Numbers, Amazon-style

In last week's Science, Stanislas Dehaene and colleagues describe the relation of cultural invention to "universal intuition" about mathematical logic:

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education (Dehaene et al. 2008:1217).

The idea is that children in Western societies have to learn that a number line is a linear representation; they begin by compressing the space devoted to large numbers:

When asked to point toward the correct location for a spoken number word onto a line segment labeled with 0 at left and 100 at right, even kindergarteners understand the task and behave nonrandomly, systematically placing smaller numbers at left and larger numbers at right. They do not distribute the numbers evenly, however, and instead devote more space to small numbers, imposing a compressed logarithmic mapping. For instance, they might place number 10 near the middle of the 0-to-100 segment. This compressive response fits nicely with animal and infant studies that demonstrate that numerical perception obeys Weber's law, a ubiquitous psychophysical law whereby increasingly larger quantities are represented with proportionally greater imprecision, compatible with a logarithmic internal representation with fixed noise (7, 20, 21). A shift from logarithmic to linear mapping occurs later in development, between first and fourth grade, depending on experience and the range of numbers tested (17-19).

They note that there's a problem testing these ideas in Western children, who are surrounded throughout their development by numbers -- in books, "elevators" and other places. Most of these numbers are small ones -- especially one through ten -- so they might naturally accentuate the ones they know.

They found when testing the Mundurucu that both adults and children tended to compress the high end of the number scale, even testing numbers between one and ten. This compression is logarithmic -- they accentuate contrasts between small numbers disproportionately. It makes sense logically -- we care more about detailed contrasts between small numbers than large numbers. They don't give an idea of which logarithm people are using; and in fact it may be different ones for different people. The important fact is the small number/large number contrast.

Dehaene and colleagues attribute this scaling to mapping at the neural level:

What are the sources of this universal logarithmic mapping? Research on the brain mechanisms of numerosity perception have revealed a compressed numerosity code, whereby individual neurons in the parietal and prefrontal cortex exhibit a Gaussian tuning curve on a logarithmic axis of number (27). As first noted by Gustav Fechner, such a constant imprecision on a logarithmic scale can explain Weber's law -- the fact that larger numbers require a proportional larger difference in order to remain equally discriminable. Indeed, a recent model suggests that the tuning properties of number neurons can account for many details of elementary mental arithmetic in humans and animals (21). In the final analysis, the logarithmic code may have been selected during evolution for its compactness: Like an engineer's slide rule, a log scale provides a compact neural representation of several orders of magnitude with fixed relative precision.

From that perspective, the Western conception of the number line appears as a very distinctive invention, capable of adjusting the logarithmic encoding to arrive at faster and more accurate mathematical conclusions about large numbers. The authors speculate that addition and subtraction (which display invariance between large and small numbers) and experience with measurement underlay the development of the linear concept in Western children.

References:

Dehaene S, Izard V, Spelke E, Pica P. 2008. Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science 320:1217-1220. doi:10.1126/science.1156540