4

Metaphysical Geometry, Alien Attractors, and the Shape of the World Soul

It is the proposal of this chapter that within any imaginary at least in part constituted by transhistorical (ideal) objects, certain distinct figures will function as constellating centers, providing the core structure for further associated sets of figures and ideas. To substantiate this proposal, I provide a detailed examination of two fundamental figurations: that of the circle/sphere and the “bifurcation set” that is the arboreal emanationist schema. No matter what the image of the circle/sphere or arborial schema actually represent, it is certain that these objects have exerted a profound influence on cognitive functioning.

As an introduction to the importance of imagery in cognitive functioning, and the historical shifts that have attended the history of the iconic, I first examine an important notion proposed by Spengler and Couliano: that the beginnings of the modern era are characterized by the iconoclastic program of the Reformation, a program that sought to suppress the phantasmatic imaginary of the Renaissance and of Hermeticism.

The idea that certain geometrical figures have a principle role to play in cognitive functioning, and therefore have an ineliminable functional role within any imaginary, is pursued in the latter part of this chapter by an examination of the ideas of contemporary mathematician René Thom, the well-known originator of “catastrophe theory” in mathematical modeling. It is Thom’s conviction that all phenomena are governed by the functioning of a limited set of geometrical “attractors,” and that the great diversity of phenomena observed in the world is the direct result of the coordinated functioning of these various attractors.

THE PROGRESSIVIST ACCOUNT OF PHILOSOPHY

The principles of explanation that underlie all things without exception, the elements common to gods and men and animals and stones, the first whence and the last whither of the whole cosmic procession, the conditions of all knowing, and the most general rules of human action—these furnish the problems commonly deemed philosophic par excellence, and the philosopher is a man who finds the most to say about them.¹

This is William James’s characterization of philosophy in his Some Problems of Philosophy (1911). And it is far from being an idiosyncratic characterization. Up until quite recently this supernal status of philosophy, this uncontested supervenience over the human sciences, would have seemed almost axiomatic. Even Nietzsche, famous for his uncompromising attack on philosophy in the West, pursued his excoriation as a philosopher. The idea that philosophy is the guarantor of “right thinking” in all lesser human pursuits, that it both provides the tools for the discovery of truth and is the final legislator in all such attempts, is inextricably linked to a particular view of its history. According to this view, a view espoused by such diverse Anglo-American philosophers as Bertrand Russell, G. E. Moore, Alfred Ayer, and W. V. Quine, philosophy has earned the right to be queen of the human sciences through a long and valiant process of ridding itself of its ties to myth and religion. This progressivist picture of the history of Western philosophy only really began to be extensively critiqued in the latter part of the twentieth century, and that principally because of the failure of the logicist project earlier in that century.

The progressivist account of philosophy is really a caricature of the complexities involved in the various developments within Western thought. The “defining moment” of Western philosophy, the moment where thought was revealed in all its abstract purity, differs according to which ideologue of the progressivist account one might ask, but is generally conceded to have occurred some time within the period beginning in the early seventeenth century and continuing up until the beginning of the twentieth century when philosophers attempted to realize the ground of philosophy in logicism. Whatever historical “moment” is championed, the principal effect of this illusory progressivist account is a rewriting of the mytho-poetic locutions of early philosophers in terms of the modern analytic tradition.

Philosophy, according to this rewriting of Western philosophical thought, finds its vocation and form in its distinction from other human pursuits: it is distinct from literature, fable, myth, and the fantastic imagery of mystical illumination. Abstract thought is revealed only when the excrescences of imagery, allusion, and metaphor are expunged; conscientious attention to this abstractive process assures a thinker of the revelation of the “pure shape” of thought itself. Yet this picture of philosophy is quite one-sided. A close examination of philosophical texts also reveals “statues that breathe the scent of roses, comedies, tragedies, architects, foundations, dwellings, doors and windows, sand, navigators, various musical instruments, islands, clocks, horses, donkeys and even a lion,”² as Michele Le Doeuff recounts in her preface to The Philosophical Imaginary. Of course some would say that in philosophy, just as in science, one must occasionally resort to figural analogies to make one’s ideas more clear. The problem with this view, however, is that the force of many ideas throughout the history of Western philosophy is intrinsically reliant on some such organizing figural center. Rather than a superfluous mnemonic device, figuration becomes a structural necessity.

Yet it is certainly true that we can discern decisive “moments” in the history of thought that pinpoint, as it were, determinative vectorial shifts. Spengler has delineated perhaps one of the most significant when he writes:

The reader will not be shocked if we speak of a Baroque, and even a Jesuit, style in psychology, mathematics, and pure physics. The form-language of dynamics, which puts the energetic contrast of capacity and intensity in place of the volition’s somatic contrast of material and form, is one common to all the mind-creations of those centuries.³

This dynamical form-language is one in which the syntax is governed not by figural forms derived from the natural world (as Spengler asserts it was for Classical philosophy), but by an etiolated, geometrical abstraction characterized by its avoidance of images and the imaginal. Following the Renaissance, Spengler suggests, the concept of God loses its “sensuous and personal traits” (as is evident in the carved images associated with medieval Gothic cathedrals, for instance) and becomes increasingly identified with infinite space.⁴ He locates this identification within the “baroque style of apprehension and comprehension,” the infinite/deity identification^*52 being one of his principal representative “objects [for] physiognomic study.”⁵ Perspectival space, the new tonalities in music, and the invention of the infinitesimal calculus are all parts of this morphology, as Spengler sees it. All are manifestations of a style of thinking that has begun to move away from the sensuous apprehension of figural form toward an increasing reliance on abstract, even geometric conceptions.

This is no more clearly appreciated than if we recall Kant’s characterization of his “revolution” in thought: “This attempt to introduce a complete revolution in the procedure of metaphysics, after the example of the geometricians and natural philosophers, constitutes the aim of the Critique of Pure Reason.”⁶

No matter what one may choose to call it, Spengler’s “Jesuit style” marks an important shift in Western consciousness. And this style is intrinsic to an understanding of the transition away from a manner of thought in which images and the imaginal are vertigral, toward one in which increasing abstraction will account for the Cartesian dream of “clear and distinct” thought. But it would be a mistake to take Spengler’s opposition of the sensuous apprehension of form and geometric abstraction as two completely different forms of consciousness with no possible contact or dialogue, for the imaginal elements of both are found interwoven in that form of consciousness Spengler calls “Magian.” Here the geometric imaginary of Arabian arabesques, of Byzantine frescoes and Egyptian hieroglyphs all become aspects of a type of consciousness that was inclusive of the Hermetic imaginary. Abstract thought and the sensuous experience of bodies are not necessarily mutually exclusive, as the previous chapter attempted to elucidate. In foregrounding the importance of the contributions of the Hermetic imaginary, we can apprehend that Magian consciousness takes the form of the dialectical resolution of the contraries of Spengler’s “Classical” and “Jesuit” thought.

THE ECLIPSE OF THE PHANTASMIC

The call to a Christian community based on ideals of purity and the removal of all traces of the Renaissance interest in the phantasmic is one of the primary characteristics of both the Reformation and of Spengler’s “Jesuit” thought. Insisting that there was no other authority than the Bible, the movements initiated by Luther and Calvin mirrored the circumstances under which the early Jewish religion sought to distinguish itself from the Canaanitic cults: by reviling graven images of all kinds. The eventual response of the Catholic Church, rather than violently rejecting the program of the Puritans, was to follow them in their zeal for a reformation in the church. As Couliano notes, it “was along the lines of severity and harshness that the Reformation developed,” such that by the time of the Council of Trent in the latter part of the sixteenth century, the Catholic Church reassigned the Holy Inquisition to be overseen by the Society of Jesus.⁷ Originally forged by the Dominicans^*53 as an instrument against the Cathar heresy, the Holy Inquisition now became the instrument and overseer of the “Jesuit” style of thinking proposed by Spengler. For Couliano, both Catholics and Protestants, Reformation and Counter-Reformation, rather than fighting among themselves, should be seen as vanquishing a common enemy: the phantasmic culture of the Renaissance. This censure of the imaginal signals a “profound change in the human imagination.”⁸ Not only that, but this concordance of Catholic and Protestant through the recognition of their common enemy in the “pagan” Renaissance is one of the principal foundations of modern Western culture itself.

This transition is not without its contradictions and anomalies, however. The emerging new science—represented by thinkers such as Bacon, Kepler, and Descartes—finds its inspiration in the pagan past, particularly in the Alexandrian notion of the micro/macrocosmic couplet and its attendant notion of the harmony of the kosmos. The participatory world formative of Hermeticism was, at this early stage in the transition from Hermetic to modern scientific thought, by no means moribund. While it is quite true that with the advent of the Renaissance we observe a new emphasis on the role of the individual, and a particular emphasis on the operations of the imaginative faculty, the vis imaginativa, we certainly do not see a fully developed “passion for the third dimension” as Spengler calls it, a Nietzschean “will” that serves to functionalize the opposition between a self and the kosmos.⁹ The theory of celestial influx that accompanied a world arranged like a tensile grid of complex polar forces in which human beings were but temporary communicative nodes still undergirded much Hermetically inspired metaphysical speculation.

