Tuesday, August 17, 2010

The Proof Is in the Unattainable

Pain killers can do funny things to people. To pass time while my sprained ankle healed, I read books that have been collecting dust for years. One of them was Michio Kaku’s Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (which I’ll tell you more about some other time).

When I went through the chapter about Charles Hinton’s search for a fourth spatial dimension, I didn’t really think much about it. But then that night, I lay awake trying to make sense of the hypercube, a concept that hadn’t been known to make me uneasy in the past. Finding the analogous unraveled cube example lacking, I set out to “prove” to myself how that the funny-looking cross was indeed an accurate representation of the unraveled hypercube. I wasn’t just going to take these guys’ word for it.

After a few hours: I’d constructed a proof* for the number of lines, sides, and angles in a hypercube. I’d unraveled the cube from its square-in-a-square “shadow” (rather than from the cube itself as Kaku showed). And I’d unraveled the hypercube from its cube-in-a-cube “shadow,” the unit tesseract (something Kaku should’ve showed so I could’ve gotten some sleep).

I’ll come to my point. This story isn’t about me attempting to reinvent the wheel in sciences I don’t understand. It’s about being obsessed with proof. When it comes to highly complicated things, I’m willing to trust the experts. However, when something seems within grasp of my level of understanding, I become more demanding.

I’ve seen this in my students too, especially in my microeconomics class last year. There were times when the students would just accept a model as an advanced concept never to be met with again. And there were other times when, believing they had the skills to master everything, they insisted on challenging every point.** Good or bad, the reasons for this remain a mystery to me. I’ll just accept it.

*It was to my satisfaction, but I don’t plan on showing it to any of the three mathematicians in my family.
**Note to self: When students won’t settle down, use the C-word (calculus). It always works.