a set. Solomon Feferman has suggested more recently that this sort of piecemeal conceptual analysis is already present in mathematics; and that this rather than any global foundation is the true role of foundational research. In this spirit, the relative consistency arguments of modern proof theory (a continuation of Hilbert’s Program) provide information about the epistemic grounds of various mathematical theories. Thus, on the one hand, proofs that a seemingly problematic mathematical theory is a conservative extension of a more secure theory provide some epistemic support for the former. In the other direction, the fact that classical number theory is consistent relative to intuitionistic number theory shows (contra Hilbert) that his view of constructive reasoning must differ from that of the intuitionists. Gödel, who did not believe that mathematics required any ties to empirical perception, suggested nevertheless that we have a special nonsensory faculty of mathematical intuition that, when properly cultivated, can help us decide among formally independent propositions of set theory and other branches of mathematics. Charles Parsons, in contrast, has examined the place of perception-like intuition in mathematical reasoning. Parsons himself has investigated models of arithmetic and of set theory composed of quasi-concrete objects (e.g., numerals and other signs). Others (consistent with some of Parsons’s observations) have given a Husserlstyle phenomenological analysis of mathematical intuition. Frege’s influence encouraged the logical positivists and other philosophers to view mathematical knowledge as analytic or conventional. Poincaré responded that the principle of mathematical induction could not be analytic, and Wittgenstein also attacked this conventionalism. In recent years, various formal independence results and Quine’s attack on analyticity have encouraged philosophers and historians of mathematics to focus on cases of mathematical knowledge that do not stem from conceptual analysis or strict formal provability. Some writers (notably Mark Steiner and Philip Kitcher) emphasize the analogies between empirical and mathematical discovery. They stress such things as conceptual evolution in mathematics and instances of mathematical generalizations supported by individual cases. Kitcher, in particular, discusses the analogy between axiomatization in mathematics and theoretical unification. Penelope Maddy has investigated the intramathematical grounds underlying the acceptance of various axioms of set theory. More generally, Imre Lakatos argued that most mathematical progress stems from a concept-stretching process of conjecture, refutation, and proof. This view has spawned a historical debate about whether critical developments such as those mentioned above represent Kuhn-style revolutions or even crises, or whether they are natural conceptual advances in a uniformly growing science.
See also CALCULUS , GÖDEL’S INCOMPLETE- NESS THEOREMS , HILBERT ‘S PROGRAM , LOGI – CISM , MATHEMATICAL INTUITIONISM , SET THEOR. C.J.P.