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

mathematical explanation: why it matters 135near the horizontal than near the vertical. To see this, consider a single stick witha fixed midpoint position. There are many ways this stick could be horizontal(spin it around in the horizontal plane), but only two ways it could be vertical(up or down). This asymmetry remains for positions near horizontal and vertical,as you can see if you think about the full shell traced out by the stick as it takes allpossible orientations. This is a beautiful explanation for the physical distributionof the sticks, but what is doing the explaining are broadly geometrical facts thatcannot be causes. (Lipton, 2004, pp. 9–10)In this sense mathematical explanation is explanation in natural sciencecarried out by essential appeal to mathematical facts. It is immediate from thequotation that one of the major philosophical challenges posed by mathematicalexplanations of physical phenomena is that they seem to be counterexamples tothe causal theory of explanation. The existence of mathematical explanationsof natural phenomena is widely recognized in the literature (Nerlich, 1979;Batterman, 2001; Colyvan, 2001). However, until recently very little attentionhas been devoted to them. Accounting for such explanations is not aneasy task as it requires an account of how the geometrical facts ‘represent’ or‘model’ the physical situation discussed. In short, articulating how mathematicalexplanations work in the sciences requires an account of how mathematicshooks on to reality, e.g. an account of the applicability of mathematics toreality (see Shapiro (2000), p. 35; cf.p.217). And this opens the Pandora’s boxof models, idealization, etc.In the analytic literature, a first attempt at giving an account of mathematicalexplanation of empirical phenomena was made by Steiner (1978b). Steiner’stheory of mathematical explanations in the sciences relied on his theory ofexplanation in mathematics (see below) and was not supported by a detailedset of case studies. The central idea of Steiner’s account is that a mathematicalexplanation of a physical fact is one in which when we remove the physicswhat we are left with is a mathematical explanation of a mathematicalfact. He discussed one single example, the result according to which ‘thedisplacement of a rigid body about a fixed point can always be achieved byrotating the body a certain angle about a fixed axis.’ Steiner’s discussion was acontribution to the larger worry of whether one could use the existence of suchexplanations to infer the existence of the entities mentioned in the mathematicalcomponent of the explanation. His reply was negative based on the claim thatwhat needed explanation could not even be described without use of themathematical language. Thus, the existence of mathematical explanations ofempirical phenomena could not be used to infer the existence of mathematicalentities, for this very existence was presupposed in the description of the factto be explained. Indeed, he endorsed a line of argument originating from