CHAPTER EIGHT: ZEROES

1. THE ZERO-SPACE

ZERO IN BABYLON AND GREECE

CHINESE NUMERALS

THE MARK OF ZERO

AL-KHWARIZMI AND FIBONACCI

1. THE ZERO-SPACE

Following the analytical model of Marcel Duchamp's With Hidden Noise, we have counted down, by the numbers, beginning from the six external faces of the cubic volume occupied by the piece of sculpture. The aspect of "fiveness" was counted in the measurement of each edge of that cube by the standard of the English inch. The piece is held together materially by four nut/bolt combinations. It was made in an edition of three, and there are three lines of ciphered letters inscribed on each exterior face of the two brass plaques, which contain the one ball of twine. This process of counting down has brought us to the dark--initially empty--space, contained and defined when the brass plaques were bolted over the once-open ends of the toroidal twine ball. To this space we accord the number "zero"; or, let us call it by the name, the "zero-space."

This zero-space, like the ball of twine itself, manifests certain unitary qualities. For one thing, it has been precisely defined, in the sense of being physically contained by elements of the sculpture. In a way, number theory also bolsters this notion because, in the set of natural numbers, zero is considered to be a form of unity.

For the primitive reckoner number is always a number, a quantity, and only a number can have a symbol. Thus it is easy to see how the zero came to be the great stumbling block for the medieval arithmeticians in the West. They found it very hard to give up the old principle of ordering and grouping, in which every unit has its own symbol, which appears when that unit is present and is lacking when that unit is absent.

[Karl Menninger, Number Words and Number Symbols: A Cultural History of Numbers, The MIT Press, Cambridge, Mass. (1969), p. 400, See also, pp. 399, 398.]

The other numbers are either odd or even, and it would seem that zero is even, but it does have some properties not shared with the other natural numbers. Zero is neutral with respect to sign; that is, all the other numbers are either positive or negative, but zero is neither. In the "group" of natural numbers, zero is the identity element for the operations of addition and subtraction, but not so for multiplication and division in which operations the identity element is one. Ambiguity about zero arises in the power series, or exponentiation, since a number raised to the zeroth (0th) power equals unity: X to the 0 = 1. But by another rule, any number multiplied by zero is zero: X times 0 = 0. So, some ask, does 0 to the 0 = 1 or 0? There is, perhaps, something to be said here for keeping it simple.

The word SIMPLE < Latin sim-plex, "once-folded" (plicare, "to fold") is one of a large group of words whose original stem is the Indo-European root sem....The German word for zero, Null, contains the other Indo-European one-form, oins > Latin unus. The Latin diminutive ending equivalent to "-ling," or the German - chen is -ulus: mus, "mouse"--musc-ulus, "little mouse." Thus from unulus > ullus, "little one, one-ling, any at all." The Indo-European negative symbol n combined with this to form the Latin word n-ullus, "none, not any." The numeral 0 obtained its name "null" because in the medieval view it was "no (numeral)," nulla (figura)....Other members of the same tribe are the English words ON-LY < "one-like," and AL-ONE < "all- one."

[Menninger, Number Words, p. 171.]

The two modern numerical glyphs we use to represent the values of zero and one are: for "zero" a closed curve or circle, and for "one" a straight, vertical line segment. The figure eight, as a double closed curve, is the graphic symbol which, when rotated ninety degrees, con-ventionally indicates "infinity." Viewed simply as graphic symbols, or theoretically, the line segment of the glyph for "one" also could be extended to infinity in either direction; that is, it may be read as a token of the name indicating infinite extension in space. And, with the glyph for "zero," in the form of a circle (or ellipse) one could go around and around forever; that is, it may be read as a token of the name indicating infinite extension in time.

For that matter, we could imagine, say, the numeral "seven," with the two "arms" of the symbol being extended infinitely. Nevertheless, the glyphs for "zero" and "one" represent the simplest illustrations of the closed curve, and the open line (which may be considered an open, "not-closed" curve). This leads us to the famous Jordan Curve Theorem: A simple closed curve in the plane separates the plane into two regions, one finite and one infinite. In other words: the inside of the curve facing the finite region = IN; the distinguishing FUNCTION of the curve itself = AROUND; and the outside of the curve facing the space in which the curve has been drawn = OUT.

In all the cosmological models of early civilizations, a wide, four-cornered planar Earth was surrounded by infinitely extensive waters, surmounted by ever higher mountain pinnacles....This flat conceptioning is manifest right up to the present in such every-day expressions as "the wide, wide world" and "the four corners of the Earth."..."Up" and "down" are the parallel perpendiculars impinging upon this flat-out world. Only a flat-out world could have a Heaven to which to ascend or a Hell into which to descend. Both Christ and Mohammed, their followers said, ascended into Heaven from Jerusalem.

Scientifically speaking (which is truthfully speaking), there are no directions of "up" or "down" in Universe-- there are only the angularly specifiable directions "in," "out," and "around." Out from Earth and into the Moon, or into Mars. IN is always a specific direction--IN is point-to-able. Out is any direction....Around the world nothing has ever been formally instituted in our educational systems to gear the human senses into spontaneous accord with our scientific knowledge. In fact, much has been done and much has been left undone by powerful world institutions that prevents such reorientation of our mis-conditioned reflexes....None of the perpendiculars to a sphere are parallel to one another. The first aviators flying completely around the Earth...having completed half their circuit, did not feel "up-side-down." They had to employ other words to correctly explain their experiences. So, aviators evolved the terms "coming-in" for a landing and "going-out," not "down" and "up." Those are the scientifically accreditable words--in and out. We can only go in, out, and around.

[Fuller, Critical Path, p. 55.]

In our relatively complex world of space and time, we can easily show how two sides of a line differ. That is because what is "easy" is posterior, complex and superficial, on the surface, relative to what is anterior, simple and important, at the core and "hard." To the theoretical, mathematical line of one dimension-- extension--let us add a second dimension: width. Now the line of distinction is more like a ribbon or band, and we may label the two different edges, so that when closing upon itself the band actually forms a three- dimensional cylinder in which "Edge A" is rejoined to the other end of "Edge A," and similarly for "Edge B." But now let us add another dimension, a space in which we can perform a little experiment to see if the marked sides really are any different. If we give the band a single twist (we have to go into four-dimensional space to do this), and rejoin the ends--where each end of "Edge A" is matched up with an end of "Edge B"--the band forms what is called a Möbius strip. The twist that we can now see in the band shows not only that "Edge A" and "Edge B" are different, but also demonstrates that they have been switched, while the topological qualities of both edge and surface become unitary.

ZERO IN BABYLON AND GREECE

Some of the logical problems and apparent contradictions associated with the idea of zero are related to understanding its necessary function in systems of numerical computation. At the root of the issue are the inevitable difficulties of attempting to talk about orders of complexity that are simpler than, and prior to, the order represented by language: the problem of trying to describe with a system that has certain qualities, an order of being that doesn't.

In the past, some historians of science concluded...that the Babylonians used the zero only in a medial position and that their zero was therefore not functionally identical with ours. But as we now know from the work of Otto Neugebauer, Babylonian astronomers differed from Babylonian mathematicians in this respect: they used the zero at the beginning and end of written numbers, as well as in a medial position.

In a Babylonian astronomical tablet from the Selucid era, now in the British Museum, the number 60 is written in the following form (the value attributed to it is assured by a mathematical relation indicated in the context):

1;0 (= 1 X 60 + 0)

--->

in which the zero sign is used to mark the absence of units of the first order....Use of the zero in the initial position enabled Babylonian astronomers to note sexagesimal fractions (that is, fractions whose denominator is equal to a power of 60) without ambiguity....Thus, at least as early as the first half of the second millennium BC[E], Mesopotamian scholars developed a written numeration that was eminently abstract and far superior to any other system used in the ancient world; they probably devised the first strictly place-value numeration in history.

Later they also invented use of the zero [the Selucid dates do not go back beyond the third century BCE]....But to the Babylonians the zero sign did not signify "the number zero." Although it was used with the meaning of "empty" (that is, "an empty place in a written number"), it does not seem to have been given the meaning of "nothing," as in "10 minus 10," for example; those two concepts were still regarded as distinct.

