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.
The latter possibility is hinted at by the fact that Mura, a now-extinct dialect closely related to Pirahã, possessed some of the cultural artefacts Pirahã does not: fables and legends and other fiction about the distant past. Perhaps it had number words as well, or a common ancestor language did so.
If the Pirahã are indeed descended from people who counted, the situation is quite a dizzying one. Not only is counting a cultural artefact, an invention that – it turns out – not every society possesses, since not every society lives a lifestyle that needs it. But counting is capable of extinction if the need for it passes away. One of the branches on the story of counting ends – for the moment at least – in a world without counting. One of the possible futures for any human culture is indeed a world after counting, in which the skill – for all its range, power and great age – has been forgotten. In which counting words have fallen silent, counting gestures have been forgotten, tallies are no longer made and beads are once more just beads.
Conclusion
So many hands and bodies and voices and minds have counted. Mentally, manually, with materials, with words, with marks, with symbols, with machines. They have counted people, possessions, food items, friends, enemies, votes, years, days, eggs, trees, coins. They have counted to pay their taxes, to ensure their survival, to administer their cities and their businesses, to measure their self-worth, to commemorate their ancestors, to buy or sell. They have counted for fun. There is always a strange alchemy to counting, which restlessly transforms one thing into another: days into tally marks; people into counters; books into magnetic tapes.
This book has described a few of those processes. There have been thousands more, in all the thousands of languages living and dead, in all the thousands of cultures and hundreds of scripts. No two are alike. The real alchemy, perhaps, is in turning all of these processes into a single thing, and calling it ‘counting’.
Counting is – this book has assumed – repeated attention to things or events, plus a way of keeping track. The different things to which attention might be paid, together with the manifold ramifications of ‘ways of keeping track’, have produced an almost inconceivable complexity and diversity of counting processes at different times and places.
Counting builds on innate abilities: the ability to estimate the relative sizes of groups of objects, and (possibly) an additional ability to recognise at a glance – to ‘subitise’ – sets of one, two, three or four. At least the first of those abilities is shared widely among the primate lineage, as well as with some other mammals and with birds; possibly with other animal groups as well. But no animal can count precisely. No animal species has been observed spontaneously using counters or tallies in the wild; none can learn more than the first few in a set of number words or number symbols. Counting uses regions of the brain innately specialised for the approximate number sense. Depending on what device for keeping track is involved, it engages other brain regions as well, including those that process spoken or written language, or those that handle fine motor control.
The ancient Stone Age environment contained several structures that could be used for the keeping-track function: structures involving a stable sequence of objects or actions. The marks produced by repeatedly striking stone on bone are found up to 3.2 million years ago; examples interpretable as tallies start to appear perhaps 70,000 years ago. Beads on strings also appear in the East African archaeology by at least 70,000 years ago, very possibly earlier. Extended or folded fingers first appear in archaeological evidence just 27,000 years ago, in Europe, but it is overwhelmingly likely that people counted on their fingers in Africa tens of thousands of years before. Vocalisations leave no archaeological trace, but there were very possibly conventionalised sets of sounds accompanying repeated actions early in the story of language; it is impossible to say when they were incorporated into speech or transformed into cardinal – ordinal, distributive, frequentative, and so on – numbers. Very nearly every documented language has some number words, and considering what they have in common enables (only) a few steps towards imagining what the earliest number words may have been like.
Number symbols – potentially a way of counting, but of most use as a way of recording and communicating the outcome of a count – appeared alongside the earliest notations recording spoken words, in contexts from the Near East, East Asia and Central America. There are constraints on the structure a set of visual symbols can have if it is to encode numbers unambiguously: each of the main possibilities has been used successfully and stably over long periods of time and large areas of space. Around a hundred distinct sets of number symbols are documented from around the world.
