The Dimensionality of Signs, Tools, and Models: An Introduction

Animal
Communication

There is good reason to use the evolutionary term of palingenesis, because animals other than homo faber apparently use symbolic communication by way of palingenesis, and there is ample evidence for suggesting that human semiotics evolved in ways similar to the evolutionary patterns of other animals. In his classic essay “Courtship-Habits of the Great Crested Grebe” Julian Huxley pioneered inquiry about symbolic communication among animals.⁶ Furthermore, his use of the term “ritualization” effectively defined the semiotic transfer between categories that makes all symbols seem discontinuous. In that essay he observed among grebes a displacement of their common habits, such as preening and aggression, into the semiotic dimension of their mating ceremony. The most spectacular part of their courtship dance is a segment in which both male and female dive and emerge with a beak full of weed. They then rapidly approach each other and leap from the water; by paddling furiously, they stand mutually erect (see Figure 1). Huxley observed that this and other parts of the dance originally served as a useful action, which then turned into a symbol, and subsequently into a ritual: “or, in other words, the change by which the same act which first served a definite purpose directly comes later to subserve it only indirectly (symbolically), and then not at all” (506). For example, the physical act of ascending the nest and of assuming a passive attitude for coition is first used by either in the dance on the open water “as a sign of the readiness to pair”:

We may say that readiness to pair is indicated precociously—it is pushed back a step. Such processes of pushing back are very common in early ontogeny; embryologists then say that the time of the appearance of the character is coeno-genetic (even though the character itself, as here, may be palingenetic). The phylogenetic change has here been precisely similar; the only difference is that the displacement affects a mature instead of a very early period of life (p. 507).

FIG. 1. Courting habits of Podiceps Cristatus. At the center of the sketch is the most dramatic part of the western grebes’ courtship ritual, where the partners rise into a “penguin dance.” From Julian Huxley, “The Courtship Habits of the Great Crested Grebe,” Proceedings of the Zoological Society of London.

has here been precisely similar; the only difference is that the displacement affects a mature instead of a very early period of life (p.507).

There is a null point in this diachronic process. For grebes repeat this passive attitude in the open water, one with another, and then resume their ordinary acts of preening or feeding:

From useful symbolism to mere ritual is the last step—one that has taken place often enough in various human affairs. It appears that these actions and attitudes, once symbolic of certain states of mind and leading up to certain definite ends, lose their active symbolism and become ends in themselves (p. 507).

This semiotic pattern of exchange with displacement, which Huxley called ritualization, has subsequently become a serviceable concept for other ethologists’ understanding of animal communication. Edward O. Wilson has explained that ritualized symbols emerge among animals when they are in conflict about how to complete an act.⁷ Their hesitant behavior about a course of action is communicated to other animals of the same species and so comes to signal the possible need to coordinate activity in the group. Many species of birds have ritualized a hesitancy, about whether or not to fly, into a crouching stance that signals a warning to others. This semiotic dimension works by virtue of logical types. Signaling depends upon the class of birds that defines its members, but it is of a different category of meaning than the ritualized crouching in preparation to leap of an individual bird. Only as it is a member of the group do the physical acts take on symbolic meaning.

Increasing degrees of ritualization among individual species provide important steps for understanding the evolution of communication displays among neighboring species. Wilson offers an instructive example of several species of dance flies that have similar courtship displays (p. 35). In one species the courtship merely involves a simple approach by the male, but in another species the male offers another insect to his carnivorous partner, apparently in order to escape falling victim himself. In still another species the male attaches threads of silk to the insect, “rendering it more distinctive in appearance, a clear step in the direction of ritualization.” And another species completely encloses the offering. Some decrease the offering but maintain the size of the cocoon. Finally, “the male of another species does not bother to capture any prey object but simply offers the female an empty balloon” (p. 35).

This kind of displacement of aggression into a semiotic dimension, by means of increasing ritualization, prompted Konrad Lorenz to follow his colleague Huxley’s example and to use it as an equation for human ritual or art:⁸ “There is hardly a doubt that all human art primarily developed in the service of rituals and that the autonomy of ‘art for art’s sake’ was achieved only by another secondary step of cultural progress” (p. 73). This telescoping of culture into nature achieves its allure by way of a similar semiotic pattern. Although Lorenz’s line of development will not be followed here, one can seize upon the pattern of means-ends variation in order to understand better the hypothetical dimension that is pointed to. For many human rituals, as well as tools and models, have bionic origins.

Presently, one of these rites, the crane dance, will be explored, but first consider more closely the ritualization among dance flies of that empty cocoon. Wilson states that this display “is so far removed from the original behavior pattern that its evolutionary origin in this empid species might have remained a permanent mystery if biologists had not discovered what appears to be the full story of its development preserved step by step in the behavior of related species” (pp. 35-36). Instead of using the empty balloon as a model for evolution among dance flies or of using it as a model for the development of human ritual and art, let us use it as an analogy for understanding the hypothetical dimension of semiosis. It might also be used by some structuralist followers of the anthropologist Claude Lévi-Strauss to argue an essentially deceptive quality in communication, the argument being that the sign deceives as much as it communicates by diverting attention from the true state of affairs.⁹ Or a Marxist literary theorist like G. Lukacs might maintain that as a commodity it has become fetishized by a process of reification.¹⁰ Those arguments will be noticed in subsequent chapters, for there exists an essential means-ends confusion in semiotic functions that must be clarified if discovery is to work.

In the context of the dance fly’s empty balloon, the end point of ritualization is a complete destruction of the material object from which it began. It is the extreme of parsimony in that natural series. By understanding the evolution of the series, one conceives the empty container to represent a type for the species. Now by a similar process of extrapolation C. S. Peirce explained in Speculative Grammar that the linguistic sign, a “symbol,” refers not to an object but rather to a general class. As Roman Jakobson summarizes in an influential essay, “a symbol, for instance, a word, is a ‘general rule’ which signifies only through the different instances of its application. . . .”¹¹ Although a “table” may be used to refer to a specific object in the room, as a sign it still refers to the whole class. In effect Peirce emptied the sign of all its material members, making it a type different from all its specific usages. By means of a similar logic, Peirce maintained that mathematics is not a study of quantities of objects, but rather it is “the study of what is true of hypothetical states of things”:

For all modern mathematicians agree with Plato and Aristotle that mathematics deals exclusively with hypothetical states of things, and asserts nothing of fact whatever; and further that it is thus alone that the necessity of its conclusions is to be explained.¹²

For Whitehead also, mathematics is the penultimate logical type: “Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.”¹³ Like the empty balloon, at its extreme the hypothetical dimension of semiotics is similarly emptied of all its specific occasions and natural dimensions. And like any other class without members it becomes a null set.

