Numbers: Their Tales, Types, and Treasures.

Chapter 2: Numbers and Psychology

2.3.THE APPROXIMATE-NUMBER SYSTEM

Although useful in some situations, an object-tracking system limited to the number four is certainly not sufficient. Very often one needs approximate knowledge of higher numbers. When a tribe of early humans met a rivaling group, they had to decide quickly whether they should stay and fight or run away if they were outnumbered by their enemies.

Consider the two collections of dots in figure 2.3. Can you say without counting which set is larger? When presented with this, and similar tasks, most adults will usually give the right answer. In figure 2.3, the collection of dots on the right is about 27 percent greater than the collection on the left—a difference that is well within the capacity of the approximate-number sense.

Figure 2.3: The approximate-number sense tells you which field contains more dots.

This strong intuition, the ability to estimate and compare approximate numbers, is another innate mental mechanism that contributes to our preverbal understanding of number. This second core-knowledge system is called the approximate-number system. Experiments have shown that it is already active in newborn babies, and it becomes more and more accurate with age and experience.

The approximate discrimination between the number of two sets can only be done when the two sets differ by a certain percentage. Babies can only distinguish two numbers when one of the sets is more than twice as large, while adults can discriminate with some certainty when one set is about 20 percent larger than the other. In all cases, the error rate decreases when the two sets differ by a higher percentage.

This shows that the approximate-number system follows the Weber-Fechner law, which applies to sensual stimuli in general. This states that the same impression of change is created when the stimulus is increased by a certain percentage. So if you are able to distinguish between the numbers 12 and 15 with an error rate of, say, 10 percent, then you will be able to distinguish between the numbers 120 and 150 with the same error rate. The Weber-Fechner law can also be stated as follows: One perceives the same difference between two collections when their numbers have the same ratio (e.g., ), or equivalently, if they differ by the same percentage (15 is 25 percent larger than 12, and 150 is 25 percent larger than 120).

Approximate knowledge of number is sufficient in most situations of everyday life. Sometimes, the approximate character is explicit. Whenever we say “about a dozen,” we usually do not care whether it is 11 or 13. Even seemingly exact numerical information is often meant in an approximate sense. When you drive at 50 mph, it could be, in fact, 48 mph or 53 mph. In particular, large round numbers in statements like “this village has a population of 500” or “the galaxy has 400 billion stars,” are automatically understood as approximate numbers. For large numbers, we have no sense of exactness. Would you expect that anyone can tell you the number of hairs on a particular dog? From our intellect, we know that the dog must have an exact number of hairs. But we don't really conceptualize that. To us, this number is a very fuzzy quantity. Moreover, the dog would continuously shed hairs and grow new ones, so the number would not remain constant even for a short time. We are perfectly happy with an approximate answer like “Could well be about 10 million,” but we would have the same reaction if the answer is twice as much or only half as much. The concept of exact number is irrelevant for this question. If you had never learned to count and had to rely completely on your approximate-number sense, you would feel that way in view of much smaller numbers: Numbers beyond the subitizing range could be perceived only in a fuzzy or approximate way.

Today, there are indeed still tribes in the Amazon jungle that have never yet encountered counting. We know about these tribes from research done by psycholinguist Peter Gordon, who visited the Pirahã Indians, and Pierre Pica (1951–), who visited the Mundurukú to carry out experiments codesigned by the French neuroscientist Stanislas Dehaene (1965–). The Mundurukú have only number words up to five and, beyond that, words for “a few” and “many.” They normally do not count at all and use the number words inconsistently, making occasional mistakes already with four or five items. They use their word for five, which literally means “a handful,” also for six to nine items. They have never heard of addition or subtraction. This makes them ideal to test hypotheses about an innate number sense. Some simple experiments were done by presenting them with various “number games” on the screen of a solar-powered notebook computer. It turned out that they were able to compare large groups of dots as skillfully, and with about the same precision, as educated people integrated into a Western culture. For example, when presented with two groups of objects that were subsequently hidden, they could compare the sum with a third number shown to them. They obviously have an inherited capacity to understand how collections of objects behave in operations like taking away or joining. But, unlike people who have learned to count, they fail with exact arithmetic beyond the number 3, being able to give only approximate answers. Dehaene concludes that the Amazon natives share our innate number sense. This number sense already provides us with an arithmetic intuition that is sufficient to master the major concepts of arithmetic, like larger-smaller relations, addition, and subtraction. Number words are not necessary for understanding these concepts in an approximate sense.

The Pirahã studied by Peter Gordon are linguistically even more restricted than the Mundurukú. The Pirahã have number words only for one and two, which are probably also synonymous with few and many. They did not even master the bijection principle (i.e., one-to-one correspondence), because when the number of objects exceeded three, they could not compare quantities exactly by this one-to-one correspondence; they always did this by estimation. Some linguists believe that these people are forever unable to grasp the concept of a number beyond two or three because they lack the essential language tools. Without suitable number words, people cannot count.

In any case, verbal counting seems to help with the integration of the approximate number representation and the discrete number sense of the object-tracking system. At age three or four, children learning to count realize that each number word refers to a precise quantity—and, as Dehaene formulates it, this “crystallization” of discrete numbers, out of an initially approximate continuum of numerical magnitudes, seems to be exactly what the Mundurukú and Pirahã are lacking.