I just had a look to the article The set theoretical multiverse by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, but I was fascinated by the possible philosophical perspective of being compelled (by mathematical practice of forcing in set theory) to consider a whole multiverse of sets as a natural "landscape" for set theory, without committing to any specific choice.
[There's a kind of abstract in the introduction of a n-category café blog post (which I haven't completely read yet) by David Corfield]
In the article it is also stated that it's possible to mimick the study of the "full multiverse" within ZFC. This is actually done in A natural model of the multiverse axioms: "we shall internalize the study of multiverses to set theory by treating them as mathematical objects within ZFC [...]"
Personally, I have more affinity for the formalistic viewpoint than for some version of platonism (multiversed or not). So, my first question (somehow dually to this one) is:
Is it conceivable that the "set theoretic multiverse principles" (which at the moment are, properly, ZFC sentences - see Hamkins and Gitman-Hamkins) could fit into a formal "multiverse theory" which is carried out in its own, i.e. not within a metatheory like ZFC, hence capturing the full-blown multiverse? Could such a theory be taken as the foundation (at least in some ZFC-flavoured sense) of mathematics?
(I will probably open one or more followup "philosophical" questions about multiverses, when I'll clarify to myself what to ask)