*Today’s post is the seventh in a year-long series called,”Computing Space,” which highlights new mapping technologies and new areas for cartographic innovation, along with stories of the lives and work of many of the mostly unknown cartographers, geographers, mathematicians, computer scientists, designers and architects who both now, and in the past, have had a hand in the development of computer cartography and its applications.*

It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely minimal, while the true determinations appear in the form of axioms. –Bernhard Riemann, On the hypotheses which lie at the base of geometry (1873)

Roughly, by a complex system I mean one made up of a large number of parts that interact in a non-simple way. In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical way, but in the important pragmatic sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole.

–Herbert Simon, Architecture of Complexity (1962)

In time those unconscionable maps no longer satisfied, and the Cartographer’s Guild struck a map of the empire whose size was that of the empire, and which coincided point by point with it…

–Borges, On Exactitude in Science (1946)

They are the names of cats. When you glanced at the title of this post I know you asked yourself the question, “Magpie and Possum, what does a bird and marsupial have to do with geographic space?” They really are however, the names of two cats, specifically the philosopher David Lewis’ cats, and as you will see, they and their parts have everything to do with geography, cartography and the ontology of the objects that make up our perceptions, both linguistic and geometrical, of our lived geographic space. We will talk more about cats later.

The visualization of geographic spaces through cartographic images took on its modern form in the *Geographia*, a book of the second century by the geographer Claudius Ptolemy. “The introduction of a grid of longitude and latitude lines as the basis for determining the location and translating the sphere into a two-dimensional map is”, writes Denis Cosgrove, “by far the most significant feature of modern cartography[1].”

For centuries this sustained the making and design of maps on paper and on parchment, the type of cartography that readers here would easily recognize. Traditional cartographic images like these however, no matter their deep historical roots, cannot represent the immaterial, mobile and transient dimensions of the wide range geographic phenomena that we see before us in our contemporary and globally defined world. Our world, unlike that of Mercator, is a world where, in many cases, distance is not the most important spatial relationship. These kinds of visualizations, as useful as they might have been in the past, simplify the complex relationship between places even when trying to describe these spaces and phenomena thematically[2]. The distances and the outlines of places, cities, and features shown on these traditional maps miss the increasingly interconnected and interdependent relationships that exist on a global scale. These kinds of connections, brought about through the increasing globalization of economic phenomena, and the ease of information flow across networks, are only partially related to geographical distance—New York is in this sense more related to London than it is to, say, Eire, Pennsylvania.

*Motto for Modern Cartography*: Contemporary mapping and geographic analysis now purposefully distort the traditional Ptolemaic forms of visualization and represent topological rather than topographical spaces[3]. Future cartography will focus on *processes *not structures…all geographic systems from rivers to glaciers, from the development of cities to agricultural land use, from natural environments to transportation and internet networks* reject any notion of equilibrium…there is nothing static in the world.
*

Connection is now more important for geographers than distance and our journey here, like my own, towards these kinds of ideas starts with topology, the mathematics of connectedness, and with an accidental event many years ago in a bookstore. I can still remember the day that I became interested in the underlying form, the logic, connectedness and structure of space. I was in graduate school, a physics student, so it was not geographic space that first held my interest, but rather, space in the abstract and purely mathematical sense. A new book called *300 Years of Gravitation* had just come out celebrating the publication of Newton’s *Principia Mathematica*. I picked up the book while in the Princeton University Bookstore, and when I opened it, all that I can recall seeing is a series of illustrations that showed something called Everett Branching Space-time. I had never seen diagrams like this before; sheets of space and time folding away from one another and branching out into every logically possible world.

This branching and sheeting 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.

Although I never looked into the Everett diagrams any further, topology came to be my main subject of study and over the next three years I devoured the classic works on the subject. In particular, Felix Hausdorff’s *Set Theory* and Nicolas Bourbaki’s *General Topology* became my close friends, as I started spending more time in mathematics than in physics departments[4]. Topology, especially in its algebraic form, would later become quite important to me in my theoretical and historical work on geography and can be formally defined as the study of qualitative properties of certain objects, called topological spaces that are invariant under a certain kinds of transformations. I have written about the intersection of some of the classic theorems like the Brouwer Fixed Point and the Borsuk-Ulam Theorem[5] and geographical problems[6]. Most of these have to do with applications in which the properties that we are interested in are invariant under a certain kind of equivalences, called homeomorphisms. To put it quite simply, topology is the study of the continuity and connectivity of continuous fields, networks and discrete objects. This is the guise under which topology appears in the geographic and cartographic sciences and is really the subject of this post.

