Real stars, as modern astronomy since Galileo very well knows, come in an amazingly rich array of colors, shapes, sizes and constitutions. The unaided human eye can see about five thousand stars, but the actual number of stellar objects is, practically, beyond counting. Even the number of galaxies is beyond the capacity of most ordinary people to imagine; and each of those galaxies--as everybody who has watched the popular astronomer Carl Sagan on television also very well knows--contains billions and billions of stars. The smaller stars appear to burn out much more slowly than do the big ones--which might seem paradoxical--eventually becoming dull brown dwarfs, or dark stars that have radiated all of their substance except for the solitary remnant of a slowly-cooling core composed primarily of solid iron. Our own sun, the nearest star, seems to be such a candidate.

Much new information about our sun was gathered during the solar eclipse of July 11, 1991, because it occurred (conveniently) directly over the observatory atop Mauna Kea on the Big Island of Hawaii: far from city lights, with clear air because distant from major sources of atmospheric pollution, located much nearer to the equator than to one of the Earth's poles, and at a high altitude, in air that is also dry, hence with few water molecules which tend to disperse celestial light.

For astronomy, as for many fields of contemporary science, there is a daily flood of new information to analyze, assess, publish and interpret. Here, we do not intend to survey even the most recent or astounding of these discoveries, and yet, astronomy is one area of study that enjoys continuity from the most archaic culture down to the present day. Some lore of celestial observances about the patterns of stars that assist navigation--even for mammals and birds during the migrations--has been programmed into our consciousness from a time long before telescopes or calendars, somewhat like the fiddler crab's motives are triggered by the periodic pulls of lunar-influenced gravity we know as tides. Astronomy thus provides an excellent formal frame of reference for questions of space and time. For example, scientists recently reported that they have found the "Oldest Thing in the Universe," a quasar (or, a "quasi-stellar," very enigmatic object) more than twelve billion light-years from Earth:

We're seeing it when it was very young. When you look back at these great distances, you're looking at very close to the beginning.

[ Donald Schneider of the Institute for Advanced Study in Princeton, with James Gunn of Princeton University and Maarten Schmidt of the California Institute of Technology, in the Astronomical Journal (September 1991), as reported by Frances Ann Burns of United Press International in the San Francisco Chronicle (August 26, 1991). ]

As if to emphasize the swiftness of our contemporary scientific riverrun, a report on efforts to define what the term "oldest" could possibly mean describes a newly-observed cosmic phenomenon--even more recent than the preceding quasar--likened to the discovery of "the original blueprints of the universe," eliciting a perspective of such scope and scale in space/time that (for some imaginative capacities or aesthetic sensibilities atrophied by negligence which have acquiesced in defining the limits of art as projections merely of their own Earth-bound tastes) it may appear excessively majestic. HOWEVER:

In what may be one of the century's milestones in astronomy, researchers using a NASA satellite to peer at the edge of the cosmos have spotted ancient imprints of "ripples in space and time" that shaped the newborn universe an instant after creation.

...The structures they see are clouds of hot gas from a time before the existence of stars and dwarfing in size and age even the largest known clusters of galaxies....[Said the discoverer] George Smoot, an astrophysicist at the University of California at Berkeley, "If we hadn't found it, I'd have about given up on gravity." The discovery, he said, seems to confirm gravity as the prime organizing force behind the galaxies and clusters of galaxies that dominate the visible structure of the universe.

"We are talking about the birth of the universe itself, the largest and most ancient things there are," Smoot said....

UC Santa Cruz astrophysicist Joel Primak compared the finding to encountering "the handwriting of God. It is the closest we are ever going to get the the beginning. At least, it is the only way I know as a professional cosmologist to get that far back into the mystery."

The measurements may rank in importance with two other great achievements in 20th century cosmology. The first came in the late 1920s from astronomer Edwin Hubble, who used the 100-inch telescope at Mount Wilson Observatory in Pasadena to measure the recession speeds of distant galaxies and concluded that the whole universe is expanding. The second was detection with a radio telescope in 1964 by Bell Laboratories physicists Arno Penzias and Robert Wilson of the microwave background radiation left over from the primordial fireball. "This is what we have been waiting for...," said [cosmologist] J. Richard Bond...."It is the most important discovery since the background was found....This has been the Holy Grail."

[ Charles Petit, "Scientists Report New Images Of Universe Just After Creation," San Francisco Chronicle, (April 24, 1992), front page, A4.]

The playful student of Duchamp might idle away some amusing hours by matching words or phrases in that report with references in Duchamp's notes or illustrations from his oeuvre, as in "ripples in space and time" = Draft Pistons, or with the Illuminating Gas, the Milky Way, the cover of View, and the Handler of Gravity.

Although difficult to conceive in any ordinarily meaningful sense of time, this birth--when the entire universe is supposed to have been almost infinitely hot and dense, compacted into a space far smaller than that occupied by a single electron--the "Big Bang" has been reckoned at one ten thousand billion quadrillion quadrillionth of a second from a hypothetical "time zero." This could be represented as a decimal with forty-two zeroes and a one after the point. Because of certain constraints imposed on quantum mechanics by the Uncertainty Principle, (or the Principle of Indeterminacy, formulated by Werner Heisenberg in 1925), to speak of earlier times is--in THIS universe, anyway--next to meaningless. In order to balance these considerations of cosmic old age, it seems only fair to cite reports about some of the newest and youngest things in the universe; we might also discover parameters for formulating a paradigm of artistic creation (or, re-creation) in the supernova phenomenon of stellar rebirth.

The standard press reports that NASA's orbiting Hubble Space Telescope has transmitted just such information petaining to the idea of rebirth on a cosmic scale:

evidence that dying stars can rejuvenate themselves when they collide with each other...[restoking] the stars' thermonuclear furnaces and enabl[ing] them to be reborn as the equivalent of one younger star, burning hotter and brighter.

These bright, young stars are known as "blue stragglers," which have been objects of mystery to astronomers because they appear in places where it is believed that star-formation ceased some ten to fifteen billion years ago.

A team led by Francesco Paresce of the Space Telescope Science Institute in Baltimore used the European Space Agency's Faint Object Camera, one of several instruments aboard the Hubble telescope, to peer straight into the heart of one such place: a globular cluster of stars known as 47 Tucanae.

[ Kathy Sawyer, "Hubble Shows Old Stars Reborn," Washington Post, reprinted in the San Francisco Chronicle (August 22, 1991). ]

Among almost 600 stars newly observed, because too faint for land-based optics, 20 are blue stragglers, a very high proportion for such a small area. One now wonders if only Tucanae play this game. Some astronomers believe that many of the larger stars will continue their accretion of mass until precipitating the transformative phenomenon called a nova or supernova: a gigantic explosion which then scatters and redistributes the stuff of the former star in space. Subsequent observations made with the Hubble Space Telescope revealed

"a new class of object in the universe," gigantic and violent star-forming clusters that were created from the wreckage of two galaxies that are colliding.

[ Kathy Sawyer, "Collision of 2 Galaxies Forming Huge New Stars," Washington Post, reprinted in San Francisco Chronicle, June 3, 1992).]

Astronomers from the University of Maryland peered through a cloud of obscuring, cosmically-bred(!) dust, into the heart of a galaxy that might as well have been named for a pal of Duchamp, the French-German, Dada-Surrealist, painter-sculptor, Jean-Hans Arp:

a peculiar galaxy known as Arp 220, about 230 million light-years away in the constellation Corona Borealis. The Hubble view shows the first details ever seen of such a stellar nursery, known as a "star-burst galaxy." It reveals six knots of brightness, each a cluster already filled with billions of suns and producing massive new stars at a furious pace....The star-forming clusters put out energy so intense that collectively it is equivalent to 500 billion suns....Such output exceeds that of a supermassive black hole, equal to some 400 billion suns, that is believed to be hiding nearby at the core of the Arp 220 galactic debris....

[ Kathy Sawyer, ibid.]

Some knots! Supernovae give birth to all that we know is essential for life itself. What is now seen as the Crab Nebula (in the constellation Taurus) is the collective name for the still rapidly-expanding remnants of the best known of these titanic events that occurred in 1054 (although it does not seem to have been recorded in any European chronicles). The Danish astronomer Tycho Brahe did record a similar supernova phenomenon in 1572, which shook the foundations of the Aristotelian belief in the "perfection" of the heavens, and asecond was seen by his pupil Johannes Kepler in 1604. According to Joseph Needham, the historian of science George Sarton

has said, in another connection, that the failure of medieval Europeans and Arabs to recognise such phenomena was due, not to any difficulty in seeing them, but to prejudice and spiritual inertia connected with the groundless belief in celestial perfection. By this the Chinese were not handicapped.

