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Feferman on Foundations pp 469–485Cite as

Gödel, Nagel, Minds, and Machines

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 13))

Abstract

This is the author’s slightly revised Nagel Lecture as it was published in the Journal of Philosophy CVI, 4 (April 2009), pp. 201–219. The permission of the Journal to republish the article is gratefully acknowledged.

S. Feferman—Revised text of the Ernest Nagel Lecture given at Columbia University on September 27, 2007. I wish to thank Hannes Leitgeb and Carol Rovane for their helpful comments on an earlier version.

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Notes

  1. 1.

    Nagel and Newman, Gödel’s Proof (New York: University Press, 1958); revised edition edited by Douglas R. Hofstadter (New York: University Press, 2001).

  2. 2.

    Gödel, Collected Works, Volume V: Correspondence H-Z,, Solomon Feferman et al., eds., (NewYork: Oxford, 2003), p. 135ff.

  3. 3.

    Scientific American, cxciv (June 1956): 71–86.

  4. 4.

    In The World of Mathematics: A Small Library of the Literature of Mathematics from A’ h-mosé the Scribe to Albert Einstein, Presented with Commentaries and Notes, Volume 3 (New York: Simon and Schuster, 1956), pp. 1668–1695.

  5. 5.

    Nagel and Cohen, An Introduction to Logic and Scientific Method (New York: Harcourt Brace, 1934).

  6. 6.

    Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Monatshefte fur Mathematik und Physik, xxxviii (1931): 173–198; reprinted with facing English translation in Gödel, Collected Works, Volume I: Publications 1929–1936, Feferman et al., eds. (New York: Oxford, 1986), pp. 144–195.

  7. 7.

    See Hilary Putnam, “Review of Nagel and Newman (1958),” Philosophy of Science xxvii, 2 (1960): 205–207, for a review of Nagel and Newman’s Gödel’s Proof in which several errors are identified, the most egregious being the misstatement of Gödel’s first incompleteness theorem and Rosser’s improvement thereof on p. 91.

  8. 8.

    Gödel, Collected Works, Volume IV: Correspondence A—G, Feferman et al., eds. (New York: Oxford, 2003), p. 1.

  9. 9.

    Gödel, Collected Works, Volume V: Correspondence H—Z, pp. 138–139.

  10. 10.

    Cf. Gödel, Collected Works, Volume II: Publications 1938–1974, Feferman et al., eds. (New York: Oxford, 1990), p. 166.

  11. 11.

    Gödel, Collected Works, Volume V: Correspondence H—Z, p. 147.

  12. 12.

    In Martin Davis, ed., The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions (Hewlett, NY: Raven, 1965); reproduced in Gödel, Collected Works, Volume I: Publications 1929–1936, p. 369.

  13. 13.

    Nagel and Newman, “Goedel’s Proof,” The World of Mathematics, p. 1694.

  14. 14.

    That was in the article, “Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes,” Dialectica, xii (1958): 280–287; reprinted with facing English translation in Gödel, Collected Works, Volume II: Publications 1938–1974, pp. 240–251. As a sign of his concern with the issues involved, Gödel worked on a revision of that until late in his life, 1972, “On an Extension of Finitary Mathematics Which Has Not Yet Been Used,” also in Gödel, Collected Works, Volume II, pp. 271–280. For the full story, see my forthcoming piece, “Lieber Herr Bernays! Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert’s Program,” in Horizons of Truth (Gödel centenary conference, Vienna, April 27–29, 2006).

  15. 15.

    Nagel and Newman, “Gödel’s Proof,” p. 1695.

  16. 16.

    Gödel’s lecture was the twenty-fifth in a distinguished series set up by the American Mathematical Society to honor the nineteenth century American mathematician, Josiah Willard Gibbs, famous for his contributions to both pure and applied mathematics. It was delivered to a meeting of the AMS held at Brown University on December 26, 1951. See Gödel, Collected Works, Volume III: Unpublished Essays and Lectures, Feferman et al., eds. (New York: Oxford, 1995), pp. 304–323.

  17. 17.

    Philosophia Mathematica, Series III, xiv (2006): 134–152.

  18. 18.

    In modern terms, consistency statements belong to the class \(\prod _{1}\), that is, are of the form \(\forall xR(x)\) with R primitive recursive.

  19. 19.

    Calculating machines are assimilated more closely to axiomatic systems in the concluding reflections of Nagel and Newman, Gödel’s Proof, p. 100.

  20. 20.

    Wang, From Mathematics to Philiosophy (New York: Routledge and Kegan Paul, 1974), pp. 324–326.

  21. 21.

    Wang, A Logical Journey: From Gödel to Philosophy (Cambridge: MIT, 1996), especially Chap. 6.

