David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Analysis 71 (1):22 - 30 (2011)
Many of our concepts are introduced to us via, and seem only to be constrained by, roughand-ready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (from numbers to numbers) – i.e. one whose value for any given input can be determined by a step-by-step calculating procedure, where each step is fully determined by some antecedently given finite set of calculating rules. We are told that we are to abstract from practical considerations of how many steps will be needed and how much ink will be spilt in the process, so long as everything remains finite. We are also told that each step is to be ‘small’ and the rules governing it must be ‘simple’, available to a cognitively limited calculating agent: for we want an algorithmic procedure, step-by-minimal-step, to be idiot-proof. For a classic elucidation of this kind, see e.g. Rogers (1967, pp. 1–5). Church’s Thesis, in one form, then claims this informally explicated concept in fact has a perfectly precise extension, the set of recursive functions. Church’s Thesis can be supported in a quasi-empirical way, by the failure of our searches for counterexamples. It can be supported too in a more principled way, by the observation that different appealing ways of sharpening up the informal chararacterization of algorithmic computability end up specifying the same set of recursive functions. But such considerations fall short of a demonstration of the Thesis. So is there a different argumentative strategy we could use, one that could lead to a proof? Sometimes it is claimed that there just can’t be, because you can never really prove results involving an informal concept like algorithmic computability..
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