Monday, April 5, 2010
Numerical Systems, Logarithmic Scales and Cognitive Spaces
This morning I came across this article on the Guardian which is basically an extract of Alex Bellos' book Alex's Adventures in Numberland. I haven't read the book yet, but the extract looks really promising. To put it short, Alex Bellos refers to the work of the french anthropologist Pierre Pica and his study on the Munduruku , an amazonian tribe which has a numerical system restricted to 5. A part from the intriguing insights on the origins of the numeric intuitions, what really fascinated me is an experiment conducted by Pica (Dehaene et al. 2008) which links numerical systems to spatial perception. Imagine a straight line with one dot on one edge and 10 dots on the other edge. Members of the tribe were asked to place a given set of random dots between 1 and 10 along any point along the line. If you do this experiment yourself, you would most probably put a set of 5 dots in the middle of the line, and a set of 7 dots somewhere near the 3/4 of the line. If you then plot your results in a xy plot (with x=number of dots in the sets and y=distance from the origin ) you would probably achieve a linear relation with a 45 degree slope (y=x). Now the results of the Munduruku were quite different, the relation is in fact not linear but logarithmic, the greater is the number of dots the smaller was the difference between sets. More interestingly, experiments on western children have shown how they do have the same "logarithmic thinking" which is however lost when they get old. Dehaene and colleagues rightly points out the evolutionary implications of this discovery and they also point its relation with Weber–Fechner law, while Alex Bellos acknowledges its implications in a wider context, among which how logarithmic perception might affect perspective. Now the combination of Weber's law and perspective has a very interesting implications in spatial cognition. Put it simply Weber's law states that relation between stimulus and perception is logarithmic. Suppose you want to buy some milk, and there are two stores in close distance, one at 100 meters and the other one at 200 meters. You would definitely choose the first one. Suppose now that the two stores are at 10,2 km and 10,3 km. Now, despite the difference between the two are the same (100 meters) you can easily go with the second stores without worrying to much. The reason is that when we compare the two we don't look at the absolute values but at their ratio. In the first case store B is the two times the distance to store A. In the second case, the ratio is 1.009, so it really doesn't make that difference for us. What are the consequence of this? Well most models of spatial perception are based on linear relation between stimulus and perception. This linear relation is assumed in models of spatial interaction which determines the backbone of many spatial analysis, and at the very same time, most agent based models do the very same thing. Suppose a very simple model, where an agent choses a location within a search neighbourhood based on some cell value. Distance and perspective really does not play any role in most cases: the agent choses the cell with the highest value. If we integrate the cost of movement the choice will be distance dependent, but there will be still no difference in the cognitive aspects of the decision making at different distances. What we really need to do is to integrate Weber's law in our submodels, so that the relation between environment stimulus and the actual perception becomes logarithmic. The results is that difference between objects at close distance have a stronger effects in the decision making process than the difference between objects at far distances. A quite intuitive concept that however, as far as I know (and I'll be glad if somebody proves that I'm being wrong) hasn't been integrated in computational and statistical models of human spatial cognition.