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Stephen Yablo

Print publication date: 2010

Print ISBN-13: 9780199266487

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199266487.001.0001

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Abstract Objects: A Case Study

Abstract Objects: A Case Study

(p.200) 8 Abstract Objects: A Case Study

Stephen Yablo (Contributor Webpage)

Oxford University Press

Numbers have many puzzling features. Their properties are mostly essential to them, but they exist in all possible worlds. Number theory seems a priori, yet it makes existence claims and existence (setting aside the Cogito) is not supposed to be a priori knowable. If-thenism can perhaps explain the felt a priority, but it makes numerical truth relative where it seems absolute. A figuralist solution is proposed: ‘2 + 3 = 5’ seems necessary, a priori, and absolute because it has a logical truth as its assertive content. A rule that associates logical truths with each arithmetical truth is given, and also a rule that associates a logical truth (modulo concrete combinatorics) with each truth about hereditarily finite impure sets. The view that emerges takes something from Frege and something from Kant; one might call it Kantian logicism. The view is Kantian because it sees mathematics as arising out of our representations. Numbers and sets are ‘there’ because they are inscribed on the spectacles through which we see other things. It is logicist because the facts seen through our numerical spectacles are facts of first-order logic.

Keywords:   existence, indispensability, metaphor, make-believe, pretense, internal, external, numbers, sets, Kantian logicism

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