Turing’s thesis, by contrast, expresses that effective computability can be analyzed in terms of properties of functions defined on strings of symbols.
Thus, the two theses provide very different analyses of the concept in question.
explanation, account - a statement that makes something comprehensible by describing the relevant structure or operation or circumstances etc.; "the explanation was very simple"; "I expected a brief account" sur le developpement des relations entre le Soudan et la Turquie et sur les relations soudano-allemandes et a affirme l'importance de realiser un partenariat entre le Soudan et la Turquie ainsi que l'echange des expertises entre les deux pays.
In consequence, claims Rescorla, Turing’s thesis fails to be an analysis of computability at all.
Rescorla refers to the well known problem, sometimes called “the semantical Halting problem”, according to which if there were no “external” constraints on the choice of such a semantics—for instance, constraints on possible denotation functions that can be used for mapping from numerals to numbers—a Turing machine would compute—under some “deviant semantics”—non-computable functions on natural numbers (Shapiro Disagreements about the proper analysis of important concepts tend to linger on indefinitely, as witnessed by many similar debates in the history of philosophy.
If one assumes, as is often tacitly done, that only one analysis of a given concept can be correct, once the latter has been properly disambiguated, then Church’s analysis and Turing’s analysis cannot both be adequate.
Gödel, for whom the problem of defining computability was “an excellent example [...] of a concept which did not appear sharp to us but has become so as a result of a careful reflection” (reported by Wang , p.
Moreover, Gödel also held that it is exactly the adequacy of Turing’s thesis as a conceptual analysis that establishes the correctness of all the other equivalent mathematical definitions of computability (Gödel , p. Even Church himself acknowledged the conceptual advantages of Turing’s definition, writing in 1937 that when it comes to conceptual adequacy of his clarification of the concept of computability:, the confluence of different notions, quasi-empirical evidence, and Church’s step-by-step argument but do not ascribe it any special merit.
Logic and computer science textbooks from the decades following the pioneering work the 1930s ignore it altogether.