On a class of meaningful permutable laws

The permutability equation G(G(x, y), z) = G(G(x, z), y) is satisfied by many scientific and geometric laws. A few examples among many are: The LorentzFitzGerald Contraction, Beer’s Law, the Pythagorean Theorem, and the formula for computing the volume of a cylinder. If we required that a permutable law be meaningful, the possible forms of a law are considerably restricted. The class of examples described here contains the Pythagorean theorem. The mathematical expression of a scientific law typically does not depend on the units of measurement. The most important rationale for this convention is that measurement units do not appear in nature. Thus, 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. As far as I know, however, there is no agreed upon formalization of this type of invariance of the form of scientific laws, even though there has been some proposals (see Falmagne and Narens, 1983; Narens, 2002; Falmagne, 2004). Expanding on the just cited references, I propose here a general condition of ‘meaningfulness’ constraining a priori the form of any function describing a scientific or geometric law expressed in terms of ratio scales variables such as mass, length, or time. We define this meaningfulness condition in the second section of this paper. In this definition, all the units of the variables are explicitly specified by the notation, as opposed to being implicitly embedded in the concepts of ‘quantities’ and ‘dimensions’ of dimensional analysis (c.f. for example Sedov, 1943, 1956). The interest of such a meaningfulness condition from a philosophy of science standpoint is that, in its context, general abstract constraints on the function, formalizing ‘gedanken experiments’, may yield the exact possible forms of a law, possibly up to some real valued parameters. An example of such a constraint is the condition below, which applies to a real, positive valued functionG of two real positive variables. It is formalized by the equation G(G(y, r), t) = G(G(y, t), r), (1) where G is strictly monotonic and continuous in both real variables. An interpretation of G(y, r) in Equation (1) is that the second variable r in modifies the state of the first variable y, creating an effect evaluated by G(y, r) in the same measurement variable as y. The left hand side of (1) represents a one-step iteration of this phenomenon, in that G(y, r) is then modified by t, resulting in the effect G(G(y, r), t). Equation (1), The only exception is the counting measure, as in the case of the Avogadro number. The results can be extended to other cases, in particular interval scales. which is referred to as the ‘permutability’ condition by Aczél (1966), formalizes the concept that the order of the two modifiers r and t is irrelevant. Many, and various, scientific laws are ‘permutable’ in the sense of Equation (1). Some examples of permutable laws are the Lorentz-FitzGerald Contraction, Beer’s law, the formula for computing the volume of a cylinder, and the Pythagorean theorem. For the Lorentz-FitzGerald Contraction, for example, written in the form L(`, v) = ` √ 1− (v c )2 in which c is the speed of light, we have L(L(`, v), s) = L(`, v) √ 1− (s c )2 = √ 1− (v c )2√ 1− (s c )2 = L(L(`, s), v) . Not all scientific laws are permutable. Van der Walls Law, for instance is not: see the Counterexample 4(f). Under fairly general conditions of continuity and solvability making empirical sense, the permutability condition (1) implies the existence of a general representation G(y, r) = f−1(f(y) + g(r)), (2) where f and g are real valued, strictly monotonic continuous functions. This is stated precisely in our Lemma 7, which is due Falmagne (2012), and generalizes results of Hosszú (1962a,b,c) (cf. also Aczél, 1966). It is easily shown that the representation (2) implies the permutability condition (1): we have G(G(y, r), t) = f−1(f(G(y, r)) + g(t)) (by (2)) = f−1(f(f−1(f(y) + g(r))) + g(t)) (by (2) again) = f−1(f(y) + g(r) + g(t)) (simplifying) = f−1(f(y) + g(t) + g(r)) (by commutativity) = G(G(y, t), r) (by symmetry) . We will also use a more general condition, called ‘quasi permutability’, which is defined by the equation M(G(y, r), t) = M(G(y, t), r) (3) and lead to the representation M(y, r) = m((f(y) + g(r)) (4) (see also Falmagne, 2012, and our Theorem 7 here). The combined consequences of permutability or quasi permutability and meaningfulness are powerful ones. For instance, suppose that meaningfulness holds, that the function G is symmetric, and that it also satisfies a certain quasi permutability condition. Our Theorem 9 states that, under reasonable solvability conditions, there are only two possible forms for G, which are: 1. G(y, x) = θyx (θ > 0) (5) 2. G(y, x) = ( y + x ) 1 θ (θ > 0) . (6)