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Did Turing have any knowledge of Babbage's work? Is it known if he ever mentioned Babbage?

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See:

[page 10, regarding Turing's paper On computable numbers... (1937)] It should be remarked that there is no evidence that Turing had read any of the scanty and sporadic literature concerned with the general theory of mechanical computation. In particular, one can be sure that if Turing had read either account by Babbage of the Analytic Engine (Chapter VIII of 'Pages from the life of a philosopher' [1864]) or the account of Menabrea [1842] translated by the Countess of Lovelace [1843] he would have mentioned Babbage's ideas. He might well have read the article on Calculating Machines in the 11th edition of the Encyclopaedia Britannica. In later years he often consulted the copy of the Encyclopaedia which he inherited from his father - but that article contains only a short and rather dismissive reference to the Analytic Engine: 'a much more powerful machine... intended to perform any series of possible arithmetical operations'. This would hardly have suggested to Turing that Babbage had in fact conceived a universal machine.

[page 15, regarding Turing machines and electronic computers] According to Randell [1980], Turing and Newman and a group of mathematicians and engineers at Bletchley Park discussed in 1942-43, out of office hours, Turing's universal machine, Babbage's plans for the Analytic Engine, and the possibilities of artificial intelligence. Turing played some part in the design of the first machines built under Newman's direction.

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