Are Mathematical Relationships Represented by Geometries in Some Functional State Space? - Williams, Andy (My University, 2022)
Human-Centric Functional Modeling is considered in this paper to be a branch of network theory which hypothesizes that all systems can be represented as having a set of functions through which the system might transition from one functional state to another. These states are referred to as “functional states” because they are described solely in terms of the functions available to transition to adjacent states, rather than being described in terms of any entities that might implement those functions. All possible behaviors of the system are then hypothesized to be described by a graph containing a network of nodes representing such functional states, where those nodes are connected by edges representing the processes through which those functional states might transition between each other. Modeling human cognition as moving through a “space of concepts” or “conceptual space” that forms the functional state space of the cognitive system and that has the capacity to represent all possible concepts and reasoning, it has been hypothesized that all information and therefore all mathematical proofs and other logic can be represented in this space. Modeling human conscious self-awareness as moving through an “awareness space” or “space of awareness's” that represents the functional state space of the consciousness, it has been hypothesized that awareness of any mathematical or other truth, whether or not yet identified or proven, can be represented as a potential awareness in that space. This paper outlines the hypotheses that in the limit of infinite ability to navigate logic through understanding or reasoning processes (infinite intelligence) mathematical logic is represented by a geometry in conceptual space, and that in the limit of infinite ability to be aware of truth (infinite awareness), mathematical truth is represented as a geometry in awareness space. This paper also outlines some of the questions that must be answered in order to validate these hypotheses.
Keywords: Human-Centric Functional Modeling; mathematics; state space