On the Abstractness of Mathematical Sets

Let us agree that x is concrete iff x is causally/active passive and abstract otherwise.  Many say that mathematical sets ('sets' hereafter: 'mathematical' as opposed to 'commonsense') are abstract objects, abstract entities, abstracta.  Why?

Argument One:  In set theory there are singleton sets, e.g. {Quine}.  Obviously, Quine is not identical to {Quine}.  The second is a set, the first is not.  Yet the difference cannot be the difference between two concreta.  Quine is a concretum.  Therefore, {Quine} is an abstractum.  This is of course meant generally: singletons are abstracta.  Now if singletons are abstracta, then all sets are. 

Argument Two:  In set theory there is a null set.  It is not nothing, but something despite having no members. Yet it cannot be a concrete something.  Therefore, it is an abstract something.  And if one set is abstract, all are.

Contra Argument One:  A statue and the lump of clay that constitute it are numerically distinct.  (For the one has properties the other doesn't have, e.g., the lump, but not the statue, can exist without having the form of a statue.)  And yet both are concrete, i.e., both are causally active/passive.  If this is possible, why should it not also be possible that Quine and {Quine} both be concrete?  One could say that Quine and {Quine} occupy the same 'plime' to borrow a term form D. C. Williams, the same place-time, in the way statue and lump do.

Contra Argument Two:  Possibly, there is a concrete atomic entity. Being atomic, it has no parts.  So why should a set's having no members rule out its being concrete?

Are any of these arguments compelling?

Comments

[TaiChi]

I don't have anything to say regarding argument one, except that it puzzles me that {Quine} is non-concrete, if a member and thus a part of {Quine} is the concrete Quine.
Regarding your response to Two, though: an atomic entity has no proper parts, but it does have a single improper part, which is identical with itself as a whole. So there is a disanalogy with the null set there. Your response to One suggests another worry. A set such as {Quine} seems somewhat plausible to regard as concrete because we can identify a plime for the set in question, and having a spatio-temporal location is paradigmatic of the concrete. But in the case of the null set, does it have a plime? If so, where and when is it? The best answer seems to be that it encompasses all non-existence, else there would be a multitude of these null sets. But then we arrive at the absurd point of saying that, whilst non-existence is causally inert and so non-concrete, the set which ranges over non-existence is on the contrary something concrete, fashioned out of nothingness.

[William]

Personally I would start with arguments for such things existing, before moving to a discussion of putative features or classifications of such things.

[David Brightly]

Not compelling, but interesting nevertheless. Neither argument is very clear and both contras address themselves to specific interpretations.

I find it hard to discern an argument in A1. A1 says that the difference between Quine and {Quine} is of a different nature to the difference between Quine and a fellow concretum, say, Carnap. This surely begs the question. CA1 puts some flesh on the argument. It suggests that Quine constitutes {Quine} the way a lump of clay constitutes a statue. Perhaps Quine is a material cause of {Quine}. But this is to see math sets as commonsense sets. It invites the question, What happens to {Quine} when Quine ceases to exist? Does it too cease to exist, or does it become the empty set, perhaps? This is like saying that if the number of apples in a bag is six then eating one causes the number six to become the number 5! Math sets, like numbers, are fixities through which change in changeable things can be expressed. This might form the basis of a better argument for abstractness: Concreta, being part of the causal nexus, are subject to change. Math sets are not.

A2 and CA2 appear to make an analogy between the part-of and member-of relations. Implicit in A2 is the premise that any concrete thing has a proper part. By analogy of member and part, the empty set has no parts at all and hence cannot be concrete. CA2 denies the premise, citing a possible atomic concrete entity as having no proper parts. Both arguments founder on a significant disanalogy between parts and members: although part-of is transitive, member-of is not: x∊y and y∊z does not imply x∊z.