On the Anti-Mechanist Arguments Based on Gödel’s Theorem

Abstrakt

DOI: http://doi.org/10.26333/sts.xxxiv1.02

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent (Gödel’s Unknowability Thesis), and that we cannot fully describe our notion of a natural number.

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Bibliografia

Anderson, A. R. (Ed.). (1964). Minds and Machines. Englewood Cliffs, NJ: Prentice-Hall.

Boolos, G. (1995). Introductory Note to *1951. In: S. Feferman et al. (Eds.), Collected Works III, Unpublished Essays and Lectures (pp. 290–304). Oxford: Oxford University Press.

Boolos, G. (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press.

Benacerraf, P. (1967). God, the Devil and Gödel. The Monist, 51, 9–32.

Berto, F. (2009). There’s something about Gödel. Hoboken, New Jersey: Wiley-Blackwell.

Bowie, G. L. (1982). Lucas’ Number is Finally Up. Journal of Philosophical Logic, 11, 279–285.

Brockman, J. (Ed.). (1995). The Third Culture. New York: Simon & Schuster.

Burgess, J. (1998). Introduction to Part III. In: G. Boolos, Logic, Logic, and Logic (pp. 345–353). Cambridge, MA: Harvard University Press.

Byers, W. (2007). How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics. Princeton, NJ: Princeton University Press.

Chalmers, D. (1995). Minds, Machines, and Mathematics. PSYCHE, 2(9).

Chihara, C. (1972). On Alleged Refutations of Mechanism Using Gödel’s Incompleteness Results. Journal of Philosophy, 69(17), 507–526.

Craig, W. (1953). On Axiomatizability Within a System. Journal of Symbolic Logic, 18, 30–32.

Davis, M. (Ed.). (1965). The Undecidable. New York: Raven Press.

Descartes, R. (1637). Discourse on the Method. Leiden. Retrieved from: http://www.gutenberg.org/files/59/59-h/59-h.htm

Feferman, S. (1960). Arithmetization of Metamathematics in a General Setting. Fundamenta Mathematicae, 49, 35–92.

Feferman, S. (1962). Transfinite Recursive Progressions of Axiomatic Theories. Journal of Symbolic Logic, 27, 259–316.

Feferman, S. (1984). Kurt Gödel: Conviction and Caution. In: P. Weingartner, C. Puhringer (Eds.), Philosophy of Science—History of Science. A Selection of Contributed Papers of the 7th International Congress of Logic, Methodology and Philosophy of Science. Salzburg: Anton Hain, Meisenheim/Glan.

Feferman, S. (1988). Turing in the Land of O(z). In: R. Herken (Ed.), The Universal Turing Machine. A Half-Century Survey (pp. 113–147). Oxford: Oxfrod University Press.

Feferman, S. (1995), Penrose’s Gödelian Argument, PSYCHE, 2(7).

Feferman, S. (2006). Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy. Philosophia Mathematica, 14(2), 134–152.

Feferman, S. (2006a). The Nature and Significance of Gödel’s Incompleteness Theorems (Lecture in Princeton, 2006). Retrieved from: http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf

Feferman, S. (2007). Gödel, Nagel, Minds and Machines [Ernest Nagel Lecture]. Retrieved from: Columbia University, http://math.stanford.edu/~feferman/papers/godelnagel.pdf

Feigenbaum, E. A., & Feldman, J. (Eds.). (1995). Computers and Thought. New York: McGraw-Hill.

Franzén, T. (2004). Inexhaustibility, A Non-Exhaustive Treatment, Wellesley, MA: A K Peters.

Franzén, T. (2004a). Transfinite Progressions: A Second Look at Completeness. Bull. Symb. Log., 10(3), 367–389.

Franzén, T. (2005). Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.

Goldstein, R. (2005). Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries). New York: W. W. Norton & Company.

Good, I. J. (1967). Human and Machine Logic. British Journal for the Philosophy of Science, 18, 144–147.

Good, I. J. (1969). Gödel’s Theorem is a Red Herring. British Journal for the Philosophy of Science, 19, 357–358.

Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I [On Formally Undecidable Propositions of Principia Mathematica and Related Systems I]. Monatshefte für Mathematik und Physik, 38, 173–198.

Gödel, K. (1951). Some Basic Theorems on the Foundations of Mathematics and Their Implications. In: S. Feferman et al. (Eds.), Collected Works III, Unpublished Essays and Lectures (pp. 304–323). Oxford: Oxford University Press.

Gödel, K. (1986). Collected Works, Volume I: Publications 1929–1936. New York: Oxford University Press.

