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

IntroductionThe essays contained in this volume have the ambitious aim of bringingsome fresh air to the philosophy of mathematics. Contemporary philosophy ofmathematics offers us an embarrassment of riches. Anyone even partially familiarwith it is certainly aware of the recent work on neo-logicism, nominalism,indispensability arguments, structuralism, and so on. Much of this work can beseen as an attempt to address a set of epistemological and ontological problemsthat were raised with great lucidity in two classic articles by Paul Benacerraf.Benacerraf’s articles have been rightly quite influential, but their influence hasalso had the unwelcome consequence of crowding other important topics offthe table. In particular, the agenda set by Benacerraf’s writings for philosophyof mathematics was that of explaining how, if there are abstract objects, wecould have access to them. And this, by and large, has been the problemthat philosophers of mathematics have been pursuing for the last fifty years.Another consequence of the way in which the discussion has been framed isthat no particular attention to mathematical practice seemed to be requiredto be an epistemologist of mathematics. After all, the issue of abstract objectsconfronts us already at the most elementary levels of arithmetic, geometry,and set theory. It would seem that paying attention to other branches ofmathematics is irrelevant for solving the key problems of the discipline. Thisengendered an extremely narrow view of mathematical epistemology withinmainstream philosophy of mathematics, due partly to the over-emphasis onontological questions.The authors in this collection believe that the single-minded focus on theproblem of ‘access’ has reduced the epistemology of mathematics to a torso.They believe that the epistemology of mathematics needs to be extendedwell beyond its present confines to address epistemological issues having todo with fruitfulness, evidence, visualization, diagrammatic reasoning, understanding,explanation, and other aspects of mathematical epistemology which