Vagueness, Topology, and Mereology
An Experiment in Bio-Biblio-Geographic Writing
The branching of space-time into different possible worlds made such an impression on me that I can still, more than 25 years distant, draw them from memory. As it turns out they were part of a radical re-thinking of the mathematics of space-time by Hugh Everett, called the relative-state formulation, which is based on what has become known as the many-worlds interpretation and lots of topology.
In studying Gödel’s work Kripke found several alternate proofs, and his lectures and unpublished manuscript, ‘Non-Standard Models and Gödel's Theorem: A Model-Theoretic Proof of Gödel's Theorem,’ have circulated widely in manuscript form. So widely in fact, that the philosopher of science Hilary Putnam felt it necessary to publish a summary of the article in 2000. Putnam showed that while today we know purely algebraic techniques that could be used to show the same thing, Kripke actually used techniques to establish incompleteness that could have, in principle, been understood by nineteenth-century mathematicians. This kind of thing, at least to me, is truly beautiful stuff. It is this kind of retrograde analysis that makes looking back at the history of geographic analysis so rewarding.
Mereotopology is composed of two parts and for logicians and mathematicians studying spatial structure at this level of abstraction these two parts are really two ways at looking at spatial entities. One of them considers part/whole distinctions, which is the job of mereology. Modern mereology is very much connected with various forms of ontology that philosophers have studied since Plato and Aristotle, and that were a bit of an obsession for medieval philosophers like Abelard and Aquinas. The problems of parts and wholes and their relationship to the identity of objects would not receive formal treatment however, until after Edmund Husserl published his Logical Investigations around 1900.
Many researchers are now looking into these kinds of representations and have started to think in terms of “geographic flows” and the kind of dynamical systems that would be required to picture these in a GIS. In mathematical terms this can be thought of as the difference between Eulerian and Lagrangian approaches. Eulerian models look at the evolution of a system, or a piece of geographic space, through time as series of discrete snapshots. Lagrangian models, on the other hand, follow some part of the system being modeled, continuously.
On the geographic but still analytical side, perhaps the most important person for me, and for many others, who have over the years become interested in the foundations of GIS, was William Warntz. During the 1970s Warntz ran the Harvard Laboratory for Computer Graphics and Spatial Analysis, and with an extremely creative group of mathematicians and programmers, took the first steps towards creating modern GIS. Warntz, and other members of the lab, produced a series of important but now largely forgotten papers called, The Harvard Papers in Theoretical Geography.
The Theseus Ship paradox, which Plato writes about in the Meno, is a problem that brings up the question of whether an object which has had all its component parts replaced remains fundamentally the same object. In geography this is important because we are constantly seeing the objects of our study change and evolve. This kind of change takes place in the material, conceptual and bio-geographical sense continuously in the real world, and is the source of many of the philosophical and analytical conundrums that confront the foundations of GIS.