Alex’s Adventures in Numberland Read online

  Tens of thousands of years ago, well before the arrival of numbers, however, our ancestors must have had certain sensibilities about amounts. They would have been able to distinguish one mammoth from two mammoths, and to recognize that one night is different from two nights. The intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of ‘two’, however, will have taken many ages to come about. This occurrence, in fact, is as far as some communities in the Amazon have come. There are tribes whose only number words are ‘one’, ‘two’ and ‘many’. The Munduruku, who go all the way up to five, are a relatively sophisticated bunch.

  Numbers are so prevalent in our lives that it is hard to imagine how people survive without them. Yet while Pierre Pica stayed with the Munduruku he easily slipped into a numberless existence. He slept in a hammock. He went hunting and ate tapir, armadillo and wild boar. He told the time from the position of the sun. If it rained, he stayed in; if it was sunny, he went out. There was never any need to count.

  Still, I thought it odd that numbers larger than five did not crop up at all in Amazonian daily life. I asked Pica how an Indian would say ‘six fish’. For example, just say that he or she was preparing a meal for six people and he wanted to make sure everyone had a fish each.

  ‘It is impossible,’ he said. ‘The sentence “I want fish for six people” does not exist.’

  What if you asked a Munduruku who had six children: ‘How many kids do you have?’

  Pica gave the same response: ‘He will say “I don’t know”. It is impossible to express.’

  However, added Pica, the issue was a cultural one. It was not the case that the Munduruku counted his first child, his second, his third, his fourth, his fifth and then scratched his head because he could go no further. For the Munduruku, the whole idea of counting children was ludicrous. The whole idea, in fact, of counting anything was ludicrous.

  Why would a Munduruku adult want to count his children, asked Pica? The children are looked after by all the adults in the community, he said, and no one is counting who belongs to whom. He compared the situation to the French expression ‘ j’ai une grande famille’, or ‘I’m from a big family’. ‘When I say that I have a big family I am telling you that I don’t know [how many members it has]. Where does my family stop and where does the others’ family begin? I don’t know. Nobody ever told me that.’ Similarly, if you asked an adult Munduruku how many children he is responsible for, there is no correct answer. ‘He will answer “I don’t know”, which really is the case.’

  The Munduruku are not alone in the sweep of history in not counting members of their community. When King David counted his own people he was punished with three days of pestilence and 77,000 deaths. Jews are meant to count Jews only indirectly, which is why in synagogues the way of making sure there are ten men present, a minyan, or sufficient community for prayers, is to say a ten-word prayer pointing at each person per word. Counting people with numbers is considered a way of singling people out, which makes them more vulnerable to malign influences. Ask an Orthodox rabbi to count his kids and you have as much chance of an answer as if you asked a Munduruku.

  I once spoke to a Brazilian teacher who had spent a lot of time working in indigenous communities. She said that Indians thought that the constant questioning by outsiders of how many children they had was a peculiar compulsion, even though the visitors were simply asking the question to be polite. What is the purpose of counting children? It made the Indians very suspicious, she said.

  The first written mention of the Munduruku dates from 1768, when a settler spotted some of them on the bank of a river. A century later, Franciscan missionaries set up a base on Munduruku land and more contact was made during the rubber boom of the late nineteenth century when rubber-tappers penetrated the region. Most Munduruku still live in relative isolation, but like many other Indian groups with a long history of contact, they tend to wear Western clothes like T-shirts and shorts. Inevitably, other features of modern life will eventually enter their world, such as electricity and television. And numbers. In fact, some Munduruku who live at the fringes of their territory have learned Portuguese, the national language of Brazil, and can count in Portuguese. ‘They can count um, dois, três, up until the hundreds,’ said Pica. ‘Then you ask them, “By the way, how much is five minus three?”’ He parodied a Gallic shrug. They have no idea.

  In the rainforest Pica conducts his research using laptops powered by solar-charged batteries. Maintaining the hardware is a logistical nightmare because of the heat and the damp, although sometimes the trickiest challenge is assembling the participants. On one occasion the leader of a village demanded that Pica eat a large, red sauba ant in order to gain permission to interview a child. The ever-diligent linguist grimaced as he crunched the insect and swallowed it down.

  The purpose of researching the mathematical abilities of people who have the capacity to count only on one hand is to discover the nature of our basic numerical intuitions. Pica wants to know what is universal to all humans, and what is shaped by culture. In one of his most fascinating experiments he examined the Indians’ spatial understanding of numbers. How did they visualize numbers when spread out on a line? In the modern world we do this all the time – on tape measures, rulers, graphs and houses along a street. Since the Munduruku don’t have numbers, Pica tested them using sets of dots on a screen. Each volunteer was presented with the figure overleaf, of an unmarked line. To the left side of the line was one dot; to the right, ten dots. Each volunteer was then shown random sets of between one and ten dots. For each set the subject had to point at where on the line he or she thought the number of dots should be located. Pica moved the cursor to this point and clicked. Through repeated clicks, he could see exactly how the Munduruku spaced numbers between one and ten.

