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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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Reals by Abstraction

Reals by Abstraction

(p.399) 15 Reals by Abstraction
The Reason's Proper Study

Bob Hale (Contributor Webpage)

Oxford University Press

The neo‐Fregean argues that principles having a certain `abstractive’ form can play a distinctive foundational role in the philosophy of mathematics––the paradigm being the reduction of elementary arithmetic to the single second‐order sentence often referred to as ’Hume's Principle’ (HP). This essay considers an abstractionist account of another mathematical domain––real analysis. The proposal has two parts: (1) an abstraction principle (’ratio abstraction’) is given, which represents the real numbers as ratios of quantities. However, this procedure is effective only if we are already given a domain of objects with a particular kind of structure. (2) A series of abstractions (starting with HP, and including a Dedekind‐inspired ’cut abstraction’) gives a domain with the required kind of structure to underpin (1).

Keywords:   abstraction principle, analysis, Dedekind, Frege, Hume’d=s Principle, quantities, ratios, real numbers

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