So the Stone Age human environment contained – at certain times and places – at least three sets of physical objects that could have been ‘wild’ number lines, that could have supported a transition from not-counting to counting: fingers, beads on strings, and marks on surfaces. It is possible that a conventional series of sounds accompanying repeated actions might have served a similar function and become the first counting words. It is also possible that unbound counters – which would be impossible to distinguish from ordinary pebbles or fragments of bone or other material in the archaeological record – also supported counting in the Stone Age world, although since they do not possess a natural order their ability to function as a ‘wild’ number line is less clear.
How did these different precursors to counting – and these different early ways of counting – interact? Which came first? Many scenarios have been proposed, but there can be no clear dénouement to the mystery story that is the origin of counting.
Some have argued that language must come first: that, for instance, no one would make tally marks without having words for the numbers tallied. Or that there may have been words for the subitisable numbers from one to four at a very early stage, before counting was later extended by fingers, beads and tallies. Others – probably representing the balance of opinion at the moment – believe that language is not essential for counting: that you don’t invent words for things you have not yet experienced, and that fingers, beads and tallies more likely came before counting words. Or perhaps that counting words evolved alongside material ways of counting.
But perhaps the search for a single scenario is itself a mistake. In the context of prehistoric art, it has been well said that development should be imagined not like a line or a ladder but like a bush. In the same way, perhaps the development of counting in the Stone Age was more like a knot garden than a footpath. More like a collection of short stories than a novel. The scales of time and distance are enormous, even taking the most conservative estimates for the rise and spread of Homo sapiens, and it is perfectly possible that, over thousands of kilometres and tens of thousands of years, different ways of counting arose and were forgotten – even multiple times – by people who never met each other, in Africa and beyond.
Whatever the truth of these matters, ways of counting with beads, tallies, fingers and words are found the world over, and it is all but certain that by the time modern humans began to populate the world beyond Africa – perhaps 70,000 years ago – they were doing all of those things. That the stage was set for the immense, global phase of the human history of counting.
INTERLUDE
The numbers
At some point in the history of counting, people started to treat counting words not just as noises-in-a-sequence, but as properties of collections of objects: as adjectives. And at some point they started to treat them not only as adjectives, but sometimes as nouns (in some languages, there are concomitant changes to the exact form of the words). That is, words like ‘three’ would now denote not only a certain property that collections of events or objects could have (‘I have three axes’); they would also denote certain abstract objects (‘three is a small number’). Not all languages do this, and in many the number words occupy an uneasy position between adjective and noun, with some – but not others – capable of being modified for case or gender, or even possessing separate plural forms (in English you can speak of ‘hundreds’ of something but not really of ‘ninety-nines’). But most – not all – cultures do appear to have a category of ‘numbers’, whose names at least are built from the counting words.
To study the relationships between numbers is called arithmetic or, in its higher reaches, number theory. To place them in correspondence with other things or ideas – beyond using them as a set of mere labels – is called numerology. Such studies almost certainly stretch back into prehistory, and they appear very early on in written texts; there can be no knowing when or where exactly they started. Some would certainly see artefacts like the tallies from Ishango or Laussel as reflecting early attempts to study number rather than merely to count.
But what are numbers?
A common intuition is that behind different ways of counting there must lie some thing which guarantees their integrity and their mutual consistency: some set of abstract objects with the right properties to account for everyone’s experiences of counting. The numbers, in other words: with a reality independent of human minds; indeed, independent of physical reality altogether, living eternally in a separate realm. In this view, the nature of things constrains the ways people can count, and the history of counting might be seen as one of striving towards the real truth about numbers. The numbers and their properties would then be another of the roots of human counting: surely the most important root, in fact, underpinning both the innate perceptions people can have of number and quantity, and the properties they can find in sets of beads, tallies, fingers and counting words.
