Is there another mathematician (likely an analyst) out there that finds this debate even more absurd with the existence of geometric measure theory? GMT bypasses all of these algebraic constructions; it finds very similar objects (currents and varifolds), but it just makes more sense to me. I never found the exterior algebra (or the Clifford algebra) to be a natural way of thinking geometrically. I do not agree that the exterior product is more natural than Jacobians and determinants. I was relieved to find that GMT cut through all of it at higher generality, at least for my purposes anyway. I don't think this belief is shared by many, since GMT is apparently notoriously incomprehensible, but hey, maybe there's someone else out there?

Yeah, while I’ve never used GMT for anything substantial I certainly like currents and densities.

That reminds me, I’ve been meaning to rewrite parts of Hormander’s epic with tools from GMT but never found the time.