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

170 johannes hafner and paolo mancosuin B III and patterns in B II . But even so, by the same token as before B IIIcannot, on pain of contradiction, be embedded into a proper subset of B II(nor can it actually be apropersubsetofB II ). Because this would contradictthe status of E II (K) as the best systematization of K (relative to KII ∗ ). Theargument is exactly the same as the one in the preceding paragraph (mutatismutandis).The upshot, then, is this. The two rival systematizations E II (K) and E III (K)turn out to be incomparable within Kitcher’s framework. Neither his corollarynor our generalization of it can discriminate between them, i.e. rank themaccording to their unifying or explanatory power. Faced with a genuine issuethat arose within mathematical practice Kitcher’s model remains silent. It failsto account for and possibly even confirm Brumfiel’s position. Yet, on the otherhand, it doesn’t succeed either in revealing, as the case might be, Brumfiel’sunderlying intuitions as wrongheaded by ranking the other systematizationas in fact more explanatory after all. If we had a well-established theory ofmathematical explanation it would also possess critical or corrective potentialvis-à-vis mathematical practice. Unfortunately, at this point Kitcher’s modelis far from well-confirmed and uncontroversial. Moreover, it doesn’t evenreach a decision concerning the given issue. There is a certain irony in thissince Brumfiel champions a kind of unification of real algebraic geometry byinsisting on proofs that exhibit a ‘natural’ explanatory uniformity. Yet, despiteits focus on unification Kitcher’s account of explanation apparently does nothave the resources to provide insight into the controversy over the ‘right’ proofmethods or at least enhance our understanding of Brumfiel’s motivations. Oneof the reasons for Kitcher’s failure may lie in the fact that his account, althoughmuch more sophisticated than Friedman’s model, still shares the latter’s basicintuition, namely that unifying and explanatory power can be accounted foron the basis of quantitative comparisons alone.¹⁷ However, in the controversyover the use of transcendental methods in real algebraic geometry the pointat issue concerns qualitative differences in the proof methods. The features ofBrumfiel’s unification which he regards as explanatory escape Kitcher’s model.So we have to conclude that even under the assumption that an account ofexplanation as unification is, in principle, on the right track, Kitcher’s modeldoesn’t tell the whole story yet. In general there is more to explanation than¹⁷ At some point Kitcher does consider a modification of his account which goes beyond merequantitative comparisons, ‘so that, instead of merely counting the number of different patterns in a basis,we pay attention to similarities among them’ (Kitcher, 1981, p.521). And he mentions the existenceof a common core pattern as another criterion that determines the best systematization. However, hemakes no attempt to precisely specify this criterion nor the ways in which it is to be balanced againstthe other criteria. So it is not clear at all how it should be implemented.