The extreme of this kind of thinking is Ferdinand de Saus-sure’s conception of la langue, which is “the sum of the verbal images stored up in all the individuals, a treasure deposited by the practice of speaking in the members of a given community; a grammatical system, virtually existing in each brain, or more exactly in the brains of a body of individuals; for la langue is not complete in any one of them, it exists in perfection only in the mass.”¹⁴ Odgen and Richards attack this as a “fantastic” notion, one that could only be arrived at by some “Method of Intensive Distraction analogous to that with which Dr. Whitehead’s name is associated.” Whitehead’s Fallacy of Misplaced Concreteness will be explored in Chapter 3.

In view of the logical dangers of extrapolation, I agree with Rudolf Arnheim that linguistic signs must be linked with percepts of different dimensions if discovery is to result.¹⁵ For Arnheim, thinking solely in a one-dimensional sequence of verbal language is “thoughtless thinking.” But thinking by means of a visual medium allows the representation of shapes in both two-dimensional and three-dimensional space. “This polydimensional space not only yields good thought models of physical objects or events, it also represents isomorphically the dimensions needed for theoretical reasoning” (p. 232). The possibilities and limitations of theoretical reasoning in different dimensions will be explored in subsequent chapters. What needs to be stressed here is that neither language nor any other one-dimensional mode is being attacked. The point is that a sign of any dimension must be undergirded by the other dimensions. In addition to the Peir-cean idea that a linguistic sign refers to a class of things rather than to a concrete object, is the corresponding fact about a visual sign constructed on a two-dimensional plane. For example, Jean Piaget and Barbara Inhelder have explained that if perception is an understanding of objects by means of direct apprehension, then representation or imagination evoke absent objects and thereby aid in perception.¹⁶ So, “one may recognize a ‘triangle’ and liken the given figure to the entire class of comparable shapes not present to perception” (p. 17). Long ago, Aristotle speculated about the same figure: “though we do not for the purpose of the proof make use of the fact that the quantity in the triangle (for example, which we have drawn) is determinate, we nevertheless draw it determinate in quantity.”¹⁷ All signs, tools, and models of discovery, although determinate in their construction, point to the indeterminate meaning of their class.

For every act of limitation implies a transcendence. In Chapter 4 we shall see how semiotic selection displaces, by suppressing, one set of alternative signs that have not been chosen, while the selection simultaneously fails to delimit a larger set. That indeterminacy is the reason why the semiotic dimension is hypothetical. In this book the boundary conditions by which signs of various dimensions evoke their indeterminate classes will be explained by the figure-ground relationship of Gestalt psychology. For three-dimensional signs both set forth and suppress the ground of their construction, two-dimensional signs similarly evoke the plane, and one-dimensional signs are points extrapolated from time, but which nonetheless evoke sequence.

Sacred Space

Figure-ground semiotics can be used to explore ancient rites of the crane dance, as well as other human rituals, where the figures are dancers and the ground is a literal site. And the hypothetical locus of semiotic discourse can be equated with the “sacred space” set aside by johan Huizinga for describing the site of transcendent acts of ritual.¹⁸ The crucial element of “play” in sacred performances of early cultures demands that a sacred space be explicitly hedged off in a “temporarily real world of its own” (p. 14). For Huizinga and others, ritual dances re-present acts of nature and so urge nature to keep the world in its right course and to remain benevolent to the dancers. E. A. Armstrong explains that the crane dance was part of a pattern of sacrificial and funerary rites that extended from the Fertile Crescent east and west to China and the Aegean; hence its counterparts are to be discerned in the minotaur’s labyrinth and in Chinese underground passages.¹⁹ Dances, in other words, are figures that require a formulaic maze, a substructural ground that evokes the cosmic happening. This classification according to structural and functional similarities allows the conjecture that crane dances are part of a totemic grammar that groups creatures of one leg (birds, animals, or humans) into an order of unipeds which signifies hobbling, lurching, and leaping in the maze. Otherwise diverse creatures can be grouped pedomorphically in this fantastic class: partridge dancers who hold one foot cocked to strike, serpent-tailed daemons of Oriental cultures, Hephaestus and Vulcan who hobbled, Tantalus which means ‘lurching’, and Oedipus which means ‘sore-foot’ or perhaps ‘one-foot’.²⁰ The totemic logic would be the hesitational stance of signaling itself, as a bird’s teetering on a limb is ritualized into a sign of warning. The emblem of this class of signalers would be the ancient morris-maze pattern of the triskelés or swastika which signifies ‘fly-foot’.²¹ The common structural pattern is an exchange—with displacement that evokes a cosmic order, but in a place and way set aside from the true order.

The rite is a cosmic map, where the symbolic space is different from the real space. Sacred space is another way of expressing the hypothetical dimension of semiotics, where the Danzenplatz represents the ground of discourse and the lurching and leaping figures represent the signs. The members of the class must leap for the hypothetical dimension, of Nature. As discontinuous signs of mediation they serve as go-betweens, as shamans that evoke the class. As members they also evoke the class: they draw back to leap higher. This is the knight’s gambit in chess: up a square and over two. The process of ritualization therefore includes both the synchronic paradox of member and class plus the diachronic paradox of ontogeny and phylogeny.

Accommodation

These ritual dances enact an ancient rhetorical trope of “accommodation” in which human language, used to evoke the sympathy of the gods, cannot hope to imitate the visionary language of the supernatural order. As John Milton phrased it in Paradise Lost:

what surmounts the reach

Of human sense, I shall delineate so,

By lik’ning spiritual to corporeal forms,

As may express them best, though what if Earth

Be but the Shadow of Heav’n, and things therein

Each to other like, more than on Earth is thought?²²

The ordinary use of accommodation has a use similar to Huxley’s ritualization: the “adaptation of a word, expression, or system to something different from its original purpose.” But in many archaic cultures, accommodation was not simply a failure of human language or acts, but rather a fear of imitating exactly objects of veneration or awe. The Anglo-Saxon ‘kenning’ (world-candle for sun, word-hoard for mind, head-jewels for eyes) is but one example of a periphrastic form that took the long way around in order to displace the equivalence betwixt word and thing. For Ortega y Gasset, metaphor originates with this form of taboo.²³ And for Sigmund Freud this form of accommodation defines the distinctive state of human beings. Mankind is a “prosthetic God” who is magnificent when he puts on those auxiliary organs, “but those organs have not grown on to him and they still give him much trouble at times.”²⁴ Furthermore,

A good part of the struggles of mankind centre around the single task of finding an expedient accommodation—one, that is, that will bring human happiness—between this claim of the individual and the cultural claims of the group; and one of the problems that touches the fate of humanity is whether such an accommodation can be reached by some particular form of civilization or whether this conflict is irreconcilable (p. 43).

