At the end of the first quarter in the twentieth century with presentations by Niels Bohr on Complementarity and Werner Heisenberg on the Principle of Indeterminacy, and following an important paper by the mathematician Kurt Gödel in 1931, the eggshell of an ideally objective view of science which we inherited from the Enlightenment began to show some serious cracks. Academic philosophy--fascinated by theoretical physics and mathematical formalism, but divorced from the responsibility of their real concerns--continued its quest for certainty, riding under the girdle of logical positivism. With an appetite whetted by Ludwig Wittgenstein's essay On Certainty, and determined to make the best of this once absolute ideal, many were ready to start cooking omelets. Even so, in the rarefied domain of number theory there are still to be found pristine, abstract truths about formal properties and relationships that offer impressive, if somewhat ethereal illustrations of eternal verity.

One such truth is our understanding of six as the first "perfect number." This fact implies an order of truth both precise and timeless, independent of anyone's opinions or beliefs and just as true for any universe we can imagine (or at least for any universe in which we count). On the other hand, a grievous error associated with the fallacy of solipsism (and which also indicates a lack of proper instruction in the discipline of mathematics) contends there are no perfect, objective truths outside the self. The issue of perfection has deep significance for the histories of both science and art.

An outstanding example of objectivity in the delineation of truth is the text by Leonard Eugene Dickson, History of the Theory of Numbers which, quite uncommonly--in all of its three volumes, with a grand total of some 1600 finely-printed pages--nowhere appears to express a subjective opinion that depends upon the part or the person of the author. As it happens, Professor Dickson appropriately begins his impeccable and monumental enterprise on the first page of text in Chapter 1, in the first volume, after first defining "aliquot divisors" ("the divisors, including unity, which are less than the number") with the following (therefore, actually the second) sentence:

A number, like 6 = 1 + 2 + 3, which equals the sum of its aliquot divisors is called perfect (vollkommen, vollständig).

[ Leonard Eugene Dickson, History of the Theory of Numbers, Chelsea, New York (1971). ]

Well, the page on which this sentence appears printed is unnumbered; and although it is the "first page" of TEXT, it should be counted as "page 3," because the other side of the leaf is numbered "4." The preceding page is blank; but its overleaf is marked with the numeral "1," and on it is printed a "Table of Contents." If we presume at all to address the issue of perfection, we feel obliged to proceed with due respect for precision. Here we have the problem of accurately describing a bound order, to be sure, quite an irrelevant issue at Cumae with its prophetic loose leaves. In this, we may see just how far we can go before the apparently firm structures of language and logic become, as Wittgenstein saw, very much like a house of cards. By the way, the next perfect numbers after 6 are, in order: 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, and 2658455991569831744654692615953842176. And, unless somehow a typographical error has crept in, or some devilish printer has pied the type, this is exactly what those numbers are, quite apart from anyone's opinion, and no matter how you or I or anyone else feels about it.

[ N. J. A. Sloane, Handbook of Integer Sequences, New York & London, Academic Press (1973), Sequence 1744, p. 144; Sloane and Plouffe, Encyclopedia, Sequence M4186. ]

Professor Dickson offers an intriguing chronicle of commentaries on perfect numbers by mathematicians and philosophers: from Euclid and the early Hebrew writers through Boethius, Alcuin, Fibonacci (also known as Leonardo of Pisa) renowned for his famous sequence related to the logarithmic growth spiral, and Luca Paciuolo de Borgo San Sepolcro (also known by the name "Luca Pacioli") who consulted with the artist Piero della Francesca--both honored members of the resplendent court attending the Duke, Federigo de Montefeltro, at Urbino. A later Italian mathematician named Cataldi noted in the Preface of his Trattato (Bologna, 1603) that Paciuolo's fourteenth "perfect" number was in fact not perfect but abundant, since the sum of its divisors exceeds the number, (for example, like the abundant number 12, whose divisors 1 + 2 + 3 + 4 + 6 = 16). Cataldi was also first to correct the often-repeated but erroneous notion that perfect numbers end alternately in 6 and 8. The subject of perfect numbers has continued to receive attention from many great mathematicians, including Descartes, Mersenne, Fermat, and Euler. Yet in the eighteenth century Johann Christoph Heilbronner was still moved to present his comments replete with theological allusions:

The fathers of the early church and many writers always held this number 6 in high esteem. God completed the creation in 6 days and since all things created by Him came out perfect, he wished the work of creation completed according to the number 6 as being a perfect number.

