Their main use, though, was in the Mayan calendar. Or rather calendars, for there were several, and they interlocked to evoke the vastness of time to an unparalleled degree. There were twenty names for days: imix, ik, akbal … These were crossed with the numbers from 1 to 13 to give an almanac cycle of 260 days, each specified by a unique combination of name and number. This system was already present in pre-Mayan texts of the first millennium BCE, and possible explanations for its unusual length range across astronomical phenomena and the duration of a human pregnancy, or possibly the length of the Mesoamerican agricultural season.

Next, and quite independently of the 260-day calendar, the year was built up from eighteen named months, each with twenty days, plus a sub-month of five days, making a total of 365 days. This, too, was an ancient system, and it provided a second way to name any given day within its cycle.

Third and finally, a neater year of just eighteen months – 360 days – known as the tun, was used as the basis for specifying dates in the so-called ‘long count’ calendar. Tun were grouped in twenties (katuun); katuun were grouped in twenties too (baktun). From a start date placed in the mythic past, the Maya could specify any day by giving the number of baktun, katuun, tun, months and days that had passed. This system was quite specific to the Maya, and indeed strongly associated with the classical period from roughly 300 to 900 CE (in Mayan terms, from 8 baktun 12 tun to 10 baktun 4 tun).

Many Mayan inscriptions began by naming the date in all three of these ways; like that on the stela at Oxwitik, dedicated on a day identified as long count 9.15.5.0.0, almanac day 10 Ahau and year day 8 Ch’en. With three different cycles in play, of 260, 360 and 365 days, there were many different possiblities for anniversaries and commemorations. It turns out, for instance, that fifty-two long years make exactly seventy-three repetitions of the almanac cycle, and this so-called calendar round was carefully observed as an anniversary. Meanwhile, it was of particular interest if the almanac day matched that of a previous event being commemorated. Furthermore, new years and the ends of katuun and baktun were treated as special anniversaries (very roughly like decades and centuries in the Western decimal count of years); half and quarter katuun were also observed in some cities.

When recording an end-of-cycle date, the final two or three number ‘slots’ in the long count date were necessarily empty, and the Maya used their sign for the adjective ‘no’ to meet the need: ‘there were 9 baktun, 15 katun, 5 tun, no months and no days’. The sign was therefore functioning as something like a zero in the dating system: one of the world’s rather few independent inventions of a symbol for zero.

There was something exuberant about the Mayan counts of days within the various cycles, and in many of the Mayan inscriptions the abundance of their numerical information was augmented by providing still more information beyond the three basic counts. Some gave the day’s position in a nine-day cycle of ‘lords of the night’; some specified the age of the moon and the length of the current lunar month: either twenty-eight or twenty-nine days in the conventionalised calendar for such things. Some mentioned yet another calendar, totalling 819 days. Others referred to dates in the mythic past that lay before the zero date of the long count: that is, during a putative previous long count which had totalled thirteen baktun. Others again evoked the far distant future, using rare glyphs for multiples like pictun (8,000 tun), calabtun (160,000 tun), kinchiltun (3,200,000 tun), and alautun (64,000,000,000 tun). One stela continued this series of ever-larger time units to twenty-four iterations: a count of years that would fill thirty-one figures in a decimal representation.

It is thanks to this near-obsession with dedicating and dating, commemorating and counting, that the history of sites like Oxwitik – now called Copán – is well understood today. Nestled in a valley at about 900 metres and watered by the Copán river, it occupied a pocket of rich agricultural land and relied heavily on maize farming. Occupied by non-Maya people as early as 1400 BCE, it was taken over in 426 CE by a new, Mayan ruler, installed by the nearby city of Tikal. This outsider king busily erected buildings and monuments, and founded a dynasty that would last for seventeen generations, eventually ruling over a population of perhaps 20,000.

His successors created temples, plazas, open-air altars, ceremonial ball courts and a complex of royal residences at Oxwitik, a group of monumental buildings whose impact was immediate and spectacular. They diverted the river, and they commissioned some of the richest and most distinctive sculpture of the Mayan period. As in other Mayan cities, society became steadily more stratified and production more specialised, with plants, animals, trees and mineral resources all exploited and traded across the region. At different times seashells, greenstone beads, cacao beans and copper bells were all used as money. The city’s pottery became a prized export.

