Psychologists who examine numerical cognition in young children and people from nonnumerate cultures have found that our default, unlearned, mode to represent cardinalities is not according to a linear mental number line, but a logarithmic one. In other words, our intuitive sense of numerosities roughly corresponds to the natural logarithm of those numbers. Linear numerical representations, such as the natural numbers and the way we place them on rulers and other linear representations, are cultural inventions. As Dehaene et al. put it: "The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education."
The research by developmental psychologist Robert Siegler indicates that children learn to make linear representations of number by prolonged cultural expose in school and in (more informal) home settings like playing board games. Siegler and Booth (2004) found that this linearity of number lines appears gradually. In one of their experiments, they gave five- to seven-year-olds an unscaled number line with 0 at the left side and 100 at the right, and asked them to place various numbers on this number line. Younger children typically placed small numbers too far to the right. For example, they tend to place the number 10 in the middle of the scale. On the other hand, they tended to underestimate the distance between the higher numbers, placing 70 far too close to 100. The older children, on the other hand, made much more linear estimations. Intriguingly, this process is repeated as children learn to deal with cardinalities up to 1000. Siegler and Opfer gave children between 7 and 11 years of age lines from 0 to 1000, and again found that the younger children tended to have logarithmic representations, and the older ones linear representations.
I wonder whether we could generalize the observation that linear numerical representations need to be relearned. In other words, is the case that we need to re-calibrate our mental representation of magnitudes each time we learn to deal with higher numbers? Leiter's blog recently pointed my attention to the following site, which gives you a sense of how much one billion dollars can buy. The shopping list of expensive and useless items (including a private island, a plane, some yachts, etc.) is impressive, and yet the billion dollars are not even halfway spent. Most of us have no idea what a billion dollars is, or indeed have an intuitive feeling of the difference between one billion and one million. It's all equally mind boggling.
If the default mode of our mental representation of numerosities is logarithmic, most of us have no idea of how much some people are making, and what a huge difference it would make if they were fairly taxed. In other words, one reason that we do not protest at the extreme wealth, is that we simply do not have an intuitive grasp of how extreme it is.
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