Church's Thesis and the Conceptual Analysis of Computability

Notre Dame Journal of Formal Logic 48 (2):253-280 (2007)
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.
Keywords Church's thesis  computation  Turing machines
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DOI 10.1305/ndjfl/1179323267
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Deviant Encodings and Turing's Analysis of Computability.B. Jack Copeland & Diane Proudfoot - 2010 - Studies in History and Philosophy of Science Part A 41 (3):247-252.
Against Structuralist Theories of Computational Implementation.Michael Rescorla - 2013 - British Journal for the Philosophy of Science 64 (4):681-707.
The Representational Foundations of Computation.Michael Rescorla - 2015 - Philosophia Mathematica 23 (3):338-366.
Copeland and Proudfoot on Computability.Michael Rescorla - 2012 - Studies in History and Philosophy of Science Part A 43 (1):199-202.

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