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AbstractionismEssays in Philosophy of Mathematics$
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Philip A Ebert and Marcus Rossberg

Print publication date: 2016

Print ISBN-13: 9780199645268

Published to Oxford Scholarship Online: January 2017

DOI: 10.1093/acprof:oso/9780199645268.001.0001

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Conservativeness, Cardinality, and Bad Company

Conservativeness, Cardinality, and Bad Company

Chapter:
11 (p.223) Conservativeness, Cardinality, and Bad Company
Source:
Abstractionism
Author(s):

Roy T. Cook

Philip A. Ebert

Marcus Rossberg

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199645268.003.0011

This paper argues that abstraction principles must be conservative in a number of distinct senses if they are to be acceptable, and uses this insight to present a solution to the Bad Company Problem. Field conservativeness provides a necessary, although not sufficient, condition for an abstraction principle to be acceptable. By reflecting on what role, exactly, is played by Field conservativeness—that it tells us what criteria an abstraction principle must satisfy if it is to be compatible with non-necessary, empirical theories—a second conservativeness condition is formulated—one that tells us what criteria such a principle must satisfy if it is to be compatible with necessary, non-empirical theories (including other abstraction principles). This latter notion turns out to be intimately connected to the suggestion that acceptable abstraction principles must be strongly stable, and the paper concludes by endorsing this constraint as a solution to the Bad Company Problem.

Keywords:   logicism, abstraction, Bad Company, conservativeness

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