  Data analysis and data mining are concerned with unsupervised pattern finding
and structure determination in data sets. The data sets themselves are
explicitly linked as a form of representation to an observational or otherwise
empirical domain of interest. "Structure" has long been understood as symmetry
which can take many forms with respect to any transformation, including point,
translational, rotational, and many others. Beginning with the role of number
theory in expressing data, we show how we can naturally proceed to hierarchical
structures. We show how this both encapsulates traditional paradigms in data
analysis, and also opens up new perspectives towards issues that are on the
order of the day, including data mining of massive, high dimensional,
heterogeneous data sets. Linkages with other fields are also discussed
including computational logic and symbolic dynamics. The structures in data
surveyed here are based on hierarchy, represented as p-adic numbers or an
ultrametric topology.
