Numbers: Their Tales, Types, and Treasures.

Chapter 2: Numbers and Psychology

2.4.GOING BEYOND THE CORE SYSTEMS

It appears that the two core-knowledge systems—the object-tracking system and the approximate-number system—constitute our innate number sense. Like the uneducated tribesmen of the Amazon jungle, we would have these faculties even without culturally driven learning opportunities. Moreover, it has been shown that even babies and some animals do have these abilities.

The two core systems give us two quite different impressions of “number.” The object-tracking system provides us with a precise mental representation of a small number of individual objects. This representation is discrete and tells us about the exact number, with 2 being perceived as fundamentally different from 3 or 1. This system gives us a precise mental model of what happens when we add or remove one item.

The approximate-number system, on the other hand, represents large numbers as a continuous quantity. It gives only an approximate and vague impression of number. There is no fundamental difference between 12 and 13, and the difference between 200 and 300 is rather a difference in the intensity of number perception, as it would be with other continuously varying quantities like size or density.

The inherited number sense represents a rather primitive knowledge and is a long way from the culturally refined understanding of number that children might have acquired already at the age of three or four. The core-knowledge systems, however, influence and guide later learning activities. Humans have the ability to go significantly beyond the limits of the core-knowledge systems and develop new cognitive capacities. For example, children in our culture soon learn to reconcile the two different impressions of number: They can apply the idea of discreteness provided by the object-tracking system to large numbers for which the approximate-number system only gives the vague feeling of a continuously varying quantity. Soon they realize that 12 and 13 are different in the same sense as 2 and 3. Even if they cannot count that far, they know that a large number, like 50, is changed by adding or taking away one item. Obviously, the idea of the discreteness of number can soon be applied to large collections.

One factor that might help in applying the idea of discreteness to large numbers is that the small numbers 1, 2, and 3 seem to be represented by both core-knowledge systems, so that we feel no discontinuity in our perception of numbers when they increase beyond the limit of the object-tracking system. Hence, for example, the idea of adding one item to obtain a new number can be easily transferred to higher numbers. Even monkeys trained to order small sets according to their size can generalize this ability immediately to larger sets of up to nine items. It appears, however, that the ability to think of larger numbers as discrete units is unique to humans and is not shared by animals.

But this still does not explain how children acquire these additional insights, and, consequently, this is a matter of ongoing research. It is probable that other core-knowledge systems help in this process—for example, systems related to social interaction and the ability to acquire language. In particular, language seems to be important for being successful in combining the mental representations from different core systems, like the discrete representation of small numbers and the continuous representation of large numbers. In this process, children develop a sense for the exact cardinality of arbitrary collections.