[ Needham, Science and Civilisation in China, III, p. 428. ]

Professor Needham, the scientist, cultural historian and sinologist, cites five separate Chinese texts describing the 1054 nova and accurately marking the diminution of brightness that, as the modern astronomer Professor Fred Hoyle has pointed out, we now know agrees very well with what must have been its actual rate of decay.

Since the Chinese records say that at its maximum apparent brightness the "guest-star" was as bright as Venus, it can easily be calculated that at the time of the explosion the star was several hundred million times as bright as our sun.

[ Needham, Science and Civilisation in China, III, p. 426 f. ]

A Jesuit priest, John Adam Schall von Bell, the first European Director of the Chinese Bureau of Astronomy,

was present as a young man in the hall of the Roman College in May 1611 when Galileo received a triumphant welcome from Clavius and his 'mathematicians' after their confirmation of his discoveries.

Later in China, in 1626, the priest published a treatise on the telescope, The Far-Seeing Optick Glass (Yuan Ching Shuo).

This book included a rough picture of the Crab Nebula, though neither Schall nor his readers knew what its later importance for cosmology would be, nor that it had come from the +1054 supernova observed only in China and Japan.

[Needham, Science and Civilisation, III, p. 444 f. ]


With the remarkable figure of Galileo Galilei (1564-1642), we find clear articulation of two complimentary modes of knowing, translated as "intensive" and "extensive," bringing these radical concepts together in such a grand fashion that--much more than the merely mechanical innovation of the telescope, improved and applied to astronomical observation by Galileo, but actually invented in the Netherlands--the implications of his thought precipitated a fundamental revolution in the way the universe itself could be imagined. The continuing importance of Galileo's views for whole systems, even for modern relativistic models of the cosmos, was emphasized by Albert Einstein, who concluded that

a close analogy exists between Galileo's rejection of the hypothesis of a center of the universe for the explanation of the fall of heavy bodies, and the rejection of the hypothesis of an inertial system for the explanation of the inertial behavior of matter [basic for the theory of General Relativity].

[ Albert Einstein, "Foreword," Galileo Galilei, Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican, translated by Stillman Drake, University of California Press, Berkeley and Los Angeles, (1953), p. xiii. ]

When Galileo was enjoying, briefly, his recognition by the most powerful institutions of information control in Western Europe, he decided to make a basic change in the style of presenting his data to the general public--one which provides a fascinating parallel to his wholistic attempts at bridging science and art, or mathematical knowledge and his own surprisingly constant spiritual faith.

He had chosen his form of expression wisely--all too well aware that the full power of the Inquisition lay behind the tenets he assailed. Not only the content, but the manner of his New Science was strange: experiment and observation. He must have been fully convinced of the effectiveness of the Platonic form in the face of such real threats. Almost of necessity then, did Galileo write in lucid Tuscan Italian. He charged himself with changing the ideas of his society. To have written in Latin would have been a sheer anomaly. Here is a revival of the Humanist spirit of Dante, Petrarch and Boccaccio, a spirit elemental in Galileo's challenge to authority. As his language was geared to communicability, so was the style and structure of his literary form. After 1611 his style turned from the systematic treatise to more truly literary forms, characterized by eloquence, persuasiveness, and rhetorical ingenuity, combining clarity with elegance and profundity with art....The form of dialogue permitted him to use almost any example or method of proof. He used many diverse arguments--literary and mathematical, psychological and logical, speculative and concrete--which are magnificently combined into an overpowering whole. As it was said about the Dialogue Concerning the Two Chief World Systems during his lifetime, if Galileo does not prove his theories absolutely, at least he completely dispells our faith in the old ones.

It is generally conceded that the administrative violence [of Galileo's trial, conviction, condemnation, and imprisonment, in 1633] was for reasons of State, acting through the Church....Galileo's social situation [and physical circumstances] derived from the almost impossible task he set for himself: the incorporation of a radical method within a traditional structure, to be effected without defacing or distorting either. The two virtual incompatibilities could only be resolved by a grand, dramatic act, on a heroic scale, directly involving nothing less than the configuration of the whole universe. Within the gigantic limits of a new and relativistically oriented universe, Galileo defined a law of an even higher order of infinity than that of material cosmology, ancient or "modern." This is the law of the supervening mathematical abstraction.

