Change

Chapter 2 of the book is on the concept of change. It’s based on a discussion of the Laplacian Paradigm, which effectively is if you can specify the complete state of a system at any given point, you can use that state + the laws of physics to calculate/predict the next state. Because this chapter is on the concept of change, he introduces calculus.

For those of us that remember (and don’t remember), calculus is a mathematical technique for dealing with infinitesimal quantities. Derivatives are used to calculate the rate of change of something and integrals are used to calculate the total amount of change over a period of time.

Derivatives can be thought of as the slope of the line at any point along the function curve. This is called the tangent line. As we zoom in on any particular curve (ie: as we take smaller and smaller time increments, for example) the slope eventually becomes that of a line. This process is called taking the limit. Integrals are calculated very much the same way, by taking smaller and smaller increments and then calculating the area under the curve for that portion, then adding them all up. As an easy rule, derivatives help to make sense of what happens when you want to divide something by zero, and integrals help to make sense of what happens when you multiply infinity by zero.

One of the interesting parts of this chapter is that Sean compares Kepler’s observations to that of Newton’s. Effectively, Kepler determined the motion of planets in orbit, but in order to do so he looked at the problem globally. Newton, in contrast, used local thinking to derive the same concepts. That is to say that he was able to derive the motion of the planets by making use of his 3 laws.

He ends the chapter by talking about continuity and infinity. Effectively, a value that is considered continuous has an infinite number of values between any two points that we can think of. We think of time as continuous (there are no discrete chunks of time). He then talks a bit about Cantor’s Theorem which is effectively that not all infinities are equal.