A few years later the same sort of questions brought up by the Everett diagrams and the many-worlds interpretation came back into my thinking through a seminar with the philosopher David Lewis, whose cats we began with, and whose lectures concentrated on the theory of modal realism. Modal realism also deals with questions surrounding the plurality of worlds, although from a much more logical and less mathematical perspective. Of all the professors that I have had the pleasure of learning from it was Lewis who had the most profound effect on me. Lewis was a mathematical and philosophical renegade, and although firmly part of academia, was always putting forth new ideas that pushed the limits. Today I still re-read his four books, *On the Plurality of Worlds*, *Counterfactuals*, *Convention*, *Parts of Classes* and his essays almost yearly, as the depth of their insights is boundless[7].

Lewis, in his seminal book *Parts of Classes* talks a great deal about his cats and the possible fusions of individual cats, classes and subclasses of the set of all the cats in the world, and the parts of cats, like tails, heads and down to their molecules, all to develop a mereological foundation for set theory. Mereology is the logical formalization of the parts and wholes of objects, classes and concepts, and the prying open of its foundations and their relationship to cartography will take up most of the theoretical ramblings on mapmaking that follow.

Getting back to our cats, Magpie and Possum, it was Lewis, who first introduced me to the subject of mereology. Mereology in Lewis’ sense is simply a formal and mathematical theory that tries to discern general principles regarding the relationships of parts and wholes that provide the starting point for most of pure mathematics. When first approaching these ideas this level of abstraction may seem to have little relationship to location and cartography, but in fact it is critically important to the foundations of modern geography and mapmaking, as these activities take place within a mereotopological formal framework. In fact it is here that modern cartographers and users of GIS find themselves, whether they realize it or not, compelled to draw conceptual parallels and distinctions with ideas that appeared in the literature on spatial and temporal reasoning, and it is here that mapmaking meets analytical philosophy, cognitive science, pure mathematics and modal logic.

We begin with the abstract objects of mathematical set theory. It is worth quoting extensively from Lewis’s *Parts of Classes* and from his rather comical discussion of his cats because it points out interesting differences in the concepts of parts and wholes found in mereology and those that can be gleamed from the more abstract notions of set theory. This difference will be extremely important in what follows as we delve deeper into some of the ontological and epistemological changes brought about by the computerization of the study of geographic and cartographic spaces since the late 1960s.

Lewis begins,

Mereology is the theory of the relation of part to whole, and kindred notions. One of these kindred notions is that of mereological fusion, or sum: the whole composed of some given parts.

The fusion of all cats is that large, scattered chunk of cat-stuff which is composed of all the cats there are, and nothing else. It has all cats as parts. There are other things that have all cats as parts. But the cat-fusion is the least such thing: it is included as a part in any other one.

It does have other parts too: all cat-parts are parts of it, for instance cat-whiskers, cat-cells, cat-quarks. For parthood is transitive; whatever is part of a cat is thereby part of a part of the cat-fusion, and so must be part of the cat-fusion.

The cat-fusion has still other parts. We count it as part of itself: an improper part, a part identical to the whole. But also it has plenty of proper parts— parts not identical to the whole—besides the cats and the cat parts already mentioned. Lesser fusions of cats, for instance the fusion of my two cats Magpie and Possum, are proper parts of the grand fusion of all cats. Fusions of cat-parts are parts of it too, for instance the fusion of Possum’s paws plus Magpie’s whiskers, or the fusions of all cat-tails wherever they may be. Fusions of several cats plus several cat-parts are parts of it. And yet the cat-fusion is made of nothing but cats, in this sense: it has no part that is entirely distinct from each and every cat. Rather, every part of it overlaps some cat.

