It has dawned on me that physicists may be willing to use something logically nonsensical just because Einstein did so, and achieved correct results.
I think if the modern version of differential geometry had been around in 1900 then Einstein would have used it instead.
I agree of course that physicists are really using their intelligence and intuition, rather than mathematiocal rigor, which is why they so seldom go astray.
I love the recent story of the puzzle as to how many rational cubic curves lie on a general quintic hypersurface in complex 4 space. The mathematicians, by brute force computation had one answer, and the physicists by relating the problem to one in quantum gravity or something, had a different prediction which popped out of a recursion formula and a differential eqaution they thought applicable.
Of course the physicists were actually right, and it led to a whole industry in enumerative algebraic geometry.
We mathematicians are merely trying to formulate precisely the intuitions physicicts seem blessed with because of their familiarity with nature. We are at a big disadvantage here.
But we do not seem to argue as hopelessly as some theoretical physicists do, because we do eventually make clear what we are saying.
My error in my previous long harangue, was not to ask precisely what Pete meant by his notation, and not to say precisely what I meant by it.
That was what I meant when I expressed confidence we would agree at some level, once we understood each other properly.
I have almost never heard a disagreement that was not found to be based on different interpretations of the same words being used.