The Idea of Counting
Piaget was fond of relating an anecdote about a child who grew up to be an accomplished mathematician. One day the future mathematician confronted a set of objects lying before him and decided to count them. He determined that there were ten objects. He then pointed to each of the objects, but in a different order, and found that - lo and behold! - there were again ten; the child repeated this procedure several times, with growing excitement, as he came to understand - once and for all - that the number 10 was far from an arbitrary outcome of this repetitive exercise. The number referred to the aggregate of elements, no matter how they happened to be acknowledged in the sequence, just so long as each of them was taken into account once and once only. Through this playful dubbing of a group of objects, the youngster had (as all of us have at one time or another) arrived at a fundamental insight about the realm of number.
We'll explore the idea of counting and the underlying principles in a sequence of a few short stories. The first one was written at my urging by my son David when in high school. It was illustrated by my other son Eli when he was 11.
The story conveys an important truth about our world and explains our ability to count the number of objects in any finite set. Using mathematical terminology, it tells us that the result of counting is invariant under a process of counting. Invariance pops up in a variety of places in mathematics, some of which are described in the article Tribute to Invariance.
(The fully illustrated story is now available as a book and a pdf file).
The story intimates that numbers are not quite like labels or names. You can count objects one, two, three ... Any object could be counted one, or two, or three, and so on. Numbers tell you nothing about the objects you count but only how many of them are out there.
Commutativity of addition stems directly from the intuitive properties of counting: to count the total in two groups of bananas one may start with either group.
There are many ways to count on fingers and 10 is not necessarily the largest number you can reach counting on fingers. It all depends on how you do that!
It's convenient to summarize and to bring to the child's attention that counting obeys certain rules that make it faster and funnier.
- H. Gardner, Frames of Mind, 10th Anniversary Edition, BasicBoks, 1993
- J. Piaget, Genetic Epistemology, W. W. Norton & Company, 1971