In a mathematical text from Susa, the scribe, obviously not knowing how to express the result of subtracting 20 from 20, concluded in this way: "20 minus 20...you see." And in another such text from Susa, at a place where we would expect to find zero as the result of a distribution of grain, the scribe simply wrote, "The grain is exhausted." These two examples show that the notion of "nothing" was not yet conceived as a number.

[Georges Ifrah, From One to Zero: A Universal History of Numbers, Viking Penguin, New York (1985), p. 381 f.]

Thus, examples of the recurring contradictions about zero were apparent in ancient times. We can appreciate this fact in the old (but possibly post-Homeric) joke in the Odyssey as a failure to distinguish between Name and Form, or between how a name is CALLED and what a name IS (what it "tokens" or "represents"). When asked by Polyphemus the Cyclops what his name was, Odysseus (always sly and wily) said that his name was called "No Man." When the two-eyed Greek hero put out the single eye of Polyphemus and escaped from his cave, the howling, blinded barbarian was asked by his fellows, WHO had hurt him; he answered "No Man," simple-mindedly, and so was ignored. Though, considering the speaker, the phrase might have sounded out of character, Polyphemus should have aswered "A man whose name is called 'No Man.'"

As far as the zero is concerned, the Greek astronomer Ptolemy was familiar with the symbol o--an abbreviation of the Greek word ouden, "nothing"--as a sign indicating a missing place, and used it in writing Babylonian sexagesimal fractions...[in a way that] the symbol o indicated not only the absence of a fractional group, but even the absence of an integer (a degree). This means that the Greco-Babylonian model already possessed a zero symbol which could have been the stimulus for the creation of a zero in the Indian numeral system-- perhaps it even influenced the form of the Indian zero, for later on the zero was written as a small circle instead of a dot.

There is documentary evidence that the Babylonian astronomi-cal writings had a significant influence on Indian astronomy in the century following Alexander the Great, during which Hellen-istic culture spread farthest to the east and deep into India.

[Menninger, Number Words, pp. 399, 398.]

CHINESE NUMERALS

The ways of writing numerals in China probably evolved from the relatively late migration of Babylonian notation systems through Hellenistic intermediaries to India, then traveling as far as China with the wave of Buddhist influence.

The contacts of Chinese culture with India, its books and its numerals, were brought about by Buddhism which took root in China in the 7th century and attained great importance there. In the 13th century the zero (Chinese ling, "gap, vacancy" [but also "fragment, fraction, small change"]) first appeared in Chinese books and has continued to be used occasionally ever since.

[Menninger, Number Words, p. 460 f. See, in Matthews: ling 4057.]

There were fundamental differences in the written forms of ideogramic Chinese and the alphabetic Sanskrit of India. In Chinese, the line segment representing the numerical value "one" is drawn horizontally, but is otherwise apparently the same. Earlier there was no numeral corresponding to zero, because the written numerals in Chinese are ideographic, and each corresponds to a specific number word. Chinese written numbers and their ranks (units, tens, hundreds...) are expressly written down, whereas the Indian--hence the Arabic and Western--system, in ordering these ranks does not express the value of the number, but rather indicates it by the place-value of the digits.

Thus the Chinese is a "named" and the Hindu an "unnamed" or abstract place-value notation, if we equate position with rank.

Although both of these systems of numerals are essentially similar in reflecting the structure of the gradational spoken number sequence, the difference between them...becomes clear in the case of a number in which one or more places are left unfilled by digits: Indian: 4 0 8 9.

Chinese: 4 thousands szu-ch'ien, 8 tens pa-shih, 9 (units) chiu. In other words, the Indian numerals need and have a zero sign, while the Chinese do not. Thus in China there was never a struggle in the popular mind against the concept of zero (and its cipher) such as there was in the West.

[Menninger, Number Words p. 459.]

The express indication of rank values in Chinese made computation extremely cumbersome; so it was always carried out on an abacus, with the aid of multiplication tables, the numerals being written in vertical columns. But when logarithmic tables came to China from India, the numerals were written in horizontal columns (running right to left). The rank values were no longer written in, so the system changed from "named" to an abstract place-value notation that required a glyph for zero, for which a round circle was employed.

