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

eyond unification 155This example was fashioned in exact analogy to the one Kitcher givesfrom Newtonian mechanics. Kitcher remarks on the difference with respect topurely logical patterns:Whereas logicians are concerned to display all the schematic premises which areemployed and to specify exactly which rules of inference are used, our exampleallows for the use of premises (mathematical assumptions) which do not occur asterms of the schematic argument, and it does not give a complete description ofthe way in which the route from [1 and 2 to 3] istogo.(Kitcher,1981, p.517f)Having defined the notion of general argument pattern we now needto capture formally the notion of explanatory store over K, E(K), whichinformally will turn out to be ‘the set of derivations that makes the besttradeoff between minimizing the number of patterns of derivation employedand maximizing the number of conclusions generated’ (Kitcher, 1989, p.432).A set of derivations is acceptable relative to K just in case every step in thederivations is deductively valid and each premise of each derivation belongsto K.A generating set for a set of derivations is a set of argument patterns suchthat each derivation in the set instantiates some pattern in the generatingset .A generating set for is complete with respect to K if and only if everyderivation which is acceptable relative to K and which instantiates a pattern in belongs to .Informally, the determination of E(K) will proceed through the following steps:(1) Select among all possible systematizations of K only the acceptablesystematizations of K, i.e. those sets of derivations that are acceptablerelative to K.(2) To each acceptable systematization selected in (1) associate the collectionof the generating sets for that systematization that are complete withrespect to K.(3) For each acceptable systematization select now a basis, where a basisis an element of the collection of the generating sets for that systematizationwhich ranks best according to the criterion of paucity ofpatterns.¹(4) Finally, rank all bases of the acceptable systematizations according totheir unifying power.¹ Here we leave out the criterion of stringency of patterns, since Kitcher does not provide a sufficientlyworked out account of how the criterion is supposed to figure in determining the unifying power of asystematization. Thus it is not at all clear how to apply it in our context. We will discuss this problemin more detail below.