[ Dickson, Theory of Numbers, Vol. I, p. 18. ]

Number theory is among the "purest" of mathematical fields, at one time thought to be least susceptible of practical application, but now thought to be valuable in the study of fractals, chaos theory, ciphers and AI (artificial intelligence). Also, psychological interest in perfect numbers has long sought to discover qualities of "perfection" that might be extrapolated from eternal realms to material manifestations. Analysis of the word PERFECT, we anticipate, will contribute a lexical clue for solving the enigma posed by Marcel Duchamp's piece of sculpture. PERFECT derives from the Latin perfectus, composed of per-, "completely," and facere, "to do," which in turn comes from the Indo-European root dhe, "to set" or "to put." What has been put inside With Hidden Noise we suppose to be a six-sided object recapitulating the abstract cubic space the piece can be imagined to fill.


A time-honored working definition of art alludes to that which has been made or done completely and perfectly, "all the way through" (to which nothing can be added, and from which nothing could be taken, without destroying its "perfect" harmony), as if to represent on a smaller scale a larger whole, cosmic, perhaps divine perfection. Similar allusions to perfection or wholeness have appeared in the traditional lore and symbolism of alchemy, drawn from the common currency bank of archetypal imagery. The Swiss physician and psychologist Carl Gustav Jung formulated his ideas of the archetype based upon evidence derived from records of some 30,000 clinical studies that he personally conducted. From this work came a profound understanding of the archetype and the idea of completeness implied by its iconic imagery. He often encouraged his patients to make artistic representations; these in turn provided the therapist with a sort of map that could be used to guide or direct the integration of the psyche. At very near the end of his life, Doctor Jung, sensitive to the unconscious as a source of creative ideas, had a dream where

instead of sitting in his study and talking to the great doctors and psychiatrists who used to call on him from all over the world, he was standing in a public place and addressing a multitude of people who were listening to him with rapt attention and understanding what he said.

[ John Freeman's "Introduction" to Carl G. Jung, Man and his Symbols, Doubleday, Garden City, New York (1964) p. 10. ]

That dream led Jung to coordinate work by several of his close associates into a publication project which sought to explain his ideas to people who had no special knowledge of psychology. At eighty-three years of age, he devoted the last few months of his life to editing a book called Man and his Symbols, completing his own key contributions to the text just ten days before he died.

The discovery that the unconscious is no mere depository of the past, but is also full of germs of future psychic situations and ideas, led me to my own new approach to psychology....In addition to memories from a long-distant conscious past, completely new thoughts and creative ideas can also present themselves from the unconscious--thoughts and ideas that have never been conscious before. They grow up from the dark depths of the mind like a lotus and form a most important part of the subliminal psyche....The ability to reach a vein of such material and to translate it effectively into philosophy, literature, music or scientific discovery is one of the hallmarks of what is commonly called genius. We can find clear proof of this fact in the history of science itself. For example, the French mathematician Poincaré and the chemist Kekulé owed important scientific discoveries (as they themselves admit) to sudden pictorial "revelations" from the unconscious. The nineteenth-century German chemist Kekulé, researching into the molecular structure of benzene, dreamed of a snake with a tail in its mouth.

[ C. G. Jung, Man and his Symbols, p. 37 f. ]


Just where the tail of this mythic, psychic serpent may emerge from is a matter that has given many people cause to ponder--in the full, weighty sense of pondere that we read repeatedly in the medieval quotations of the verse from the biblical Wisdom of Solomon. The modern approach to study of the psyche that emerged within the profession of medicine toward the turn of the twentieth century brought with it some unfortunate extra baggage, amounting to a deep bias of pathology. One of the consequences was the extension of a medical model focusing on physical disease and dysfunction into the domain of the mind, to describe and characterize the activities of imagination, intellect, consciousness, dreaming, pondering, and so on.

It was, perhaps only to be expected that the prevailing beliefs and practices of somatic therapy would be, without a second thought, projected onto the model of a "healthy" or "unhealthy" mind, thereby perpetuating many camouflaged biases of class, caste, education, religion, or "taste," together with all manner of ego- and ethnocentricities.

