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The Philosophy of Mathematical Practice$
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Paolo Mancosu

Print publication date: 2008

Print ISBN-13: 9780199296453

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199296453.001.0001

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Beyond Unification

Beyond Unification

(p.151) 6 Beyond Unification
The Philosophy of Mathematical Practice

Johannes Hafner

Paolo Mancosu (Contributor Webpage)

Oxford University Press

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.

Keywords:   mathematical explanation, Philip Kitcher, unification, real closed fields, Gregory W. Brumfiel

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