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Abstraction and Infinity$
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Paolo Mancosu

Print publication date: 2016

Print ISBN-13: 9780198746829

Published to Oxford Scholarship Online: January 2017

DOI: 10.1093/acprof:oso/9780198746829.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 24 October 2021

In good company? On Hume’s Principle and the assignment of numbers to infinite concepts

In good company? On Hume’s Principle and the assignment of numbers to infinite concepts

(p.154) 4 In good company? On Hume’s Principle and the assignment of numbers to infinite concepts
Abstraction and Infinity

Paolo Mancosu

Oxford University Press

In this chapter Mancosu generalizes some worries, raised by Richard Heck, emerging from the theory of numerosities (discussed in Chapter 3) to a line of thought resulting in what he calls a ‘good company’ objection to Hume’s Principle (HP). The chapter is centered around five main parts. The first takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where Mancosu shows that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which one can derive the axioms of second-order arithmetic. All the principles he presents agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section Mancosu offers a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, the fifth part makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

Keywords:   Abstraction principles, Hume’s Principle, neo-logicism, cross-sortal identities, finite, infinite, good company objection, analyticity

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