Perhaps 20,000, or even 25,000 years before the present. A woman poses, holding up the horn of a bison. The mother of children, she is pregnant again, expecting her new child in a month or two, and with her free hand she gestures to her abdomen. She turns her head to her right, to look along the length of the engraved horn; as she does, her hair falls across her shoulder.

The horn itself is engraved with thirteen short parallel marks. A decoration? Perhaps. A count? Very probably. But what was she counting?

Something the Stone Age environment contained in large numbers was the bones of animals, many of them bearing scratches and cuts from butchery done with stone tools. During the Middle Stone Age, from as early as 90,000 years ago, some human groups also began to work bone, making bones into pointed tools and other artefacts.

The combination of accidental scratches and cuts on bones, together with the increasingly skilled working of bone, led some humans to mark bones with deliberate scratches; and, eventually, to use those marks as a way to record or communicate information. This, again, is a world of tantalising clues and ambiguous evidence; once the original context is gone, it is seldom possible to be confident what information an artefact was meant to communicate. But, on the one hand, a series of neat, symmetric, cross-hatched incisions on a bone tool looks irresistibly like decoration, and raises questions as to what it might have conveyed about a person’s identity, or a group’s. And, on the other hand, a series of roughly parallel scratches on a bone (or other) surface looks irresistibly like a tally, the outcome of a counting process: one scratch for each object or each event in some set or sequence.

The site at Blombos provides, as well as shell beads, some of the most intriguing of the very early evidence for deliberate marking of bone. Working bone was, indeed, a regular activity here: twenty-eight bone tools – awls, spear points and a retoucher – have been recovered from the early archaeological deposits. The age of these bone artefacts is over 70,000 years, and at least two of them bear possible deliberate engravings. One has eleven incisions parallel to its long edge, while a more superficial oblique line intersects six of them. These are certainly not random marks, and microscopic analysis shows that they were produced by the same stone point in a single session. Their design was intended from the outset and made deliberately.

As with the bead strings from Blombos, these earliest marked surfaces most likely bore no numerical meaning, if indeed they carried any meaning at all. Scratches could have been made on bones for years or generations as an object of play and experiment before they meant anything. But making a series of marks on a surface is, at the same time, another ‘wild number line’: another artefact in the human environment capable of being eventually put to numerical use. And like the beads on a string, it could potentionally function as a way of counting even in a culture that contained no other way of doing so.

Archaeological evidence for patterns of deliberate scratches on bone and stone – and on shell – is present sporadically throughout the African Middle Stone Age, and towards the end of that period it becomes possible to be confident that counting was involved. Archaeologists emphasise the fact that a proportion of the marks and scratches that have been interpreted in these ways are the results of natural phenomena such as gnawing or of damage during excavation, or are interpretable as non-deliberate butchery marks rather than patterns made deliberately. They note that some patterns of scratches may have been made in order to make the surfaces of tools easier to grip. They acknowledge that meaningless doodling by Stone Age people is also a possibility, as is decoration without any specific information content.

But the consensus does seem to be that the step from decoration to symbolism was taken by Stone Age people, and that some of the marks on very ancient artefacts recorded information for the people who made them. That information could have served various different functions for the hunter-gatherer societies in which the artefacts were made, and some of it may well have been numerical information. In particular, where there is a sequence of similar marks in a repeated pattern, regularly spaced and running along a well-defined path, it is not unreasonable to see this as evidence of counting, and where there is subdivision of the marks into (irregular) groups, that possibility becomes all the stronger. Some scratches really are tallies.

For instance: between 20,000 and 25,000 years ago a group of humans lived by what is now Lake Rutanzige in the Democratic Republic of the Congo. Theirs was a more settled community, perhaps, than those of their predecessors, and they relied on spear and harpoon to hunt fish in the lake. They used quartz to tip their tools; one that survives is the fibula of a mammal about 10cm long, with a quartz fragment fixed in a cavity in one end. Its exact purpose as a tool is uncertain: engraving or tattooing are possibilities. It is probably the most-discussed artefact in prehistoric mathematics, however, because as well as its functional characteristics it bears sixteen groups of engraved lines, arranged in three columns. For what it is worth, the groups contain 11, 13, 17 and 19 marks in the first column, 3, 6, 4, 8, 10, 5, 5 and 7 in the second and 11, 21, 19 and 9 in the final column. These are well-defined, clearly grouped marks that can scarcely be interpreted as accidental or decorative. Interpretations of the bone have focused, understandably, on the assumption that these are tallies, made as part of a counting process and in their finished state communicating the end-points of the counts to others. Potentially, communicating them to the present day.

