Plato just won’t go away.
I’ve been thinking about whether the truthmakers for modal truths could involve abstract possible worlds, but this requires backing up a bit in order to consider the status of abstract objects in general.
Defining what it is to be "abstract" is not trivial. Lewis made this point in On the Plurality of Worlds. His breakdown of several ways to approach the question is followed by Gideon Rosen in this brief SEP article on Abstract Objects. The most often used methods are to define abstract objects in terms of what they are not, and then work out the idea using examples. Abstract objects are neither physical nor mental – usually they are thought of as unchanging and causally inert (the potential role of abstract objects in a theory of causation is something to come back to, however). Numbers and universals (“redness”, “roundness”) are paradigm examples.
This SEP article on Platonism in Metaphysics, by Mark Balaguer, summarizes the state of abstract objects in modern metaphysics. He surveys the landscape by seeing how the main candidates for abstract object status (mathematical objects, properties, propositions, possible worlds) fare under Platonism and its main rivals (nominalism, immanent realism, conceptualism). It’s an exceptionally reader friendly article, and was a helpful review for me.
One item I thought was interesting is that, according to Balaguer, the strongest argument for Platonism is a truthmaker argument. The things denoted in literally true statements (“3 is prime”) must exist as abstract objects in order to make the statement true. Alternative accounts of how these statements work or attempts to deny their truth all have problems and objections.
Of course, Platonism has strong objections. Balaguer singles out the epistemological issue as the biggest problem: if abstract objects are non-physical (and exist outside of space-time) how can we have knowledge about them? Different accounts have been proposed regarding how this can work; objections have been lodged, and the debate continues.
There is a running “meta-theme” here which I’ve been thinking about as I’ve tried to survey different topics in metaphysics (ontology, causality, modality). These metaphysical questions are difficult, and simple solutions obviously don’t work or the debates would have ended long ago. What this means to me is that the common presumption that something like physicalist monism should be the “default” metaphysical position is unfounded. More “extravagant” metaphysical systems need to be weighed in the quest to find a better mousetrap for explaining how the world works.