Chapter

Reals by Abstraction

Bob Hale

in The Reason's Proper Study

Published in print March 2001 | ISBN: 9780198236399
Published online November 2003 | e-ISBN: 9780191597565 | DOI: http://dx.doi.org/10.1093/0198236395.003.0016
Reals by Abstraction

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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

Chapter.  11657 words. 

Subjects: Philosophy of Mathematics and Logic

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