On Computable Numbers, with an Application to the Entscheidungsproblem. TURING, Alan Mathison.
A fine set, not ex-library, of arguably the single most important theoretical work in the history of computing. In this paper Turing introduced the concept of a ‘universal machine’, an imaginary computing device designed to replicate the mathematical ‘states of mind’ and symbol-manipulating abilities of a human computer. Turing conceived of the universal machine as a means of answering the last of the three questions about mathematics posed by David Hilbert in 1928: (1) is mathematics complete; (2) is mathematics consistent; and (3) is mathematics decidable. Hilbert's final question, known as the ‘Entscheidungsproblem’, is concerned with whether there exists a definite method, or ‘mechanical process’, that can be applied to any mathematical assertion, and which is guaranteed to produce a correct decision as to whether that assertion is true or not. The logician Kurt Gödel had already in 1931 shown that arithmetic (and by extension mathematics) could not be both consistent and complete. Turing showed, by means of his universal machine, mathematics is undecidable. To demonstrate this, Turing came up with the concept of ‘computable numbers’, which are numbers defined by some definite rule, and thus calculable on the universal machine. These computable numbers would include every number that could be arrived at through arithmetical operations, finding roots of equations, and using mathematical functions like sines and logarithms - every number that could possibly arise in computational mathematics. Turing then showed that these computable numbers could in turn give rise to uncomputable ones, ones that could not be calculated using a definite rule, and that therefore there could be no ‘mechanical process’ for solving all mathematical questions, since computing an uncomputable number was an example of an unsolvable problem. Turing's idea of a ‘universal machine’ was given the name "Turing machine" by Church. The concept of the Turing machine has become the foundation of modern computer science. Origins of Cyberspace 394. Richard Green Library (Christie's sale 2013, lot 326). Erwin Thomas Library T61 and T62. In: Proceedings of the London Mathematical Society, Vol.42: pp.230-265 and Vol.43: pp.544-546 ("A Correction"). The complete volumes offered in near contemporary cloth with gilt spine lettering, completely clean and fresh throughout - a fine set. N° de réf. du libraire 2091