Added to the walls around the same time – and involving much more effort, time and preparation than the simple schematic animals and signs – were dozens of stencilled hands. The technique was to dilute pigment – red clay or charcoal – with water, and blow it from the mouth, possibly using a tube. A hand pressed against the wall could thus make a striking stencilled image: a red or black halo with a clear, sharp image of a hand centred in it in the colour of the rock itself. If a single puff of pigment didn’t work, it could be repeated, or the image retouched by dabbing on extra pigment by hand. The result was a dense ensemble of hands stencilled in various parts of the cave.

Stencilled hands are one of the iconic images from the art of the ice age, and they are known in three dozen caves across what are now Spain, France and Italy. They are familiar beyond that cultural area too, for instance in the art of Aboriginal Australia. The French cave, now known as La Grotte Cosquer, has more than sixty hand stencils, one of the largest collections known.

The hands are distributed unevenly around the cave walls, exclusively on the eastern and northeastern sides of the cave: the right side as you go in. They are concentrated near the shaft and the vault at the eastern end of the main chamber: that is, in the more dangerous section of the cave. Many are in positions that would have required some climbing. There is a hint, perhaps, that a set of mainly red hands leads around the main chamber towards the shaft, and a large set of black ones stands beside the shaft itself, including on a stalagmite above it.

Most appear to be left hands (assuming the hand was held with the palm against the wall to stencil it). If the people were, like people today, mostly right-handed, perhaps they were using the more agile right hand to hold a torch or a supply of pigment, or to cling to a support when the position was a difficult one.

Whose hands were they? The stencilling process is a mechanical one; it records the characteristics of the maker’s hand automatically, unaffected by questions of style or artistic convention. The hands bring close – in a sense, deceptively close – the individuals who made them: they are the signs of individual bodies, even of individual identities tens of thousands of years ago. They feel almost like signatures. The makers were adults: there are no child hand stencils at Cosquer. Some were over two metres tall, unless they stood on each other’s shoulders or built scaffolding in the cave to reach the higher spots on the walls.

The hands stencilled were generally large and robust, though some of those by the shaft show a slimmer wrist; finger ratios and lengths are often – not always – consistent with those of modern women. Certainty is impossible, but both men and women seem to have been involved. There are what appear to be pairs of stencils from the same hand: one with an unusually long forefinger; another with a thicker ring finger. But just three of the stencils have been directly dated by archaeologists so far, and it is impossible to say how many people’s hands are represented at Cosquer, or what period of time was involved. Is the stencil scheme the work of a single day? Of a lifetime? Or a record of visits to the cave by many people over hundreds of years, as mysterious to each other as they are to visitors today?

And the evidence for counting? Some of the stencilled hands are incomplete. That is, certain of their fingers are shortened. Of the fully legible hand stencils at Cosquer, rather more than half, in fact, show less than a complete hand. These incomplete hands are, for some reason, much more common among the red stencils than the black. Incomplete hand stencils are known from a few other caves in France and Spain, but most of those known to exist are at Cosquer.

How do you produce a stencil of part of a hand? Suggested explanations have included that the hands in question were really incomplete: that these hunter-gatherers suffered from frostbite, from disease, or practised ritual mutilation. But their sheer number – and the impracticality of living a Palaeolithic life with a mutilated hand – count against these possibilities, and the balance of opinion seems to be that the simpler explanation is right: that the incomplete stencils were made by folding down certain fingers. Modern experiments have confirmed that it is possible to make stencils like those at Cosquer by folding down one or more fingers and pressing your hand against a rock surface. The bent fingers prevent the hand from quite touching the rock, resulting in a characteristically slightly fuzzy outline at those points: a characteristic that can be seen in the stencils at Cosquer.

Given four fingers and a thumb, there are in all thirty-two possibilities for folding down some, none or all of them. At Cosquer just five of those possibilities were used, namely those patterns in which the extended digits are thumb alone; thumb plus index finger; thumb, index and middle; thumb, index, middle and ring; and all five fingers. It is impossible that this was a matter of chance. Moreover, one of the patterns – folding down just the little finger – is relatively hard to make. This set of five hand shapes was deliberately chosen by the stencil makers at Cosquer.

Why? As with other prehistoric signs and proto-signs, the more general question of the meaning of Palaeolithic hand stencils has been a vexed one for over a century; one scholar aptly says that here modern observers are ‘uninvited guests in a system of communication designed for others’. But, for the set of hand stencils at Cosquer, it does seem possible to go further: to say that the five hand positions that were used meant something, and that the twenty-seven others were left out because in this system they had no meaning. And it is almost impossible to avoid the speculation that the hand positions recorded had something to do with counting. They form a sequence consisting of extending the digits one after another; they are one of the easiest ways you can count to five on your fingers. They are a good candidate for the earliest counting vocabulary recorded: a vocabulary not of spoken words but of gestures; a sign language.