Invoking a contemporary metaphor, one might say that in terms of the Alexandrian mentalité human beings were regarded more as “receivers” than “transmitters.” In the sixteenth and seventeenth centuries equivalent metaphors perhaps would be the camera obscura (a chamber that receives light) and the magic lantern (a chamber that transmits light). Whatever the choice of metaphor, I would maintain that the valorization of a privileged cogito (Spengler’s term is the Faustian ego) was still largely absent from considerations of humankind’s place in the natural order. Therefore the world was still a place of interpenetrative communication, where the borders between self and other, res cogito and res extensa, were to a large degree mediated by a decidedly diaphanous and osmotic division. In this imaginary, this worldview, the body was influenced by the stars, the humors reacted to the natural elements, and the human mens reflected the movements of the deity after which it was modeled.

Yet it was the very force of such an imaginary that would eventually lead to its eclipse. Key concepts of this pneumatically mediated worldview came to be refined such that Renaissance theories of “natural magic” were perhaps the first step in shifting the balance toward the command-obedience consciousness that largely characterizes the science of the modern era. Figurations from the Renaissance phantasmic imaginary would pass through the barrier of the emerging scientific culture of modernity, becoming in their passage distorted into the figures of the technological dream of a “new world order,” as Faivre has recently suggested.

The seventeenth-century “metaphysical catastrophe” (as Gilbert Durand reprovingly calls it¹⁰) that was the advent of modern consciousness is a catastrophe perhaps more in the sense of René Thom’s concept of catastrophic change whereby two geometrical vectors are enfolded one into another, a process consequently wrought with dialectical tensions (for which idea see later in this chapter). The cusp of this catastrophe may be characterized as a paradoxical moment where the tensions between the Hermetic and mechanical philosophies began to define a new object—modern consciousness.

An example of the paradoxes inhering in this transition can be observed in the spiritual practices initiated by the founder of the Jesuit Order, Saint Ignatius of Loyola. His Spiritual Exercises are to all intents and purposes a reworking of the mnemotechnical processes of the “pagan” Renaissance for spiritual, rather than temporal, ends. The “memory theatre” favored as an intellective archival device by Renaissance thinkers is, in the hands of Saint Ignatius, redeployed to produce an internal phantasmic theatre whereby the supplicant is encouraged to observe (i.e., create a phantasmic double of) themselves participating in a scenario reliving the torments of Christ or the terrors of eternal damnation, for example. The instrument of the Exercises is clearly derived from Renaissance imaginal mnemotechnics, but it is put to completely new uses, in active opposition to the phantasmic imaginary of the Renaissance. The very same operational theory was used by the Jesuits in their attempts at converting the Chinese—to convert the unbelievers one must use their instruments against them: learn the language, understand the “mind-set,” use their arguments against them.

The exercises of Ignatius are perhaps the last example of a way of thinking that assumed the necessity of imagery and the imaginal in human cognition. The new science adopted the Pythagorean faith in number and mathematical description, but increasingly rejected the wealth of imagery associated with the Hermetic worldview. The disappearance of the emblematic books, at one time so popular (and instrumental in assuring the maintenance of the hieratic, “hieroglyphic language” of Hermetic “commonplaces”¹¹), is a good indicator of the process of abstraction from scientific and philosophical discourse of a once rich imaginary.

The lexicon (imagicon? figuricon?) of images was never completely erased, however. In fact, from my viewpoint this is a cognitive impossibility. Human cognition relies on images at a very “deep” level (as a moments reflection on the imagery of “profundity” clearly demonstrates). Certain images are almost ubiquitous in their appearances across disciplines. The fact that they do appear in several quite different discourses strongly suggests these images are important organizing centers—“strange attractors”—of human cognition. It should therefore be possible to create a sort of “philosophical bestiary” in which the ubiquity of these pivotal images is demonstrated not only across time frames in the West, but across cultural/national boundaries as well. Images/figures of this kind may well constitute the semantic primitives that structure the “language of thought” as contemporary philosopher Jerry Fodor imagines it.¹² It is possible, following Fodor’s supposition, that a limited number of recurring figures are instrumental in the development of systematic thought in general, providing crucial semantic content in the form of basic imagistic relationships. Certainly such figures lay behind much of Renaissance interest in the pneumatic/phantasmic language of the soul and mnemotechnical rhetorical devices.

Such primitive figures I will call (for reasons that will become clear toward the end of this chapter) attractors. These figures are not of the same order as, for instance, the figures examined by Michele Le Doeuff in her Philosophical Imaginary; they are not as imaginally rich as Condillac’s talking statue or Kant’s island of truth. Yet they are as potentially rich, in fact more so. This is because they operate at a more primal level, as the initial “engines,” so to speak, that generate ideal objects. They may be regarded as abstract, geometrical figures that have generated and organized certain discourses in the history of human thought. At least, that is my hypothesis.

THE INFINITE CIRCLE

The concept of the infinite is surely one of the oldest known to humankind, and it has had an unarguable determining influence on many different cultures. For Spengler, infinite space was the ne plus ultra ursymbol of Faustian man. Modernity began when the Aristotelian horor infini, the fear of an actual infinite, was transformed into a yearning for the infinite, a yearning clearly seen not only in the spires of the Gothic cathedrals, but (if Spengler had lived to see it) the interstellar probes of our own time.

Yet since the time of Zeno of Elea the infinite was a notion quite ill-defined, a matter for philosophical discussion and religious meditation rather than precise definition. For mathematicians and logicians this state of affairs was remedied with the work of Richard Dedekind in the late nineteenth century. In 1872 Dedekind’s definition of an infinite set was published in Stetigkeit und irrationale Zahlen: “A system S is said to be infinite when it is similar to a proper part of itself; in the contrary case S is said to be a finite system.”¹³ In other words, a set of elements S is said to be infinite if the elements of a proper subset of S’ can be put in a one-to-one correspondence with the elements of S. It is this isomorphic alignment between two series that is the basis for Cantor’s “diagonal argument,” the paradox of “Hilbert’s hotel,” and many others.

Early in the twentieth century, mathematicians Stephen Banach and Alfred Tarski produced a remarkable extension of a paradox first noted by the German mathematician Hausdorff in regard to the congruence of separate parts of the surface of a sphere. Extending the paradox into three rather than two dimensions, Tarski and Banach demonstrated that there is a way of dividing up a solid sphere, a sphere the size of the sun for example, such that no two parts of this sphere will have points in common—that is, they are entirely, geometrically separate—yet without any modification of these parts, they may all be fit into a similar sphere the size of a pea.

This monstrous paradox was demonstrated in the following manner. Divide the sun-sized sphere S into as many parts as you like with the condition that the separate parts are finite (i.e., an integer less than infinity—the number of parts could be conceivably as close to infinity as is tractable). The separate parts can then be denoted as S¹, S², S³, . . . , Sⁿ. The sum of these parts constitutes the sphere S. Now do the same with a sphere S’ the size of a pea, such that you get a series S’ ¹, S’², S’ ³, . . . , S’ ⁿ. Tarski and Banach then propose that if the two spheres have been divided in a suitable manner so that all parts of the sphere S and all parts of the sphere S’ are congruent (that is, they are of the same size and shape), one can order the twin series such that S¹ is in an isomorphic (one-to-one) relationship to S’ ¹ and so on. Now, this process of congruence will encompass not only all the tiniest portions of the pea-sized sphere, but also the most tiny portions of the sun-sized sphere as well. The paradox is derived from the fact that each part of the sun-sized sphere is completely congruent with those of the sphere the size of a pea—each part of the two series being of the same size and shape—and that there is a one-to-one relationship between the two series. In other words, not only can a sphere the size of the sun be rearranged such that it will be exactly the size of a pea, but a ball of Plasticine the size of a pea can be rearranged so that it completely fills the entire universe!¹⁴

The mathematical problem of “sphere packing” (building lattice structures composed of spheres) has produced very real and practical uses in the contemporary world of information technology and computer-aided design (CAD),¹⁵ and is strongly related to the “transfinite” mathematics of Banach and Tarski’s paradox. What is interesting about these mysterious infinite and transfinite mathematical paradoxes is that they have, relatively speaking, only a very short history. It is generally conceded that Georg Cantor (1845–1918) is the person chiefly responsible for the renewal of interest in the idea of the mathematical infinite in the late nineteenth century. He was not the only individual who was interested in the notion of the infinite series at that time, but it was Cantor who produced the famous “diagonal argument”—the idea of which led not only to Banach and Tarski’s paradox, but Gödel’s theorem as well—and the realization that there are not one, but several infinities: “Cantor’s paradox.”

It is in the very nature of the concept of the infinite that nothing about it can be “finally” decided. When considering the infinite series, something will always have to be left unconsidered, unseen—a mysterious excess must be acknowledged, but never known. It is this unknowable beyond that Spengler recognized as the very pattern of Faustian humankind’s desire: always searching, increasing, exploring, extending—a mathematics of the infinitesimal calculus as opposed to the mathematics of the Classical age. Rather than the closing scenes in the history of the infinite, the set theoretical works of the late nineteenth and early twentieth centuries are but morphological transforms, new scenes in a drama that attempts to delimit the labyrinthine beast of infinity.

Yet the nature of the beast is not to be found in its mathematical descriptions. As much as it may temporarily satisfy the mathematical mind, the infinite will always be a metaphysical necessity before it becomes a device for the production of equations. To explore further this assertion, we need to have recourse to another paradox of the nineteenth century.

We owe to Jakob Steiner (1796–1863), in an unpublished manuscript, some of the earliest work concerning the geometric transformation now known as “inverse geometry.” Steiner’s discovery was this: imagine two points P and P’ lie on a ray (a straight line drawn) from the center O of the circle C with a radius r = 0. Now if the product of the distances OP and OP’ is r², then P and P’ are said to be inverse to each other with respect to C. The consequence of this is that for every point outside the circle there is a corresponding point inside the circle. But there is a further consequence of this state of affairs best illustrated with another paradox first attributed to Bernard Bolzano (1781–1848): The inside of any and every circle contains one more point than the portion of the plane outside the circle, regardless of how big or small the circle is. As there is also no outside point P’ corresponding to P when P coincides with the center O in Steiner’s illustration, we can appreciate the connection between these two formulations. Not only that, but the very same considerations obtain when this geometry is extended into three dimensions, defining the “inverse point” in regard to a sphere.^*54

Bolzano recorded many other important properties of infinite sets in his posthumous Paradoxien des Unendlichen (1850). He rigorously demonstrated, for example, that there are just as many real numbers between 0 and 1 as between 0 and 2, and that there are as many points in a 1-inch line segment as in a segment 2 inches long. He seems to have early realized that correspondences exist between the members of an infinite set and a proper subset thereof. Bolzano maintained that there was a link between ontology and a fully developed science based on the objective logical entailment that obtained between the world and scientific theories.^†55 The “circularity” of the idea of the infinite was clearly something that needed to be curtailed if logical paradoxes (and hence, “ill-formed” descriptions of the world) were to be avoided.

Steiner and Bolzano’s use of circular diagrams is a comparatively late manifestation of an image that has a very long history in the West: the image of the wheel or circle. Banach and Tarski’s sphere is a contemporary revisualization of Parmenides’s το εν reality (the One) considered as a giant, unitary, all-encompassing sphere. None of the recent mathematical examples are direct acknowledgments of the ineluctable necessity of invoking such forms, yet they nevertheless participate in the same imaginal field that yields their earlier metaphysical equivalents.

Why do we use the symbol of a circle to denote zero in the West? Is it just another example of the arbitrary nature of signs? Both the concept of, and the symbol for, zero entered Western mathematical thought from India via Arabic mathematics. The word itself is a contraction of the Medieval Latin zephyrum, from the Arabic cifr, which gives us the word cipher. In Arabic it literally means “empty,” a direct rendering of the Sanskrit sunya, “void.” The association with some form of secret writing is first found in the French (chiffre) and Italian (cifra) equivalents. The mysteriousness of the notion of “nothing” having a determinate effect—like a sort of secret code, a cipher—is similar to the mystery inherent in Bateson’s concept of information (difference) having effect/affect in the phenomenal world.

“Nothing” and “something” also seem to be a binary pair. In Nagarjuna’s metaphysics sunyata (the void) is the very ground from which any figure must emerge. According to Gestalt psychology, this figure/ground discontinuity is one of the most primitive (in the sense of “first”) operations behind our appreciation of contrast, of difference. In Hellenistic thought, the phenomenal world was preceded by chaos, which has the literal meaning of a “cavity” or “to gape.”^*56 From this formless ground, reality emerged. Related concepts in Sanskrit include kha and purna. Kha has the meaning of “the hole in the nave of a wheel through which the axle runs” (as well as providing the proto-Indo-European root word for “chaos”), and purna, quite paradoxically, means “plenum.”¹⁶ Of course from a metaphysical point of view, the idea that the plenum and the void are related is not paradoxical in the least, for “the implication [is] that all numbers are virtually or potentially present in that which is without number; expressing this as an equation, 0 = x – x, it is apparent that zero is to number as possibility to actuality.”¹⁷ This is particularly the case in the aforementioned metaphysics of Nagarjuna, a metaphysics that—not coincidentally, according to the vertigralist viewpoint—has often been compared to that of Parmenides in the West.

Coomaraswamy’s thesis in his paper “Kha and Other Words Denoting ‘Zero’ in Connection with the Metaphysics of Space” is that the Indian mathematicians consciously chose their terms for the mathematical concept of zero from an already preexisting body of related metaphysical terms found in sacred literature (the Rig Veda, Upanishads, Mahabharata) and Hindu and Buddhist philosophy. To demonstrate this connection, Coomaraswamy asks the reader to prepare a diagram similar to those of Jacob Steiner mentioned earlier in this chapter here: Draw two concentric circles (in Sanskrit a cakra or mandala) of any radii (but one much less than the other) on a piece of paper and draw a line out from the center of the concentric circles to the circumference of the outer circle.

With the exception of the centre, which as point is necessarily without dimension, note that every part of our diagram is merely representative; that is, the number of circles may be indefinitely increased, and the number of radii likewise, each circle is filled up becoming at last a plane continuum, the extended ground of any given world or state of being.¹⁸

For Coomaraswamy this diagram represents the logical relationship between “zero, inconnumerable unity and indefinite multiplicity”: the blank surface (sunya) is zero because it has no number; the central point (bindu in Sanskrit) is the “inconnumerable” unity, as there cannot be a second center for the two concentric circles; and either of the circumferences represents an infinite number of points. In Sanskrit, this last is ananta, the endless (anta means “end”), the exact equivalent of unendlich in German.

A further connection in this interesting matrix of symbols is that the symbol of the ouroboros (a serpent devouring its own tail; an extremely ancient symbol found in many alchemical texts—perhaps of Egyptian origin¹⁹) is inherent in the name of the man who reintroduced and foregrounded the concept of sunyata within Buddhist metaphysics, Nagarjuna. The name literally means “Arjuna of the serpents” (Arjuna is the name of the prince who is in dialogue with the god Krishna in the Bhagavad Gita). The Buddha himself is often referred to as “King of the Nagas”—king of the serpents. Here we recognize again the serpent as symbol of knowledge, or more precisely, gnosis. Whether tempting Eve in the Garden or inspiring the pythoness at Delphi, the serpent seems to be a symbolic recognition of gnosis across many cultures. The reason for this, I think, is that the serpent or dragon is a symbol that represents the central nervous system including the spine and brain. This symbolism relies not only on the resemblance between the CNS and a serpent or mythical dragon, but also, and most importantly, on the circuit that it represents experientially: the pneumatic circuit of kundalini (in Vedic yoga) or the circuit of the du and ren channels (known as the “Great Waterwheel” in Daoist alchemy) of traditional Chinese pneumatology.

The concept of circularity within gnosis is inherent in the elaborate cosmo-mythology of the various Gnostic sects. As Hans Jonas has demonstrated, the theoretical content of Gnosticism, the worldview it proposes, contains the process and ends of gnosis within itself. It is a self-referential system that moves around the central axis of the process of a mysterious knowing—gnosis—the loss of which is the very cause of the phenomenal world and the regaining of which is the culmination of the Gnostic path. Gnosis, or rather the lack of it,^*57 is the cause of the creation of the cosmos through the actions of an ignorant and arrogant Demiurge. And it is gnosis that guarantees the completion of this spiritual/cosmological/historical cycle: true knowledge of the actual nature of reality, of secret names, rulers, and mediatory levels of existence, enables the Gnostic to attain salvation.²⁰ Salvation is but the culmination of a process in which we find the “last things answering to the first, the reversal of the fall, the return of all things to God,” as Jonas notes.²¹ This is a metaphysic of “pure movement and event” in which the ontological conception of knowledge (gnosis) holds to a dissolving of matter such that, as with Hegel, “substance is subject.”²²

We should also note the circular form, the cyclical nature of time in Gnostic thinking. This links the astronomical/astrological meditations of the Alexandrian thinkers with that of the Gnostic sects. The precession of the heavens was thought to have started from some initial state (that is, pattern of stars) that obtained at the moment of creation, and the stars would one day cyclically return to this same state/pattern—the so-called Great Cycle. This notion probably dates from “before the Magi,” as Doresse says, and gave rise to a cult of time in many ancient Oriental and Middle Eastern cultures. In these cults time was represented by an anthropomorphic figure who was both the generator of and ruler over humankind.²³ This cyclical, eschatological notion, combined with the concept of celestial influx that engineered moments of isomorphism between the heavens and human physiology, would have logically led to the fixing of the macro/microcosmic couplet, probably in early Babylonian times.

It seems to me that we observe again the periodic return of the importance of the cyclical form in the centrality given the idea of circular process within cybernetics, particularly in regard to the study of living systems. Wiener’s notion of “feedback,” where the self-regulatory process is sustained by the continual exchange of energy in the form of a loop between organism and environs, mirrors the overall “shape” of both cyclical conceptions of time and the Gnostic process where “last things answer to the first . . . and all things return to God.” In fact, aspects of the Gnostic worldview seem to undergird the central contribution of Wiener’s cybernetic theory, for just as Gnosticism conceives of cosmological events and processes repeated and reflected within the individual self, so too does cybernetic theory consider the process of feedback as occurring both within organisms (their internal environment) and between organisms and their world (outside environment.)

