to model "multiple extended quantities" (Riemann)
The terminology of "multiple extended quantities" was introduced by B. Riemann, synonymly to a "continuous manifold". It is conceptually based on two essential attributes: "continuity" and "multiple extension".
The history of manifolds is the attempt to build a mathematical structure to model the whole (the continuum) and the particular (the part) to put its combination them into relationship to describe motion, action etc. (Helmholtz, Riemann, Poincare, Lie and others).
From the paper from E. Scholz below we recall the two conceptual design strategies for a "continuum":
Design of an "atomistic" theory of the continuum (which to H. Weyl's opinion contradicts to the essence of the continuum by itself)
Strategy II: develop a mathematical framework which symbolically explores the "relationship between the part and the whole" for the case of the continuum.
The later one leads to the concept of affine connexion, based on the concept of a manifold, which were developed during a time period of about 100 years.
The concept of manifolds leads to the concept of co-variant derivatives, affine connexion and Lie algebra to enable analysis and differential geometry, but (according to H. Weyl, Scholz 1) ...a .." truly infinitesimal geometry ... should know a transfer principle for length measurements between infinitely close points only."
Our alternative definition of the energy (inner product) Dirichlet integral (see Braun K., "A new ground state energy model"), which is rotation invariant with respect to infinitely close points, is proposed to build a truly infinitesimal geometry, which then would lead to a "principle of general contra-variance".