BUZZ. :-D

A (non-degenerate) coordinate transformation is an isomorphism - a one-to-one, onto mapping which preserves structure.

But there is no isomorphism from the linear one-dimensional world of black-and-white to the multi-dimensional world of color.

In fact, black and white naturally embeds only as a proper subset of the entire color space.

There is exactly the same amount of information in my upside-down world as the right-way up one.

For example, if I saw a black and white picture of a tree, and then saw a color picture, a reasonable response might be, "Oh, it's red, I thought it would be green."

The color picture just gave me new information that couldn't have possibly been conveyed just in black and white, without cheats like writing "This tree is red" on the image in a black marker. :-D

(The dimensionality of the color space is hard to work out. Potentially you could mix in an arbitrary number of pure colors, each with an arbitrary frequency! But it's certainly higher than linear.)