There are no abstract objects

In Theodore Sider, John Hawthorne & Dean W. Zimmerman (eds.), Contemporary Debates in Metaphysics. Blackwell (2008)
Abstract
I explicate and defend the claim that, fundamentally speaking, there are no numbers, sets, properties or relations. The clarification consists in some remarks on the relevant sense of ‘fundamentally speaking’ and the contrasting sense of ‘superficially speaking’. The defence consists in an attempt to rebut two arguments for the existence of such entities. The first is a version of the indispensability argument, which purports to show that certain mathematical entities are required for good scientific explanations. The second is a speculative reconstruction of Armstrong's version of the One Over Many argument, which purports to show that properties and relations are required for good philosophical explanations, e.g. of what it is for one thing to be a duplicate of another.
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Meghan Sullivan (2012). The Minimal A-Theory. Philosophical Studies 158 (2):149-174.
Stephen Schiffer (2006). Replies. [REVIEW] Philosophy and Phenomenological Research 73 (1):233–243.
Stephen Schiffer (2006). Replies. [REVIEW] Philosophy and Phenomenological Research 73 (1):233-243.
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2009-05-15
Cross-posted from http://mleseminar.wordpress.com/
...

This week we discussed Cian Dorr’s ‘There are no abstract objects’, which isn’t currently available online, but is in ‘Contemporary Debates in Metaphysics’. Here’s the handout instead.

As we had Cian on the spot for this meeting, the discussion mostly took a question-and-answer format. So here are what I recorded of some questions and some answers, with a few that I didn’t get time to ask thrown in at the end. Apologies if my paraphrases of Cian's answers misrepresent him!

Q: What about people who would resist the paraphrase strategy (p.37) because they think that counterpossibles are all vacuously true (Williamson takes this line in The Philosophy of Philosophy).
A: Nominalism/anti-nominalism are both contingent theses. But even if you think that nominalism is necessary if true, there will be certain kinds of truths like ‘there are possibly some things with a number-like structure’ which can be used to ground the relevant counterfactua ... (read more)