  In this paper we extend the traditional framework of noncommutative geometry
in order to deal with spectral truncations of geometric spaces (i.e. imposing
an ultraviolet cutoff in momentum space) and with tolerance relations which
provide a coarse grain approximation of geometric spaces at a finite
resolution. In our new approach the traditional role played by $C^*$-algebras
is taken over by operator systems. As part of the techniques we treat
$C^*$-envelopes, dual operator systems and stable equivalence. We define a
propagation number for operator systems, which we show to be an invariant under
stable equivalence and use to compare approximations of the same space.
  We illustrate our methods for concrete examples obtained by spectral
truncations of the circle. These are operator systems of finite-dimensional
Toeplitz matrices and their dual operator systems which are given by functions
in the group algebra on the integers with support in a fixed interval. It turns
out that the cones of positive elements and the pure state spaces for these
operator systems possess a very rich structure which we analyze including for
the algebraic geometry of the boundary of the positive cone and the metric
aspect i.e. the distance on the state space associated to the Dirac operator.
The main property of the spectral truncation is that it keeps the isometry
group intact. In contrast, if one considers the other finite approximation
provided by circulant matrices the isometry group becomes discrete, even though
in this case the operator system is a $C^*$-algebra. We analyze this in the
context of the finite Fourier transform.
  The extension of noncommutative geometry to operator systems allows one to
deal with metric spaces up to finite resolution by considering the relation
$d(x,y)<\epsilon$ between two points, or more generally a tolerance relation
which naturally gives rise to an operator system.
