How to Map a Sandwich:
Potential Theory,Topological Existence Theorems, and the Changing History of the Ontology of Cartographic Objects
In the 1960s and 1970s the most important work being accomplished in mathematical cartography had to do with the topological properties of surfaces and their relationship to geographical and spatial analysis. The Harvard Laboratory for Computer Graphics and Spatial Analysis was a hotbed of such work and was led into new areas by the ideas of the theoretician William Warntz. While most other researchers in the field where looking at the numerical properties of surfaces Warntz’s approach centered on understanding their topology. He recognized that the most important properties of surfaces from a mathematical point of view had nothing to do with numbers but rather their invariance under transformations. Warntz described the relationship of the topological properties of a surface to cartography in a number of important papers that adopted a terminology and methodology built on the work of the mathematician Arthur Cayley (1859). Warntz was particularly interested in mapping thematic surfaces and adopted a macrogeographical theoretical perspective that led not only to fundamental mathematical breakthroughs but also yielded philosophical insight into the nature of the objects described by the “science” of cartography. This paper focuses on one particular aspect of the work of Warntz and one of his students at the Harvard Laboratory; existence theorems. Existence theorems contain a statement of existential quantification such as “there is” and prove the existence of a particular set of mathematical objects. They do not however contain any directions of how such objects might actually be constructed algorithmically or numerically.
The researchers at the Lab published two very important works on existence theorems in the influential and now largely forgotten series the Harvard Papers in Theoretical Geography. We will provide a close reading of two of these papers, “The Sandwich Theorem: A Basic One for Geography”, and “Geography and an Existence Theorem: A Cartographic Solution to the Localization of Sets of Equal-Valued Antipodal Points”, in order to show how the Lab used a mathematical approach that was underexploited in cartography and in doing so changed the accepted notions of the nature of cartographic objects.
This shift in the nature of what constituted geographical and cartographic objects is discussed in this study within the framework of Thomas Kuhn's Structure of Scientific Revolutions. Kuhn provides an example in his analysis of the development of the theory of relativity in the beginning of the 20th century of the type of profound conceptual shifts that took place in cartography in the 1960s and 70s. These shifts were not simply dramatic changes in beliefs about the world or even in scientific and geographic methodology, but rather in the very concepts that define the structure and formal properties (topological and transformationally invariant) of the objects of inquiry. In this way Kuhn’s framework and lexicon provides us with a solid philosophical and historical framework in which to discuss the same type of radical shifts that took place at the foundations of mathematical cartography. These changes in the conceptual framework of cartographic science redefined the nature of geographical objects (what is it that is mapped) and laid the foundations for the development of topological data structures and modern GIS.