Who is G. Spencer Brown and where is that marvelous music coming from?
Text by Walter Barney, Illustrations by Tom Cervenak
"The enclosed manuscript," wrote the author. "was commissioned by the Saturday Review and didn't get published because the magazine went broke. I think your readers will dig its subject and its craziness.
"There really was a conference at Big Sur at which G. Spencer Brown discussed his calculus with a group of far-out scientists. What went on at the conference is described in the text. I did not, however, report on what Watts, Lilly, von Foerster, Bateson, et al. had to say about Brown and the Laws of Form because once I tumbled to what Brown was up to, I became more interested in using the laws of form than in listening to talk about them. My text is a song of appreciation for the heroic work Brown performed, working out the formal relationships which structure our universe. Or, more accurately, by which we structure our universe. Most magazine articles are Apollonian; this one is Dionysian. Why not?"
The reader of this curiously exhilarating and absolutely perplexing work is hereby forewarned.
WHEN THE CREAM of America's hip scientists, or precision hipsters, depending on the focus of the moment, gather to learn a calculus at heaven — a precise, mathematical discipline that purports to show how we as individuals construct our universes — under the guidance of the scientist/magus who devised it, one result is -
A clap of thunder of which this report is the merest grace-note. The reverberations will be heard throughout the land's unofficial networks and possibly in the official ones as well.
G. (for George, Virgil's Georgics, songs of husbandry, hence husband, male principle) Spencer (the dispenser) Brown (seventh color, seventh Tarot key, the Chariot, the mind, the new man): G. Spencer Brown, also known as James (the King James version) Keys (key, a wedge, prob. akin to OE cina, crevice, female principle, though the male key, speaks for the female that supports its existence); Brown/Keys, a wise man from the east, that is to say England, so he's thoroughly Western, may be the first hierophant of the mind since Newton to insist on the tandem yoking together of male science and logic with female art and religion, with the intellectual clout to command a serious audience.
A few years ago, Brown published a book, Laws of Form, which revealed what had never been seen before, an arithmetic calculus for Boolean algebra, which was designed for logic (and is now the logic of computers, 1 and 0, on and off, light and darkness) and whose arithmetic had never been investigated until Brown came along.
Bertrand Russell called the book a rare achievement in mathematics; and indeed the arcane relationship of logic, algebra and arithmetic might seem to be of interest only to professionals. However, Alan Watts and John Lilly, neither of them principally mathematicians, read into the book something of a different order, something notable about the way we experience our experiences. In the words of William Blake, quoted by Brown in the book: "Tho' obscured, this is the form of the Angelic land."
And so Watts, himself a guru of renown, and Lilly, psychedelic speaker with dolphins, invited some of their friends to read Brown's book and then to gather at Esalen Institute with psychiatrist George Gallagher as gracious host and Brown himself present to explain what he meant in the hard parts and to initiate the New World shamans in the magic of his most potent spell. All of which came to pass under the last full moon before the vernal equinox, to the resonant rhythms of the brontobooming Big Sur surf.
Inside, seated on a standard motel carpet in a standard motel rectangle, Brown faced: Gregory Bateson, anthropologist and ecologist of the mind: Karl Pribram, mapper of the brain; Ram Dass, formerly the professional Dr. Alpert and lately suffusing devotion as a friendly traveling lotus flower: Claudio Naranjo, Gestalt therapist in the meditational form of a cloud of bees around an orange tree in blossom; Heinz von Foerster, Viennese choreographer of thought processes whose lucid review of Laws of Form in The Last Whole Earth Catalogue had introduced the book to many of those present; Charles Tart, psychologist of altered states of consciousness; Douglas Kelly and Ted Guinn, mathematician-yogis; Stewart Brand, cataloguer of the whole earth; Kurt von Meier, professor of art and mythology who saw Brown from the start as a magician and who had arrived with a green suitcase full of his own spells, ready to swap with the visiting master; others, variously interested in cybernetics, numbers, and inner space, including myself, nominally an observer and hence, by Heisenberg's principle, bound to be a performer since I could not observe the conference without changing it and being changed by it.
