Infinity and the mind

A fascinating book (but probably not for non-math nerds) that I read many moons ago is Rudy Rucker’s Infinity and the Mind (review). There are actually different types and levels of infinity, and whole mathematics of infinities… pretty mind-blowing stuff.

Even the simple notion of infinity is hard to wrap your mind around. But here is a neat little trick I learned a long time ago, that I’ve never forgotten. A line segment is packed with an infinite number of points, right? Say, a one-inch line segment. You would think that a line segment twice as long has more points in it than the shorter one, right? Maybe… twice as many points? Well, this is wrong, as can be seen in the following diagram: [missing]

If you take the two line segments, L1 and L2, and curve them into concentric circles, then you can see that for every single point on the outer, larger line segment L2, it corresponds to a unique point on the inner line segment L1–the same radius line passes through both. If L2 had more points than L1, you could find at least one point on L2, that does not correspond to a unique point on L1… but you can’t. Every point on L2 connects by the radius line, to a point on L1.

So, the longer line segment L2 does not have “more” points than the shorter segment L1. But it does not have the “same number” of points, either–that’s the thing about infinity. Each segment has an infinite number of points, but this is not a fixed number, so they can’t be the “same”. That is why 2 x infinity = infinity. If infinity was a certain number, then this equation would not be true–but it is.