Incompleteness, mechanism, and optimism

Bulletin of Symbolic Logic 4 (3):273-302 (1998)
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Abstract

§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 ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent.

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Stewart Shapiro
Ohio State University

Citations of this work

Arithmetical Reflection and the Provability of Soundness.Walter Dean - 2015 - Philosophia Mathematica 23 (1):31-64.
On the philosophical relevance of Gödel's incompleteness theorems.Panu Raatikainen - 2005 - Revue Internationale de Philosophie 59 (4):513-534.
Mechanism, truth, and Penrose's new argument.Stewart Shapiro - 2003 - Journal of Philosophical Logic 32 (1):19-42.
Penrose's new argument.Per Lindström - 2001 - Journal of Philosophical Logic 30 (3):241-250.

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References found in this work

Computing machinery and intelligence.Alan M. Turing - 1950 - Mind 59 (October):433-60.
Minds, Machines and Gödel.J. R. Lucas - 1961 - Etica E Politica 5 (1):1.
Transfinite recursive progressions of axiomatic theories.Solomon Feferman - 1962 - Journal of Symbolic Logic 27 (3):259-316.
God, the Devil, and Gödel.Paul Benacerraf - 1967 - The Monist 51 (1):9-32.

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