Oh, no! Not math again!
Among those who try to imagine alien first contact, many believe that mathematics will be the basis of initial communication. This is based on the perceived universality and inevitability of mathematics. They see math as so fundamental any intelligence must not only discover it, but must discover the same things we’ve discovered.
There is even a belief that math is more real than the physical universe, that it may be the actual basis of reality. The other end of that spectrum is a belief that mathematics is an invented game of symbol manipulation with no deep meaning.
So today: the idea that math is universal and inevitable.
We’re still motoring through numeric waters, but hang in there; the shore is just ahead. This is the last math theory post… for now. I do have one more up my sleeve, but that one is more of an overly long (and very technical) comment in reply to a post I read years ago. If I do write that one, it’ll be mainly to record the effort of trying to figure out the right answer.
This post picks up where I left off last time and talks more about the difference between numeric values and how we represent those values. Some of the groundwork for this discussion I’ve already written about in the L26 post and its followup L27 Details post. I’ll skip fairly lightly over that ground here.
Essentially, this post is about how we “spell” numbers.
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.