One-to-one correspondence: counting without number words

0:00Dusk. A herd is coming back through a gate, and somebody has to know before dark whether all of them are in. He has no word for twenty-three. He has no word for any number at all, because nobody has invented them yet. And he is going to answer the question exactly. Not roughly, not by having a look at them: exactly. The animals are not going to wait while he works something out, and in any case there is nothing to work out. He has no arithmetic either.

0:34What he has is a pot, and a heap of pebbles. In the morning, as the animals go out, he drops one pebble into the pot for each animal that passes. One animal, one pebble. Another animal, another pebble. He never counts the pebbles. He never counts the animals. He only keeps the two of them in step. By the time the last one is through the gate, the pot holds the herd. Not a number for the herd — the herd itself, in stones.

1:07And notice that nothing here can go wrong through miscounting, because nothing is being counted. The only mistake available is dropping two stones for one animal, and that is a slip of the hand rather than of the head. In the evening the whole thing runs backwards. As each animal comes in, he takes one pebble out of the pot. One animal, one pebble. Out, and out, and out again.

1:37Every pebble that leaves is a promise kept. The animal that put it there has come home. And the matching is exact for a physical reason rather than a careful one. A pebble cannot half leave the pot. An animal cannot half come through a gate. The pairing is as exact as the objects are separate, which is to say completely. So at dusk he looks in the pot, and the pot has the answer ready.

2:08If it is empty, every pebble has been claimed, and every animal is back. He can go to sleep. If two pebbles are still sitting in the bottom, two animals are still out. Not roughly two. Not around two. Exactly two, and he takes a lamp and goes looking. He does not even have to remember the morning. The pot remembered the morning for him. The question is answered, and no number was ever spoken.

2:39Now here is the part worth slowing down for. Ask him how big his herd is, and he cannot tell you. The pot said two were missing. It never said two out of twenty-three. Two pebbles could be left from a herd of twenty-three with twenty-one back. Or from a herd of forty with thirty-eight back. Or from a herd of two with nothing back at all. Counting through every herd up to forty, thirty-nine different mornings end with two pebbles in that pot.

3:11He knows exactly what he needed to know, and nothing whatsoever beyond it. The method answers one question and refuses the other. Which tells you something about what counting actually is. Counting is not saying the names. The names are one way of recording a pairing. They are not the only way, and they arrived very late. Look at your own hand for a second. Every finger has three joints, and the thumb can reach all of them.

3:44Four fingers, three joints each. That is twelve on one hand. Not ten. Systems that group by twelve instead of by ten were built on exactly that habit. Ten is a choice with a reason behind it, not a law of nature. And it is not a small thing that the pairing came first. The names were invented to record something people had been doing for a very long time already.

4:15But pebbles have a problem. A pot gets knocked over. Stones get borrowed, or lost, or used for something else entirely. The record lasts exactly as long as nobody disturbs it. So cut the marks into bone instead. One notch for each thing, exactly as before: the same pairing, made permanent. And permanence changes what a record is able to do, because a mark that survives can be looked at again. Compared. Rearranged.

4:50A pot of stones can only ever answer today's question. A notched bone still answers it a season later, and it can be set down beside another bone. The oldest such record we have was found in the Lebombo mountains, on the border of South Africa and Eswatini. It is about thirty five thousand years old. It carries twenty nine notches. Distinct, deliberately cut, and all the same size as each other.

5:22That total sits close to the moon's cycle, which is why it is read either as a counter for the phases of the moon or as a menstrual calendar. And here is the honest difficulty with choosing between them. Both readings add one notch a day over the same span. Two different things, followed for the same number of days, leave behind a record you cannot tell apart. The bone records a period. It does not say which one.

5:55A second bone, found near the source of the Nile in what is now the Democratic Republic of Congo, is dated to around twenty thousand years before the common era. Its notches are cut in three columns, and unevenly — none of the neat matched rows of the older bone. And one of those columns falls into groups. Eleven. Thirteen. Seventeen. Nineteen. Sit with those four for a moment. They are the primes between ten and twenty. Every single one of them, and nothing else from that range.

6:35Which is worth testing, because there is an easy way to get it wrong. The four climb by two, then by four, then by two again. Follow those gaps and the next one after nineteen is twenty one. But twenty one is three sevens. It is not prime at all. Odd numbers and prime numbers travel together for a while and then part company — at fifteen, in fact, which is three fives.

7:04Follow the property instead of the gaps, and the next three after nineteen are twenty three, twenty nine and thirty one. So why does that column matter so much? A plain tally is a record. It reports how many there were. It makes no claim at all about the numbers themselves. But a tally that has been grouped by a property of numbers is doing something else entirely. Somebody sorted those notches by primality — or the grouping is a coincidence, and we genuinely cannot be certain which.

7:42Another of the columns is read as showing doubling: a total, then twice it, then twice that again. If those readings are right, the bone is not keeping track of livestock. It is saying something about arithmetic. One last thing, and it is a crack that this whole business opens up. The numbers all of it produces are the counting numbers. One, two, three, four, and onwards for ever. They begin at one, and nothing lives to the left of that.

8:17Add any two of them and you land back inside. Seven and three make ten, and ten is there. Multiply two of them and you land back inside as well. But try taking away. Seven take away three is four, and four is there. That is fine. Three take away seven is not there. And five take away five is not there either, because there is no nought in this set.

8:43So subtraction sometimes walks straight off the left end and finds nothing waiting. That gap is real, and filling it is the next thing that has to happen.