Local Uniqueness and Global Non-Uniqueness in Philosophy: A Categorical, Axiomatic, and Sheaf-Theoretic Metaphilosophy

This work proposes a structural explanation for the persistent non‑convergence of philosophy. Rather than treating philosophical disagreement as a sociological accident or a failure of rationality, the paper argues that philosophy functions as an interpretation system: a domain in which multiple, internally coherent but mutually incompatible frameworks coexist without a shared meta‑level procedure for adjudication. To articulate this claim, the paper introduces three “Axioms of Uniqueness”—logical, metaphysical, and methodological—and shows that philosophical systems typically satisfy logical uniqueness internally while systematically violating metaphysical and methodological uniqueness at the global level. The paper then develops a unified categorical, axiomatic, and sheaf‑theoretic framework for modeling philosophical systems. Philosophical theories are treated as objects in a category P, with morphisms representing translations or reinterpretations. The absence of terminal objects, limits, exponentials, and subobject classifiers in P explains why no globally unique philosophical system can exist. A presheaf‑theoretic analysis further shows that locally coherent philosophical systems cannot be “glued” into a global section, demonstrating a structural failure of the sheaf condition. Historical case studies—from Plato and Descartes to Kant and Nietzsche—are reinterpreted as concrete instances of local uniqueness combined with global non‑uniqueness, illustrating the structural pattern across the history of philosophy. The result is a new form of metaphilosophy: a categorical and axiomatic science of non‑uniqueness, offering a unified explanation for the internal rigor and external incompatibility of philosophical systems.