Alex's Adventures in Numberland - Alex Bellos [6]
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