Church's Thesis and Hume's Problem

We argue that uncomputability and classical scepticism are both re ections of inductive underdetermination, so that Church's thesis and Hume's problem ought to receive equal emphasis in a balanced approach to the philosophy of induction. As an illustration of such an approach, we investigate how uncomputable the predictions of a hypothesis can be if the hypothesis is to be reliably investigated by a computable scienti c method. 1. RELATIONS OF IDEAS AND MATTERS OF FACT Following an ancient tradition, David Hume boldly divided the objects of inquiry into two kinds: relations of ideas and matters of fact (Hume, 1984). Relations of ideas embrace all mathematical and logical inquiry, whereas matters of fact are the principal concerns of empirical science and daily life. The view that mathematics concerns relations of ideas has two important consequences. First, mathematical questions can be answered independently of all empirical data and second, the ideas upon which such questions depend can be scanned all at once by the \mind's eye," resulting in certainty concerning their relations. After a brief discussion of this happy situation in mathematics, Hume turned to the apparently more problematic case of inquiry concerning matters of fact. Here, a general law covers a potentially unbounded stream of empirical data that can refute it at any time in the future, so there is no time by which certainty about the law can be achieved. In sharp contrast with the concepts exhaustively scanned by the mathematician's inner eye, the unbounded observations relevant to an empirical generalization always outrun the scope of the empirical scientist's outer eye. Hume's problem is both an observation and a challenge. The observation is that inductive inferences can never be certain or demonstrative in the way that mathematical inferences were supposed to be and the challenge is to explain what is good about them, other than that we habitually draw them. Church's thesis is the proposition that the intuitively algorithmically computable functions are all recursive, or equivalently, Turing computable. Turing showed that the thesis follows from locality conditions and nite bounds on the resources and perceptual abilities of a human following an algorithm. The locality and perceptual boundedness assumptions are virtually identical to those invoked in classical arguments for inductive skepticism. For example: I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares. I shall also suppose that the number of smbols which may be printed is nite. If we were to allow an in nity of symbols, then there would be symbols di ering In this paper, induction will always refer to empirical inquiry rather than to mathematical induction. For an extended discussion of the relevance of Turing's argument, cf. (Sieg, 1994).