Just posting regarding an old paper that I think has some relevance: `Richness and Reflection’ contrasts motivations for reflection principles from both a multiversist and universist perspective.
I think this issue merits further examination. More generally, there are many positions between the `extremes’ of a one-universe perspective and an almost-anything-goes first-order perspective. Justification of new axioms in set theory is going to have a slightly different flavour for each.
Abstract: A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, the status of reflection principles as axioms is left open for the Zermelian.