Much more prosaic is Lévi-Strauss’s “bricoleur,” the odd-job man who accommodates himself to a task by using a limited repertoire of means at his disposal for whatever contingency.²⁵ He explains that the old meaning of the word “bricoleur” referred to “some extraneous movement: a ball rebounding, a dog straying, or a horse swerving from its direct course to avoid an obstacle” (p. 16).

The semiotic meaning of accommodation therefore involves the awareness that any sign, tool, or model evokes a discontinuity when it has been transferred from one category of use to another. As a form of leaping, swerving, or lurching, as a form of accommodation, that is, the sign always evinces torsion. Shakespeare’s Timon of Athens once said, “All is oblique.” Torsion will be an important term in this study. Semiotic torsion is observed most clearly in things such as Found Art or in fine art such as Picasso’s “Baboon with Young,” which features a recognizable toy automobile welded onto the torso of a baboon. Torsion is that “push-pull” phenomenon which one experiences as he sees in a painting the semblance of a landscape and then the medium of painted brushstrokes. A sign is like any bounded line that exhibits a torsion between separation and connection, difference and sameness, Snout and his Wall.

When a sign, tool, or model accommodates itself to the task of connecting two categories that were originally separate, it always evinces a torsion that indicates its jointure is not absolutely fitting. Torsion will therefore be used in subsequent chapters as one of the increments in arriving at the six categories of signs. If ritualization defines the diachronic process, if accommodation means an imperfect semiotic fitting, and if torsion defines the observed discontinuity in a sign, then two elementary principles emerge from the construction of any dimensional sign. First, most cultures have celebrated Aristotle’s bionic dictum, expressed by Alexander Pope as “First follow Nature.” Second, every imitation accommodates some Platonic awareness that the model is a poor imitation, a distortion of the hypothetical dimension, in this case, Nature. These principles may be illustrated by Figures 2 and 3. According to Joseph Needham, the Chinese junk looks strange to Western eyes, because one does not recognize the natural origin of its design.²⁶ Junks were modeled after the order palmipeds, webbed-footed birds such as pelicans, petrels, and swans, whose buoyancy is buffeted both by water and by air. Most of the outlandishness disappears as one observes the high tail, the bluff breast, the eyes, and the face, indeed, the seaworthiness of a water bird hidden in the outline. The other illustration is an Elizabethan shipwright’s, Matthew Baker, who superimposed a fish upon the hull in order to visualize the designer’s maxim: “a cod’s head and a mackerel’s tail.” Two different kinds of animal helped determine shape, but one was airborne, and the other was submerged.

FIG. 2. In The Design Of This Chinese Ocean-Going Junk One Can Discern A Water Bird As Its Bionic Model. From Liu-Chhiu Chih Lüeh (1757). Harvard Yenching Library.

FIG. 3. In this technical drawing of a sixteenth-century hull one can see a submerged fish as its bionic model. From the Matthew Baker manuscript in the pepysian library. By permission of the master and Fellows, Magdalene College, Cambridge.

Ships are particularly vivid examples for introducing principles of accommodation, because they are radicals of semiotic transfer. As a transporting vehicle, a ship exchanges information from a source to a recipient, and vice versa, by traversing two dissimilar media, the land and the sea. As a sign it seems to be autonomous, independent upon the sea, while really remaining dependent upon its surrounding contexts, its ports. The ancient cult of the ship springs from this seemingly magical independence. The most familiar personification of this cult is the daemon Charon the Porter. In the theory of logical types a ship as container is superior to its contents, but the periphrasis of synecdoche, of substituting the container for the thing contained, was useful when a ship was ritualized from utilitarian needs to funerary rites.²⁷ The pattern is one of exchange-with-displacement, wherein the ship’s bionic origin combines with its supernatural end to transport stuffs from this world to another.

The Grecian cult of the independent ship was responsible for the beginnings of scientific discovery, according to Ortega y Gasset:

. . . when the Greeks set themselves to dreaming, they would dream of ships, capable in themselves, without a pilot, of bringing the sailor safe to port. These are the mysterious ships of the Phoenicians, and when one begins to study them, there on the coast of Ionia, in Miletus, a society of men emerges presided over by Thales, which calls itself The Forever Sailors (Siempre Navigantes); these celebrate their scientific sessions in a boat on the high seas. And thus, in one way or another, the idea of the ship enters into the most profound and moving depths of the ancient soul. Hence its cult of the ship and of opportunity, for opportunitus means nothing more or less than the way which leads us to the portus, or port. ²⁸

According to Herodotus, Thales was a merchant of Phoenician extraction; it is therefore logical that the first Greek model of scientific discovery depended upon a ship’s semiotic qualities. For Thales wrote, “the earth is supported by water on which it rides like a ship.”²⁹ And for George Thomson, Thales’ adage was in keeping with an even older idea, of “the waters that are beneath the earth,” the Babylonian apsu.³⁰ As a ship is to water, so earth is to the apeiron: the semiotic figure, limited in itself, accommodates an indeterminate ground by imitation and distortion. In this book, therefore, accommodation is to be construed as a general term for a series of transformations that result in a semiotic complex of display and displacement. In a later section the transformations will be divided into three operations, where the dimensional aspects of any sign are (1) Highlighted, (2) Torqued, (3) Suppressed.

Means-Ends
Conversions

A useful way to define the terms ‘sign’, ‘tool’, and ‘model’ is by way of means-ends analysis. As the epigraphs attest, a conversion from means to ends is a variant of semiotic exchange and displacement. One agrees with Nietzsche that it is difficult to discern, for example, the neolithic origins of clay in finished pottery, or of copper powder in elegant bronze, or of flax in linen, or indeed of grapes in wine. On the other hand, Thoreau has his followers who claim that means have been ritualized into unnatural prominence so that, for example, industrialization exaggerates a necessary mode of maintaining social goods into an overriding end, just as the Prussian state made a fetish of militarism.³¹ The ethics of “ought” is part of a philosophical corrective to instrumentalism that must be understood.³² Yet the more neutral task of defining the categories of transference and their observed torsion will take precedence here.

For John Dewey, a sign occurs as an “expressed” need, and it is converted from the very materials that seem to block immediate action. “Etymologically, an act of expression is a squeezing out, a pressing forth. Juice is expressed when grapes are crushed in the wine press . . . .”³³ The deflection of raw material into a medium or expression converts obstacles into means or media by way of thinking or of “reflection.” The formal cause of pressure, coming both from without, in the environment, and from within the body, converts the blockage into a sign, a Matterhorn.

