Differentialis

Differentialis is a term used in speculative mathematics and some science-fiction contexts to describe a generalized framework for studying differentials and their interactions across diverse mathematical and computational structures. The idea is to extend the traditional calculus of differentials to encompass continuous, discrete, and hybrid systems within a common language.

Origin and etymology: The name derives from the Latin differentialis, relating to differential calculus, and is used to signal a focus on change and rates of change across structures.

Core ideas: A differential object carries a degree-one operator d with d^2 = 0, and maps between objects preserve the differential structure. The framework seeks to unify differential operators on smooth manifolds, graphs, and discrete systems by tracking differential information along various paths. Some formulations use category-theoretic language to relate objects and their differentials via functors or natural transformations.

Status and usage: Differentialis is not a widely standardized theory in mainstream mathematics; it appears in niche expositions and speculative literature as a conceptual framework rather than a settled discipline.

Applications and related ideas: Potential applications include geometry, dynamical systems, and theoretical computer science, particularly in models blending continuous change with discrete transitions. Related concepts include differential geometry, algebraic topology, and category-theoretic approaches to differential calculus.