The nested rectangular array as a model of data

Data, like electricity and gravity, are part of the world in which we live. Some occur naturally, as in the genetic code, while most occur as a consequence of language and social organization. The search for a theory of data, which begins with the choice of a model, is as important and interesting as the development of theories in physics, economics, and psychology. Most models of data are collections, such as the unnested array of APL, the one-axis nested list of LISP, and the set, which is nested but lacks the properties of order, repetitions, type, and multiple axes inherent in rectangular arrangement. Nested rectangular arrays have all these properties. The existence of simple, universally valid equations in both set theory and linear algebra suggests that equally simple equations may hold for all arrays. The principles of nested collections developed in set theory apply with few changes to the nesting of arrays. A one-sorted theory of arrays, in which type is preserved for empty arrays, provides an algebra of operations interpreted not only for data but also types of data.