It is possible to lay signs, tools, and models in a sliding scale of means and ends. Charles Morris said that “something is a sign only because it is interpreted as a sign of something by some interpreter. . . . Semiotics, then, is not concerned with the study of a particular kind of object, but only with ordinary objects insofar (and only insofar) as they participate in semiosis.”³⁴ Hence, a sign is construed here to be anything that has been ritualized from one category of use toward a new end so that it is seen by an interpreter as accommodating the place of something else. Although the definition usefully includes the several terms so far isolated—ritualization, accommodation, and torsion—it does not yet discriminate among the different functions of signs, tools, and models. Now let us follow one example in order to define signs, tools, and models in terms of means-end conversions.

To Hamlet the skull of Yorick is a memento mori, a sign to him about existence. But a skull as a three-dimensional sign need not necessarily betoken the futility of earthly pleasures. Consider the way that Goethe “turned” a skull to his own ends. Helmholtz mentions this anecdote about discovery in “Goethe’s Scientific Researches”: “A fortunate glance at a broken sheep’s skull, which Goethe found by accident on the sand of the Lido at Venice, suggested to him that the skull itself consisted of a series of very much altered vertabrae.”³⁵ Devoid of intentionality, the skull was insightfully interpreted as being in itself a fused symptom of physiological development. Goethe saw the torsion in the sign by decomposing it and then reconstructing it on a different plane of signification. Or consider the skull that Pericles found, and its subsequent use as a prophetic sign of subsequent events. Plutarch tells how one day the popular leader brought from his farm to Athens a ram’s head that had only one horn. A diviner told Pericles that the single horn growing strangely out of the skull was sign or token of fate, indicating that of the two factions in the city, his and that of Thucydides, the leadership of Athens would eventually devolve upon the owner of the farm where the sheep was found. Anaxagoras, however, took to the tools of natural philosophy, dissected the skull, and gave a physiological explanation for the distorted growth of a single horn. For this answer he was much admired, until Pericles subsequently came to power. Plutarch’s summary is so clearly honed as to be a parable about an ancient linguistic theory:

And yet, in my opinion, it is no absurdity to say that they were both in the right, both natural philosopher and diviner, one justly detecting the cause of this event, by which it was produced, the other the end for which it was designed. For it was the business of the one to find out and give an account of what it was made, and in what manner and by what means it grew as it did; and of the other to foretell to what end and purpose it was so made, and what it might mean or portend. Those who say that to find out the cause of a prodigy is in effect to destroy its supposed significance as such, do not take notice, that, at the same time, together with divine prodigies, they also do away with signs and signals of human art and concert, as, for instance, the clashing of quoits, fire-beacons, and the shadows of sun-dials, every one of which has its cause, and by that cause and contrivance is a sign of something else.³⁶

This passage drives a wedge into the question of antecedents and consequents with which we opened the chapter. As we shall see in Chapter 4, the structural question “Where did the skull come from?” is logically different from the prophetic question, “What is it for?” But to question the form and function of signs is to make of them an illogical composite. All three examples verify our starting point, that signs are limited objects which are enjoined to stand for something else; meaning stands apart from their original composition, as the means vary from the ends.

However, unless we limit our definition to hand-held tools, the examples serve also to define tools at large. Although a tool might be considered to be any object “turned” to a new kind of manual operation, the definition should include instruments of more distant extension that are still under control of human hands. For instance, one of the first tools was a “dawn stone,” with a rudimentary percussed edge, but the class of cutting tools should include such distal instruments as plow, keel, stylus, intaglio cylinder—all probes which figuratively describe furrows of significance upon their grounds. Beginning as hand-held and dependent upon the kinesthesia of body movement, tools become more and more distant from manual labor as they are applied more widely to different ends. By extending the definition, however, one has overlapped into the category of models. Although skulls could be placed with the class of container tools, they were interpreted as three models for the way the world performs. So let us define models as tools whose utility has been extended and so transformed that they are no longer means to a different end but are mirrors of the problematic end itself. For, the increasing application of an effective tool to wider areas of endeavor renders it a model by reason of its near universality: the principle of the Egyptian astronomer’s gnomon gradually becomes transformed into the cosmic law of the Great Pyramid; the rope knotted lengthwise into ratios of 3-4-5, used as measuring tools by Mesopotamian surveyors, becomes a Pythagorean model for the chord music of the spheres; a device such as a watch becomes a model for a mechanistic universe. Each of these examples will be discussed in subsequent chapters.

In other words, the wider application of an exosomatic instrument to the world implies that the laws which had governed the working of a tool have become so useful at large that, by synecdoche, they come to substitute for the world. When a tool is ‘turned’ from its intended use and contemplated instead of applied, the arbitrary connection between a tool and its referred function is transformed so that it is no longer a means to a different end. Seen as reflections of the end itself, the principles by which a tool are constructed may be construed as hieroglyphs, omens, signatures, symptoms, laws, or models of higher function. Our definitions therefore are perspectivist; depending upon the oblique perspective given a semiotic object, it becomes sign, tool, or model.

To recapitulate, a model functions as a simpler substitute for a problem less easily understood: “No substantial part of the universe is so simple that it can be grasped and controlled without abstraction. Abstraction consists in replacing the part of the universe under consideration by a model of similar but simpler structure.”³⁷ A model tends to reflect an end state, while ignoring the means to a different end. This complex accords with Herbert A. Snow’s description of means-ends analysis.³⁸ For him all problem solving is divided among process descriptions and state descriptions, means and ends, recipes and blueprints: “To construct a circle, rotate a compass with one arm fixed until the other arm has reached its starting point. . . . A circle is the locus of all points equidistant from a given point” (p. 479). All organisms survive by combining the two, which we verbalize as means-ends analysis: “Given a desired state of affairs and an existing state of affairs, the task of an adaptive organism is to find the difference between these two states, and then to find the correlative process that will find the difference” (p. 479). One imagines a blueprint of the future state and then seeks out a recipe as a means. Because spatio-temporal axes are intertwined here, we shall diagram, in the next section, the several dimensional possibilities of signs. Then these six possibilities of semiotic discourse can display different torsions of means and ends.

Semiotic Dimensions

One of the far-reaching issues which confronts the several disciplines of science and the humanities is a distortion that can result from the transfer of a successful discovery from one of those disciplines to serve a need in a field for which the new method was not originally designed. In the history of ideas, for example, the successes of Newtonian mechanics, of Darwinian evolution, of information theory, might be gauged by their widespread misapplication in remote disciplines. The complete substitution of the model for a different problem produced cross-categorical sports such as the interiorization of Newtonian space inside the head of Lockean psychology, the transformation of natural selection into an economic principle justifying Herbert Spencer’s dog-eat-dog theories, and the chess-playing computer that would subjugate man. I do not intend to demean the cross-categorical method. On the contrary, it is clear that in many cases the transfers have been intellectually enlightening or socially effective. Think of William Harvey’s application of a hydraulic principle to the heart’s circulation of blood, or of the ethologists’ redirection of natural selection toward a better understanding of territory, or of geneticists’ and physiologists’ transfers of information theory to their disciplines. In the examples where distortions have occurred, methods were thought of as universal panaceas; where they were effective, revolutionary models were understood to be special cases of wider phenomena which the original designers had not imagined. The study of models in itself is a special case of a semiotic study that highlights the limits both of remaining within strict categories and of transcending them.

As the study of signs regardless of particular disciplines, semiotics allows the arrangement of symbolic discourse into a new category of Euclidean dimensions, one that classifies instruments according to their use in space. Because semiotics does not limit itself to the science of linguistic signs, but includes signs outside of language within its purview, models of scientific discovery therefore can be considered as a special case of the more inclusive question of the dimensional locus of man’s artificial extensions. Let us begin by comparing models with linguistic signs and hand-held tools. By exercising variations upon a pioneering diagram of semiotics drawn by C. S. Peirce, we can better understand the hypothetical dimension of symbolic discourse, no matter whether it be a hammer that exists in three dimensions, a pictographic model on a two-dimensional plane, or a one-dimensional linguistic sign (Figure 4).

FIG. 4. An often-used semiotic triangle designed to show an imputed relation between symbol and referent.

In The Meaning of Meaning Ogden and Richards appropriated this diagram in order to stress the indirect relation that exists between a symbol or verbal expression and the object or referent which is referred to.³⁹ And Umberto Eco provides an excellent analysis of a possible “referential fallacy” in the diagram, which he reveals by comparing Ogden and Richards’ diagram with a similar construction of C. S. Peirce’s and one of Frege’s.⁴⁰ For Ogden and Richards most of the misunderstanding that language transposes to physics or philosophy arises from the indirect relation between symbol and referent, or signifier and signified. “Every great advance in physics has been at the expense of some generally accepted piece of metaphysical explanation which has enshrined itself in a convenient, universally practised, symbolic shorthand” (p. 14). The diagram serves as a specific reminder that thought, directed by knowledge of the conventions of language, yokes the dissimilarities into unity. Since for Ogden and Richards language is a “ready instrument,” the handy telescoping along the dotted line is more important than the relative inaccuracy of neglecting to include the interpreter’s thought processes: “Thus such a shorthand as the word ‘means’ is constantly used to imply a direct simple relation between words and things, phrases and situations” (p. 14). It is precisely that neglect of the mental process which causes many unnecessarily inaccurate intrusions of semiotic conventions upon the satisfactory solution of a problem.

In Figure 5 we can stress the dimensional aspects of the linguistic sign by locating at B the signifier as a pictorial expression that exists on a two-dimensional plane when written. If the linguistic sign that we are examining is the word “piston,” we shall enclose it with two slashes //piston// in order to isolate the two-dimensional visual image of the signifier. Similarly, let C represent the signified three-dimensional object, whose physical properties can be referred to as ///piston///. For Ogden and Richards, it was thinking along the relational axes, here depicted as y and z, that unified the linguistic triangle at A. Since mentation is here considered a process rather than a thing, let A represent the one-dimensional relation that temporally processes the linguistic sign along y and z. When we wish to highlight the thought process that allows the handy shortcut from B to C, we shall refer to the process as /piston/, which thus reveals its nonexistence in space and thereby provides a reason for the indirect relation along x. For thought seems unneeded, not being seen, in the seemingly direct relation existing between thought and thing.

FIG. 5. Diagram depicting the semiotic relation between the written word “piston” and its imputed object. The slashes number the pertinent dimension.

Let us initially propose that linguistic signs exist primarily in a one-dimensional mode, since in the time that they are being used one reads from the two-dimensional signifier to the three-dimensional object. Reasons for this designation will be presented further on, and in Chapter 3 at large. Consider that even though all signs are in time, many are not linguistic, but some are two-dimensional pictographs, such as international road signs, or three-dimensional, such as Hamlet’s reading of Yorick’s skull.

For purposes of comparison, consider in Figure 6 how tool usage can be adapted to the semiotic triangle. In performing a utilitarian task such as hammering, one again assumes the indirect relationship between symbol and object, that //hammer//, means ///hammer///. Hence, draw once again the dotted line along x as a reminder of the arbitrary shortcut that is taken. But consider that while one is hammering, no direct relation is made with the pictorial images of the word //hammer//. While I am driving a nail, I do not think, “Hammer, hammer, hammer!” Hence y is similarly dotted in order to suggest the subsidiary state of that axis in tool usage. It might be argued that z is also arbitrary, since the relation between a ///hammer/// and its employment is not limited to the designed structure of the object. A hammer may be misappropriated for an indefinite number of tasks other than that of driving nails. However, for the purposes of the semiotic triangle the present issue is not the different ends for which an instrument such as a hammer might be employed. Rather, the point here is that in the act of hammering anything a direct relation exists with A and C along z as thought impels the object, as /hammer/ wields the ///hammer///. Before evaluating the differences between linguistic signs and tools, one must realign the semiotic diagram in order to observe its fittingness for accommodating a model of discovery.

FIG. 6. Diagram depicting the act of hammering, in which no relation exists with the visual image of the written word //hammer//.

One difference between models and either linguistic signs or tools is crucial to discern. We have construed a linguistic sign to be a one-dimensional set of two-dimensional signifiers connected with three-dimensional objects by virtue of an arbitrary and indirect convention that we have sketched as a dotted line along x. We have stated that a tool also maintains the indirect link at x, and although it has not been depicted in Figure 6 above, we stipulated that the relation between ///hammer/// and purpose is arbitrary. But a model of discovery functions as a simpler hypothetical substitute for a problem less easily understood. That is, a linguistic sign links arbitraries together by a remembered convention, while a model predicts similarities between itself and a problem that is too complex to grasp in its totality. A model, furthermore, is a link between a theory and experiment. When we construct a dimensional reduction that models a joint or a synapse, for instance, we are examining a connecting principle, a “link” that has evolved in the spatio-temporal enterprise as a model of that enterprise.

In order to make the comparison of the diagrams more symmetrical, let us continue to use a three-dimensional object for modeling, as we have with ///piston/// and ///hammer///, although we recall that models, like signs and tools, may be designed to work in other dimensions, such as two-dimensional blueprints or one-dimensional chemical equations. A serviceable model with which to display and displace a universe is a ///sphere///. The semiotic triangle in Figure 7 must be modified so that the arbitrary linkage at x is retained, where there is no inherent similarity between //sphere// at B and ///sphere/// at C that substitutes for a universe. Since, however, the ///sphere/// incorporates or will incorporate features that characterize the universe to be comprehended, we should depict an axis of thought that would show the relationship of similarity, if not identity, between model and problem. If we construct C’ to represent the tacit end of the problem, in an n–dimensional problematic universe, that displacement into an explicitly hypothetical dimension would more graphically depict the displacement that has taken place once we substitute a simpler model for a more complicated problem. In drawing the tangent x’ from the model at C to the projected and unknown end at C’, we mean that one solves the indirect relation at x’ by thinking along z and z’.

FIG. 7. Diagram depicting the use of a 3D model, such as a sphere, to substitute for a problem of partially understood properties at C’.

The discussion has focused so far upon the differences and similarities shared by tools, signs, and models in a semiotic triangle which was arranged so that the three different kinds of ex-osomatic instruments had a common three-dimensional point at C. That means one can interchange the examples in Figures 5, 6, and 7. Pistons and spheres are used as tools, while as cranks and discs they once served as parts of La Mettrie’s model of a human machine. And of course the diagrams are meant to remind us that each of the objects has a linguistic signifier which is arbitrarily attached. By showing that three-dimensional objects may be freely interchanged in the several semiotic triangles, have we slurred distinctive differences among signs, tools, and models? On the contrary, by emphasizing the dimensionality of signs, tools, and models, we shall see that traditional definitions in actuality make distinctions where there is no difference.

Now let us vary the semiotic triangle in order to highlight signs, tools, and models that work from a two-dimensional plane. (The diagrams that have been sketched of the semiotic triangle are, of course, two-dimensional examples.) Consider that in Figure 8 we have substituted a two-dimensional pictograph for the three-dimensional model of Figure 7; that is, for an example of a two-dimensional model we have replaced the sphere with a projectional map. In Figure 8, therefore, point C has been deleted from consideration, for we are no longer solving along x’, from a three-dimensional model at C to a hypothetical dimension at C’. Neither are we preoccupied with the pictorial images, the signifiers which constitute the figures of a linguistic sign for the word “map” at point B. Hence y and p are axes of suppressed interest in Figure 8 (y representing mentation about the letters and p representing the axes of indirect relation) because connection between signifiers of //map// and the projected blueprint of a universe is capricious. Our concern is with the assumed similarities along r. In other words, thought is focused along q and z’ to yield the indirect relationship at r between two-dimensional model and tacit end of the problem, or Δ AB’C’.

FIG. 8. Diagram depicting the use of a 2D model, such as a road map.

If in Figure 8 one wished to consider “map” as a linguistic sign, he could then work with Δ ABC. Similarly, if he wanted to understand the related dimensions of a map as a two-dimensional tool such as a blueprint, he could consider Δ AB’C. But notice that as a two-dimensional pictographic sign, as distinguished from the linguistic sign at Δ ABC, the maplike features of the plane are not distinguishable from a two-dimensional tool at Δ AB’C. If the features of a two-dimensional sign and a two-dimensional tool are congruent, we have simplified our problem in defining them. As a final consideration of Figure 8 we must observe that this triangle does not explain the reasons for the choice of a two-dimensional model and the corresponding displacement of three-dimensional considerations. For instance, Helmholtz once asked a popular audience to play a figure-ground game, to imagine a country of flatlanders who were limited to a two-dimensional plane so that the audience could better understand its experiential limits of being forced to exist in a Euclidean frame of three dimensions. This convenient combination of display and displacement, of observed distortion, will be considered in Figure 10 and in Chapter 2.

Now consider a diagram of one-dimensional signs, tools, and models. We allowed earlier that a linguistic sign is primarily one-dimensional, even though its pictorial images lie on a two-dimensional plane, and, as images, might lie in a heavy book composed of a certain amount of ink and might refer to a three-dimensional object, ///piston///. In the act of reading, however, the pictorial and kinesthetic dimensions are subsumed into the movement of mentation, a process which cannot be described except retrospectively in spatial terminology. Let us describe in Figure 9 a tangent from point A to point C’ that represents the thinking process of memory and prediction which comprises the method of reading toward the analytic point of any passage of linguistic signs. The axis z’ therefore ‘means’ that searching toward a hypothetical dimension which characterized z’ in Figures 7 and 8. All are one-dimensional probes of a thinking process that moves back and forth from model to hypothesis. Is there then a difference between a linguistic sign and a one-dimensional tool or model? As soon as a one-dimensional sign is set to work by thought, it is indistinguishable from anything else we might call a one-dimensional tool or model. For we are discussing that category of semiotic exchanges that includes any series of notations, whether it be a mathematical equation for a chemical process, a score in music, or a passage from Oliver Twist; all refer to the same kind of hypothetical point of attention at C’, reached by remembering and predicting along z’. Thus all three instruments have features which are congruent along Δ ABC’ in Figure 9.

FIG. 9. Diagram illustrating the act of reading, from linguistic sign to imputed meaning.

Notice that one is unable to depict a point at A’, which would represent pure process, for images pictured in some kind of space are required in any one-dimensional construct. In Chapter 3 a one-dimensional sign will be studied as a semiotic point of reference extrapolated from specific instances in time. Hence it is technically zero-dimensional in a Euclidean sense. So Figure 9 allows the visualization of that process at point A. As we recall from Figure 5, B represents the two-dimensional aspects of the linguistic sign, or tool, or model. As signifier, it is decoded from a conventional memory bank in order to probe C’. Just as in Figure 1 the axis x was construed to be an indirect relation between signifier and signified, so in Figure 9 the axis r “means” an indirect relation between signifier and tacit problem. What does it mean that one-dimensional signs, tools, and models are congruent at Δ ABC? It has the same significance as the observation that two-dimensional signs and tools are congruent along Δ AB’C’ in Figure 8. The verbal distinctions between the three kinds, as well as their dictionary definitions, are meaningless in operation.

One-dimensional probing along the axis of z’ may be characterized by the relational unit (n+1). If random mentation and random history may be considered as a succession of discrete events without semblance of causation, mathematics and history seem to have begun with the desire to extrapolate an orderly sense of twoness separate from specific objects or events in time. History and mathematics may be said to begin intuitively with the desire to transcend time. The relational sequence (n+1) is therefore a serviceable mnemonic for representing the mentation process along the vertical axes of the depicted triangles. Thinking along an arithmetical continuum of (n+1), where there is no finite analytical boundary and where there is no temporal limitation, is a serviceable way to describe the orderly accretion of evidence that builds to the hypothetically indefinite end for which signs, tools, and models substitute.