Galileo's primary assumption is that "human understanding can be taken in two modes, the intensive, or the extensive." This is the crux of his argument for the scientific method and the supremacy of mathematics. He implies by it his own raison d'être,and upon the basis of this idea (as the principal one of eight points) he was tried and convicted in 1633.


Galileo concludes from this that the key to all valid human knowledge is the understanding of mathematics, for any such knowledge is God-like, no matter how infinitely inferior to God, which cannot be said for any other category of knowledge. [James Ackerman here added the comment, "Exactly Leonardo's view."]

[ Kurt E. von Meier, "Neoplatonism and Platonism in Galileo's Cosmology," unpublished paper, University of California, Berkeley (May 17, 1957), pp. 24 f., 18 f., 32. ]


In 1913, on the day the Armory Show opened, a New York newspaper, published the scurrilous cartoon showing Duchamp's Nude Descending a Staircase, No.2, with the mocking insinuation that it might have been hung upside down. Not that this caused overmuch philosophical concern for Duchamp, who was inclined to take a creative and flexible stance on issues of orientation. Evidence for this assertion is the complex three-in-one painting, Réseaux des Stoppages, or Network of Stoppages (1914) as it has been described by the gallery director Julien Levy, "The Man Who Organized The First Surrealist Exhibition In America":

The body of the canvas is printed with greens and ocher, the colors of the famous "Nude Descending a Staircase." Part of the painting is half-erased, but one can distinguish what seems to be multiple female figures grouped like the composition of Czanne's "Les Baigneuses." If these are what they seem, then the picture should be hung vertically with the nudes upright, but it remained impossible to determine the top and the bottom of the picture--although when I saw it hung vertically, I found Duchamp's signature in the lower righthand corner, with the date 1911. But there is another signature one sees in pencil with a date, 1913, when the picture is hung upside down. Should not this signature apply to the design drawn over the whole canvas in pencil, which can be recognized as a rough sketch for "La Marie," executed in glass (1915-1923)? The Glass is Duchamp's major work. There is a third signature when the canvas is hung horizontally. This would apply to the strange series of lines, circles, and numbers, the lines traveling toward a large bull's-eye of circles which would be the "Réseaux des Stoppages." This portion is of a later date--would it be the year Marcel stopped painting? I argue that the canvas was probably made up of the souvenirs of the first painting Duchamp ever attempted, of the project for the cumulative work, and the postscript when he would last set brush to canvas. The whole had been launched upon the public without explanation, to make what effect it could be means of its own mysterious worth, wound up to run by its own inner springs for as far as it might carry.

[ Julien Levy, Memoirs of an Art Gallery, G. P. Putnam's Sons, New York (1977), p. 23 f. ]

Rather than attempting to represent multiple points of view in space/time as had the Cubists Braque and Picasso, with a fixed orientation of the painting's frame--or, for that matter, as had Galileo, long before them and in a much larger frame--Duchamp entices the regardeur of Network of Stoppages to participate in a kinetic act, literally changing his own frame of reference, to appreciate Duchamp's self-referential epitome of his painterly career (in terms of the three Aristotelian "unities" of beginning, middle, and end), authenticating each space/time point of view with his signature.

With Hidden Noise, as a revolutionary expression of sculpture, might also be displayed in a similar inverted fashion. Even though it may call to mind unsavory images of some dead brute with legs sticking up in the air, the rightness of our theory about the proper orientation of With Hidden Noise is proven by reflecting upon the physiological metaphor we use when politicians and true-believers speak of security as the "legs we stand on."

For many otherwise intelligent people, the psychological metaphor of security has now become entangled with the science fiction scenario of Star Wars, a confusing nightmare of topsy-turvy misrepresentations in which even the generals supposed to be in charge don't know which way is up. Still, dramatic flip-flops, such as the inadvertent "heels-over-head" exhibition of an upside-down work of art, rarely happen nowadays, you might say, which makes the June 23, 1986 issue of Time magazine so provocative. One photograph showed the official Star Wars logo appearing on a wall behind General James Abrahamson. The design features the "shield of Ajax," presumably copied by some anonymous Department of Defense staff artist from a well-known piece: a 16th century B.C. Mycenaean dagger in the Athens Museum. However--this must surely give the citizens of the world (or, anyway, thinking people) reason to pause--the shield as shown in the logo is upside down!