Having developed symbols to express the contents of each column, [the innovative abacus-user] had to invent a symbol for the numberless content of the empty column--that symbol became known to the Arabs as the sifr; to the Romans as cifra [or, ciphra]; and to the English as cipher (our modern zero).

Prior to the appearance of the cipher, Roman numerals had been invented to enable completely illiterate servants to keep "scores" of one-by-one occurring events--for example, a man would stand by a gate and make a mark every time a lamb was driven through the lamb-sized gate....Since one cannot see "no sheep" and cannot eat "no sheep," the Roman world seemingly had no need for a symbol for nothing. Only an abacus's empty column could produce the human experience that called for the invention of the ciphra--the symbol for "nothing."...The discovery of the symbol for nothing became everything to humanity. the cipher alone made possible humanity's escape from the 1700-year monopoly of all its calculating functions by the power structure operating invisibly behind the church's ordained few.

[Fuller, Critical Path, p. 32)

In China, however, awareness of the special nature of emptiness can be seen in the famous initial poem of Lao Tzu's Tao Te Ching:

The Tao that can be told is not the eternal Tao.
The name that can be named is not the eternal name.
The nameless is the beginning of heaven and earth.
The named is the mother of the ten thousand things.
Ever desireless, one can see the mystery.
Ever desiring, one can see the manifestations.
The two spring from the same source but differ in name;
this appears as darkness.
Darkness within darkness.
The gate to all mystery.

[Lao Tsu, Tao Te Ching, a new translation by Gia-Fu Feng and Jane English, Vintage Books, New York(1972).]

The form itself manifests in as many ways as there are ways of distinction. As in the Tao Te Ching, we start with the first proposition, "The way, as told in this text, is not the eternal way, which may not be told." The eternal way may not be told because it is not susceptible to telling. It is too real for that. It manifests in as many different ways or different expressions as there are in the beings to which it manifests....And when one looks at a cow in a field and somebody says "What is it doing?" I say, "Well, I think it is contemplating Reality." And they say, "Don't be ridiculous, how can a cow contemplate Reality?" "Why not?" I ask. "What else does it have to do all day? What else has it to do? The being is contemplating Reality...what else COULD it be doing? But the Form, as it is apparent to a cow--although it is the same Form, it is the Way without a Name [the "nameless Tao"]--how it manifests to a cow, is not how it manifests to me. How it is expressed to a cow is not how it is expressed to me.

[Keys, AUM Conference Transcript, p. 93.]

THE MARK OF ZERO

For those of us who speak and read and write in modern American English, the origin of our Germanic language may be found in the late middle ages. Special instruction is required, in addition to a mere word list, in order to be able to read Beowulf, the poems of Caedmon, the Venerable Bede, or even Chaucer's Canterbury Tales. The written form of the letters we have adopted for writing English is, with a few modifications, essentially Roman and hence considerably older than our language. The system we use for writing numbers is apparently even older. By tracking the path of "zero" we may trace the history of our numerals back through India, the Hellenistic expansion, and the later kingdoms of Babylon, who revived their traditions of numerical reckoning originally inherited from the earlier kingdoms of Babylon, who in turn, probably got it from Sumer, who for all we know, may have got it from Susa in Elam or from the culture-bringer Oannes from the legendary island of Dilmun that some archaeologists have sought to identify with the modern Bahrain, while others have seen in the figure of Oannes a sidereal messenger from the Once and Future Star.

[See George Michanowsky, The Once and Future Star: The Mysterious Vela X Supernova and the Origin of Civilization, Barnes and Noble New York (1979).]

The attentive reader may note that we have frequently cited an excellent study by the distinguished German scholar, Karl Menninger, that appeared as Zahlwort und Ziffer, when published by Vandenhoek and Ruprecht in 1958. As intelligently translated by Paul Broneer for the MIT Press edition, this work, having been done once with integrity and style, ought not to need doing over. Like that work by other teachers and scholars upon whose inspired and devoted accomplishements so much of our present work depends, we acknowledge with gratitude these prior contributions to knowledge and understanding, metaphorically hoping to build upon the solid base of the great unfinished pyramid by laying our own new block in the fresh course under construction. Another way to look at this process involves the metaphors of spinning yarn, tying twine, knotting webs, and weaving, by which we hope to following some threads--with particular attention to "symbolic and mythical" interpretations--in order to suggest relationships and connections with Duchamp's exemplary work of art, With Hidden Noise. Here, a few of Karl Menninger's words from the "Preface" to to the original edition seem to be pertinent, especially in relation to our present exercise:

