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 also satisfy this conclusion."
So if you say "2+2 = 4", you're actually saying that if there were a system that behaved like the natural numbers (which is logically possible, so long as the axioms are consistent), then in that system two plus two would equal four.
See Hellman's "Mathematics Without Numbers" for the classic defense of this kind of position.
Thanks for the answer! But I am still confused regarding the ontological status of "2" under many of the philosophical positions. Or, better yet, the ontological status of the real numbers field R. Formalism and platonism are easy: under formalism, R is a symbol that has no referent. Under platonism, R exists in the HTW. If I understand your preferred position correctly, it says: "any system that satisfies axioms of R also satisfies the various theorems about it". But, assuming the universe is finite or discrete, there is no physical system that satisfies axioms of R. Does it mean your position reduces to formalism then?
Thagard (2012) contains a nicely compact passage on thought experiments: