There are plenty of accounts of what's going on with mathematics that don't have mathematical terms referring to floaty mathematical entities
Could you list the one(s) that you find convincing? (even if this is somewhat off-topic in this thread...)
What he argues is that this means that the meaning of a term depends on stuff outside your head, which is a bit different
That is, IIUC, the "meaning" of a concept is not completely defined by its place within the mind's conceptual structure. This seems correct, as the "meaning" is supposed to be about the correspondence between the map and the territory, an not about some topological property of the map.
Have a look here for a reasonable overview of philosophy of maths. Any kind of formalism or nominalism won't have floaty mathematical entities - in the former case you're talking about concrete symbols, and in the latter case about the physical world in some way (these are broad categories, so I'm being vague).
Personally, I think a kind of logical modal structuralism is on the right track. That would claim that when you make a mathematical statement, you're really saying: "It is a necessary logical truth that any system which satisfied my axioms would...
Thagard (2012) contains a nicely compact passage on thought experiments: