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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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On Frege’s Applications Constraint

On Frege’s Applications Constraint

Chapter:
15 (p.311) On Frege’s Applications Constraint
Source:
Abstractionism
Author(s):

Paul McCallion

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199645268.003.0015

As Benacerraf famously observed, the natural numbers may be reduced to sets in many different ways and these various set-theoretic reductions seem to be equally appealing. There is often more than one abstractionist route to a particular mathematical theory. Are all abstractionist reductions equally appealing? An adequate answer to this question ought to be sensitive to Frege’s claim that definitions of numbers must build in their applicability. This paper examines how Frege’s claim might be motivated and how it relates to the issue raised by Benacerraf.

Keywords:   natural numbers, real numbers, complex numbers, neo-logicism, abstraction principle, applicability, application constraint

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