On the illegitimate use of Taylor’s theorem in the definition of dual and hyper-dual functions

We present a foundational critique of a widespread practice in the kinematics and automatic-differentiation literature: defining the extension of a real function to the algebras of dual numbers D and hyper-dual numbers H D via a “truncated Taylor series”, on the grounds that powers of the nilpotent part vanish. This derivation is a category error : Taylor’s theorem is an analytic statement about a remainder tending to zero over an Archimedean ordered field; it cannot be invoked over rings with nilpotent elements. The correct formulation is purely algebraic (jet evaluation), and the general canonical form of the holomorphic (in the Scheffers sense) extension contains an arbitrary family of functions that the “Taylor” derivation tacitly excludes. We provide the correct statements, explicit counterexamples on D and H D , and discuss how the error propagates to nilpotent algebras of higher index (multicomplex, truncated polynomial). The critique does not target the Study–Kotelnikov–Scheffers algebraic tradition, nor the author’s own operator-based formulations previously published in this journal, in which the dual/multidual extension is introduced by a differential transform on the nilpotent algebra, thereby sidestepping the fallacy.