Azimuthal Theories

Published March 15, 2019.

The New Yorker map of the United States went on to reflect the New Yorker’s notions in the very sizes the several areas appeared to have on the map. Our Mercator world maps have done much the same thing for our Western world image. Some say the Mercator world map is so popular because it shows the correct angles essential for navigation (even though its shapes are almost as badly distorted as its areas). But if you use a map not for navigating but for placing and comparing at a glance different parts of the world, shapes and areas are more important than angles. Moreover, areas are more important than shapes, because they have cultural implications. What is objectionablea bout the Mercatorw orld map in fact is not that it distorts the shape of North America, nor even that it shows Greenland so large-our conception of Greenland makes little difference. Rather, it is that it shows India so small, and Indonesia, and all Africa. (I call such a world map the “Jim-Crow projection” because it shows Europe as larger than Africa.) (Hodgson 1963, 228)

Prelude: Perception

Here is an obvious fact: train tracks running off into the distance appear to converge but do not in fact converge. Why is this? One might start talking about perspective. Think of a canvas placed between the tracks and an observer. Draw lines from her to the ends of a close tie and the angle is quite obtuse; draw again to the ends of a distant tie and the angle is acute. In the former case the lines pierce the canvas at distant points and in the latter in tight pairs. This is just perspective, one might conclude, the sort of thing Euclid taught us.

But we do not see with a canvas placed at a close remove: we see with our eyes. Fair enough, but the principle of the thing holds. Perhaps we should instead imagine a sphere encasing the head instead (our eyes are more or less spheres) and end up with a non-Euclidean geometry of the visible. (As, apparently, Thomas Reid did, cf. Daniels 1972.) This model too oversimplifies—it does not, for example, give us an account of binocular vision—though perhaps there is something to it: the points at a distance strike closer together on the retina and so appear closer.

Why, however, should two points striking closer on the retina mean that they appear closer? There is surely no deep necessity here: there could be a creature with eyes like ours who had isometric vision. For them, train tracks would not diminish in apparent size as they got further away. This child’s drawing perspective is odd, but is not inconceivable. Moreover, as Hatfield (2011) informs us, modern perceptual psychology suggests that apparent space is not so neatly mathematical as the above models would have it (cf. Wagner et al. 2018). The apparent size of objects diminishes “less rapidly than would be predicted by the diminution in visual angle angle” (47). Our visual experience is more like a Roman fresco than a Rennaissance painting.

The early days of architectural painting.
The early days of architectural painting.

If geometry, then, does not determine apparent diminuition of size with distance, we should wonder what does. In recent, unpublished work Hatfield suggests a broadly evolutionary explanation. The closeness of distant points on the retina is relevant, but relevant only insofar as with the compressed retinal distance comes decreased available visual information. Think of our fantastical isometric friend again. Suppose she is looking at, say, the Bayeux tapestry rolled out lengthwise away from her. It would appear to her as a perfect, quite long rectangle. The farther regions, further up in her visual fields, however, would likely be blurred or pixelated as fewer pieces of information (fewer stimulations of rods and cones) had to take up equivalent apparent area.

The human visual system, then, diminishes far off objects to maintain a relative constancy of information-per-area, this being a useful feature. This system has other attractions: allowing us to estimate relative size and distant and orient ourselves in our environment. As a consequence, Hatfield suggests, we should think of this phenomenon not as misrepresenting the size of distance objects, but presenting objects in a way that makes salient certain features of the objects and their arrangements. (That is, though our vision does indeed represent distant objects with less apparent size it does not represent them as having less physical size.)


Something similar clearly holds for maps. In the cartographic case, the organizing scheme of the representation owes itself to the intentions of a designer, rather than to evolution, but the goal is the same: to compress a complex three-dimensional arrangement in a way that makes it easy to extract useful information. The Mercator projection, for instance, preserves angles but, like our vision, has objects closer to the equator diminish in apparent size. Again, it is not that the Mercator projection represents, say, that Greenland is larger than Brazil, but that it does represent Greenland more prominently than it does Brazil.

The Mercator Projection, credit Wikimedia.
The Mercator Projection, credit Wikimedia.

Here, I am disagreeing verbally with Hodgson (1963). I do not think that the Mercator projection “shows Europe as larger than Africa” (228).1 But I take his point as valuable and want to reframe it slightly. He offers an explanation of the popularity of the map, despite its potentially misleading foreshortening, in that “it flatters our egos” (229). Importantly, he understands this foreshortening as, in effect, offering less information, a lower resolution, for regions nearer the equator.