Doctor Carl Jung developed and continually modified his notion of the "collective unconscious" in order to serve as an adjunct to diagnostic and analytic procedures within the context of his own practice. The hypothesis did accommodate a holistic approach, and appears to have been applied with real generosity and compassion. Yet, the existence of such an entity as the collective unconscious was never happily resolved, much less proven. Subsequent work by others in the field appears to have fallen short of providing us with a truly objective definition of the phenomenon, despite the anecdotal success of some therapies, and the generation of intriguing theories.

The concept of the so-called Unconscious for many has come to obscure more than enlighten. The special weakness demonstrated by the Western medical model in dealing with the issue of "consciousness" is shown, in particular, by the use of that very term by one of its early popularizers (but not, as we shall see, its inventor), Sigmund Freud. The crux of the problem with Freud's pseudo-science of psychoanalysis is that "its data, method, theory are indissolubly one." This lays the way open for practicing very bad medicine, and it flatly cannot be science--at least not in any conventional, modern, or universal sense.

Dependence of the data on the theory separates psychoanalysis from all true sciences. What Freud thought free associations are not free. The nondirective torture of Catholic inquisitors extracted mea culpa's of previsioned heresies: the communists secure confessions of expected deviations and disloyalties. Interpretations of chaotic dreams are still controlled by theory, and that theory was in the head of Freud. Change this, and you have changed the method and the data. This is the curse of all attempts to understand things social. They are essentially sharing. What we seek to understand is coupled back through us, so that we ourselves change the thing we seek to understand.

[ Warren S. McCulloch, "The Past of a Delusion," Embodiments of Mind, p. 282 f. See also, Collected Works, Volume II, p. 761 f. This paper was read before the Chicago Literary Club, January 28, 1952. ]

In his scathing, scintillating critique of Freud, Professor Doctor McCulloch details the background and early training of the psychoanalyst, pinpointing the purloined, unacknowledged perceptions of other, more honest philosophers who came before the Viennese pretender: Leibnitz "who introduced the notion of the Unconscious a sort of calculus of knowledge," Emanuel Kant, in his Practical Reason, Samuel Butler's Unconscious Memory, Fichte, Schelling, Lessing, Schopenhauer, Lipps, and Eduard von Hartmann's The Unconscious,

a key notion and inspiration of Janet, whom Freud said he never knew, and of the Nancy School, where Freud had studied....Since Freud told several people he had never read von Hartmann, we may find the link nearer to Freud's specific introduction of the Unconscious in Breuer's knowledge of Herbart's psychology, wherein the unconscious specifically appears with repression and resurgence of ideas rising into consciousness [which] Herbart had derived from von Hartmann.

In The Future of an half-truths told with bad intent, that beat all the lies he could invent, he [Freud] arrogantly prophesies the time when God will fade away and leave the stage to man, made in Freud's image. Freud's delusion has no future, but it has a past.

[ McCulloch, Embodiments of Mind, p. 295, 288, also p. 294; See also, Collected Works, p. 773 f. Here I am delighted to acknowledge Professor Heinz von Foerster who, along with inspiration and encouragement, most graciously conveyed to me the four volumes of McCulloch's Collected Works. (KvM)]

Not only did Sigmund Freud fail to give credit, but without a shred of untainted, unvitiated clinical evidence, he then managed to concoct one of the most egregious quackeries of modern medical practice:

a tissue of unverified and unverifiable hypotheses....Freud's doctrine may be even more demoralizing for it is a way of saying you are not responsible for these, your acts, because they are nothing but the inevitable doings of mere matter, fixed in your infancy, or at your birth determined by Unconscious Memory.... When you...are emotionally persuaded that it is universal, you too may spread the gospel of material determination and demoralize mankind for gold.

These things I had to say of Freud who said worse of himself because he thought they would substantiate his theory that man's depravity is causally determined by his past in simple ways, by motives buried in forgotten memories waiting to be revealed by his, Freud's, psychoanalysis....One of the cornerstones of Freud's delusion is the belief that we forget no single jot or tittle of what at any time has happened to us. By calculations that began naïvely with the senior Oliver Wendell Holmes and are today best handled by the physicist [Heinz] von Förster, man's head would have to be about the size of a small elephant to hold that much. His body could not eat enough to energize its mere retention even if we suppose a single molecule of protein would serve as trace. Actually, the mean half-life of a trace in human memory, and of a molecule of protein, is only half a day. Some few per cent of engrams do survive, presumably because we recreate the traces in our heads, but that is all fate leaves us of our youth. Where written words remain to check our senile recollections they often prove us wrong. We rewrite history, inventing the past so it conforms to present needs. We forget, as our machines forget, because entropic processes incessantly corrupt retention and transmission of all records and signals.

