## Some prefer the name Turing-Church thesis.

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[I]t is difficult to see how any language that could actually be runon a physical computer could do more than Fortran can do. The ideathat there is no such language is called Church’s thesis.(Geroch and Hartle 1986: 539)

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Nachum Dershowitz and Yuri Gurevich and (independently) Wilfried Sieghave also argued that the Church-Turing thesis is susceptible tomathematical proof. Sieg focusses on Turing’s argument I,offering an axiomatized version of it in his 2002 and 2008 (for acritique of Sieg’s approach see Shagrir 2006). In their 2008Dershowitz and Gurevich offer

Turing discussed a thesis that is closely related to Turing’sthesis, namely *for every systematic method there is acorresponding substitution-puzzle* (where‘substitution-puzzle’, like ‘computable by Turingmachine’, is a rigorously defined concept). He said:

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Notice that the Turing-Church thesis does not entail thesis M; the truth of the Turing-Church thesis is consistent with the falsity of Thesis M (in both its wide and narrow forms). A thesis concerning effective methods - which is to say, concerning procedures of a certain sort that a can carry out - carries no implication concerning the extent of the procedures that are capable of carrying out (since, for example, there might be, among a machine's repertoire of atomic operations, operations that no human being who is working effectively is able to perform). The above-mentioned evidence for the Turing-Church thesis is not also evidence for Thesis M.

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The Church-Turing thesis is a thesis about the extent of effectivemethods, and therein lies its mathematical importance. Putting thisanother way, the thesis concerns what a *human being* canachieve when working by rote, with paper and pencil (ignoringcontingencies such as boredom, death, or insufficiency of paper). Thethesis carries no implication concerning the extent of what*machines* are capable of achieving (even digital machinesacting in accordance with “explicitly stated rules” (Gregory 1987)).Because, among a machine’s repertoire of basic operations, theremay be operations that a human being working by rote with paper andpencil cannot perform.

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A significant recent contribution to the area has been made by Kripke(2013). A conventional view of the status of the Church-Turing thesisis that, while “very considerable intuitive evidence” hasamassed for the thesis, the thesis is “not a precise enoughissue to be itself susceptible to mathematical treatment”, asKripke puts it (2013: 77). Kleene gave an early expression of this nowconventional view:

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connectionist models … may possibly even challenge the strongconstrual of Church’s Thesis as the claim that the class ofwell-defined computations is exhausted by those of Turing machines.(Smolensky 1988: 3)

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Thesis M itself admits of two interpretations, according to whether the phrase 'can be calculated by a machine' is taken in the narrow sense of 'can be calculated by a machine that conforms to the physical laws (if not to the resource constraints) of the actual world', or in a wide sense that abstracts from the issue of whether or not the notional machine in question could exist in the actual world. The narrow version of thesis M is an empirical proposition whose truth-value is unknown. The wide version of thesis M is simply false. Various notional machines have been described which can calculate functions that are not Turing-machine-computable (for example, Abramson (1971), Copeland (1997), (1998c), da Costa and Doria (1991), (1994), Doyle (1982), Hogarth (1994), Pour-El and Richards (1979), (1981), Scarpellini (1963), Siegelmann and Sontag (1994), Stannett (1990), Stewart (1991); Copeland and Sylvan (1999) is a survey).