Squaring the Cartographic Circle

Matthew Edney

I just received a really silly book that I am compelled to discuss:

Groarke, Paul Vincent. 2018. How to Draw a Simple Map of the Earth: A Philosophical Examination of the Unmappability Thesis. np: rebelliON booKS.

It’s not very long, with just 106 small pages, excluding an appendix that largely repeats the text, and including many pages of diagrams. It’s self-published so I didn’t know quite what to expect. I began skimming through and found myself fascinated by the author’s attempted critique and remedy. I just had to write up my immediate reactions. The book is deeply misguided and, in the process, it reveals the functioning of some deeply rooted convictions of the ideal of cartography. Let me first explain the argument and then I will turn to the ideal. (I place page references in {..}.)

A Supposed Refutation of the Unmappability Thesis

I am not going to go into great detail, as the book does not deserve that.

In style, this is very much a work that hopes that if an assertion is repeated enough times, readers will be persuaded. The core assertion, restated over and over, is that there is a thing that Groarke calls the “unmappability thesis”:

The UNMAPPABILITY THESIS holds that it is not possible to map the surface of a spheroid accurately in 2-dimensions. {12}

Groarke insists that this is a basic tenet of “the literature” of mathematics and cartography but, as he gives no citations to any of the literature, it is impossible to know what portions of the literature of cartography (at least) he has read. He also insists (repeatedly) that it is the job of philosophy to evaluate such basic axioms, not through the application of formal logic but rather through some kind of empirical evaluation. He also sustains a difference between idealism and pragmatism, he being a pragmatist {29}.

From his pragmatic position, Groarke argues against the “divisibility fallacy.” This is the fallacy that underpins Zeno’s paradoxes: you know, the ones where it’s impossible for an action to be completed, say for a runner to finish a race, because the actor must first undertake half of the action, which first requires half of that first half to be completed, and so on. For mapping, this is tantamount to claiming that slicing a curve into ever smaller portions will always produce a curve. (Groarke calls this the idealist position.) But just as the runner has a length of stride that pragmatically overcomes the ever smaller distances left to be travelled, so too “at some point” the subdivided curve “yields a line without a discernible curve.” The “pragmatic conclusion” is that “such a line” is “hypothetically—i.e., theoretically—curved, but physically—i.e., empirically—straight.” Groarke’s system thus depends upon the willingness of the individual human to see no difference between an arc and its chord {29}.

Groarke undermines his claim to philosophical rigor by admitting in several places {e.g., 19, 23} that he had thought, as soon as he had learned of the unmappability thesis, apparently almost viscerally, that the thesis is in fact incorrect. He also states incorrectly that the unmappability thesis is only “a conjecture, which derives its authority from the brute fact that no one has been able to draw an accurate 2-dimensional map of the earth’s surface” {20, also 30, 31}. By such an “accurate” map, Groarke means a map of the entire earth that is at once conformal and equal area.

Quick aside, why the “thesis” is not simple conjecture: in any map projection, one can define the scale factors at any point (the rate at which scale is changing) along both the meridian (fm) and the parallel (fp). For a map to be conformal (shape preserving), the ratio between the two scale factors at every location on the map must be the same: fm ÷ fp = 1. For a map to be equal-area, the product of the two scale factors anywhere on the map is constant: fm × fp = 1. The only way for both properties to be valid for every point on a map is if fm = fp = 1 uniformally for all points, i.e., scale is always constant, which means that the original surface is completely flat.

Most of Groarke’s book is an explanation of how to make a world map that is both conformal and equal area. And he does so by, supposedly, mapping each small place (not point! {34}) as flat (i.e., the plane tangent to the earth at any point). Each place is supposedly separately projected, and he refers accordingly to the work as the ubiquitous projection, justified in his introduction by references to some philosophical principle of ubiquity. He fails to see the mathematical impossibility of his claim because he rejects the use of differential calculus as it “resists philosophical scrutiny” and is too “abstract” {37}; he even recasts fundamental mathematical issues that are manageable through the application of the calculus to “linguistic issues” {34}.

But Groarke’s actual process is actually quite different and quite idiosyncratic. He takes small portions of an equal-area sinusoidal projection of the world and somehow reconfigures them to be like a conformal Mercator projection, and then reassembles all the small portions as a map “without distortion”:

Groearke’s fig. 5, showing the reassembly of Eurasia. I really don't understand the placement of the "skeleton"; why do the lines (meridians?) not align as they seem to be on the diagrams provided for other continental outlines. Groarke does not provide enough explanatory detail for his work to be undertaken and verified by others. Reproduced here under academic fair use.]

Groearke’s fig. 5, showing the reassembly of Eurasia. I really don't understand the placement of the "skeleton"; why do the lines (meridians?) not align as they seem to be on the diagrams provided for other continental outlines. Groarke does not provide enough explanatory detail for his work to be undertaken and verified by others. Reproduced here under academic fair use.]

Groarke’s terminology is confusing to say the least, the diagrams are very small and difficult to read, so it is not exactly clear just how he undertakes the reconfiguration and reassembly. And I completely lost the thread in the section entitled, “Squaring the Map” {72ff}. But it is clear that his methodology is to take small sections of only the continental coastlines from one projection and reconfigure them on another, as in the image above. He claims it is an intuitive process: “The map essentially put itself together” {48}.

