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

142 paolo mancosufor instance Guldin’s program in the 17th century (Mancosu, 1996, 2000) andBolzano’s work in geometry and analysis (Kitcher, 1975)). Steiner’s model ofexplanation, to be discussed below, although not relying on the Aristotelianopposition, aims at characterizing the distinction between explanatory andnon-explanatory proofs.The opposition between explanatory and non-explanatory proofs is notonly a product of philosophical reflection but it confronts us as a datum frommathematical practice. A mathematician (or a community of mathematicians)might find a proof of a certain result absolutely convincing, but nonethelesshe (they) might be unsatisfied with it for it does not provide an explanation ofthe fact in question. The great mathematician Mordell, to choose one exampleamong many, mentions the phenomenon in the following passage:Even when a proof has been mastered, there may be a feeling of dissatisfactionwith it, though it may be strictly logical and convincing; such as, for example, theproof of a proposition in Euclid. The reader may feel that something is missing.The argument may have been presented in such a way as to throw no light onthe why and wherefore of the procedure or on the origin of the proof or why itsucceeds. (Mordell, 1959, p.11)This sense of dissatisfaction will often lead to a search for a more satisfactoryproof. Mathematicians appeal to this phenomenon often enough (see Hafnerand Mancosu (2005) for extended quotations from mathematical sources;and of course, their joint paper in this volume) as to make the project ofphilosophically explicating this notion an important one for a philosophicalaccount of mathematical practice. But explanations in mathematics do notonly come in the form of proofs. In some cases explanations are sought in amajor conceptual recasting of an entire discipline. In such situations the majorconceptual recasting will also produce new proofs but the explanatoriness ofthe new proofs is derivative on the conceptual recasting. This leads to a moreglobal (or holistic) picture of explanation than the one based on the oppositionbetween explanatory and explanatory proofs (in Mancosu (2001) I describein detail such a global case of explanatory activity from complex analysis;see also Kitcher (1984) and Tappenden (2005) for additional case studies).The point is that in the latter case explanatoriness is primarily a propertyof proofs, whereas in the former it is a property of the whole theory orframework and the proofs are judged explanatory on account of their beingpart of the framework. This captures well the difference between the twomajor accounts of mathematical explanation available at the moment, those ofSteiner and Kitcher. Before discussing them, I hasten to add that other modelsof scientific explanation can be thought to extend to mathematical explanation.