Is platonism regarding arithmetic consistent with the multiverse view in set theory? 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.
 A: 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


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*Solomon Feferman, Is the continuum hypothesis a definite mathematical problem? Exploring the Frontiers of Incompleteness (EFI) Project, Harvard 2011-2012. 


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:


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*Nik Weaver, Predicativity beyond $\Gamma_0$, 2005. 


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.
A: 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.