[ McCulloch, Embodiments of Mind, pp. 289, 298, 292. ]


Jung illustrated the notion of the archetype with two powerful similes. One of these involves the spectrum of visible light: from the high-frequency/energy (short wavelength) blue (the ultra violet) end, to the low frequency/energy (long wavelength) red end. Closure of the linear spectrum into a circular ring or wheel is made possible because of the unique properties of the color violet, whose frequencies mostly transcend the upper limits of human perception. This may remind us of the color wheel, since violet can be produced by mixing the hues of red and blue from either end of that linear spectrum, in order to close the circle, ring, or wheel. For violet is cerebral and, in a sense, imaginary and magical; perhaps that is why violet (or purple) claims a symbolic spiritual primacy among colors of the rainbow.

Because the archetype is a formative principle of instinctual power, its blue is contaminated with red: it appears to be violet....Although it can be admittedly no more than an analogy, I nevertheless feel tempted to recommend this violet image to my reader as an illustrative hint of the archetype's affinity with its own opposite. The creative fantasy of the alchemists sought to express this abstruse secret of nature by means of another, no less visual, symbol: the Uroboros, or tail-eating serpent.

[ Carl G. Jung, "The Spirit of Psychology," Spirit and Nature {Papers from the Eranos Yearbook, Volume 1; Bollingen Series XXX}, Pantheon Books, New York (1954), p. 423. ]

The TV personality and cultural historian James Burke discusses several instances of scientific insight coming about through dreamlike or visionary experiences, including Albert Einstein's account of riding on an apparently static beam of light, leading the great physicist to develop the concept of relativity in which the speed of light is treated as constant.

Einstein's dreamlike experience is echoed by other descriptions of the same kind of event. August Kekulé, the discoverer of the benzine ring which typifies the mechanism or structure by which groups of atoms join to form molecules that can be added to other molecules, wrote of gazing into the fire and seeing in the flames a ring of atoms like a serpent eating its own tail.

[ James Burke, The Day the Universe Changed, Little, Brown, Boston (1985), p. 303. Others are Newton, Archimedes, and Gutenberg. ]


In the posthumously published text of Jung's Man and his Symbols (p. 38) is an illustration showing a page from the 1861 edition of Kekulé's Textbook of Organic Chemistry, identified as a representation of a benzene ring. Among the most fundamental molecules in nature, benzene (C6 H6) has six carbon atoms arrayed with six hydrogen atoms in a hexagonal ring. But in that version of his textbook, Kekulé only labeled the carbon atoms, implying the hydrogen, but actually showing only the single and double linear bonds. Next to this illustration there appears on Jung's page the age-old symbol of the Uroboros, the ring-forming serpent, from a 3rd century BCE Greek manuscript. This archetypal image, like that of the alchemist's garden, expresses the basic idea of a recurring cycle.

And now the garden reveals one of its greatest mysteries: this seed, which is sown again, will begin the cycle anew. Here we have the phenomenon of the rotatio, the cycle, another typically alchemical concept. This cycle is the seal and true image of alchemy. There is one figure that is never absent from the old alchemical manuscripts of Egypt, scantily illustrated as they are, and that is the Uroboros, the serpent with a head like a dragon and a scaly body, biting its own tail. This whole metamorphosis, this growth and passing, this eternal cycle, is the first great lesson we find in our garden.

[ Rudolf Bernoulli, "Spiritual Development as Reflected in Alchemy and Related Disciplines," Spiritual Disciplines {Eranos Yearbooks, Volume 4; Bollingen Series XXX.4}, Pantheon Books, New York (1960), p. 307. ]

The alchemical papyri have come to be regarded as basic documentary evidence upon which the academic study of alchemy may proceed in the light of recent scientific and historical interest both in its psychological aspects, and in the origins of holistic spiritual ideas generally.

