The choice of the definition matters: numerical versus categorical variable
They say: A variable is called categorical if each observation belongs to one of a set of categories. A variable is called quantitative if observations on it take numerical values that represent different magnitudes of the variable (Agresti and Franklin, p.25)
I say: Not all definitions are created equal. The definition you stick to must be short, easy to remember and easy to apply. My suggestion: first say that we call numerical variables those variables that take numerical values and then add that all other variables are called categorical. In the paragraph immediately preceding the above definition, the authors have this idea. However, they choose a less transparent definition. Not a big deal, but in a book that is 800+ pages this matters.
In case of more complex notions the choice of the definition becomes critical. Definitions not only give names to objects but they also give direction to the theory and reflect the researcher’s point of view. Often understanding definitions well allows you to guess or even prove some results.
For the benefit of better students you can also tell the following. Math consists of definitions, axioms and statements. Definitions are simply names of objects. They don’t require proofs. Axioms (also called postulates) are statements that we take for granted; they don’t require proofs and are in the very basement of a theory. Statements have to be proved.