  This paper shows how gauge theoretic structures arise naturally in a
non-commutative calculus. Aspects of gauge theory, Hamiltonian mechanics and
quantum mechanics arise naturally in the mathematics of a non-commutative
framework for calculus and differential geometry. We show how a covariant
version of the Levi-Civita connection arises naturally in this commutator
calculus. This connection satisfies the formula $$\Gamma_{kij} + \Gamma_{ikj} =
\nabla_{j}g_{ik} = \partial_{j} g_{ik} + [g_{ik}, A_j].$$ and so is exactly a
generalization of the connection defined by Hermann Weyl in his original gauge
theory. In the non-commutative world $\cal N$ the metric indeed has a wider
variability than the classical metric and its angular holonomy. Weyl's idea was
to work with such a wider variability of the metric. The present formalism
provides a new context for Weyl's original idea.