This book is written for the lover of intellectual and cultural history, but the professional historian will discern many things in it not previously expressed. Of course the plain knowledge of the number series and number symbols...has to be cited first, but ethnology and ethnography contribute a most colorful addition: the history of language, of culture, and of politics. There are few things of this world in which these branches of of research meet each other in such an exciting and fertile manner as the concept of number. The area of its symbolic and mythical interpretation is not even included.

With this overwhelming wealth of detail it became difficult to pursue the great art of following the threads in this closely woven fabric, to separate them without destroying the fabric itself....It is not often that the lover of numbers becomes acquainted with the intrinsic connection of his special area with cultural history; it is equally rare that the friend of the history of culture becomes aware of the relationship between his field and the life of numbers. I hope that the present work will serve both groups to gain the insight and the joy which comes from all knowledge of creative intellectuality in the diversity of men and peoples.

[Menninger, "Preface," Number Words, p. v.]

Having chosen the concept of number as the primary organizing principle for our analysis of Marcel Duchamp's piece of sculpture--so unusual in its material and structure: hard brass and soft twine, and with its "zero space"--clearly of central significance for us is the concept of "zero," including its symbolic and mythical aspects, and how its numerical representation came to Western Europe.

Our numerals do not have the same origin as the phonetic alphabet we use in writing and printing. Our letters are not akin to our numerals. One would naturally be inclined to suppose, in all innocence, that the human mind, when it took the trouble to record its ideas and concepts, would have devised similar systems of writing words and numbers, "seven" and "7." But this did not happen, neither in our western culture nor anywhere else in the world. The early system of writing numerals is everywhere the older of the two sisters....Our language is Germanic, our writing [earlier] is Roman, our numerals [earlier still] are Indian.

[Menninger, Number Words, p. 54 f.]

The case of zero is special; in Sanskrit, it was called shunya, "void, empty" (sometimes written sunya or shunya-bindu, "empty dot") after its physical meaning: "the position (originally on the counting board) is empty." Our modern custom of indicating a missing word or line of verse by a row of dots goes back to this Indian practice. From a much later date than other Indian numerals, the inscription containing the earliest true zero known thus far has been identified in India and, delightfully, concerns the dedication to a temple of both flowers and the land upon which flowers may be grown in the future.

This famous text inscribed on the wall of a small temple near Gvalior (near Lashkar in Central India) first gives the date (AD 870 in our reckoning) in words and in Brahmi numerals. Then it goes on to list four gifts to a temple, including a tract of land "270 royal hastas long and 187 wide, for a flower garden." Here, in the number 270, the zero first appears as a small circle...in the twentieth line of the inscription it appears once more in the expression "50 wreaths of flowers" which the gardeners promise to give in perpetuity to honor the Divinity.

[Menninger, Number Words, p. 400 f.]

AL-KHWARIZMI AND FIBONACCI

In 773 there appeared at the court of the Caliph al-Mansur in Baghdad a man from India who brought with him the writings on astronomy (the Siddhanta) of his compatriot Brahmagupta (fl. ca. AD 600). Al-Mansur had this book translated from Sanskrit into Arabic (in which it became known as the Sindhind). It was promptly disseminated and induced Arab scholars to pursue their own investigations of astronomy.

One of these was al-Khwarizmi...who was probably the greatest mathematician of his time, [and who] wrote among other things a small textbook on arithmetic in which he explained the use of the new Indian numerals, as he had probably learned them himself from Indian writings. This was around AD 820.