Wiener is one of a small number of thinkers who, inspired by the example of certain mechanical devices, attempted to delineate an apparently “new” fundamental principle of nature. Yet the principle of feedback was used in the construction of the steam engine long before it was given theoretical treatment. James Watt patented the flyball governor device in 1782. In this device two metal balls attached to a vertical spindle are set in upward motion by the centrifugal force imparted by the spindle, this force counteracted by the force of gravity that tends to keep the balls close to the same shaft that imparts their upward motion. Increases and decreases in the speed of the spindle force the balls up and down. This device is attached to a valve between the boiler and steam engine so that if the speed exceeds a set value the valve closes, and if it falls below this value the valve opens. The principle behind this action, the achievement of “homeostasis” through “feedback,” is the same principle behind contemporary thermostats and many other “selfregulating” devices.

It was not until the following century that James Clerk Maxwell published the first theoretical treatment of such a device in his 1868 paper, “On Governors.” In the twentieth century, eighty years after Maxwell’s paper, Norbert Wiener invested the principle of feedback with an explanatory power that neither Maxwell nor Watt could perhaps have imagined. Like Claude Shannon, Wiener endeavored to equate nineteenth-century thermodynamic conceptions of equilibrium and entropy with the dynamics of information exchange as exhibited by computing devices of the late 1940s and early 1950s. In Wiener’s new jargon, Watt’s flyball governor becomes one of the first examples of mechanical “telltales” or “monitors”: “It is the function of these mechanisms to control the mechanical tendency toward disorganization; in other words, to produce a temporary and local reversal of the normal direction of entropy.”²⁴

It is the extension of this fundamental concept of Wiener’s to living organisms that allows thinkers such as Michel Serres to consider the phenomenon of life as a “temporary and local” reversal of entropic decay. What is intriguing in this twentieth-century isolation of a “fundamental principle” that seemingly pertains to both mechanical devices and living organisms is the centrality afforded “information exchange” between organism and environment. In this cybernetic view, life is somehow defined by the constant “monitoring” of the exchange of signals both within the internal organismic economy of an individual and between the environment and the individual. It is the circularity inherent in this conception that may well demonstrate the necessary recrudescence of quite ancient notions of cyclic procession, perhaps figured most famously in the image of the ouroboros.

The foregoing discussion would seem to indicate that the figure of the circle was a deep, metaphysical structure long before it entered Western and Arabic mathematical notation. Coomaraswamy’s diagram, however, has another significant feature at its center. The central point of the twin circles, the bindu, represents the plenum of all possible numbers represented on either circumference (as each circumference is composed of an infinite number of points). In terms of his diagram, any possible radius drawn from the center “points to” a point on both concentric circles; therefore each point on the concentric circles has its origin in the central bindu. But this central point is also nothing, zero—as the very notion of a point is that of a dimensionless, spaceless quantum, as Euclid early maintained.^*58 Also, as Coomaraswamy says, “The mathematical infinite series, thought of as both plus and minus according to direction, cancel out where all directions meet in common focus.”²⁵

Coomaraswamy notes that the Hermetic dictum, “as above, so below” is illustrated by the fact that any radius connects the center with any number of points on the two circumferences. The diagram, then, can be read as an image of the relationship between individual selves and the cosmos.

He notes that it is in connection with the symbol of a wheel (specifically the wheel of the solar chariot) that one finds the most significant ideas that are later manifested in mathematical concepts. The concept of the solar wheel is found in a number of very ancient Hindu texts that see “the Year as an everlasting sequence [that] is thought of as an unwasting wheel of life . . . in which all things have their being and are manifested in succession; ‘none of its spokes is last in order,’ [as the] Rig Veda [says.]”²⁶ In terms of contemporary thought, the idea of an “unwasting wheel of life” being “manifested in succession” is akin to Clausius’s first law of thermodynamics: the energy of the universe is constant; it may transform—into matter (Einstein’s matter/energy equation), for instance, manifest in succession—but the total amount of energy in the universe is always conserved, unwasted.

HIERATIC SPACE

Coomaraswamy points out that the use of verbal terms for space (kha, sunya, and so on) as symbols for zero represent not the physical space of the post-Newtonian era, but a “principial space without dimension,” although this space is the very matrix from which dimension arises.²⁷ Here we can appreciate the correspondence between this conception of space and the primeval chora of Plato’s Timaeus—and both chaos and chora find their root term in kha. Not only that, but according to the Upanishads the locus of this space is within. Coomaraswamy quotes the Maitri Upanishad: “What is the intrinsic aspect of extension is the supernal fiery energy in the vacance of the inner man.”²⁸

Furthermore, according to the Chandogya Upanishad, this space is in the heart, “where is deposited in secret all that is ours already or may be ours on any plane (loka) of experience.”²⁹ It is important to recognize the ease with which we unthinkingly invoke our contemporary (that is, post-Newtonian) concept of absolute space whenever we encounter spatial terms or terms of locatedness. Newtonian space is like the “space” inside a jar or room. This was/is a corollary, in scientific terms, of the (then) comparatively new perspectival space of the Renaissance. Objects within this space no longer varied in size or location in the picture plane according to their importance—the hierarchic space, in other words, of medieval allegory. Renaissance perspectival space was the representation of an absolute space (as we would now say) where objects were situated according to the mechanical exigencies of (for example) Albrecht Dürer’s viewing frame. This frame was an upright wooden easel divided by evenly spaced strings into a grid. This grid corresponded to a similar grid inscribed on the paper on which the artist would draw the image viewed through the easel. When situated behind this easel, the artist would remain very still while attempting to draw those parts of the scene divided by the grid before him in the corresponding squares of his paper grid—ultimately producing the illusory space of single-point (“vanishing point”) perspective.

For Plato and the thinkers whose work is recorded in the Rig Veda and Upanishads, locus was another word for “place”—things had their place (locus), and they were arranged in a particular order (relationship to each other) until they exchanged places through the impressing of some force.

The Stoic philosopher Simplicius recounts the words of Theophrastus in explication of this concept.

Space is not a reality by itself but is defined by position and order of the bodies according to their nature and faculties, as is the case with animals and plants and all non-homogenous bodies which either have souls or are without souls but have the nature of a structure. . . . Thus each, being in its proper place, is said to have a specific order, especially as every part of a body desires and strives to occupy its own place and position.³⁰

As a consequence, movement for the ancients was only explicable as an exchange of loci.^*59 In contrast, Newton’s absolute space was a space akin to that of the Renaissance painters, an attempt to replace the relational space of the ancients—a space that was a predicate of individuated beings—with one that better supported his conception of relative motion, a motion relative to absolute, empty space.

A hieratic space that enfolds a plurality of relations—a “field” plenum as opposed to an empty container—lies behind the reception of the mathematical concept of zero in the West.^†60 Coomaraswamy quotes Bhaskara, an early twentieth-century mathematician, describing his conception of ananta:

This fraction of which the denominator is zero, is called infinite quantity. In this quantity consisting of that which has cipher for its divisor, there is no alteration, though many be added or subtracted; just as there is no alteration in the Infinite Immovable at the time of the emanation or resolution of worlds, though hosts of beings are emanated or withdrawn.³¹

Bhaskara clearly was fully aware of the ancient metaphysical background to his mathematical thoughts. According to Coomaraswamy, it is by no means surprising that mathematical terminology should have been generated from previous metaphysical terms and ideas, as it is in the very nature of Indian thought to proceed from the general, the universal, to the particular. He notes that the Indian classification of traditional literature upholds this relationship to traditional knowledge (as principally expounded in the Vedas). The sciences of grammar, astronomy, law, medicine, architecture, and so on are all identified as Vedanga, the “limbs or powers of the Veda,” or alternatively as Upaveda, “accessory with respect to the Veda.”

At this point one should also note that the traditional Scholastic division of learning into the quadrivium (arithmetic, geometry, astronomy, and music) and trivium (grammar, rhetoric, logic), the combination of which produced the seven “liberal arts,” reflects a similar relationship in the West. Here the acquisition of knowledge is associated with the celestial archons, the seven stars or their personifications—an atavistic recollection of the Gnostic program.

THE SHAPE OF THE ANIMA MUNDI

The circle and its center, according to the above examination, represent the interpenetration of the world and the self. We find this particular figuration across a wide range of Western texts. In Dante’s La vita nuova, we find Love, the “glorious Lord,” saying, “I am as the center of a circle, to which the parts of the circumference stand in equal relation; but thou not so.”³²

In Truth’s Golden Harrow (see chapter 4), Robert Fludd describes the Philosophers’ Stone in this manner: “It is the essential or formal centre and circumference, the beginning and the end, the all in all . . . the middle and central soul of the sun.”³³ (One should note in passing the striking parallel with the Vedic concept of the bindu and Coomaraswamy’s note that the chakra is most often associated with the solar wheel or chariot.)