  When American adults were given this test, they placed the numbers at equal intervals along the line. They recreated the number line we learn at school, in which adjacent digits are the same distance apart as if measured by a ruler. The Munduruku, however, responded quite differently. They thought that intervals between the numbers started large and became progressively smaller as the numbers increased. For example, the distance between the marks for one dot and two dots, and two dots and three dots were much larger than the distance between seven and eight dots, or eight and nine dots, as the following two graphs make clear.

  The results were striking. It is generally considered a self-evident truth that numbers are evenly spaced. We are taught this at school and we accept it easily. It is the basis of all measurement and science. Yet the Munduruku do not see the world like this. Stripped of a language of counting and number words, theyvisualize magnitudes in a completely different way.

  When numbers are spread out evenly on a ruler, the scale is called linear. When numbers get closer as they get larger, the scale is called logarithmic.* It turns out that the logarithmic approach is not exclusive to Amazonian Indians. We are all born conceiving numbers this way. In 2004, Robert Siegler and Julie Booth at Carnegie Mellon University in Pennsylvania presented a similar version of the number-line experiment to a group of kindergarten pupils (with an average age of 5.8 years), first-graders (6.9) and second-graders (7.8). The results showed in slow motion how familiarity with counting moulds our intuitions. The kindergarten pupil, with no formal maths education, maps out numbers logarithmically. By the first year at school, when the pupils are being introduced to number words and symbols, the curve is straightening. And by the second year at school, the numbers are at last evenly laid out along the line.

  Why do Indians and children think that higher numbers are closer together than lower numbers? There is a simple explanation. In the experiments, the volunteers were presented with a set of dots and asked where this set should be located in relation to a line with one dot on the left and ten dots on the right. (Or, in the children’s case, 100 dots). Imagine a Munduruku is presented with five dots. He will study it closely and see that five dots are five times bigger than one dot, but ten dots are only twice as big as five dots. The Munduruku and the children seem to be making their decisions about where numbers lie based on estimating the ratios between amounts. When considering ratios, it is logical that the distance between five and one is much greater than the distance between ten and five. And if you judge amounts using ratios, you will always produce a logarithmic scale.

  It is Pica’s belief that understanding quantities approximately in terms of estimating ratios is a universal human intuition. In fact, humans who do not have numbers – like Indians and young children – have no alternative but to see the world in this way. By contrast, understanding quantities in terms of exact numbers is not a universal intuition; it is a product of culture. The precedence of approximations and ratios over exact numbers, Pica suggests, is due to the fact that ratios are much more important for survival in the wild than the ability to count. Faced with a group of spear-wielding adversaries, we needed to know instantly whether there were more of them than us. When we saw two trees we needed to know instantly which had more fruit hanging from it. In neither case was it necessary to enumerate every enemy or every fruit individually. The crucial thing was to be able to make quick estimates of the relevant amounts and compare them, in other words to make approximations and judge their ratios.

  The logarithmic scale is also faithful to the way distances are perceived, which is possibly why it is so intuitive. It takes account of perspective. For example, if we see a tree 100m away and another 100m behind it, the second 100m looks shorter. To a Munduruku, the idea that every 100m represents an equal distance is a distortion of how he perceives the environment.

  Exact numbers provide us with a linear framework that contradicts our logarithmic intuition. Indeed, our proficiency with exact numbers means that the logarithmic intuition is overruled in most situations. But it is not eliminated altogether. We live with both a linearand a logarithmic understanding of quantity. For example, our understanding of the passing of time tends to be logarithmic. We often feel that time passes faster the older we get. Yet it works in the other direction too: yesterday seems a lot longer than the whole of last week. Our deep-seated logarithmic instinct surfaces most clearly when it comes to thinking about very large numbers. For example, we can all understand the difference between one and ten. It is unlikely we would confuse one pint of beer and ten pints of beer. Yet what about the difference between a billion gallons of water and ten billion gallons of water? Even though the difference is enormous, we tend to see both quantities as quite similar – as very large amounts of water. Likewise, the terms millionaire and billionaire are thrown around almost as synonyms – as if there is not so much difference between being very rich and very, very rich. Yet a billionaire is a thousand times richer than a millionaire. The higher numbers are, the closer together they feel.

  The fact that Pica temporarily forgot how to use numbers after only a few months in the jungle indicates that our linear understanding of numbers is not as deeply rooted in our brains as our logarithmic one. Our understanding of numbers is surprisingly fragile, which is why without regular use we lose our ability to manipulate exact numbers and default to our intuitions judging amounts with approximations and ratios.

  Pica said that his and others’ research on our mathematical intuitions may have serious consequences for maths education – both in the Amazon and in the West. We require understanding of the linear number line to function in modern society – it is the basis of measuring, and facilitates calculations. Yet maybe in our dependence on linearity we have gone too far in stifling our own logarithmic intuition. Perhaps, said Pica, this is a reason why so many people find maths difficult. Perhaps we should pay more attention to judging ratios rather than manipulating exact numbers. Likewise, maybe it would be wrong to teach the Munduruku to count like we do since this might deprive them of the mathematical intuitions or knowledge that are necessary for their own survival.

