In this paper, we use noncommutative differential geometry to formalize a framework for discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a \(C^*\) -algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function C(M) over a manifold M can be approximated by a direct limit of \(C^*\) -algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the d-lattice in \(\mathbb {R}^d\) and for the torus \(\mathbb {T}^d\) .