  22. 22.

    Kreisel, “Which Number-theoretic Problems Can Be Solved in Recursive Progressions on \(\prod ^{1}_{1}\) Paths through O?” Journal of Symbolic Logic, xxxvii (1972): 311–334, see p. 322; italics added.

  23. 23.

    Lucas, “Minds, Machines, and Gödel,” Philosophy, xxxvi (1961): 112–137.

  24. 24.

    Lucas, “Minds, Machines, and Gödel: A Retrospect,” in P.J.R. Millican and A. Clark, eds., Machines and Thought: The Legacy of Alan Turing, Volume 1 (New York: Oxford, 1996), pp. 103–124.

  25. 25.

    Penrose, The Emperor’s New Mind (New York: Oxford, 1989).

  26. 26.

    Penrose, Shadows of the Mind (New York: Oxford, 1994).

  27. 27.

    Penrose, “Beyond the Doubting of a Shadow,” Psyche, II, 1 (1996): 89–129; also at http://psyche.cs.monash.edu.au/v2/psyche-2-23-penrose.html.

  28. 28.

    Shapiro, “Mechanism, Truth, and Penrose’s New Argument,” Journal of Philosophical Logic, xxxii (2003): 19–42.

  29. 29.

    Lindström, “Penrose’s New Argument,” Journal of Philosophical Logic, xxx (2001): 241–250, and “Remarks on Penrose’s ‘New Argument’,” Journal of Philosophical Logic, xxxv (2006): 231–237.

  30. 30.

    Shapiro, “Incompleteness, Mechanism, and Optimism,” Bulletin of Symbolic Logic, iv (1998): 273–302.

  31. 31.

    Cf. Penrose, “Beyond the Doubting of a Shadow,” p. 137ff.

  32. 32.

    Feferman, “Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy”.

  33. 33.

    Cf. Feferman, “Gödel’s Program for New Axioms: Why, Where, How and What?” in Gödel ’96, P. Hajek, ed., Lecture Notes in Logic, vi(1996): 3–22, and Feferman, “Open-ended Schematic Axiom Systems (abstract),” Bulletin of Symbolic Logic, xii(2006): 145, and Feferman and Thomas Strahm, “The Unfolding of Non-finitist Arithmetic,” Annals of Pure and Applied Logic, civ (2000): 75–96.

  34. 34.

    Cf. Saunders Mac Lane, Categories for the Working Mathematician (Berlin: Springer, 1971), and Feferman, “Categorical Foundations and Foundations of Category Theory,” in R.E. Butts and J. Hintikka, eds., Logic, Foundations of Mathematics and Computability Theory, Volume I (Dordrecht: Reidel, 1977), pp. 149–165, and Feferman, “Enriched Stratified Systems for the Foundations of Category Theory,” in G. Sica, ed., What Is Category Theory? (Monza, Italy: Polimetrica, 2006), pp. 185–203.

  35. 35.

    Cf. Michael Beeson, Foundations of Constructive Mathematics (Berlin: Springer, 1985).

  36. 36.

    Feferman, “Does Mathematics Need New Axioms?” Bulletin of Symbolic Logic, vi (2000): 401–413.

  37. 37.

    As indicated in “Are There Absolutely Unsolvable Problems?”.

  38. 38.

    Cf., for example, Efraim Fischbein, Intuition in Science and Mathematics (Dordrecht: Reidel, 1987); George Lakoff and Rafael E. Núñez, Where Mathematics Comes From (New York: Basic, 2000); Paolo Mancosu, Klaus Frovin Jørgensen, Stig Andur Pedersen, eds., Visualization, Explanation and Reasoning Styles in Mathematics (Dordrecht: Springer, 2005); and George Polya, Mathematics and Plausible Reasoning, Volumes 1 and 2 (Princeton: University Press, 1968, 2\(^{nd}\) edition) among others. In addition, there is a massive amount of anecdotal evidence for the nonmechanical essentially creative nature of mathematical research; for a small sample with further references, cf., for example, Philip J. Davis, and Reuben Hersh, The Mathematical Experience (Boston: Birkhäuser, 1981), and David Ruelle, The Mathematician’s Brain (Princeton: University Press, 2007), and Benjamin H. Yandell, The Honors Class: Hilbert’s Problems and Their Solvers (Natick, MA: A.K. Peters, 2002).

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    Solomon Feferman

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  1. Institut für Informatik, Universität Bern, Bern, Switzerland

    Gerhard Jäger

  2. Department of Philosophy, Carnegie Mellon University Dept. Philosophy, Pittsburgh, Pennsylvania, USA

    Wilfried Sieg

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Feferman, S. (2017). Gödel, Nagel, Minds, and Machines. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_17

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