On 11/23/07, Dan Asimov <dasimov@earthlink.net> wrote:
I've often wondered if it is somehow a mistake for mathematics to distinguish between i and -i.
There is absolutely no mathematical way to distinguish between the two. Perhaps they should only be referred to as a pair, and never one at a time?
Opinions?
--Dan
"Pooh looked at his paws. He knew that one of them was the right, and he knew that when you had decided which one of them was the right, then the other one was the left, but he could never remember how to begin." [A.A.Milne (1926); http://www.everfree.ca/2007/01/pooh_on_your_shoe_kid.html] And, using elementary physics at least, there's also no way to tell the difference. But the existence of symmetry doesn't (necessarily) imply that it's a good idea to factor out the symmetry group! One situation where something analogous does routinely occur is classical projective geometry; and it's a significant nuisance when this discipline is applied to computer graphics, etc. [In practice it can be repaired by retaining the sign when normalising homogeneous coordinates --- easily overlooked in a coordinate system based on points, the only subspaces which happen not to be orientable.] Some authors have attempted to repair the omission --- see e.g. Jorge~Stolfi \sl Oriented Projective Geometry \rm Academic Press (1991); www.dcc.unicamp.br/~stolfi --- but the extra complexity doesn't engender any apparent mathematical interest, merely engineering practicality. Fred Lunnon