Similarly, the lines of indirect relation that we have drawn as dotted lines at the bases of the triangles may be described as a unit of (n-1). We use the relation (n-1) because the bases of the Δ ABC represent a conventional Euclidean sequence in which point, line, plane, and sphere are defined by twisting neighbors ninety degrees. Turning a line at right angles to itself reveals a point; rotating a plane yields a line; a sphere unveils a plane. Martin Gardner says, “In every space of n dimensions the ‘mirror’ is a surface of n–1 dimensions.”⁴¹ Each point in the original Δ ABC progresses from a lesser to a greater geometrical dimension, so each axis could be represented as turning out of itself by an increment of one twist. By increasingly indirect reversals, one works toward the tacit end of the problem at C’ which is ‘mirrored’ by a model of fewer dimensions.

By combining the two dissimilar categories—the mathematical thought that is extrapolated from time (n+1) plus the geometrical progression of (n-1)—we seem to leap from point A to point C’ by (n±l). This combination is useful to realize because (n+1) reminds us that mentation is self-consciously included in discovery, while (n-1) implies that the mirror of mind has been deleted. Following Ogden and Richards, we infer that on the one hand mentation seems outside the semiotic bases of geometrical relation, yet on the other hand we know that mentation makes the connections. So the combination (n±l) means a complex dialectic between model and tacit problem via a mentation process that is both within and without the modular system being described. From the outside we see the model as an object with geometrical qualities; seen from within, the model is processional, that is, arithmetical. That twofold perspective prompts most of the problems about “meta” languages, but we must ignore them for the more pressing task of definition.

Any instrument, whether tool, sign, or model, may be said to manifest obliquity or torsion, since it serves as a dialectical interchange between an object of Euclidean dimensions and a hypothetical problem. The dialectic occurs diagrammatically as thought probes along the vertical axes of our triangles and exchanges characteristics with the basal points so that B ‘means’ C. We have observed, however, that the exchange is not exact but in all cases indirect. For the features of one dimension have been suppressed and displaced in order to display qualities in another dimension. This functional obliquity, this dialectical system of exchange and displacement, must be examined with some care, for its torquing quality is the very medium of sleight of hand, of deception, and it bears on the potential illusoriness of model building. Consider that the most ancient recorded lie in Western literature is Odysseus’ to Polyphemus. Captured by the Cyclops and then asked his name, Odysseus answered with winged words, “Nobody.” So when Polyphemus was blinded by the wily mariner and shouted to his neighbors for help, “Nobody is killing me by force or treachery,” then Odysseus and his crew escaped to their ship. This diversionary pun draws attention to the ordinarily tacit displacement of the signifier by the signified. The pun probably is as old as the consciousness that a linguistic sign could serve not only to represent a concrete object, as in proto-writing, but could also refer to an abstract concept, as an ideograph. “Nobody” exists ‘in’ the same hypothetical dimension with Pegasus, with the square circle, with any thought experiment; all are possibilities which are pointed to by signs, tools, or models of a lesser dimension.

Polydimensional
Semiotics

Given the functional relativism that governs semiotic definitions, it becomes necessary to reorient the classes of objects according to torsion. For we can now use what we have learned about the apparent torque from one dimension to another in order to isolate the exact number of semiotic objects possible in Euclidean space. In this context there are only six possible ways to represent semiotic objects; that small number offsets the relativism of indefinite possibilities of use. In Figure 10 we have constructed a diagram that depicts the several dimensions of an object as if they were seen simultaneously from different perspectives.

The nearest position on any axis depicts the dominant dimension of a semiotic object; that view highlights the dimension of primary importance. For example, a neolithic pot exists primarily as a three-dimensional object. The right-hand view suggests the relational axis that is torqued from one dimension to another. To continue the example of the neolithic pot, it is obvious that the pot wall curves away from a two-dimensional plane toward containment. But the two-dimensional plane upon which designs of any kind are executed tends to be dematerialized as does the three-dimensional pot itself, when one reads the meaning of the described figures on it. Or consider that a highly symmetrical flaking on a flint knife turns attention away from function toward the elegance of two-dimensional patterning. This perspectivist twist of attention back and forth from function to pattern will demand discussion further on. All of the sides furthest from the viewer are meant to suggest the dimension that is suppressed in a semiotic object. For instance, the one-dimensional aspect of a neolithic pot is suppressed, even though in making it the pot’s symmetry depended upon a spinning potting wheel or the regular movement of skilled hands. Let the furthest extreme of the box diagram, which also cannot be depicted, be the assumed horizon of a sign, tool, or model; that is, let that extreme side represent the semiotic function of an instrument, the hypothetical dimension which it probes at C’. The conceptual horizon of a neolithic pot is indefinite: it means holding grain; it means the ceremony of the harvest as well as the ritual of the grave; it means Ingres’ La Source, or the water pot of the woman of Samaria. Its allusive powers are limited only by the constraints of imagination. We can therefore plot A as the class of three-dimensional objects such as pots and knives which torque two-dimensionality and suppress one-dimensionality.

FIG. 10. A modified Guilford cube that shows the six dimensions categorized in this book. To decipher their components, follow the implicit axes of x, y, and z. category a, for instance, represents that category of instruments whose dimensionality features three dimensionality (3D) highlighted (H), two dimensionality (2D) Torqued (T), and one dimensionality (ID) Suppressed (S).

Let B represent all instruments which are primarily two-dimensional, such as the cave paintings of Lascaux, or Egyptian hieroglyphs, or international road signs, in other words, that class which exists primarily on a two-dimensional plane, which torques three-dimensionality and suppresses one-dimensionality. On all such planes the illusion of depth is achieved by the familiar torsion of the figure slanted so as to achieve seeming independence from ground. The image is obviously static; one-dimensionality cannot be expressed pictorially. We will later discuss the semblance of movement in certain two-dimensional art forms, but for now we are considering utilitarian semiotic objects, not fine art.

Let C represent the one-dimensional class of instruments, such as abstract notches on primitive tally sticks, or a letter from the phonetic alpahabet where A stands for a sound, or an Arabic numeral such as 2, or a musical notation for C#. In this class a point arrested from time is highlighted, with two-dimensionality torqued as one turns from the pictorial image to its meaning, while the three-dimensional aspects, the material components such as the iota of ink that comprises the figure and the elements of paper that comprise the ground, are suppressed.

Given the combination of 3-2-1 dimensions, and given the three variations of “highlight,” “torque,” and “suppress,” there are merely a total of six possible combinations that complete the class of semiotic instruments:

What kinds of semiotic objects might satisfy the remaining combinations of D, E, F?

Examples will be more readily forthcoming once we consider that this exercise is derived from Galileo’s classification by rank of the several kinds of art forms. Erwin Panofsky reports that in a letter to an artist friend, Galileo had argued that painting, not sculpture, should be ranked more highly. One passage is worth presenting in its entirety, because Galileo’s ranking of art forms according to their dimensionality will not only provide us with examples for the remaining possibilities, but also will help explain the relationship of art forms to utilitarian semiotic objects.

