Church's Thesis and the Conceptual Analysis of Computability

Notre Dame Journal of Formal Logic 48 (2):253-280 (2007)
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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
University of California, Los Angeles

Citations of this work

The Physical Church–Turing Thesis: Modest or Bold?Gualtiero Piccinini - 2011 - British Journal for the Philosophy of Science 62 (4):733-769.
The Representational Foundations of Computation.Michael Rescorla - 2015 - Philosophia Mathematica 23 (3):338-366.
Against Structuralist Theories of Computational Implementation.Michael Rescorla - 2013 - British Journal for the Philosophy of Science 64 (4):681-707.

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References found in this work

Function and Concept.Gottlob Frege - 1960 - In D. H. Mellor & Alex Oliver (eds.), Properties. Oxford University Press. pp. 130-149.
X*—Mathematical Intuition.Charles Parsons - 1980 - Proceedings of the Aristotelian Society 80 (1):145-168.
Mechanical procedures and mathematical experience.Wilfried Sieg - 1994 - In Alexander George (ed.), Mathematics and Mind. Oxford University Press. pp. 71--117.

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