As an umbrella term, "structuralism" has to cover realist and nonrealist versions, while also carrying through the theme of its name nontrivially (for there is a trivial way to make structuralism true: just make the meaning of the word "structure" sufficiently ambiguous and vague and voila! everything can be made into an example of a pattern, even an empty or trivial pattern (with no or only one object of the pattern)). Can this be done by:
- Using a truth-functional semantics for the domain of discourse? Where the meaningfulness of component terms is read primarily down from the encompassing truth-conditions rather than up from individual-object referents of the terms.
- Propositionalism? Where the primary objects of mathematics are propositions, either "objectively" as abstract truth-theoretic objects or fictionalistically, as assertoric functions in fictional discourse.
Then the meaning of "2," for example, comes from the truth-conditions for propositions involving that very number, e.g. 2 + 2 = 4 is part of its "essential character," etc. (indeed, since 2 by 2 is 4 no matter which main-sequence arithmetic operator we use, this is arguably quite essentially characteristic of 2).