Abstract
In recent years it has been convincingly argued that the Church-Turing thesis concerns the bounds of human computability: The thesis was presented and justified as formally delineating the class of functions that can be computed by a human carrying out an algorithm. Thus the Thesis needs to be distinguished from the so-called Physical Church-Turing thesis (or Thesis M), according to which all physically computable functions are Turing computable. The latter is often claimed to be false, or, if true, contingently so. On all accounts, though, thesis M is not easy to give counterexamples to, but it is never asked why—how come that a thesis that transfers a notion from the strictly human domain to the general physical domain just happens to be so difficult to falsify (or even to be true). In this paper I articulate this question and consider several tentative answers to it.
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Notes
This claim is traditionally characterized as a thesis (as opposed to a theorem) because it relates a mathematically defined notion (e.g., that of a Turing machine) with an intuitive, un-precise notion—that of effective computability. In recent years this characterization has been challenged, e.g. by Black (2000), on the grounds that many legitimate mathematical theorems involve intuitive notions without being demoted to the status of theses. This debate need not concern us here.
As argued in Dresner (2003), during the decade following 1936 Turing has changed some of his views of the formal model he created, among which may be the exact content that should be read into his own thesis. As Piccinini notes (2003), for example, in Turing (1947) he talks of his mathematical model as capturing the scope of all digital computing machines (rather than computing humans). However, this does not bear upon the original content and justification of the Thesis, as articulated above.
In conversation.
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Acknowledgements
This research was supported by The Israeli Science Foundation (Grant No. 153/2004). I would like to thank Gualtiero Piccinini and Oron Shagrir for their comments.
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Dresner, E. Turing-, Human- and Physical Computability: An Unasked Question. Minds & Machines 18, 349–355 (2008). https://doi.org/10.1007/s11023-008-9104-8
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DOI: https://doi.org/10.1007/s11023-008-9104-8