ABSTRACT
Measurement brings numbers into science. More formally, measurement is the assignment of numbers or mathematical objects to empirical or qualitative entities. The way a sub-area of science accomplishes measurement impacts the form of its mathematical representation. Mathematical science routinely applies powerful methods of mathematical inference to the representation to infer relationships among the measured objects. The problem of sorting out which of these relationships belong to the sub-area of science under consideration and which are “just mathematical” or perhaps belong to a different sub-area of science is called the meaningfulness problem. Although measurement and meaningfulness are considered important in various parts sciences, their presentations in the literature are usually informal and superficial. This is somewhat surprising given the numerous debates in the scientific literature about what can and cannot be concluded from specific mathematical models and data sets. Considerations involving measurement and meaningfulness are also essential in providing a proper foundation for various techniques in science that make inferences based on invariance and symmetry.
