Deriving meaningful scientific laws from abstract, 'gedanken' type, axioms: five examples

A scientific law can be a faithful representation of the physical world only if its form is invariant with respect to changes in the unit or units. This is referred to as the ‘meaningfulness condition.’ This condition is powerful. If we require it, the mathematical form of a scientific or geometric law may be derivable from some abstract constraint, possibly verifiable by a thought experiment or a trivial argument. We discuss five examples of such abstract constraints in this paper: In each case, just one or a couple of meaningful mathematical representations are possible. In this paper, we derive the possible meaningful representations in five examples. These results are obtained under some general conditions, in addition to those listed in 1–5. Other meaningful representations may be possible under different additional conditions.