That intuition is not universal, however, and it resists satisfactory proof or disproof. If the numbers live outside time and space, just how do they ‘get in’ and affect the world or become perceptible to human minds? And is it really credible that not just 1, 2 and 3 but 2,938,239,856,837,641 existed from the beginning of time waiting for someone to have a use for them? Surely there must be a less extravagant way to account for what happens when people count.
Others would say that numbers have no reality except as the structures of things in the world: that numbers are abstractions from experiences that people have, and that their consistency over time and space is provided by the consistency of those experiences. People’s intuitions about small numbers, in this view, are memories of experiences with small collections of objects, while their shared ideas about larger numbers are the consequence of all playing the same arithmetical games by the same rules. There is no need to suppose that 2, 3, or any other number has an existence separate from the properties of collections of objects or sequences of events. Thus, if there are constraints on human counting, they are the ones provided by the accidents of human physiology and by the kinds of environment in which human beings have lived over the millennia.
Others, again, would say that whatever the truth about the real nature of numbers, human brains are, as it happens, hardwired to view numbers in a particular way: to recognise certain kinds of structures in experiences, or even to impose those structures on them. In this view, at least some of the constraints on the ways human beings count arise from the accidents of how their brains are organised; which are in the end the consequences of what was adaptive in the evolutionary past, not necessarily of what was true. This view opens up the rather staggering possibility that the numbers are to some degree a shared illusion.
There is no consensus about such matters. Really convincing evidence for or against any of these positions has proved to be extremely elusive. In the ancient Greek philosophical tradition, Plato was on one side (numbers are real), Aristotle on another (numbers are abstractions from experience). Among their successors, no later philosopher has come up with an argument that convinces everyone of one position or the other. Thus, it remains a vexed question how far human counting is constrained by the givens of human physiology, how far it is constrained – if at all – by the nature of things, and how far it is freely chosen and transmitted as a matter of human culture.
Luckily, for the purpose of exploring the many ways human beings have in fact counted, there seems no need to choose between these positions. This book will try to keep an open mind.
PART 2
Branches
3
Counting with words and symbols
in the Fertile Crescent
Walk out of Africa, following the fertile land. It seems that always meant the Nile valley – the only way to cross the Sahara – and then the hospitable strip of the Levant between the Arabian Desert and the Mediterranean. Then cut through the Anatolian mountains and, turning right, follow the Tigris or the Euphrates, perhaps all the way down to the sea: the modern Persian Gulf. The route was important for humans, and for animals and plants, expanding out of Africa over tens of thousands of years. The whole route – now sometimes known as the Fertile Crescent – became an important site for human culture. It lies between three continents – Africa, Asia, Europe – and indeed between three tectonic plates, whose meeting and movement created the great valleys and mountains of the region: the Dead Sea Rift, the Tigris and Euphrates valleys, the Anatolian and Caucasus mountains. It is a region, consequently, of enormous variation: marshes, deserts, slowly shifting river valleys and mountains.
One branch of the story of counting flourished in the Fertile Crescent, from the beginning of writing around 5,000 years ago, through to the Hellenistic period more than three millennia later. In Sumer, Assyria and Egypt, from the earliest writing to the final flourishing of the Egyptian script under Greek domination, and from the world’s first city to the last centuries of the Pharaohs, counting was always present, used for administration, for boasting, for keeping track, for proving who owned what and who had paid whom.
The region was of course a patchwork of different cultures speaking different languages and doing things – including counting – in different ways. But it is fair to identify words and symbols as consistently the important features of counting in this place and time. This is a place where something more specific can be learned about number words: the oldest number words for which there is secure documentation come from Sumer, in Mesopotamia. And this was also one of the places where a crucial new item was added to the human repertoire of ways of counting: number symbols.
Sumer: Counting symbols
Squat on the ground in the sunlight, in the greatest city in the world. Shape a lump of clay into a flat tablet, to fit in the palm of your hand. Take a reed, scratch a drawing in the clay: a goat. Turn the reed on end and press holes into the clay: ‘dish, min, esh’. It means ‘three goats’. Five thousand years later, it will still mean ‘three goats’. The bright light of writing changes everything.
