  In this paper, we use the language of noncommutative differential geometry to
formalise 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
$\mathbb{T}^d$.
