Full Idea
Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model.
Gist of Idea
Universalist Structuralism is based on generalised if-then claims, not one particular model
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
A Reaction
There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator.