Wordtrade LogoWordtrade.com


Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences



Divine Knowledge: Buddhist Mathematics According to Antoine Mostaert's Manual of Mongolian Astrology and Divination by Brian G. Baumann (Brill's Inner Asian Library: Brill Academic Publishers) In an original and compelling examination of traditional mathematics, this comprehensive study of the anonymous; Manual of Mongolian Astrology and Divination (published by A. Mostaert in 1969) takes on the fundamental problem of the post‑enlightenment categorization of knowledge, in particular the inherently problematic realms of religion and science, as well as their subsets, medicine, ritual, and magic. In the process of elucidating the rhetoric and logic shaping this manual the author reveals not only the intertwined intellectual history of Eurasia from Greece to China but also dismantles many of the discourses that have shaped its modern interpretations.

Excerpt: One morning almost ten years ago now, in working on the translation of the manuscript that would become the subject of this book, I noticed that time was expressed not in terms of duration, which is what I was expecting, but, rather, as occasions (M. uciral). The day, for instance, was defined not in terms of 24 hours as I would have defined it and as we find it defined in textbooks, but as the moment of dawn. Then, just as the sun was coming up outside my window, I realized that the distinction between the two ways of defining time was not arbitrary but significant. There exists in nature an absolute distinction between instant and duration such that every one thing can be known in two opposing ways. When one is valid the other is completely imperceptible. Ultimately, however, events in nature are undefined by physical, rational, or finite means but are in truth or apparently metaphysical, irrational, and infinite. Nature is ultimately void. At that same time all the implications of this distinction between instant and duration came to me in a rush. These implications, it was clear, were the first principles of all knowledge, the very essence of science, what in antiquity was often known as 'mathematics.'

Not being much of a scientist I did not know if this distinction between instant and duration was actually true in nature. So I went to the library and found that sure enough, physics recognizes a distinction between 'true or apparent' time and 'mean' time. This knowledge alone would have been enough for me to analyze the text accordingly, that is, according to the terms of 'true or apparent' time I had realized. Still, recognition of this distinction in Buddhism was most important. Not being much of a student of Buddhism I did not know if it was known. When I went back to the library I learned that not only was the distinction known in Buddhism, it was essential, regarded as the one true thing in nature.

In writing my dissertation my intention had been to limit the discussion of the first principles of knowledge to their relevance for the text I had set out to translate and explicate. I would write a separate book that more fully explained the fundamental principles of knowledge later. Following the completion of my dissertation, however, I happened across two important discoveries that made it necessary to incorporate much of the discussion of the first principles of knowledge into this book. First, I read the Bible as a source of mathematics. When I did I found that not only did the Hebrew tradition stand as the basis of the Western scientific heritage, which in my dissertation I had attributed to the ancient Greeks, but more importantly in Second Isaiah were all the key terms of determinism laid out just as I had discovered them. These terms of determinism were the perfect complement to the terms of an indeterminate universe expressed in Buddhism. Second, while in my dissertation I had correctly conceptualized genesis as a momentary, subjective or experiential phenomenon, I subsequently found genesis not only in the text I was studying but also in the shared mathematical history of the peoples of Eurasia. These discoveries made it necessary to broaden the scope of my commentary. Instead of using the Greek and Enlightenment traditions merely as a foil in discussing concepts in Buddhist mathematics relevant to my specific text, it was now possible and indeed essential to integrate oriental and occidental traditions and in so doing describe and define the parameters of science. These parameters of science are empirical in nature and historically verifiable. There is nothing philosophical about them any more than the Copernican theory was considered philosophy before it became the dominant discourse in Western Civilization or Einstein's theories were philosophical before they came to be acknowledged as correct.

Even while discussing the first principles of knowledge, the main purpose of this book is to explicate, transcribe, and translate an anonymous manual of Mongolian Buddhist mathematics. In joining two distinct objectives, the manual is made to serve as the modicum of a scientific text. However, using a single text as an example of the whole, strains the individual quality of each end. In particular it is the elaboration of the first principles of knowledge that suffers from the deference given to a single text of Mongolian Buddhist mathematics. A general discussion of the first principles and history of science remains to be written. While in my opinion explication of the text does not suffer unduly but rather is enhanced from deference given to the principles of science — indeed such deference makes it only that much more clear that the text itself is designed as a means of understanding the universe as if in a grain of sand — still, the history of Mongolian mathematics remains to be written as well.

This book contains six main sections: 1) a prolegomenon 2) transcription; 3) translation; 4) additional lists; 5) glossary; 6) and a word index. In the translation, to reduce repetition between the various sections of the book and to facilitate reading for those already familiar with Mongolian Buddhist texts, relatively few footnotes are given. Rather, when it comes to noteworthy terms, the Mongolian form is given in parentheses and may be looked up in the glossary. Between the translation and the glossary is a section of additional lists. These lists, in particular the list of omen protases, are important for comparison with other sources and overall give a unique perspective on the text and its genre. The word index, an elaboration of those of L. Ligeti (a la Monuments préclassiques), organizes the basic Mongolian word forms alphabetically according to their morphology. Occasionally in so doing the alphabetical order between a basic word form and the derivation of another is disrupted. Also, included as a derivation of the basic word forms is the attributive -tu/-tü form which could just as well have been listed separately.

Though aiming to describe foreign traditions without reference to any absolute standard, the rubric of cultural relativism, itself, is not value free. The discourse of the academy determines what can and cannot be said about the various aspects of traditional mathematics. Those standards are imbedded in a certain scientific discourse, one in which traditional mathematics is anathema, the antithesis of science. But there is an irony to this, for while aspects of traditional mathematics such as ritual, magic, supernatural beings, and so forth are held as arbitrary and conventional, they appear universal. Conversely, one thing known with certainty is that Modern Science, where universals are clearly universal, itself, is not. Modern Science, as the term itself implies, has a very short, well-documented history that goes back to the Enlightenment in Europe.

This basic disconnect between Modern Science and traditional mathematics hints of a profound misunderstanding. Thus, while in discussing aspects of traditional mathematics in a culturally relative way, not only do we discuss the subject from a point of view in which its reason for being is incomprehensible, but from a point of view that expressly prohibits understanding. One example is taken from myth. While myth gives a semblance of etiology, it is, rather, a sophisticated rhetorical trope serving purposes all its own. Though common sense allows us to distinguish myth from etiology, the rhetoric of cultural relativism does not. For this cultural relativism leaves us no better enlightened than Social Darwinism. When it comes to judging a mathematical text which turns out to be a Buddhist divination manual, from the standpoint of cultural relativism, though we are not allowed to say so, there is no avoiding the perception that we are dealing with an apparently inferior discourse on a subject only vaguely similar to something in our own culture.

While cultural relativism stifles negative judgments, it stifles positive judgments as well. In this it sterilizes learning, rendering it irrelevant or of trivial significance to the world today. By promoting relativism over objectivity, consensus forms in elitist cliques. Without a proper basis for comparison, speculation is unfalsifiable. Thus, though not in any way so offensive, and much more beneficial in the way of offering useful information, cultural relativism patronizes foreign traditions and shields Modern Science and Western Civilization from the threat of their influence.

Within the past twenty years, as the pitfalls of these evolutionary and relativistic approaches have become increasingly obvious, methods for avoiding them have grown more sophisticated. Often looking at a problem in a new way is beneficial, and more often than not scholars in their individual approaches carefully avoid miring themselves in a morass of methodology. However, the problem of understanding mathematics' reason for being has not gone away and continues to create distortions. While the safest way to circumvent the problem is to focus on minute particulars, this is often taken to the point of pedantry, whence the basic questions behind the study of these subjects are easily forgotten.

Much more detrimental has been the rise of field specific jargon, which makes cross-disciplinary researches all but mutually exclusive, a trend which points to the even greater divide between scholarship and the general reading public. Another negative trend is an abiding fear or outright rejection of universals. Finally, also detrimental is the tendency to conceive of science primarily in terms of understanding nature and not in terms of ordering chaos. These negative trends and misunderstandings have prevented scholars from correctly conceptualizing the first principles of knowledge. Because of this, traditional mathematics continues to be treated as a peripheral subject, its relative importance to Eurasian Studies underappreciated, its rightful place at the center of all learning unrealized.

The method of explicating the Manual does not arise as a means of avoiding the pitfalls of these evolutionary and relativistic approaches but, rather, stems from a key discovery concerning knowledge. Instead of representing time in terms of duration, as in defining the day 24 hours, in the Manual time is represented in terms of events or occasions, as in defining the day from sunrise. Though the difference might seem arbitrary, empirically there is an absolute distinction between the instant an event occurs and the duration between its succession. When one realizes the significance of this, one sees that in the empirical distinction between its 'true or apparent' and `mean or abstract' representations, opposing perceptions of time lead to opposing perceptions of nature, around which arise opposing world systems. Thus the meaning of time became the absolute standard for analyzing knowledge systems. Objective, value free, empirical in nature, and historically relevant, it brings valid scientific principles to the study of knowledge and so allows one to understand science and religion in the same way one understands other subjects, music, economics, language, and so on.

Knowledge, however, as a subject for study, is unlike any other subject. To study knowledge objectively breaks a taboo that serves the vital purpose of protecting established order where or whatever it may be. To objectively study knowledge is to superordinate science and religion, to forego philosophy and piety, and, so, to blaspheme. Blasphemy, when it is grounded, is very real and carries serious consequences, which are not to be taken lightly. For this reason those with power will eventually have to judge whether the objective study of knowledge is ultimately worth pursuing or better left alone. Hopefully those with power will be possessed of understanding as well. In the meantime, while blasphemy is the nature of the beast, it need not instigate disorder or unrest. In the present study, blasphemy is incidental and imparts no malice or disrespect towards any tradition. Assessments do not indicate opinion and often run contrary to the commentator's philosophy, which is a completely separate issue.

The concept of the void not only as primordial chaos but as the ultimate all-pervasive quality of nature lies behind the Buddhist philosophical notion of emptiness which holds that everything within the cultural matrix is illusory and only the void real. This notion is checked by that of quantum physics which defines chaos within determined limits as laws of nature expressing probabilities of outcomes, not certitudes (Prigogine 1997: 4). These opposing understandings raise a question of ultimate reality: Does chaos exist within order or does order exist within chaos? This question is akin to that of the chicken or the egg. However, as in the case of the chicken or the egg, the problem is not intractable. While the void can be experienced, a perfectly determined universe can not. To say chaos exists within an ultimately certain universe expresses a priori belief, while to say that order exists within an ultimately chaotic universe expresses the true or apparent reality.

For mathematics, as for any individual, taming the void depends upon fixing a point of orientation — any point will do. By fixing an arbitrary point one perceives an endless array of distinct points. But there is no rationality. Rationality comes by conceiving of a universal. In this, the movement of the sun from point to point — sunrise, sunset, sunrise, sunset — is universal, as is the full moon, the new moon, and all the various phases of the planets and stars. By these successive points the parameters of a rational universe come into focus, a time and space continuum in three dimensions.

In this transformation from one and undifferentiated to multiplicity and order, it is the arbitrary term that is crucial, for without some arbitrary action on the part of the observer, there is no knowledge. All remains chaos. Yet, all is not arbitrary, for through the arbitrary one finds objective reality, the universals which lead to rationality. Hence, it is the arbitrary term that links order and chaos inextricably, symbiotically. And so, through language, society, culture and so on, arbitrary elements are passed from one to another and from generation to generation. These are essential to forming the matrix of rational order into which we are born.

