In the late 80's I wrote a long essay (about 120 pages) on the "semantic foundations of mathematics." I was quite sure at the time that the failed attempts at "founding" math (by the intuitionist school, formalist school, etc.) could be empirically resolved by resorting to ostension-based semantics based on the hypostasis of extension-based empirical data. When I got into this idea, it was a time when semantic-based parallel processing was getting tossed around in the artificial intelligence world, so I figured they'd go hand in hand.

The result of it was this, that a platonic-pythagorean world of numbers need not be postulated. Those elusive sets of Cantor's infinities, Russell's paradox, Banach-Tarski's ball, the Axiom of Choice, etc. etc. need not be postulated either, i.e., in the universe of mathematical discourse. The background ontology of the foundations of math gets nominalized to syntactically defined relationships and predication over semantically well-defined objects founded on an empirical lexicon. You could say that the approach is a synthesis of intuitionism and formalism.

Do sets exist? Do numbers exist? Yes they do, according to this theory, as empirical extrapolations, in the semiotic world of language. Mathematical grammar and universe of discourse make it possible to concoct objects such as the Klein Bottle, in the same way that synthetic a posteriori statements,

*a la*Kant, are possible without any controversy whatsoever. This does not imply that Klein bottles exist as empirically ostensible objects.
As an aside, Quine's virtual theory of sets does away with sets altogether, reducing the axiomatic syntax of set theory to the predicate calculus, for example:

x ∈ A is equal by definition to (∃x) Ax.

The statement A ∩ B = ∅ is equal by definition to ~(∃x)[Ax ∧ Bx].

And so on.