Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

In analysis, one solves a problem by starting at the end, as if the answer had already been obtained, and then works back wishfully toward the beginning, hoping to find a path to the given assumptions. It’s what kids in school think of as working backward from the answer to figure out how to get there. Synthesis goes in the other direction. It starts with the givens, and then, by stabbing in the dark, trying things, you are somehow supposed to move forward to a solution, step by logical step, and eventually arrive at the desired result. Synthesis tends to be much harder than analysis because you don’t ever know how you’re going to get to the solution until you do. The ancient Greeks regarded synthesis as carrying more logical force, more persuasive power, than analysis. Synthesis was considered the only valid way to prove a result; analysis was a practical way to find the result. If you wanted a rigorous demonstration, you had to do synthesis. That’s why, for example, Archimedes used his analytical method of balancing shapes on seesaws to find his theorems but then switched to the synthetic method of exhaustion to prove them.