There is a common source for the above images of the circle and bindu, “centre and circumference.” The adage Deus est sphaera cuius centrum ubique, circumferentia nullibi first appears, according to Koyré,³⁴ in the pseudo-Hermetic Book of the 24 Philosophers, an anonymously compiled text of the twelfth century, but it assuredly antedates his ascription. The idea of the sphere of Being (in its theological [“God”] and metaphysical [the “Real”] forms) is of course first found in the West in Parmenides. It is considerably later, however, that we find a redescription that approximates the Hermetic adage.

The saying is usually first attributed to Nicholas of Cusa (1401–64), where it is used to intuit the notion of infinite space. Koyré admires the “boldness and depth” of Nicholas of Cusa’s cosmological thought when he transfers to the universe itself the pseudo-Hermetic conception of God as “a sphere of which the center is everywhere, and the circumference nowhere.” Of course the conception is only “pseudo-Hermetic” for Koyré because it is not explicitly found in the Hermetica as they have been collected. Other than that, it is quite in keeping with Hermetic thought. We find the alchemist Michael Sendivogius (a considerable influence on Robert Fludd and others) stating, “Nature, then, is one, true, simple, self-contained, created by God and informed with a certain universal spirit. Its end and origin are God. Its unity is also found in God, because God made all things.”³⁵

The image of the circle/sphere and bindu are not explicitly described, but the formative structure, the attractor, is certainly present when Sendivogius notes the “true, simple, self-contained” character of Nature. “Nature” for Sendivogius was equivalent to our modern conception of the “Real,” and should not be confused with the contemporary usage of the word nature to mean the natural landscape. It should also be borne in mind that what Spengler calls “Magian” consciousness is one of the determining characteristics of the alchemical/Hermetic pursuit. The Faustian (modern) conception of a world governed by dynamical forces is the antithesis of a consciousness that sees the essence (“nature”) of the world as truth, singular (“simple”), and self-contained (spherical).^*61

Nicholas of Cusa was no doubt strongly influenced by the works of Plotinus. Contemporary scholarship has demonstrated that the influence of Plotinus extended throughout the medieval period into (and beyond) the Renaissance, a trajectory that was until recently unacknowledged. Scholars have for a long time accepted that the honorific “the Philosopher” was applied by medieval scholars to Aristotle. It is now thought that when some medieval thinkers refer to “the Philosopher” they are actually referring to Plotinus and not Aristotle—in fact this famous epithet may have originally been applied to Plotinus in the first place.³⁶ It is likely, for instance, that Plotinus’s image of the soul as being like a mirror was determinant in the medieval conception of the mind (mens) as the “mirror of nature.”

It is in the Sixth Ennead (6.9.8) that we encounter Plotinus’s use of the circle/sphere attractor: “Every soul that knows its own history is aware . . . that its movement, unthwarted, is not that of an outgoing line; its natural course may be likened to that in which a circle turns not upon some external but on its own centre, the point to which it owes its rise.”³⁷

Plotinus, first, wants to distinguish his conception of the soul’s movement from that of the atomists who hold that all movement is linear; and second, he wants to establish a limit to its trajectory (as an outgoing line, under ideal circumstances, and in Euclidean space—the “Alexandrian” space of Plotinus’s times—continues on forever). For him the soul’s circular movement “about its source” makes it “divine in virtue of that movement.” He further states that the soul is not a circle in the sense of a geometrical figure, but in the sense that it both enfolds, and is enfolded by, the Primal Nature. Consequently the soul “owes its origin to such a centre and still more that the soul, uncontaminated, is a self-contained entity.”³⁸ It is this circular soul that is part of the “centre of all centres, . . . just as the centres of the great circles of a sphere coincide with that of the sphere to which all belong.”³⁹

What is this “sphere to which all belong”? Consider, first, a contemporary usage of the word sphere: we talk about a “sphere of influence” and we may be involved in the “political sphere” or something may be “outside our sphere of interest.” The meaning of the term in these colloquialisms is clear: sphere means a delimitable space containing a collection of things. The Sanskrit word loka (our “locus”) literally means “world”—and what is a world but a delimited set of things? The world for Plotinus is the “closed world” (delimited set) of the Aristotelian/Ptolemaic universe, the system of concentric spheres represented by the armillary sphere that stood in every medieval scholar’s study.

Plotinus’s whole, his “sphere to which all belong,” is the great sphere of the anima mundi, and the “great circles” of the universal sphere are individual souls and their life trajectories, a series of theoretically infinite events/beings constellated around the central, universal soul (the anima mundi). It should be observed that this figuration necessarily implies that the Great Soul and all lesser souls are like fractal dimensions of each other, spheres within spheres, or the infinity of two-dimensional souls (great circles) that together compose the three-dimensional anima mundi. Like the circle and the sphere, the anima mundi and individual souls share only a difference of dimension. In terms of its geometric imagery, the “great circles” of the sphere are those that we would now call the geodesics, an imagery that strongly links Einstein’s eventual conception of the universe as the hypersurface of a hypersphere (as considerations of the effects of geodesics in non-euclidean space aided Einstein in his reconceptualization of gravity and hence space-time) with the Plotinian conception of the universe. In other words, the insistence of the attractor of the circle/sphere was formative in the shaping of both Einstein’s and Plotinus’s conceptions, irrespective of their completely different temporal and cultural milieu.

Furthermore, it requires very little imagination to realize the close proximity between Plotinus’s vision of the kosmos and that of the contemporary theory of the big bang generation of space-time. Apart from the fact that contemporary accounts factor in time as the fourth-dimensional coordinate, and regard the universe as dynamical, the big bang theory resembles in its broad outline the emanationist schema of Plotinus: a geometrical point of infinite “mass” expanded equally in all directions, an image that naturally conjures up the concept of a sphere. And Einstein, of course, regarded the universe as a closed system in the shape of a hypersphere. For Plotinus the expansion from a supernal point (the Primal Nature) happened outside of time, therefore there was, technically speaking, no “expansion” at all. As with Parmenides, the One Sphere—self-contained Being—just is, by definition.

It also requires very little imagination to perceive that beyond these textual sources there must be an attractor that is both subsistent and insistent, perhaps one of a very few generators of logical objects that have helped to create the geometrical imaginary. The geometrical image of circle and central point, an image inscribed by every child, compass in hand, who has studied geometry, is a glyph representing a number of important metaphysical relationships. It is these relationships that are important, and not the fact that this particular symbol is found in many cultures and at different times. As Coomaraswamy indicates, this primal image serves merely as a mnemonic to ground and inculcate the aforementioned metaphysical relationships. It is not surprising, then, that we should find this particular image repeated in many forms within the imaginary of Hermeticism. Nor should it be surprising to find it transcending the barrier of the scientific revolution to reemerge in contemporary figurations describing the origins of the universe or its ultimate shape. It is the richness of the Hermetic imaginary, as well as the importance of its central ideal objects, that makes the periodic reappearance of the metaphysical relationships held within its figurations an almost foregone conclusion.

As a relatively simple geometric figure the circle/sphere was able to pass through the iconoclastic program of the Reformation without resistance. It is one of a small number of such figures within the geometrical imaginary, the simple outlines of which successfully masked the richness of its metaphysical associations and connections. In further support of this idea, I will briefly examine an equally important figure within the geometrical imaginary.

ARBOR VITAE

Why does a neuron look like a tree or perhaps a rhizome? The key to this seemingly inane question lies in the notion of “looks like.” A thoroughgoing positivist would pronounce that we are misled by the phrase looks like: a neuron actually looks like what it is, a neuron. But this, I think, would miss the point entirely. It is by no means accidental that we say a neuron resembles a tree; rather it is directly contingent upon the fact that we are immersed in a history of symbolization constellating around the image of a tree. In short, we think it looks like a tree because we think it has something to do with thinking!

The foregoing needs to be clearly explained. It is possible to hypothesize that the neuron would never have been found if it did not in some manner resemble a tree, plant, or rhizome. Imagine rooting around in an encephalon without knowing what you were looking for, idling about, as it were. What is inside the brain? you are asked. “Nothing but mush,” you would most likely reply. Now consider that you already know that human knowledge, human thinking, looks like a tree—how would you fare if you vaguely hoped to find the physical substrate for human thought in the brain? I wager that you would find the neuron within minutes.

Louis Rougier in his La Métaphysique et le Langage, in a remarkably concise passage, attempts to sum up the entirety of Western thought (and that of the Platonic/Aristotelian East).

For centuries Byzantines, Syrians, Jews, Arabs, and the Latin West learnt [sic] the same logical system, the same rudimentary ontology comprising “the tree of Porphyry,” that is to say, the ontological hierarchy of species and genus, the theory of categories, the theory of transcendentals, that of substance and accidence, form and matter; creating a common intellectual outlook which consisted of deducing the structure of reality from the analysis of language, discussing concepts instead of observing the facts of experience. They held as adequate the conceptual division which discursive thought imposes on things by explaining the world in terms of essences, predicate, substances and accidents.

