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

138 paolo mancosuFirst, in the direction of a better understanding of the applicability of mathematicsto the world. Indeed, understanding the ‘unreasonable effectiveness’ ofmathematics in discovering and accounting for the laws of the physical world(Wigner, Steiner) can only be resolved if we understand how mathematicshelps in scientific explanation. Second, the study of mathematical explanationsof scientific facts will serve as a test for theories of scientific explanation,in particular those which assume that explanation is causal explanation. Apromising start has been made by Batterman through an examination of whathe calls asymptotic explanation (Batterman, 2001, Ch.4). Such explanations‘illuminate structurally stable aspects of a phenomenon and its governingequations’, (p. 59) using highly sophisticated mathematical manipulations.Third, philosophical benefits might also emerge in the metaphysical arenaby improved exploitation of various forms of the indispensability argument.Whether any such argument is going to be successful remains to be seenbut the discussion will yield philosophical benefits in forcing for instance thenominalist to take a stand on how he can account for the explanatoriness ofmathematics in the empirical sciences.5.2 From mathematical explanations of scientific factsto mathematical explanations of mathematical factsSince we have been discussing indispensability arguments I will take my startfrom there. In an interesting note to her paper Leng says:Given the form of Baker and Colyvan’s argument, one might wonder whyit is mathematical explanations of physical phenomena that get priority. For ifthere are, as we have suggested, some genuine mathematical explanations [ofmathematical facts] then these explanations must also have true explanans. Thereason that this argument can’t be used is that, in the context of an argument forrealism about mathematics, it is question begging. For we also assume here thatgenuine explanations must have a true explanandum, and when the explanandumis mathematical, its truth will also be in question. (2005, p.174)This comment reflects the general use to which indispensability argumentshave been put. The main goal is to provide an argument for platonism inmathematics but no attention is truly given to the different kind of mathematicalentities we are postulating. From this point of view the existence of the naturalnumbers is on a par with the existence of a Mahlo cardinal or of a differentiablemanifold. It is, however, reasonable to ask whether mathematical explanationscan be used not as arguments for realism in mathematics tout court but rather