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Minds and Machines

, Volume 13, Issue 1, pp 23–48 | Cite as

Alan Turing and the Mathematical Objection

  • Gualtiero Piccinini
Article

Abstract

This paper concerns Alan Turing's ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the ``mathematical objection'' to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do.

artificial intelligence Church-Turing thesis computability effective procedure incompleteness machine mathematical objection ordinal logics Turing undecidability 

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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Gualtiero Piccinini
    • 1
  1. 1.Department of History and Philosophy of ScienceUniversity of PittsburghPittsburghUSA

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