Decisions, Structure, and Natural Law
The object here is to show that our thinking processes and our physical forms and those of all things that exist, are a result of response in nature to influences as stimuli, brought about by natural occurrences. The ideas are developed through a generalization of the role judgment plays in decision making. Judgment serves as the basic link between our conscious awareness and the stimuli of the natural world. The mathematics used to represent natural laws is derived from stimulus-response theory and this in turn from the representation of judgment as it is used in decision-making. The representation of discrete judgment as a principal eigenvalue problem is generalized to the continuous case through Fredholm theory. Solving the resulting fundamental functional equation, which is a necessary condition for the existence of a solution, gives rise to damped periodic oscillation. The Fourier transform of the real valued solution has a perturbed inverse square representation that poses a question raised on occasion in science about the full accuracy of exact inverse square laws of gravitation, optics and of electric charges. The Fourier transform of the complex valued solution is a linear combination of Dirac type distribution of impulsive functions representing how the brain must operate to respond to external stimuli. A generalization is made to a functional equation in operator form with its solution. These solutions describe all forms that exist in nature as anything that responds to influences. These considerations that originate in the mathematics of judgment, serve as a unifying approach to our understanding and to creating tools for modeling and solving complex physical and behavioral problems.