[He] was also the author of a book showing how to solve equations and problems derived from ordinary life, entitled Hisab aljabr w'almuqabala, "The Book of Restoration and Equalization." ....Its translation into Latin, Algebra and Almucabala, in the 12th century, then ultimately gave its name to the discipline of algebra. The original of al-Khwarizmi's book on arithmetic is lost, but it...made its way to Spain...and it was there, at the beginning of the 12th century, that it was translated into Latin by the Englishman Robert of Chester, who "read mathematics" in Spain. Another Latin translation was produced by the Spanish Jew, John of Seville. Robert's translation is the earliest known introduction of the Indian numerals into the West. The manuscript discovered in the 19th century begins with the words, Dixit Algoritmi: laudes deo rectori nostro atque defensore dicamus dignas "Algoritmi has spoken: praise be to God, our Lord and our Defender." At about the same time (ca. 1143) an epitome of this book was written which is now in the Royal Library in Vienna.

[Menninger, Number Words, p. 411.]

The Codex of the Salem Monastery--now in the University Library, Heidelberg--contains fifteen pages written in abbreviated Latin and must be reckoned, together with the lost text of al-Khwarizmi, as one of the oldest manuscripts in the West describing computations with Indian numerals. It begins, Incipit liber algorithmi...(Here begins the book of Algorithmi...), and in the opening lines we also find repeated the very phrase, Omnia in mensura et pondere et numero constituisti, we have seen derives from the Apocryphal Book of Wisdom, and expressing the idea that everything is constituted of measure, weight and number. The manuscript was composed around the year 1200, and documents the spread, by that time, of al-Khwarizmi's book in the Germanic part of Europe. The text also reveals a fascinating reference obviously derived from Plato concerning the origin of number, although Plato focussed on the Unity, and did not actually mention zero.

The algorism of the Salem Monastery correctly interpreted the new numerals and used them for computations, but they still created such confusion in the mind of their author that he appended the following mystical interpretation:

Every number arises from One, and this in turn from the Zero. In this lies a great and sacred mystery--in hoc magnum latet sacramentum--: HE is symbolized by that which has neither beginning nor end; and just as the zero neither increases nor diminishes / another number to which it is added or from which it is subtracted / so does HE neither wax nor wane. And as the zero multiplies by ten / the number behind which it is placed / so does HE increase not tenfold, but a thousandfold --nay, to speak more correctly, HE creates all out of nothing, preserves and rules it --omnia ex nichillo creat, conservat atque gubernat.

In this way the zero acquired its profound "significance" and began to represent something. But the learned men too were not sure whether the zero was a symbol, a numeral, or not. According to the name Null which they gave to it, it was not; and so the medieval writers would frequently present the "9 digits," to which they would add one more, which was called a cifra.

One of the greatest and most prolific mathematicians of the middle ages was Leonardo of Pisa (1180 to ca. 1250), also popularly called "Fibonacci," after whom the famous number sequence is named. In order to learn computation, he traveled to a trading post in Algeria where his father was governor. As he recounts the story:

Ubi ex mirabili magisterio in arte per novem figuris Indorum introductus. (There I was introduced by a magnificent teacher [perhaps an Arab instructor] to the art of reckoning with the nine Indian numerals.

Leonardo did not approach the new methods of computation superficially, as "just another procedure," but tested them thoroughly and came to regard them as a vast improvement. It was in this spirit that he wrote his great Book of Computations, the Liber Abaci of 1202, which prepared the ground for the widespread adoption of the Indian numerals and the new operations in the West. He introduced the new numerals in the following words:

Novem figure Indorum he sunt 9 8 7 6 5 4 3 2 1. Cum his itaque nouem figuris, et cum hoc signo 0, quod arabic cephirum appellatur, scribitur quilibet mumerus. (The nine numerals of the Indians are these: 9 8 7 6 5 4 3 2 1. With them and with this sign 0, which in Arabic is called cipherum [cipher], any desired number can be written.)

[Menninger, Number Words, p. 425.]

The notion of counting, we may recall, is one of the things that we may "take as given"--together with the phenomenon of language--as primary functional expressions of human consciousness. The idea of a numeral--and the symbolic system of discreet numerals--should be distinguished from the more profound and historically much older concept of number itself. Number need not be marked at all; or, as on many Paleolithic bones, it may be reckoned by tally strokes; again, in a more sophisticated way, using an abacus, it may be indicated by the grouping and positioning of beads. The concept of a numeral belongs to the even more abstract and symbolic context of a notational system. Of course, numerals still merely represent a conventional way of making marks intended to be taken for signs indicating values that correspond to numbers. Conventions are just agreed-to terms, rules--like "rules of the road"--not necessarily determined by laws, whether of logic or nature. For example, the laws of human institutions only have determined that, in the vertical arrangement of traffic lights on the streets of one predominantly Irish suburb of Chicago, instead of being red as elsewhere, the top light is green.

