Well, classical Euclidean geometry relies on the parallel postulate, although it can't itself be proven. Abstract mathematicians have put together alternate non-euclidean geometries that are consistent but don't rely on the parallel postulate - kind of along the same lines as Nelson here "rejecting" induction just to see what would happen. I've always wondered what might happen if you "allowed", in the same sense, a square root to be a real rather than imaginary number. Maybe nothing, but I always wondered why nobody ever tried.