It is this common mental attitude which led to the belief in the existence of a universal reason and its aptitude for elaborating a philosophia perennis.⁴⁰

Keeping in mind provisos concerning the adequacy of grand generalizations, the foregoing passage seems to suggest that the central figure of Porphyry’s “tree” may be fundamental to any thoroughgoing analysis of the development of Western thought. Yet Rougier probably gave little thought to the origins of this key image. From where exactly did this image arise? I do not mean by this question where did the image of Porphyry’s particular tree originate—the history of its dissemination in the West is reasonably clear. We know that if it was not for Boethius who wrote about it in his translation of (and commentaries on) Porphyry’s Isagoge (itself an introduction to Aristotle’s Categories), then the subsequent development of the concept of Western rationality and logic as outlined by Rougier would perhaps have never occurred in the way it has. But why a tree in the first place? And is there some intrinsic connection with rationality and logic quite apart from the fortuitous history of translation, commentary, and adoption of rhetorical figures in the early medieval period?

I propose that the attractor of the tree is another generator (of logical objects) that subtends many cognitive operations. That the disputatio so beloved of the Scholastic philosophers, the arguments per genus et differentium between Peter Abelard and William of Champeaux of the twelfth century, partakes of this form (via Boethius) is not, to use the locution of medieval logic, an accident of history, but an ineliminable characteristic of it. I am not disputing the fact that the “tree of Porphyry” was a mainstay of medieval learning, that his Isagoge wherein it is found constituted one of the central texts by which the successive glosses of littera, sensus, and sententia were taught and consolidated in the developing institution of the university in the twelfth century and onwards. This would constitute a rather standard history of the textual, rhetorical development of the Western intellectual tradition, demonstrating indeed the “construction” of “rational man” after the manner suggested by Michel Foucault.^*62 Even if this interrelated series of events and texts, individuals and institutions had never occurred (as unlikely as that seems), the treelike object would inevitably appear somewhere in our intellectual development.

While the figures of the tree attractor and the circle and/or nested-circles attractor are not logically connected as organizing figures (that is, they represent two different strategies for cognitive organization), their appearance as figurative mnemonic devices are historically coincident. Images of both attractors first appeared in the West in the seventh century. In his De natura rerum, Isidorus Hispalensis (Isadore of Seville) explicitly called for the use of rotae (circles, wheels) to illustrate his ideas, a request that was to have an influence that extended greatly beyond the illuminated pages of his work.⁴¹ Even in pre-Carolingian times, Isadore of Seville’s text was often called the Liber rotarum, rather than its original title. One of the main uses of the rotae in this work was to demonstrate and explain the calculations of the “wheel of the seasons,” “wheel of the month,” the “wheel of consanguinity” (i.e., of kinship), and “wheel of the planets.” The turning of each of these wheels ultimately derives from the precession of the heavens. In his Sententiarum Isadore of Seville noted that individuals were “another world, created from the universality of things in abbreviated fashion.”⁴² His recognition of the micro/macrocosmic couplet is an important key to understanding the logic of the almost universal use of rotae illuminations in medieval times. Information about planetary periods was widely disseminated in “handbooks” of the medieval period, but it was Isadore of Seville who used the form of a circle in his rota planetarum to illustrate astronomical calculations. One of the main ecclesiastical uses of astronomical calculations was to date Easter (a moveable feast) in any given year. This particular calculation gives us the first historical appearance of the word computus in the sense of a mathematical procedure: computus was the name given to all “Easter-dating” material and consequently is another link in a series of associations that paradigmatically links thought itself to celestial observation.

Isadore of Seville was also the progenitor of the use of arbores in his manuscripts. Perhaps the most striking feature of Scholastic philosophy is its insistence on the art of drawing distinctions, and the image that seems to most naturally illustrate this dichotomizing procedure is that of a branching tree. The two strongest textual authorities for this image were Boethius’s “tree of Porphyry” and the Tree of Life (lignum vitae) and the Tree of the Knowledge of Good and Evil (lignum scientiae boni et mali) of the Garden of Eden mentioned in Genesis.⁴³ Probably the earliest use of an arbor is found in Isadore of Seville’s Liber etymologiarum sive originum, in which one finds a number of “consanguinity trees” to illustrate genealogy. This arboreal genealogical structure would be expanded to encompass a large number of discourses in the centuries to come such that his Etymologies would have a profound effect on the way knowledge was imagined. It would be no exaggeration to say that we know that knowing looks like a tree because Isadore of Seville first demonstrated it to be so.

While it might seem “natural” to use an arbor to illustrate the points of bifurcation in the Scholastic dichotomizing process, Isadore also used the circle diagram to illustrate contrariety in a wide number of disciplines (this structuring is still put to use today in the “color wheel” shown to every undergraduate art student, for example). In this schema the rays drawn from the center of the circle represent divisions of the original point, and rays traveling in opposite directions from the center represent contraries or opposites. The use of the circle diagram to represent contrariety does not seem so immediately “natural” as that of the arbor to represent dichotomies, but if one considers the originally Pythagorean idea that contraries are necessary aspects of an original unity, then it does not seem so surprising. The central idea in both structuring processes is that what needs to be imagined is the digital choice between two opposing states of affairs. This is the “essence” of computus, of computation.

At any given time there is only a—usually quite small—denumerable set of solutions to any given problem. These solutions may be different in any given age or context, but that does not matter. The fact remains that the choice of solution is essentially computational: a choice between “yes” and “no,” “on” or “off.” Both Daoist and Epicurean sects, for example, saw the pursuance of an effective philosophical life as a choice between two opposing worlds—one could either choose to enter into politics and the world of social interaction or wholly retreat into nature where only like-minded individuals would gather together into small self-sustaining communities. The Epicureans, and most neo-Daoists, chose the latter course. While complex considerations informed this decision, this is essentially an example of action determined as the result of a simple binary choice.

The dualistic structure of Gnosticism and Hermeticism is another case of a structural doubling that may be schematized as a binary series:

			Phenomenal World		True World
			Manifest Meaning		Secret Meaning
			Analogy		Anagogy
			Body		Spirit

This twin series is by no means exhaustive in regard to the structuring series implicit in both Gnosticism and the core correspondences of Hermetic thought. Rather it is the beginning of a strictly logical process that organizes itself around a primordial division: the One (originary synchrony) divides into Two (primal diachrony), a necessary first step in the generation of all extensive sets. This primary division is then subject to further logical divisions, following what we nowadays would call a cladistic “decision tree” model.

This is perhaps best illustrated if we consider the construction of a computer program. Nearly all computer programs, no matter which particular “programing language” is used, follow the basic structures and rules of formal logic. All “machine code” is designed to instantiate a logic of binary decisions that follows the form of: “If (a certain input is received) then proceed to line B; if not (the aforementioned input) then proceed to line C,” and so on. This simple structure of “If . . . then/If not . . . then” is repeated throughout the length of the program, and this program can be designed to produce anything whatsoever, from a digital image to the complex computations of lattice structures in an artificial crystal. At the core of the functioning of contemporary computers we find Aristotle’s seemingly unassailable metaphysical law enacted: the “law of contradiction” predicated on the contraries of being and nonbeing, here transformed into its cybernetic version of information/absence of information.

The question naturally arises as to why such a division should obtain, why it is the very beginning of logical structures. I believe that here we can adopt Jung’s explanatory mode (although this particular analysis, I think, is not found in Jung) and find the answer in humankind’s earliest perceptions. Elaborate mythologies surround the two principles of light and darkness in nearly every culture, from the “primitive” to the most contemporary. These two principles are probably the engine behind the privileging of the function of sight in Western culture, and paradoxically, its denigration in certain philosophical formulations.

Every child eventually comes to distinguish between two worlds. At first these two worlds manifest a certain degree of “overlap,” of shared orientation—the world of participation mystique—later they are cognized as being quite distinct. Representatives of this primal series are the worlds of light and darkness, inside (the body) and outside (the body). The mechanism by which these two worlds are affected is of the simplest kind: the opening and closing of the eyes, the subjective reflection of the experience of day and night, eating and defecating, skin as container, and so on. Eventually the principle of bifurcation asserts its determinative influence and we recognize its operation not only inside us (in the discriminatory process), but outside us as well—in the shape of a tree, the bifurcating course of a river and its streams, in the necessity of a horizon between earth and sky.

THE SEVEN CATASTROPHES OF THE WORLD

It was perhaps in the early Pythagorean and neo-Pythagorean communities that the notion of number was first given a central role as the formative function behind phenomena. For the Pythagoreans, number was much more than the notation necessary to record the results of the addition, subtraction, or multiplication of heterogeneous objects, rather, number was directly constitutive of all phenomena (an idea found so risible by Aristotle in his De anima). Despite Aristotle’s ridicule, this notion has had a great effect far beyond the confines of the early Pythagorean communities. We in fact find the idea of the determinative functioning of number appearing throughout Western history. Saint Augustine, for example, held that the human mind is an image of the Trinity, an idea that would be echoed by Jacob Boehme centuries later: “All is in Man, both Heaven and Earth, Stars and Planets; and also the number Three of the Deity.”⁴⁴

The macro/microcosmic couplet as expressed by Boehme was the “essence of his teaching” as Nicolson says.⁴⁵ And his trinitarianism should not be glossed over too quickly. It is highly likely that when Boehme said that the number three of the deity was also in humankind, his meaning was that the number three was actually constitutive of human beings, a hypostasis made flesh. This was no arbitrary resemblance for Boehme, but an unmistakable expression of the divine language of forms in nature.