For Hodgson, this serves as a metaphor for Western, European historical understanding. As he elaborates (229-231), in this understanding there is the mainstream, those civilization and historical forces leading to us, now, and everything else. The former, Greece, Rome, etc., are foregrounded, elaborated in some detail, while everything else recedes into the distance. The possibility that the foregoing discussion highlights is that such a narrative could be spun so as to mislead without misrepresenting. By virtue of providing comparatively more information to a certain region, one inflates its apparent size relative to the unloved countries.


What I am interested in here are the practical, non-alethic, non-epistemic features of representation. What are those? Well, consider a belief, say my belief that my colleagues respect me. We can evaluate this belief for its truth or falsity, or its justification: do I know this, or is it merely a pleasant delusion. One can also think about what this belief does for me. Certainly, I would be much more on edge around my colleagues were I to lack this belief. This practical evaluation can come apart from the others. It can be good that I believe something (good for me, even) despite its being false or my belief’s being unjustified.

A useful point made here by Joshua Smart is that while we evaluate beliefs or claims for truth or falsity we do not evaluate maps or pictures for simple, binary truth or falsity. Rather, a picture encompasses many claims and can be accurate or inaccurate. And while the accuracy of a picture depends on the truth of its associated claims, it is not a simple function of them. Consider a map that swapped the places of Africa and South America: it could represent very many things truly (for example, the internal geography of the continents), but would as a whole be highly inaccurate. Correspondingly, a map could be riddled with small errors and still be on the whole accurate. As Smart further notes, whole theories or systems of beliefs are like pictures in this regard; they can be accurate or inaccurate to greater or lesser degrees and this is not determined simply by how many of their constituent claims or beliefs are true or false.

Besides, but not unrelated to, the accuracy of a belief-system is its practical value, what it does for or to the believer. For example, our mental cartography hugely exaggerates the apparent area of our homes and places of work. I have beliefs, thousands of them, likely, about the contents of my bookshelf and very few at all about those in the nearest library (though it has many more books).2 This egocentricity is obvious and, as I take the previous discussion to forebear, not necessarily a misrepresentation.


These remarks were prompted by Mills (2007)’s “White Ignorance.” The concern there is to begin to chart the dynamics of white ignorance (roughly: false and missing beliefs resulting from and germane to social racial injustices). The essay canvases a litany of failures, from ignorances that are purely epistemic (where one fails to properly weigh the evidence in one’s possession) to morally-based ignorance (where one fails to collect or even destroys evidence). I am just dipping my toe into the whole literature, so I do not have anything useful to say in general. Here I wanted to make the small point that there seems to be a relevant species of doxastic failure (call them failures of emphasis) where ones beliefs do not necessarily misrepresent or omit crucial details but simply misguide.


Daniels, Norman. 1972. “Thomas Reid’s Discovery of a Non-Euclidean Geometry.” Philosophy of Science 39 (2). Philosophy of Science Association:219–34.

Hatfield, Gary. 2011. “Philosophy of Perception and the Phenomenology of Visual Space.” Philosophic Exchange 42 (1):3.

Hodgson, Marshall. 1963. “The Interrelations of Societies in History.” Comparative Studies in Society and History 5 (2). Cambridge University Press:227–50.

Mills, Charles. 2007. “White Ignorance.” Race and Epistemologies of Ignorance 247. State University of New York Press Albany:26–31.

Wagner, Mark, Gary Hatfield, Kelly Cassese, and Alexis Makwinski. 2018. “Differentiating Between Affine and Perspective-Based Models for the Geometry of Visual Space Based on Judgments of the Interior Angles of Squares.” Vision 2 (2). Multidisciplinary Digital Publishing Institute:22.

  1. While a geographic amateur might, on first glance, conclude from the map that Europe is larger than Africa, this is to misread what the map represents rather than to read, correctly, its misrepresentation.

  2. An aspect of beliefs that is not well captured by this metaphor is not merely their presence but their salience. I might waste my time memorizing the catalog of my local library to the point that I had as many beliefs about their books as a librarian working there. Still, her beliefs about those books would be more salient to her (come up to awareness and use more frequently) than my corresponding beliefs are to me.