Johannes Hafner

The subject of mathematical explanation has received increasing attention in recent years. Philip Kitcher is a well-known defender of an account of (scientific as well as) mathematical explanation as theoretical unification. This chapter tests Kitcher's model of mathematical explanation by means of a case study from real algebraic geometry. The elementary theory RCF of real closed fields represents a unification of many scattered theorems that are proved within different real closed fields. Yet, Gregory W. Brumfiel, in his work on semi-algebraic sets, decidedly rejects RCF (together with the Tarski–Seidenberg transfer principle) as a preferred framework for proofs because such proofs are, according to Brumfiel, in general not explanatory. Instead he aims at proofs that may use non-elementary methods, but exhibit a ‘natural’ uniformity that proofs within RCF in general do not. Hence this case study shows that, as it stands, Kitcher's model of explanation does not tell the whole story.