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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.

It is natural to wonder whether results like these are real. There have, after all, been some notorious hoaxes in the field of ‘clever animals’: in the world of animal numeracy it is best not to mention Clever Hans, the German wonder horse who amazed the world in the 1890s with his accurate responses to arithmetical questions. Last of a long line of calculating horses (there is an illustration from 1594 of a ‘bay horse in a trance’ doing arithmetic, possibly the same animal referred to as the ‘dancing horse’ in Shakespeare’s Love’s Labour’s Lost), Hans naturally turned out to be responding to prompts from his trainer and questioner, who would move his head or his back when the right answer was in view. Hans and his like cast a long shadow over serious studies of number abilities in animals.

But recent experimental work is of quite a different kind, and it does seem that the animals really do recognise different numbers of objects as is claimed. It is easy enough to run experiments ‘blind’, with the experimenters unable to see the stimuli or barred from communicating with the animals. It is harder, but still in most cases possible, to make sure the animals are not responding to non-numerical cues, such as the total amount of foodstuff rather than the actual number of items (hence the business with opaque buckets for the macaques’ fruit slices). Indeed, some animal species do fail when such controls are put in place: cats, for instance, in one study turned out to be relying on the visual cue of total surface area when choosing between two quantities of food, not on the number of separate food items. (The same was true for lizards, a group not noted for its number sense.) Salamanders seemed to be assessing the amount of movement they could see, not the number of fruit flies presented to them.

Despite such failures, for the majority of species tested the results stand: even when density, shape, area and arrangement are controlled or randomised, animals continue to be capable of selecting the larger number of items. Indeed, some will do so spontaneously, even when the task does not require it. There is evidence of spontaneous assessment of number in chicks, in dogs, and perhaps in a few other species as well. There are circumstances in which number is positively preferred over other parameters when animals make a choice. Chicks, for instance, if spatial and numerical cues conflict, will follow the numerical information. So will monkeys, preferring number to colour, surface area or shape as the basis for a choice.

Finally, the assessment of number by animals is not only a visual experience. Monkeys can be trained to compare a sequence of sounds to a set of visual items and correctly choose the visual array that matches the number of sounds. They are just as accurate at this as when they match visual with visual stimuli. So – approximate and fuzzy though it is – number seems to be not simply widespread and spontaneous in the animal world, but strikingly abstract too. It has been argued that number is one of the primary features of the world as many animals experience it, guiding decisions and forming part of the information an animal’s brain encodes about a situation automatically, ‘just in case’.

Why do animals have an approximate number sense? Traits evolve because they confer an advantage; because individuals that have them are more likely to survive, to thrive, and ultimately to reproduce. It is fairly easy to imagine situations in which natural selection would favour individuals who behave in line with judgements of number: those who join the larger rather than the smaller group of their own species, becoming safer from predators as a result. Those who flee from the predators only if they are numerous enough to pose a real threat. Those who choose the larger pile of nuts to forage in. Those who flock with the larger group of potential mates. Getting any of those judgements right will – for certain species – confer selective advantage in the long run. Many of these behaviours have been observed in the wild: in untrained animals, and in completely natural settings.

In the more aggressive world of carnivore societies, lions, hyenas and wolves are all thought to assess how many members of their own species they hear calling, and decide whether to respond or not depending on whether they have a numerical advantage over the rival group. A classic experiment demonstrated this by playing recordings of intruders to female lions and tracking their decisions whether to respond. Chimpanzees, too, have been observed to attack a neighbouring band only if the numbers are on their side. This ability to assess numerical odds correctly – defenders versus intruders – may be crucial for a group’s success, and it is perfectly plausible that it is selected for in the long run.

In all of these situations, interestingly, it makes sense to be more accurate about small numbers than large ones. The difference between one and two apples matters more than that between twenty and twenty-one. Being outnumbered two to three matters more than by eleven to ten. The difference between a shoal of five and one of three is important; that between shoals of fifteen and sixteen is negligible. Natural selection is a plausible explanation not just for the presence of the approximate number system in many species, but for the fact that it is approximate rather than exact.

