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The Reason's Proper StudyEssays towards a Neo-Fregean Philosophy of Mathematics$
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Bob Hale and Crispin Wright

Print publication date: 2001

Print ISBN-13: 9780198236399

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198236395.001.0001

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On the Harmless Impredicativity of N= (Hume's Principle )

On the Harmless Impredicativity of N= (Hume's Principle )

(p.229) 10 On the Harmless Impredicativity of N= (Hume's Principle)
The Reason's Proper Study

Crispin Wright (Contributor Webpage)

Oxford University Press

To play its intended role in the derivation of the axioms of arithmetic (Frege's Theorem), Hume's Principle has to be taken to be impredicative, i.e. the cardinality operator has to be applicable to concepts under which numbers themselves fall. Michael Dummett's main objection to Crispin Wright's neo‐Fregean approach to arithmetic is that this impredicativity makes the principle subject to a vicious explanatory circularity, and hence, that it cannot justify and explain the use of the range of singular terms it is meant to introduce. Having discussed two ways of spelling out this objection––and responded to them––Wright maintains that, owing to the generality ascribed to arithmetical notions by Frege, impredicativity is integral to any reasonable Fregean account of the concept of cardinal number––and consequently, that Dummett's objection somehow has to rest on a misconception. Lastly, Wright gives a positive account of how Hume's Principle can fulfil its explanatory role by presenting a scenario in which a rational subject––competent in a suitable higher‐order logic, but innocent of the concept of cardinal number––employs the principle to acquire a grasp of the conditions for identity between numbers (of concepts) and the applications of the cardinality operator needed in the proof of Frege's Theorem, via an induction on the complexity (the ”rank”) of numerical terms.

Keywords:   arithmetic, Dummett, Frege, Hume's Principle, impredicativity, numerical terms, Wright

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