From Mathematical Foundations to Systematic Philosophy

Abstract

This paper uses category theory~\cite{maclane1971,lawvere2003} as a unified language to construct a complete theoretical chain from mathematical foundations to systematic philosophy. We rigorously prove four core propositions: \begin{enumerate} \item The whole is a function, the parts are subfunctions. \item To exist is to be a spacetime function. \item Existence is a function unified field. \item The essence of ``the whole is greater than the sum of its parts'' is that the organic is greater than the mechanical. \end{enumerate} Furthermore, we formalize metabolism as the causal-dynamical mechanism by which a system maintains its ``existence function'' (causal closure), proving that metabolism is a universal principle for any non‑equilibrium ordered structure to sustain its causal closure. On this basis we propose the Zhu–Liang metabolicon – the minimal organic unit that maintains its own causal closure – and prove that the unified field is isomorphic to the inverse limit of metabolicons at the level of sections. Finally we merge the unified field and the metabolic‑causal principle into the ``unified metabolic‑causal field'', revealing that all levels from quantum to civilization follow the fundamental laws: whole before parts, relations define entities, metabolism sustains causality. All proofs are based on standard mathematical tools (category theory, Markov categories, information theory, dynamical systems), providing a rigorous mathematical foundation for holism. An appendix uses quantum entanglement as an example to demonstrate the application of the unified metabolic‑causal field framework to concrete physical phenomena.

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