Counting in twos, and what number-names reveal about a culture's base

0:00Here are six words from one language, for the numbers one to six. Urapon. Ukasar. Ukasar-urapon. Ukasar-ukasar. Ukasar-ukasar-urapon. Ukasar-ukasar-ukasar. You do not speak this language. Neither do I. Look at how the words are built rather than what they sound like. There are only two words in there. Everything else is those two, repeated. By the end of this you will be able to look at number words in a language you have never heard and say what size of group that culture counts in.

0:40Take the fifth. Ukasar-ukasar-urapon. Ukasar is two and urapon is one, so the name is two, and two, and one. Which is five. The name is not a label for five — it is an addition that comes out at five. Check the others. Three is ukasar-urapon: two and one. Four is ukasar-ukasar: two and two. Six is three ukasar: two and two and two. So every name is as many twos as will fit, then a single one if the number is odd. One rule, and it generates all six.

1:20A rule that generates names runs backwards too: give it a number and it hands you the name, give it a name and it hands back the number. Run every number up to two hundred out to a name and back again, and all two hundred return to where they started. The rule is not a pattern in six words. It is complete — which matters, because of what people get wrong about lists like this.

1:48The words stop at six. Past six there was a single word for any quantity at all. Tempting to read that as: they could not count past six. But the rule does not stop. Ask it for seven and it answers at once — four words. What ran out was the vocabulary, not the arithmetic. Now the part that is genuinely strange. Here is a second list, from a community in South America. Tokale. Ahage. Ahage tokale.

2:21Different words entirely. Same construction: two base words, and the rest built by repeating them. And a third, from southern Africa. Xa. T'oa. Then t'oa-t'oa, t'oa-t'oa-t'a, t'oa-t'oa-t'oa. Three communities. Three continents. No known contact between any two of them. Compare the base words and not one spelling is shared between any pair of the three lists. So whatever they have in common, it is not vocabulary. Be exact about what is shared, because this is where such arguments go soft.

3:02Throw the spellings away and keep the build: how many twos, and how many singles. There are four values all three lists name, and on every one of them the three builds are identical. Five is two twos and a single, in all three. Six is three twos, in all three. And they part company at exactly one place. Three. Two of them build three as two-and-one. The third gives three a word of its own, made of nothing.

3:33So the agreement is real and not total — which is what makes it worth explaining rather than assuming. Three groups, impossibly far apart, with the same idea. What does that establish? One suggestion is common ancestry — the idea travelled with people long ago and stayed put. That is a theory, not a finding. And the difficulty with it: counting in twos is not an obscure idea. You have two hands, things come in pairs, and two is the first grouping anybody reaches for. Three groups finding it independently needs no special explanation.

4:18Shared structure is evidence of a shared problem as much as of a shared past. So keep the question open. What the three lists do prove is smaller, and better: that this is an idea people find. The South American list, as usually set out, has five numbered lines where the others have six. Read each name by the rule, and compare the answer with the number sitting next to it.

4:47The first three agree. The fourth is labelled four, and its name is ahage-ahage-tokale — two, two, one. That is five. The fifth is labelled five, and its name is three ahage. That is six. Two labels, each wrong by exactly one, in the same direction. Which tells you what happened: the line for four was dropped, and everything below it slid up a place. You did not need the language. The rule found it.

5:19All of that was grouping in twos. But nothing in the idea requires two. Count in groups of any fixed size, give the group a sign, and build bigger numbers out of that. Four sizes keep turning up: two, five, ten and twenty — two because things come in pairs, the rest because of what is on the ends of your arms and legs. The Roman numerals. One mark, two marks, three marks — a plain tally.

5:50Then five gets a sign of its own, and four, six, seven and eight are built around it — four of the first ten leaning on one group sign. The tally stops at three marks, and the group takes over. Why three? Why not run the marks to nine and start grouping there? Nine pictures. Look at each only as long as it takes to know how many things are in it.

6:17Six hold a definite number of objects. The others are bunches, with no count to give. Of the six, four hold one, two, three and four things. You did not count those. You looked, and you knew. The other two hold five things and nine. For those, something else happened — you counted, whether or not you noticed. The line falls between four and five, and it falls there for almost everybody.

6:47That limit is not a curiosity. It is the reason grouping exists. A row of tally marks has to be read by counting — one look per mark, all the way along. So replacing each group of five marks with one sign is not a tidy-up. It cuts the row into pieces small enough to be seen instead of counted. Which is what the Roman tally does when it stops at three and reaches for a sign.

7:17Though notice: of the four historical group sizes, only the smallest is at or below what the eye takes in. Five, ten and twenty are past it. So even the grouping has to be learnt. It is just far less to learn. Now the honest accounting, because grouping does less than it looks like. Suppose fives are your only bundle, and you must write one thousand three hundred and forty-five.

7:47It divides by five exactly: two hundred and sixty-nine groups, nothing left over. So the numeral is two hundred and sixty-nine signs long. Against one thousand three hundred and forty-five marks, that is five times shorter — which is exactly what a group of five buys you, and no more. Try the biggest size on the list. Twenties gives seventy-two signs — eighteen times more than a person can see at once.

8:19Grouping once divides the work by the size of the group. It does not change what kind of thing you are holding. The same bill comes due in the words, not just the marks. Take four ukasar and a urapon — that is nine — and add three ukasar and a urapon, which is seven. The answer is sixteen, and sixteen is a small number. Its name here is eight ukasar — eight words in a row, with nothing to tell you where you are.

8:52And if you cannot see a five, you certainly cannot hear an eight. So the name has the tally row's defect exactly: quick to build, slow to read. Which is what to take from all three lists. They found the first idea. They did not find the second. Grouping bought a real thing: a way of writing a number shorter than the collection it stands for. And it bought it by exactly one factor — the size of the group, and nothing more.

9:25Which leaves the bill unpaid, because you do not want to be five times better off. You want a number a thousand times bigger to cost a little more, not a thousand times more — and no single group size can do that, however large. There is one move left, and it is obvious once you see the shape of the problem. You have groups. Group the groups. Give that its own sign, and then group those.

9:54Do it, and the length of a numeral stops keeping pace with the size of the number. That is the idea every system you use is built on, and it is where this goes next.