The Ishango bone.

agefotostock / Alamy Stock Photo.

But what were the makers of this artefact counting? It seems impossible to say. What possessions might have needed counting? What resources managed? What events tracked across time? One hypothesis is that you might need to count to keep track of the traps you have set for animals. Another is that you need to count food items in order to ensure survival over the winter, particularly if the region’s climate is becoming colder or drier. Trade, gambling or ritual are other possibilities, and the distribution of surplus at feasts is another. But all of these ideas make assumptions about the society in question that no one can test.

Much ingenuity has been exercised in order to discern some regularity in the tallies, in the hope that this will reveal something about their meaning. There are 168 marks in total – sixty in each of the outer columns and forty-eight in the middle one – and it is tempting to imagine an attempt to divide up some commodity that came in twelves. Other readings focus on the presence of prime numbers only in the first column, pairs of numbers bracketing ten and twenty in the third, and relationships of multiplication by two in the middle column. It is not impossible that these counts arose from playing a game of some kind; a calendar, on the other hand, seems hard to fit to these numbers, though it has been tried.

The bone from Lake Rutanzige (it is now normally known as the Ishango bone) illustrates the sophistication that African tallies could have by the later Stone Age; it also illustrates the mysteries that will always surround this kind of evidence. It seems at least reasonably clear that the bone comes from a situation in which the individual marks meant something, that the point of the engraving was not the pattern as a whole but its details. But certainty that these marks were intended to represent numbers is elusive; confidence about what the numbers counted is quite unavailable.

Homo erectus spread out of Africa into Eurasia more than a million years ago, and by half a million years before the present, tool-making hominids could be found from the tip of Africa to Spain, England and northern India. From perhaps 100,000 years ago waves of modern humans (Homo sapiens) began to leave Africa: some successfully, some not. Some travelled along the coast into Asia, others through the Levant into Europe. Eventually their descendants swamped the Homo erectus descendants they found – Denisovans in Asia, Neanderthals in Europe, whose capabilities and cultural sophistication are a matter of debate – and spread further: to China, Australia, Japan, Indonesia, eastern Russia, North America, South America, as well as migrating back into North Africa. Humans had reached Australia by possibly 65,000 years ago, and dominated Europe from perhaps 40,000 years ago.

The trickle of tallies and possible tallies becomes a much stronger flow during the later period, especially in the prehistoric archaeology of Europe. Artefacts of this kind have been found at over forty different sites, enough to give an impression of a widespread and perhaps continuous tradition over a long span of time. Nearly all of the marked objects are small; many are hand-sized or smaller portable objects, and their survival and discovery are a matter of chance. It seems highly likely that they are the remnant of a much larger number of objects that once existed, and which very possibly included engravings on perishable materials such as wood, which do not survive in the archaeological record.

For the objects that do survive, microscopic study and painstaking replication by modern experimenters have yielded a wealth of information about how the marks were made. Lines could be made with a single stroke or with multiple strokes, ‘notches’ made by moving the cutting edge to and fro, ‘microcups’ by rotating a point. Tools could wear down, break or be replaced; they could be shifted in the hand or resharpened. These marks record in an unusually direct way the individual actions of those who made them: the individual strokes of a blade on a bone or another surface. They give the irresistible sense of bringing the thoughts of prehistoric people close to the present.

Yet those thoughts remain tantalisingly elusive, because the reason why the marks were made remains uncertain. The history of the modern answers to that question would, indeed, be a study in its own right. The first archaeologists to discover such objects during the nineteenth century frequently read them as counts of animals taken in the hunt. That view gave way during the following century to a plethora of different interpretations, including that they were records of the numbers of participants in rituals or for use in games. Popular in the 1980s was the reading of sets of scratches or notches as lunar calendars, on the analogy of tally calendars from native North America. One after another, these readings have fallen victim to a lack of direct evidence, and to the fact that once it is removed from its original situation, a symbol may mean almost anything.