. . . the farther removed the means by which one imitates are from the thing to be imitated, the more worthy of wonder the imitation will be. In ancient times those actors who could tell a whole story exclusively by means of movements and gestures were more highly appreciated than those who expressed it viva voce in tragedy or comedy, because the former used a means very different and a mode of representation quite divergent from the actions represented. Will we not admire a musician who moves us to sympathy with a lover by representing his sorrows and passions in a song much more than if he were to do it by sobs? And this we do because a song is a medium not only different from but opposite to the [natural] expression of pain while tears and sobs are very similar to it. And we would admire him much more if he were to do it silently, with an instrument only, by means of dissonances and passionate musical accents, for the inanimate strings are [of themselves] less capable of awakening the hidden passions of our soul than is the voice that narrates them. For this reason, then, what will be so wonderful in imitating “sculptress nature” by sculpture itself, in representing that which is relieved by relief itself? Certainly nothing or very little, and the most artistic imitation will be that which represents relief on its opposite, which is the plane. In this respect, therefore, painting is more wonderful than sculpture. ⁴²

Of course, the means-ends argument really suggests that wordless music is most wonderful among the several arts, because it imitates by the most indirect means. In other words, music, as we noted earlier, belongs to that class of one-dimensional models which imitate the hypothetical dimension with little recourse to the two-dimensional plane and the three-dimensional object.

In accord with Galileo’s line of thought let us test music as an example of class F, where the one-dimensionality of movement and tempo is highlighted, with the three-dimensional axis torqued into the apparent shape of harmonic structure, and the two-dimensional plane suppressed. It is three-dimensional obliqueness in one-dimensional art forms that allows literary and music critics to speak of those processional modes in static terms of structure. Correspondingly, the semblance of one-dimensionality in painting allows art critics to speak of apparent motion therein. Music differs from the musical note in class C as the means differ from the end. Similarly, painting as a fine art inhabits class E. Its one-dimensional torque, that provides the semblance of movement in a static representation of nature, differs from a utilitarian pictograph of two dimensions as music differs from musical notation. A stylized representation of a bison at Lascaux, for instance, is a partial sign of a larger meaning derived from the whole composition of bison, deer, and other animal signs on the wall; the composition refers to a hypothetical dimension whose meaning we do not know exactly but which has to do with the paleolithic cult of the hunt. Other art forms, such as mime or ballet, provide examples that fit the three-dimensional requirements of class D. As William Butler Yeats said in “Among School Children”:

O body swayed to music, O brightening glance,

How can we know the dancer from the dance?

The diagram helps to explain the relation of utilitarian semiotic objects to fine art. If then the different ways in which objects are torqued determine their category. The neolithic pot which was placed in class A becomes a sample of fine art in D as the potter incorporates the semblance of one-dimensional movement into the pot. He provides it with delicate S-shaped curvature, makes its walls extremely thin, incorporates swirling lines on the plane; it seems to defy gravity; it dances.

Galileo’s definition of art is in the classical tradition of imitatio that began with Aristotle. Art imitates nature. His further stipulation, that music and painting imitate by dimensions opposite to their original, accords with our definition of model building. Models used in scientific discovery are simpler dimensional reflections or imitations of a hypothetical dimension that is probed by an object of dimensions different from the point at C’. As imitations, therefore, rather than utilitarian tools which are means to a different end, models seem to fit with imitative art forms in classes D, E, F.

But how can that be so? The significance of the divergent grouping occurs when one separates semiotic objects, as we have seen, according to arbitrary but handy axes of relation on the one hand, or to axes of relative similitude on the other. In the first group consisting of A, B, C, one thinks ‘through’ the object to a different conceptual horizon ‘in’ a hypothetical dimension. That is, in all cases the object is interpreted as standing in the place of something else. A code is required for thought to process the relation from //piston// to ///piston///. But as we noticed before, the one-dimensional coding process of mentation is suppressed. The coding seems extrinsic to the handy yoking along x, where //piston// means ///piston///. In class A and B one-dimensionality is suppressed; in class C, though it is highlighted, one looks through it, as a point extrapolated from movement, to its referred meaning. But in the classes that include fine arts and models we note that a semblance of one-dimensionality is represented in the object. That is, in classes D, E, F, the coding process of mentation (n+1) is torqued into the object so that it seems self-referring, understandable on its own terms. As Galileo observed, the wonder of fine art depends upon an awareness of divergent means in the construction. We see both the imitation and the limitation imposed by a different medium. The same twofold awareness occurs in model building, except that we do not simply admire the handiwork, but we always recall the built-in code which displaces at the same time that it imitates the hypothetical problem. It incorporates both (n+1) and (n-1), the twofold perspective of knowing within and without.

We have seen that according to Figure 10 there is a dimensional grammar of six possible semiotic representations of objects. All six variables are limited to the conventional world of three space, but all point to a hypothetical dimension which includes an indefinite number of possible problems beyond Euclid. Semiotics, therefore, can be a useful method for instructing students in the basic structure and function of the professional implements to which they will choose to apprentice themselves. Although semiotics cannot substitute for the professional training of working practically with signs, tools, and models in a particular discipline, it can help to delineate the possibilities and pitfalls of symbolic discourse, an activity which is central to a liberal education. This classification system is therefore educationally valuable, because it shows graphically the difference in usage among signs, tools, models, and art forms. The six classes will provide the format for our discussions in the following chapters.

In diagrammatically cataloging the several possible ways that displacement activity functions in symbolic discourse, we have seen that torsion occurs as soon as coding makes one thing stand for another. Hence we see that the axiom of torsion, expressed first by Euclid, in reversing a line, a plane, etc., is not singly a geometrical phenomenon, but is also a radical of semiotic representation. This cataloged axiom could be useful, therefore, in working with a premise shared by scientists beginning with Francis Bacon, that all living activity is coded.

J. Z. Young, the neuroanatomist, however, has warned against the possible confusion that might occur when associating human semiotic conventions with other coded forms found in living activity. For instance, Young says that as symbols of the genetic code “the polynucleotide molecules serve to arrange the parts and actions of organisms so that they correspond to, or represent, their environment” (italics his).⁴³ So, for Young, an analogy with human communication codes is both advantageous and severely problematic, because one uses a speech code that is later in the evolutionary process in order to understand a hereditary code which is biologically more fundamental. Our diagrammed catalog should therefore be helpful in distinguishing torsion in human semiotic codes from the various kinds of foldings in the arrangements of DNA molecules. Similarly, the catalog might serve to separate the Euclidean geometries by which we construct signs, tools, and models from the generation of more or less symmetrical forms found in other living activity.