There are two kinds of Platonism. Both hold that there are properties. But they differ as to the grounding relation that holds between predication and property possession (I will also assume that what goes for properties goes for relations, but sometimes formulate things just in terms of properties for simplicity). Both agree that if there is a property of Fness (there might not be if F is gerrymandered or negative, on sparse Platonisms), then x is F if and only if x instantiates Fness. Deep Platonism further affirms:
- If there is a property of Fness, then the fact that x is F is grounded in the fact that x instantiates Fness.
- If there is a property of Fness, then the fact that x is F partly grounds or explains the fact that x instantiates Fness.
Deep Platonism faces two problems. The first is the Regress Problem. For if Deep Platonism is true, then "instantiates" seems non-gerrymandered and positive, and so it should correspond to a Platonic entity, the relation of instantiation. Then, the fact that x instantiates Fness will be grounded in the fact that x and Fness instantiate instantiation. But this leads to a vicious regress where each instantiation relation is grounded in the next.
The second is the Creation Problem. Everything that exists and is distinct from God is created by God. If the properties are all distinct from each other, then at most one is identical with God, and hence all but at most one property are created by God. But explanatorily prior to creating anything, will have multiple properties such as that he is able to do something and that he knows something. But how can he have those properties when there is at most one property at this point in the explanatory story?
Both problems have Deep Platonist solutions. For instance, one might say that "instantiates" is the unique non-gerrymandered and positive predicate that has no Platonic correspondent, or one might say that (1) has an exception in the case of instantiation. And one might say that God has at most one property, say divinity, and he is identical to that property. (But this, too, seems to lead to exceptions for (1), or perhaps an implausible view of what predicates correspond to properties. For there sure seem to be many other non-gerrymandered and positive predicates, like "is wise" and "is powerful", that apply to God.)
But Shallow Platonism has a particularly neat solution to both problems. There either is no regress, or if there is a regress, it is an unproblematic forward regress: because x is F, x and Fness instantiate instantiation, and because of that x, Fness and instantiation instantiate instantiation, and so on. Forward regresses are not at all problematic. And while it may be explanatorily prior to the creation of properties (or of all but one property) that God is wise, it is not explanatorily prior to the creation of properties that God instantiates wisdom.