In geographical analysis and modern cartography we are worried about keeping track of different kinds of objects and fields that inhabit our lived space and that have different dimensional and topological structure. We are concerned about objects that divide themselves, come together and have parts that change (in other words most of the physical and social world). So for example we have zero-dimensional points for cities, one-dimensional lines for roads and other networks, two-dimensional polygons for regions and territories, three-dimensional spaces for the earth itself, and four-dimensional space-time structures when we add in temporality. We want to know things about these topological spaces like, for example, if a forest is contained in a spatial region, how many trees are part of it, and perhaps if its spatial extent causes the forest to overlap some other region of space or some other fusion or conglomeration of real or abstract of geographic objects. Some of these things are parts of others, fusions or sums in the sense of Lewis’ grand cat fusion. A fusion of all the cities in the state of New York not only contains the cities but also the roads, the telephone lines, and depending on how we define “city” my bicycle in my apartment on the Upper West Side. All of this must somehow be understood and be captured in our algorithms and data structures.

These are very different questions than those brought up by less formal methods of defining spaces and those based on things like set theory, which have typically been used to conceptualize objects. In his book *Parts of Classes*, Lewis draws this distinction out quite explicitly by introducing the notion of subclass.

We could equivalently define the cat-fusion as the thing that overlaps all and only those things that overlap some cat. Since all and only overlappers of cats are overlappers of cat-parts, the fusion of all cats is the same as the fusion of all cat-parts. It is also the fusion of all cat-molecules, the fusion of all cat-particles, and the fusion of all things that are either cat-front-halves or cat-back-halves. And since all and only overlappers of cats are overlappers of cat-fusions, the fusion of all cats is the same fusion of all cat-fusions.

Theclassof all cats is something else. It has all and only cats as members. It has no other members. Cat-parts such as whiskers or cells or quarks are parts of members of it, but they are not themselves members of it, because they are not whole cats. Cat-parts are indeed members of the class of all cat-parts, but that is a different class. Fusions of several cats are fusions of members of the class of all cats, but again they are not themselves members of it. They are members of the class of cat fusions, but again that is a different class.

To keep track of all this in a computer is quite different from the drawing of lines and the printing of topographical information in traditional cartography, where boundaries and regions are less fuzzy. In a computational framework we need to understand the deep mathematical things about the world’s spatial structure, like the overlap of roads with regions, the temporal extent of events and how boundaries are spatially related to the regions they bound. It might have been Nick Chrisman, in his insightful article from 1978 called, ‘Concepts of Space as a Guide to Cartographic Data Structures,’ who first pointed out the deep conceptual and epistemological connections between the mathematical structure of space and the data structures of computer mapping[8].

As cartographer’s begin to add more complex forms of data into their analysis, the lines between big data visualization and traditional cartography has become blurred. Objects that are better looked at with partial differential equations and that are temporally changing, along with non-discrete objects, like continuous fields come into the mereological mix. When built into in spatial analysis platforms, like in a GIS, these kinds fusions have both profound mathematical and philosophical import that is just beginning to be sorted out by people like myself who are interested in such questions.

My thinking about geographical mereology, and a great place to start for anyone interested in approaching these theoretical conundrums, began with the writings of Roberto Casti and Achilles Varzi, a professor of philosophy at Columbia University. Varzi’s two books, *Holes and other Superficialities* and *Parts and Places: the structures of spatial representation* treat in great detail the various formal and mathematical systems of mereology and topology. In his books and papers Varzi gives various formalizations of spatial mereology and discusses logical structure of each of these systems and their philosophical import. One of the deepest conclusions of mereology is that there can be, just like in geometry, a large group of different axiom systems, which are all consistent with each other. The important thing proposed by Varzi, at least from a geographic perspective is that none of them is alone strong enough axiomatically and logically to contain a full theory of spatial objects. It is here that topology comes into play and provides the link to a full formal theory of GIS, something he and others call mereotopology[9].

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 (what is it for something to be) 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[10].

Classical mereology takes as its foundation the fact that any theory of spatial representation, geographic or otherwise, must consider the structure of the entities that inhabit the space. For geographers this is a critical point as one could doubt the usefulness of representing space either logically or mathematically independent of the entities that are in it.

The second part is that of connection and continuity. How are the various types of entities connected to the space they inhabit and to each other? This is the territory of topology, which studies the mathematics of connection. We can begin asking mathematical and ontological questions like, “What is the difference between the cup and the glass spread all over the floor after we drop it?” These kinds of questions are important to geographers, as they give us insight into how events are connected physically, and how they retain or loose their material identity over time.

