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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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Definitions of Numbers and Their Applications

Definitions of Numbers and Their Applications

Chapter:
17 (p.332) Definitions of Numbers and Their Applications
Source:
Abstractionism
Author(s):

Bob Hale

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199645268.003.0017

Everyone agrees that the applicability of mathematics is of enormous importance, and at the very least demands explanation. But should the very possibility of application be somehow built into definitions or explanations of the fundamental notions—for example, the notions of natural or (finite) cardinal number and real number, in the case of arithmetic and analysis—of the mathematical theories which are so widely and successfully applied? Frege’s apparent insistence on such a tight connection between definitions and applications has been labeled Frege’s Constraint. I review the constraint, and raise some obvious questions about it. I then outline some definitions of natural and real numbers which conform to Frege’s Constraint, and some which violate it. I discuss a recent attempt by Crispin Wright to show that Frege’s Constraint should be respected in defining cardinal numbers, but not when it comes to defining real numbers, and try to show that it is unsuccessful. I then consider whether Frege himself offered any convincing justification for his constraint, with largely negative results. Finally, I propose an alternative approach.

Keywords:   neo-Fregean, Frege, application constraint, natural numbers, real numbers

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