The Universe of Discourse

Imaginary units, revisited
Last night, shortly after posting my article about the fact that i and -i are mathematically indistinguishable, I thought of what I should have said about it—true to form, forty-eight hours too late. Here's what I should have said.

The two square roots of -1 are indistinguishable in the same way that the top and bottom faces of a cube are. Sure, one is the top, and one is the bottom, but it doesn't matter, and it could just as easily be the other way around.

Sure, you could say something like this: "If you embed the cube in R³, then the top face is the set of points that have z-coordinate +1, and the bottom face is the set of points that have z-coordinate -1." And indeed, once you arbitrarily designate that one face is on the top and the other is on the bottom, then one is on the top, and one is on the bottom—but that doesn't mean that the two faces had any a priori difference, that one of them was intrinsically the top, or that the designation wasn't completely arbitrary; trying to argue that the faces are distinguishable, after having made an arbitrary designation to distinguish them, is begging the question.

Now can you imagine anyone seriously arguing that the top and bottom faces of a cube are mathematically distinguishable?

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