Closeness

In the “real” world you can cover a distance. You can visually tell how close together two points are to each other; even if the points are far away from you, your brain can process the visual information so you can tell the distance between points.

In the truly wonderful and beautiful world of math, these things are not as easily true. Xenos’ Paradox tells us that you can never transverse the distance between points. You can try to get from one point to another point by dividing the distance between the two points in half, and then divide the half closest to the other point in half and so on…to infinity. Essentially, you never reach the other point! There are an infinite number of points between any two points.

On the Cartesian coordinate system, which uses a flat plane, you can calculate the distance between two points and therefore judge the closeness between two points. However, Xeno’s Paradox applies here too.

In mathematical terms, if you try to evaluate the closeness of two points independent of the Cartesian coordinate system which exists only on a flat plane, it is difficult to do. Dividing the distance in half does not work. Imagine two sets of two points out in space where there is not even any type of surface such as a flat plane or the surface of a sphere. How can you tell which points are closest together?

What a great thing to think about!