Waxaklahun-Ubah-K’awil (his name may mean ‘Eighteen are the Images of K’awil’) took the throne in July 695 (specifically, 9.13.3.6.8, 7 Ahau, 1 Mol). Over several decades, he built temples, a tomb for his predecessor and father, and a huge court for the ceremonial Mayan ball game. And he commissioned a remarkable series of stelae for the main plaza of Oxwitik. Each depicted the king performing a ritual, perhaps engaged in trance, accompanied by supernatural beings; each was adorned with a text giving its date and the dates of the acts of ancestors or divinities which it echoed and commemorated. Together, they turned the great plaza into something like a sacred space.

A first series of six stelae was completed by 9.15.0.0.0 (22 August 731); five years later, the king began a new set. The new stela at the northern end of the plaza bore sculpture of fantastic beauty and complexity, portraying the king masked and accoutred for ritual and flanked by eight manifestations of a single god. The side bearing the inscription gave the date using not the dot-and-bar number symbols of many Mayan inscriptions, but an alternative system. Each number up to 12 (including zero) was associated with one of the gods who were used to name the days in the almanac. For numbers between 13 and 19, the ten and the smaller number were combined into a single figure, echoing the structure of the Mayan number words.

This list of gods (specifically, the set of pictures of their heads) was a common way of writing numbers, second only to the bar-and-dot symbols. It was capable of huge artistic elaboration, and on occasion the gods could be depicted as complete anthropomorphic figures; this was how they appeared on the new stela. Combined with the set of zoomorphic divinities associated with the different units of time, the result was a complex, naturalistic scene of number-gods carrying time units. One of the pinnacles of Mayan sculpture, it is among the most elaborate and beautiful of all visions of counting.

If the king was the owner of time and the controller of ritual, he was also the monopolistic wielder of force, both in the form of the labour used to build temples, monuments and the rest, and in the form of war waged on his neighbours, in a continual jockeying for control of resources, trade, tribute and prestige. The Mayan kingdoms were never unified into a single state, and they spent much of their energy in strife against one another.

Fifty kilometres to the north of Oxwitik lay Quiringa, apparently a client or a tributary city. Its new king was inaugurated in 724 CE under the supervision of Waxaklahun-Ubah-K’awil. Over the following decade, something went wrong with this relationship: aggression on one part or the other, or a desire for independence by the ruler of Quiringa. Possibly with help from another larger city in the region, king K’ak’ Tiliw Chan Yopaat of Quiringa succeded in capturing Waxaklahun-Ubah-K’awil. On 3 May 738, he beheaded him.

Naturally, there are monuments at both cities giving the date. At Oxwitik the description states that the king’s ‘breath expired in war’ and lamented that the city now had ‘no pyramid, no altar, no cave’: that building work was at a standstill, as was ritual access to the underworld. The defeat was felt almost as a cosmic disaster.

There would be four more kings at Oxwitik, but the ninth century was a period of decline across the whole Maya world, for reasons that probably included both internal tensions and overexploitation of the environment. Populations declined, dynasties collapsed, cities were abandoned. Oxwitik never really recovered from its disaster, and the last king was dead by 820. Buildings crumbled, and the river channel shifted, cutting a swathe through the ruins. By the tenth century the whole valley was abandoned.

A few Mayan books from the tenth to the twelfth century survive, giving a vivid sense that astronomical and calendrical information was still being transmitted, centuries after the peak of Maya culture. The 260-day almanac cycle was retained for centuries (it is reportedly still in use today), but the long count was gradually abandoned. Dates came to be given in a simplified form, stating the number of katun but omitting the largest unit, the baktun. This was a system that would repeat every 256 years.

The bar-and-dot numerals were also abandoned, although further south in the Andes the Mixtec and Aztec cultures would adopt related systems and transform them for their own uses. One consequence of these changes is a lasting uncertainty about how exactly the classical Mayan calendar correlated with calendars elsewhere in the world: which day of the Western calendar corresponded to its start. It was around 11 August 3114 BCE, but corrections of one or two days have been proposed from time to time, and certainty is elusive.

