On Jun 18, 2013, at 5:24 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> You seem to be looking for the ghost in the machine. All the parts of thought work together - visualizations, interpretations thereof, acquired stratagems, abstraction (in the sense I gave earlier), recognition of the places where previously acquired abstractions can be employed, > tactile senses, auditory perhaps, language, simple facst that can be recalled... > > I wouldn't expect to find "mathematics" any more in "chain of thought" than I would in any other part, taken in isolation.
I am not looking for the ghost in the machine. We don't have to dig that deep to realize that mathematics requires justification. An argument of truth with some sense of certainty. You didn't give me a lot to go on with your visual example so let's assume that you thought randomly about different curves till you happened on the right ones. Ridiculous? Then imagine that you weren't even thinking about the problem at all and someone showed you something in an entirely different context and a bell rung in your head (the ghost in the machine rings the bell). That happens to me all the time. I don't care who the ghost is. I don't care how you finally got to the curves you did. I care about how you recognized that this was a proof that e^pi is greater than pi^e. I care about your sense of certainty and process of justification. That is mathematics.
Back to the use of visualizations in school. Two scenarios...
1. I show the students your visual proof that e^pi > pi^e and I ask them "Do you see that e^pi > pi^e?"
2. I show the students your visual proof that e^pi > pi^e and I ask them "How does this prove that e^pi > pi^e?"
Only one of those scenarios is verifiably mathematics. In the other scenario you have no clue what the student's experience was.