Notre Dame Journal of Formal Logic

Church's Thesis and the Conceptual Analysis of Computability

Michael Rescorla

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.

Article information

Source
Notre Dame J. Formal Logic Volume 48, Number 2 (2007), 253-280.

Dates
First available in Project Euclid: 16 May 2007

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1179323267

Digital Object Identifier
doi:10.1305/ndjfl/1179323267

Mathematical Reviews number (MathSciNet)
MR2306396

Zentralblatt MATH identifier
1139.03027

Subjects
Primary: 00A30: Philosophy of mathematics [See also 03A05] 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 03D10: Turing machines and related notions [See also 68Q05]

Keywords
Church's thesis computation acceptable notation Turing machines

Citation

Rescorla, Michael. Church's Thesis and the Conceptual Analysis of Computability. Notre Dame Journal of Formal Logic 48 (2007), no. 2, 253--280. doi:10.1305/ndjfl/1179323267. http://projecteuclid.org/euclid.ndjfl/1179323267.


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