R. Hansen says: >You used an interesting word, illustrate. Don't these illustrations in math serve essentially the same purpose as do the illustrations in a story book? And what is the crux of a story book? The story, or the illustrations? If you had to choose to keep only one, which would it be? The story or the illustrations?
My my, you do go on and on with these silly analogies, and silly distinctions based thereon. Now, if the analogy sucks, who cares about the distinctions based on it? That's what your inadequate analogies are -- essentially mental pictures -- that you yourself are basing some line of argumentation on. Now in the case of the "4 quad. sloped-line" pictures I was alluding to, they could be useful for mathematical instruction. In the case of the story book analogy, its just a useless artifact of a flimsy argument.
Everybody learns mathematics with the aid of pictures, even you. Now I'm supposing you pretend that you didn't, or that you needn't have. I hope nobody is convinced of your word games here. I'm not.
Even David Hilbert's proofs based on his famous axioms for geometry were found to have strayed at time into relying on geometric intuition. To suppose that we can simply base everything on formalism past the age of 8 or so is utter fantasy.