The texts of the magic papyri are always authentic invocations and prayers, hallowed and made magically effective by their use in various cults. These are the texts underlying the commentaries of the literary authors. Macrobius, for example, endeavors to explain why the sun is represented as serpent. Like the snake, which sheds its skin and becomes young again, the sun grows old when it sets and rises again with new youth....Fire and the pneuma as wind, life breath, or spirit make the snake the beast of Helios. Its longevity, its power to rejuvenate itself and to grow at the same time, only to end by devouring itself make it seem a fit symbol for the Aeon as life span of the cosmos, rejuvenating itself from spring to spring, only to consume itself inwardly in the end. The snake biting its tail signifies the cosmos itself. Macrobius ascribes the invention of this symbol to the Phoenecians [as the eponymous Phoenix was a self-renewing beast, and purple dye a major trade commodity]. As the snake devours itself, so does the world feed on itself and return to itself. Here we must recall the Heraclitan-Stoic doctrine of the development of the world from an original element, fire, to which the cosmos returns after the cosmic year has run its course. Since this cosmic year is the Aeon, the life span allotted to the cosmos, the uroboros, the snake biting its tail, became the symbol of the Aeon, especially when bearing the inscription hen to pan, meaning that all things are one, to wit, that they arose from the One and will return to the One, so that the Aeon extends from the One to the One.

[ Hans Leisegang, "The Mystery of the Serpent," The Mysteries {Papers from the Eranos Yearbook Volume 2; Bollingen Series XXX}, Pantheon Books, New York (1955), pp. 220 ff. See Hans Dieter Betz, editor, The Greek Magical Papyri in Translation, Including the Demotic Spells {Volume 1: Texts}, The University of Chicago Press (1986), XII. 274, p. 163; and VII. 579-90, p. 134 with drawing. The spell indexed as (PGM) XII. 202, p. 161 directs "Taking an air-colored jasper, engrave on it a snake in a circle with its tail in its mouth..." "Air-colored jasper" could be light blue, cloudy grey, or purple/violet, but such colors would probably have led to such stone being called "porphyry."]

Today, this emblem is widely recognized, even though there is no entry, sub verbum in the usually redoubtable American Heritage Dictionary. Of course, the Greek word is defined in Liddell and Scott's Lexicon as "devouring its own tail." Erich Neumann, himself a distinguished Jungian analyst who studied with the master during the 1930s, has the iconic image of the Uroboros displayed on the cover of his book, The Origins and History of Consciousness. Neumann also supplies several excellent illustrations of the theme, from a mappa mundi on a cuneiform tablet and a now-lost bowl from the Mesopotamian Mandaeans to the drawing of a 5-year-old English schoolgirl. The author interprets the Uroboros as a fundamental archetypal image of creation and consciousness, in the earlier theoretical stages said to be associated with the alimentary canal and the phenomenon of cannibalism, and later with the process of self-formation, or what Jung called individuation.

When the universal principle of opposites no longer predominates, and devouring or being devoured by the world has ceased to be of prime importance, the uroboros symbol will reappear as the mandala in the psychology of the adult....The uroboros, traceable in all epochs and cultures, then appears as the latest symbol of individual psychic development, signifying the roundedness of the psyche, life's wholeness, and perfection regained. It is the place of transfiguration and illumination, of finality, as well as the place of mythological origination.

[ Erich Neumann, The Origins and History of Consciousness, Bollingen Series XLII, Princeton University Press (1970), p. 36 f. ]


Professor Kekulé's story is one of the most famous and frequently-cited examples given for the importance of visions and dreams in the work of science. The symbol of the Uroboros persisted in alchemical texts through the Middle Ages and the Renaissance, down to Kekulé's own time...quite possibly through the agency of his teacher, Professor Justus von Liebig, also a very great figure in the history of modern chemistry. The date of the 1861 page reproduced by Jung shows what is described as a benzene ring structure four years before other sources say Kekulé while working on another edition of his textbook in 1865, is supposed to have entertained the fiery, alchemical vision of integrated wholeness which allowed him to conceptualize the benzene ring, that archetypal image of the Uroboros.

Friedrich August Kekulé (1826-96) is regarded by historians of science as being among the most distinguished of Professor von Liebig's numerous pupils who themselves became famous in the world of chemistry. Possibly some alchemical references graced the transmission of teachings about the historical sources of modern chemistry, although this aspect of the discipline has received attention only in more recent academic approaches. Such references may yet be discovered somewhere in the notes on Liebig's lectures which Kekulé took as a student; they are still extant, covering 346 pages with a neat and closely written script, including many small illustrative sketches.

