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.
At a glance: Subitising
A curious pendant to the approximate number sense is the way people can recognise small numbers at a glance. Some experiments seem to show this as a separate ability from estimation; collections of one, two, three and four items appear to be distinguished immediately, without the effects of size and ratio that characterise the approximate number sense. Babies just a few hours or days old, for instance, have been observed to tell apart collections of two and three objects but not collections of four and six. At a few weeks, they can tell apart sets of one, two, three or four items, all equally fast and equally accurately. But their performance drops sharply if the number of objects becomes larger. For adults, the limit of this ability – ‘subitising’ – seems consistently to be four, with the difference between three and four recognised just as reliably as that between one and four, and the ratio effects characteristic of the approximate number sense appearing only when the numbers are larger than this.
Like the approximate number sense, subitising is cross-modal: heard or even felt stimuli can be shown to be distinguished in a similar way. (Many readers will have experienced the difference between counting musical beats and ‘feeling’ them, which surely has something to do with the limits of subitising in the rhythmic domain.)
Subitising. You know there are three sheep without thinking ‘one … two … three’.
imageBROKER.com GmbH & Co. KG / Alamy Stock Photo.
But, in another difference compared with the approximate number system, subitising seems to stop working when your attention is distracted. While judgements of approximate number are formed ‘in the background’ of other tasks, subitising requires you to focus on the objects in order to know how many of them there are.
Subitising, in fact, seems to be very much about tracking and paying attention to the individual objects. Many researchers have suggested it works something like a set of slots or pigeonholes in your working memory: you can fill them one at a time, but when all the slots are full, you lose track and fall back on estimation. For that reason, this particular ability is sometimes called the ‘object file system’.
