The Calculus Diaries - Jennifer Ouellette [5]
Today, vacuum technology is commonplace, and the coin-and-feather experiment is a staple of physics demonstrations. I had witnessed one such demonstration, so I knew from experience it was true that objects fall at the same rate regardless of mass—or did I? I hadn’t built and performed the experiment myself. How could I know it wasn’t some kind of trick, or a mistake in the experimental setup?
Alan pondered a moment, stroking his beard, and then pointed out that I need not take the matter on faith. It would become obvious to me why this was so if I allowed him to walk me through the equation.
I resisted. Alan persisted: “It’s not real math; it’s just algebra.” He wore me down eventually, and he was right. On his office whiteboard, he patiently demonstrated how the little m—for the mass of the object, as distinct from a big M for the mass of the Earth—on each side of the equation effectively cancels out, making an object’s mass irrelevant to the rate of acceleration. And I had my first epiphany that math might actually be relevant to my life: Among other advantages, mathematics can help us better grasp the more counterintuitive notions in physics. It is certainly possible to do so without the benefit of algebra or calculus, but when I saw that equation worked out, some final piece of insight clicked into place that I hadn’t even realized was missing. Math finally had a meaningful context.
So began a gradual lowering of my knee-jerk defenses against numbers and abstract symbols, and the start of a grudging appreciation for the role of mathematics in the “real world.” It was a tantalizing glimpse into a whole new way of looking at reality, and for the first time in my mathephobic life, I wanted to learn more. Armed with a few books,3 a DVD lecture series from the Teaching Company, and the support of my physicist spouse, I set out to discover what I’d been missing all those years.
Once I started delving into calculus, I realized that this seemingly arcane subject is applicable to everything from gas mileage, diet and exercise, economics, and architecture to population growth and decline, the physics behind the rides at Disneyland, the probabilities associated with shooting craps in a Vegas casino—even the I Ching. In fact, one could argue that we all do some form of calculus all the time, without realizing it. A baseball outfielder has to estimate where the ball is likely to land after the batter hits it. Whether he knows it or not, his brain is calculating the trajectory of that ball, then sending a signal telling the outfielder where to place himself in order to make the catch. Lurking somewhere in that process is a calculus problem. Or two.
Even lowly worms do calculus, according to a University of Oregon biologist named Shawn Lockery. He’s studied roundworms to figure out how they use their sense of taste and smell to navigate as they forage for food. He compares the approach to the game of hot-and-cold one might play with a child, in which one says, “You’re getting warmer (or colder)” to help said child home in on the target. Roundworms do this, too, changing direction in response to feedback, but they get their feedback by calculating how much the strength of different tastes—in this case, salt concentrations—is changing. In calculus terminology, the worms take a derivative to figure out how much a given quantity is changing at a certain point in space and time, and adjust their behavior accordingly.
If worms can do calculus, human beings simply have no excuse for avoiding it. I think scientists have a valid point when they bemoan the fact that it’s socially acceptable in our culture to be utterly ignorant of math, whereas it is a shameful thing to be illiterate.