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

mathematical explanation: why it matters 141completely independent of mathematical explanation of mathematical facts(indeed for Steiner the former is explicated in terms of the latter). Secondly,the vicissitudes with holism recounted above have their analog in the recentdevelopments in the philosophy of mathematics. Quine originally used theindispensability argument to argue that we should believe in sets becausethey do the best job in tracking all our commitments to abstract objects. ForQuine the appeal to empirical science was essential. Maddy’s realism drops theconnection to empirical science and tries to obtain the same conclusion justby focusing on pure mathematics. In Chapter 4 of Maddy (1990) wefindalengthy discussion of theoretical virtues, including explanatory ones, that playa role in ‘extrinsic’ justifications for axiom choice in set theory. However, theproblem of mathematical explanation was not singled out but rather dealt withat the same level of other theoretical virtues (verifiable consequences, powerfulmethods of solution, simplification and systematization, strong intertheoreticconnections, etc.). And although Maddy herself gave up the attempt in favorof ‘naturalism’ (see Maddy (1997)), mathematical explanation can still playan important role in this debate. For those who believe that her realismcan be revived perhaps the detour through indispensability arguments thatappeal to mathematical explanations might provide a more persuasive typeof argument than the other varieties of ‘extrinsic’ justifications mentionedin 1990. Moreover, those who are persuaded by the ‘naturalist’ approach ofher latest book will as a matter of fact have to welcome investigations intomathematical explanation as they are part and parcel of the kind of work themethodologist in this area ought to carry out. So both these options call for anaccount of mathematical explanations of mathematical facts.5.3 Mathematical explanations of mathematical factsThe history of the philosophy of mathematics shows that a major conceptualrole has been played by the opposition between proofs that convince but do notexplain and proofs that in addition to providing the required conviction that theresult is true also show why it is true. Philosophically this tradition begins withAristotle’s distinction between to oti and to dioti proofs and has a rich historypassing through, among others, the Logic of Port Royal written by Arnauld andNicole, Bolzano, and Cournot (see Harari (2008), Kitcher (1975) and Mancosu(1996, 1999, 2000, 2001)). This philosophical opposition between types of proofalso influenced mathematical practice and led many of its supporters often tocriticize existing mathematical practice for its epistemological inadequacy (see