A Formal Language for System Incommensurability: Introducing Axiomatic System Dynamics (ASD)

This paper introduces Axiomatic System Dynamics (ASD), a novel, formally self-consistent framework for describing the evolutionary processes of diverse complex systems. ASD is founded on three fundamental axioms which establish a unified, action-driven causal architecture. By utilizing the formal syntax of "State-Action-State Increment," ASD reduces distinct phenomena to Generalized Constitutive Relations that possess a unified structure. This formalization creates a critical bridge between observed phenomena and their mathematical expressions, directly addressing the challenge of incommensurability by providing a common semantic language for models across scientific domains. The framework's efficacy is demonstrated through three progressively complex applications. The analysis of Newtonian Mechanics re-interprets inertia as an Internal Action. The study of material constitutive relations validates the use of the Generalized State Parameter to formally model path dependence and system memory in Non-Markovian systems. Finally, the application to soil consolidation illustrates ASD's power in logically integrating multi-scale, multi-mechanism coupled processes. ASD provides a powerful conceptual tool for advancing the goal of scientific methodological unification.