Penelope Maddy’s “Realism in Mathematics” has a plausible account of simple informal mathematical concepts, rooted in the fact that intuition is often less an intrinsic feature of the mind and more a learned process. In particular, parsing sense-inputs and distinctions between objects are developed over an extended process of using those senses to interact with the world. (Apparently small children and people who gain sight after being blind their whole lives can’t intuitively distinguish squares and triangles — they have to individually count the corners, because they haven’t had the visual exposure to those shapes.)
Maddy argues, I think accurately, that the basic informal concept of “set” develops from a similar process. This seems to be a physicalist more than platonist realism. I’m basically persuaded that at least finite sets are about as real as ordinary physical objects; which is to say “not entirely” (I’m not sure composites do anything ontologically that the arrangement and relationships of their parts can’t) but if I change my mind on one I’ll change my mind on the other.
With regards to non-set mathematical objects (numbers, functions, and so on) she seems to treat them as basically properties of sets, in a way that splits the difference between Frege’s analysis of the natural numbers, a sort of vaguely platonist maybe-structuralism, and ZFC constructions of objects (which she treats as convenient bookkeeping rather than ontology). It mostly counts as a canonical-if-true account as far as I’m concerned.
However, she says that numbers are properties of sets and there are no objects which are numbers, which seems like a bigger revision of the usual mathematical picture than what she talked about in the introduction. She doesn’t really cover the set of natural numbers, which is odd to me. I see why she eventually landed on a “thin realism” that’s basically indistinguishable from nonrealism.