When the "Arabic" (Indian) numerals were first written in the West, the sequence ran from 9 to 1, and 0-- reading from left-to-right in decreasing value; this followed merely the appearance of the numerals in the Arabic manuscript, which had been written following the Semitic convention of reading right-to-left. Curiously, the direction for writing and reading numerals changed in the conventions of both languages: soon enough in the West the sequence increased in value reading left-to-right, the same way in which numerals are written in modern Arabic even though the letters and words in the language continue to be written and read from right-to-left. This is a clue to the somewhat different way in which we read the letters making up a word or a numerical expression representing the value of a number (when the value is greater than that which can be represented by a single integer): the "reading" of the written "number" must take in the whole array of component integers all-at-once, as it were, in order to interpret the place-value, and so determine the correct value of the "first" numeral read (the one written furthest to the left) or, indeed, that of any other given numeral in the expression. Yet this may not be so different from the feeling of English-speaking students who, in learning German, must become accustomed to the syntactical convention of placing verbs at the end of a sentence.

Despite some initial reluctance, the adoption of a notational system composed of an abstract set of numerals distinct from the letters of any alphabet proved to have enormous consequences for late medieval Western Europe. The price paid in this divorce was to be reckoned in the collapse of the mystical systems of gematria, such as the Qabala, in which--since antiquity--both Greek and Hebrew accorded conventional numerical values to specific letters, inevitably sacrificing some of the poetry of interconnectedness, and undermining a whole level of meaning for the letters of each written word.

Viewed another way, the introduction of Arabic numerals had the immediate and salutory effect (for those who adopted the system) of freeing NUMBER from NOUN or NAME, and VERB or WORD. That is, separate notational systems for counting or TOLLING, and for "recounting" or TELLING, liberated number reckoning from what had become a convoluted, dogmatically artificial, institutionalized set of unremittingly self-referential associations. Even though there remain to this day, in the conventions of the Arabic language, a precise correspondence between numbers and letters of the Arabic alphabet, according to the abjad system, the introduction of Arabic numerals at the end of the middle ages in Europe there and then greatly relieved the objective function of computation from the superstitious cramps of mystical constipation.

The...Europeans' adoption of Arabic numerals and their computation-facilitating "positioning-of-numbers" altogether made possible Columbus's navigational calculations and Copernicus's discovery of the operational patterning of the solar system and its planets. Facile calculation so improved the building of the ships and their navigation that the ever-larger ships of the Mediterranean ventured out into the North and South Atlantic to round Africa and reach the Orient. With Magellan's crew's completion of his planned circumnavigation, the planet Earth's predominantly water-covered sphericity was proven. The struggle for supreme mastery of human affairs thus passed out of the Mediterranean and into the world's oceans."

[Fuller, Critical Path, p. xxi.]

It is 1492. Columbus sails the ocean blue. Antonio de Nebrija presents the Gramtica, the first grammar of a modern European language, to Queen Isabella. "But what is it for?" Isabella asks. "Your Majesty, language is the perfect instrument of Empire, he replies.

It is 1573. King Philip II of Spain declares that the conquest will be referred to as the pacification.

The cutting edge of language is a knife.

hell * heaven * hell

_________________________________

When the cacique Hatuey is tied to the

stake, a Franciscan attempts to

persuade him to convert to Christianity.

The friar explains to the cacique that if

his soul is saved, he will go to heaven

and be spared the eternal torments of

hell.

Hatuey asks if there are any

Spaniards in

heaven.

Assured by the friar that there are,

Hatuey replies: I prefer

hell.

From Brief Account of the Devastation of the Indies, 1542.

Bartolomeo de las Casas

[Deborah Small, with Maggie Jaffe, 1492 What is it like to be discovered?, Monthly Review Press, New York (1991), n.p.]