In Mathematics

So much for individual memes. What about clusters of memes?

In many ways, mathematics is a very compelling example. At present, with global communication, it's hard to distinguish the locus from the single gigantic cluster that occupies it, but in the past there were different clusters. People even used to use completely different bases! The Babylonians used base 60, and the Mayans used base 20. Even English-speakers inherit a bit of base 12 from somewhere in the Anglo-Saxon past—that's why “eleven” and “twelve” aren't compound words like “thirteen” and the rest, why feet are twelve inches, and why there are special names for a dozen and a gross.

If that's not a profound enough difference for you, when I was reading about base 60, somewhere or other I discovered that the Babylonians didn't use a multiplication table to multiply, instead they learned the squares and applied some version of the following identity.

ab = [ (a+b)² − (a−b)² ] / 4

Anyway, the point here is the similarities, not the differences. If they'd had enough time to develop, the different clusters would have converged on more or less the same mathematical truth. In fact, in some cases, they did! For example, numerous results in number theory and modular arithmetic were developed independently in China and the West, as Hua emphasizes in Introduction to Number Theory.

At a smaller scale, I'm sure most of my little discoveries about math and numbers have been made many times before by other people. Consider the idea of risk, for example—it would be mind-boggling if nobody had ever had that idea before. From that perspective, convergent evolution looks a lot like the idea that there's nothing new under the sun.

I have one more piece of evidence that mathematics is an example of convergent evolution. People often imagine that if we ever have a chance to communicate with an alien civilization, it will be mathematics that lets us begin to understand each other. In Contact and Diaspora, the key was prime numbers, if I remember correctly; in Macroscope it was set theory, as you can see in The Macroscope Signal. There are many other examples, that's just what comes to mind right now.

People also imagine that the key will include science, particularly the fundamental constants of nature and the spectral lines produced by pure elements. Basically, the assumption is that mathematics and science are both strongly implied by the environment, i.e., the universe, or physical world.