tual or philosophical difficulties with their use. This debate among the so-called constructivists, like Wittgenstein, who believed that in mathematics “everything is algorithm”, and the formalists like Hilbert, has left a large body of philosophical literature that has deeply analyzed the ontology of mathematical objects. 
In the fields of geography and cartography, these theorems entered into early computer systems through the construction of practical algorithms that calculated particular sets of objects useful in geographic analysis. Two important papers that can be seen as case studies in the use of constructivist forms of existence theorems in early computer cartography were published in the series Harvard Papers in Theoretical Geography by William Warntz and his associates at the Harvard Lab for Computer Graphics and Spatial Analysis in the late 1960s and early 1970s. This series of papers developed algorithmic constructions of many existence theorems and two of the most interesting, because of the sheer complexity of the mathematics, the Borsuk-Ulam Theorem and the Ham Sandwich Theorem, were applied to real world geographic problems .
Besides using existence theorems, mathematical cartographers would also begin to re-conceptualize on a more general level questions about the use of pure mathematics and its role in defining the diagrammatic logic of maps. In an early lecture, later written as a discussion paper for the Michigan Inter-University Community of Mathematical Geographers, Warntz says that, "More than ever before geographers are using the tools of calculus, probability, topology, symbolic logic, the various algebras, geometries, for example, are being taken more literally than ever before." He elaborates on these comments by explaining to the reader that something as abstract and foreign to geography as Venn diagrams are being taken, "in a far more literal sense than they were originally intended and by substituting real space and attendent phenonema for ideal space and by insisting on the utilization of all geometric properties involved as well as just the topological ones, geographers can reinterpret, add to, and refine the conventional concepts in the methodology of uniform regional geography and provide it with a basis in logic." 
Many geographers at the time would push the concept logic form and notions from set theory further into geographic analysis and not just in the sense of a useful analogy. In a paper written for one of the classic compilations texts from early years mathematical geography called, The Philosophy of Maps, Warntz and others like Waldo Tobler, and William Bunge, would change not only the vocabulary used in analysis but would also alter the very form of its expression. In an article in the collection, called Some Elementary and Literal Notions About Geographical Analysis and Extended Venn Diagrams, Warntz would say that, "Maps showing regional classification can be regarded as logic diagrams. Mapping of sets is a general mathematical concept. Geographical mapping is merely a special case of this."  Warntz here sees almost a mereological or mereotopological relationship between the spatial extent of Venn diagrams and their isomorphic counterparts of geographic regions.
John VennIt is quite remarkable that the two systems of logic that Warntz draws on in this paper, Venn diagrams and existential graphs, are both visual and not symbolic logical systems. Most of the work done in logic during the 20th century has focused on symbolic systems with little research, at least until quite recently, on the heterogeneous reasoning of the type Warntz is advocating. He says that, "It is part of our purpose here to extend the use of such diagrams to the mapping of geographical regions by making use of properties already inherent in Venn diagrams but as yet unutilized... We intend to apply spatial properties literally to real spatial distributions on the earth's surface..."
Venn diagrams can grow to extremely complex forms depending on the number of sets one is dealing with and recent research on 