When doing mathematics and providing models that satisfy a given theory, we differentiate between standard and non-standard models. Now, assume you are a platonist and believe that the objects described by these models do really exist.
Now would these standard models be "standard" because of some pragmatic or epistemic reason? For instance, it could be that the standard model of arithmetic is simply the one we are most interested in or most familiar with (or perhaps easiest to work with for some reason). It could just be, as the synonymous usage of "intended model" might suggest, something we intend to talk about for various pragmatic or epistemic reasons.
But it is also possible that the standard models are in some sense metaphysically privileged, perhaps because they are more fundamental than the non-standard models. This sort of an understanding is suggested by David Lewis when he sketches his solution of the Plus-Quus problem (found in "New Work for a Theory of Universals"; see here for a brief description of the problem). His solution involves invoking degrees of naturalness among properties. Some properties are perfectly natural, these are the most fundamental. Other properties can, however, be more or less natural than one another. Part of Lewis's theory of reference (sometimes called "reference magnetism") has it that naturalness makes properties "reference magnets", in the sense that--- ceteris paribus ---the most natural of a range of possible referents for a predicate is the best candidate for reference and the one most likely to be referred to. His solution, then, is to say that "plus" (our normal addition function) is a better candidate for reference than "quus" (which matches our addition function up until some number n no one has calculated then delivers the result 52 for any number >n) because it is more natural. Thus it seems that Lewis's solution metaphysically privileges the mathematical function (since naturalness is a metaphysical notion). A similar attitude might be taken towards standard models, that they (or the concepts they use) are more natural than non-standard models.
Are there philosophers who have explicitly discussed the metaphysical status of standard models? Are there philosophers whose views on mathematics seem to suggest something of the sort?