Conceptual differential calculus part ii: Cubic higher order calculus

TL;DR

This paper extends a conceptual framework for higher order differential calculus by introducing cubic calculus based on hyper-cubes, providing new insights into foundational mathematical structures.

Contribution

It develops a new higher order differential calculus using hyper-cubes, expanding the local linear algebra framework from Part I to cubic structures.

Findings

Introduces full and symmetric cubic calculus versions

Provides a new perspective on foundational issues in differential calculus

Generalizes local linear algebra to higher orders

Abstract

Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cube, and so we qualify this calculus as '' cubic ''. More precisely, we define two versions of conceptual cubic calculus: '' full '' and '' symmetric cubic ''. The theory thus initiated sheds new light on several foundational issues.