There is no demonstration that the process actually preserves equal-areaness and gains conformality, beyond his assertions. Moreover, Groarke admits that his map of continental coastlines would indeed reveal “interruptions and conflations” if the continental interiors and seas were also mapped within and between the coastlines, but he blithely states that “those kinds of inaccuracies can clearly be corrected by projecting the areas in which they occur, independently, and reassembling the map” {58}. Overall, Groarke’s claim for having made a distortion-free map is dramatically overblown. He has done nothing of the sort.

Some of the Ideal’s Preconceptions, Revealed

The ideal of cartography comprises many preconceptions that determine how scholars and the public understand maps and mapping. Not least is the conception that there is a universal and singular endeavor of cartography. The preconceptions are ubiquitous and largely taken for granted. Many scholars are breaking away from those preconceptions, but they nonetheless remain bound to others. What intrigues me about Groarke’s work is that he baldly states some of the ideal’s constituent convictions: in the context of his entirely misguided exercise, those convictions readily appear to be the nonsense that they are.

Groarke is especially indebted to the ideal’s observational preconception, which among other convictions holds that all maps are necessarily grounded in observation and measurement and that the default map, indeed the first map, is a fine-resolution plan of the environment as experienced by the individual. Thus:

In many ways, we experience the surface of the earth as a flat planar surface. It is accordingly unclear why it is impossible to map it in two dimensions. {19}

A further element of that preconception is the conflation of the act of observation to survey a map, and the act of looking at a map:

Philosophically, a map generally has a point of origin—essentially a starting point—which represents the place [we] stand, notionally, in looking at a map. … The telling observation here is that the maps are accurate at the point or line of origin. This is enough, in itself, to defeat the unmappability thesis. {35}

At the same time, the ontological preconception holds that just one geometry underpins all maps, which is necessarily the same as the geometry of the world. Groarke refers to the manner in which one can consider the world to be a series of places joined by vectors, like a survey plan, and the goal of the map is to recreate those places and vectors {40, 73}. Any and every map:

The properties of the local maps—here distributed along the shorelines—remains the same, whether we are dealing with a local map or a global map. … Philosophically, this is what we want in a map. {59}

This conviction is evident in Groarke’s refusal to worry about how to define when the arc and the chord of the curve combine: “There is no need to set out exact conditions under which the curve is no longer discernible, which will vary with the circumstances” {29}. As an idealized process, cartography is independent of scale. Groarke can therefore treat any portion of the world to be a flat place regardless of size: the survey of a place, or a very long chunk of a continental coastline (that’s already been projected). What Groarke takes to be a well thought-out “philosophical pragmatism” is just another unexamined set of beliefs about the supposed nature of cartography.

These points are rarely so openly stated as in Groarke’s book, but they are nonetheless common in the cartographic literature. “Maps” are held to be a universal and unambiguous category of phenomena, definable by certain criteria: based on measurement and observation, having both ontological and pictorial relationships to the world (even if those two relationships can be contradictory), etcetera.

Groarke’s entire project rotates around one of the several paradoxes innate to the ideal. (I discuss the background in chapter 5 of my forthcoming Cartography: The Ideal and Its History.) I refer specifically to the ambiguity of map scale, which is at once a feature of all maps even if it is highly variable on some maps. The idea of map scale as a universal attribute of all maps developed only in the nineteenth century. In 1802–3, Pierre-Alexandre-Joseph Allent, creator of the numerical ratio (what in English has come to be called the “representative fraction”), admitted that the ratio of map distance to ground distance was perfect for maps and plans of precise areas where the world might as well be treated as flat, and that the same ratio was a permissible approximation for more regional maps of territory (at scales in the order of 1:50,000–1:100,000). But, Allent averred, the ratio was quite meaningless for maps of extensive regions and of the whole world, because scale varied across the surface of such maps. Yet by 1900, even the grandest professors of geography thought that every map must have the representative fraction. In the post-war era, academic cartographers saw the representative fraction as the single metric that defines the very nature of a map.

Map scale can be understood as a defining characteristic of “the map” only if all maps are indeed grounded in detailed observation and measurement, in plane surveys where the ratio is valid. Thus Groarke’s insistence, indeed his visceral certainty, that it is should be possible to map three dimensions onto two “without distortion”:

I was troubled by the sweeping nature of the blanket statement that it was not possible to map the surface of a sphere accurately in 2 dimensions without distortion, which goes against the evidence of our senses. It was easy to see the distortion on global maps, but the distortion disappeared as you reduced the size of the area that was mapped. I wondered whether there was any meaningful distortion in maps of local areas. {30}

Conversely, working from the local to the global, if maps are accurate “at their point of origin” then they should be accurate all over {35}.

What Groarke makes explicit is a position left implicit by most map scholars: all maps must be the same. Faced with a profound difference, Groarke tries to rescue the ideal by demonstrating the error of the cartographers. He should have turned his philosophical gaze instead on the entire ideal. Breaking with the ideal requires us to celebrate rather than explain away the differences between maps. There is more than one way to conceptualize the world: early peoples as well as modern peoples made world maps as well as local maps, and the two are quite distinct. (There are many other kinds of maps.) There are a whole series of maps of places, properties, and landscapes that rely, even today, on plane geometry in ways that are fundamentally distinct to the cosmographical geometry of latitude and longitude that underpins regional and world mapping. Only within the high idealizations of the twentieth century have projective geometries been deployed to yoke the other geometries together, to give the impression that there is just one geometry to cartography.

Groarke’s insistence on the unitary nature of “the map” demonstrates that cartography is a myth. There are multiple mappings and the definition of “map” is utterly ambiguous.

update: I modified some of the language, nothing much (25 May and 29 November 2018).