The eastern end of the Fertile Crescent, now sometimes called Mesopotamia, contains in itself much of the region’s diversity. Near the rivers, annual flooding makes the land fertile. Marshes skirt the coast. There are resources ranging from fish, clay and reeds, wild barley and wheat to wild sheep, goats, cattle, pigs and camels. The potential for hunting and fishing is great. People have been in the region for many millennia.
As the climate improved from its ice age low, this was one of the regions in the world – the first – where humans took control of their food supply by domesticating cereals: selecting, sowing, harvesting, storing. Settlement moved from northern Mesopotamia to the plains beside the rivers; by about 7000 BCE, villages dependent on agriculture existed throughout the Near East, wherever there was enough rainfall to sustain it, and the aeons-long dependence on hunting and gathering faded.
By late in the fourth millennium BCE, the plains in the south of what is now Iraq were becoming densely populated, supported by fertile soil and increasingly elaborate irrigation. Settlements became larger, and the first cities arose.
To its inhabitants, the greatest of these was simply ‘the city’: ‘Uruk’. It lay in the very south of Mesopotamia, on the Euphrates, near the Persian Gulf. It may have formed from two settlements on the two sides of the river, and by 3000 BCE or so it covered 600 acres or more; two and a half square kilometres. The population may have approached 40,000. The world had, literally, never seen anything like it.
Physically, it was dominated by a temple complex covering 75 acres: ceremonial areas set on platforms, some built of stone, their walls decorated with mosaics of white, black and red clay and visible for long distances over the plain. Culturally, the city dominated its surrounding villages, consuming much of the food they produced and setting up a specialisation of labour among producers: fishers, farmers, gardeners, hunters, herders; but also weavers, potters, metalworkers and other kinds of specialist. Timber, stone and metals all had to be imported, and long-distance networks of barter existed as a result.
All of this required organisation in various senses. There was a ruler and some sort of city assembly, though the working of these institutions in the fourth millennium are very obscure. To organise people and things on this scale required language, of course, and the inhabitants of Uruk and its area spoke a language called Sumerian, which has no known relatives and whose grammar and even set of sounds are not completely understood.
And it required counting.
Sumerian had a set of number words: dish or ash, min, esh, leemu, ya. This primary set from one to five was compounded to form words for six through nine: ya-ash becoming yash; ya-min becoming imin, and so on. Beyond that, Sumerians counted in a decimal system, with ten being hu, twenty nish, thirty ushu, and so on up to sixty: gi or esh. The limited evidence suggests that for larger numbers, 60 was used as a base, so that you would have said the equivalent of ‘sixty, ten, three’ for 73 and ‘two-sixty, forty, two’ for 162. Shar appears to have meant 3,600 (that is, 60 times 60) and (some scholars say) shar-gal 216,000 (that’s 60 times 60 times 60).
Indirectly, the use of five and ten in the set of counting words suggests an interest in the fingers used as a set of five, as in some – but by no means all – other languages and language families. It is very likely that Sumerians counted on their fingers at times, but there is no direct evidence of the fact: no hand prints like those from Cosquer. Equally, it is possible that they sometimes made tally marks, but none have been found from early contexts.
More probable is that counters were used in the Mesopotamian world. Unmarked pebbles cannot be confidently identified as counters in the archaeological record, but from as early as 8000 BCE, small clay objects appear at sites in Syria and Iran, and one attractive interpretation is that at least some of them, at least some of the time, functioned as counters. Thousands have been found all over the Near East from the Mediterranean to Persia, and it seems very likely that those with more elaborate shapes or those pierced for – perhaps – stringing, had other uses and meanings besides number (these become more common after about 4000 BCE). Yet, as with beads from much earlier periods, they witness at least to the presence in the human environment of objects whose manipulation could form an ordered sequence and come to be used to count. Exactly when and how frequently that happened in this case seems likely to remain obscure.
