‘Counting’ can seem like an unruly grab-bag of almost totally unrelated actions; a label covering a huge set of very different cultural practices. The range of different activities called counting seems too wide for comfort, and at least superficially, it is not clear what they all have in common; or even whether they have anything in common at all.
Almost any definition of counting is problematic, but one of the best is attributed to the seventeenth-century German philosopher Gottfried Leibniz. It says that counting is repeated attention. Counting is what happens when you think ‘this … this … this … this’, and have some way of keeping track.
The means of keeping track might be a set of words or a set of symbols; it might be a set of tally marks or a set of beads on a string. There are several other possibilities. But if you are paying repeated attention to objects or events and you have some way of keeping track of that process, then you are counting.
It is in the different ways of keeping track, in fact, that much of the enormous diversity of human counting takes place.
Counting is different from measuring, which is concerned with comparing one object with another, although symbols that record the outcome of a count have a very long history of being repurposed to record the outcome of measurements as well. Counting is different from calculation, although it turns out that nearly every method of counting has at one time or another been adapted in order to do at least simple arithmetic: to add together two counts or to take one away from another.
Counting is also different from using number words or number symbols as a handy set of labels. A phone number is not the result of anybody counting anything, and ‘prisoner two-four-six-oh-one’ does not necessarily stand at the end of a line – real or imaginary – of 24,601 people. Although he might.
Counting has a less clear boundary in the direction of machines that count: visitor counters at the doors of shops and museums, for instance. It seems eccentric to insist that those machines are not really ‘counters’, yet they break any claim that counting has to involve conscious, human attention. Perhaps not every boundary can be quite distinct, particularly at a time when counting machines are changing the world so rapidly.
Animals do not spontaneously count. They certainly pay attention to features of their environment one after another, but no wild animal has been spotted devising a way to keep track of that repeated attention. And even the brightest specimens of the most promising species struggle to use counting techniques invented by humans – words, symbols – after the first few numbers. On the other hand, animals do display some of the capabilities that underlie human counting: an ability to estimate the relative size of groups of objects, in particular. In biological terms, counting didn’t come from nowhere, although it does seem to be unique to humans – at least on this planet.
Counting does not have one single history. There are several different ways to keep track of the things you pay attention to; they have different advantages and disadvantages that become important in different situations. Counting with words, with gestures, with symbols, using machines: each has risen and fallen and risen again at different times and places. There is no way to take the world’s ways of counting and arrange them in a line, from worst to best or from most primitive to most sophisticated.
The story of counting is shaped, instead, like a tree. It has several roots, many branches, and innumerable twigs and leaves. Counting has grown and travelled with the human species, ramifying into very nearly every culture past and present. Sometimes it is possible to follow a single branch for some distance: sometimes a branch turns out to cross, to touch (or nearly touch) other branches. The number symbols that were invented in India, and now dominate the world, are like this. It is possible to follow them from their origin through their long – and ongoing – peregrination across the world, and to watch their interaction with many other traditions of counting along the way.
Elsewhere there are groups of branches – perhaps, better, the twigs within a single branch – with something definite in common. A preference for counting devices in East Asia: rods, the abacus and microchips. A preference for words and gestures in Oceania.
This book is shaped like a tree, too. First, there are two chapters about the roots of counting: the features of human cognition and anatomy, and of the Stone Age environment, that make it possible for humans to count, and that provide the most basic, pervasive and enduring ways to keep track of different objects or events. Humans have innate abilities that are relevant to counting, as well as perhaps an innate habit of spontaneously focusing on quantity. And the earliest available ways of keeping track of a count are counters, fingers, tally marks and words: technologies that will turn up again and again in the different branches of the world’s story of counting.
Then there are six chapters about different branches of counting’s story, organised as a world-wrapping journey that follows the great human spread out of Africa: to the Near East, Europe, South and East Asia, and on into Oceania and finally the Americas. These emphasise what is most characteristic in each part of the world: the invention and use of number symbols in the Fertile Crescent, for instance, or counters and counting boards in Europe. One chapter is about the Indian number symbols, and necessarily spans the world in pursuit of their story. Further east, the book emphasises counting machines in East Asia and counting words in the Pacific. Different choices could have been made; no part of the world has an exclusive preference in its ways of counting.
The stories told in each chapter emphasise the local and the personal: narratives of specific people actually counting, for specific reasons. Some illustrate novelties and turning points, but most are about the way things usually were, the kind of events so common they are seldom written about or remembered. Within each chapter, the illustrations are often arranged by their date, but they are still branches on a tree, not stops on a highway, with ‘later’ often meaning different but seldom meaning better (or worse).
The story of counting has the tree’s property that a closer look always brings more structure into view. That property comes to a head in the Americas. They were the last major landmass to be populated, and their languages and cultures are famously diverse, with dozens of different human groups across millions of square kilometres. Ways of counting in the Americas span the whole range from beads to tallies to words to symbols, with no clear, continentwide preference. So by way of an epilogue, this final chapter presents something more like a tree in its own right, a microcosm of the world’s counting in a 15,000-kilometre journey from the Alaskan Arctic to the Amazon basin.
But first, the roots.