Lilly introduced Brown as "more of an enigma than Carlos Casteneda." Inasmuch as the form of an enigma manifests the enigma, Brown fit the description perfectly.
Dressed in sweater, slacks and scruffy tennis shoes, speaking in clipped tones and telling academic jokes in which he imitated people who were even more English than he, he came on like the archetypal American fantasy of an Oxbridge don; yet we knew that he also published poetry and fine letters under the pun name of James Keys, that he had worked with R.D. Laing as a psychotherapist, that he was supposed to be a world record holding glider pilot and a chess half-blue (that fit, at least), that he had written sports correspondence and, latest of all, that he had become a publisher, where his interests lay in dictionairies of music. So the riddle posed by his appearance was that he was exactly what he appeared to be, and yet he was not.
Watts had set the bearing for the conference: We were to steam ahead on the course of jnana yoga, to approach the calculus with a clear precise intellect, to embark on the head trip. This was a calculus for logic, an intellectual tool: the task was to learn to use the tool. Everyone was quite clear that we were going to discuss the use of the tool, rather than actually using it, a process which brought up an interesting problem.
Laws of Form is written in injunctive language. It is a set of instructions with accompanying comments. The language is not about anything; it speaks directly for itself. The language I am using here is descriptive; I am talking about the calculus. The language of the calculus jumps out of the page and gives orders — do this. There is nothing mysterious about this language, any cookbook uses it when it tells you how to make eggplant parmigiana without quite trying to tell you what eggplant parmigiana is like. You don't understand eggplant, you eat it. However, if you don't understand the phrase "Cut the eggplant into half-inch slices," you might as well open a can of tuna. We had to learn the language of Laws of Form before we could use the calculus.
Yet we all knew that language; it was English and Brown uses English very precisely. He speaks slowly, sometimes pausing for long seconds to come up with just the right word, and he enunciates clearly. He recalled for us the etymologies of the words he used, so that there should be no confusion. Still, what he said remained oracular since, as he explained very carefully, one of the principles of the laws of form was that you could not explain how you use them, you could only use them. Or, as one translator of the Tao Te Ching rendered the expression that serves as an epigraph to Laws of Form, "The Tao that can be named is not the eternal Tao."
So we were trying to understand something that could not be merely understood, but which must be embraced with the mind — a perfect jnana yoga exercise. What better way to spend the rainy beginning of the week than by stretching the mind on an impossible task, taking time out occasionally to treat the body in the Esalen saunas and baths where strong and lissome masseuses kneaded out muscular knots and thus un-knotted the mind as well. (For further relaxation the conferees had the chance to hang out with a bunch of interesting people, namely each other, and the friendly natives of Esalen.)
Brown stayed for the first two days of the conference, presenting his riddle in one form after another. At the heart of the variations was an internal contradiction in and of the calculus itself, an arithmetical manifestation of the self-referential statement in logic — where the contradiction shows itself in sentences like "This statement is false." If the statement is true it's false, and if it's false it's true, a situation which so upsets logicians that they ignore it completely on the authority of Russell and Whitehead who labeled such statements un-logical and banished them under the grand name of the Theory of Types.
Far from trying to deny paradox its reality, Brown insists that contradictions grow out of the system itself as we apply the rules that make it go. The rules of ordinary arithmetic lead inexorably to the mind-bending dimension of imaginary numbers (√-1, the square root of minus one), just as the rules of grammar inevitably produce the paradoxical statement.
Brown's first instruction in Laws of Form is to draw a distinction. The distinction, by its nature, creates two states, which we label the marked state and the unmarked state. The label for the distinction, "cross," ᒣ , is also — primarily — an instruction to cross the boundary. From this trinity (i.e. the two states and the boundary between them) we can construct an entire universe, Brown tells us. In fact he suggests that is exactly what we do as human beings; this is our experience of life, crossing between the marked state and the unmarked state. Or,
The Tao begot one. (cross)
One begot two. (the unmarked state and the unmarked state)
Two begot three. (one and two together)
And three begot the ten thousand things.
