18 June 2013

Miscellaneous thoughts related to base-twelve

If you haven't already, read my previous posts, "The case for base-twelve" (Part 1 and Part 2)

FINALLY we can get this whole
counting thing sorted out...

So no, I don't think switching to base-twelve is a feasible idea. I do lament the fact that it's not what we're using right now, though. It strikes me as a more elegant system, and it's a shame that happenstance has pretty much dictated that it'll never be the system we use. Unless, maybe, all of civilization collapses and the society that rises from the rubble happens to use it... But wishful thinking will get me nowhere.

One way I console myself over the fact that a modern, real-world base-twelve system will never be is to imagine a fictional society that uses it. Helpfully, I already happen to have a fictional society on hand. I can't recall if I've alluded to it before, but basically my friend and I have an imaginary continent in the North Pacific, on which I have an imaginary country called Morsenia. The whole thing sounds silly—and it is—but it's basically a collaborative thought experiment that gives us a mental playground for all sorts of hypotheticals. It's constantly-evolving geopolitical fan-fiction, more-or-less. Most of the concepts explored are political and historical, but in developing our countries' stories there's plenty of room for mulling on all sorts of subjects, like language, religion, and biogeography. All-in-all it's a fun excuse to learn a bunch of new things in a variety of subjects as you go about the business of building a believable country and culture.

In the past Morsenia was much more of a Mary Sue; I'd basically take all the things I think are good or cool ideas, say my country did them, and things would inevitably work out pretty well (so they were a wealthy, high-tech nation with a huge standing army filled with tilt-rotor aircraft and digital camo; but also they were somehow isolationist and relatively peaceful). I've made a lot of progress griming it up and throwing wrenches into the works to make it a little more believable (now they're poor and use decades-old weapons and equipment), but I still definitely use it as an intellectual testing ground for ideas I think might work well in real life. Base-twelve is one of those.

The case for base-twelve, Part 2

(If you haven't already, read Part 1, where I explain why we're going through all this trouble.)

An important thing to understand before we move on: what we tend to call a "number" is actually comprised of several different aspects. It has a value, which is the count of things it represents; it has a symbol, which is made up of the numerals 0-9 which can be arranged in single- and multi-digit fashion; and it has a name. In day-to-day life, when we don't bother with anything other than the standard base-ten system, the three aspects seem essentially tied to each other. However, once we start considering other systems we find that the three aspects are actually separate. In base-ten the symbol "10" coincides with this many things: (o o o o o o o o o o), but in another system it wouldn't (and it may or may not be called "ten").

In light of those distinctions, I'm going to employ a certain formatting scheme in this post that will hopefully make it easier to compare base-ten with other systems. If I write out the name of a number with letters then I'm referring to the value. So by "ten" I mean a group of this many things: (o o o o o o o o o o). On the other hand, if I were to write "10" then I'm referring to the symbol formed by the digits themselves, and not a specific value or count of something. It gets a little less clear when we get to the name, since I'm already using the spelled-out word to indicate the value. But generally, if I put the word for the number in quotation marks I'm probably talking about it specifically as a name rather than as a representation of the value. A little confusing, but hopefully it'll make sense in context.

Now, if we were planning to switch to any system between base-one and base-nine we'd be all set for symbols: we'd have a few extra hanging around that we wouldn't need anymore. But since we're talking about switching to a system higher than base-ten, we're gonna need a couple more single-digit numerals to fill in the new gap before we reach "10" (which, remember, is twelve things in base-twelve). For now, we can just swipe a couple symbols from the Greek alphabet*. Doing so could give us a series of number symbols that looks like this:  

1, 2, 3, 4, 5, 6, 7, 8, 9, γ, ψ, 10

Two values that were formerly represented by double-digit symbols, ten and eleven, are now represented by single-digit symbols, and the symbol "10" has been reassigned to represent a dozen of something.

The case for base-twelve, Part 1

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This is going to be another of my pet rants (like this one)—the kind that people who know me in real life may have heard five or six times now. I'm going to go into waaaaay more detail than I ever have in person, though. So maybe getting it all out of my system on the internet will mean they won't have to hear it anymore? (HAH!)

This topic's a favorite of mine because it challenges some of our most basic assumptions, and because the solution I advocate for is at once sensible yet terribly impractical. Basically my point is that instead of ten, we should be counting to twelve.

I'm by no means the first person to say as much but, even so, I don't think it occurs to most people that there's even an option to count to anything besides ten. It's just not the sort of thing you'll encounter unless you're involved in computer science (nope), are really into mathematics/number theory (nope), or are just a huge dork with too much time on your hands (ding-ding-ding!). In fact, I don't want to alarm you, but the ideas in this series of posts may very well BLOW YOUR MIND. :O (They had that effect on me when I was first introduced to them, anyway.)

To start, we need to understand that the symbol "10," regardless of how many things it actually represents, is very important. As the first number with more than one digit it basically says, "Okay, at this point we've finished one 'set' of things, and now we're starting over with a new 'set.'" So a number like "27" basically means, "two complete 'sets,' plus seven."

The big question I'm asking here is, Why is this many things: (o o o o o o o o o o) considered a full set? In other words, Why is ten the first two-digit number?

The truth is that there's no real reason—perhaps other than the fact that we happen to have ten fingers to count on. But you may notice that that's not a very good reason. If human beings were like Homer Simpson, then eight might have been considered a full set and thus represented by "10;" all-in-all, it's troublingly arbitrary.

And so while the way we count seems incredibly natural to us, it's actually just one of many (infinite, really) possible systems. The one we use is called base-ten, or decimal. You could hypothetically have a system based on any value (base-ten, base-seven, base-five-hundred, whatever)—it just means you start two-digit numbers when you've reached the amount named after "base-". They wouldn't all be equally convenient, however. And as you'll see, that's one of my main points in discussing all this: some systems are more useful than others and in my opinion the one we use, base-ten, is not the best. Base-twelve (duodecimal, or dozenal) is far superior.

 "What's so bloody great about twelve?" you may be asking yourself. Well, I'm happy to explain...