Pierre Pica is a linguist who was a student of Noam Chomsky. His work has been focused on the Munduruku, an indigenous group of about 7,000 people in the Brazilian Amazon. The Munduruku language has no tenses, no plurals and no words for numbers beyond five. Whenever he returned home from the Amazon, he found that he had lost his sense of time and number. He forgot appointments. He was disoriented by simple directions. He had trouble getting used to the straight lines and angles of Paris. When Pica asked them to place numbers (represented by images of dots) on a number line, he found that they put lower numbers further apart, and larger numbers closer together. This matches what mathematicians call a logarithmic scale. > 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 of 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 molds our intuitions. The kindergarten pupil, with no formal math education, maps numbers out logarithmically. By the first year at school, when the pupils are being introduced to number words and symbols, the graph is straightening. And by the second year at school, the numbers are at last evenly laid out along the line. > Imagine a Munduruku is presented with five dots. He will study them 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 by estimating the ratios between amounts. In 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 relative amounts. > 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 100 meters away and another 100 meters behind it, the second 100 meters looks shorter. To a Munduruku, the idea that every 100 meters 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 intuitions. 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 linear and a logarithmic understanding of quantity. For example, our understanding of the passing of time is often logarithmic. I remember the years of my childhood passing a lot more slowly than the years seem to fly by now. Yet, conversely, yesterday seems a lot longer than the whole of last week. > 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, and that is why without regular use we lose our ability to manipulate exact numbers and default to our intuitions, judging amounts with approximations and ratios. Source: [[Here's Looking at Euclid]]