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Minds, Machines and Gödel1

  • J. R. Lucas (a1)

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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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. 2099–123; and Popper, K. R.: “Indeterminism in Quantum Physics and Classical Physics”; British Journal for Philosophy of Science, Vol.1 (1951), pp.179–88. The question is touched upon by Rosenbloom, Paul; Elements of Mathematical Logic; pp.207–8; Nagel, Ernest and Newman, James R.; Gödel's proof, pp. 100–2; and by Rogers, Hartley; Theory of Recursive Functions and Effective Computability (mimeographed), 1957, Vol.1, pp. 152 ff.

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. 202–6; and “On not worshipping. Facts”; The Philosophical Quarterly, April 1958, p. 144.

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.

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.

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

1 A paper read to the Oxford Philosophical Society on October 30, 1959.

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