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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

Chapter:
(p.151) 6 Beyond Unification
Source:
The Philosophy of Mathematical Practice
Author(s):

Johannes Hafner

Paolo Mancosu (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199296453.003.0007

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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