There are especially seven forms in nature, both in the eternal and external nature; for the external proceed from the eternal: The ancient philosophers have given names to the seven planets according to the seven forms of nature; but they have understood thereby another thing, not only the seven stars, but the sevenfold properties in the generation of all essences: There is not anything in the Being of all beings, but it has the seven properties in it, for they are the Wheel of the centre, the cause of Sulphur, in which Mercury makes the boiling in the anguish-source.⁴⁶

Boehme stresses an important consideration in regard to the “seven influences”: the stars merely reflect the influence of the seven forms of nature, mysterious formative principles that account for the “law of seven” (note, too, the appearance of the “Wheel of the centre” again). Can there be anything to this “seven-ness” apart from the quaint repetition of archaic cultural remnants?

Contemporary mathematician René Thom insists that there must be. For Thom there are seven elementary catastrophes that describe all forms, all phenomena in the world. I suggest that we find in Thom’s work the reappearance of certain key aspects of the Hermetic imaginary, certain signature “ways of thought” that have resurfaced on the other side of the barrier of the “scientific revolution” and of scientific thought in general.

In his Structural Stability and Morphogenesis, first published in the early 1970s, Thom proposes to describe the deep mathematics that demonstrate the existence of seven geometric principles of the phenomenal world. What is immediately intriguing about this work is that it proposes not a quantitative analysis of a system, but a qualitative one. Thom’s theory of catastrophes makes no pretension to be predictive in the manner of Newtonian science. In this way it is a theory akin to Darwin’s theory of natural selection: it is an attempt to bring together a large number of disparate phenomena under the one explanatory umbrella—in Thom’s case, the fundamental geometrical structures that govern all major changes in any system. And just as Darwin’s theory of evolution is not—cannot be—predictive, so with Thom’s theory of catastrophes.

A catastrophe is Thom’s name for a major change in a system (described topologically) that occurs following a minor change in the external variables used to describe that system. His theory of catastrophes, however, is much more than an exercise in applied topology. Despite his—admittedly not very strong—protestations to the contrary, catastrophe theory makes a very strong metaphysical claim. Everything in the phenomenal world is a type of dynamical system that can be described by one of seven “catastrophe sets” (i.e., the parameters describing the system that indicate governance by one of the seven catastrophes). The geometrical shapes of the seven “elementary catastrophes” bear an iconic relationship to their names: the “cusp,” the “fold,” the “swallowtail,” the “wave” (hyperbolic umbilic), the “hair” (elliptic umbilic), the “mushroom” (parabolic umbillic), and the “butterfly” catastrophe set. Each of these is a recognizable “shape” imaging the topological distortions that the system undergoes when one or more of its variables are “tweaked.”

Paradigmatically, a system shifts from one relatively homeostatic (stable) state to another—the beginning and end states are attractors, and the shift process is the topological space Thom calls a catastrophe. Morphogenesis, the creation and mutation of forms, is “described by the disappearance of the attractors representing the initial forms, and their replacement by capture^*63 by the attractors representing the final forms.”⁴⁷ Concerning “attractors,” Thom admits, “little is known of the topological structure of structurally stable attractors. . . . It seems to me, however, that this is the essential geometric phenomenon intervening in a large number of morphogenetic processes, such as changes of phase in physics and the phenomenon of induction in embryology.”⁴⁸

Thom’s attractors, like the catastrophe itself, cannot be observed directly. They are organizing fields (“morphogenetic fields”) occurring in a four-dimensional space. All that can be “observed” is a succession of three-dimensional catastrophes: an evolving catastrophe set. Nonetheless Thom, through numerous examples, believes that we are forced to infer the presence of these organizational geometries. “The essential point of view advanced here is this: the stability of every living being, as of every structurally stable form, rests, in the last analysis, on a formal structure—in fact, a geometrical object—whose biochemical realization is the living being.”⁴⁹ And Thom means every form. He sees only two basic divisions between structures: static forms like stones or geological formations and metabolic forms that include, for example, a jet of water, a plume of smoke, a dancing flame, and a human being!⁵⁰ This geometrical idealization allows him to say, in the spirit of Synesius, that “every animal is topologically a three-dimensional ball.”⁵¹ “Complicated forms in the world, like a cloud or a tree, are not governed by one catastrophe, but rather are the result of the accumulation of numerous elementary catastrophes.”⁵²

What is important though, is that these catastrophes can be described independently of the physical substrate that may instantiate them: “One of the basic postulates of my model is that there are coherent systems of catastrophes (chreods) organized in archetypes and that these structures exist as abstract algebraic entities independent of any substrate.”⁵³

This allows Thom to make some surprising claims for his studies in morphogenesis. He insists that we must “accept the idea that a sequence of stable transformations of our space-time could be directed or programmed by an organizing center consisting of an algebraic structure outside space-time itself.”⁵⁴

Thom borrows the terminology of the theoretical biologist C. H. Waddington when he introduces the term chreod. A chreod (creode) is Waddington’s term for what he calls the “canalization of development” of a zygote, where its histogenesis is governed by “a restricted number of end states among which there are few if any intermediates.”⁵⁵ A chreod then is a “pathway of developmental change” involving the actions of a “considerable number of . . . systems, and these are interrelated by some feedback connection in such a way that the developing system, if diverted to a minor extent from the creode, has a tendency to return to it.”⁵⁶ For Thom a chreod differs from a morphogenetic field only in the “privileged role allotted to time,” that is, it is the term for a developmental description of the field.⁵⁷ All creatures “tend along some convergent funnelled route”—a chreod—and the maintenance of the creature relies on a small number of regulatory functions such as sleeping, eating, respiration, reproduction, and so on. Each of these functions is “breaking hypersurfaces in the global model” (i.e., the organism considered in its totality) and each begins with a chreod that creates first the organs that correspond to the function and then the “function fields” of the nervous system. The function fields “create organizing centers of global functional fields (archetypal chreods) in our mental organization.”⁵⁸

Following the logic of his geometric imaginary, Thom proposes that the psychological organization of organisms pivots around a series of “dualistic” states. “In some sense the nervous system is an organ that allows an animal to be something other than itself, an organ of alienation.”⁵⁹

He describes how at some point a predator “becomes” its prey, its mind dominated and hence alienated from itself by the image of that which it seeks. Suddenly the predator “jumps from the surface corresponding to the prey to its own surface,” this jump corresponding to the perception catastrophe, “an instantaneous cogito.”⁶⁰ Here Thom proposes the geometrical equivalent of the quantum jump of subatomic particles, the “moment” of the perception catastrophe being characterized as the “non-moment” (unquantifiable “jump”) between the alienated mind of the predator-become-prey and the predator proper. Thus for Thom the dichotomy between self and other is resolved in the collapse of one geometric state into another, a process that continually repeats itself.

The details of Thom’s theories are outside the scope of this chapter; it is sufficient to recognize the implications of the language he chooses in his explanations. The notion of an “archetypal chreod” is quite clearly a topologist/geometer’s way of acknowledging and reformulating the Jungian concept of a mental “archetype.” The fact that the “meaning” of the chreod, as Thom says, is not inherent in its geometry, but depends on its manifestation over time, closely ties it to the empty categories of Jung’s archetypes of the collective unconscious that only become meaningful when “filled” with content by the individual psyche.

The idea of “attractors”—since adopted by contemporary chaos theory—is a curious acknowledgment of the ancient progenitors of Thom’s catastrophe metaphysics. There is at once a hint of the erotic ontology of the Hermeticists and a nod to the “occult quality” of Newton’s mathematics of universal gravitation. The fact that, as Newton said, objects are “attracted” to the center of the Earth, and that the solar system itself is subject to the complex attractions of the celestial bodies occupying it, is but further support for the contention that Newton himself was heir to an ancient tradition (revived and reinterpreted primarily by Ficino) that envisaged the relationships in the phenomenal world as instances of an “erotic” lawfulness. In fact, it was Newton’s insistence on the use of the term attraction that prevented his theories from being accepted on the continent for almost a century after the publication of his Principia. The notion that some form of erotic force was inherent in dead matter was found laughable by scientists in France.^*64

Thom’s insistence on making the topology of fields of force the subject matter of his all-inclusive metaphysics links him with the Stoic tradition that saw the all-pervasive pneuma as a kind of tensile, topologically deformable field of activity. As Sambursky notes, the concept of the pneuma “fulfilled the functions of [the contemporary notion of the] physical field by its tensile qualities and by its capacity to give bodies a coherent structure with well-defined physical properties.”⁶¹ This description of the pneuma also describes, in skeletal form, Thom’s estimation of the functioning of the geometry of catastrophe sets.