[ See, John Read, Through Alchemy to Chemistry: A Procession of Ideas and Personalities, Harper and Row, New York (1963), p. 173. For Kekulé's dream we have borrowed extensively from this study. ]


Fortunately, we have Kekulé's own words describing the two separate visionary experiences that he felt were so powerfully significant. In a speech many years after the event, he described how his "long, long thoughts" culminated in a flash of inspiration during a visit to London:

One fine summer evening I was returning by the last omnibus ...through the deserted streets of the metropolis, which are at other times so full of life. I fell into a reverie, and lo, the atoms were gamboling before my eyes! Whenever, hitherto, these diminutive beings had appeared to me, they had always been in motion; but up to that time I had never been able to discern the nature of their motion. Now, however, I saw how, frequently, two smaller atoms united to form a pair; how larger ones even embraced two smaller ones; how still larger ones kept hold of three or even four of the smaller; whilst the whole kept whirling in a giddy dance. I saw how the larger ones formed a chain, dragging the smaller ones after them....The cry of the conductor: "Clapham Road," awakened me from my dreaming; but I spent a part of the night in putting on paper at least sketches of these dream forms. This was the origin of the Structurtheorie.

[Read, Through Alchemy to Chemistry, p. 173, f. ]

That text will have been Kekulé's Theory of Molecular Structures, which he published in 1858, and which provided two solid foundations for modern organic chemistry in postulating the quadrivalency of the carbon atom and the capacity for carbon atoms to link together to form chains. But apparently many substances called aromatic compounds, related to the simple hydrocarbon benzene, still could not be explained by it. Kekulé had been a student of architecture before Liebig's inspiring lectures directed his mind to the study of chemistry, and the power to formulate imaginary structural relationships was a faculty that characterized his most renowned contribution to his chosen scientific field. By 1865, having been appointed to the chair of chemistry at Ghent, Kekulé was in the process of writing a textbook of organic chemistry when he experienced a second compelling vision, which also featured imaginary serpent-like dancing atoms.

"I was sitting, writing at my text-book; but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together; all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis."

And so this vision of the Ouroboros Serpent, the "tail-eater" of Greece and ancient Egypt, a symbol "half as old as time," brought across the wide ocean of time to Kekulé the solution of one of the most baffling and most important problems of organic chemistry. "Let us learn to dream, gentlemen," Kekulé used to say to his students, "and then perhaps we shall learn the truth...but let us beware of publishing our dreams before they have been put to the proof by the waking understanding."

[ Read, Through Alchemy to Chemistry, p. 179-180. ]

Nowadays, the benzene molecule is so familiar and basic that it is usually represented without indicating either the carbon and hydrogen atoms or their bonds, but simply as an elongated hexagon. Indeed, polycyclic molecules composed of benzene with its nestled hexagonal rings closely resemble the structure of a bee's honeycomb.

The conception of the benzene ring has been called the crowning achievement of the linking of carbon atoms. Kekulé's fundamental ideas of organic molecular structure, that is to say, his conceptions of the carbon-chain and the carbon-ring, have led to developments in pure and applied chemistry that stand unsurpassed in the whole history of science. Later researches have shown that all organic Nature is based upon the carbon-chain and the carbon-ring, and that life itself depends upon the capacity of carbon atoms to link together so as to form the molecular chains and rings of acyclic and cyclic compounds, respectively.

[ Read, Through Alchemy to Chemistry, p. 181. ]

Overwhelming indirect evidence has confirmed Kekulé's vision, upon which much of the theoretical structure of organic chemistry has been erected. Recently, scientists at an IBM laboratory in San Jose, California, provided an even more impressive kind of evidence by graphically demonstrating photograph-like images--the first time ever anyone has actually seen direct evidence of individual benzene rings.

The group used a "scanning tunnel microscope," of a type invented in 1981 that won its creators...the 1986 Nobel Prize in physics. ...The instruments have sparked a revolution in understanding the atom-by-atom structure and chemical behavior of surfaces of many materials. The special microscopes use exceedingly sharp needles, typically of tungsten, that are delicately traced across the substance under study. When the tip is within about an atomic diameter of the surface--a distance of a few billionths of an inch--electric current jumps (or, in physics terms, tunnels across) the gap. Sensitive instruments that record the size of the current allow computers to recreate the shapes of the clouds of electrons that surround individual atoms and molecules, and to produce the image....Benzene presented a particular challenge because it does not ordinarily conduct electricity, and because individual molecules are not still enough under ordinary conditions to be detected by the probe....The image shows each ring with three prominent bright spots. Each bright spot corresponds to a bond, one of the electric ties that hold the atoms together.

