Identifying finite cardinal abstracts

Sean C. Ebels-Duggan*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of (Cook and Ebert 2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2.

Original languageEnglish (US)
JournalPhilosophical Studies
DOIs
StateAccepted/In press - 2020

Keywords

  • Abstract objects
  • Abstractionism
  • Cross-sortal identity
  • Neo-Fregeanism
  • Philosophy of mathematics
  • Structuralism

ASJC Scopus subject areas

  • Philosophy

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