Estimation and humans

The University of Tübingen, Germany, 2008; a laboratory. Electric light on wood and plastic. A human being – let’s call her Miriam – is being tested. She sits in a chair and looks at a computer display. It shows her a pattern of dots, then another, flashing each one up too fast to count. Do the patterns contain the same number of dots, or not?

Miriam completes dozens of repetitions of the task, and much of the time her answers are correct. She is more likely to be wrong when the number of dots is larger, or if the difference between the two sets of dots is small. Her ability to estimate numbers, in other words, turns out to be just like that of any other primate.

The approximate number system also exists in humans, although its operation is sometimes harder to see, overlain as it often is by a deeply learned impulse to count precisely using words, symbols, or some other device. If explicit counting is made impossible, though, estimation comes into its own. If you glance at a flock of twenty birds and a flock of thirty, you are likely to know that one is bigger than the other, even if there is no time to count them or even form an estimate of the actual number. In the lab, counting can be suppressed by keeping the stimuli brief, or by giving subjects tasks that interfere with verbal counting, such as asking them to read aloud at the same time.

Estimation. You know which flock has more birds without counting them.

kristof lauwers / Alamy Stock Photo.

Under those conditions, the same features show up as in animals. Humans have a sense of how many, and it is an approximate sense: it deals in estimates, and those estimates become less accurate as the numbers become larger or the ratios between them become smaller. Everyone can tell two objects from three, or twenty from thirty. No one can tell a hundred objects from a hundred and one without explicitly counting them. Compared with other species, though, humans perform rather well. While even the best monkeys’ limit of discrimination is a ratio of around six to five, adult humans have been observed managing ratios as small as eight to seven or even ten to nine.

As in animals, the human ability to estimate number is robust when experimenters control for confounding factors like the size or the density of the stimulus. It exists across cultures, and it works not only on the numbers of objects seen but on the numbers of sounds heard or physical taps felt.

An advantage for experimenters working with humans is that they can be asked more complicated questions and set more complex tasks than are feasible with animals. Thus in some ways the human approximate number system is better described than that in animals, with a dense tangle of different – occasionally contradictory – experiments and results. Estimation has an upper limit, for instance, beyond which a set of visible objects is perceived as having not a number but a texture. It shows, intriguingly, an effect called adaptation, shared with other types of sensory stimulus. If you put your hand in warm water, other objects will feel colder than they really are for the next few minutes. In the same way, if you look at a collection of a hundred dots, even for less than a second, then other, smaller collections of dots will look less numerous than they really are for the next few minutes. You will underestimate their number by up to a factor of two. The same effect occurs in reverse: if you put your hand in cold water, things will feel warmer than they are for the next few minutes; and if you look at a group of ten dots, other, larger collections will appear more numerous than they really are, until the effect wears off. The midpoint, the set of dots that is apparently ‘normal’ enough that it doesn’t affect the perception of subsequent sets one way or the other, is around fifty. Rather wonderfully, this effect, too, is not limited to the visual mode of perception. In one experiment, subjects were asked to tap their fingers either quickly or slowly, and then to judge the number in a sequence of flashes or an array of dots. Slow tapping caused them to overestimate, quick tapping to underestimate.

The sense of number is already present a few hours after birth, albeit in only quite a fuzzy form, distinguishing, for instance, three items from one. But by four days old, babies can distinguish three-syllable words from two-syllable ones. The ratio of discrimination continues to narrow throughout childhood. And like animals, humans seem to form judgements about number spontaneously, even if they are not needed, and even if they actually interfere with the task at hand. People can’t help processing the world in terms of how many, any more than they can help processing it in terms of colour and shape. Despite there being no obvious sense organ devoted to number, it is just as much one of the senses as vision, hearing and smell. Different people have it to a different degree, and those differences are at least in part genetically determined; one study with twins found around 30 per cent of variance in the approximate number sense was inherited.

