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2007 Church's Thesis and the Conceptual Analysis of Computability
Michael Rescorla
Notre Dame J. Formal Logic 48(2): 253-280 (2007). DOI: 10.1305/ndjfl/1179323267

Abstract

Church's thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing's work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church's thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing's work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these syntactic entities with numbers. I argue that, in providing this correlation, we must demand that the correlation itself be computable. Otherwise, the Turing machine will compute uncomputable functions. But if we presuppose the intuitive notion of a computable relation between syntactic entities and numbers, then our analysis of computability is circular.

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Michael Rescorla. "Church's Thesis and the Conceptual Analysis of Computability." Notre Dame J. Formal Logic 48 (2) 253 - 280, 2007. https://doi.org/10.1305/ndjfl/1179323267

Information

Published: 2007
First available in Project Euclid: 16 May 2007

zbMATH: 1139.03027
MathSciNet: MR2306396
Digital Object Identifier: 10.1305/ndjfl/1179323267

Subjects:
Primary: 00A30, 03A05
Secondary: 03D10

Rights: Copyright © 2007 University of Notre Dame

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Vol.48 • No. 2 • 2007
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