—Tao Te Ching, 42
The trinity also corresponds to the organism-environment field, separated by our awareness of a contact boundary, upon which Fritz Perls built his theory of Gestalt therapy.
The mathematical process set in motion by drawing the first distinction leads the paradoxical expression when the calculus reaches the Boolean equivalent of equations of the second degree, which have imaginary roots just like their counterparts in ordinary arithmetic. The square root of minus 1 is a logical paradox; if the "1" is negative, it can't be a square, and if it's a square, it can't be negative. Nevertheless, designers use √-1 in equations while building space ships that real men then fly in. Brown, in his days as a computer engineer, used the imaginary Boolean values to devise a safety system for British Railways that operates by counting and uncounting the number of wheels passing a given point. Real trains use this system, big wheels run by faith and little wheels run by the grace of God.
We are taught to regard contradictions, logical falsities, as something bad, which is absurd, Brown says, a holdover from the nursery where truth is good and lies are bad. To Wittgenstein's observation that a tautology (equals sign) says nothing, Brown adds that a contradiction says everything. He relates that logical statement which is true if it's false and false it it's true to the basic oscillations of our life, breathing in and breathing out, yin and yang, alternating current.
A sense of contradiction or paradox that says everything and embraces that which cannot be said pervaded the conference. For all his supposed inability to talk about his handbook of the formal structure of the cosmos, Brown did expound it brilliantly. For all the absurdity of discussing that which one could only perform, the conferee did begin to grasp the idea of the calculus. And for all their determination to cling to the intellectual approach, the mystically acquainted soon began picking their way through the teacher's five-layered model of Creation with the help of the Tarot deck and Brown's textual references to Saint Dionysius the Areopagite and the Gospel According to St. Thomas. So when Brown left, there remained one group, guided by mathematician Kelly, tuning into the jnana frequencies and mapping Brown's cosmos into their own, and another, underground movement, looking for a ticket to ride on Brown's spaceship itself. Heinz von Foerster was the bridge between them.
On Wednesday afternoon, Kurt von Meier made the contradiction explicit by observing that we were not yet doing what Brown had told us to do: we were not following the injunctive language and performing the calculus. Not even Kelly has performed it, technically speaking, von Meier insists, although he has talked intriguingly about it. Much chalk dust has flown from the Qabalistic expressions on the blackboard, but the runic glyphs have been incanted in terms like "Feed this expression into that term in this equation and you get this result," etc. Phonically, an invocation of pronouns and prepositions; at no time a formal recitation in the terms of the sacred text itself, as transmitted by Brown in Laws of Form. Von Meier reminds us that Brown has said all along that we must pay attention to the form; to perform the calculus, we must respect the form precisely. Von Meier's insistence that the mathematical expressions be read out — their "names called" exactly as they were written, named, and called by Brown, is all but drowned in a sea of groans. No one wants to play this game, on grounds that it would take too long. Von Meier replies that it may take a week to recite a sacred text such as the Mahabarata, yet this ritual is performed; when called for, precisely or not at all.
Does it really help to say
a cross b cross cross all a cross b cross 2 expresses the same value as a
when we see the expression?
Although Brown indicates on pp. 91-92 of Laws of Farm that such a process has its limitations, von Meier seriously suggests that we chant the equations as mantra.
Von Foerster hears the request and in his Middle European politeness he honors it. Next morning, as a prelude to his discussion of " A construct of reality," he performs Brown's two basic axioms. The first states "The value of a call made again is the value of the call." Everyday manifestation: if I call Fred once, and then call him again, I am still referring to a single person, Fred. Manifestation in the calculus:
or "cross cross equals cross."
