🤖 AI Summary
Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.
📝 Abstract
Modeling has become a widespread, useful tool in mathematics applied to diverse fields, from physics to economics to biomedicine. Practitioners of modeling may use algebraic or differential equations, to the elements of which they attribute an intuitive relationship with some relevant aspect of reality they wish to represent. More sophisticated expressions may include stochasticity, either as observation error or system noise. However, a clear, unambiguous mathematical definition of what a model is and of what is the relationship between the model and the real-life phenomena it purports to represent has so far not been formulated. The present work aims to fill this gap, motivating the definition of a mathematical model as an operator on a Hilbert space of random variables, identifying the experimental realization as the map between the theoretical space of model construction and the computational space of statistical model identification, and tracing the relationship of the geometry of the model manifold in the abstract setting with the corresponding geometry of the prediction surfaces in statistical estimation.