Body parts and tally marks: counting before numerals

0:00Every number system needs a sequence to count along. The two oldest ones we know about are made from the only two things every human already has. A body. And something to scratch. Neither needs to be bought, agreed, or carried. And each of them solves exactly half the problem, and fails at the other half. A body needs no agreement at all — but a body ends. Notches never end — but a row of notches has no shape, so reading one means counting it all again.

0:36Between them, those two failures set the job every later system is trying to do. Start with the body, and watch where the count goes. One, two, three, four, five — the fingers of one hand. Then it does not stop, the way ours usually does. It carries on up the arm — wrist, forearm, elbow, shoulder. Across to the head. The ear. The eye. And then the nose, right on the middle line of the face.

1:07That is where the count turns around. Then the other eye, the other ear, the other shoulder, and out along the far arm to the far hand. Every one of those places has a name, and the naming is what makes this a sequence rather than a gesture. Now count the stations rather than the objects. One side carries thirteen: five fingers, and eight more places up the arm and across to the eye.

1:37The nose is one more, and it belongs to neither side. Then the far side carries thirteen again — the same thirteen, in reverse. Thirteen, then one, then thirteen. You do not have to count the picture to know the total. Add them. Twenty-seven. And the middle station, the nose, is the fourteenth — one past a whole side. The shape of the walk gives you both numbers. That mirror is worth staying with, because it does real work.

2:11Take any station and its partner on the other side. One and twenty-seven. Two and twenty-six. Thirteen and fifteen. Every single pair adds to twenty-eight. So you never memorise the far side at all — it is the near side, subtracted. And there is exactly one station the mirror leaves where it is. The one that pairs with itself. Fourteen. That is the nose, and it is the only one, because the walk has an odd number of stations.

2:44An even body would have no middle to turn around on. Here is what this system gets for free. A counting sequence needs a fixed order, agreed before anybody starts. For an invented sequence that agreement is work: somebody decides, everybody remembers. For this one, the body decides. The elbow is above the wrist and below the shoulder, on everybody. Two people who share the convention always stop in the same place — there is nothing left to disagree about.

3:21And it does not matter which order you pick the objects up in. Take six things in any order at all — every order stops at the same station. No equipment, no writing, no agreement. That is a lot for nothing. So what is wrong with it? Count your goats — up the arm, across the face, down the other side. You reach the far hand. Twenty-seven. And there are still goats.

3:50The ceiling is twenty-seven, because that is where the body stops. A herd of thirty beats it by three. Not by a lot. But a system that fails at thirty is not going to run a harvest. And the failure is not arithmetic. The counter simply ran out of body. There is an obvious repair, and people used it. At the far hand, start again — and remember you went round once.

4:20Twenty-seven, and then a second body takes you to fifty-four. A third to eighty-one. Which should sound familiar, because it is the same move as running the alphabet on past z. Something to hold the number of complete rounds, and something to hold the leftover. That idea comes back later in a far more powerful form. But it needs a second person, or a memory, or a mark. Which brings us to the other system.

4:52A surface, and something sharp. For every object you count, you cut one line. One goat, one notch. Another goat, another notch. If that sounds familiar, it should — it is the stick method, exactly. One thing, one mark, nothing left over on either side. So mathematically these are not two methods. They are one method, done in two materials. Count the same herd with sticks and with notches and you get the same answer, every time.

5:25It cannot come out differently, because both are the same pairing. So the difference between them is not arithmetic at all. The difference is that a heap of sticks can be kicked over. It has to be guarded, and it cannot be left out for a season. A cut in a bone stays cut. It survives the night, the winter, and the person who made it. And it has no ceiling — the marks go on for as long as the surface does.

5:55So notches fix the exact thing the body could not do. They go on forever, and they last. Which is why the oldest counting objects we have are not drawings or words. They are bones with cuts in them. Two of them are worth knowing about. One was found in southern Africa, and it is estimated at around forty-four thousand years old. The other came from central Africa, and its age is given as a range — twenty to thirty-five thousand years.

6:31Now: which is older? You might think you cannot say, because one of them is a range. But you can, and here is the test. The southern bone's youngest estimate is still older than the central bone's OLDEST estimate. The two ranges do not touch, so the answer is settled: the southern one is older, by somewhere between nine and twenty-four thousand years. If the ranges had overlapped, no honest answer would have been available at all.

7:05The southern bone carries twenty-nine notches. And twenty-nine is a number that makes people sit up. A cycle of the moon runs about twenty-nine or thirty days. Twenty-nine notches is right at the bottom of that range. So the bone may have been a record of days rather than of things — a calendar rather than a count. May have been. Nobody knows, and the marks cannot tell us. But notice what the possibility depends on.

7:36Counting days needs a record that survives the night, and outlasts the counter, and starts again next month. Sticks could never have done that. A bone can. So the notch wins on both counts? Not quite, and here is the flaw, measured rather than asserted. Writing a tally costs one cut per thing counted. Twenty-nine things, twenty-nine cuts. That is as cheap as it gets. Now read it back. You cannot glance at twenty-nine identical marks and know there are twenty-nine.

8:13You go along them one at a time — which is the entire count, done again. Reading costs exactly what writing cost. Twenty-nine. And that never improves. Four hundred marks take four hundred looks. Quick to write and slow to read is a strange combination for something you make in order to read later. So what would fix it? Something that lets the eye take in a run of marks at once.

8:44Suppose every group of five marks were replaced by a single new sign. Not five marks with a line through them — one new sign, standing for five. Then twenty-nine stops being twenty-nine things to look at. It becomes five of the new signs, and four marks left over. Nine things. Twenty of the twenty-nine looks, gone. And four hundred marks come down from four hundred looks to eighty. That is the whole of what comes next, and you can see why it had to come.

9:18Look at what these two systems asked for between them. The body wanted a sequence that does not run out. The bone wanted a row you can read at a glance. Neither one could give you both. Everything after this is somebody trying to have them at the same time.