Von Forster, the Viennese, puts it to music, thusly:
Axiom 2 states "The value of the crossing made again is not the value of the crossing." If I decide to call Fred and then change my mind, Fred is not called. Or,
expressed verbally as "cross cross all equals ." Heinz sings:
Later that day, he and I are sitting in a bright yellow panel truck belonging to one of the Sufi wanderers who have penetrated the parliament of high-flying pundits (crashed the party), eating dried fruit and talking about the craziness of reciting the laws of form as though they were a musical score for some primeval Mass. We decide in our high spirits to collaborate on a new version of axiom 2, since two voices might be able to handle the cross-under-a-cross more easily. We sing:
Wow! Mini-satori. I heard the music of the laws of form, and I grabbed the astonished professor in an ecstatic bear hug, babbling that we had finally done what we had been told to do, we had performed the calculus; we had tripped over the Lost Chord, stumbling like Finn again over the deep and secret resonances of intelligence, mathematics and music. In Brown's terms, I had crossed back into the form, the state in which experience embraces paradox and tautology and logic does not exist. In the unmarked state, inside the form, where there are no distinctions, there are no paradoxes. One way to "understand" (stand under) the calculus was, paradoxically, to accept it on faith and do it and see what the results were. I had heard similar suggestions in Sunday school.
Operating in the ecstatic mode, I naturally separated myself from the discipline of intellectual yoga. In gratitude to Heinz, I offered him my own construct of reality, an apple seed, and went off looking for someone to play with, full of faith and eager to perform good works. In the dining lodge at Esalen, I providentially crossed paths with two musicians, otherwise strangers to me but fellow travelers in the form. Five minutes of spiralling them the magic of the laws and we were making music, archetypically plugged in, as it were, to the great and wonderful cosmic Wurlitzer, or whatever it is that Aristotle recognized as the prime organ of all motives. The musicians picked up on the notation right away, and with elan we began to decipher the structure and dynamics of the divine script. The length of the crosses, for example, can indicate a time scale. (It doesn't have to represent that, but it can if we choose to let it.) Some expressions even had five lines, just like a musical staff:
The letters of the alphabet, used as variables in the algebra. can by their very shape suggest a melodic line how do you sound your a? Operating in perfect harmony, my new friends and I rounded up some other players, found rehearsal space in a beautiful house overlooking the Pacific and began to compose celestial music.
Meanwhile, the conference flowered under Lilly's tonic attention. Fire of the spirit, the return of the California sunshine, and the Esalen sulphur baths had incubated the eggheads to hatching thresholds. Alan Watts calculated, on an imaginary abacus, that they were inventing the Glass Bead Game, right there in the South Coast Motel; the introduction to Hermann Hesse's novel is so close to a commentary on the laws of form that the book could be called, in retrospect, prophetic. Here is Hesse:
These rules, the sign language and grammar of the Game, constitute a kind of highly developed secret language, drawing upon several sciences and arts, but especially mathematics and music (and musicology) and capable of expressing and establishing interrelationships between the content and conclusions of nearly all scholarly disciplines."
-- Das Glassperlenspiel, translated as The Glass Bead Game, and also Magister Ludi
It was the graphically expressed interrelationships that enabled us to make music out of Brown's mathematics, and the other conferees were finding that they could make their own music in their own disciplines. Not even Hesse had invented the Glass Bead Game (he wrote about it, instead), possibly because he did not have access to Brown's calculus, the form.
The form itself expresses — the form. There is nothing mysterious about the calculus, it is obvious, Brown had told us. He states in his introduction to Laws of Form, "A major aspect of the language of mathematics is the degree of its formality. Although it is true that we are concerned, in mathematics, to provide a shorthand for what is actually said, this is only half the story. What we aim to do, in addition, is to provide a more general form in which the ordinary language of experience is seen to rest." He has written the laws of form, not the laws of thought. "Form" here may also mean "shape," and among other things, Brown is talking about the shape of the book, which is oblong, like most books, with 6x9-inch pages, about the typography, about the pagination, which links content to Qabala, about the exact words he uses, just those words and no others, about the organization and structure of the book as a whole. Brown determined very carefully that Laws of Form has 12 chapters, as does Only Two Can Play This Game, the book he published under the mask of James Keys, though both books have an almost comic number of introductions, prescripts, notes, prefaces, indices, appendices and the like. Twelve chapters, 12 signs of the Zodiac, 12 the Hanged Man, associated with Hermes, the messenger of the gods, the god of in-formation.
