The real action is in the very large but finite

Photo by Markus Spiske on Unsplash

Numbers seem so simple. We learn to count at a very early age, and after a while counting seems boring to most people, compared to what else is going on. For transcendence we think of gods and other manifestations of the infinite — straight lines, lives, and universes that we imagine can go on forever — leaping over the very large but “merely” finite.

It’s only recently, with the advent of our digital technology, that we have been able to explore and use very large but finite numbers, and the results are surprising and even fascinating. It turns out that, to me at least, our ideas of the infinite are boring by comparison.

Modern encryption algorithms depend on very large, but finite, prime numbers that are essentially impossible to guess. And these days we can literally “own” large finite numbers; that’s what cryptocurrencies like Bitcoin are, along with digital “wallets” in which to store them and “block chain” systems to keep track of who owns them. A quick look at some of the internet addresses for your personal web pages, like those for your bank accounts, will show you some very large numbers in daily use. Here’s an example: 317867036CA2021039809 Try guessing that! By the way, the letters C and A in it are part of the number; it’s written in the base-16 (hexadecimal) number system, in which the 16 digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

And consider the power of numeric positional notation itself. That’s what enables billions of people each to have a different, memorizable, phone number consisting of a country code of up to 3 or 4 digits, followed by a 10 digit phone number within that country code.

The power of large numbers is also shown by counting subsets. Any set of n objects has 2 to the nth power subsets. The numbers of subsets increase very fast as the size of the set increases — in fact, exponentially so:

Powers of 2

Among other things, that means that in an organization of just 32 people, there are over 4 billion ways to form a committee!

Considering that the number of nerve cells in our brains is in the billions, the number of nerve cell networks they can form by connecting with each other, which is very roughly comparable to the number of subsets of nerve cells, is truly mind boggling. And that’s essentially how many experiences and ideas we can have in our lifetimes.

So forget the infinite and enjoy the finite!

Studies language, cognition, and humans as social animals

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