Five-Dimensional Mathematics: A Mathematical Framework for Systems Science

Traditional mathematics is confined to linear superposition of homogeneous units and cannot char-acterize nonlinear synergy, emergence, or systems involving human subjective initiative. Based on theFive-Dimensional Ontology (boundary, structure, reserve, direction, intensity), this paper constructs afive-dimensional mathematical system with intensity units as primitives and five-dimensional synergy asthe core operation. We prove the five-dimensional minimal completeness theorem, define the synergydegree Γ and synergy coefficient κ, establish a five-dimensional dynamic evolution theory, and show thattraditional mathematics is a degenerate case (κ = 1) for isolated, non-interacting systems. Througheveryday examples such as a shaky table stabilized by a wooden block, scattered vs. concentratedeggs, and army coordination, we quantitatively validate the framework. Systems are further classifiedinto three types according to subjective initiative (none, weak, strong), corresponding to three opera-tion types (Type I passive-passive, Type II passive-active, Type III active-active). Traditional addition,which assumes κ = 1, can only describe quantitative change (entropy increase) and fails to capture thequalitative leap from dispersion to synergy. For strong subjective initiative systems, we introduce therest quality index Q rest . The framework unifies mechanical stability, social synergy, physical limits, andhuman games, providing a new mathematical foundation for general systems theory.