(Brown's remark that his own account of the laws of form expands indefinitely, but that since the book must end somewhere, he will break off with the words "and so on," suggested to one musician that the placing of this dieresis in the book is a clue to the placement of the octave, and hence the structure of the musical scale.)
The form of the algebra and the function of the notation are the same; as in the myths, from Barbarella's invisible key to John Barth's Chimera, the key to the treasure is itself the treasure. The cross itself, ᒣ , is the simplest example, dividing one space from another, marking out one quadrant of the Cartesian coordinates on which imaginary numbers were first mapped. It is no coincidence, it is part of the great cosmic scheme that this cross is half of the Christian cross, that it looks like the number 7, lucky seven, that brown, as noted above, is the seventh color.
Laws of Form, a mathematical analog of Finnegan's Wake, full of puns and etymological niceties, itself manifests Brown's cosmos. Brown writes always and ever as Qabalist, magus, spinning out one single exquisitely resilient gossamer skein on which to string all the glassy, cool, translucent beads of Western civilization's transcendental sweat. Wow!
In Only Two Can Play This Game, the coyotetrickster-wizard comments on how the mathematician looks inside himself to observe mathematical functions "as a spectacle or play (theatre and theorem have the same root) put on for his benefit in the holocosm where numbers exist and evoked by the particular way he learned, through his initiation into mathematics, to conjure with certain symbols." Evocation, initiation, conjuring; Keys even put in the italics himself, giving the whole cosmic show away (though ultimately, even a magician cannot give away anything, let alone his most lofty and arcane knowledge).
Within Brown's treatise on mathematics and logic, we may also discover myth and religion. Frequent references to God, Paradise, heaven and angels are explicit in Only Two Can Play This Game and in Brown's conversation, and implicit in Laws of Form, the second chapter of which has the rhythms of Genesis. That is not to say that Brown is a mathematician who also happens to be religious. Rather, the bridge of his mathematics formally unifies science, knowledge, logic with art, myth, religion and poetry. An avatar of James Joyce, he functions as the conscience (consciousness: from scire, to cut through) of the race — that is, as our awareness. Like the fugitive from Plato's cave, he guides us from the world of the merocosm, marked as a world of apartness where people write and read magazine articles, through the tunnel of paradox to the holocosm, the world in which all is one. We oscillate between these two states -- or rather, Brown says, we would if we could. Motion, he says, is illusory and apparently we exist in both states and reflect back and forth in the infinite illusory space of barbershop mirrors. Life is not one state or the other, but comprehensively the male mode of either/or distinction-drawing bound in eternal union with the female embrace of both/and unification.
This Pythagorean evocation of myth and religion in a rigorous mathematical framework is bound to have its effect on Western science, whose own practicioners, not to speak of its victims, have often called attention to its lack of contact with humanity. Humanity is male and female and Brown is expressing in integrated terms the argument that is made in partisan form by the movement to focus women's consciousness.
The conference that had proceeded on intellectual lines closed on the other side of the looking-glass. Sunday morning, the seventh day, and Kurt von Meier in white suit and silken robes shines with the sunrise in a ceremony before the fireplace on the Esalen sundeck. With Dakini and child by his side von Meier expresses the union of the teachings of the scientist/magus Brown and those of other masters of alchemy and grace, principally the Tibetan vajrayana yoga, the path to enlightenment from which there is no turning back, of Chogyam Trungpa, Rinpoche. In his audience is Karl Pribram, who the day before had covered the blackboard with notes on the structure of the brain, now busily taking notes on the relationship of the laws of form and the visualizations of the sun, moon and stars formalized in the signs of the Zodiac. Many of us partake of a sacrament, amanita, some say the food of the gods.
And so Watts, a founder of the Society for Comparative Philosophy and himself a guru of renown, leads a simple, silent meditation. Then the Brown Cross Chorale two flutes, dulcimer, viola, temple bell and vox human, - performs Consequence 1 of the laws of form (p. 31), our delighted reflection of Brown's gift, a little off-key and out of time to sublunar ears, perhaps, but infallibly the music of the spheres.
PACIFIC SUN — Week of September 6-12, 1973
Below is a copy of letter from Tim Leary to Cliff (Walter) Barney: