R Hansen says: >It appears that we are talking about the exact same process, but he is clinging to this notion that because the rationalization involves a visualization then it is a visual rationalization, aka visual thinking.
I'm not clinging to anything that entire conferences and books on mathematical visualization are not devoted to. It is very funny to see you cling to the notion that you simply know better than not just me, but large numbers of practicing mathematicians.
R Hansen says: >But you would have even done this even if I placed the two bars next to each other without any mention of 4 and 5. This is what I am calling "rat-sense". This is not rational. >If however, in this example you justify to yourself the general association of each bar with its length in units (a measure) and then justify to yourself that the visual comparison of length is equivalent to the abstract comparison of length (a measure), then it is rational.
Nonsense. Measure itself is based on intuitive notions. This is true of the most formalized notions of measure, which will still reduce at best to intuitions underlying the axioms of the "foundation" of your choice.