  Interest in the mathematical abilities of those who have no words or symbols for numbers has traditionally focused on animals. One of the best-known research subjects was a trotting stallion called Clever Hans. In the early 1900s, crowds gathered regularly in a Berlin courtyard to watch Hans’s owner, Wilhelm von Osten, a retired maths instructor, set the horse simple arithmetical sums. Hans answered by stamping the ground with his hoof the correct number of times. His repertoire included addition and subtraction as well as fractions, square roots and factorization. Public fascination, and suspicion that the horse’s supposed intelligence was some kind of trick, led to an investigation of his abilities by a committee of eminent scientists. They concluded that, jawohl!, Hans really was doing the math.

  It took a less eminent but more rigorous psychologist to debunk the equine Einstein. Oscar Pfungst noticed that Hans was reacting to cues in von Osten’s body language. Hans would start stamping his hoof on the ground and stopped only when he could sense a build-up or release of tension in von Osten’s face, indicating the answer had been reached. The horse was sensitive to the tiniest visual signals, such as the leaning of the head, the raising of the eyebrows and even the dilation of the nostrils. Von Osten was not even aware he was making these gestures. Hans was clever at reading people, certainly, but was no arithmetician.

  Many further attempts were made in the last century to teach animals to count, not all for the purposes of circus-like entertainment. In 1943 the German scientist Otto Koehler trained his pet raven Jakob to select a pot with a specified number of spots on its lid from a selection of pots with a variety of numbers of spots on their lids. The bird could perform this task when the number of spots on any one lid was between one and seven spots. In recent years, avian intelligence has reached more impressive heights. Irene Pepperberg of Harvard University taught an African grey parrot called Alex numerals from 1 to 6. When shown an assortment of coloured blocks he could reply, for example, how many blue blocks there were by squawking the English number word. So renowned had Alex become among scientists and birdlovers that when he died unexpectedly in 2007, his obituary appeared in The Economist.

  The lesson of Clever Hans was that when teaching animals to count, supreme care must be taken to eliminate involuntary human prompting. For the maths education of Ai, a chimpanzee brought to Japan from West Africa in the late 1970s, the chances of human cues were eliminated because she learned using a touch-screen computer.

  Ai is now 31 and lives at the Primate Research Institute in Inuyama, a small tourist town in central Japan. Her forehead is high and balding, the hair on her chin is white and she has the dark sunken eyes of ape middle age. She is known there as a ‘student’, never a ‘research subject’. Every day Ai attends classes where she is given tasks. She turns up at 9 a.m. on the dot after spending the night outdoors with a group of other chimps on a giant tree-like construction of wood, metal and rope. On the day I saw her she sat with her head close to a computer, tapping sequences of digits on the screen when they appeared. When she completed a task correctly an 8mm cube of apple whizzed down a tube to her right. Ai caught it in her hand and scoffed it instantly. Her mindless gaze, the nonchalant tapping of a flashing, beeping computer and the mundanity of continual reward reminded me of an old lady doing the slots.

  When Ai was a child she became a great ape in both senses of the word by becoming the first non-human to count with Arabic numerals. (These are the symbols 1, 2, 3 and so on, that are used in almost all countries except, ironically, in parts of the Arab world.) In order for her to do this satisfactorily, Tetsuro Matsuzawa, director of the Primate Research Institute, needed to teach her the two elements that comprise human understanding of number: quantity and order.

  Numbers express an amount, and they also express a position. These two concepts are linked, but different. For example, when I refer to ‘five carrots’ I mean that the quantity of carrots in the group is five. Mathematicians call this aspect of number ‘cardinality’. On the other hand, when I count from 1 to 20 I am using the convenient feature that numbers can be ordered in succession. I am not referring to 20 objects, I am simply reciting a sequence. Mathematicians call this aspect of number ‘ordinality’. At school we are taught notions of cardinality and ordinality together and we slip effortlessly between them. To chimpanzees, however, the interconnection is not obvious at all.

  Matsuzawa first taught Ai that one red pencil refers to the symbol ‘1’ and two red pencils to ‘2’. After 1 and 2, she learned 3 and then all the other digits up to 9. When shown, say, the number 5 she could tap a square with five objects, and when shown a square with five objects she could tap the digit 5. Her education was reward-driven: whenever she got a computer task correct, a tube by the computer dispensd a piece of food.

  Once Ai had mastered the cardinality of the digits from 1 to 9, Matsuzawa introduced tasks to teach her how they were ordered. His tests flashed digits up on the screen and Ai had to tap them in ascending order. If the screen showed 4 and 2, she had to touch 2 and then 4 to win her cube of apple. She grasped this pretty quickly. Ai’s competence in both the cardinality and ordinality tasks meant that Matsuzawa could reasonably say that his student had learned to count. The achievement made her a national hero in Japan and a global icon for her species.