I guess I don't even really understand the objection. That's how ALL mathematics works. You specify some axioms or a construction and then reason about objects that satisfy those constraints. Some of them like the complex numbers turn out to be particularly useful.
But it's not fundamentally any different than what we do with the natural numbers. Those just feel more familiar to you.
But it's not fundamentally any different than what we do with the natural numbers. Those just feel more familiar to you.