VII

Notes on a Family of Edges

Attempting in his Philosophical Investigations to survey the variety of language-games that loosely belong together, Ludwig Wittgenstein spoke of their “family resemblances.” With this metaphor in mind, let us make a severely brief conspectus of several critical edge-types, where different mental and material purposes at once appear, most of them of great importance for humans. Among properties of all edges two stand out: an edge connects separate points (the vertices of Euler’s formula) and an edge may also separate two areas (such as the inside of one country from the inside of another country). The former draws a line connecting two endpoints, as when one state boundary starts (say, on its Western side and where that boundary ends), while the latter divides areas projecting away from the line itself and linking that face with the “solid angle” or corner of each figure. (Face, surface, and side here mean virtually the same thing.) This fact indicates that the concept of shape develops eventually, after PoincarĂ© created an algebraic approach, into a numerical method of measuring a point-source topology. What began in Euler’s two great discoveries modulates into extremely sophisticated mathematics in our own time.
Lines such as longitude and latitude drawn on the surface of a sphere are entirely abstracted from the sailor’s material ocean, if as in earlier times they were used for navigation. Using a sextant, by relating such lines to speed and distance, the sailor could identify his location on a blank spherical expanse. Edges in that case are geometrical boundaries of sections of the manifold. Nowadays we use satellites to broadcast such positions to oceangoing ships, and the advances still indicate edges of a larger space, reducing it to specific place. If we reduce the scale of our concern, we find the neurophysiologist examining and describing edges as minute thresholds, boundaries or junctures between different sides of membranes, as more generally with the study of cellular biology, where only by using microscopic instruments can we begin to see what actually goes on in the network of bodily interconnections (as for instance the functions of the system of blood circulation). In essence the edge articulates the system, and such concerns are topological, because the networks and channels inside the body—we might call them the biological wiring—effectively give that body its functioning shape and overall bodily character.
The list of familiar and unfamiliar edges might go on forever, it would seem, when looking under “edge” in the Oxford English Dictionary, but I am choosing examples that involve our planet, exemplary types for the reader to ponder beyond mere mention. For us every edge is a paradox, since Earth ideally can have no edges. The scientific and instrumental advances we have recently made have reduced the electronic world to a shrinkable sphere, but these measures of the edgeless round Earth have the further effect of electronically collapsing our ideas of time, a collapse psychologically critical, because our world-time seems to be going faster and faster. Time, not space, is the primary scene of mental endeavor. Yet edges still exist on spatial manifolds or surfaces.

Horizon

It might seem strange, and I hope it does, to begin with an edge that is not quite an edge, namely the horizon. Usually the word horizon means “as far as the eye can see,” while more precisely it means the lateral line beyond which we cannot see some region of the planet, a line blocking that region beyond our visual perception. The visual array of things we see before us depends on many factors, but shows in every case that the horizon is ultimately a perceptual phenomenon. The original Greek for horizon, kyklos, means a separating or dividing circle, where sky meets earth, but much of the time on dry land we cannot perceive a distinct horizon-line of the planet, because buildings, trees, and irregular outcroppings of Earth obscure the dividing line itself. But what happens if we move toward that line? It moves away from us, like a friendly animal backing off when we try to pet it. Horizon, as its original Greek name suggests, is undoubtedly a most unusual scrolling line or boundary.
On the plane of movement the horizon moves always ahead of us, when we reach out to its edge, as if attempting to see beyond it. Oceangoing sailors knew this. The open sea they were accustomed to call “the offing.” For them the horizon is a perceptual edge that keeps moving away from the observer, as the observer travels toward it. Owing to the curvature of Earth, one might ask whether horizon is not a prototype of all abstracted mathematical edges, which exist only as conceptual forms, floating before our minds in a free and open space? They fix only a relative edge.
It pays to consider the phenomenon of horizon at this point. Joseph Conrad’s sea stories do suggest something of that experience. When Captain MacWhirr in Typhoon scans the horizon from the Nanshan’s bridge, he cannot be sure which quarter of the Northeast to examine closely, so he mutters in his usual flat tone, “There’s some dirty weather knocking about . . . Go and look at the glass.” The falling barometer warns him of something wrong about the oncoming weather, a storm “knocking about” somewhere beyond the wide extent of a vague, remote horizontal line, for no angle of horizontal vision from the bridge will tell the Captain and the mate where the dirty weather is coming from. A blank horizon gives them no clear bearings on the typhoon, a difficulty increased because the cyclonic storm itself will be slowly rotating on its way toward them. The unknown is about to test their courage to the utmost. In another story, based partly on his own experience, Conrad describes a similar if yet more metaphysical horizon. The Shadow Line narrates a vicious retreating game for a young captain whose whole life shrinks to its own cyclical Vichian periodicity. He must cross a double horizon, interior as much as exterior, which Conrad calls the line between periods of unexamined youth and bitter experience, assumed success and threatening doom; throughout that tale Conrad not unexpectedly alludes to a Shakespearean drama of indecision. In his memoir, The Mirror of the Sea, Conrad writes of such uncertainty wrought by Nature, “in this ceaseless rush of shadows and shades, that, like the fantastic forms of clouds cast darkly upon the waters on a windy day, fly past us to fall headlong below the hard edge of an implacable horizon,” and yet, on another occasion—the story being his novel Victory, where Axel Heyst is enamored of islands—there is a supreme quietude pervading the edge of vision: “The islands are very quiet. One sees them lying about, clothed in their dark garments of leaves, in a great hush of silver and azure, where the sea meets the sky in a ring of magic stillness.” The contrasts could not be more extreme, while with another writer, whom Conrad greatly admired, Stephen Crane, the horizon may be dangerously occluded. In a way we are lost when the horizon disappears and of the four men in his story, “The Open Boat,” he writes the immortal words: “None of them knew the color of the sky.” Like Conrad, Crane is fascinated by the dividing horizon, in his poems as well as his fiction. The Red Badge of Courage dramatically shows the horizon disappearing into a general blur, for the chaos of war virtually blocks any sense of planetary edge, which leaves only the foreshortened immediate space extending just beyond the young soldier’s fragile body—a body denied horizon.
Perceptually, despite its geometry, the horizon is an edge without a final or fixed occluding limit, as if its Galilean relativity never ceases, since, as it stretches sidelong at some distance along our line of sight, its position depends mainly on how high above the surface of the earth or sea we are standing. Distance perception is here a relative matter. On a high hill, like Robinson Crusoe scanning the horizon, we might see as far as fifteen or more miles, on a sand dune less than four miles, and in Conrad’s sailing days at sea sailors climbed to the topmost yards, or a “crow’s nest,” to extend their lines of sight. Thus we may think of our being blocked by the horizon, and yet this word creates an illusion, for in fact we are simply not able to follow our earth’s curvature over its edge, which forever shifts ahead of us, as we move toward it. We could better say that we are seduced by the horizon. There seems to be an elastic, even vague topological limit to our phrase, “as far as the eye can see.”
The same situation pertains on land. If we drive across wide flat country (in Kansas, let’s say), the boundary circle of our perception yields the same kind of rolling visual terminus, and like the fleeing runner it never quite stands still. Like the hero of Hitchcock’s North by Northwest, perceptually as well as physically we are moving toward or away from a visual circle that retreats from our advance into a dangerous nowhere, a field of stubble, all around us, which we recognize as an ominously boundless perimeter. This condition of being unprotected is life-threatening. As already stated, the Romans therefore called the sacred bounds of a city the mundus, or world, to inscribe their protected and often walled civic environment. Even to speak of the mundus is to define a kind of dream.
Perception and belief play such a large part in orienting the science of whatever we think we know that we may forget our perspectival situation as thinkers. My favorite example is the peasant poet John Clare, who, when still a child, went off running one day in search of the horizon, before he got lost and gave up, in tears. Clare was to become a brilliant self-taught ornithologist and perhaps he thought he might like a bird outrun the distance between himself and Earth’s edgeless edge. As a grown man he would speak of his territory, the familiar fens and marshes surrounding the village of Helpston in early nineteenth-century England, as “my knowledge.”
Horizon and its recession are the outer (and probably also the inner) edge of all life and human perception. The world we personally know limits our horizon, and events like sunrise and sunset are prior mindsets built upon our unaided macroscopic perceptions. We are not so much wrong scientifically to think that the sun is rising, as we are imaginatively right; we can agree with the Teacher of Ecclesiastes or his followers, and we understand Ernest Hemingway, whose “Big Two-Hearted River” describes a flowing that forever quests its horizon. Gnomic authors like these, however different their worlds, equally raise the question of the gap between scientific knowledge located “out there” and personal knowledge located “in here”—in our conscious and finally determining relation to horizon. Sunrise and sunset ground John Clare’s intense yet unanalyzed “knowledge,” and no matter how abstract we may wish to be, we still do not normally picture our planet as rotating toward the sun’s rays, as dawn approaches, or slowly falling away at the end of the day.
There is always a gap between immediate and analyzed experience, and the case of the horizon is a reminder that much of our belief in decisive habitual judgments is potentially illusory, and here the horizon marks a strange dividing line both on and off earth, for it seems halfway to accept the idealized fact that because spheres have no edges, the horizon promises a different world beyond, a moveable feast defining the earth’s ambiguously retreating boundaries. Longitude and latitude share this function, for they mark the conceptual edges of temporal and spatial navigation over a surface-manifold, where no final terminus can be found. Horizon is finally an edge that says, “sometimes edges simultaneously exist and do not exist, except in mind’s decision, a decision also making and marking an edge, but always within our thinking.” Wisely, the mathematician and philosopher, Gilles Chatelet, was wont to speak of the horizon as a “precious hinge” allowing a sparkle to remain in our metaphysical rise beyond common, domesticated rationality.
Beckoning to our senses, Chatelet’s horizon, he says, “impregnates” our sphere with the signs and far reaching signals of life.

Perceptual Edges

On the largest scale we may imagine a theoretical wholeness to our planet and then onward to outer space, and given great powers of topological imagination, it remains for the artist such as Dante, poet of The Divine Comedy, to convey something of the detailed texture of which our lines and points are schematic indices. Dante is famous for his precise visual images, in experience those Proustian memories that define where and who we are.
Using geometric projections our sphere may schematically be mapped onto flat surfaces, whose projected lines of demarcation include idealized edges such as the longitude. Paradoxically such edges are powerfully functional concepts or ideal mathematical objects, whereby particular locations on our planet or in astronomical space may be indicated on a map. We know that such mapping represents a real world, even when the map produces ideal marks on what might seem a striking cartoon. The map sketches the scale model of a larger whole, a principle satirized in Lewis Carroll’s Silvia and Bruno, where the map is the same size as the countryside it represents and the farmers must lift it up, to see where they are going. In short, maps and much of topology train the explorer to notice scales and relative (though not absolute) sizes, so that that maps reach far beyond the confines of our own planet. Similarly, the great epics of literature explore an ever widening sense of cosmic and terrestrial space, at times ending in mystery, as with the metaphysics of light in authors like Dante and Milton. In more realistic literature, as in philosophy and physics, imagery is the handmaiden of perception. In this tradition Parmenides wrote his philosophic poem on knowledge and deceptive appearances, around which Plato wrote intricate dialogues, establishing the Theory of Ideas.
Maps are one thing, of course, and our actual experience is another. An image may not render what Henry James ironically called “the real thing”—indeed we may feel that images attempt to deceive us or try more seriously to depict a private perception, similar to Wittgenstein’s “private language.” Images leave us with a broader question, how can we know or judge if our imagery is true to some perceived fact? Can we train our perceptions? Perhaps we are born with them and need only stimulate their first infantile use.
Perceptual theory is the field where all such matters are central concerns. In trying to understand the origins of our ideas of order, we study the directly apprehensible boundaries between and within all separable objects, boundaries we find we can distinctly observe, a view that earlier psychophysicists did not fully develop. As a great nineteenth-century forerunner, Helmholtz had argued that regarding what he called the facts of perception, “in the real there must exist some or other relations, or complexes of relations, which specify at what point in space the object appears to us. Of their nature we know nothing,” and he concluded that “spatially different perceptions presuppose a difference in factors.” This psychophysical observation sought direct experience of the environment, but about the real perceptual “conditions” Helmholtz added the crucial reminder, “of their nature we know nothing.” Such questions were not to remain unexamined in psychophysical research, however, and in particular we will shortly consider an example of experimental advance. Helmholtz was expecting to find the changing variables such as the sizes of objects, to which our perceptual apparatus would respond and neurally decode into useful information.
This new approach was to lead inevitably into a development of Gestalt thinking. It aspired to a directly realistic view of perceptual theory, a method not always reducible to mathematical expression. Gestalt approaches relate and analyze the particular image of a particular shape or texture, to a larger surrounding background. This might lead to remote questions in philosophy, such as: Is it possible that before there is concept, there is percept? After all, we commonly train viewers to notice certain features of the environment, perhaps by attending to the exact quality of their sensations, in which case concept precedes percept. Much of Merleau-Ponty’s interest in art is meant to question the concept / percept relation, wedded as he was to analyzing the embodiment of thoughts. Our thoughts, on his view, are always to a degree psychosomatic. On the other hand, our perceptions, while often flawed as measuring instruments, belong to us the perceivers, and with that fact we are forced to begin. We are forced by nature to rely on information immediately available to us, as we move about in our world. Every particular image, every particular shape, interacts with and within a larger surrounding background—the assumption developed in Gestalt psychology.
Suppose then, for the moment, that we return to our exemplary case in visual experience, the horizon. “As far as the eye can see” might be the oldest phrase for an immediate perceptual datum, the reality of an edge, and we acquire skills in noting and using edges of all sorts, as we have said. We ask: how exactly does the visual system reach out toward the far horizon, processing the array of objects and textural arrays accumulated between that psychophysical boundary and the viewer? Children begin to acquire hand-eye skills at about the age of six months, largely because, becoming more mobile, they begin to see and feel how objects are initially defined by their edges. They move, to see. In principle, one might as well place the neurophysiology of such depth perception and depth constancy on a theoretical and experimental shelf—brain research locates different regions focused on specific mental tasks. How can all this work? As if our senses were designed correctly for the task of decoding where edges start and end, their adequacy in processing the visual is not something that can be replicated easily under controlled laboratory conditions. The “conditions” Helmholtz referred to appear in all areas of a surrounding world, and their belonging participates in the ecology of perception.
To pursue such a vision was the destiny of one remarkable perceptionist. Over many years James J. Gibson at Cornell University pioneered study of what he early called The Perception of the Visual World, the title of his first book. Significantly, Gibson was fascinated by works of visual art such as paintings and film, and especially with the former he was aware of their inaccuracies and their accidental flourishes. With his colleagues, among them the distinguished expert in child development studies, his wife Eleanor Gibson, he focused attention on perceptual powers exercised in the real world surrounding us, our living ecological surround. Most of his final thoughts from this perspective are presented in his late text, The Ecological Approach to Visual Perception. A legendary story goes that he became interested in the “visual cliff” phenomenon, whereby virtual neonates would instinctively back away from the appearance of a dangerous declivity (reproduced in the lab at Morrill Hall), because one of his own children had crawled away from the edge of the Grand Canyon, having looked over the precipitous drop. Something at the South Rim was triggering an aversive response. He asked how perceiving the visual array of any perceived texture was objectively related to movement, movement being the chief non-laboratory condition f...