The meaningfulness condition, applied to scientific or geometric laws, requires that the mathematical form of an equation does not change when we change the units of its (ratio scale) variables. This makes sense: any mathematical model or law whose form would be fundamentally altered by a change of units would be a poor representation of the empirical world. Suitably axiomatized, the meaningfulness condition becomes a powerful tool in the derivation of the exact form of a scientific or geometric law, on the basis on some general intuitive properties. The idea that the mathematical form of a scientific equation should not depend on the units of its variables is also the focal concept of dimensional analysis. However, in dimensional analysis, this invariance concept is embedded in the matrix notation. It is not an axiom, as it is here. The difference is a critical one: in dimensional analysis, the invariance is not taken advantage of as much as it could be.
In this talk, I will give seven examples of meaningful representations of empirical laws. But because my time is limited, I will only prove two of the representation theorems. One of them is a new proof of the Pythagorean Theorem (to be added to the 370 proofs that exist according to the last edition of Elisha Scott Loomis’ book). The other one is the meaningful derivation of Beer’s Law.