This method now seemed to suit everyone. It was widely used, not only to count sheep, but also to set a seal on all kinds of agreements. Cereals such as barley or wheat, wool and textiles, metal, jewels, precious stones, oil and pottery also had their tokens. Even taxes were controlled by tokens. In short, at the end of the fifth millennium, in Uruk, any contract in due and proper form had to be sealed by a hollow-sphere envelope with its clay tokens.
All this worked wonderfully well, but then one day a new idea emerged, so brilliant and so simple that one wonders why no one had thought of it before. Since the number of animals was inscribed on the surface of the sphere, what was the point of continuing to put the tokens inside it? And what was the point of continuing to make spheres? You could simply draw a representation of the tokens on an arbitrary piece of clay – for example, on a flat tablet.
This came to be called writing.
I’m back in the Louvre. The collections of the department of Near Eastern Antiquities bear witness to this story. The first thing that strikes me when I see these sphere–envelopes is their size. These clay spheres that the Sumerians created simply by turning them around their thumbs are scarcely any larger than ping-pong balls. As for the tokens, they are no bigger than a centimetre.
Stepping further into the museum brings us to the first tablets. Their numbers grow and they quickly come to fill whole display slots. Over time, the writing became more precise and took on its cuneiform appearance, comprising small notched wedges in the shape of a nail, with a stem and a head. Following the disappearance of the first Mesopotamian civilizations at the beginning of the modern era, most of these pieces had lain quietly for centuries beneath the ruins of those deserted towns before they were unearthed by European archaeologists from the seventeenth century onwards. They were only gradually deciphered during the nineteenth century.
These tablets are not very large either. Some of them are the size of simple visiting cards, but covered with hundreds of tiny signs which are crammed together one above the other. Mesopotamian scribes did not waste the slightest portion of the clay when they wrote. The museum’s explanation cards placed alongside these pieces enable me to interpret these mysterious symbols. They concern livestock, jewels or cereals.
Next to me, some tourists are taking photos with their tablets – this is an amusing nod to the carousel of history where writing appears on so many different media, from clay to paper, by way of marble, wax, papyrus and parchment, and which, in a final jest, has endowed electronic tablets with the shape of their earthenware ancestors. The face-to-face encounter of these two objects has something particularly touching about it. For all we know, in five thousand years’ time these two tablets may find themselves next to each other again, but on the same side of the glass.
Time has passed. We are now at the start of the third millennium BC, and another step has been taken: numbers have been freed from the objects that they count. Previously, in the case of sphere–envelopes and the very first tablets, the counting symbols depended on the objects in question. A sheep is not a cow, so the symbol for counting a sheep was not the same as the one for counting a cow. Every object that could be counted had its own symbols, just as it had had its own tokens.
But all that was over and done. Numbers had acquired their own symbols. In other words, eight sheep were no longer counted using eight symbols each denoting a sheep, but instead the figure eight was written down followed by the symbol for a sheep. And to count eight cows, you simply had to replace the symbol for a sheep by that for a cow. The number itself stayed the same.
This stage in the history of thought was absolutely fundamental. If I had to assign an actual date to the birth of mathematics, I would be sure to choose this moment: the moment when numbers came into existence in their own right, when numbers became detached from reality to observe it from a greater height. Everything before that had simply amounted to a gestation period. Bifaces, friezes, tokens – all these amount to preludes in the scheme that led towards the birth of numbers.
From that point on, numbers became an abstract concept, and that is what gave mathematics its identity: it is the science of abstraction par excellence. The objects that mathematics studies do not have a physical existence. They are not material, they are not made of atoms, they’re purely ideas. And yet these ideas are remarkably effective when it comes to making sense of the world.
It is undoubtedly no accident that the need to be able to write down numbers acted at the time as a determining factor in the emergence of writing. While other ideas could be transmitted orally without any problem, it seemed on the other hand difficult to establish a system of numbers without moving to a written notation.
Can we, even today, dissociate the idea we have of numbers from our written representation of them? If I ask you to think of a sheep, how do you see it? You probably represent it as a bleating animal with four legs and a woolly back. You would never think of visualizing the five letters of the word ‘sheep’. Yet if I talk now about the number one hundred and twenty-eight, what do you see? Do the 1, the 2 and the 8 take shape in your brain and come together as though they are written in the intangible ink of your thoughts? Our mental representation of large numbers appears to be definitively linked to their written form.
This example came out of the blue. Although for everything else, writing was just a means of transcribing something that had previously existed in the spoken language, now, in the case of numbers, it was the writing that would dictate the language. Think about it: when you pronounce ‘One hundred and twenty-eight’, you are just reading 128: 100 + 20 + 8. Above a certain threshold, it becomes impossible to speak about numbers without the written medium. Before they were written down, there were no words for large numbers.
To this day, some indigenous peoples still only have a very limited number of words for designating numbers. For example, the members of the Pirahã tribe of hunter-gatherers who live on the banks of the Maici River in Amazonia only count to two. Above that, they use a single word signifying ‘several’ or ‘many’. Again in Amazonia, the Munduruku only have words for up to five – that is to say a hand.
In our modern societies, numbers have invaded our everyday lives. They have become so omnipresent and indispensable that we often forget just how brilliant the idea is, and that it took our ancestors centuries to provide us with evidence of this.
Throughout the ages, very many procedures for recording numbers have been invented. The simplest of these involves writing down as many signs as the desired number – for example, a sequence of bars side by side. We often still use this method, for example to count the points in a game.
The oldest known indication of the probable use of this procedure dates back to well before the invention of writing by the Sumerians. The so-called Ishango bone (actually two baboon bones) was found in the 1950s on the shores of Lake Edward in what is now the Democratic Republic of Congo and dates back some twenty thousand years. The bones are 10 and 14 centimetres long, and feature a large number of more or less regularly spaced incisions. What was the role of these incisions? It is likely that this was a first counting system. Some people think the bones are a calendar, while others extrapolate an already highly advanced knowledge of arithmetic. It is hard to know for sure. The two bones are now on display in the museum of the Royal Belgian Institute of National Sciences in Brussels.
This method of counting using a mark for each unit added rapidly reached its limits as soon as there was a need to manipulate relatively large numbers. Packets were introduced to achieve a greater speed.
The Mesopotamian tokens could already represent several units. For example, there was a particular token for representing ten sheep. This principle was retained at the time that writing came in. There are symbols to denote packets of 10, of 60, of 600, of 3,600 and of 36,000.