Sometimes there is, I think, a resistance to Platonism about abstract objects. Here's my hunch about the cause of resistance: only substances can exist independently of other entities, such as physical objects, and perhaps immaterial substances, such as souls, if such there be. But abstract entities aren't substances; therefore, they can't exist independently of other things. But Platonism entails that they *can* exist indepedently of susbtances; therefore, Platonism is false.
I think this line of reasoning is flawed. To see this, consider the following distinction. It's a truism that everything exists. However, not everything that exists is instantiated. So, for example, the property of being a Wal*Mart one cubic foot larger than any currently existing Wal*Mart is a property that *exists*, and yet it isn't *instantiated* anywhere. Lots and lots of properties exist that aren't instantiated; in fact, many properties *can't* be instantiated (e.g., being a married bachelor). Others are actually yet contingently instantiated (e.g., the property of being the President of the United States), and some are necessarily instantiated (e.g., the second-order property of being a property is necessarily instantiated by all properties). Thus, there's a distinction between existing and being instantiated. Many things both exist and are instantiated; many others exist and are not.
In light of this distinction, I think I can put my finger on the mistake in the line of reasoning underwriting the anti-Platonistic intuition of many: they plausibly think that:
1. No property can be *instantiated* without an ontologically prior substance in which to inhere.
and conflate that with the claim that:
2. No property can *exist* without an ontologically prior substance in which to inhere.
Unfortunately, while (1) is at least plausible (although notice that it's false: recall the example of the second-order property of being a property: it's instantiated, and yet by another *property* -- not a *substance*), (2) seems outrageous.
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