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Finite Approximations to Lie Groups; The Extensions of a Group.
University Press; Noordhoff 1938; 1938, Princeton; Groningen - First edition, the extremely rare offprint issues, and with a fine provenance, of the two papers on group theory written by Turing at Princeton under the supervision of John Von Neumann. Offprints of Turing’s papers are extremely rare in institutional holdings, and even more so in commerce. We have located only two other copies of the first offprint and one of the second: a copy of each in the Turing Archive at King’s College, Cambridge (AMT/B/13 and AMT/B/14), and a copy of the first in the Max Newman collection (now at Bletchley Park). As far as we are aware, the latter is the only other copy of either offprint to have appeared in commerce. Turing’s undergraduate tutor at King’s College, Cambridge was the group theorist Philip Hall; this probably explains why the first three of Turing’s published papers were on group theory. His first paper "was a small and easily stated advance on a recent theorem of von Neumann’s. The latter had defined two notions of ‘left’ and ‘right’ periodicity but had missed the fact that they are equivalent. Turing proved this, and it appeared as a two page paper in April 1935 [‘Equivalence of left and right almost periodicity,’ Journal of the London Mathematical Society, Vol. 10, 1935]. By coincidence von Neumann arrived on a sabbatical visit from Princeton that month and proceeded to lecture on the subject; it is from this time that the two must have been acquainted" (Welch, p. 3). The next two years were spent working on his great paper on the ‘Entscheidungsproblem,’ including the introduction of the ‘universal Turing machine’. The Entscheidungsproblem had been solved slightly earlier by Alonzo Church, and with both Church and von Neumann at Princeton it was natural for Turing to visit Princeton, which he did in 1937. Once there, and under von Neumann’s direction, Turing attacked "a problem suggested by the emigrĂ© Polish mathematician S. Ulam: that of asking whether continuous groups could be approximated by finite groups, rather like approximating a sphere by polyhedra. Von Neumann had passed the problem to Alan, who successfully dealt with it by April, when it was submitted. This was fast work, although as he had shown that continuous groups could not in general be approximated in this way, it was a rather negative result" (Hodges, pp.129-130). The actual result proved by Turing is described by Max Newman as follows. "If a (connected) Lie group L can, for arbitrary ? > 0, be ?-approximated by a finite group whose multiplication law is an ?-approximation to that of L, in the sense that the two products of any two elements are within e of each other, then L must be both compact and abelian" (Newman, p. 258). The second offered paper "concerns the problem of determining the extensionsof a given group G by a given group H inducing given classes of automorphisms" (Welch, p. 10). "This was an important contribution to the theory of extensions of a not necessarily Abelian group as it stood at that time" (J. L. Britton in Collected Works: Pure Mathematics, p. xi). On the strength of these two papers von Neumann offered Turing a post-doctoral position at Princeton, but he declined and returned to Cambridge in the summer of 1938 to resume his Fellowship. Although there are no ownership marks, these offprints are from the working library of Robin Gandy. "[Gandy’s] interest in [mathematical logic] commenced through his wartime acquaintance with Alan Turing as far back as 1944. His twin interests in the foundations of science and mathematics were fostered by Turing. These interests moulded a career which always seemed larger than his publication list, although that grew in diversity with the years to include some seminal papers. Initially, he read widely and published little, but around 1960 he started to produce the major results which made him an international figure in the theory of abstract recursion and in effective, descriptive set theory. There were also major conceptual innovations such as the theory [Attributes: First Edition]
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