Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

introduction 11terms of their consequences. Realism in Mathematics aims at providing both anaturalistic epistemology that replaces Gödel’s intuition as well as a detailedstudy of the practice of extrinsic justification. It is this latter aspect of theproject that leads Maddy, in Chapter 4, to quite interesting methodologicalstudies which involve, among other things, the study of the following notionsand aspects of mathematical methodology: verifiable consequence; powerfulnew methods for solving pre-existing problems; simplifying and systematizingtheories; implying previous conjectures; implying ‘natural’ results; strongintertheoretic connections; and providing new insights into old theorem (seeMaddy, 1990, pp. 145–6). These are all aspects of great importance for aphilosophy of mathematics that wants to account for mathematical practice.Maddy’s study in Chapter 4 focuses on justifying new axioms for set theory(V = L or SC [there exists a supercompact cardinal]). In the end, her analysisof the contemporary situation leads to a request for a more profound analysisof ‘theory formation and confirmation’:What’s needed is not just a description of non-demonstrative arguments, butan account of why and when they are reliable, an account that should help settheorists make a rational choice between competing axiom candidates. (Maddy,1990, p.148)And this is described as an open problem not just for the ‘compromiseplatonist’ but for a wide spectrum of positions. Indeed, on p. 180, sherecommends engagement with such problems of rationality ‘even to thosephilosophers blissfully uninvolved in the debate over Platonism’ (p. 180).In Naturalism in Mathematics (1997), the realism defended in Realism inMathematics is abandoned. But certain features of how mathematical practiceshould be accounted for are retained. Indeed, what seemed a self-standingmethodological problem in the first book becomes for Maddy the key problemof the new book and a problem that leads to the abandonment of realismin favor of naturalism. This takes place in two stages. First, she criticizes thecogency of indispensability arguments. Second, she positively addresses thekinds of considerations that set-theorists bring to bear when considering newaxioms, the status of statements independent of ZFC, or when debating newmethods, and tries to abstract from them more general methodological maxims.Her stand on the relation between philosophy and mathematics is clear andit constitutes the heart of her naturalism:If our philosophical account of mathematics comes into conflict with successfulmathematical practice, it is the philosophy that must give. This is not, in itself,a philosophy of mathematics; rather, it is a position on the proper relationsbetween the philosophy of mathematics and the practice of mathematics. Similar