. To the modern reader the association of “he” with “computer” might sound strange but one must remember that back in the 1930s the term computer referred to a human being who performed computations. Moreover, the expression “state of mind” referred to distinct states in the course of the computation. In this sense Turing was developing a precursor to the modern computer. Tellingly, Turing wrote that “the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares” [12, p. 249]. That “tape” is equivalent to what we refer to as codedinstructions, or stored programs today, such as those stored in FORTRAN. With this in mind, a Turing machine consists of a finite set of quintuples of the form \( p \alpha \beta X q \). Here the machine encounters the symbol \( \alpha \) in state \( p,\) transforms it into the symbol \( \beta \) in state \( q,\) and the machine then moves to a state to the right, left, or same position as the state where it began according to whether \(X\) is R, L, or N. Turing called a real number “computable” if such a machine could produce a sequence of binary numbers representing the expansion of that number into 0s and 1s starting with an empty tape. This whole approach, even as seen in this brief description, appears to be entirely theoretical. But there was a practical side to Turing as well. This universal machine provided a model for what we call a “stored program” computer today. While at Princeton he even designed an electro-mechanical binary multiplier and gained entrance to the graduate-student machine shop in the physics department to try to build the relays himself. Although he was unable to complete this project before returning to England, once back in Cambridge he was awarded funds to construct a special-purpose analog computer to calculate the zeros of the Riemann zeta-function. However, World War II broke out and that machine too was never built. Alan Turing in America – Cryptography The play Breaking the Code and the movie The Imitation Game popularized Alan Turing’s decisive role in deciphering codes produced by the German encrypting/decrypting machines. In fact, his interest in cryptography may have been sparked in Princeton when he set out to build the binary-multiplier machine. In any event, Turing spent the war years at Bletchley Park, a country mansion housing Britain’s leading cryptanalysts. He was assigned to work with naval communications and, using information supplied by Polish mathematicians, was intimately involved with building machines that could decipher messages sent from the German military. That project enabled the British Admiralty to defeat German submarine maneuvers aimed at strangling Britain’s shipping routes in the Atlantic. Not only did Turing pursue his studies on logic and ideas on computing after arriving back in Cambridge, England, but he also published two conventional papers. However, all of this activity came to a grinding halt shortly after Great Britain declared war on Germany in 1939, whereupon Turing was asked by a British governmental office for help with breaking codes. German scientists at the Lorenz engineering company, adept at mathematics and its applications, had built a machine called Enigma that automated the processes of coding and decoding messages. Enigma required the work of only three people: an operator who typed the coded message, a wireless operator who tapped out the enciphered message in Morse code, and a receiving operator who typed the message and handed it to the intended target.
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