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By Root 603 0

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