 According to concept structuralism, what matters in modern axiomatic mathematics aren’t really objects, especially not problematic abstract objects.  Rather, crucial are mathematical concepts, e.g., the concept “natural number system” (or “model of the Dedekind-Peano axioms”, “progression”); similarly for the concept “complete ordered field” etc. (In Ketland 2015, Other Internet Resources, such concepts are explicated further in terms of intensional propositional functions, while Isaacson and Feferman leave their nature more open.) More precisely, what matters in the end is what follows from those concepts, in the sense of what can be derived from the corresponding axioms. It is true, as concept structuralists might admit, that often the way in which we reason in mathematics involves talking about objects falling under the relevant concepts. But as they would add, such talk can be explained away in the end (e.g., by taking up a formalist position). In this or similar forms, concept structuralism seems again a fairly widespread view among mathematicians and logicians, even though it has not been very prominent in the structuralism debates until recently.