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Mathematics of the brain: generalization from Eigen vectors to eigen functions.

Thomas Saaty

In this paper, we provide an overview of the mathematics of the brain and present the generalization from Eigen vectors to Eigen functions. Initially we summarize some of the mathematics of derived priority scales involved in the multicriteria decision process, and give the corresponding generalization to the continuous case. Then we discuss of density and approximation and how we can generalize from discrete to continuous judgments. The solutions of the functional equation w(as)=bw(s) are given in the real and complex domain and then for quaternions (non-commutative) and octonions (non-commutative and non-associative). In the end we present the consequences and evidence for validation: a) The Laws of Nature are written in the Workings of our Brains, b) the Weber-Fechner Law of Psychophysics, and c) The Brain Works with Impulsive Firings.