On a meaningful axiomatic derivation of the Doppler effect and other scientific equations

The mathematical expression of a scientific or geometric law typically does not depend on the units of measurement. This makes sense because measurement units have no representation in nature. 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. This paper formalizes this invariance of the form of the laws as a meaningfulness axiom. In the context of this axiom, relatively weak, intuitive constraints may suffice to generate standard scientific or geometric formulas, possibly up to some numerical parameters. We give several example of such constructions, with a focus of the Doppler effect and some other relativistic formulas. When properly formalized, the invariance of the mathematical form of a scientific or geometric law under changes of units becomes a powerful ‘meaningfulness’ axiom. Combining this meaningfulness axiom with abstract, intuitive, ‘gedanken’ type properties such as associativity, permutability, bisymmetry, or other conditions in the same vein, enables the derivation of scientific or geometrical laws (possibly up to some parameter values). In the last section of this paper, I will show how, in the context of meaningfulness, the axiom L(L(λ, v), w) = L(λ, v ⊕ w) (1) yields specific numerical expressions for the function L and the operation ⊕. Equation (1) is an abstract axiom representing the mechanisms conceivably involved in the Doppler effect of the Lorentz-FitzGerald Contraction (Feynman, Leighton, and Sands, 1963, Vol. 1). The operation ⊕ represents the relativistic addition of velocities. The left hand side of Equation (1) formalizes an iteration of the function L. The equation states that such an iteration amounts to adding a velocity via the relativistic addition of velocities operation. A. Motivating the meaningfulness condition The trouble with an equation such as (for example) L(`, v) = ` √ 1− (v c )2 , (2) ∗I am grateful Chris Doble, Jean-Paul Doignon, and Louis Narens for their collaboration on various part of my work in this area. I also thank Don Saari and C.T Ng for their comments. 1 MBS TECHNICAL REPORT 17-02