Such a distinct, well-defined ability should surely have a dedicated part of the brain to perform it, although the fact that it is found in such a wide range of animal species raises questions about whether they can possibly all use comparable brain regions for the purpose. The classic technique of functional magnetic resonance imaging – which can show in real time which areas of the brain are being activated – has progressively narrowed the region in which approximate number processing takes place: to the neocortex (the outer layer of the brain); to the parietal lobes (upper back areas of the neocortex); to the interparietal sulci (grooves running along the side); and finally to the horizontal segments of the interparietal sulci. Reliably, in people from different cultures, adults as well as children, this is the specific brain region that first activates when numerical information is being extracted from the world, even before it is converted into words or arithmetic is done with the numbers: tasks which use different distinct parts of the brain, and which are indeed culturally dependent, depending on whether you do arithmetic using – or imagining – spoken words, written symbols, or an abacus, for instance. Numbers presented in any format – including as spoken words or as written number symbols – activate this same part of the brain.

There are similar results for animals, with monkeys also using a region of the interparietal sulcus, and crows a specific part of their very differently organised brains. It was long predicted, based on simulations, that there would be individual brain cells or clusters of cells specialised for the detection of different numbers. In 2002, a team successfully located individual cells in the brains of macaques whose firing was associated with the number of elements in a visual display. Certain cells were indeed associated with certain numbers. Present the monkey with two objects, and one set of brain cells would fire. Present it with three, and a different set fired. Even completely untrained animals turned out to have such ‘number neurons’. In 2015, similar cells were found in the brains of crows, increasing their activity in response to numerical stimuli, and responding preferentially to particular numbers.

The number neurons turn out to do just what would be expected if they are responsible for the approximate number sense. They respond only to numerical information, not to other features such as whether the number is presented visually or as sound or rhythm. And they work approximately. Each one is tuned to respond to a particular number – say four – but also responds more weakly to neighbouring numbers on either side – say three, five and six. The precision of the tuning decreases as the numbers get larger: smaller numbers have more precisely tuned neurons, larger ones more fuzzily tuned ones, creating the possibility for making mistakes, and accounting for the observed limits on animals’ abilities to discriminate between a number and its neighbours.

It is an exciting set of results, and it confirms what had been suspected from neural network simulations: that you do not need a large brain or a large set of dedicated brain cells to detect numbers in your environment. Computer simulations involving as few as twenty-five dedicated cells have been shown to do it successfully, and with a few hundred cells some of the more complex features of extracting number from a visual display can be replicated. Simulations also hint that forming representations of numbers is something a network of a few hundred brain cells may start to do spontaneously, given the right combination of input and reinforcement: in other words, it is an easy capacity to acquire, a potential that even the simplest brains might have.

So it looks very much as though humans inherit from their animal ancestors an ability to detect, and encode approximately, numerical information from the environment, from a range of different types of stimulus. But it is not yet clear just how ancient the ability is.

The fact that humans and macaques use the same part of their brains for the approximate number sense suggests very strongly that the ability was present in the common ancestor of primates, about twenty-five million years ago, and was passed down to humans through the line of pre-human hominids. The data for other mammals are too incomplete to make any similar claim about their common ancestor. But the fact that the approximate number sense turns up with a relatively high degree of precision – comparable to that of monkeys – in some birds, is intriguing. The last common ancestor of birds and mammals – a reptile-like animal – lived about 320 million years ago. The brains of the two lineages have evolved separately ever since, though to some degree in parallel: the bird brain is of quite a different design from a mammal’s, having evolved for low weight (they have nearly twice as many neurons as a primate brain of the same mass), despite providing many similar functions in response to similar pressures from the environment. It is conceivable that that common ancestor could already detect and distinguish quantities, but perhaps rather more plausible that the approximate number sense has evolved independently in the two lineages. This suggestion is strengthened by the limited data for reptiles, which usually seem to have more restricted numerical abilities; it indicates, perhaps, that birds acquired their numerical skills after they split from the dinosaurs around 150 million years ago.

And it is the same story with the fish – and still more so the invertebrates – that have shown numerical abilities. Although it has been suggested that numerical abilities go right back to a common ancestor half a billion years ago, it is more often postulated that the different groups have evolved their numerical estimation abilities independently: testament as much as anything to the strength and constant presence of the environmental pressures that make it useful to have an approximate sense of number.