The problem with abstract objects is that their existence seems required to provide truthmakers for our propositions about them. Nominalist and other deflationary accounts can’t adequately meet this requirement. Yet how could abstract objects be real if they aren’t part of our concrete world? In modern discussions, abstract objects are causally inert by definition. If they somehow exist in some platonic realm, how could we know them in the absence of any causal connection? (A previous post on abstract objects is here).
Well, we have been considering here a model of causality that incorporates abstract modal realism. The concrete world is a causal network of events which are actualized possibilities. The set of possibilities available to be actualized in an event is constrained by preceding or adjacent events but the outcome isn’t fully determined prior to a new actualization. Unactualized possibilities may be considered “abstract” in that they are non-concrete yet real, and abstract seems as good a term as any for this mode of being (see note on terminology at end of post).
So a certain kind of abstract entity does enter into causal connections. Specifically, the concrete events of the world make contact with abstract possible events. Then the question is can we use this theory to make sense of our seeming knowledge of abstract truths, such as the prototypical logical and mathematical ones?
Well, I don’t have a developed model of how our macroscopic brain/body system would accomplish this. But given the centrality of modality to our reasoning, and given an independently motivated theory that we, as natural systems, exist in a web of actualized possibilities, we can try to connect the dots. The idea would go something like this: we have a direct primitive acquaintance with possibility, which we leverage into knowledge of idealized abstract truths through a process of counterfactual analysis. It is this notion of “primitive” acquaintance which makes this process not just a matter of conceptual or psychological construction, but a matter of reaching toward metaphysical truths.
Terminological note: it is easy to get misled by terminology here and I have probably been sloppy at times. Often in discussions of modal realism (See SEP article on Actualism), a distinction is drawn between the “actual” and “mere” possibilia. Unless one subscribes to David Lewis’ model of modal realism, where possible worlds are all concrete and the term actual is an indexical, I suggest using actual and concrete interchangeably and ask the reader’s forbearance to not misread the fact that possibilities are “non-actual” as saying they don’t exist. They exist, but are non-concrete, hence abstract.