Mathematics as a Cognitive Interface to Physical Complexity: Abstraction, Mechanistic Comprehension, and the Limits of Human Intuition

Modern physics has developed through increasingly abstract mathematical representations. From statistical mechanics to quantum theory, quantum field theory, and effective field theory, physical reasoning increasingly relies on state spaces, operators, symmetries, fields, representations, probability amplitudes, and effective degrees of freedom. Existing philosophy of science has examined mathematical explanation, structural representation, model-based inference, and scientific understanding. This paper proposes a complementary cognitive mechanism: the mathematization of advanced physics may be partly understood as a representational response to the effective complexity of lower-order physical descriptions exceeding the capacity of human reasoners to sustain explicit mechanistic representations. Building on the abstraction-threshold framework of {LiLi2026}, we introduce the concept of a mechanistic-comprehension threshold. For a given problem and representation, the lower-order description may require a reasoner to maintain too many degrees of freedom, interactions, dependencies, scales, and state relations to remain cognitively tractable. Higher-order mathematical abstraction can then become advantageous by compressing and re-objectifying these relations into manipulable structures. Mathematics thereby functions not merely as a language of physics but as a cognitive interface to physical complexity. The framework distinguishes cognitive tractability from mechanistic transparency. A mathematical representation may substantially increase the former without guaranteeing a corresponding increase in the latter. We therefore propose an explanatory abstraction window: a problem-dependent region in which complexity is sufficiently compressed while enough recoverable lower-order organization remains to support mechanistic understanding. Statistical mechanics provides a clear case of complexity-induced mathematization; quantum theory illustrates the separation between empirical and structural success on the one hand and contested mechanism or ontology on the other; effective field theory shows that selective suppression of lower-order detail can also be physically principled rather than merely cognitively convenient. Finally, artificial intelligence provides a possible counterfactual cognitive architecture. Comparing representations generated by agents with different memory, search, and parallel-processing capacities may help distinguish features of physical theory driven primarily by physical structure from those partly contingent on human cognitive architecture.