In terms of the thesis of this chapter, it is important to understand the fact that Thom must ground his abstract geometry in concrete images and examples. The impossibility of picturing objects in a world of greater dimensions than three (a cognitive limitation of the human Umwelt, according to Uexküll), necessitates such an exigency. This was a fact well acknowledged by the popular explicators of the concept of the fourth dimension in the late nineteenth/early twentieth centuries.^*65 A truly fourth-dimensional object would in theory be impossible for us to recognize—we are only allowed, as it were, approximate glimpses of what can only be for us entia rationis.

Thom’s notion of the central nervous system as being an “organ of alienation” has undeniable Gnostic resonances. The activities of the “breaking hypersurfaces” of chreodic canalization (part of the global organization that defines any organism) pictures the relationship between part and whole as a continuing series of recognitions and misrecognitions, of knowledge and its absence. Furthermore this chreodic activity, this geometrical description, is the very definition of the organismic itself. In other words, self is knowing or gnosis: substance is subject. One is strongly reminded of the legend above the portal to Plato’s Academy: “Let no man ignorant of geometry enter here.” Thom has adopted this logic quite literally, for just as Plato opposes “ignorance” to “geometry” (geometry being equivalent to knowledge), so too does Thom suppose that the final description of the Real resolves itself into the geometrical.

According to Thom “coherent systems of catastrophes” (i.e., chreods) constellate into structures that become “abstract algebraic entities independent of any substrate.” He furthermore posits that the transformations of our space-time may be directed or “programed” by another “algebraic structure” that is itself outside our space-time entirely (refer to quotation above). Rather surprisingly we find contemporary topology here pressed into the service of a Gnostic mythology. While couched in his own peculiar terminology, it is quite clear that Thom is proposing that the kosmos was demiurgically created by a “system” of which all creatures are microcosmic equivalents. As Jonas is careful to point out, the transcendence of the god of the Gnostics is stressed to the extreme. Jonas even says, “Topologically, he is transmundane . . . ontologically, he is acosmic.”⁶² While this entity is ineffable and “unknown,” some Gnostic “systems” (Jonas, like Couliano after him, is careful to use this important locution) note that one of the transmundane entity’s secret names is “Man,” a term that Jonas states is significant, yet does not pursue.^*66 This again is evidence of the principle of circularity operating within Gnostic mythology, the abstract principle behind the micro/macrocosmic couplet, where “man” becomes structurally equivalent to “god” or creator. For Thom, both humankind and the transcosmic creator are alike “abstract algebraic entities,” a formulation that reifies and recapitulates the Timaeic account of creation through the artifice of geometry. Furthermore, if “every animal is topologically a three-dimensional ball,”⁶³ then so, following his own logic, is the absconditus creator.

CONCLUSION

In 1956 an article appeared in the Psychological Review called “The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information,” by George Miller. In this paper, Miller expounded his view that there is a quantitative limitation to our cognitive abilities. Specifically he said that an individual’s ability to distinguish absolutely between a series of stimuli went through some sort of “breakdown” when the number of items to be distinguished reached seven.

There seems to be some limitation built into us either by learning or by the design of our nervous systems, a limit that keeps our channel capacities in this general range. On the basis of the present evidence it seems safe to say that we possess a finite and rather small capacity for making such . . . [judgments] and that this capacity does not vary a great deal from one simple sensory attribute to another.⁶⁴

In other words, seven seems to be the “catastrophe point” for all sensory capacities. Miller notes that this is the reason why telephone numbers, for example, were originally limited to only seven digits—anything greater than this and most people would not be able to remember them. Telephone numbers are commonly greater than seven digits nowadays, but Miller’s observations still hold. This is because Miller recognized that we are clearly able to recall information comprising elements greater than seven in many circumstances, and we do this by associating discrete sets of seven elements or less, a process called “chunking.” Each “chunk” cannot contain more than seven elements, but a number of chunks can be associated together so that we have a large number of elements connected together. That is why we can remember large telephone numbers—we break them up into chunks: the area code (three digits), the exchange (three digits), the telephone number itself (four digits.)

In considering Miller’s famous paper, we meet again the “magic number seven” that seems to have guided so much of the thought of the prescientific era. Miller suggests that there is no mystery to the continuing recurrence of this number as an organizational strategy; it is simply that we are “hardwired” with this particular cognitive limitation. In this view, Thom’s seven elementary catastrophes are also an expression of the limits of our cognition: there may be more “catastrophes”; it is just that we would be hard-pressed to discern them. Arrangements of seven then become a figuration (a recurring form exhibiting the same functional relationships) that itself organizes elements within any imaginary: the “magic number seven” is an attractor just like that of the circle/sphere and arborial schema.

Is this the reason for the preponderance of the number seven throughout Western culture? After all, even ancient peoples knew that there were not just seven planets. The sun and moon were part of the seven, but they certainly do not resemble the small, twinkling “wandering stars”^*67 that are Venus, Mars, Jupiter, Saturn, and Mercury. Did they include the sun and moon because somehow they felt that they needed seven stars and that this was the only possible “upper limit”? Why is Ursa Minor (a constellation of signal importance in Chinese and much of Middle Eastern mythology) considered to be composed of seven stars and not elements of the other constellations that lie so close to it? Did early astronomers/astrologers look for patterns of seven because they were cognitively inclined to do so?

In attempting to answer this question we find ourselves in a position to understand the nature of the metaphysical geometries that have been the subject of this chapter. I do not think that we are limited by structural, or in contemporary parlance, “hardwired” cognitive limitations but rather that we participate in a complex^†68 number of interrelated causal influences—societal, historical, and cognitive factors—all of which considered are but discontinuously observed aspects of a particular ideal object. These discontinuities would all resolve into a single object if it were possible to observe it from a fourth-dimensional perspective. There is only a “lucky number seven” limitation intrinsic to our cognitive makeup to the extent that we, as cognizing subjects, are subject to such an attractor that generates and organizes ideal objects. On the surface this may seem very like Richard Dawkins’s idea of the meme, a pattern of behavior that replicates itself rather like a virus throughout a society and over a period of time, but the resemblance is only superficial. Dawkins’s memes are relatively trivial patterns with a limited “life span”; an attractor, in its very definition, is possessed of a profound (deep) geometry that lies outside of temporal considerations.

René Thom believes that his geometrical models of dynamical processes produce the first “rigorously monistic model of the living being, . . . reduc[ing] the paradox of the soul and body to a single geometrical object.”⁶⁵ While his system of description (geometry) is certainly a monistic model, it is clear that a vision of reality that stresses the importance of discontinuous breaks as opposed to linear, mechanical dynamics (the Newtonian paradigm) can hardly be described as “monistic.” But there is a definite continuity of thought perceivable behind Thom’s project that aligns his ideas with ideas much more ancient than those of twentieth-century topology. And just as the logic of Gnostic anthropology/cosmology imagines a circulatory system subtending the relationship between the human and the divine, so too does Thom’s conception “combine causality and finality into one pure topological continuum.”⁶⁶

In his essay “Is God a Mathematician?” Hans Jonas notes in regard to living organisms:

As a physical body the organism will exhibit the same general features as do other aggregates: a void mostly, crisscrossed by the geometry of forces that emanate from the insular foci of localized elementary being. But special goings-on will be discernible, both inside and outside its so-called boundary, which will render its phenomenal unity still more problematical than that of ordinary bodies, and will efface almost entirely its material identity through time. I refer to its metabolism, its exchange of matter with its surroundings . . . the material parts of which the organism consists at a given instant are . . . only temporary, passing contents whose joint material identity does not coincide with the identity of the whole which they enter and leave, and which sustains its own identity by the very act of foreign matter passing through its spatial system, the living form.⁶⁷

In another essay Jonas notes that the Cartesian “machine metaphor” breaks down when considering the idea of organismic metabolism, for metabolism is the “constant becoming of the machine itself,” a notion that is diametrically at odds to the conception of the machine as a stable collection of structures or parts. Metabolism “builds up” and “continually replaces the very parts of the machine itself,” thus undermining and countering the classical definition of the machinic. Jonas proposes that a better analogy is that of a flame as

the permanence of the flame is a permanence, not of substance, but of process in which at each moment the “body” with its “structure” of inner and outer layers is reconstituted of materials different from the previous and following ones, so the living organism exists as a constant exchange of its own constituents, and has its permanence and identity only in the continuity of this process, not in any persistence of its material parts.⁶⁸

Jonas has (perhaps unconsciously) borrowed the same exact metaphor from Wiener: “The individuality of the body is that of a flame rather than a stone, of a form rather than that of a bit of substance.”⁶⁹

We should recall that Thom proposes—in a much stronger way—that a geometrically governed homeomorphology operates among such seemingly diverse forms as a flame, a jet of water, and a living organism. In simple terms, each of these is governed by, and is beholden to, a geometrically describable form/potentiality before it can become a material entity. And I think that it is clear that if we were to but substitute “pneumatic influences” for “material exchanges” in the above passages, we would have a fair approximation of the Hermetic conception of process in the world. If we include Thom’s geometrical description of this economy of exchanges, then we can also apprehend the proximity of his imaginary to that of the Hermetic, this imaginary itself adumbrating Norbert Wiener’s conclusion, “We are not stuff that abides, but patterns that perpetuate themselves.”⁷⁰