Occasionally, though, there is more context to help. Palaeolithic Europe was a place and time where cave art – representations of people and animals drawn on walls – had begun to appear. In a rock shelter at Laussel in the Dordogne, people engraved a series of human figures into the limestone. One has now fallen to the ground (or been ‘excavated’ by the heroic methods of the nineteenth century) and been subsequently removed. Not all of the figures can be easily read, but there are certainly two women among the group as well as an adolescent, and – the one now removed – a pregnant woman holding a bison horn in her right hand.

This figure bears what is very probably a tally: the horn has a series of thirteen marks on it. The figure is about 40 centimetres high, and traces of red pigment – ochre – indicate that it was once coloured in whole or in part, perhaps mimicking the use of ochre on actual human bodies in this culture. The period famously produced a large number of depictions of women, notably in the form of small portable statuettes. Compared with later ice age art they are not particularly schematic: indeed, some archaeologists have felt that their details make them portraits of individuals rather than representations of ideal types, still less of goddesses (the old designation ‘Venus’ still lingers around some of them). At the least, they realistically represent different physiological situations such as youth, age or pregnancy. To this end, they tend to neglect details such as hands, feet and hair.

The woman of Laussel.

Musée d’Aquitaine, Bordeaux, Gironde. Japhotos / Alamy Stock Photo.

The woman of Laussel is a superb specimen of this art. Unusually, her hair is clearly depicted; as is one of her hands, complete with its fingers, which gestures towards her abdomen as though to draw attention to her pregnancy. Collarbone, hips and navel are all clearly drawn or suggested, and her pose is realistic, with one shoulder slightly higher than the other, because of the movement of the arm to hold the horn. She turns her head to look along the length of the horn. On the other hand, her face is not even sketched, and her feet are hardly suggested; and the proportions of the body have been somewhat distorted in order to make the pose work. She looks, for all that, more like a real person than the figment of an artist’s imagination.

The marks on the horn she holds seem to form a single series. Unlike those on the Ishango bone, there is no internal structure provided by grouping of the marks: that is, the gaps between them are all about the same size. The marks themselves are not quite identical; the second from the end is more like a Y than a single stroke (there is a mark of similar shape on the woman’s left hip), and differences in length and width might make the subsequent eleven into two groups of four and one of three. But this may be to over-interpret the image.

Who was she? What does the image mean? And what was she counting? As with prehistoric art generally, there have been many attempts to interpret the woman of Laussel, ranging from a record of animals hunted to a magical icon; perhaps even a request for a certain number of animals. The horn has been seen as a drinking horn, a musical horn or a horn of plenty. A reading as a ‘lunar calendar’ has been tried, too: thirteen – the number of marks – is after all the number of lunar months in a year, and the horn does look something like a crescent moon. Recent interpretations have circled around the idea of a fertility calendar or an obstetric calendar, linking the woman’s obvious pregnancy to the possibility that the horn reports a count of days or months. As the museum of Aquitaine, where the figure is held today, puts it, ‘unfortunately, the deeper significance of this art will probably remain unknown to us.’ Prehistoric tally marks, indeed, will always tantalise.

Cosquer: Counting by hand

The north coast of the Mediterranean, 27,000 years ago. In the long slope between the ice and the sea, men and women enter a cave. They carry charcoal, ochre and pine-stick torches. A tunnel leads to a huge chamber, too big for the torches to light completely. Stalagmites, stalactites, pillars and concretions on the walls add to the weird, other-worldly feel of the place. The people climb and balance. One takes a mouthful of pigment, presses a hand high on the wall and sprays the pigment out, creating a stencil.

Handprints from Cosquer.

SOPA Images Limited / Alamy Stock Photo.

So the Stone Age environment included beads on strings and marks on surfaces, each of which could be made, used, experimented and played with both before and beyond the point when they were used to keep track of how many of something: used to count. It also had available the human hand, which provided a more limited but more immediate option for keeping track of processes or things.

Compared with speech, gesture was late to be recorded in writing, and late to be studied seriously by people interested in the ways and origins of language. But it is now regularly speculated that gestures actually preceded spoken language in the ancient human past. Certainly, gestures seem to be basic and obvious – both to oneself and others – and are made spontaneously by human children, even before they learn to control their vocal apparatus. (It is the same story with other primates, which generally show much more spontaneity and flexibility with their gestures than their vocalisations, and have much more success learning new ones.) Perhaps early hominids pointed and pantomimed through gestures – to direct attention, to gain desired behaviour from others and to facilitate cooperation – even before they acquired speech.

