Structuralism without structures

Philosophia Mathematica 4 (2):100-123 (1996)
Authors
Geoffrey Hellman
University of Minnesota
Abstract
Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- and fourthorder number theory. The mathenatics of many structures of central importance in functional analysis, measure theory, and topology can be recovered within essentially these frameworks.
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DOI 10.1093/philmat/4.2.100
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Plural Quantification Exposed.Øystein Linnebo - 2003 - Noûs 37 (1):71–92.
Modal Structuralism and Reflection.Sam Roberts - forthcoming - Review of Symbolic Logic:1-38.
Modal Structuralism Simplified.Sharon Berry - 2018 - Canadian Journal of Philosophy 48 (2):200-222.

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