[...] it should be noted that the heuristic principle of my construction of undecidable number theoretical propositions in the formal systems of mathematics is the highly transfinite concept of ‘objectivemathematical truth’ as opposed to that of ‘demonstrability’ (cf. M. Davis, The Undecidable, New York 1965, p. 64, where I explain the heuristic argument by which I arrive at the incompleteness results), with which it was generally confused before my own and Tarski’s work”.
27 Note that in the case of, e. g., second-order logic the situation is different – one does not have here the completeness phenomenon.
28 Add that this formulation is consciously rather vague – e.g. it is not specified here in which formal theory (or theories) formal proofs should be constructed and what should be the underlying logic. A similar but stronger thesis was formulated by Barwise (1977, p. 41) under the name “Hilbert’s Thesis” where he wrote: “... the informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic. Following the sug[g]estion of Martin Davis, we refer to this view as Hilbert’s Thesis”. This thesis says that first-order logic is the logic of mathematics.
29 Note that in the case of Church Thesis the formal framework is precisely specified: the intuitive notion of computability should be captured by one specific formal model of computation.
←52 | 53→
The Status of Church’s Thesis
Co-authored by Jan Woleński
The Church’s Thesis can be simply stated as the following equivalence: (CT) A function is effectively computable if and only if it is partially recursive.30
Thus (CT) identifies the class of effectively computable or calculable (we will treat these two categories as equivalent) functions with the class of partially recursive functions. This means that every element belonging to the former class is also a member of the latter class and reversely. Clearly, (CT) generates an extensional co-extensiveness of effective computability and partial recursivity. Since we have no mathematical tasks, the exact definition of recursive functions and their properties is not relevant here. On the other hand, we want to stress the property of being effective computable, which plays a basic role in philosophical thinking about (CT).31
A useful notion in providing intuitions concerning effectiveness is that of an algorithm. It refers to a completely specified procedure for solving problems of a given type. Important here is that an algorithm does not require creativity, ingenuity or intuition (only the ability to recognize symbols is assumed) and that its application is prescribed in advance and does not depend upon any empirical or random factors. Moreover, this procedure is performable in a finite number of steps. Thus a function f ∶ Nn → N is said to be effectively computable (briefly: computable) if and only if its values can be computed by an algorithm. Putting this in other words: a function f ∶Nn →N is computable if and only if there exists a mechanical method by which for any n-tuple (a1, . . . , an) of arguments, the value f (a1, . . . , an) can be calculated in a finite number of prescribed steps. Three facts should be stressed here: (a) no actual human computability or empirically feasible computability is assumed in (CT); (b) functions are treated extensionally, ←53 | 54→i.e. a function is identified with an appropriate set of ordered pairs; (c) the concept of computability has a modal parameter (“there exists a method”, “a method is possible”) as its inherent feature.
Typical comments about (CT) are as follows:
(i) Church’s thesis is not a mathematical theorem which can be proved or disproved in the exact mathematical sense, for it states the identity of two notions only one of which is mathematically defined while the other is used by mathematicians without exact definition.32 (Kalmár 1959, p. 72)
(ii)While we cannot prove Church’s thesis, since its role is to delimit precisely a hitherto vaguely conceived totality, we require evidence that it cannot conflict with the intuitive notion which is supposed to be complete; i.e. we require evidence that every particular function which our intuitive notion would authenticate as effectively calculable is [...] recursive. The thesis may be considered a hypothesis about the intuitive notion of effective calculability; in the latter case, the evidence is required to give the theory based on the definition the intended significance. (Kleene 1952, pp. 318–319), Kleene 1967, p. 232)
(iii) This is a thesis rather than a theorem, in as much as it proposes to identify a somewhat intuitive concept phrased in exact mathematical terms, and thus is not susceptible of proof. But very strong evidence was adduced by Church, and subsequently by others, in support of the thesis.
It [(CT) – and other similar characterizations – our remark, R.M. and J.W.] must be accepted or rejected on grounds that are, in large part, empirical. [...]. Church’s Thesis may be viewed as a proposal as well as a claim, that we agree henceforth to supply previously intuitive terms (e.g., “function computable by algorithm”) with certain precise meaning. (Rogers 1967, p. 20)
These three quotations shed some light on several problems raised by (CT). Firstly, we can and should ask for evidence for it. 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
