  This paper lays the foundations of an approach to applying Gromov's ideas on
quantitative topology to topological data analysis. We introduce the
"contiguity complex", a simplicial complex of maps between simplicial complexes
defined in terms of the combinatorial notion of contiguity. We generalize the
Simplicial Approximation Theorem to show that the contiguity complex
approximates the homotopy type of the mapping space as we subdivide the domain.
We describe algorithms for approximating the rate of growth of the components
of the contiguity complex under subdivision of the domain; this procedure
allows us to computationally distinguish spaces with isomorphic homology but
different homotopy types.
