This is the first appearance of Turing's seminal paper introducing the concept of a universal problem-solving machine. In it the mathematician shows how his hypothetical machine (subsequently known as the Turing machine) could replicate anything done by a human being following a set procedure or by any other mechanism.
Turing had originally conceived of his paper as a response to the last of the great German mathematician David Hilbert's three major questions concerning mathematics - is it complete as a system? Is it consistent?, and is it decidable? By 1930, when Turing was entering Cambridge, the young Czech philosopher Kurt Gödel had answered the first two questions in the negative, but the third, the decision-problem (Entscheidungsproblem) remained unanswered. To answer it required finding a process, or proving none existed, to decide if any particular mathematical statement is true or not. As it happened the American mathematician Alonzo Church had independently proved that there is no solution to the Entscheidungsproblem. Turing's similar result however, arrived at by the simple concept of a rudimentary machine, is now vastly more famous: the Turing machine can be thought of as a computer program, and the mechanical task of interpreting and obeying the program as what the computer does. "So the universal Turing machine is now seen to embody the principle of the modern computer: a single machine which can be turned to any task by an appropriate program. The universal Turing machine also exploits the fact that symbols representing instructions are no different in kind from symbols representing data - the 'stored program' concept of the digital computer. However, no such computer existed in 1936, except in Turing's imagination..." (Alan Hodges, ODNB).
The second paper contains a few corrections to Turing's earlier paper of the same title, following comments by Alonzo Church in his review.
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