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

introduction 19in metaphyics when discussing fruitfulness and naturalness of concepts; andMacLarty shows how structuralism in mathematical practice can help usevaluate structuralist philosophies of mathematics. Indeed, the whole book isan attempt to expand the boundaries of epistemology of mathematics wellbeyond the problem of how we can have access to abstract entities.But coming now to the analytic developments related to Maddy’s work,I should point out that whereas Maddy has limited her investigations to settheory we take a much wider perspective on mathematical practice, drawingour case studies from geometry, complex analysis, real algebraic geometry,category theory, computer science, and mathematical physics. Once again, webelieve that while set theory is a very important subject of methodologicalinvestigation, there are central phenomena that will be missed unless wecast our net more broadly and extend our investigations to other areas ofmathematics. Moreover, while Maddy’s study of set-theoretic methodologyhas some points of contact with our investigations (evidence, fruitfulness,theory-choice) we look at a much broader set of issues that never comeup for discussion in her work (visualization, purity of methods, explanation,rigor in mathematical physics). The closest point of contact between herinvestigations and this book is probably the discussion of evidence in Avigad’sintroduction. There is also no explicit commitment in our contributions tothe form of mathematical naturalism advocated by Maddy; actually, the spiritof many of our contributions seems to go against the grain of her philosophicalposition.Let me conclude by coming back to the comparison with the situation inphilosophy of science. I mentioned at the beginning that recent philosophyof science has thrived under the interaction of traditional problems (realismvs. instrumentalism, causality, etc.) with more localized studies in the philosophiesof the special sciences. In general, philosophers of science are happy withclaiming that both areas are vital for the discipline. Corfield takes as the modelfor his approach to the philosophy of mathematics the localized studies in thephilosophy of physics, but decrees that classical philosophy of mathematicsis a useless pursuit (see Pincock (2005)). As for Maddy, she gets away fromtraditional ontological and epistemological issues (realism, nominalism, etc.)by means of her naturalism. What is distinctive in this volume is that weintegrate local studies with general philosophy of mathematics, contra Corfield,and we also keep traditional ontological and epistemological topics in play,contra Maddy.Hopefully, the reader will realize that my aim has not been to make anyinvidious comparisons but only to provide a fair account of what the previoustraditions have achieved and why we think we have achieved something worth