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

156 johannes hafner and paolo mancosuThe quality of a generating set is inversely proportional to the number ofpatterns it contains.Thus, with paucity we have the criterion needed to proceed through step3. In order to rank the bases in step 4 we need one more thing. Define theconclusion set of a set of arguments ,C(), to be the set of sentences whichoccur as conclusions of some argument in . Kitcher concludes by giving aqualitative assessment of the unifying power of a systematization . Assumingthat we can even give precise numerical values to the number of patterns in thebasisandthesizeofC() then the degree of unification of a systematizationis directly proportional to the size of C() and inversely proportional to thenumber of patterns.Exemplification. In order to connect these definitions to our example, letK be the set of true sentences of calculus. Among such truths are ‘thetangent line passing through the parabola 2x 2 + 3x + 1 at point (1, 6) is(x − 1)7 = (y − 6)’ and ‘the tangent line passing through the curve x 3 atpoint (1, 1) is (x − 1)3 = (y − 1)’. Obviously we have an infinitude of suchtruths which can be unified by the argument pattern we have given. Eachinstance of the argument pattern gives rise to a derivation that is acceptablerelative to K. Let be the entire set of derivations in the calculus whichhave as conclusions all the truths mentioned above (C()). Then , i.e. thesingleton set containing our general argument pattern is a generating set for .Moreover, is complete with respect to K.6.2 How does Kitcher’s model fit concrete cases?In order to discuss the application of Kitcher’s model to concrete cases it isimportant to gain an insight into the sort of examples that Kitcher thinks hismodel can handle. The most extended discussion of such examples is given byKitcher in his book The Nature of Mathematical Knowledge (1984). In Chapter 9,‘Patterns of mathematical change’, one of the patterns discussed is ‘systematization’.This discussion is important for our goals as it is presented by Kitcher asan exemplification of the claims contained in ‘Explanatory unification’ (1981).Kitcher divides ‘systematizations’ into two major groups: systematizationsby axiomatization and systematizations by conceptualization. In the case ofsystematization by conceptualization he mentions the improvement obtainedby Viete’s algebra by treating systematically classes of equations of the samedegree, say, degree 3, in contrast to the case by case analysis typical ofCardan’s Ars Magna. Similarly, Lagrange—through his analysis of resolvent