Identifying finite cardinal abstracts

• Published in: Philosophical studies Vol. 178; no. 5; pp. 1603 - 1630
• Main Author: Ebels-Duggan, Sean C.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer 01.05.2021; Springer Netherlands
• Subjects: Education; Epistemology; Ethics; Metaphysics; Philosophy; Philosophy of Language; Philosophy of Mind; Abstractionism; Abstract objects; Philosophy of mathematics; Cross-sortal identity; Structuralism; Neo-Fregeanism
• ISSN: 0031-8116; 1573-0883
• DOI: 10.1007/s11098-020-01503-1

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.