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

eyond unification 157equations and permutation of equations—represents a huge step forward inour understanding of why the solution of certain equations can be reduced tothe solution of equations of lower degree:In both these cases, one adopts new language which allows for the replacementof a disparate set of questions and accepted solutions with a single form ofquestion and a single pattern of reasoning, which subsume the prior questions andsolutions. Generally, systematization by conceptualization consists in modifyingthe language to enable statements, questions, and reasonings which were formerlytreated separately to be brought together under a common formulation. The newlanguage enables us to perceive the common thread which runs through ourold problem solutions, thereby encreasing our insight into why those solutionsworked. This is especially apparent in the case of Lagrange, where, antecedently,there seems to be neither rhyme nor reason to the choice of substitutions andthus a genuine explanatory problem. (Kitcher, 1984, p.221)In ‘Explanatory unification and the causal structure of the world’ (1989), weare also presented with several examples. While many of the examples thereare only meant to bring home the point that there are non-causal explanations,some of them are relevant to our discussion. Let us mention in particular,Kitcher’s mention of Galois theory which shows ‘why equations in theseclasses [2, 3, and4] permit expressions of the roots as rational function of thecoefficients’ (Kitcher, 1989, p.425).We have spent so much time reviewing these examples given by Kitcherbecause they allow us to say something more precise about his formal modelsof unification. Recall that Kitcher starts from a consistent class of statementsK closed under logical consequence. Then E(K) is the best systematization ofK. But here immediately a problem arises. For while this approach might fitsome simple cases of axiomatization it does not, as it stands, apply to all casesof axiomatization and, we claim, to most cases of systematization by conceptualization.The problem is this. Let us envisage first a case in which Kitcher’sapproach would apply well. Suppose we have an alternative axiomatizationto Euclid, given by Euclid ∗ , which systematizes exactly the same body K ofsentences and uses different axioms (already in K). Then we could comparethe two axiomatizations with the tools given by Kitcher. We skip here overa number of other problems such as the fact that it is not clear what rolethe argument patterns are playing here as both systematizations will requireargument patterns of arbitrary complexity, unless we reduce the notion ofargument pattern to that of being a logical argument that only makes use ofthe axioms of either Euclid’s axiomatization or of Euclid ∗ ’s axiomatization.Be that as it may, any axiomatization that needs axioms formulated in a richerlanguage than those of the class K of sentences being systematized will end