Abstract
We sketch the historical and conceptual context of Turing's analysis of algorithmic or mechanical computation. We then discuss two responses to that analysis, by Gödel and by Gandy, both of which raise, though in very different ways. The possibility of computation procedures that cannot be reduced to the basic procedures into which Turing decomposed computation. Along the way, we touch on some of Cleland's views.
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References
Ackermaun, W. (1928), 'On Hilbert's Construction of the Real Number', in Van Heijenoort (1967), pp. 493–507.
Church, A. (1936a), 'A note on the Entscheidungsproblem', Journal of Symbolic Logic 1, pp. 40–41, pp. 101–102.
Church, A. (1936b), 'An Unsolvable Problem of Elementary Number Theory', American Journal of Mathematics 58, pp. 345–363.
Cleland, C. (1993), 'Is the Church-Turing Thesis True', Minds and Machines 3, pp. 283–312.
Cleland, C. (1995), 'Effective Procedures and Computable Functions', Minds and Machines 5, pp. 9–23.
Cleland, C. (2001), 'Recipes, Algorithms, and Programs', Minds and Machines 11, pp. 219–237.
Davis, M. (ed.) (1965), The Undecidable, Hewlett, NY: Raven Press.
Gandy, R. (1978), 'Church's Thesis and principles for mechanisms', in K. J. Barwise, H. J. Keisler, and K. Kunen, eds., The Kleene Symposium, Vol. 101 of Studies in Logic and Foundations of Mathematics, New York: North-Holland, pp. 123–148.
Gandy, R. (1988), 'The Confluence of Ideas in 1936', in R. Herken, ed., The Universal Turing Machine: A Half-Century Survey, Oxford: Oxford University Press, pp. 55–111.
Godel, K. (1931), 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems', in Van Heijenoort (1967), pp. 596–616.
Godel, K. (1958), 'Uber eine bisher noch nicht benutze Erweiterung des finites Standpunktes', Dialectica 12, pp. 280–287. Also published in Volmune II of Kurt Godel: Collected Works, Oxford University Press, 1990, S. Feferman et al., eds., pp. 240–251, with commentary by A. S. Troelstra,pp. 217–241. A later version of an authorized translation (1972), On an extemsion of finitary mathematics which has not yet been used, is reproduced in pp. 271–280.
et al., eds., pp. 369–371.
Godel, K. (1990a), 'Remarks Before the Princeton Bicentenial Conference on Problems in Mathematics', in S. Feferman, J. J. W. Dawson, S. C. Kleene, C. H. Moore, R. M. Solovay, and J. van Heijenoort, eds., Kurt Godel: Collected Works, Vol. II, New York: Oxford University Press, pp. 150–153. With introductory material by Charles Parsons, pp. 144–150.
Godel, K. (1990b), 'Some Remarks on the Undecidability Results', in S. Feferman, J. J.W. Dawson, S. C. Kleene, G. H. Moore, R. lvi. Solovay, and J. van Heijenoort, eds., Kurt Godel: Collected Works, Vol. II, New York: Oxford University Press. With introductory material by Judson Webb.
Godel, K. (1995), 'Some Basic Theorems on the Foundations of Mathematics and Their Implications', in S. Feferman, J. J. W. Dawson, W. Goldfarb, C. Parsons, and R. N. Solovay, eds., Kurt Godel: Collected Works, Vol. III, New York: Oxford University Press, pp. 304–323. With introductory material by C. Boolos, pp. 290–304.
Hilbert, D. (1904), 'On the Foundations of Logic and Arithmetic', in Van Heijenoort (1967), pp. 129–138.
Hilbert, D. (1927), 'The Foundations of Mathematics', in van Heijenoort (1967), pp. 464–479.
Minsky, M. (1967), Computation: Finite and Infinite Machines, Englewood Cliffs, NJ: Prentice-Hall.
Penrose, R. (1989), The Emperor's New Mind, London: Oxford University Press.
Rogers, H. (1967), Theory of Recursive Functions and Effective Computability, Series in Higher Mathematics, New York: McGraw-Hill.
Scott, D. (1967), 'Some Definitional Suggestions for Automata Theory', Journal of Computer and System Sciences 1, pp. 187–212.
Sieg, W. (1994), 'Mechanical Procedures and Mathematical Experience', in A. George, ed., Mathematics and Mind, Oxford: Oxford University Press.
Turing, A. (1936), 'On Computable Numbers With an Application to the Entscheidungsproblem', Proc. London Math. Soc. 42, pp. 230–265. Corrections ibid. 43 (1937), pp. 544–546. Also published in Davis (1965), pp. 155–222.
Turing, A. (1986), 'Report on the Automatic Computing Engine', in B. Carpenter and R. W. Doran, eds., A.M. Turing's ACE Report of 1946 and Other Paper's, Cambridge: Cambridge University Press, pp. 106–124.
van Heijenoort, J., ed. (1967), From Frege to Godel: A Sourcebook in Mathematical Logic, 1879– 1931, Cambridge MA: Harvard University Press.
Wang, H. (1974), From Mathematics to Philosophy, International Library of Philosophy and Scientific Method, New York: Humanities Press. See especially pp. 324–326 and notes there to.
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Israel, D. Reflections on Gödel's and Gandy's Reflections on Turing's Thesis. Minds and Machines 12, 181–201 (2002). https://doi.org/10.1023/A:1015634729532
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DOI: https://doi.org/10.1023/A:1015634729532