There are two kinds of theorems

In mathematical study there are two kinds of theorems, which serve very different purposes. Math instruction follows the same pattern. Students are often very puzzled by this, and rightly so, because it's never explained, or at least I've never seen it explained. There is this crucial, critical piece of mathematical methodology which is never made explicit, students just have to figure it out on their own, and many of them never do. When we do mathematics, we construct a simplified model of some phenomenon. For example, Euclidean geometry is a simplified model of how shapes and lines actually work. In formal geometry, things are simple: lines have no thickness, and three or more lines might all intersect at the exact same point. There are perfect circles, where every point is the exact same distance from the center, and there are perfect rectangles with perfectly straight sides and perfectly equal angles. Real shapes aren't like this. Nobody can draw an infinitely thin line. Nobody…