On Jun 21, 2013, at 12:46 AM, Joe Niederberger <firstname.lastname@example.org> wrote:
> R Hansen tries attempts gambit: >> I guess I should have been more specific. I meant to say that the other 9 pictures also relate those quantities, but not in a way that would make an argument that e^pi is greater than pi^e. > > Oh! Please show me those 9 OTHER pictures! > > (Or, in my paranoid conspiracy theory moments, might I suspect Robert may come up with 9 non-nonsensical pics and only one that rings true, just to prove his point?) > > Stop right there, mister! Suppose I ask you you to picture a nice 5 dresser drawer Chippendale cabinet, made of walnut. Will you call up in your mind's eye a hairy amoeba?
Let's cut the chase. You cannot visualize a mathematical proof without connecting the dots. It is impossible. I don't think any mathematically minded individual will agree with you that you managed to pull off a proof without connecting the dots.
> Or maybe vision is a fundamental sense that factors > into much of our thinking. Fundamental sense belonging to perhaps the majority of humans, by virtue of their genetic inheritance, and grounding our thought processes. The ability to picture a Chippendale dresser drawer, or a picture relating pi, ln(pi) and 1/e, not so. To suppose the two are at all alike, is a category error.
Virtually every living animal, human or otherwise, possesses the sense of vision, with pretty much equal acuity and resolution (within their species), yet very few humans (and no animals) could make sense of your picture. Thus, vision is obviously not the key ingredient. Mathematical reasoning is the key ingredient. Connecting the dots is the key ingredient.
I don't see a world of algebrists and world of geometers, just like I don't see a world of arithmetic and world of mathematics. How many students of algebra would be stumped by your visual proof and how many students of geometry would be stumped by a symbolic proof? Hardly any. Note, "students" = "students successfully progressing through the art".
Telling me that the algebrist possess a different sense of mathematics than the geometer is like telling me that the pianist possess a different sense of music than the violinist or that the sculpturer possess a different sense of art than the painter. Each pair is using the same sense. They will have to develop strategies unique to their medium but they are feeding the same sense. Musicians generally play more than one instrument. Artists generally work in more than one medium. Mathematicians generally understand visual proofs as well as symbolic ones. And they all know arithmetic.:)
I give you credit for not connecting the dots when you presented your picture, even though you claim (foolishly imo) that there aren't any dots to connect. That is the problem when teachers use visualizations poorly. There is no math left. They will work through algebra problems from start to finish and say "See? X = 3." and the students will nod their heads. Then, when the students fail their algebra exams, the teachers throw their hands in the air and say that it is the X's and Y's fault. So they proceed to use visualizations instead and repeat the same mistake all over again. They will work through the problem from start to finish and say "See? Segment A is longer than Segment B." and the students will nod their heads, then fail their visual algebra exams.
I know you are a believer in Dehaene, and by analogy, the "Chariots of the Gods" was a damn interesting story, and if it were true, it would be pretty cool as well. But open your eyes (PUN INTENDED) and look around you. Vision cannot be a key ingredient to mathematics any more than breathing can. If it were, we wouldn't have any trouble teaching it. These theories that share the common theme that there is some dark force suppressing the innate mathematical genius of our youth are getting old and absurd. My comparison to "belief in psychics" wasn't that far off.