Beschreibung
First edition, two journal issues in original printed wrappers, of Turing's papers on Church's type theory. "Church's type theory is a formal logical language which includes first-order logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory. It is particularly well suited to the formalization of mathematics and other disciplines and to specifying and verifying hardware and software" (Stanford Encyclopedia of Philosophy). Turing's interest in logic and computation began when he attended the lectures of Max Newman at Cambridge in 1935. This led, two years later, to the publication of his great work 'On computable numbers, with an application to the Entscheidungsproblem,' which included the introduction of the universal Turing machine. Following this, "most of Alan's efforts were directed towards a new formulation of the theory of types. Russell had regarded types as rather a nuisance, adopted faute de mieux in order to save Frege's set theory. Other logicians had felt that a hierarchy of logical categories was really quite a natural idea, and that it was the attempt to lump together every conceivable entity into 'sets' that was strange. Alan inclined to the latter view. He would prefer a theory with the way in which mathematicians actually thought, and which worked in a practical way. He also wanted to see mathematical logic used to make the work of mathematicians more rigorous . . . His own efforts to bridge the gap began with an attempt 'to put the theory of type into a form in which it can be used by the mathematician-in-the-street without having to study mathematical logic, much less use it . . . The type principle is effectively taken care of in ordinary language by the fact that there are nouns as well as adjectives. We can make the statement 'All horses are four-legged', which can be verified by examination of every horse, at any rate if there are only a finite number of them. If however we try to use words like 'thing' or 'thing whatever' trouble begins. Suppose we understand thing to include everything whatever, books, cats, men, women, thoughts, functions of men with cats as values, numbers, matrices, classes of classes, procedures, propositions . . . Under these circumstances what can be made of the statement 'All things are not prime multiples of 6'. What do we mean by it? Under no circumstances is the number of things to be examined finite. It may be that some meaning can be given to statements of this kind, but for the present we do not know of any. In effect then the theory of types requires us to refrain from the use of such nouns as 'thing', 'object', etc., which are intended to convey the idea of 'anything whatever'. The technical work of separating mathematical 'nouns' from 'adjectives' was based upon that of Church, whose lectures he had followed at Princeton, and who published a description of his type theory in 1940. Part of Alan's work was done in collaboration with Newman through correspondence; their joint paper [i.e. I] being received at Princeton on 9 May 1941. It must have crossed the Atlantic just as the München was captured. Alan produced a further paper of a highly technical nature, 'The use of dots as brackets in Church's system,' and submitted it just a year later (Hodges, Alan Turing, pp. 215-6).[Paper II] "shows once again Turing's ability to reason about important issues in computer science before there were digital computers to reason about. In this case, Turing essentially studies an important aspect of programming languages, a syntax for trees. In short, Turing's dots gave him a way to think about the order of operations in a structure that was more intuitive to him, to prepare him for planned further work on Church's lambda-calculus" (Lance Fortnow, in Alan Turing: his work and impact, pp. 227-8). Two complete journal issues in original printed wrappers. Very good. Bestandsnummer des Verkäufers ABE-1651228937746
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