Logical and Geometric Inquiry Available to Purchase

This paper proposes a framework for quantifying logical and geometric inquiry through specific interpretations of Bayes’ Theorem and Information Theory. In logical inquiry there are a countable number of possible discrete answers that define the inquiry, and Bayes’ Theorem serves to move the observer posing the question along a trajectory in a hyberbolic figure in a manner suggested by Rodriguez. For N=3, this plane is a hyperbolic triangle whose angles sum to zero — the smallest possible value in the hyperbolic plane where the sum of the angles of a triangle must sum to a positive number less than pi. In euclidean space, the hyberbolic figure becomes a multi‐dimensional simplex or polyhedron described by Shannon in his paper on a geometrical perspective of channel capacity. A theory of geometric inquiry requires that one consider an observer who conjointly possesses an objective reality space Θ and a physical or measurable space X. It is discussed how the matching of these spaces characterizes the ability of an observer to distinguish its posited objective reality. A simple functional form I is suggested as a measure of the degree of distinguishability for an observer. This form corresponds to the trace of the Fisher information matrix of p(x|θ) over θ∈ Θ. The origin and precise specification of the requirements that give rise to the specified functional form are unknown and represents an important area of future study with clues suggested in the work of Balasubramanian. At the same time, the question is asked regarding the nature of the metrics and probability distributions arising when an observer balances prior ignorance and prior knowledge through the extremizing of a functional J (p,∇p) = I + λH over probability densities p. The functional I is the a priori ability of the observer to distinguish pure space, H is the prior ignorance of the same observer over the same space, and λ is a scalar Lagrange multiplier ostensibly needed to balance units, but having additional interesting properties. Explicit solutions are derived for optimal p in both one and in general in N dimensions for λ = 0 and λ ≠ 0. In particular, the distributions that result when λ ≠ 0 include gaussian densities satisfying the functional form of distributions defining the elements of the Fisher Information matrix of pure‐space as discussed by Rodriguez which possesses negative curvature when spatial uncertainty exists. Although only inquiry is discussed, a formalized conjoint theory of inquiry and control has significant implications regarding the engineering and design of intelligent systems that operate cybernetically.