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
