R Hansens syas: >You are also confusing what I am saying with rigor. I don't care about rigor. I also don't care if it is geometric or algebraic. All I care about is backed with REASON.
>Appealing to one's rat-sense of continuous is not the same thing as appealing to one's abstract sense and theory of continuity.
No, I think you are confusing abstraction with correctness. Rigor is the accepted price of increasing confidence in correctness, but one can be correct with less (Gauss, Euler, Newton...) Rigor today is associated almost exclusively with deductive thought (this, therefore that..etc.) from primitives that are as divorced from everyday "reasoning" as possible. Hence all the angst in trying to "derive" numbers (which most people accept as obvious) from things supposedly even more primitive.
But everyday "reasoning" can also be correct, and it can be visual, or visual in part.
Pictures can be abstract as well, if I think of an infinitely long, 1 dimensional Euclidean line, that's a very abstract picture. But its still a picture, and I can think about that; I can reason about it, visually.
I can think about line segments and compare lengths and distances etc. If I see in my mind a curve that is constantly "falling off" another straight line (the log curve versus a line of constant slope 1/e), I can correctly deduce the distance between those at any x value will increase as x increases, even though that does not constitute a rigorous proof. But neither does it rely on any "abstract" sense of distance that is not itself visual.
(I've mentioned before that continuity seems to be the norm in intuition, one need not know the modern definition of continuity to get the above correct. Many mathematicians made correct conclusions before any theory of continuity existed. When you bring up such matters as "the theory of continuity", of course I'm going to think you are confusing proof and rigor with more general reasoning.)
Usually, the visual thinking will be mixed with other facts and modes of thought. Since we don't know how thought works, its pretty difficult to say more that that, but its certain that the visual apparatus of our brains is involved.
There's quite a lot of thinking involved in vision per se, such as suppression of that uninteresting details and even insertion of details that do not originate on the retina. We may as well call all that "vision" -- that "stimulus + processing" is what we experience as vision. And, I doubt I see exactly as a rat does. I've mentioned this before as well, artists and photographers *see* differently than untrained people, and so do visually oriented mathematicians.