PART 1
Roots
1
Number sense before counting
Can animals count? Do humans inherit from animal ancestors a ‘sense of number’, or even something more? The answer is a complex yes-and-no. Even the most gifted animals cannot learn to wield number words or number symbols beyond the first few, to perform calculations or to work an abacus. Yet many species do display a pair of abilities related to counting.
On the one hand, there is an ability to estimate which is more of two groups of items. The items might be chunks of food, predators, or members of the animal’s own species; they might even be sounds or taps on the head rather than visible objects. The ability to make these kinds of judgements shows some consistent properties – and limitations – across the different species in which it has been found.
Well-designed experiments with humans, suppressing the more sophisticated ways of counting that nearly all have access to, can show that the ability to estimate in this sense is also present in Homo sapiens. You, too, can judge which is the larger flock of birds or the more numerous plate of cookies without actually counting them: and you can still do so when confounding factors like the shape or density of the flock are controlled for. This, surely, is one of the innate abilities that humans build on when they count in Leibniz’s sense of paying repeated attention and keeping track.
On the other hand, most humans also share a sense that for very small numbers – up to about four – recognition is both immediate and exact. If you see three sheep in a field, you just know there are three: it doesn’t feel like an estimation process, but it doesn’t feel like a counting process either. It is more like pattern recognition, working at a glance: but it functions even if the objects are not presented in any special pattern or arrangement.
So it has often been suggested that there is another innate ability that sits alongside estimation and deals specifically with the smallest numbers. Sometimes called subitising, because it happens subito, suddenly, it has long been dogged by controversy, with some experts unconvinced that the evidence proves it exists at all. Experiments resist replication; results can be explained in more than one way. Perhaps subitising is just what estimating looks like when the quantities are small. If it is real, though, it is another ability that underlies human practices of counting the world over, and that can do something to explain why those practices have the characteristics they do.
These two capacities might be called proto-counting. Humans inherit them from the distant evolutionary past, and they lie at the root of what human beings do when they count. Although they are about animals, they are an important part of the human story of counting.
Estimating and the approximate number system
The Cayo Santiago, Puerto Rico, 1999: a tree-filled island amid the Caribbean waters. A rhesus macaque, foraging for food, spots something unusual. Two humans have approached. Each displays a coloured, opaque bucket; tips it sideways to show it is empty; places it on the ground. The first human places slices of apple in the bucket while the monkey watches; the second does the same. Both humans then turn and walk away. And wait.
After a few moments, the macaque goes to investigate. The humans watch, observe, record. The macaque can’t see the contents of the buckets from any distance. But still it approaches, for preference, the bucket into which it has seen more pieces of fruit being placed.
Experiments like this one have been repeated with many species, and with similar results. It is not just monkeys that show a sense of number. Mealworm beetles can distinguish between different numbers of potential mates; cuttlefish can tell one prey item from two, two from three, and so on up to at least five. Certain spider species show a preference for settling with just one of their kind, rather than with none or with two or three. From frogs that count the pulses in their croaks to guppies that choose the larger shoal to swim with, and from parrots that select the greater number of food items to African elephants that can learn to choose between stimuli consisting of up to ten elements, something like counting seems to be everywhere in the animal world, present in nearly every species that has been tested for it. Cuttlefish, salamanders, barn owls, domestic chickens, New Zealand robins, pigeons, rats, bears, lions, hyenas, dogs, wolves, a dozen different primates … Forest, ocean and savannah seem to be teeming with numbers.
Something like counting can exist, then, without language, and without much – for some species, without any – training. Without a large brain; without a vertebrate nervous system.
Something like counting, but not really counting itself. The right word might be estimating; the technical term often used to describe animals’ judgement of numbers is the approximate number system. What it does not provide is precision. It shows – and this is the same in every species tested – a characteristic pattern of errors, with discrimination becoming less accurate as the quantities get bigger. Rhesus monkeys can tell one from two, two from three, three from four, four from five … but start to fail from five upwards. Rats that learned to press a lever a given number of times, from four up to twenty-four, became markedly less and less precise in their responses as the number increased: by the top end of the range they would merely produce a spread of numbers around the target. It is a common observation that when testing the accuracy of animals’ number sense, the size of the numbers matters.
In the same way, for the purpose of telling one number from another, the distance between them also matters: responses are always faster and more accurate if the difference is larger. Two and four are easier to tell apart than two and three.
Putting these two effects together, the best description of the approximate number system is that it is governed by a ratio. Most species seem to have a ratio above which they can reliably tell one number from another, while below it they become rapidly less accurate. For fish, the ratio is about two to one: so, for example, they can tell fifty objects from twenty-five, or two hundred from one hundred. Dogs and crows can do rather better and are accurate down to ratios of about three to two; some birds do better still with a limiting ratio of four to three. For rhesus monkeys, the ratio is perhaps six to five: they can tell, for instance, twelve items from ten or twenty-four from twenty. Estimates vary, and much depends on the type of task being carried out and, of course, the amount of training the animals have received. And different individuals are better or worse at it than others. An experiment with zebrafish found that of eight fish tested, some could only tell three from two, but others learned to tell four from three or even five from four.
One way to think of this is that if animals have anything like a mental ‘number line’, it does not have the numbers evenly spaced. Instead, the smaller numbers are widely separated, but the larger ones are more and more crowded together and hard to distinguish. No species on Earth can tell one hundred items from one hundred and one.
