Turning Hard Stuff into Easy Stuff - The Infinite Power of the Infinite Magnifying Glass - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act I

2. The Infinite Power of the Infinite Magnifying Glass

Calculus is the most powerful weapon of thought yet devised by the wit of man.

—W. B. Smith, Infinitesimal Analysis

2.1. Turning Hard Stuff into Easy Stuff

2.1.1Oh, There I Am!

First, a joke about mathematicians. I didn’t come up with this joke, but I don’t know who did. Ready, set, joke:

As a psychology experiment, a mathematician is placed in a room with a sink, a cooking pot, and a stove. He is asked to boil a pot of water. He takes the empty pot, fills it with water from the sink, sets it on the stove, and turns it on. Next he is led into a room with a sink, a pot full of water, and a stove. Again he is asked to boil a pot of water. He takes the pot and dumps the water out in the sink. He then announces, “I have reduced the problem to the previously solved problem.”

As goofy as the mathematician’s behavior is, the joke makes an important point. It shows us that there are two ways to solve any problem — not just mathematical problems, but problems in anything we’re trying to do. Here are the two ways to solve a problem:

1.Solve the problem from scratch.

2.Solve a small part of the problem, and then notice that the rest of the problem is like something you already know how to do. Then do that.

To put it another way, problems are hard only when we don’t know what to do. Once we know what to do, we can put ourselves on autopilot and just relax until we’re done. For example, at some point each of us has been lost in some place we’re not familiar with and trying to get home. How did you get home? Usually you don’t just suddenly stumble into your own backyard and say, “Oh, I’m here.” That is, you don’t solve the problem of being lost all at once. What usually happens is that you stumble upon some other place that you’re familiar with. You say, “Oh! There’s that methadone clinic with the stained-glass windows! I know how to get to Grandma’s house from here.” By stumbling upon a familiar location, you’ve managed to reduce the problem to one you’ve already solved in the past, and the rest is easy. That’s all of mathematics! One of the best ways to gain a visceral understanding of this is to invent calculus. Let’s do that now.