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.

An encyclopedia entry which covers various revisionary conceptions of which macroscopic objects there are, and the puzzles and arguments that motivate these conceptions: sorites arguments, the argument from vagueness, the puzzles of material constitution, arguments against indeterminate identity, arguments from arbitrariness, debunking arguments, the overdetermination argument, and the problem of the many.

This chapter reviews several varieties of grounding skepticism as well as responses that have been proposed by grounding enthusiasts to considerations raised by grounding skeptics. Grounding skeptics, as I conceive of them here, are theorists who belong to one of the following two schools of thought. “Old-school” grounding skeptics doubt the theoretical utility of the grounding idiom by denying one of its presuppositions, viz., that this notion is at least intelligible or coherent. “Second-generation” grounding skeptics call into question the theoretical (...) utility of the grounding idiom for other reasons; their skeptical doubts tend to focus on one of the following three purported theoretical virtues grounding enthusiasts ascribe to their idiom: (i) its alleged power to unify an apparently heterogeneous collection of phenomena; (ii) its alleged power to capture and/or elucidate the distinction between the fundamental and the non-fundamental; or (iii) the alleged metaphysical (as opposed to mind-dependent, epistemic, or psychological) utility of grounding claims. Grounding enthusiasts have already formulated responses to many of the objections described in this chapter. At this point, however, it is fair to say that the state of the literature is still evolving, and no conclusive judgment can therefore be reached as of yet as to whether grounding enthusiasts or grounding skeptics have gained the upper hand in these debates. In the meantime, though, grounding skeptics continue to maintain that the classification of factual and/or nonfactual connections under the rubric of grounding does not really help us illuminate the nature of the connections at issue; instead, from the view of the grounding skeptic, we are better off studying these various connections separately and in their own right. (shrink)

In this paper I return to the mid-20th-century debate between Quine and Carnap on the status of metaphysics questions with an eye toward advancing contemporary debates about whether naturalists can coherently undertake substantive metaphysical inquiry. Following Huw Price, I take the debate between Quine and Carnap to hinge, in part, on whether human inquiry is functionally unified. However, unlike Price, I suggest that this question is not best understood as a question about the function(s) of descriptive discourse. This goes along (...) with rejecting a “linguistic conception” of the starting point of metaphysical inquiry, which, although shared by Quine and Carnap, Price gives us no good reason to think is mandatory for naturalists. I sketch two reasons naturalists have to reject a particular manifestation of this linguistic conception in Quine’s work—his criterion of ontological commitment. Finally, I show how these reasons can help us identify the grains of truth in some recent critiques of “mainstream metaphysics of mind.”. (shrink)