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Minds, Machines and Gödel1

Published online by Cambridge University Press:  25 February 2009

J. R. Lucas
Affiliation:
Merton College, Oxford
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Gödei's Theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines. So also has it seemed to many other people: almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely until he could see the whole argument set out, with all objections fully stated and properly met. This I attempt to do.

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Copyright © The Royal Institute of Philosophy 1961

References

page 112 note 2 See Turing, A. M.: “Computing Machinery and Intelligence”: Mind, 1950, PP. 433–60, reprinted in The World of Mathematics, edited by James R. Newman, pp. 2099–123Google Scholar; and Popper, K. R.: “Indeterminism in Quantum Physics and Classical Physics”; British Journal for Philosophy of Science, Vol.1 (1951), pp.179–88.Google Scholar The question is touched upon by Rosenbloom, Paul; Elements of Mathematical Logic; pp.207–8Google Scholar; Nagel, Ernest and Newman, James R.; Gödel's proof, pp. 100–2Google Scholar; and by Rogers, Hartley; Theory of Recursive Functions and Effective Computability (mimeographed), 1957, Vol.1, pp. 152 ff.Google Scholar

page 117 note 1 Gödel's original proof applies; v. § i init. § 6 init. of his Lectures at the Institute of Advanced Study, Princeton, NJ., U.S.A., 1934.

page 117 note 2 Mind, 1950, pp. 444–5; Newman, p. 2110.

page 118 note 1 For a similar type of argument, see Lucas, J. R.: “The Lesbian Rule”; PHILOSOPHY, 07 1955, pp. 202–6Google Scholar; and “On not worshipping. Facts”; The Philosophical Quarterly, April 1958, p. 144.Google Scholar

page 119 note 1 In private conversation.

page 119 note 2 Theory of Recursive Functions and Effective Computability, 1957, Vol. I, pp. 152 ff.

page 120 note 1 Godel's original proof applies if the rule is such as to generate a primitive recursive class of additional formulae; v. § I init. and § 6 init. of his Lectures at the Institute of Advanced Study, Princeton, N.J., U.S.A., 1934. It is in fact sufficient that the class be recursively enumerable, v. Rosser, Barkley: “Extensions of some theorems of Godel and Church”, Journal of Symbolic Logic, Vol. 1, 1936, pp. 8791.CrossRefGoogle Scholar

page 120 note 2 Op. cit., p. 154.

page 120 note 3 University of Princeton, N.J., U.S.A. in private conversation.

page 121 note 1 See, e.g., Church, Alonzo: Introduction to Mathematical Logic, Princeton. Vol. I, § 17, p. 108.Google Scholar

page 125 note 1 Mind, 1950, p. 454; Newman, p. 2117–18.

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