[ Charles Petit, "How IBM Proved a 123-Year Old Theory," San Francisco Chronicle (July 18, 1988). ]


And so, Kekulé's internal experience of enlightenment finally received, after one hundred and twenty-three years, its confirmation by an event of exoteric enlightenment: the objective collection of measureable, countable, recordable, transmittable, publishable data. The realm of this enlightenment is what we call the first knowledge, the "reconstruction of the form of all science."

The bodhisattva, or candidate for awakening, must first learn the form of all representative knowledge, including western science, sufficiently to be able to turn the page of any significant enough book and say, "I know what is coming next." At some stage during the course of this first discipline, which may last many [pages or] years, the knowledge of what the whole manifest world actually is begins to dawn. This is the second knowledge. Every language is constructed so that one cannot help saying what it really is all the time, but at this stage one begins to say it more and more consciously because one cannot actually believe it.

The first knowledge, including as it does the natural science of physics and the divine science of mathematics, is plainly publishable, and is published. The second knowledge is also publishable and, although rarely, is also published. But nobody can believe it. Even a fully awakened one does not believe it: he knows it; and before he knew it even he did not believe it, because, like everybody else, he could not see how it could be so. Its publication makes no sense to those who do not already know it, who therefore always corrupt it on the ground that the enlightened one "obviously could not have meant this."

[James Keys, "Coda on Enlightenment," unpublished ms. (1976), p. 5 f. I am personally grateful to Mr. Keys for the direct communication of related material. The publication of his frequently brilliant (albeit quixotic) text, Only Two Can Play This Game in 1972 , with the support of the late Mr. Arthur Ceppos of the Julian Press, New York, provided a key pendant to G. Spencer Brown's Laws of Form. (KvM) ]

Thus the nineteenth-century Uroboric enlightenment of Kekulé is reciprocated in the twentieth-century high-tech "enlightenment" of the archetypal, hexagonal, magical basic diamond of life, the carbon atom in a fundamental, organic, cyclic arrangement. Perhaps it was appropriate for the enlightenment to occur in the Almaden labs; since almaden is the Spanish for quicksilver, cinnabar (mercuric sulfide) the alchemical Mercurius, Hermes, the instructor and conductor of souls. Several Chinese alchemists took to eating gold, and mercury in its cinnabar ore, pursuing their theories in quest of yet more gold.

The thought linkage thus established between aurification and immortality was destined to have nearly twenty centuries of life, taking on in due course the formulation that all the other metals, rusting and corroding, suffered from the same illness as mortal man, so that the philosophers' stone would be the supreme medicine of men as well as of metals. Both of them it would cause to put on incorruptibility, its essential tendency being to transmute "imperfect" things into "perfect" ones.

[ Joseph Needham, Science and Civilisation in China, Volume V, Chemistry and Chemical Technology, Part 2 "Spagyrical Discovery and Invention: Magisteries of Gold and Immortality," (1974), p. 14. ]

By the mouths of the old almaden mines, the magic metal was tungsten--which name comes from the Swedish, where it means "heavy stone." That benzene ring enlightenment came about by subversive tunneling of electromagnetic energy between teeny tungsten tips and the adamantine aura of atomic carbon, across Joyce's "ginnandgo" the Ginnungga-gap cleaving the easy outside from the hard-core inside, the yawning abyss between what we imagine to be our real world of time and the eternal space, the heart of which we call the void but which is without either name or form.

Thus the beginning and the end are made the same, and it is the realization of the beginning in the end and the end in the beginning that brings the final knowledge of what it is and how it came about....Enlightenment, complete awakening, perfect consciousness, the complete answer to all questions, the end of all mystery, the knowledge of the Great Secret that confers divinity upon its vessel and enables its vessel to confer divinity upon another, was once honoured in the East.

[ Needham, Science and Civilisation, Volume 5, Part 2, p. 4-5. For the "the ginnandgo gap between antediluvious and annadominant," see James Joyce, Finnegans Wake, The Viking Press, New York (1939) p. 14. Joyce was apparently inspired by the name given to "the interval of timeless formlessness between world aeons" in the Icelandic Eddas. See, Joseph Campbell and Henry Morton Robinson A Skeleton Key to Finnegans Wake, The Viking Press, New York (1944, Compass edition 1960), p. 46.

Among other symbolic and metaphorical references, this gap is the hiatus that appears in systems of linear chronological reckoning, between "B.C." (Before Christ) and "A.D." (Anno Domini). The American Heritage Dictionary notes, "Literally, 'in the year of the Lord.'...When neither abbreviation appears, the time is assumed to be after Christ." Current usage now prefers: CE (Common Era), and BCE (Before the Common Era), the latter abbreviation appearing in the 3rd, although not in the 2nd edition of the A.H.D. (or, AHD). We drop the periods as a matter of style. The "gap" comes about primarily as a bookkeeping anomaly, because there was no "year zero" reckoned as having transpired between the two eras, thereby a producing a disparity between the set of natural numbers in their continuity from negative to positive integers, and the marking (or, the naming of the marking) of time. This may lead to trivial--and for some people serious--problems, consequences of short-sighted programming set to confound actuarial computer data in the year 2000. ]

Employing such circumlocutions illustrates the contention that there is no native name or word for an awakened one in Western languages. Is the very idea of Enlightenment--the bodhicitta--technically unspeakable for us? As the term bodhicitta is rendered by Garma C.C. Chang in The Hundred Thousand Songs of Milarepa, the text which the sculptor Brancusi kept with him at all times, it is:

...the aspiration to Buddhahood; the determination to practice all the virtuous deeds that lead toward Buddhahood; the enlightened insight into immanent Reality; the great compassionate Vow to serve, benefit, and deliver all sentient beings.

[ The Hundred Thousand Songs of Milarepa, translated and annotated by Garma C.C. Chang, Volume 2, University Books, Secaucus, New Jersey (1962), p. 692. ]

What, then, is the meaning of "Buddha," "Enlightenment," and "Buddhahood?" As set forward in simple words by the great teacher, the late, Very Venerable Kalu, Rimpoche, holder of the Karma Kagyu Lineage:

When the meanings are considered: after the awakening from the sleep-like ignorance, mind is enlightened in the two knowledges of knowing how everything actually is and how everything appears; this state is called Awakened Enlightenment (Buddha).

[ The Very Venerable Kalu Rimpoche, The Foundations of Buddhist Meditation, Kagyu Kunkhyab Chüling, 725 West 14th Avenue, Vancouver, British Columbia, Canada (1972), p. 31. ]

Dante also hoped to transcend the dilemma of appearance and Reality, singing about his gleaming vision of Enlightenment, ineffable though it was. Just so, the alchemist who practiced the "secret" inner visualization, accompanied by disciplined breathing and the blessings of Divine Energy, could hope to embody the Uroboros. This selfsame emblem supplies the cryptographer's key for the poet's words wafting magically through time on the sine waves of sound, or for the iconic vision of the artist revealed by and through the crystalline lens of a Thaumaturgus opticus, all at once, in a blinding flash, a gleam of pure light.

The enlightenment that comes through the western mode, although exceedingly difficult, is, as might be expected, fully communicable and intelligible, but it cannot be attained without first mastering the most advanced ideas in western science. Here the Secret is eventually reunited with itself at its very outermost level of manifestation, and the ultimate Key to the complete understanding of the nature and origin of the whole world thus partakes of and participates in the extremes of both realities, the inmost of the inner and the outmost of the outer, and it is in the final uniting of the beginning and the ending that the circuit is closed and the snake in the emblem eats its own tail.

[ Walter Arensberg, The Cryptography of Dante (A Borzoi Book), Alfred A. Knopf, New York (1921), p. 5. ]

The quality of sweetly poignant ineffability felt by artists, poets and visionaries who have by Grace received a glimpse of eternal Unity only to find themselves once again back on Earth with the rest of us ordinary human beings was beautifully expressed by a young girl in 13th-century Germany, Mechthild von Magdeburg, who sang of the vision she experienced before she was 13 years old:

[ Translated by Jane Hirshfield, in Stephen Mitchell, editor, The Enlightened Heart: An Anthology of Sacred Poetry, Harper and Row, New York (1989), p. 66.