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.

On the basis of these origins, counting has developed in a vast variety of ways at different times and places, its roots supporting many, many branches. Humans faced with different things to count and different reasons for counting have been hugely creative in finding new ways to count, and in adapting the old ways. Counting practices have always been devised in response to particular needs at particular times and places; they have always emerged in the course of specific interactions between people and their world. Some have been transmitted to neighbouring cultures and subsequent generations, but more have been short-lived, persisting only briefly before being transformed again or replaced by new experiments and inventions. Feedback has been an important force: once people gained new technology – a new way of counting – they invariably found new things that they could do with it; perhaps things they never previously knew they wanted or needed to do. And those new functions sometimes, in turn, drove new innovations in ways of counting.

In a lot of the situations where people have wanted to count, they have also wanted to do at least simple arithmetic, so there has been an overlap between the technologies used to count and those used to calculate. A lot of devices for keeping track have been adapted, more or less drastically, to provide ways of doing arithmetic. At least sometimes, considerations of their efficiency and effectiveness for calculation have affected the choice of which technology to use for counting itself. The shift from counting board and Roman numerals to paper and Arabic numerals in medieval Europe is one such case; the rise of the suanpan and soroban in East Asia is another. On the other hand, there have been plenty of times and places when counting and calculation used separate technologies.

Over time, manual and verbal ways of counting have replaced one another; spoken and written numbers have gained or lost in importance, more concrete or more purely mental ways of counting and calculating have advanced and retreated. The accidents of history have given prominence to one way of counting, then to another. The simplest ways of counting (fingers, beads) have never really vanished; neither has the innate ability to estimate. Nor have the limitations of the subitising system, and their consequences for the devices people build and the notations they devise.

The story of counting is a dense tree, with several roots and nearly an infinity of branches. Its end points – as the tree stands today – include elaborate routines with unmarked counters on marked surfaces, highly developed sets of number words, successful – and successfully exported – sets of number symbols, and electrical machines whose internal states are, for some purposes, taken to represent numbers. They include smaller sets of words, sometimes accompanied by gestures, tally sticks and written representations of number words. They include a world in which there is no counting whatever.

What will counting look like in the future? The tree will continue to grow and change. Some branches will burgeon with new growth; others, not so much. Over time, those patterns will themselves change: human ingenuity is most certainly not finished with counting yet. Almost nothing can be said with confidence about the detailed forms that counting will take in the future. But they will continue to be manifold, various, surprising and fascinating: there will always be much that is new in the story of counting, as well as much that is old.

Notes on sources

This is a work of synthesis, based largely on survey articles and outlines of the handbook/companion type, supplemented for the detailed case studies by publications discussing and/or reproducing specific sources. What follows is a list of the principal sources used for each section of the text, including the sources of all direct quotations; it does not list exhaustively the minor or supplementary sources consulted, nor attempt to specify the authorities standing behind every assertion.

A high proportion of the topics addressed have been the subject of disagreement among specialists, and in some cases that disagreement is ongoing. In such cases I have attempted to follow the weight of recent opinion; that is, I have not intentionally presented outdated or minority interpretations. Alternative views are in certain cases mentioned within these notes.

Confusions and errors in the text are naturally my responsibility, not that of the scholars cited below, to whom grateful acknowledgement is made and from whom indulgence is craved. In a project of this kind, reach necessarily exceeds grasp, perhaps by some distance; one of its premises is that the task of synthesis nevertheless has value.

ABBREVIATIONS

BNP

Cancik, Hubert,

et al

., eds.,

Brill’s New Pauly

(Leiden, 2012).

CEWAL

Woodard, Roger D.,

The Cambridge Encyclopedia of the World’s Ancient Languages

(Cambridge, 2004).

CWP

Bahn, Paul and Colin Renfrew, eds.,

The Cambridge World Prehistory

(Cambridge, 2014).

EHAS

Rashed, Roshdi, ed.,

Encyclopedia of the History of Arabic Science, Volume 2: Mathematics and the Physical Sciences

(New York, 1996).

HMC

Campbell, Jamie I. D.,

The Handbook of Mathematical Cognition

(New York, 2005).

OCD

Hornblower, Simon and Antony Spawforth,

The Oxford Classical Dictionary

(3rd edition, Oxford, 2005).

OHHM

Robson, Eleanor and Jacqueline Stedall, eds.,

Oxford Handbook of the History of Mathematics