Archaeological research argues that this shield type was ungainly and inefficient, and had long been obsolete by the time of the Trojan War. What does this tell us about its selection as an emblem for the modern military project? As shown on the Star Wars logo, the shield ought to cause our confidence to shudder if it is meant to symbolize the would-be invincibility of the so-called space shield, because whichever way the shield is meant to face in its action setting as depicted, the Department of Defense version blocks only three out of five space-blasts that the logo shows. Sixty per cent efficiency?! The editors of Time, in replying to a reader's letter lamely alluded to the notion of non-directionality in space. Nevertheless, in any imaginable interaction with ballistic missiles, we presume that a system of defense requires as one of its major functions, a precise orientation in terms of what is going up and what is coming down.

Ronald Reagan's Star Wars proposal may seem to us merely an expensive, if dangerous, piece of science fiction....Unfortunately the Star Wars or Strategic Defense Initiative (SDI) project has been analyzed and criticized largely from a technical standpoint. Critics make a good case that Star Wars is technically unfeasible, costs astronomical sums, and is likely to be destabilizing because it will strengthen calls for a first-strike attack against the Soviet Union [or other foe] and will undermine existing arms limitation agreements. Significant as these criticisms are, they stop short of the wider economic, political and ideological significance of Star Wars....In other words, Star Wars and its associated Strategic (fifth-generation) Computing Program may very well be working, even as more and more people recognize that Star Wars cannot succeed in its official purpose.

The billions of dollars in [non-competitive] contracts for SDI would be criticized as improper government intrusion into the free market place if the object were the provision of non-military services, [for] the same companies that have benefitted most from Pentagon contracts are the chief beneficiaries of Star Wars deals. Star Wars...pressures government to maintain a permanent war economy which provides a stable flow of guaranteed profit for [Pentagon contractors, and] by subsidizing the efforts of these companies to enter high-technology production it is contributing to restructuring the U.S. economy. At the moment [1987], 42 per cent of all US research and development funding comes from the Pentagon. This will only grow as Star Wars grows. Pentagon officials have been explicit about the sort of control [over research and development] this provides.

The opposition [to Star Wars] would also be strengthened by deflating the notion of a technical fix for defense, especially the very idea of "defense" for the world's leading imperial power. And Star Wars opposition would, finally, be strengthened by pointing to the ways that SDI can work militarily against the majority of the people living in the Third World.

[ Vincent Mosco, "Star Wars is Already Working," Science as Culture, Pilot Issue (Radical Science Series, No. 21), Free Association Books, London (1987), pp. 12 ff. ]

Some five years after Mr. Mosco set down these thoughts, and despite vociferous objections from scientists with profound--and unanswered--questions about any such program's efficacy, and following the virtual collapse of Communism as a global threat--a palpably contrived sales pitch for America's war industry, as it now appears--and the disintegration of the Soviet Union (but with a caveat on potential threats from China) Star Wars related projects still command billions of dollars as the largest item in the "defense" budget, with abundant evidence of stupifying waste and colossal fraud.

In its quest for a space-based defense against nuclear attack, the U.S. government invested at least $7.7 billion in Star Wars projects that never got off the ground, according to a review of government reports and audits. Of the government's $29 billion investment so far in the Strategic Defense Initiative, as Star Wars is formally known, fully a quarter went to projects later cast aside as unneeded, unworkable, or unaffordable....

The collapse of those efforts has not slowed Star Wars spending, however. This year [1992], the program will consume $4.15 billion, making it the biggest item in the defense budget even though the threat that led to Star Wars' creation--Soviet nuclear attack--has receded.

[ Associated Press, dateline Washington, "Star Wars Projects That Died Cost the U.S. $7.7 Billion," San Francisco Chronicle (May 26, 1992). ]

Whoops! It's going to cost a lot more money than we thought yesterday.

The Star Wars defense system will cost $37 billion over five years, about $10 billion more than the Bush administration estimates, a congressional report said yesterday [May 27, 1992].

The Congressional Budget Office said costs for Star Wars...would be about $8 billion a year.