The Mayan people described and counted longer spans of time than any culture before or since, and their sculpture endures as a monument to their skill and imagination. Theirs was a world pervaded by numbers as dates, timescales, commemorations; even as gods. On the great stela at Okwitik they left a memorial of their king, their city, and one of the supreme expressions of the power and value of counting.

Pirahã: Lost count

Xagiopai, in lowland Amazonia, on the banks of the Maici River, 2004.

A member of the Pirahã tribe faces an anthropologist. The anthropologist sets out a row of batteries on the ground and asks his interlocutor to make another row matching it one-to-one.

What will happen?

Deep in South America, 10,000 kilometres from Agaligmiut and 3,000 from Copán, the Pirahã live – today – beside a tributary of a tributary of the Amazon. They number about four hundred, and they are hunter-gatherers. Their villages – each of about ten or fifteen adults – have some contact with Spanish-speaking traders, and they have a history of more than two hundred years of contact with Brazilians as well as their neighbours the Kawahiv. But they have rejected assimilation into the mainstream culture of Brazil, and remain monolingual in the Pirahã language. Linguist Daniel L. Everett, who spent over six years living with the Pirahã, writes:

The Pirahã are some of the brightest, pleasantest, most fun-loving people that I know. The absence of formal fiction, myths, etc., does not mean that they do not or cannot joke or lie, both of which they particularly enjoy doing at my expense, always good-naturedly.

Their language is the last surviving member of its family. It is famous for a number of unusual features, which threaten to break several otherwise attractive generalisations about human languages. In some ways, it is a complex language, with intricate verbal morphology and a rich five-way system of syllable types distinguished by weight. On the other hand, its set of sounds is one of the smallest in the world; Pirahã women use just seven consonants and three vowels (men have one more consonant). It is the only documented language with no terms for colours; it has remarkably few words for time, and it possesses the simplest set of pronouns known (which appear in any case to be borrowed from another language). It has no perfect tense: no way for a verb to express specifically the completion of an action described.

The cultural factor thought to lie behind these various features of their language is that the Pirahã people communicate only about subjects within their immediate experience: things that have been seen or recounted by someone now living. They completely avoid talking about the abstract or the second-hand. Everett’s example:

‘I prefer whole animals to portions of animals’. (Literally ‘I desire [a] whole animal[s], not piece[s].’) Sentences like this one cannot be uttered acceptably in the absence of a particular pair of animals or instructions about a specific animal to a specific hunter. In other words, when such sentences are used, they are describing specific experiences, not generalizing across experiences.

Other consequences of this are that abstract structures like long genealogies or complex kinship relationships are absent from the Pirahã culture. Kinship terms refer only to relatives a person knows, never to those who died before that person was born (Everett ‘could not find anyone who could give the names of his/her great-grandparents, and very few could remember the names of all four grandparents’). There are, furthermore, no fiction, no creation stories, no myths nor any other stories about the ancient past.

And there is no counting.

First, the Pirahã language has no distinction between singular and plural. No part of speech is marked to show a distinction between ‘one’ and ‘many’, so – unlike in most languages – you don’t have to answer the one/many question every time you utter a sentence: indeed, you couldn’t if you wanted to.

Second, the Pirahã have no counting routines, no counting practices, no habit of putting objects in a sequence. Counting with fingers or with counters is unknown here. And Pirahã has no counting words. Early investigation reported the terms hói, hoí and bá a gi so as meaning something like ‘one–two–many’. But further investigation has revealed that these terms are not part of a counting sequence, and they are not number words at all; they mean respectively a ‘small size or amount’, a ‘somewhat larger size or amount’, and ‘cause to come together’ or ‘many’. None is used consistently to refer to a specific number. The language has no other terms for quantity such as ‘all’, ‘every’, ‘most’, ‘each’ or ‘some’, nor words for items in a sequence such as ‘first’ or ‘last’. Hói often denotes a single one of something, but it doesn’t have to; it can mean as many as six. Similarly, hoí can denote groups from two up to ten objects, depending on context.

During the Brazil nut season, there are regular visits to the Pirahã villages by river boats, a contact that has probably lasted more than two hundred years, with Pirahã men collecting nuts and storing them to trade. But the trade is not numerate. The Pirahã, despite this long contact, have not learned the Portuguese number words and barter with – reportedly – little regard for the quantity of goods involved. ‘Someone can ask for an entire roll of hard tobacco in exchange for a small sack of nuts or a small piece of tobacco for a large sack.’ ‘In this “trade relationship” there is no evidence whatsoever of quantification or counting or learning of the basis of trade values.’

Anthropologist Peter Gordon, in 2004, involved members of the Pirahã community in a series of experimental tasks, to learn more about what quantity did and did not look like in their world:

I sat across from the participant and with a stick dividing my side from theirs, I presented an array of objects on my side of the stick … and they responded by placing a linear array of AA batteries (5.0cm by 1.4cm) on their side of the table.

He emphasised, when describing this scene, that the Pirahã clearly understood the tasks and were trying hard at them.

Matching a small number of items – two or three – often produced a one-to-one match or something very close to it. Matching larger numbers of items produced more approximate representations, with the numerical match falling off as the numbers became larger. For the Pirahã, it seems, a set of nine objects and a set of ten are distinguishable, but only just: perhaps three-quarters of the time.

From simple rows of batteries, Gordon moved on to ‘clusters of nuts matched to the battery line, orthogonal matching of battery lines, matching of battery lines that were unevenly spaced, and copying lines on a drawing’. A final task involved watching nuts placed in a can and then taken back out one by one, with the experimenter asking at each stage whether there were any nuts left in the can.

For these more complex tasks, some of which involved transferring counts across time or space, the representations produced by the Pirahã were still more approximate, with exact matches produced rarely or not at all for the larger sets of objects. The range of their representations became wider as the sets became larger, showing the classic characteristic, in fact, of the approximate number sense. A colleague later performed related experiments, presenting the Pirahã with two rows of objects and asking if they were equal in number. The results confirmed still more directly that small numerical differences simply were not perceived – were not experienced. Indeed, with no way to perceive or to name exact quantities, it appeared that exact numbers simply were not concepts for the Pirahã.

Like all humans, the Pirahã experience number approximately, with differences among adjacent numbers less and less perceptible as the numbers become larger. Unlike nearly all adult humans, however, they do not overlay this perception with any other way of counting.

The Pirahã and their remarkable language caused a good deal of surprise when first reported in journals of anthropology and linguistics. Debate will doubtless continue about what exactly certain words in the Pirahã language mean or do not mean, but the fact remains that no one who has visited the tribe has ever claimed to see them spontaneously counting anything. Two further teams of anthropologists repeated certain of the number-matching tasks. One thing that emerged from this contact was a desire of some of the Pirahã to learn counting words. Everett responded to their request to learn to count in Portuguese, and reported that after eight months of daily effort none had learned to count to ten. A second attempt was more successful. A researcher who spent several months in the Pirahã village of Xagiopai trained its inhabitants in one-to-one matching tasks and invented Pirahã-like words for numbers up to ten. The result was – in that village alone – a significantly more exact set of responses to some matching tasks by the end of the period. This certainly confirmed that the Pirahã are capable of learning to count – that there is no question of an inherent inability here – but it does not seem to change the fact that, in their language and culture as originally documented, number is experienced only as approximate. It is not certain quite how rare this view of quantity is among humans: it has been claimed of a number of other, mainly Amazonian groups that their languages contain no number words, but none has received anything like the systematic attention paid to the Pirahã.

The Pirahã represent a remarkable case within the range of possibilities for human language and culture, and a quite astonishing branch on the story of human counting. The knowledge that there is a living culture with no counting at all is a salutary reminder that counting – in the sense of repeated attention plus a way to keep track of it – is not inevitable for human beings. It is not clear just what sequence of events led to the existence of an Amazonian tribe with no counting practices, but the prevalence of counting words across the Americas makes it scarcely credible that the Pirahã somehow represent a line of humans that never learned to count in the first place. It seems all but certain that they are descended instead from ancestors who did count.