On Jun 15, 2013, at 3:00 PM, Joe Niederberger <email@example.com> wrote:
> Of course I can. I take it *you* cannot, nor can you apparently widen the scope of your imagining beyond yourself.
Then prove it. Surely you accept that everyone on this list understands to some degree and acknowledges abstract analytical (non visual) thinking. I think the task is up to you to prove that this mysterious non abstract visual thinking exists and is up to the task of mathematics. I have found no evidence of it. Sure, there are plenty of classrooms where teachers stand in front of students and display a visual, ask the students if they "see" and the students nod their heads. But they aren't seeing math. Visualization is a lot of things to mathematics but the source of generalized abstraction in thought it is not.