PFUSRC-112 On the Three Faces of Mathematics and the Ontological Self-Awareness of Physics — A Paradigm Critique from the Kakeya Conjecture, Feynman Path Integral, and the Fictionality of Lines and Surfaces

The correlation between mathematics and physics stands as the most fundamental yet long-neglected foundational issue in contemporary theoretical physics. The physics community generally takes mathematics as a universal descriptive language of nature, yet rarely conducts categorical discrimination on its functional positioning: whether mathematics participates in physical research as instrumental computing tools, abstract axiomatic analogies, or ontological structures isomorphic to cosmic reality. This paper proposes a tripartite classification of mathematics: Instrumental Mathematics, Abstract Mathematics and Ontological Mathematics. It argues that physics can only achieve authentic identification of objective physical reality by structural matching with Ontological Mathematics. Instrumental Mathematics acts as the operational tool of physical calculation; Abstract Mathematics provides self-consistent logical mapping references; only Ontological Mathematics constitutes the intrinsic structural skeleton of physical ontology. Three representative paradigm deviations are analyzed as critical evidence: the Kakeya Conjecture’s century-long categorical rupture from physical rigid-body motion problems to pure abstract set-theoretic dimension propositions; the widespread misinterpretation of Feynman path integrals’ equivalent computational framework as authentic ontological depiction of quantum microcosms; and the long-standing misrecognition of zero-measure mathematical fictions (points, lines, surfaces) as fundamental constitutive units of physical spacetime. The text demonstrates that loss of ontological self-awareness leads physics into three recurring errors: overreliance on mathematical formal complexity, conflation of abstract geometric mappings with real cosmic architecture, and treating artificially defined axiomatic fictions as objective physical entities. Finally, three core self-awareness principles for physics are established, taking PFUSRC’s 11-dimensional triple coaxial 45° bicone topology as a standard exemplar of Ontological Mathematics. Rigid boundaries are drawn between physical reality and mathematical models, correcting the prevalent cognitive bias that mathematical self-consistency equals physical truth.