Yesterday I was re-watching Arachnids in the UK, the fourth episode of the latest season of Doctor Who, and a somewhat goofy idea popped into my head about how to respond to the charge that sometimes stories are just ‘too improbable’ to enjoy — or to have happened at all.
That certainly is an accusation that seems to apply in many cases. In order for some story to have happened at all, certain events had to happen just so and in the right order. It’s easy to shake your head and think, “Yeah, right. As if that could actually ever happen.”
For many years I’ve had a generic response to that accusation, but yesterday I realized it can be justified mathematically!
Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the Theory of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: countable and uncountable. It also connects with the Smooth or Bumpy post, which considered differences between the discrete and the continuous.
This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including Star Trek). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)
So let’s consider Cantor.
You probably have some idea of what infinity means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!
As a simple example, a small circle is infinite in the sense that you can loop around and around the circle forever. At the same time, your entire path along the circle is bounded in the small area of the circle. Compare that to the straight line that extends to infinity. If you travel that line, you follow a path that goes forever in some direction.
What if we draw a larger circle outside the small circle. If there are an infinite number of points on the small circle and an infinite number of points on the large circle, does the larger circle have the same number of points as the small one? [The answer is yes.]
To understand all this, we have to first talk a bit about numbers.