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A Subject With No ObjectStrategies for Nominalistic Interpretation of Mathematics$
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John P. Burgess and Gideon Rosen

Print publication date: 1999

Print ISBN-13: 9780198250128

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198250126.001.0001

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A Mixed Modal Strategy

A Mixed Modal Strategy

(p.146) C A Mixed Modal Strategy
A Subject With No Object

John P. Burgess (Contributor Webpage)

Gideon Rosen (Contributor Webpage)

Oxford University Press

Develops a variant of the strategy of the preceding chapter, drawing on ideas of so‐called mathematical structuralism, according to which statements about ‘the’ system of real numbers are really statements about all systems of objects and relations of the right kind. A structuralist reading of standard mathematics is no more nominalistically acceptable than a straightforward reading because there generally are not any systems objects and relations of the right kind among the concrete. But the structuralist reading can be modified to make it nominalistically acceptable by bringing in modal logic, and combining it with Leśniewski's mereology and Boolos's plural quantification or both. The end result is a kind of partial vindication of logicism, in that all relevant mathematical assumptions are reduced to assumptions of a grand logic, including modality, mereology, and plural quantification.

Keywords:   Boolos, Leśniewski, logicism, mereology, modality, modal logic, plural quantification, structuralism

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