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?
My first opinion is that this is a great question! :-) My second opinion is that topos theory gives a way to make Dan's insight precise. Galois theory and the theory of Riemann surfaces do this too, but in a more limited, less mind-blowing way. With topos theory we can distinguish between two kinds of two-element sets: those for which it's possible to distinguish between the elements, and those for which it's not possible. The relevant geometrical model is a double cover of the circle: locally, each fiber consists of two points, but if you go around the loop you find that the two points have traded places. This is just the standard Riemann surfaces picture of the two-valuedness of the square root function. But with topos theory, this kind of ambiguity gets elevated to the level of ontology, which is the sort of thing I think Dan had in mind. FYI, what little I know of topos theory I picked up "on the street" as it were, so others who know more should do me the favor of alleviating my ignorance. Jim