Ordinarily, when we measure things, we do not think about the scientific principles we are applying. These are the topics here.
Given any two people x and y, we can observe that there is a binary relation on pairs of peopleWe say that "taller than" is an empirical relation for height of persons. We can define several empirical relations on the same set: "taller than" and "much taller than". Empirical relations need not be binary, i.e., we can define a realtion on a single element of a set, or on collections of elements (eg. "is tall")
- x is taller than y, or
- y is taller than x
We can think of these relations as mappings from the empirical, real world to a formal mathematical world. We have entities and their attributes in the real world, and we define a mathematical mapping that preserves the relationships we observe.
Formally we define measurement as a mapping from the empirical world to the formal, relational world. Consequently, a measure is the mumber or symbol assigned to an entity by this mapping in order to characterize one of its attributes.
We say that the real world is the domain of the mapping and that the mathematical world is its range. When we map the attribute to a mathematical system, we have many choices for the mapping and the range. We can use real numbers, integers, or even a set of non-numerical symbols. The empirical relation must also be mapped into a mathematical relation.
The requirement that the measurement mapping must preserve the relations in the empirical world is called the representational condition.