Gödel, K. (1995). Collected Works III, Unpublished Essays and Lectures. Oxford: Oxford University Press.

Hofstadter, D. R. (1979). Gödel, Escher, Bach, an Eternal Golden Braid. New York: Basic Books.

Horsten, L., & Welch, P. (Eds.). (2016). Gödel’s Disjunction. The Scope and Limits of Mathematical Knowledge, Oxford: Oxford University Press.

Kemeny, J. G. (1959). A Philosopher Looks at Science. Princeton, NJ: D. Van Nostrand.

Koellner, P. (2016). Gödel’s Disjunction. In: L. Horsten, P. Welch (Eds.), Gödel’s Disjunction. The Scope and Limits of Mathematical Knowledge (pp. 148–188). Oxford: Oxford University Press.

Koellner, P. (2018a). On the Question of Whether the Mind can be Mechanized I: From Gödel to Penrose. The Journal of Philosophy, 115(7), 337–360.

Koellner, P. (2018b). On the Question of Whether the Mind can be Mechanized II: Penrose’s New Argument. The Journal of Philosophy, 115(9), 453–484.

Krajewski, S. (1983). Philosophical Consequences of Gödel’s Theorem. Bulletin of the Section of Logic, 12, 157–164.

Krajewski, S. (1988). Twierdzenie Gödla a filozofia. Studia Filozoficzne, 6–7(271–272), 157–177.

Krajewski, S. (1993). Did Gödel Prove That We Are Not Machines? In: Z. W. Wolkowski (Ed.), First International Symposium on Gödel’s Theorems (pp. 39–49). Singapore: World Scientific Publishing Co.

Krajewski, S. (2003). Twierdzenie Gödla i jego interpretacje filozoficzne – od mechanicyzmu do postmodernizmu. Warsaw: IFiS PAN.

Krajewski, S. (2004). Gödel’s Theorem and Its Philosophical Interpretations: From Mechanism to Postmodernism (A Book Summary). Bulletin of Advanced Reasoning and Knowledge, 2, 103–108.

Krajewski, S. (2007). On Gödel’s Theorem and Mechanism: Inconsistency or Unsoundness is Unavoidable in Any Attempt to ‘Out-Gödel’ the Mechanist. Fundamenta Informaticae, 81(1–3), 173–181.

Krajewski, S. (2012). Umysł a metalogika. In: M. Miłkowski, R. Poczobut (Eds.), Przewodnik po filozofii umysłu (pp. 619–647). Kraków: WAM.

Krajewski, S. (2012a). Emergence in Mathematics? Studies in Logic, Grammar and Rhetoric, 27(40), 95–105.

Krajewski, S. (2015). Penrose’s Metalogical Argument Is Unsound. In: J. Ladyman et al. (Eds.), Road to Reality with Roger Penrose (pp. 87–104). Kraków: Copernicus Center Press.

La Mettrie, J. O. de (1748). L’homme-machine, Leiden.

Lewis, D. (1969). Lucas Against Mechanism. Philosophy, 44, 231–233.

Lewis, D. (1979). Lucas Against Mechanism II. Canadian Journal of Philosophy, 9(3), 373–376.

Lindström, P. (2001). Penrose’s New Argument. Journal of Philosophical Logic, 30, 241–250.

Lindström, P. (2006). Remarks on Penrose’s “New Argument”. Journal of Philosophical Logic, 35, 231–237.

Lucas, J. R. (1961). Minds, Machines, and Gödel. Philosophy, 36, 112–127.

Lucas, J. R. (1968). Satan Stultified: A Rejoinder to Paul Benacerraf. The Monist, 52, 145–158.

Lucas, J. R. (1970). The Freedom of the Will. Oxford: Oxford University Press.

Lucas, J. R. (1970a). Mechanism: A Rejoinder. Philosophy, 45, 149–151.

Lucas, J. R. (1996). Minds, Machines and Gödel: A Retrospect. In: P. Millican, A. Clark (Eds.), Machines and Thought (pp. 103–124). Oxford: Oxford University Press.

Lucas, J. R. (1997). The Gödelian Argument. Truth Journal. Retrieved from: http://www.leaderu.com/truth/2truth08.html

Lucas, J. R. (1998). The Implications of Gödel’s Theorem [talk given to the Sigma Club]. Retrieved from: http://users.ox.ac.uk/~jrlucas/Godel/goedhand.html

Lucas, J. R. (2000). The Conceptual Roots of Mathematics. An Essay on the Philosophy of Mathematics. London, New York: Routledge.

Matiyasevich, Y. V. (1993). Hilbert’s Tenth Problem, Cambridge, MA: MIT Press.

