magnitudinen

Magnitudinen is a theoretical framework for examining magnitude as a foundational concept across disciplines. It seeks to unify various size-measuring constructs under common axioms.

A Magnitudinen function M assigns a nonnegative real number to elements of a space, generalizing familiar size measures such as vector norms. It satisfies non-negativity, M(0) = 0, homogeneity M(a x) = |a| M(x) for scalars a, and the triangle inequality M(x + y) <= M(x) + M(y). In Euclidean space, M equals the Euclidean norm; for data vectors, M_p is defined as M_p(x) = (sum_i |x_i|^p)^{1/p}. In non-linear spaces, M is defined via Minkowski functionals or gauges.

Magnitudinen emphasizes invariance under scaling, coordinate-free interpretations, and the separation of magnitude from direction when appropriate. It supports decomposition of quantities into magnitude and directional components in models that require such a split.

Applications include physics for comparing field strengths, geometry and robotics for measuring lengths and distances, computer science for embedding similarity, and data analysis for ranking features and assessing vector magnitude.

See also: Magnitude, Norm, Metric, Dimensional analysis.