By a ‘scientific law’, we mean an equation in which the variables represent physical or geometrical quantities such as mass, length, (absolute) temperature or time duration. Examples are the Ideal Gas Laws, Coulomb’s Law, the Lorentz-FitzGerald Contraction, Beer’s Law, or van der Waals’ Equation (see e.g. Hix and Alley, 1958), the last three playing important roles in this book. Examples in geometry are many, such as the Pythagorean Theorem and the equations of a parabola and of the volume of a sphere. As suggested by these examples, we restrict consideration to laws linking ratio scale quantities. In such cases, which represent the large majority in science and geometry, fixing the units suffices to specify the variables in an equation1. The second word in the title of our book, ‘meaningfulness’, refers to a fundamental invariance property that is satisfied by practically all scientific or geometrical laws. Stated informally, it is that: the mathematical form of a law is not altered by a change of units, for example from meters to centimeters or miles, or from kilograms to grams or pounds. There are three important reasons for requiring meaningfulness in the formulation of scientific laws. The first two are obvious. 1) Permitting non-meaningful laws would result in a scientific Tower of Babel, in which the mathematical form of a model would depend upon the particular measurement units favored by a community of scientists, promoting confusion. 2) More importantly, as the units of a scientific variable have no representation in nature, any non-meaningful expression would be a poor representation of reality. 3) Taking meaningfulness as an axiom of a scientific theory may lead to weakening, or even replacing, the other axioms, resulting in a refocusing of the theory, and conceivably, leading to a deeper understanding of the basic mechanisms at work.