More or less. It should be emphasized that realism in ontology implies that the structure of the natural numbers “exists” in the same sense that a traditional Platonist would say that the number 2 does.

A similar comment goes here: an antirealist in truth value would say that the structure doesn’t exist except insofar as it is exemplified by some instantiation.

Antirealism in truth value is best illustrated by set theory. Zermelo-Fraenkel set theory can be formally augmented by either asserting or denying the axiom of choice (ZFC: Zermelo-Fraenkel with Choice, ZF!C: Zermelo-Fraenkel without Choice). Both set theories are equally valid and neither one is any “more right” than the other. A realist in truth value would say that though both are logically consistent, the axiom of choice is “really” either true or false.

Even more radically, intuitionistic and orthodox mathematics are equally right. Orthodox mathematics is pretty much what most people work with, while intuitionism is the logical underpinning of the constructivist program. This is a clearer example because orthodox set theory leads to results which are false in intutionistic mathematics (e.g.: the Banach-Tarski paradox), while intuitionism gives results which are rejected by the orthodoxy (e.g.: all functions from **R** to itself are continuous). Both bodies results on **R** are seen by non-realists in truth value (there are subtler choices than antirealism) to be equally valid descriptions of the structure **R**, while a realist in truth value would say that there is only one “real” set of real numbers (only one structure which is the “proper” set of true statements).

This is an (understandable from what I’ve said) oversimplification of traditional Platonism. Properly, Platonism fails to consistently account for what properties a given object of a theory “should” have. More to the point, if there is only one “real” number 2 and that number “is” (for purposes of argument) the set {{{}}}, then 2 has the property of containing the set {{}} and not containing the set {}. A Structuralist would say that to even talk about the natural number 2 having these properties is ill-founded, since it only exists as a slot in the natural number structure, which has no means of talking about set-theoretic containment.

Realism in ontology but antirealism truth value would boil down to “structures exist, but are not necessarily unique”. This is a pretty scarce area because of a number of thorny issues that arise very quickly. More densely populated is what I’d call realism in truth value but obtainist in truth value. This would say that both ZFC and ZF!C set theory structures exist, but in the “real world” only one obtains, which puts it in a preferred position. Basically (assuming choice holds), theories of ZF!C are just as valid as those of ZFC in the abstract, but there are no models of ZF!C in the real world. Of course, maybe the constructivists are right and neither of these two specific structures obtain. Still, the point is that many structures exist (ontologically realist) but that only one describes the real world (non-realist in truth value).

Traditional Formalism comes closest to antirealism in ontology (the abstract idea of the natural numbers is just an abstract idea) but realism in truth value (every well-formed formula has a definite truth value, independant of the investigator).

Nominalism tries to be antirealist about mathematical ontology, and it may well eliminate “numbers” from science, but it ends up being very realistic. The basic objects are points and regions of spacetime, which end up becoming the place-fillers for a self-defined structure. Those who claim to have eliminated mathematics from physics, even in principle, have a very shallow concept of mathematics indeed. By translating everything into statements about “real” points and regions of spacetime and “real” relations between them, physics itself becomes a structure which, by its very definition, exists. Ultimately it’s realist (in its way) in ontology and truth value.