 
Summary: KEVIN T. KELLY AND OLIVER SCHULTE
CHURCH'S THESIS AND HUME'S PROBLEM
ABSTRACT. 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 mat
ters of fact are the principal concerns of empirical science and daily life. The
view that mathematics concerns relations of ideas has two important conse
quences. 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
