This example, one we have studied quite thoroughly with many children, was first suggested to me by a mathematician friend who quoted it as the point of departure of his interest in mathematics. When he was a small child, he was counting pebbles one day; he lined them up in a row, counted them from left to right, and got ten. Then, just for fun, he counted them from right to left to see what number he would get, and was astonished that he got ten again. He put the pebbles in a circle and counted them, and once again there were ten. He went around the circle in the other way and got ten again. And no matter how he put the pebbles down, when he counted them, the number came to ten. He discovered here what is known in mathematics as commutativity, that is, the sum is independent of the order.
I believe that what the boy has discovered is more basic than commutatitivity and of which commutativity of integer addition is just a consequence.
Why are we at all able to count? It is not that we are wired in a certain way. Any group of objects is somehow endowed with an attribute that some researchers call Number, while others Numerosity - to distinguish from other uses of the word "number". (The terminology is rather loose, e.g., for Lakoff and Núñez, numerosity is the ability to make consistent rough estimates of the number of objects in a group.)
From the time of Plato and Aristotle, philosophers and later on cognitive scientist and even mathematicians have been opining on the seat of that attribute. I believe that it is of the same fundamental existence as other properties of objects - color, location, movement - in our world. It is here, it is available, and we use it all the time starting with a very early age. (Some animals do too.) I shall call it Number.
A number associated with a group of objects is unique for that group and, if so, is independent of the relative positions of the objects in the group. This is what the boy Piaget wrote about has found out. Most people learn counting and do not stop to think what makes it possible. The boy did, and this I believe was the first step in his becoming a mathematician.
The nature of the objects in a group is also inessential, if only for the numerosity in the sense of Lakoff and Núñez. We may replace one object in a group with any other object without affecting the associated number. Given two groups, if we replace an object at a time from one group with an object from the other and eventually replace all the objects, the process establishes whether the same number is associated with the two groups. This is the basis for the concept of the 1-1 correspondence.
We can count objects in a group because there is indeed a unique number associated with that group. Commutativity is a consequence of that fact. For, if the group is split into two, its associated number does not change. One can count the objects in the subgroups in any order, sum up the results, and get exactly the same result as before. Associativity is another consequence of this number/group association. The group needs to be split into three subgroups whose elements may be counted in several ways, all leading to the same result.
- B. Butterworth, What Counts: How Every Brain is Hardwired for Math, The Free Press, 1999.
- S. Dehane, The Number Sense, Oxford University Press, 1997.
- R. Hersh, What Is Mathematics, Really?, Oxford University Press, 1997.
- G. Lakoff, R. E. Núñez, Where Mathematics Comes From, Basic Books, 2000
- J. Piaget, Genetic Epistemology, Columbia University Press, 1970.