An Intellectual Close-up of Counting in Research

Counting requires that the group within which it is done be sep­arable. If it is said that there are more people in Chicago than in New York, it is understood that the areas that are officially demarcated as Chicago and New York are known without ambi­guity, relative to their various suburbs, that only people living within the respective areas are counted, and that the number of people so counted within Chicago is more than the number of people so counted in New York. An assumption made is that the entities counted—in this case, men, women, and children—are discrete. (A pregnant woman due to deliver a baby in a moment is counted as one, not two.)

The theory of numbers involved in counting, done by rote by most civilized persons, has been found worthy of analysis by some of the greatest mathematicians. Suffice it for our purpose to record the three basic rules involved:

  1. If a set A of discrete entities and another set B of discrete entities are both counted against a third set C of discrete entities, then if A and B are counted against each other, they will be found to have the same number. We may formalize this relation (mathematically) as

If A = C and B = C, then A = B.

  1. Starting with a single entity and adding continu­ally to it another entity, one can build up a series (or set) of entities that will have the same number as any other collection whatsoever.
  2. If A and X are two collections that have the same number, and B and Y are two other collections that have the same number, then a collection obtained by adding A to B will have the same number as the number obtained by adding X to Y.

In terms of mathematics, we may state this as

If A = X and B = Y then A + B = X + Y

These apparently simple, to most of us obvious, rules charac­terize the cardinal use of numbers, which form the basis of count­ing; particularly familiar is rule 3. Let us say, for some purpose, that the total population of Chicago and New York City together needs to be measured. To do it, we do not require that all men, women, and children of both these cities be huddled together in a place where we do the counting from start to finish. We do the following:

  1. The population of Chicago is found; it is noted as a number, say Nc.
  2. The population of New York City is found; it is noted as a number, say
  3. The population of Chicago + the population of New York City is found as Nc +Nn.

This very simple procedure, obvious to most of us, is an appli­cation of rule 3 of counting above. Though the other two rules are even more fundamental, this one, by virtue of being “mathe­matical,” presents explicitly the fundamental principle of the use of cardinal numbers, namely, counting.

Source: Srinagesh K (2005), The Principles of Experimental Research, Butterworth-Heinemann; 1st edition.

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