# Is this a legitimate way to reframe structuralism in the philosophy of mathematics?

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:

1. 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.
2. 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).

• I don't think so. Domains of discourse are antithetical to the idea of structuralism as reducing objects to derivative placeholders, although I am not sure what "read down from the encompassing truth-conditions" means. Truth conditions in what terms? Propositionalism is also a mismatch because structuralists do want something (even if fictionalized) standing behind linguistic/descriptive content to which it can be said to refer. Feb 26 at 1:49
• Better ways to reframe structuralism could be predicate functor logic (PFL) or combinatorial logic (CL) on a surface syntactic (potentially shallow) level though... Feb 26 at 21:33