Geographical space is much different than any abstract notion of *bare space,* which is infinitely extended and is an infinitely divisible continuum. This conception of space has proved enormously fruitful in providing a framework for the physical sciences. Geographical space on the other hand is different and is divided into regions and populated with many kinds of objects. Regions themselves can be treated as abstract objects and their existence is entirely dependent on the existence of other more concrete objects. As soon as space is partitioned like this the mathematical continuum loses its purity but acquires a degree of richness that is represented by sets of relations where space itself is composed of discrete and identifiable objects. It is these complex conceptual connections that mereotopology sets out to explore.

Formal mereotopological treatments, which really form the basic ontology of today’s GIS, have their roots in some of the debates surrounding the axiomatic foundations of geometry that took place at the turn of the last century. In the midst of all the problems stemming from the discovery and applications of non-Euclidean geometry, logicians like Alfred Tarski and Stanislaw Lesniewski, wrote classic papers on the mereology of objects. One of Lesniewski’s in particular, “Foundations of the General Theory of Sets,” from 1916[11], started me thinking about some of the problems of integrating time in geographic analysis as a continuum rather than a series of discrete values. A kinematics of cartography is the way I like to look at it[12].

Both the structure of space, and its temporal properties, are profound areas of current research at the most abstract levels of GIS theory and for researchers in artificial intelligence, and we will explore them more deeply in the coming weeks. Next time in the * Computing Space* series we will look at some of the computational tools currently being developed at the forefront of artificial intelligence and how they that might help us grasp the complexity of geographic space. We will discuss Neural Networks, Monte Carlo Simulations, the recent victory of Google’s DeepMind program Alpahgo, and what the ancient game of go has to teach us about spatial searching and cartography.

[1] Denis Cosgrove, *Geography and Vision: seeing, imagining and representing the world*, (London: IB Taurus, 2008)

[2] Ola Söderström, “How Images Assemble the World,” *New Geographies* 4: Scales of the Earth (2011) 113-120

[3] Peter Taylor, “A new mapping of the world for a new millennium,” *The Geographical Journal* 167 (2001) 213-222

[4] Felix Hausdorff, *Set Theory*, translated by John Aumann (New York: Chelsea Publishing, 1962) and Nicolas Bourbaki, *General Topology*, (Paris: Hermann, 1962).

[5] For more on the relationship of the Brouwer Fixed Point Theorem and Borsuk-Ulam see Jiri Matousek, *Using the Borsuk-Ulam Theorem: lectures on topological methods in combinatorics and geometry* (Berlin: Springer-Verlag, 2003).

[6] See my article, “How to Map a Sandwich: Surfaces, Topological Existence Theorems and the Changing Nature of Modern Thematic Cartography, 1966-1972,” *Coordinates* 7 (2009) 1-19 http://www.stonybrook.edu/libmap/coordinates/seriesa/no7/a7.htm

[7] For more on David Lewis see Daniel Nolan, *David Lewis* (Chesham, UK: Acumen Publishing, 2005)

[8] Nick Chrisman, “Concepts of Space as a guide to cartographic data structures,” *Harvard Papers on Geographic Information Systems* 6 (1978).

[9] Roberto Casati and Achille Varzi, “Ontological Tools for Geographic Representation,” in *Formal Ontology in Information Systems*, edited by N. Guarino (Amsterdam, IOS Press, 1998) 77-95.

[10] The modern form of mereology takes its beginnings from Edmund Husserl’s Third Logical Investigation. See Edmund Husserl, *Logical Investigations*, translated by J.N. Findlay (New York: Routledge, 2001)

[11] Stanislaw Lesniewski, *Collected Works*, volume 1 (Dordrecht: Kluwer Academic Publishers, 1991) 129-173.

[12] The integration of time into GIS has spawned a large body of literature both from a technical and philosophic perspective. For good summaries see the work of Antony Galton particularly his *Qualitative Spatial Change* (Oxford: Oxford University Press, 2000) and “Space, Time, and the Representation of Geographical Reality,” *Topi: an international review of philosophy* 20 (2001) 173-187.