Infinitesimelle

Infinitesimelle is a fictional mathematical and philosophical construct that generalizes the notion of an infinitesimal by incorporating a context-dependent scale parameter. The term blends infinitésimal, from calculus, with the French diminutive suffix -elle, signaling a small, context-bound quantity.

Definition and framework: An infinitesimelle δx_c is assigned to a given context c that encodes resolution, environment, or observational scale. For a function f, changes are expressed as δf_c ≈ f'(x) δx_c, but δx_c depends on c and may not commute with limits. In the limit of a classical scale (c → ∞ or c → a standard reference), δx_c reduces to a traditional infinitesimal or to zero, recovering standard calculus.

Properties and rules: In the fictional framework, δx_c is additive across disjoint contexts and obeys a scale-dependent product rule, δ(fg)_c = f δg_c + g δf_c + R_c, where R_c captures higher-order, context-dependent corrections. The approach supports multiscale modeling where local changes are measured at different resolutions.

History and reception: Infinitesimelle was introduced in speculative mathematics and appears in certain fictional and thought-experiment treatises to address observer-dependent change. It has limited formal endorsement; critics argue it overlaps with existing multiscale and nonstandard analyses and can complicate foundations.

See also: infinitesimal, nonstandard analysis, multiscale analysis, scale calculus.