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

Studia Semiotyczne 34 (1):243-266 (2020)
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Abstract

This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski : 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.

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Paula Quinon
Lund University

Citations of this work

Implicit and Explicit Examples of the Phenomenon of Deviant Encodings.Paula Quinon - 2020 - Studies in Logic, Grammar and Rhetoric 63 (1):53-67.

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References found in this work

Logical foundations of probability.Rudolf Carnap - 1950 - Chicago]: Chicago University of Chicago Press.
Computing machinery and intelligence.Alan M. Turing - 1950 - Mind 59 (October):433-60.
Philosophy of logic.Willard Van Orman Quine - 1970 - Cambridge: Harvard University Press. Edited by Simon Blackburn & Keith Simmons.

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