Quasi-Concrete Objects

One of the central claims of Charles Parsons' remarkable new book Mathematical Thought and Its Objects is that abstract objects come in two flavors: pure and quasi-concrete. Independently of the epistemological consequences of this division, we can ask how cogent and well-motivated it really is. Abstract objects are introduced using the standard negative tests: "an object is abstract if it is not located in space and time and does not stand in causal relations" (1). Quasi-concrete objects are abstract objects that have an additional feature:

What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete "representation," such that different objects of the kind in question are distinguishable by having different representations (34).

A central example of this sort of objects is expression-types. Each token represents a type and we individuate a given type by the tokens that represent it. It is clear how the token is concrete and also fairly clear how its representation of the type is intrinsic. One could say that the token stands in the relation it does to the type solely in virture of its intrinsic features.

Still, the situation is less clear with a second central case:

Although sets are in general not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership (35).

The first objection that Parsons notes to this proposal is that the representation relationship is too different because "one element can hardly represent the set as a whole". But it seems to me that a more serious objection focuses on the intrinsic nature of the membership relation. For concrete objects do not stand in any intrinsic relationship to sets. That is, a concrete object is not the member of a set solely in virtue of its intrinsic features. If we drop this intrinsic-ness test, then the motivation for carving out the quasi-concrete objects escapes me.

It is true that an impure set stands it an essential relation to its members, and so we might say that it also stands in an intrinsic relation to its members. But this it to reverse the direction of representation that Parsons originally invokes.