Stewart Shapiro (Columbus, Ohio)
Computability, Open-texture, and the Church-Turing Thesis

According to Friedrich Waismann, a predicate P exhibits open-texture if there are possible objects p such that nothing in the established use of P, or the non-linguistic facts, determines that P holds of p or that P fails to hold of p. In effect, the sentence Pp is left open by the use of the language, to date. Waismann focused on empirical predicates; his target was a crude phenomenalism. In this paper, I argue that the concept of open-texture can be extended, fruitfully, to at least some mathematical predicates, in a more or less straightforward manner.

The notion of computability, at least as it was understood in the 1930's and a bit beyond, is the central case study, although other cases are not hard to find. My conclusion is that we should think of Church's thesis as a sharpening of the once open-textured notion. To echo Waismann, to ask if the notion of computability we have today, which is more or less identified with Turing computability, recursiveness, etc., is the same as the original target notion of effective computability is "to operate with too blurred an expression" . There are some interesting ramifications concerning the intuitive, or informal, notions of proof and provability, and the extent to which theses like Church's are susceptible of proof.