Relative measurement and its generalization in decision making why pairwise comparisons are central in mathematics for the measurement of intangible factors the analytic hierarchy/network process

According to the great mathematician Henri Lebesgue, making direct comparisons of objects with regard to a property is a fundamental mathematical process for deriving measurements. Measuring objects by using a known scale first then comparing the measurements works well for properties for which scales of measurement exist. The theme of this paper is that direct comparisons are necessary to establish measurements for intangible properties that have no scales of measurement. In that case the value derived for each element depends on what other elements it is compared with. We show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. Such measurements, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale. They are necessary to use when intangible factors need to be added and multiplied among themselves and with tangible factors. To derive and synthesize relative scales systematically, the factors are arranged in a hierarchic or a network structure and measured according to the criteria represented within these structures. The process of making comparisons to derive scales of measurement is illustrated in two types of practical real life decisions, the Iran nuclear show-down with the West in this decade and building a Disney park in Hong Kong in 2005. It is then generalized to the case of making a continuum of comparisons by using Fredholm’s equation of the second kind whose solution gives rise to a functional equation. The Fourier transform of the solution of this equation in the complex domain is a sum of Dirac distributions demonstrating that proportionate response to stimuli is a process of firing and synthesis of firings as neurons in the brain do. The Fourier transform of the solution of the equation in the real domain leads to nearly inverse square responses to natural influences. Various generalizations and critiques of the approach are included.