Other caves with hand stencils display different vocabularies, using finger combinations that are not seen at Cosquer. Many locations show complete hands only, or record a more restricted set of folded-finger gestures. Perhaps the language of hand gestures differed from place to place; perhaps chance survival of fragile images in difficult environments has played a role. That does not detract from the consistency and suggestiveness – to say no more – of the evidence at Cosquer.

The gestured numbers, if that is what they are, do not all appear with the same frequency. The complete hand is by far the most common gesture recorded; those showing smaller numbers of fingers are progressively less common, with the interesting exception that four fingers is less common than three. A sign for zero (logically, you would fold down all five fingers and stencil just your fist) does not appear at all. The signs never seem to appear in relationship to each other – sequences of numbers, pairs – nor do they occur in two-handed groups that might suggest counting to numbers beyond five. If this was a number vocabulary, in other words, it ran only from one to five.

As with other prehistoric counts, it is impossible to be sure what was being counted. But the proximity of images of animals, including animals with signs on them such as barbs that might be schematised weapons, means that a strong possibility is a count of animals taken. Another would be a count of visits to the cave.

Nor is it certain why people chose to record these gestures by stencilling in this particular location. To turn a gesture into a stencil is to extend it beyond your body and – potentially – beyond your own lifetime, to transmit it to people you have not met and to generations not yet born. Yet to place the stencil on the wall of a deep cave requires those who wish to see it to visit that location. Like certain tallies, these counts were not portable, could play no part in exchange with other groups in other locations. Their position in a dark cave at the end of a long tunnel hints at a special purpose, but anything from a game or a joke to an important ritual is equally possible.

It is not clear how long this use of Cosquer went on: years, centuries, or even longer. But eventually it ended and the cave, it seems, was neglected for millennia. It was then used again, around 19,000 years ago. The walls began to become a palimpsest, made by people who knew nothing of each other beyond what was recorded on the walls of the cave. This later period saw some of the hand stencils destroyed, scratched or overwritten with new signs, changing their meaning in ways that are just as impossible to recover as is the original significance of the cave and its decorations. A wealth of paintings in black were added at this stage: over 150 images of horses, bison, deer as well as seals and – not yet extinct – great auk.

The cave was abandoned once again, and as the ice began to retreat 16,000 years ago, the sea rose towards its entrance. Some time around 5000 BCE the cave mouth was closed by water, along with those of many other caves along the coast of what is now France, in the area of Marseilles and Cassis. Others, surely, had been used by prehistoric people. By about 500 CE, the sea had reached something like its present level, with the cave mouth more than 30 metres below the surface. Sea water destroyed perhaps three-quarters of the paintings, stencils and engravings, and the cave remained unknown until its rediscovery by a professional diver in the 1980s. Its art has been documented since 1991; it is the only painted cave known today whose present entrance is under sea level. It is also the only location with evidence of prehistoric counting in this particular form: an astonishingly lucky survival. Without it, the finger-counting of the Stone Age would be gone for ever.

Counting words

A final form in which counting may have existed in the African Stone Age is in words. Spoken words leave no trace of themselves, of course, and it would be many millennia before people devised ways to record speech in writing. There is no amazing survival like the cave at Cosquer to report what Ice Age voices sounded like. The clues are plentiful, as are the theories: but certainty is not to be expected. The mystery is once again part of the story.

Language arose in the human species almost certainly in Africa, after the split from the chimpanzee lineage (six or seven million years ago) and before the dispersal to populate the rest of the world (seventy-odd thousand years ago). Within that very broad range, almost every possible hypothesis has been tried, from an early or gradual origin and a period when language was shared among several hominid species to a late and/or sudden appearance, part of a package of modern behaviours whose most obvious result was the exodus from Africa itself. More gradualist scenarios seem to be in favour at the moment, and genetic evidence taken from certain relatively isolated African groups speaking what seem to be languages of very ancient lineage might seem to support that, providing a minimum date for the origin of language of perhaps 130,000 years ago. Much effort has been made to argue that certain developments visible in the archaeological evidence – such as more complex tools – required language for their development or transmission, or that symbolic behaviours such as the use of ochre as a pigment or the manufacture of beads and the marking of surfaces – as at Blombos – are functionally equivalent to language; but these claims have been denied with equal force, and it seems fair to say that there is no consensus as yet. Claims for a ‘language gene’ or set of genes have proved inconclusive, and if a change in human anatomy was involved – such as subtle change to the way the brain was organised – the archaeological remains cannot confirm it.

