On Nov 11, 2012, at 12:57 PM, Joe Niederberger <email@example.com> wrote:
> Robert Hansen says: >> There is no such thing as a "common sense view of continuity". > > This didn't take long to find via Google: > http://www.maa.org/pubs/Calc_articles/ma004.pdf > > In that paper we see Leibniz coming very close to the modern concept, but still no cigar. How could that be if he had no understanding of continuity *prior* to the modern formulation? Well, of course he had an intuition, that was based on something entirely other than the modern formulation.
That "intuition" you speak of is not common sense, otherwise we would see it everywhere. I don't see a shortage of common sense in the world, so why is mathematics so difficult for most? The "intuition" you speak of is the underpinnings of analysis and formal reasoning.
> > Here's another paper that uses the pencil idea to illustrate the inutitive notion: > http://www.math.harvard.edu/~nasko/documents/topology_and_continuity.pdf > > Note the author quickly points out that the formal definition captures cases that are completely outside the purview of the intuitive notion: "functions that do not jump at an isolated points as above, but infinitely often in a dense manner". Completely unintuitive and non-commen-sensical. > > That the views two lead to different phenomena only reinforces the fact that the common sense view and the formal view are not at all the same animal. > > Joe N
What I am doing right now is analysis. I am questioning our "common" concept of "common sense". I started digging deep into this a couple years ago when I was reviewing a visualization for the concept of local linearity. I asked myself "What exactly does this visualization do for a student?" I realized that while I saw the analogy between the visualization and my understanding of local linearity, the student would only see the visualization. That was when I began to realize how much we embellish these visualizations with our understanding.
Another example, visual proofs. If a student saw a visual proof in the same manner as we see a visual proof, then wouldn't they see non visual proofs as well? Again, we embellish these concrete examples with our own understandings. The student doesn't see what we see. They don't see the "proof", unless the see "proofs" to begin with.
I am not saying that visualizations are wrong, just that their use and interpretation are generally wrong. If I wanted to teach you about small engine repair then I would put a small engine on the table and start teaching. Likewise, when I teach about something like continuity I am going to draw all over the board. But these are examples to teach to.
Our sense of the concrete is not the basis of our ability to reason. In fact, that is the problem. If concrete sense progressed naturally into reasoning then this would be a very different world.