Nagel, E., & Newman, J. R. (1989). Gödel’s Proof. New York: New York University Press.

Nagel, E., & Newman, J. R. (1961). Answer to Putnam (1960a). Philosophy of Science, 28, 209–211.

Neumann, J. von (1966). Theory of Self-Reproducing Automata. Urbana: University of Illinois Press.

Penrose, R. (1989). Emperor’s New Mind. Oxford: Oxford University Press.

Penrose, R. (1994). Shadows of the Mind. Oxford: Oxford University Press.

Penrose, R. (1996). Beyond the Doubting of a Shadow. PSYCHE: An Interdisciplinary Journal of Research on Consciousness, 2(23).

Penrose, R. (1997). The Large, the Small and the Human Mind. Cambridge: Cambridge University Press.

Penrose, R. (2006). Lecture at Gödel Centenary Conference. Vienna.

Penrose, R. (2011). Gödel, the Mind and the Laws of Physics. In: M. Baaz, Ch. H. Papadimitriou, H. Putnam, D. Scott, Ch. Harper (Eds.), Kurt Gödel and the Foundations of Mathematic: Horizons of Truth (pp. 339–358). Cambridge: Cambridge University Press.

Post, E. (1941). Absolutely Unsolvable Problems and Relatively Undecidable Propositions—Account of an Anticipation. In: M. Davis, The Undecidable (pp. 340–433). New York: Raven Press.

Presburger, M. (1929). Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus de 1er Congrès des Mathématiciens des Pays Slaves (pp. 92–101). Warsaw.

Putnam, H. (1960). Minds and Machines. In: S. Hook (Ed.), Dimensions of Mind: A Symposium (pp. 138–164). New York: New York University Press.

Putnam, H. (1960a). Review: Nagel and Newman, Gödel’s Proof. Philosophy of Science, 27, 205–207.

Putnam, H. (1995). Review of The Shadows of the Mind. Bulletin of the American Mathematical Society, 32(3), 370–373.

Raatikainen, P. (2005). On the Philosophical Relevance of Gödel’s Incompleteness Theorems. Revue Internationale de Philosophie, 59(4), 513–534.

Reinhardt, W. N. (1986). Epistemic Theories and the Interpretation of Gödel’s Incompleteness Theorems. Journal of Philosophical Logic, 15, 427–474.

Rodriguez-Consuegra, F. A. (1995). Kurt Gödel. Unpublished Philosophical Essays. Boston: Birkhauser Verlag.

Rosenbloom, P. (1950). Elements of Mathematical Logic. New York: Dover.

Rucker, R. von (1982). Infinity and the Mind. Boston: Birkhäuser.

Searle, J. R. (1990). Is the Brain’s Mind a Computer Program? Scientific American, 1, 26–31.

Shanker, S. G. (Ed.). (1988). Gödel’s Theorem in Focus. London: Croom Helm.

Shapiro, S. (1996). The Limits of Logic. Aldershot: Dartmouth.

Shapiro, S. (1998). Incompleteness, Mechanism, and Optimism. Journal of Philosophical Logic, 4, 273–302.

Shapiro, S. (2003). Mechanism, Truth, and Penrose’s New Argument. Journal of Philosophical Logic, 32, 19–42.

Shapiro, S. (2016). Idealization, Mechanism, and Knowability. In: L. Horsten, P. Welch (Eds.), Gödel’s Disjunction. The Scope and Limits of Mathematical Knowledge (pp. 189–207). Oxford: Oxford University Press.

Slezak, P. (1982). Gödel’s Theorem and the Mind. British Journal for the Philosophy of Science, 33, 41–52.

Smart, J. J. C. (1959). Professor Ziff on Robots. Analysis, 19, 117–118.

Smart, J. J. C. (1961). Gödel’s Theorem, Church’s Thesis, and Mechanism. Synthese, 13, 105–110.

Turing, A. (1937). On Computable Numbers, With an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, s2–42(1), 230–265.

Turing, A. (1939). Systems of Logic Based on Ordinals. Proceedings of the London Mathematical Society, s2–45, 161–228.

Turing, A. (1950). Computing Machinery and Intelligence. Mind, 59, 433–460.

Wang, H. (1974). From Mathematics to Philosophy. New York: Routledge and Kegan Paul.

Wang, H. (1993). On Physicalism and Algorithmism: Can Machines Think? Philosophia Mathematica, 1, 97–138.

Wang, H. (1996). A Logical Journey. From Gödel to Philosophy. Cambridge, MA: MIT Press.

Webb, J. C. (1980). Mechanism, Mentalism, and Metamathematics. Dordrecht: Reidel.