Conceptual differential calculus part ii: Cubic higher order calculus

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.