Whenever language did arise, there may well have been times and places when humans had words for certain things, but no counting words. Counting words and verbal counting routines are not, in fact, absolutely ubiquitous among cultures living today. Despite the fact that virtually all modern humans are capable of acquiring counting words if they are taught them, not all in fact do so, and not all take the sequence beyond a few words or attach any great importance to it. The value placed on counting varies from community to community, and it does not correlate with, and should not be identified with, some such abstraction as ‘sophistication’ or ‘civilisation’. It is quite unimportant to some. Thus there is no positive reason to assume that counting words were part of the ‘ground floor’ of human vocabulary; nor that they must have appeared at any particular stage in the development of language. It is quite possible, indeed, that beads, tallies and even numerical hand gestures were made by people who had no way of counting in words.

On the other hand, nearly all of the roughly 7,000 languages spoken today – or extinct recently enough to be documented – do contain at least some counting words: some terms that can give an answer to ‘how many’ that is more precise than ‘few’, ‘some’ or ‘lots’. It has seemed to many a reasonable speculation that at least some of the people of the African Stone Age therefore had verbal counting routines alongside – or instead of – their other ways of counting.

Perhaps, then, certain human groups possessed ritualised sets of sounds that they made to accompany repeated actions, and over time those sounds acquired something of the character of a ‘wild’ number line. Perhaps those sounds eventually came to be thought of as words just like other words: came to be used as adjectives describing the how many property of groups of things or sequences of actions. Perhaps other sets of words were then derived from them to answer related questions like how many times or which in a sequence. (The range of ways the list of counting words may be further modified is longer than you might think. In English there are not just cardinal numbers one, two, three and ordinals first, second, third but also sequences like once, twice, thrice; or single, double, triple; or single, twofold, threefold. Other languages have similar ranges of options, known by names such as the ‘distributive’ and the ‘frequentative’ numerals: they are used to answer questions like how many each or into how many parts. They are widespread enough among documented languages to suggest that some selection of these types of number word may have occurred in very ancient languages.)

There is no way to know what these ancient number words sounded like. Attempts to reconstruct vocabulary and phonology from several tens of thousands of years ago are occasionally made, but they have not found acceptance by mainstream linguists, and for good reason. It took just a thousand years for Latin to turn into Italian (and Romansh, French, Catalan, Spanish, Portuguese …). Multiply that degree of change – in sounds, structures and meanings – by seventy or more, and it is clear that the noise will overwhelm the signal. It is quite impossible for precise information about the sound or structure of language to survive from the time before humans left Africa.

Perhaps some Stone Age humans went further in verbal counting. If your set of counting words has just five or seven or ten members, say, it is quite acceptable for them to be arbitrary, unrelated sounds. But beyond a certain point, in order not to make intolerable demands on people’s memories, in order to be learned by children and retained by adults, there will be a need for some sort of internal structure. The norm today is to reuse a fairly small set of words over and over, employing multiplication and addition to reach higher numbers. (‘Seventeen’ meaning seven and ten, for instance; ‘twenty-four’ meaning two tens and four.) A number that regularly occurs as a multiplier in such a system is called a ‘base’; it is perfectly possible, and even quite common, for a system of number words to have more than one base. Again, this feature is not ubiquitous, but it is common enough in modern languages for it to be plausible that some of those who counted in Stone Age Africa may have used counting words with this kind of internal structure.

Once again, it is not possible to know what number bases were used in the distant past. Ten is by far the most popular base in the world today, and it is tempting to suppose a cause-and-effect relationship with human anatomy: the ten fingers of both hands. It has been many times pointed out that in a lot of languages the word for five derives from the word for hand, or that the word for digit also means number. But it is not clear that creatures with five digits on each of their four limbs must always arrive at 10 as a base for counting. It is perfectly possible to arrive at 4 as a base, by counting fingers but not the thumb, or to arrive at 6 by incorporating the wrist; or indeed to arrive at a base-5 system by using just one hand. Counting toes as well will give you base twenty, and it is equally possible to arrive at base 12 by using the thumb to count the three joints of each of the four fingers … History and ethnography in fact show a wide selection of other bases (2, 3, 4, 5, 6, 7, 12, 15, 20 …) being used at various times and places. The prevalence of 10 today is in part the effect of happenstance, the result of its adoption by a small number of successful languages and language families. There is, in short, no good reason to project onto the distant past today’s widespread preference for tens.

A cautious hypothesis, then, is that the hunter-gatherer groups of the African Stone Age had just as much variation in their sets of counting words as the cultures to be observed today. That is, some had none, some had short lists of counting words with little or no internal structure, and some had larger sets of counting words structured with number bases. (Many may have had the feature called ‘grammatical number’: a distinction between singular and plural forms for certain kinds of word; or even between a larger set of options such as singular, dual (two items), trial (three) and plural. But not all need have had this feature, since not absolutely all living languages do.) The different systems were transmitted culturally; they were not biologically determined. And they must surely have changed over time, perhaps developing out of all recognition over hundreds of generations.

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

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