Did Church and Turing Have a Thesis about Machines ?

This article draws attention to a central dispute in the interpretation of Church’s Thesis. More precisely, we are concerned with the Church-Turing thesis, as it emerged in 1936 when Church endorsed Turing’s characterization of the concept of effective calculability. (The article by Sieg in this volume details this history. It is valuable also to note from Krajewski, also in this volume, that the word ‘thesis’ was used only in 1952.) This controversy has a scientific aspect, concerning the nature of the physical world and what can be done with it. It has a historical aspect, to do with the ‘confluence of ideas in 1936’. We shall focus on the historical question, but it is the continuing and serious scientific question that lends potency to the history. The principal protagonist in this matter is the philosopher B.J. Copeland, who when writing with his colleague D. Proudfoot for a wide readership in Scientific American, denounced a prevailing view of Church’s Thesis as ‘a myth’ [Copeland and Proudfoot 1999]. Copeland has made similar assertions in numerous leading articles for journals and works of reference, e.g. [Copeland 2000; 2002; 2004]. What is this myth? It is that the Church-Turing thesis places any limitation on what a machine can do. On the contrary, according to Copeland and Proudfoot, ‘Church and Turing claimed only that a universal Turing machine can match the behavior of any human mathematician working with paper and pencil in accordance with an algorithmic method—a considerably weaker claim that certainly does not rule out the possibility of hypermachines.’ ‘Hypermachines’ are defined by Copeland to be physical machines that outdo Turing