R Hansen says: >Let's cut the chase. You cannot visualize a mathematical proof without connecting the dots.
(We have not been talking about proofs per se, we've been talking about visualization, and whether or not one can use visualization to think about math problems. Also, "connect the dots" means ...???)
The fact, rather than your opinion of it, *is*, that by creating a mental picture, based on things I already knew (about log, d/dx log, pi, etc.) -- by the very act of forming that picture clearly, I was able to notice something new that I did not know before. Namely that that picture also allowed me to conclude that e^pi > pi^e. Its a fact, I was there. Once I had the picture, it can be turned into a proof very easily. In your "connect the dots" parlance, I was able to connect the dots this particular way only after I had the picture in mind. The visualization was an essential part of my thought process.
That the visualization approach is a *distinctive* approach from other sorts of mathematical thinking, is easily seen. Just google the question, "which is greater, e^pi or pi^e?", and look at the answers. IF math thinking is just math thinking, (and then one just reverse-engineers the pictures) then somebody else would have found this proof. (No doubt, someone else has, but I haven't seen it googling around...)
Its very easy to see, but the vast, vast majority don't look for the answer through simple visualization. That's why they don't see *this* simple solution. Without the picture they don't connect the dots *this* way.
(By the way, for interested readers here, there *is* also a way to visualize this fact even more directly, without using log at all. Sadly, I didn't find *that* one.)
R Hansen says. >Thus, vision is obviously not the key ingredient.
It as *a* key ingredient, and has factored in much mathematical thought through the ages, (with the latest 100 some years showing a bit of bias against it that is somewhat new in the history of math.) It does not suffice alone as your own argument shows. It can even be substituted for, but there are *extremely few* blind-from-birth mathematicians.