For the purpose of counting, or as a wild number line in existence before counting, the fingers form a sequence with a stable order: and they are always, literally, to hand. As with beads on strings or marks on surfaces, people could flex their fingers in sequence for many generations before beginning to use them to keep track of objects or events. Thus they are another possible bridge from playing and experimenting to counting.

Another advantage is that the digits of one hand take you just beyond the subitising limit of four. If the first four finger gestures can be distinguished and recognised using the subitising capacity, a single hand provides just one more item in the sequence: a whole-hand gesture easy to make and recognise. Perhaps this could have been a first step from the small numbers – recognisable at a glance – to what comes beyond them.

In a similar way, extending or flexing the fingers of one hand might act as a bridge from counting for oneself to communicating with others. A finished gesture does not persist the way a mark or an artefact does, but it can be shown to someone else, and perhaps become an answer to the question ‘how many’: a ‘number word’ in a certain sense.

To check how plausible these speculations are, it is not possible – unfortunately – to observe people today using gestures in isolation from other ways of counting: because there seems to be no culture that uses gesture as its only way to count. It is possible to observe young children, who learn early to extend one finger after another, and – in a modern, number-saturated environment – learn early to associate those gestures with other ways of counting. There is convincing evidence that children learn number gestures before they learn number words, and indeed that in the early stages of learning they may be more accurate when using gestures for small numbers than when using words: even that their acquisition of gestures precedes and facilitates the acquisition of words.

So widespread is finger counting, indeed, that its traces persist in adults’ handling of number and calculation. Errors of size five – a whole hand – are more common in adults’ arithmetic than errors of other sizes, even when words or written symbols are being used rather than fingers: which may suggest that people continue to associate numbers at least in part with hand gestures. Finger gnosia – the ability to tell your fingers apart when they are touched but you can’t see them – is a good predictor of arithmetic ability: better in fact than general intelligence. And adults on average count and calculate more slowly if they are required to do something else with their fingers at the same time.

As well as this modern evidence, there are also – remarkably – direct traces of the use of the fingers in the Palaeolithic, from certain human groups which found a way to make their gestures endure.

During the last ice age (roughly 30,000 to 10,000 years ago), a strip of land in what is now the southern fringe of Europe was occupied by humans. Although much of the continent was technically a polar desert, the inhabited area, beside the Mediterranean Sea, was not impossibly cold, with temperatures up to 12 or even 15° Celsius in the summer and relatively mild, survivable winters. Sheltered valleys had woodland with both conifers and broadleaved trees, and there was a range of land animals as well as maritime fauna: enough to support a human population. Comparable kinds of tools existed from northern Spain to the Russian plains – knives, arrows and spears – as did comparable artistic traditions. Little is certain about these people’s lifestyle: it was heavily based on hunting, and they may have followed animal herds from place to place during the year.

Lifestyles were certainly shaped by the climate, and natural shelters were often used. One was a cave near the southern coast of what is now France, a few kilometres southeast of Marseilles. It opened at the foot of a limestone cliff with an entrance wide enough for two or three people to walk abreast. The entrance tunnel climbed gradually over a distance of 175 metres until it opened into a large chamber, 60 metres across and in places three times the height of a person. At the far end, there was a dome 30 metres high and a shaft nearly as deep. The slope of the land meant the chamber was far below the surface, and there was only one entrance. It was an isolated, mysterious place, and except for torchlight an utterly dark one.

No one lived there. Fires were lit to provide light, not to cook. Torches were carried in; charcoal from them fell scattered on the floor. And the walls, by the flickering light, were decorated with paintings and engravings. The earliest were made more than 27,000 years ago.

Finger-marks were made in the soft calcite that covered the walls. People went to every part of the cave, even – perhaps especially – to the most inaccessible parts of the roof, covering dozens of square metres with simple lines made by the fingers. Curves, zigzags, scrolls intersecting one another. They formed a background showing – apart from anything else – that humans had been here, had made the place their own.

People also drew and scratched on the hard parts of the walls: pictures of a bison, perhaps of some horses. They drew quickly, in minutes, with simple schematic lines, designs up to a metre or more long. They added signs to some: long barbed lines superimposed on the animals, angles or chevrons, zigzag bands. Were they icons of the hunt? of weapons? or representations of animal footprints and faces?