[ "Higher Cost Estimate For Star Wars System," dateline Washington, San Francisco Chronicle (May 28, 1992), tucked in on page D7. ]

But is it art? Having so little to do with objective science, or even (in the terms it professes) with reality, this mega-boondoggle is every bit as fanciful as the George Lucas Star Wars movies from which the would-be "Strategic Defense Initiative" derived its more popular tag. Using kind and gentle terms, it is now recognized as a deeply flawed con job and charade by internal experts such as David S. C. Chu (assistant secretary of defense for program analysis and evaluation), John E. Pike (director of space policy at the Federation of American Scientists, a private group in Washington), and by almost every self-respecting scientist not securely stuffed into the Pentagon's pocket. Less kind and gentle language would doubtless point to the whole program's cynical cruelty and perilous deceit.

[Secretary Chu's critique is] "saying that any system deployed in this century isn't going to work, that the current plan is a procurement disaster in the making," Pike said. Leaders of the [Star Wars] effort, he added, "basically want to decide what to build before they test it, and hope they guessed right."

[ William J. Broad, dateline Washington, "High-Level Criticism of Star Wars: Top Pentagon analyst warns of risk from rushing to put it in place," New York Times, reprinted in the San Francisco Chronicle (June 2, 1992). ]


Stars have always been among the most significant of objects to be represented, in art and in both the older oral and the much younger written traditions of human culture. For example, the Diamond Sutra--one of the central texts of Buddhist India summarizing the teachings of Gautama succinctly set forth in three thousand lines--mentions stars as the first way (among nine) that the Whole, the adamantine essence (of both the text and the Prajaparamita teachings it represents in an epitomized form) may be visualized. In addition, other traditional Buddhist practices find perhaps unconscious (but still surprising) parallels in Duchamp's aloof philosophical stance.

An important instance of this is the principle of artistic integrity rooted in ethical precepts of "right action," as it were, based on some almost mathematically formulated, pristine model. Of course, we can also understand this in terms of a Western ethical tradition as well. Duchamp seems to have been very conscious and purposive about the bargain--for which he was willing to serve as an intermediary--between (on the one hand) what classical antiquity would have personified as the Muse, and (on the other hand) Posterity: the rest of the world of mortal beings, in both the age present and (more importantly, for Duchamp) in the ages to follow. Such was the clarity and power of his understanding the function of an artist, that the probability of ultimately transmitting the fullest qualities of his vision to citizens of the future depended upon an honest bargain being struck (and kept) with the sources of inspiration, whether personified as the Muse (which later he may have embodied as Rrose Sélavy), or only imagined as an abstract ideal in the way a mathematician might conceive a theorem that could prove the consistency of a calculus.

To his idiosyncratic essay (written in a style that abjures capital letters), davidantin appended the following comment on one of the notes from Duchamp's Green Box that began, "Take a Larousse dictionary and copy all the so-called `abstract' words...."

i am now quite convinced what he was interested in was a complete underlying logical calculus of language or mind in spite of his own and everyone elses insufficient competence to carry it out

[ davidantin, "duchamp and language" is the notation of an improvised talk given 12 april 1972 on the occasion of the duchamp festival at us irvine arranged by moira roth. Reprinted in D'Harnoncourt and McShine, pp. 99-115; see also their note 2, p. 115, relating to p. 111. ]

What might it take to invent a theorem proving the property of consistency in a calculus? A tautology is consistent, but it is not sufficiently complex enough to fit the definition for a "system" of calculating, a calculus. Complexity, in turn, requires qualification by a proof of completeness--otherwise a would-be "system" might simply multiply its complexities whenever an inconsistency arose; of course, this would lead to an ever greater number of eventual contradictions, and, however elusive, an infinity of inconsistencies.

G. Spencer-Brown has presented theorems which prove both consistency and completeness for the "Primary Arithmetic" in Laws of Form (p. 24). Kurt Gödel addressed the issue by considering a system of a minimum complexity on the order of what is conventionally called "arithmetic," the arithmetic of natural numbers, which is technically an algebra. That it is an algebra can be appreciated by any computer programmer who soon learns that the natural numbers can be represented by values in a Boolean algebra, which are even simpler, namely, the two elements zero (0) and one (1). Algebras can be thought of as calculi of formal relationships that include a variable function; and all algebras have an underlying order of arithmetic. In Laws of Form, Spencer Brown proposes the Primary Arithmetic as an illustration of that (eternal) complete and consistent arithmetic underlying all "Boolean" algebras, whether of computers, wiring circuits, formal logic, or any other dialectical system. But there are many ways to characterize or define "systems," a theme dear to the heart of that branch of mathematics known as Group Theory.

