Lógos and Máthma 2. Roman Murawski

We take the standard position that the implication from recursivity to computability (every recursive function is computable) is obvious and the opposite implication from computability to mathematical definition of effective calculability, i.e., recursivity (every computable function is recursive), has a sufficient justification.33 Secondly, one can ask for the fate of (CT) in some logical framework, in particular, in intuitionistic or constructive systems (see Kleene 1952, pp. 318, 509–516, Kreisel 1970,McCarty 1987), but we ←54 | 55→entirely neglect this question.34Thirdly, there are various special problems, mostly philosophical, we believe, related to (CT). Does this thesis support mechanism in the philosophy of mind or not (see Webb 1980)? How is it related to structuralism in the philosophy of mathematics (see Shapiro 1983)? We also neglect this variety of questions, except eventual parenthetical remarks aimed at exemplification. Fourthly, and this is our main concern in this paper, there arises the problem of the status of (CT). We split this topic into two subproblems. (CT) can be considered from the point of view of its function in mathematical language or various conceptual schemes. The second subproblem focuses on the character of (CT) as a statement or sentence. To be more specific, we note that one of the views considers (CT) as a definition. This gives an illustration of the former subproblem. However, independently whether (CT) has the status of a definition or not, it is captured by a sentence. Now, we can ask whether this sentence is analytic or synthetic, a priori or a posteriori .This provides an illustration of the latter subproblem. Although both subproblems are closely related, their separation, even relative, makes the analysis of the status of (CT) easier.

      The following views about the function of (CT) can be distinguished :35

(A)	(CT) is an empirical hypothesis,
(B)	(CT) is an axiom or theorem,
(C)	(CT) is a definition and
( D)	(CT) is an explication.

      Ad. (A) (CT) can be considered as referring to human (possibly idealized) abilities. Hence Church’s Thesis is connected with questions concerning relations between mathematics and material or psychic reality. The last view is represented by Post, who regarded the notion of computability as a notion of a psychological nature (Post 1965, pp. 408, 419; page-reference to the first edition):

      [...] for full generality a complete analysis would have to be made of all the possible ways in which the human mind could set up finite processes for generating sequences. [...] we have to do with a certain activity of the human mind as situated in the universe. As activity, this logico-mathematical process has certain temporal properties; as situated in the universe it has certain spatial properties.

      Post maintained that (CT) should be interpreted as a natural law and insisted on its empirical confirmation (similarly DeLong 1970, p. 195). However, one should be very careful with such claims. In fact, we have no general and commonly accepted psychological theory of human mental activities, even as far as the matter concerns ←55 | 56→the scope of computation. Hence, it is very difficult to think about (CT) as an empirical hypothesis about the human mind and its abilities. Eventually (CT) can be considered as related to mechanism in the philosophy of mind (see above) and even as confirming this view. Although we do not deny that (CT) has an application in philosophy, we do not think that this thesis as a tool for a philosophical analysis of the mind body problem functions as a genuine empirical hypothesis. We think that the use of (CT) for supporting mechanism in the philosophy of mind is very similar to deriving (or not) indeterminism from the principle of indeterminacy. We have to abstain from further remarks concerning this interpretation of (CT), because it would require a longer metaphilosophical discussion.

      Ad. (B) One should distinguish two understandings of axioms or theorems. Firstly, axioms can be considered as principal metatheoretical assumptions. When Kreisel (1970) considers (CT) as a kind of reducibility axiom for constructive mathematics, he uses the concept of axiom in this sense. Yet, we are inclined to think that his comparison of (CT) with “hypothesis” ofV = L in set theory is rather misleading. In the case of V = L, there is no need to use quotes and say “hypothesis”, because we have to do with a set-theoretical axiom, whereas Kreisel’s analysis does not result in an axiomatic system of set theory. This leads to the second use of the term "axiom", referring to an element of an axiomatic system. A similar problem arises with respect to the category of being a theorem. Mendelson (1990, p. 230) writes: