Dimensionality in Nature and Culture
Poems are made by fools like me, but only an algorithm can make a tree.
— California Institute of Technology, rest room graffiti
In putting the languagedominated theory of metaphor behind us, I would suggest that what remains of Fernandez’s quality space is a model of culture as semiospace that must be rather like the analytical concepts of “phase space” and “Hilbert space” that figure importantly in physics and, particularly for the latter concept of Hilbert space, in quantum mechanics.^{5}
The first lesson I would like to draw from Roger Penrose’s erudite presentation of these truly mindboggling concepts is that Fernandez’s terse assertion that culture “is a quality space of `n’ dimensions or continua” now acquires new meaning and substance. The fact that mathematical creatures like phase space and Hilbert space exist and, moreover, have great explanatory power for what goes on in the physical, “real” world gives a cachet to the notion of a “quality space” of culture. Without these stunning models from mathematical physics, we would, I think, be inclined to take Fernandez’s work on metaphor metaphorically, that is, to fall back into the old habit of anthropologists of regarding the elegant geometrical compositions of structural analysts like Leach and LéviStrauss as mere teaching tools, helpful illustrations of what a languagebound intelligence is up to. Penrose’s exposition offers a promising alternative, for while the “spaces” he describes are highly abstract (far more so than the pedestrian structural models anthropologists devise), I’m sure he would claim that they are nonetheless “real” in the sense of describing the physical properties of matter.^{6}
The outstanding contribution Penrose makes to the present discussion may be drawn from his words of encouragement to his nonmathematical readers, whom he knew would be doing mental cartwheels trying to visualize phase space: Don’t even try; it’s impossible to visualize and wouldn’t help much if you could. A space that one cannot visualize — now that is a tall order for most of us, anthropologists included, who, even if we’ve had umpteen years of schooling, still cling to a core of commonsense beliefs about what we will (and can) see when we open our eyes in the morning. It seems more a Zen exercise than a science project, and for that very reason is helpful in disrupting our habitual pattern of thought just enough to let the seed of doubt, and imagination, slip in: suppose there are “spaces” that we can’t visualize . . . and suppose that culture is such a space. . . . That is what I propose. The quality space of culture, or semiospace, is dimensional, is a “world” (hence the tremendously suggestive power of the term, “Umwelt”, the “worldaround”). The fact that we don’t get up in the morning, go to the window, and see vistas of semiotic dimensions stretching off into the distance does not mean that they don’t exist. It takes something like Penrose leading us on a forced march through some of the thorn bush of contemporary mathematics to alert us to the fact that dimensionality, our old clearasday, straightasanarrow acquaintance from high school geometry classes is in fact an elusive, difficult, and tremendously complex subject.
I will discuss aspects of that complexity presently, but would first like to note that it is probably our commonsense beliefs, augmented by a little high school geometry, that have made us illdisposed toward those who would introduce mathematics into a discussion of social relations or cultural values. For there is a dominant belief (and here the myth of America with its ambivalencefraught stereotype of science reasserts itself) that the truths of mathematics apply to a pristine, cutanddried, artificial world of straight lines, right angles, and perfect circles but do not fit the convoluted, emotional, real world of people’s lives. Our collective psyche again dips into the deep well of American myth, and draws forth the sentiment, both cherished and crippling, that the rational world of scientists and mathematicians is cold and even cruel — like the scientists and mathematicians themselves — and cannot describe or explain the emotion and subtlety of human experience.^{7}
But what is this supposed “complexity of dimensions” I insisted on earlier? Perhaps theoretical physicists and cosmologists have abandoned their rulers and compasses, but why should that ameliorate the hopelessly sterile attempt to draw lines and boxes around people’s lives, the attempt so dreaded by humanistic anthropologists? Whatever the highpowered physical theory behind them, aren’t lines just lines and boxes just boxes? Well, no. There is a major problem with this retreat into common sense, which hits us in the face as we slog through the Penrose passages above: the world served up to us by our common sense (and its fellow travelers, elementary math and science courses) is simply not realistic. It is a rough gloss of how things actually work, enough to get by with on a daytoday basis, but one that lets us down hard from time to time.
Concerning the dimensionality of lines, for example: We all know that a straight line is the shortest distance between two points, and that if you have two straight lines going in different directions in something called a “plane” you have a twodimensional, “flat” figure. Add another straight line travelling away from the first two, out of the “plane,” and you have a threedimensional solid. The problems — big problems — start dropping out of the sky right away and landing on top of this tidy view. For starters, it’s hard, or actually impossible, to find a straight line. The ruler, or “straight edge,” on your desk lets you down right away because of the little detail that the space on which you inscribe the line (the paper, blackboard or football field) happens to be curved, so that if you extend your line a bit, say a few billion light years, it actually curves back on itself or describes a weird, saddleshaped hyperbola. For that matter, your ruler itself, just lying there in your hand taking up space, is also curved, but just by, as Maxwell Smart says, “that much.” You might object that this is something of a cosmic quibble; you aren’t measuring light years in drafting a floor plan or even composing a structural model in anthropology, and your trusty ruler does a pretty good job. True enough, but notice that even this tiny discrepancy has upset the applepie, smilebutton reality of common sense: things are supposed to be what they seem, not almost what they seem. What has happened here is what I described earlier as a critical attribute of myth: it introduces a vertigo, a tilt to everyday life that, be it ever so slight, still serves to announce its presence, to crack open the gates of The Dreamtime.
But worse is to follow. Let us say you opt to give up the high ground of cosmic distances, and let Einstein have his curved space, bent light rays, elastic rulers, and clocks that speed up or slow down depending on who is reading them. You take your nice, solid, nonelastic ruler and retreat into the practical world, where the cosmic scale does not alter the results you expect to find. Once there, however, you find there is very little to measure with your “straight edge” except other straight edges that have been put there earlier by someone who made them with his “straight edge.”
Looking around you in this practical, real world and discounting all the “artificial” straight edges of buildings, sheets of paper, pieces of furniture, tennis courts, and even the video borders on your word processing screen, you see the silhouettes of hills and mountains, the outlines of clouds, the path of a river, the branches of a tree, the veins of a leaf, and so on. “Real” lines are almost invariably curved, and therein lies the rub: how will you measure them with your ruler? Also, a quick inspection of some of these “real” lines shows you that there is great variation in their amount of curvature or twistedness. For example, there is the rare, straightasanarrow line described by — what else? — an arrow.^{8} While the people in white coats at the National Bureau of Standards wouldn’t be happy with its “straightness,” preferring their lasers instead, the lowly arrow serves as a good, practical index of what a straight line should look like. Then there are gently curving lines that still have an obvious directionality about them: the graceful track a skier makes down a gentle slope, for example. But then things, as always, get complicated and messy: many, many “lines” we encounter wander so much that they lose all sense of directionality and begin to fill up a lot of the area traversed, like an ant on a sidewalk or a fiveyear old crossing the yard on an errand, for example (see Figure 3.1). A very interesting variety of this kind of complicated line is the great number of lines that describe boundaries: the silhouette of a mountain range that marks the boundary of earth and sky, or a coastline that separates land and sea.
When you proceed to measure one of these complicated lines with your ruler, some very peculiar, noncommonsensical factors rapidly come into play. Take the path of the ant on your sidewalk, for example. You’ve watched the little creature for a few minutes, then gone into the house and come back with your trusty camcorder, which is a deluxe model equipped with a superfast tape speed for ultraslowmotion effects (you use it to study and improve your tennis serve). After taping the electrifying event of the ant crossing your sidewalk for a few minutes (your neighbors are now looking at you like Richard Dreyfuss’s neighbors in Close Encounters, when he was tearing up his nice suburban yard to use in his sculpture of the aliens’ mountain rendezvous), you go back inside and put the tape in your VCR. Taking several pieces of transparency the size of your TV screen and a finetipped felt pen, you sit down beside the set with the VCR remote. You place a transparency over the screen and, for the first run, set the equipment to play at normal, reallife speed. When the ant appears on the screen, you position the tip of your pen over its image and trace its path as it wanders across the screen. Because you are new to this ant path tracing business, on this first effort you miss many of its twists and turns, glossing over them with gently curving lines that lack the squiggles of the ant’s motion.
To improve your chart, on successive runs with fresh pieces of transparency you increasingly slow the motion, until you reach the molassesworld of Sunday football’s “instant replay,” and even resort to the “coach’s clicker” freezeframe button on your remote, so you can catch the ant in midstride. Even though the transparencies are now piling up, you are still not content: your TV has a “zoom” function, which you crank up to the max, so that the ant now looks like a Volkswagen driven by a lunatic, crazily veering across an endless patch of concrete. At this highpowered setting, with the set zoomed to the max and the tape at ultra slow motion, you make one more chart, feverishly hitting the freezeframe control so you can follow the ant’s every motion with your pen (see Figure 3.2).
Exhausted at the end of this session, you spread out your transparencies (there are now a couple of dozen) and proceed to inspect them, trusty ruler in hand. The differences between early and late charts are considerable: gentle, meandering lines gradually become fRenétic, staticlike squiggles. While you are contemplating this disparate collection, your longsuffering spouse or roommate comes in and says, “Well, now that you’ve spent the day on this stupid project, tell me: How far did the ant travel? How long is your line?” And these are tough questions to answer. If you sit down with your ruler and the stack of transparencies and begin to measure the first chart, the one with the gentle curves, you can probably come up with a fair approximation by, say, marking off oneinch gradations on the antline and then adding these up. Of course, there will be plenty of places where the antline curves within the oneinch distance, but you’ve already discounted Einstein’s curved space, so why not the ant’s curved path?
In this way, you can come up with one answer to your companion’s question. But if the two of you are standing there surveying the charts spread out around you, it will be clear to both of you that you have only flirted with an answer to your companion’s question. Suppose you take your final, ultrafinegrain transparency and begin to measure it. It will be clear from the outset that you cannot get away with marking this line off in oneinch segments, since whole mountain ranges of squiggles will lie within some of those segments. So you break out a draftsman’s ruler marked off in sixteenthinch gradations (and a good magnifying glass) and begin the tedious chore of measurement. Even at this scale, there will still be plenty of squiggles between the marks whose lengths are glossed over by the measurement procedure. Still, you come up with a result. The problem is, the length of the line you now announce to your companion is different from the first measurement you gave: it is considerably longer, because of all those squiggles.
Which is the better measurement? How far did the ant really travel? Again, not easy questions. If you’re after an ant’s eye view of its travels, perhaps your last, finegrained chart is a more accurate representation. But note the extremely important consideration that this detailed chart does not represent what you, or any other human standing beside you, saw as you watched the ant’s motions on the sidewalk. The better representation of the human’s eye view is your first, and most impressionistic or sensory chart with its gentle, meandering line. After all, there is nothing “natural” or authoritative about trying to represent the ant’s eye view of its travels (which, if you could actually bring that off would “look” absolutely otherworldly and unintelligible to those other, human eyes). The crew cuts at the Bureau of Standards, for instance, might find even the ant’s scale of things much too indiscriminate for their purposes, and insist on trotting out their lasers and micrometers in order to nail down the length of the squiggles in Angstrom units. That procedure would yield something on the scale of a microbe’s eye view (if the microbe happened to possess organs) as it clung, say, to the ant’s left antenna. A measurement at this microbial scale would yield a much larger result (in fact, very much larger since it begins to approach infinity) than obtained from your initial, carefree foray into what now turns out to be an impenetrable thicket of ant path measurements.
Dismayingly, the result of this last measurement is substantially different from your first results obtained with an ordinary ruler, since lots of those previously glossedover squiggles now get figured into the final result. And, depending on whose equipment you choose to use, subsequent measurements you might make would differ from all these results. The unwelcome conclusion to be drawn from this little brush with an obsession neurosis (sometimes called “laboratory science”) is that the “same” line, which you actually drew with your very own felt pen, has different lengths depending on how you go about measuring it.
Benoit Mandelbrot, mathematician and cult figure, makes these same points about the length of a line, more eloquently and with far deeper mathematical understanding than I, in his famous essay, “How Long is the Coast of Britain?” (see his The Fractal Geometry of Nature). Basically, Mandelbrot’s answer to his own question is: as long as you want to make it. You can fly along the coast in an airplane and record the air miles traveled; you can drive along it on a coastal highway and take odometer readings; you can walk along it on a foot trail or even jump from rock to rock at the shoreline; or you can set a mouse — or our ant friend — to traversing it pebble by pebble. The coastline of England, like the lines we have been examining, has an infinite number of lengths. Mandelbrot’s surprising finding is not just mental sleightofhand done with smoke and mirrors to distract us from the hardedged reality of things; it is a mathematically correct description of reality — as real as it gets, the Platonists among us might say. Our difficulty in reaching the same conclusion, our reluctance to take this paradoxical stuff seriously, is due to the fact that the smoke and mirrors involved here are frantically deployed, not by the evil scientists and mathematicians of The Dreamtime, but by our own common sense in a hopeless attempt to cling to a simple vision of a world in which an ant crawling on a sidewalk covers a certain, measurable distance, in which a line has a length. Myth and reality: Is there any way to unscramble these categories we habitually separate? If so, which is the source of delusion?
Mandelbrot has even more disturbing news in store for us than this business of the indeterminacy of a line, and it bears directly on the matter of dimensionality. Even though the inoffensive line has become a treacherous serpent in our grasp, we might hope to salvage something of our common sense (and our faith in school day memories of Euclid) by supposing that however long lines may be, they still serve to mark off and define the onetwothree dimensions of line, plane, and solid which we see, like the nose on our face, right before our very eyes, stretching out around us. We might grant Penrose his infinite dimensions of phase space and Hilbert space in the same, savethewomenandchildrenfirst spirit in which we granted Einstein his curved space and elastic rulers, while insisting that for all practical purposes (a phrase already sounding hollow as a gourd) we see and walk around in a world of three clearcut dimensions. Yet Mandelbrot has denied us even that practical refuge, in a stunning theoretical development that has transformed contemporary mathematics and, amazingly, inspired a large cult following of spinoffs of his work. His thesis, like that in the essay on the coastline of England, is brilliantly simple: There are no clearcut boundary markers that separate lines, planes, and solids from one another as elementary dimensions of space; instead there are any number of transitional or fractional dimensions (hence the popular term, “fractal”) that connect the three classical dimensions and that are fundamental to a description of physical objects.
Examples may be found in our nowtreasured archive of ant path charts. Take one of the finegrained, zoomed closeup charts in which your felt pen line wanders all over the place and fills up a lot of the chart (see Figure 3.2). Tack this chart to the wall and inspect it at a “normal” reading distance of eighteen inches or so. Although your line is definitely more squiggly and messy than those found in your old high school geometry textbook, it is still, just as Euclid said, a onedimensional line meandering over the twodimensional plane of the transparency sheet. But now step backward a few feet and examine the chart again. Much of the squiggly detail has begun to blur together because the resolution power of your onboard optical equipment has been pushed past its limit, so that some individual line segments now form clots or islands surrounded by relatively open areas transected by lines that still retain their individual identity. In Mandelbrot’s terms, this simple change of perspective has altered the geometry of the chart from that of a onedimensional line on an open plane to that of an object of a fractional dimension, say 1.2 or 1.3, which is transitional between line and plane.
This highly original perspective on geometry also allows our experiment to proceed in the opposite direction: instead of stepping back from the chart, you zoom in on it, probably with the help of a magnifying glass or lowpowered microscope. Now your felt pen lines grow and expand to fill much of the visual field; their onedimensionality is again seen to be an ephemeral, contingent attribute and you are back to contemplating an object of dimension 1.2 or 1.3. Dimensionality does not lose its mathematical stature as a fundamental property of things, but it does lose its onetwothree, pigeonhole determinacy. The scale or region of space involved in the observation of a particular object now becomes a primary criterion in describing its physical properties. Mandelbrot, in essence, chucks out the ruler and compass we were fretting about earlier, and installs in their place the zoom lens, particularly in its modern form of adjustable coordinates on the screen of a computer monitor (as hundreds of thousands of Mandelbrot set trekkies will attest).
Mandelbrot’s arguments, when allied with the works of chaos and complexity theory partly inspired by his pioneering discoveries, lead to conclusions as unsettling, and as productive, as those Penrose highlighted in the field of theoretical physics. If a line has no fixed length and if the dimensionality of a figure depends on the scale selected for its observation, it becomes exceedingly difficult for anyone — mathematician, anthropologist, politician, evangelist — to maintain that the elementary truths of existence, in this case the physical properties of objects as apprehended by the human mind, are fixed, unambiguous propositions that affirm the central tenet: things are a certain way. It is this tenet, applied to the areas of social relations and cultural values, that has been responsible for the strident denial of a Dreamtime world of mythic virtuality, and for a very great many horrible things done to people by other people in the name of the way, the truth, and the light.
Theoretical physics as presented by Penrose acts conjointly with Mandelbrot’s mathematics in eroding this cherished but vicious dogma of certainty and fixedness. The “nuts and bolts” of existence turn out to be true phantoms: elusive “virtual particles” that may be here, there, or both places at once, and that inhabit, together with their shadowy companions, the dynamic, vectordriven, many dimensional worlds of “phase space” and “Hilbert space,” worlds, as Penrose says, that we have no hope of ever “seeing.” Contemporary mathematics and physics thus seem to have developed rigorous models of physical reality that correspond on crucial points, and that together paint a very different picture of the “real world” from that usually presented by “normal” folks (and by notsonormal folks, like anthropologists).
From my vantage point as a distinct outsider to fractal geometry and mathematical physics (their anthropologist, if you will), I find it astonishing that these fields have for so long pursued topics of the sort most anthropologists and other social thinkers have not only ignored but shrilly rejected: the virtuality or multiple possibilities of experience; the coexistence of incompatible, contradictory states of being; the complexity and importance of dimensionality in a cultural system; and the indispensable role of the knower (the sentient presence) in fixing the properties of the known. Quantum mechanics and relativistic cosmology, as I noted earlier, have been around for most of the century. And, while fractal geometry and the hacker cult of the Mandelbrot set are quite recent, nonEuclidean geometry and nonlinear mathematics, developed by giants in the discipline — Gauss, Bolyai, Riemann, Cantor — are even older than the field of quantum mechanics.
The truly impressive, and bewildering, thing to me in the peculiar conjunction of the success of contemporary physical theory and the widespread rejection of its findings by laymen and social thinkers alike is that the theory, while smacking of “weirdness” all the way, miraculously serves up an image of the world that is far more familiar, and really far more comprehensible, than that of earlier theories of both physical and social reality. Classical mechanics presented us with a clockwork world in which everything followed in a perfectly determinate manner from what came before, provided only that you confined your attention to actual clocks, projectiles, ball bearings on inclined planes, or, best of all, the distant stars and planets. If, however, you foolishly let your attention wander to the swirl of cream in your cup of coffee, the movement of clouds across the sky, the traffic clogging the freeway around you, or the tropical storm bearing down on your town, then classical mechanics, along with William Burroughs’s Nova Mob (colloquially known as “civilization”), abruptly check out of the Mind Motel.^{9}
When we stop to look at the things around us, to smell the roses,^{10} we see a world of ebb and flow, of twists and turns, of pushes and pulls, of change that is sometimes gradual and subtle and sometimes bewilderingly dramatic, but overall, a world of great diversity, vitality, and, as basic a property as all the rest, a world of much confusion. To say that it is otherwise, that the subtlety and turbulence of everyday life may be harnessed by the deterministic shackles of a science that is not really science (a “science” that flows from the dark forces of the Dreamtime) is, perhaps, to express a wistful longing for certainty in an uncertain life, or, all too often, to foist off a lie concocted by those who do or should know better, who find it convenient to bend the mercurial human spirit of their subjects, employees, followers, or students to the yoke of an order that can be inscribed in the report cards, spreadsheets, and law books of a blighted civilization.
The clockwork world of classical mechanics parallels that of Euclidean geometry and linear equations, and, just as it has been superseded by a world of quantum mechanics and relativity, so the old mathematics has yielded, decades ago for the most part, to the inroads of nonEuclidean geometries (with an emphasis on the plural), set theory, and nonlinear equations. Here, too, as with theoretical physics, the seeming offthewall weirdness of contemporary mathematics actually describes a much more livable, believable world than that promulgated by our old high school textbooks. The great problem with the public perception of mathematics in the United States is the very widespread and accurate sentiment that the math most of us learned in school has almost no relevance to our daily lives. Euclid’s compulsively tidy, axiomatic world of points, lines, triangles, and circles is hardly to be found when we raise our eyes from the text and confront the things around us, whose shapes are of the meandering, fragmented, complex kind we encountered in our ant path exercise. The things in our lives, like our lives themselves, are not measurable or determinate in the Euclidian or commonsense meaning of those terms. Remarkably, it turns out not to matter all that much if we acknowledge that we cannot fix the length of our ant path, or even of the coastline of England (except, perhaps, to tourism promoters who advertise “x hundreds of miles of lovely, pebblestrewn, cloudshrouded beaches”). And whether we want to believe that we can come up with some kind of satisfying answer to vexing little problems of the ant path variety, we know we cannot answer apparently straightforward questions of the most pressing urgency, like “When will the Big One level L. A. (and issue in the postapocalyptic world of Blade Runner)?” or “When will a hurricane finish leveling Miami?”
The tremendous appeal of chaos and complexity theory lies in its restoration of the familiar world around us as the object of scientific inquiry: for the first time those of us who are not mathematicians (which, after all, is almost all of us) have an immediate grasp of what the subject matter is and why it is important, if not how to go about modeling it in the difficult nonlinear equations of complexity theory. Earthquakes, weather patterns (including hurricanes), traffic flow, the shapes of plants and insects, even what the stock market is up to, all become the subject matter of what James Gleick, in his popular work on chaos theory, has fittingly called a “new science.” These subjects replace the hopelessly artificial, impoverished ones of lines, planes, triangles, and circles, which we all somehow knew, as we suffered through Mr. Dork’s geometry class, were way off base.
The most important aspect of this revolution in mathematics and science and, particularly, of its impact on the Dreamtime world of a global, Americanized psyche, is that it installs unpredictability and undecidability as distinguishing features of the physical world, which must be accommodated rather than arbitrarily expelled. It is first necessary to know the inherent limitations of earthquake or hurricane prediction before those dramatic, turbulent events can be put into a framework of “reallife” phenomena like daily weather patterns, the smell of coffee (or a rose), or the neural events associated with your reading these words. Where earlier mathematics carved out a small, precise area for itself and pretended its deductive power in that area could be extended to the actual physical phenomena of daily experience, contemporary mathematics abandons the deterministic posture as regards those events and considers them on their own terms, in all their complexity and changefulness.
As I have noted, most of this is old news in the mathematical community, as evidenced in a work by Morris Kline, Mathematics: A Cultural Approach, published over thirty years ago and detailing developments much earlier still.
The very fact that there can be geometries other than Euclid’s, that one can formulate axioms fundamentally different from Euclid’s and prove theorems, was in itself a remarkable discovery. The concept of geometry was considerably broadened and suggested that mathematics might be something more than the study of the implications of the selfevident truths about number and geometrical figures. However, the very existence of these new geometries caused mathematicians to take up a deeper and more disturbing question, one which had already been raised by Gauss. Could any one of these new geometries be applied? Could the axioms and theorems fit physical space and perhaps even prove more accurate than Euclidean geometry? Why should one continue to believe that physical space was necessarily Euclidean?
At first blush the idea that either of these strange geometries [Gauss’s and Riemann’s] could possibly supersede Euclidean geometry seems absurd. That Euclidean geometry is the geometry of physical space, that it is the truth about space is so ingrained in people’s minds that any contrary thoughts are rejected. The mathematician Georg Cantor spoke of a law of conservation of ignorance. A false conclusion once arrived at is not easily dislodged. And the less it is understood, the more tenaciously is it held. In fact, for a long time nonEuclidean geometry was regarded as a logical curiosity. Its existence could not be denied, but mathematicians maintained that the real geometry, the geometry of the physical world, was Euclidean. They refused to take seriously the thought that any other geometry could be applied. However, they ultimately realized that their insistence on Euclidean geometry was merely a habit of thought and not at all a necessary belief. Those few who failed to see this were shocked into the realization when the theory of relativity [with its curved space] actually made use of nonEuclidean geometry. . . .
Perhaps the greatest import of nonEuclidean geometry is the insight it offers into the workings of the human mind. No episode of history is more instructive. The evaluation of mathematics as a body of truths, which obtained prior to nonEuclidean geometry, was accepted at face value by every thinking being for 2000 years, in fact, practically throughout the entire existence of Western culture. This view, of course, proved to be wrong. We see therefore, on the one hand, how powerless the mind is to recognize the assumptions it makes. It would be more appropriate to say of man that he is surest of what he believes, than to claim that he believes what is sure. Apparently we should constantly reexamine our firmest convictions, for these are most likely to be suspect. They mark our limitations rather than our positive accomplishments. On the other hand, nonEuclidean geometry also shows the heights to which the human mind can rise. In pursuing the concept of a new geometry, it defied intuition, common sense, experience, and the most firmly entrenched philosophical doctrines just to see what reasoning would produce. (563^{_}577)
Kline’s balanced, not to say charitable, account of the late development and reluctant acceptance of a new paradigm for conceptualizing space is a fitting point to conclude, on a similarly balanced note, that the seemingly obvious ideas we have of dimensionality, of the lines and boxes in our lives, warrant close scrutiny outside the hermetic realms of contemporary mathematics and physics. In simply going about our daily lives, we are not immune to the déracinement, or sense of uprootedness, conventional mathematicians experienced toward the end of the nineteenth century when they began to confront the stunning implications of a nonEuclidean geometry that could no longer be treated as a mental diversion, but had to be accepted as a representation, however bizarre, of the physical world. And our daily lives, as I argue throughout this work, wander, like our ant path, in and out of the shallow depth of field of common sense, in and out of an enveloping Dreamtime consciousness. The critical question before us now is how this new understanding of dimensionality may be applied to the cultural world around us, and particularly to an anthropology of the cultural productions of the American Dreamtime, in the form of popular movies.
If mathematicians and physicists have taken two thousand years to come up with the concepts embodied in nonEuclidean geometries and quantum mechanics, then anthropologists may perhaps be forgiven for having spent their first meager century engaged in the sorts of butterflycollecting activities Edmund Leach criticized three decades ago. Nevertheless, as Kline and numerous others have pointed out, once the spurious certainties of Newtonian mechanics and Euclidean geometry had been unmasked (a process substantially completed by the 1930s), there followed a diffuse but widespread acknowledgement among intellectuals that the world had shifted underfoot, that the solid ground of science and moral order had given way to a morass of unknowns and, worse, unknowables. What was happening in anthropology, that upstart new “science of humanity,” while this major transformation in worldview was underway? Did anthropologists, like Newton’s pygmies standing on the shoulders of giants, absorb the new intellectual climate purportedly inspired by the mathematicians and physicists and proceed to build it into their tentative theories of culture (literally from the ground up, since they were just starting work on their own disciplinary edifice)?
Actually, no. You see, a funny thing happened on the way to anthropology.
This is not the place to attempt to chart the parade of isms and social movements that accompanied and embodied the fundamental change of perspective ushered in by such unlikely revolutionaries as Gauss, Riemann, Einstein, and Heisenberg. It suffices to note that the grand themes, the basic principles and problems of aesthetics, of political and moral discourse, of philosophical debate and literary creation have changed in ways that would have been unthinkable to educated persons of the nineteenth century. The place of anthropology and the other social sciences in this time of ferment and change is one of the major paradoxes of contemporary intellectual history, a monstrous curiosity that leads repeatedly into scandal. For the role anthropology has played in pursuing the implications the new perspective of Einstein, Riemann, Schrödinger, and company holds for the cultural world has been essentially that of Uncle Remus’s tarbaby: “De tarbaby, he jus’ sit dere, and he don’ say nuthin’!”
In truth, that assessment is too charitable, for anthropologists had a great deal to say about the nature of culture, both before and after World War II, but almost nothing they said indicated an understanding of the tremendous dynamism and multiplicity of their subject. In a staggering absurdity, while the world was coming apart at the seams and would never be the same again, pre and early postwar sociologists and anthropologists labored mightily and produced a grand theoretical scheme, “structural functionalism,” that proclaimed, according to various versions, that societies were like organisms, possessing a morphology (structure) of parts that all nicely worked together (function), or were “integrated systems” whose institutional subsystems articulated to form cohesive, stable wholes.^{11}
The easy successes of structuralistfunctionalist arguments indicate that here, in the stormtossed world of postwar, postcolonial, and preGodonlyknowswhat uncertainty, the line supposedly separating anthropology as scientific discourse from anthropology as Dreamtime “science” is, as so often the case, perilously thin and crooked (we already know something about such lines). As I have discussed in other essays,^{12} the anthropology of myth must often be interpreted as anthropology as myth, for the images we anthropologists conjure up of our disciplinary Other, the “native,” have disconcerting resemblances to ourselves. It is far from established that in doing anthropology we are engaging in some sanitized, intellectual undertaking that is heavenandearth removed from what we normally think and say about the other people in our lives. As I have argued here and elsewhere, the impetus and process of doing anthropology are so compelling precisely because they are also the bases of “doing humanity”: what we think and say as degreed, bonafide social scientists is intimately tied to how we think of people as a function of being human ourselves. Thus in employing the Dreamtime “science” of structuralfunctionalism to describe and analyze “natives,”we contribute more to the rapidly growing myth of humanity than to some (mythical!) body of carvedinstone, objective fact.^{13}
