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Incompleteness, Mechanism, and Optimism

Published online by Cambridge University Press:  15 January 2014

Stewart Shapiro
Stewart Shapiro*
Affiliation:
Department of Philosophy, Ohio State University at Newark, Newark, Ohio 43055, USAE-mail: shapiro+@osu.edu
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Extract

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?

A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.

I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine, M, and claims that the output of M consists of all and only the arithmetic truths that a human (like Lucas), or the totality of human mathematicians, will ever or can ever know. We assume that the output of M is consistent.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 4 , Issue 3 , September 1998 , pp. 273 - 302
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Benacerraf, P., God, the devil, and Gödel, The Monist, vol. 51 (1967), pp. 9–32.Google Scholar
[2] Boolos, G., Introductory note to [11], in [12], (1995), pp. 290–304.Google Scholar
[3] Boyer, D., J. R. Lucas, Kurt Gödel, and Fred Astaire, Philosophical Quarterly, vol. 33 (1983), pp. 147–159.CrossRefGoogle Scholar
[4] Chalmers, D. J., Minds, machines, and mathematics, Psyche, vol. 2 (1995), no. 9, http://psyche.cs.monash.edu.au/v2/psyche-2-09-chalmers.html.Google Scholar
[5] Detlefsen, M., Hilbert's program, D. Reidel Publishing Company, Dordrecht, 1986.Google Scholar
[6] Dummett, M., The philosophical significance of Gödel's theorem, Ratio, vol. 5 (1963), pp. 140–155.Google Scholar
[7] Dummett, M., Reply to Wright, The philosophy of Michael Dummett (McGuinness, B. and Oliveri, G., editors), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994, p. 329–338.Google Scholar
[8] Feferman, S., Arithmetization of mathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.CrossRefGoogle Scholar
[9] Feferman, S., Transfinite recursive progressions of axiomatic theories, Journal of Symbolic Logic, vol. 27 (1962), pp. 259–316.Google Scholar
[10] Feferman, S., Turing in the land of O(z), The universal Turing machine (Herken, R., editor), Oxford University Press, New York, 1988, pp. 113–147.Google Scholar
[11] Gödel, K., Some basic theorems on the foundations of mathematics and their implications, in [12], (1951), pp. 304–323.Google Scholar
[12] Gödel, K., Collected works III, Oxford University Press, Oxford, 1995.Google Scholar
[13] Hofstadter, D., Gödel, Escher, Bach, Basic Books, New York, 1979.Google Scholar
[14] Kreisel, G., Which number theoretic problems can be solved in recursive progressions on paths through O?, Journal of Symbolic Logic, vol. 37 (1972), pp. 311–334.Google Scholar
[15] Kripke, S., Wittgenstein on rules and private language, Harvard University Press, Cambridge, Massachusetts, 1982.Google Scholar
[16] Lakatos, I., Proofs and refutations (Worrall, J. and Zahar, E., editors), Cambridge University Press, Cambridge, 1976.Google Scholar
[17] Lakatos, I., Mathematics, science and epistemology (Worrall, J. and Currie, G., editors), Cambridge University Press, Cambridge, 1978.Google Scholar
[18] Lucas, J. R., Minds, machines, and Gödel, Philosophy, vol. 36 (1961), pp. 112–137.Google Scholar
[19] Lucas, J. R., Minds, machines, and Gödel: A retrospect, Machines and thought: The legacy of Alan Turing, Volume 1 (Millican, P. J. R. and Clark, A., editors), Oxford University Press, Oxford, 1996.Google Scholar
[20] McCarthy, T. and Shapiro, S., Turing projectibility, Notre Dame Journal of Formal Logic, vol. 28 (1987), pp. 520–535.Google Scholar
[21] Penrose, R., The emperor's new mind: Concerning computers, minds, and the laws of physics, Oxford University Press, Oxford, 1989.Google Scholar
[22] Penrose, R., Shadows of the mind: A search for the missing science of consciousness, Oxford University Press, Oxford, 1994.Google Scholar
[23] Penrose, R., Beyond the doubting of a shadow: A reply to commentaries on ‘Shadows of the mind’, Psyche, vol. 2 (1996), no. 23, http://psyche.cs.monash.edu.au/v2/psyche-2-23-penrose.html.Google Scholar
[24] Putnam, H., Minds and machines, Dimensions of mind: A symposium (Hood, Sidney, editor), New York University Press, New York, 1960, pp. 138–164.Google Scholar
[25] Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[26] Shapiro, S., Epistemic and intuitionistic arithmetic, Intensional mathematics (Shapiro, S., editor), North-Holland Publishing Company, Amsterdam, 1985, pp. 11–46.Google Scholar
[27] Smullyan, R., Gödel's incompleteness theorems, Oxford University Press, Oxford, 1992.Google Scholar
[28] Tennant, N., Anti-realism and logic, Oxford University Press, Oxford, 1987.Google Scholar
[29] Tennant, N., The taming of the true, Oxford University Press, Oxford, 1997.Google Scholar
[30] Turing, A., Systems of logic based on ordinals, Proceedings of the London Mathematical Society, vol. 45 (1939), pp. 161–228.Google Scholar
[31] Turing, A., Computing machinery and intelligence, Mind, vol. 59 (1950), pp. 433–460.Google Scholar
[32] Wang, H., From mathematics to philosophy, Routledge and Kegan Paul, London, 1974.Google Scholar
[33] Webb, J., Mechanism, mentalism andmetamathematics: An essay on finitism, D. Reidel, Dordrecht, Holland, 1980.Google Scholar