Before asking how many objects there are, we must define what we are counting. This choice must be made explicit at the outset. In a population census, for example, major political and economic issues may depend on it (see the next article in this feature).
Once this has been clarified, the mathematical notion of a set comes into view. But finding the number of elements in a set is not always easy. If, for example, you want to know how many chocolates are in a box you were given, you need only count them one by one to find out how many there are (we would say that you have counted the chocolates). By contrast, to determine how many students attend a school, we may prefer to find the number of students in each class and then add these figures together. When the objects to be counted are not already organized into smaller groups, we can choose to group them as a preliminary step. Finally, some sets cannot readily be described, even by a mathematical model. We may have to forgo an exact count of their elements and seek an estimate instead, using statistical methods, for example.
Counting objects by listing them lies at the very foundation of the concept of number. We take one object, then another, making two; one more makes three, and so on. In Mathématiques d’ailleurs (Seuil, 1998), Marcia Ascher writes that this "ability to count is a human universal linked to language". Enumeration can continue indefinitely, and number names are often built around arithmetic properties so that there is no limit to the numbers we can name. In French, for example, most number names are based on the decimal numeral system.
Using combinatorics
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When a collection is too large for each of its elements to be considered individually, whether because this is impossible or simply too laborious, we can try to exploit a particular way of organizing them. This is one of the aims of combinatorial analysis. We then use properties of sets with finitely many elements.