In our efforts to establish for the present text a suitably grand theoretical "loom" or frame of reference, we hope to acknowledge the scientific disciplines of cosmology and astrophysics--together with sub-atomic physics, and everything in between those extremes--as they reflect the attempts by human consciousness to come to terms with, to know, and to understand our own species in its continuous interaction with all being. For, what we see expressed by such endeavor is a seamless web uniting biological and cosmic evolution, as in experimental attempts to provide clues about the earliest stages of the genesis of the material world by staging head-on collisions of electrons and protons travelling at nearly the speed of light, or by the observation and study of distant, colliding galaxies, which practices were begun when the heroic Oculist Witness Galileo espied the moons of Jupiter.

What we can perceive, measure, and record of such micro- and macrocosmic events must be integrated into human consciousness no less so (but neither with any greater imperative) than our perceptions of theater, music, flower arranging, or any of the other arts. Any one of these "soft" activities undertaken by an accomplished practitioner may illustrate--in the end, with the same order of legitimacy as for any of the "hard" sciences--those underlying laws governing all formal interactions and thus determining (besides helping us to describe or explain) why things are the way they are.

Such fundamental formalities are traditionally expressed by creating art, performing music, dancing, making love, or by serving up the fruits of a virtuoso performance as a short-order cook. In their most abstract form, following the conventions of our present literate, logical culture, they are most clearly, pristinely and objectively recognized by the axioms, theorems, equations and other technical expressions of mathematics. To talk AT ALL about the simplest orders of being with their eternal formalities, however, we must necessarily use symbols and systems that come into existence only later, as innovations of the more complex and time-factored world in which we conventionally live. That is why both mathematics and meditation (as well as music or mixing up a recipe for a chocolate cake, for that matter) proceed by following to very special rules of injunction. To teach these procedures to others we must "take as given" some things, such as natural language, the ability to count, and the way in which we represent numbers--but we must take care, especially with numbers, because some exist and some do not. This may become more clear when we remember, as the polymath G. Spencer Brown has reminded us, discussing non-numerical mathematics:

Mathematics is not exclusively about number. Mathematics is, in fact, about space and relationships. A number comes into mathematics only as a measure of space and/or relationships. And the earliest mathematics is not about number. The most fundamental relationships in mathematics, the most fundamental laws of mathematics, are not numerical.

[ Transcript of the presentation by James Keys/G. Spencer Brown at the AUM Conference, Esalen Institute (March 19, 1973), edited by Clifford Barney and the present author, privately published and now available on the web at ]

Although it may not be immediately obvious why it should be so, according to the structural laws that apply to such formalities, counting may be introduced into a system that enjoys an order of complexity sufficient for a time of the second order, that is, what we commonly know as "duration." In the still simpler order, that of the first time (the "future") there is only oscillation. Even in counting, some numbers may be "real"; others, with their unique personalities, may be "imaginary," "irrational," or "transcendental," and so forth.


Iconic images of stars can be sorted out most simply according to the number of their points. Since one of the most obvious ways to proceed with an orderly method is "by the number," let us entertain a thought-experiment considering stars symbolically, and counting them by the number of their salient points. Really dark stars, of course, are invisible. Seen with the naked eye, most stars appear to be gleaming points of light; it is the atmospheric perturbation of their light that makes them twinkle, producing the effect of scintilation. Having reached the highest order of Paradiso, but one, Dante records having seen such a gleam of pure white light, as best he was able to describe it in Canto XXXIII of the Divine Comedy. The deep, divine, sense of "to gleam, to sparkle" is revealed by the etymology of the word FORM, deriving from the Indo-European root mer-bh, meaning "gleam." Some cognate American English words relating to form and shape [e.g. MORPHOLOGY] are based on the Greek morphe, which once implied outward appearance and beauty.

