G ödel's proof of the consistency of his axiom of constructibility relative to that of the usual Zermelo—Fraenkel axioms for set theory touched off a methodological discussion about the acceptability of a theoretical statement, so to speak, within a discipline where such things are by no means common, most people probably reckoning that most twentieth century work is the continuation ad perfectionem of the nineteenth century revolution in rigour. The purpose of this paper is twofold: it is, first, to suggest that what are called the foundational aspects of mathematical research offer workable material for methodological discussion; and second, to see whether a particular account of the evolution of the empirical sciences, due to Imre Lakatos, covers also these parts of mathematics.