"A heavy warning used to be given that pictures are not rigorous; this has never had its bluff called and has perma- nently frightened its victims into play- ing for safety."
I do not claim that a visualization of a proof or of a mathematical conclusion is less rigorous. In fact, my claims have nothing to do with rigor at all. I claim that when you show students a visualization they will see either the visualization or the mathematical reasoning and the only way to know which it is they see is to ask them to explain the reasoning behind the visualization.
This could probably be easily proven. Show students two "visually appealing" proofs, one mathematically correct and the other not, even though it "visually" looks correct. I suspect that you will find that the students lacking mathematical reasoning sense will judge both to be correct while the students that have the sense will see the flaw. I am not saying that the first set of students can't be taught mathematics or will never get the sense. I am only pointing out that a visualization of a mathematical argument still requires the same sense of mathematical reasoning to see it as does a mathematical argument in any other medium.
Another example of this is data visualization. Data visualization appeals to our non mathematical senses. These are the senses Dehaene writes about and confuses with mathematics. When I can create a data visualization I can create it in such a way that it is mathematically correct but invokes a response completely at odds with the data being represented. For example, Richard posted some time ago a graphic of a bag three times as large as another bag, by volume (or something to that effect). Not a good representation to feed to our Dehaene rat-sense of spatial area. Our rat-sense does not recognize it as three times as big. It requires actual mathematical sense to make sense of this visual contradiction. Obviously, when you visualize data you strive to avoid this situation. You choose graphical methods that look "rat-sense" correct rather than mathematically correct.
The bottom line is that rat-sense is not mathematical reasoning nor is it a possible substitute for mathematical reasoning. Visualizations appeal to rat-sense. The mathematical argument behind the visualization can only appeal to the sense of reasoning. Saying that a visualization is appealing does not tell me which occurred.
On Jul 10, 2013, at 10:15 AM, "Dave L. Renfro" <firstname.lastname@example.org> wrote: