We examine the relation between predicative recurrence and strictly finitistic tenets in the philosophy of mathematics, primarily by focusing on the role of numerical notations in computing. After an overview of Wittgenstein's ideas on the “surveyability” of notations, we analyze a subtle form of circularity in the usual justification of the primitive recursive definition of exponentiation (Isles 1992), and suggest connections with recent works on predicative recurrence (Leivant 1993b, Bellantoni & Cook 1993).