Well, classical Euclidean geometry relies on the parallel postulate, although it can't itself be proven. Abstract mathematicians have put together alternate no

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.