The Anti-Mechanist Argument Based on Gödel’s Incompleteness Theorems, Indescribability of the Concept of Natural Number and Deviant Encodings
PDF

Keywords

the Lucas-Penrose argument
the Church-Turing thesis
Carnapian explications
natural numbers
computation
conceptual engineering
conceptual fixed points
conceptual vicious circles
deviant encodings
structuralism

Abstract

DOI: http://doi.org/10.26333/sts.xxxiv1.11

This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski (2020): this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics.

PDF

References

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

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 (Eds.), Collected Works, Volume III, Unpublished Essays and Lectures (pp. 290–304).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

Button, T., Smith, P. (2012): The Philosophical Significance of Tennenbaum’s Theorem. Philosophia Mathematica, 20(1), 114–121.

Cappelen, H. (2018). Fixing Language: An Essay on Conceptual Engineering. Oxford University Press.

Cappelen H., Plunkett D. & Burgess A. (Eds.). (2020). Conceptual Engineering and Conceptual Ethics. Oxford University Press.

Carnap, R. (1950). Logical Foundations of Probability. Routledge and Kegan Paul.

Copeland, J., Proudfoot, D. (2010). Deviant Encodings and Turing’s Analysis of Computability. Studies in History and Philosophy of Science, 41, 247–252.

Cuneo T., Shafer-Landau, R. (2014). The Moral Fixed Points: New Directions for Moral Nonnaturalism. Philosophical Studies, 171, 399–443.

Dean, W. (2014), Models and Computability. Philosophia Mathematica, 22(2), 143–166.

Eklund, M. (2015). Intuitions, Conceptual Engineering, and Conceptual Fixed Points. In C. Daly (Ed.), The Palgrave Handbook of Philosophical Methods (pp. 363–385). London: Palgrave Macmillan.

Feferman, S. (1995). Penrose’s Gödelian Argument. Psyche: An Interdisciplinary Journal of Research on Consciousness, 2, 21–32.

Gödel, K. (193?), Undecidable Diophantine Propositions. In S. Feferman et al. (Eds), Collected Works, Volume III, Unpublished Essays and Lectures (pp. 164–175). Oxford University Press.

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

Halbach, V., Horsten, L. (2005). Computational Structuralism. Philosophia Mathematica, 13(2), 174–186.

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

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, 173–181.

Krajewski, S. (2020). On the Anti-Mechnist Arguments Based on Gödel’s Theorem. Studia Semiotyczne, 34(1), 9–56.

Lucas, J. R. (1961). Minds, Machines and Gödel. Philosophy, 36(137), 112–127.

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

Lucas, J. R. (1990). A Paper to Read to the Turing Conference at Brighton on April 6th, 1990. Retrieved from: http://users.ox.ac.uk/~jrlucas/Godel/brighton.html

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

Maddy P. (2007). Second Philosophy. A Naturalistic Method. Oxford University Press.

Makovec, D., Shapiro S. (Eds.). (2019). Friedrich Waismann. The Open Texture of Analytic Philosophy. New York: Springer.

Nagel, E., Newman J. R. (1958). Gödel’s Proof. New York University Press.

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

Penrose, R. (1989). The Emperor’s New Mind: Concerning Computers, Minds and The Laws of Physics. Oxford University Press.

Penrose, R. (1994). Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press.

Piccinini, G. (2010). Computation in Physical Systems. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.

Piccinini, G. (2015). Physical Computation: A Mechanistic Account. Oxford University Press.

Plunkett D., Cappelen, H. (2020). A Guided Tour of Conceptual Engineering and Conceptual Ethics. In: H. Cappelen, D. Plunkett, A. Burgess (Eds.), Conceptual Engineering and Conceptual Ethics (pp. 1–26). Oxford University Press.

Post, E. (1941). Absolutely Unsolvable Problems and Relatively Undecidable Propositions—Account of an Anticipation. In M. Davis (Ed.), The Undecidable (pp. 338–433). Hewlett, N. Y.: Raven Press.

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. (1980). Models and Reality. Journal of Symbolic Logic, 45(3), 464–482.

Putnam, H. (1995). Review of The Shadows of the Mind. Bulletin of the American Mathematical Society, 32(2), 370–373.

Quine, W. V. O. (1970). Philosophy of Logic. Harvard University Press.

Quinon, P. & Zdanowski, K. (2007). Intended Model of Arithmetic. Argument from Tennenbaum’s Theorem. In S. B. Cooper et al. (Eds.), Computation and Logic in the Real World (pp. 313–317). Berlin: Springer-Verlag.

Quinon, P. (2014). From Computability Over Strings of Characters to Natural Numbers. In A. Olszewski, B. Brożek, P. Urbańczyk (Eds.), Church’s Thesis, Logic, Mind & Nature (pp. 310– 330). Warsaw: Copernicus Center Press.

Quinon, P. (2018). Taxonomy of Deviant Encodings. In: F. Manea, R. Miller, D. Nowotka (Eds.), Sailing Routes in the World of Computation (pp. 338–348). Berlin: Springer-Verlag.

Quinon, P. (2019). Can Church’s Thesis be Viewed as a Carnapian Explication? Synthese, Online First.

Quinon, P. (2020). Implicit and Explicit Examples of the Phenomenon of Deviant Encodings. Studies in Logic, Grammar and Rhetoric, 63(76), 53–68.

Rescrola, M. (2007), Church’s Thesis and the Conceptual Analysis of Computability. Notre Dame Journal of Formal Logic, 48(2), 253–280.

Shapiro, S. (1982). Acceptable Notation. Notre Dame Journal of Formal Logic, 23(1), 14–20.

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. (2013). Computability, Proof and Open-texture. In A. Olszewski, J. Wolenski, R. Janusz (Eds.), Church’s Thesis After 70 Years (pp. 420–455). Berlin: Walter de Gruyter.

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

Turkle, S. (1978). Psychoanalytic Politics: Jacques Lacan and Freud’s French Revolution. New York: Basic Books.

Turkle, S. (1984). The Second Self: The Second Self: Computers and the Human Spirit. MIT Press.

Turkle, S. (2011). Alone Together: Why We Expect More from Technology and Less from Each Other. New York: Basic Books.

Turkle, S. (2015). Reclaiming Conversation: The Power of Talk in a Digital Age. London: Penguin Press.

van Heuveln, B. (2000). Emergence and Consciousness: Explorations Into the Philosophy of Mind via the Philosophy of Computation [Unpublished Ph.D. thesis]. State University of New York, Binghampton.

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