A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a multiverse view of set theory. Since different models of ZFC can have a different arithmetic (that is, model of the natural numbers), I wonder whether platonism regarding arithmetic is consistent with the multiverse view.
The view you are suggesting is something close to what is held by Solomon Feferman, who holds that the objects and truths of arithmetic have a definite nature that is not shared when one moves up to higher-order objects, such as the collection of all sets of natural numbers. Feferman has long been known for the view that the continuum hypothesis is inherently vague, in a way that arithmetic is not, and this seems to be basically what you are talking about. See for example his article
There are several other papers for the EFI project exploring similar issues.
One interesting aspect of the view is the idea of using classical logic in the lower more-definite realm, and intuitionistic logic in the higher realm, where assertions such as the continuum hypothesis may have a less definite meaning. Nik Weaver has pointed out in the comments below that he had first proposed this dichotomizing idea in his 2005 article:
Finally, let me criticize your use of the term Platonism to imply a kind of singularist view of mathematical existence, whereas I have argued that it should instead imply only a kind of realism or definite existence. With this idea, the multiverse view itself is a kind of Platonism, where one gives up on the uniqueness of the existence of mathematical objects, but not on their objective existence. For example, on the multiverse view in set theory, there are many different concepts of set, each giving rise to its own set-theoretic universe, which are just as real as the set theory claimed by the universists.
I'm still uncertain of its appropriateness here, but since Joel asked, here is a quote from my book that discusses this issue:
The two kinds of independence … in geometry and number theory offer us strikingly different paradigms. In both cases there is broad agreement about the correct interpretation of the independence results. For instance, no one nowadays would consider it meaningful to ask whether the parallel postulate is "really true" in some universal sense; it simply holds in some two-dimensional geometries and fails in others.
In contrast, although the arithmetical expression of the consistency of $PA$ is independent of $PA$, it is still widely regarded as true. To take a more extreme example, consider the formal system $MC$ = $ZFC$ + "measurable cardinals exist". Few would suggest that the sentence ${\rm Con}(MC)$ which arithmetically expresses that $MC$ is a consistent system might lack a well-defined truth value. Yet ${\rm Con}(MC)$ is presumably independent of $PA$, indeed, presumably even independent of $ZFC$.
… Should we suppose that the continuum hypothesis, for example, has a definite truth value in a well-defined canonical model? Or is there a range of models in which the truth value of the continuum hypothesis varies, none of which has any special ontological priority?
Forcing tends to push us in the latter direction. It creates the impression that there is a range of equally valid models of $ZFC$, and that one can always pass to a larger model in which the value of $2^{\aleph_0}$ changes … In an influential series of recent papers, Hamkins has vigorously argued for the position that there is no canonical model of $ZFC$, a position that he calls "the multiverse view".
… A picture emerges [from discussion omitted here] of a mathematical universe which is composed of countable structures that have absolute properties and which includes a range of countable models of $ZFC$ in which the truth values of questions like the continuum hypothesis can vary. Thus, with regard to independence phenomena, if we take "set theory" to be the theory of surveyable collections then it has an absolute meaning and behaves like number theory, but questions like the continuum hypothesis cannot even be posed; if we take it to be the theory of individuals in some model of $ZFC$ then it has a variable meaning and behaves like geometry.
(Now that the question has been bumped up to the front page:) Re-reading the answer by Hamkins, I notice that he did not respond to the question directly. I think that both views are possible (and I personally lean toward the negative answer but I am not a professional logician).
To give an affirmative answer, one would have to accept the distinction elaborated in the other answers, following (in alphabetical order) Feferman and Weaver: there is a difference between integers and higher-order sets; the former can be seen as somehow fixed, and the latter as varying from one universe to another. One still has to account for the countable models of the Gitman-Hamkins "toy model" of the multiverse, where the integers themselves vary from univese to universe. To this, one would have to reply that all these models are non-standard. Furthermore, Hamkins mentioned that the toy model may have spurious properties that are not obligatory for a "generic" (in a generic sense) multiverse.
Giving a negative answer is more straightforward IMHO. Here one can accept the toy model at face value, and point out that the integers vary from universe to universe, and therefore are not "fixed" (defeating any Platonism/realism about the integers). Furthermore, the toy model (built via ultraproducts) has a natural predicate of standardness that turns it into a model of an axiomatic theory of nonstandard analysis such as BST (a variant of IST). More precisely, for a pair of adjacent universes, the smaller one is the standard core (in the sense of nonstandard analysis) of the larger one.