When we use ordinary language, as in our "consideration" of Duchamp's piece of sculpture, most of us would overlook the origins of the verb to CONSIDER, which is cognate with SIDEREAL (relating to the stars or constellations), since both derive from the Indo-European root sweid, meaning "to shine." That word root, from the ancient Indo-European language of six to eight thousand years ago, has also given us the word DESIRE. In Latin, the verb considerare was a term of augury, meaning "to examine," or "to observe the stars carefully." This is what the Chinese may have in mind, etymologically reading the ideogram for TEMPLE as "a place for observing the flight of birds," which, in the night sky, for the Chinese--as for poets--are stars.

Even though astronomers think that binary stars may be at least as common as solitary ones, two-pointed representations of stars are uncommon. But there is a close analogy in the tradition of Tantric art in India where, instead of star points, one might count the petals of lotus blossoms. This is the method for systematically ordering the so-called chakras, which may be understood as centers of psychic energy. Except for the uppermost, the seventh, wholistic and all-encompassing Sahasrara chakra, the other six are within the body (albeit the subtle, not the gross body). The highest of those is known as the Aja Chakra, symbolized by a two-petalled lotus blossom, sometimes represented as the "third eye," or as the duality that manifests as form and energy, called bindu. We might also imagine the two pointed "star" as the propeller tips that so struck Brancusi, Duchamp and Lger the Paris Air Show.

[ See for example, Ajit Mookerjee, Tantra Asana, Ravi Kumar Gallery, New Delhi (1971), "Aja Chakra" (p. 160), "Sambhu Yantra," plate 26; and Mookerjee, Tantra Art: Its Philosophy and Physics, Kumar, (1966).]

A three-pointed star can be seen as the hood or hubcap ornament on any Mercedes-Benz vehicle. Four points hath a square, the cardinal directions of the compass, and so forth, including the four principal means of transmitting the secrets of the teachings (about which we promise more to follow). The five-pointed star in Sufi traditions symbolizes the idea of Mankind Perfected, insan al-kamil. Some superstitious people conjure up diabolical associations when it is shown inverted, although the meaning is simply and obviously Mankind as yet falling short of Perfection. ('Twas ever elusive!) The interlaced six-pointed star has been known as the Seal of Solomon long before the declaration of the state of Israel; while snowflakes (and the tight-packing of a beeswax honeycomb) exemplify ubiquitous hexagon-based structures in nature. Seven-pointed stars find favor with many law-enforcement agencies--sometimes pointing up, sometimes down. The star with eight points is a central image for the Huichol people of north-central Mexico, who illustrate it resplendently in their extraordinary and magical, peyote-inspired embroidery, beadwork and yarn painting. An octagonal star also provides the basic pattern for a visualization exercise in use by a distinguished contemporary mystical school; as well, it happens to be the core shape of a standard nuclear reactor. The "ideal" form described both by Duchamp's sculpture, With Hidden Noise--that of a cube in space--naturally has eight vertices.

Imagine the eight-pointed star considered as a sort of die: a model, guide, or mental template spinning in eternal, mathematical space. This form, the essential exterior shape of "Totor's" sculpture (to use the nickname given Duchamp by H. P. Roch), contains the object of our analytical mind's desire which we are prepared to examine carefully, as if it were some sort of AUGURY, since this word, as well as AUTHOR, and AUTHORITY, all come from the same root. Following our initial "Pretext" we have introduced some historical, biographical, critical and interpretive issues, addressing twelve interrelated aspects of Duchamp's work, corresponding to a model, with the precision of an imaginary die--its mathematical form as pure for us as it was for Plato--illustrated by the essential properties of a cube in space: namely, the twelve edges of an eternal, ideal Platonic cube, since V - E + F = 2, after a theorem of the great Swiss mathematician Leonard Euler, which, with a similar observation by René Descartes, established the foundations of modern topology. So, let us now proceed with our analysis by the numbers, imagining the sculpture With Hidden Noise to be filling this ideal cube--as if it were perfectly contained enclosed in a glass box--projecting its eight vertices minus twelve edges plus six faces to equal, in interactive duality, the complementarity of current Philadelphian space/time.

* * *

Let us consider sidereal forms
morphological gleaming desires,
as the stars of the Diamond Sutra,
ephermeral fulfillments of wishes,
confused transformations of dying...
or reborn, with a new world conceived:
desideratum for some new distinction.
"The conception of the form lies
in the desire to distinguish."
Or, "The Stars That Play With
Laughing Sam's Dice."