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

158 johannes hafner and paolo mancosuup being incomparable to an axiomatization that only uses sentences comingfrom K. This is because Kitcher’s model only deals with systematizations ofK and the arguments in K must have as premises and conclusions statementsfrom K. The problem we would like to point out can be explained as follows.Kitcher is very explicit about the fact that the new unification provided byLagrange (or Galois) in the theory of equations must use a richer language, newconcepts, and new properties of these concepts. But where earlier on we hadto account for a set K of sentences formulated in language L(K) nowwehaveasetK ∗ of sentences formulated in language L(K ∗ ). We would still like tosay that we have a better explanation of K offered by the new systematization.But Kitcher’s model seems to force us to compare only systematizations of Kamong each other and such a systematization has to appeal to sentences fromK only. Thus, in order to make Kitcher’s system more adequate to the actualsituation we face when making evaluative judgements of explanatoriness wemodify his model in such a way that a systematization of K can appeal to a classof sentences larger than K. Indeed, such a move also received textual supportfrom Kitcher himself, who seems to have recognized such need in his 1989article in Section 4.4 (p. 435).The new definition would thus require the following modifications:A systematization of K is any set of arguments which derive some sentencesin K from other sentences of K ∗ ,whereK ∗ is a consistent superset of K(possibly identical to K) and where K ∗ can be rationally accepted by thosewho accept K.A set of derivations is acceptable relative to K just in case the conclusion ofeach derivation belongs to K and every step in each derivation is deductivelyvalid and each premise of each derivation belongs to K ∗ ,whereK ∗ is aconsistent superset of K (possibly identical to K) and where K ∗ can berationally accepted by those who accept K.The advantage of this modification, to reiterate the point, is that it allowsus to exploit Kitcher’s machinery in a variety of situations which are verycommon in mathematics and in science.6.3 A test case from real algebraic geometryIn his monograph on partially ordered rings and semi-algebraic geometryGregory W. Brumfiel contrasts different methods for proving theorems aboutreal closed fields. One of them relies on a decision procedure for a particularaxiomatization of the theory of real closed fields. By this method one can find