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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).

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  • 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.
    – Conifold
    Feb 26 at 1:49
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    Better ways to reframe structuralism could be predicate functor logic (PFL) or combinatorial logic (CL) on a surface syntactic (potentially shallow) level though...
    – Double Knot
    Feb 26 at 21:33

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What you propose sounds really similar to Hellman's work from the 80s, who defends a mathematical structuralism without structures using modal logic. Specifically, from Structuralism in the Philosophy of Mathematics:

Hellman’s “modal structuralism” is meant to be a systematic development of Putnam’s modalized if-then-ism. [..] For Hellman, a sentence such as “2+3=5” is analyzed as follows: Necessarily, for all relational systems M, if M is a model of the Dedekind-Peano axioms, then 2M+3M=5M. To avoid the non-vacuity problem, he adds the following assumption: Possibly, there exists an M such that M is a model of the Dedekind-Peano axioms. [..] As Hellman makes clear, his goal is to develop a form of “structuralism without structures” (Hellman 1996), since the existence of abstract structures [..] is replaced by the modal aspects of his position [..] Yet it is also not meant to rely on possibilia, i.e., possible objects existing in some shadowy sense. [..] the modalities at issue [..] are meant to be basic, i.e., the relevant possibilities and necessities are not reducible to anything further. On the other hand, they are specified precisely in terms of laws of modal logic (those of the system S5).

The main references are:

  • Hellman, Geoffrey, 1996, “Structuralism Without Structures”, Philosophia Mathematica, 4(2): 100–123. doi:10.1093/philmat/4.2.100
  • Hellman, Geoffrey and Stewart Shapiro, 2019, Mathematical Structuralism (Elements in The Philosophy of Mathematics), Cambridge: Cambridge University Press. doi:10.1017/9781108582933
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