'Object' and 'criterion of identity' are philosophical terms of art whose application lies at a considerable theoretical remove from the surface phenomena of everyday linguistic usage. This partly explains their highly controversial status, for their point of application lies precisely where the concerns of linguists and philosophers of language merge with those of metaphysicians. This chapter explains the possession of determinate identity‐conditions. It argues that the distinction between 'abstract' and 'concrete' objects is itself a highly controversial one, and although it (...) has indeed been argued that this distinction turns ultimately upon differences between the criteria of identity governing objects of these two broad categories. It examines the role which criteria of identity play in our talk about objects of the least controversial varieties. The chapter concerns the problem of identity over time and the paradoxes to which identity criteria often appear to give rise when time is brought into the picture. (shrink)

Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articulating (...) and defending the idea of such objects, and outlines some problems that these strategies face. (shrink)

Bob Hale; XII*—Grundlagen §64, Proceedings of the Aristotelian Society, Volume 97, Issue 1, 1 June 1997, Pages 243–262, https://doi.org/10.1111/1467-9264.00015.

Bob Hale; XII*—Grundlagen §64, Proceedings of the Aristotelian Society, Volume 97, Issue 1, 1 June 1997, Pages 243–262, https://doi.org/10.1111/1467-9264.00015.

In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.