In the light of the rationality afforded by the matrix around him, the philosopher looks at the world and seeing only the arbitrary terms, says there are no universals. To the extent that we share conceptions in common with others, we can be said to have merely reached an agreement as to the nature of the phenomenal world. But the philosopher does not see that without universals, the very rationality by which he repudiates universals could not exist and that what we share with others everywhere is, if not universals, then the void. 0. Neugebauer, as skeptical as anyone of attempts to reach `synthesis' and convinced that specialization is the only basis of sound knowledge, wrote that even so, "In the study of astrology, ancient mathematics, one is compelled to the universal, synthetic, by the nature of the thing itself.”

In traditional mathematics, a moment of rendering order from chaos came with 'the first point of Aries.' This occurred when the vernal equinox entered the constellation Aries around 1000 BC, under which it continued to fall until the time of the Christian era. The meaning of 'the first point of Aries' as a point of rendering order over chaos has been dulled, however, by a number of factors. One is due to the metaphysical nature of a point. Defined as the point where the path of the sun reaches the intersection of the ecliptic and the celestial equator,' the point taken as the first point of Aries contains three aspects of two opposing phenomena. While the intersection of the ecliptic and the celestial equator define a sidereal phenomenon, the vernal equinox is tropical. These phenomena are divergent. Their coincidence is for but a moment's time. In this the 'first point of Aries' is often associated solely with the vernal equinox and then confused with the equinoxes in general and so occasionally given as a term for the autumnal equinox as well, which itself is occasionally referred to as the 'first point of Libra.'

This confusion stems from the fact that the zodiac to which the point belongs is not well attested until the 5th century BC. When the zodiac does become well attested, its signs have been rationalized away from the constellations themselves into 12 divisions each of 301. In this the beginning or zero point of the sign is sometimes given to reside in the mathematical center of the house, a point at which the vernal equinox fell during the 6th century BC but detached from any specific star. Eventually the Greek mathematician, Hipparchus (fl. 146-127 BC), redefined the point relative to his own time to a position east of the star Piscium. As Hipparchus' work, itself, has been lost, his legacy is preserved by later mathematicians such as the Greek geographer, Strabo (c. 63 BC), and the great Egyptian astronomer, Ptolemy (fl. AD 127-151), whose Almagest, which often refers to and quotes Hipparchus, maintains his standardization. Modern astronomers still consider this the zero point for celestial references even though the vernal equinox now falls under Pisces (and in another 600 some years will move into Aquarius.

The ecliptic is the great circle across the sky marking the apparent path of the sun. The celestial equator marks the projection of the earth's equator onto the sky. These circles intersect on the eastern and western horizons, the points of sunrise and sunset on the vernal and autumnal equinoxes.

1995: 23]). Modern historians of science have thus co-opted the point and associated it with the formalization of Western astrology which they roughly posit as having taken place between the eras of Hipparchus and Ptolemy, an unspecified period in the 1' century BC when the vernal equinox was perceived to have been in transition between Aries and Pisces (Bakich 1995: 266). Thus 'the first point of Aries' has been distended beyond a millennium and across one twelfth of the sky.

In actuality, however, the term 'first point' has nothing to do with the vernal equinox per se, nor with the constellation Aries, but rather refers precisely to the designation of a specific moment in time by which order is derived from chaos. While in the Kalacakratantra the 'first point of Aries' is given by Asvini, junction star 13 Arietis (Scheratan), the first nakshatra in its series of 27, the Manual gives an even earlier 'first point.' Here the first nakshatra in its series of 28 is Kerteg, the Pleiades, under which the vernal equinox fell from around 2300 BC to 1800 BC. A possible etymology for the Sanskrit term for the Pleiades, Krttika, derives from the verb kart 'to cut' (Burgess 1859: 328). More assuredly the symbol of the asterism in both the Manual and the oldest Indian sources is as a razor. As a razor the Pleiades separates order from chaos by fixing a point against the void, hence allowing the distribution of time and space. In Indian mythology Indra separates earth from sky by slaying the chaos dragon, Vrtra (Gonda 1989: 4-5). In the Theogony Kronos, the god of time, uses a trenchant instrument to separate his mother Gaia, the earth, from his father Ouranos, the sky (Wender 1976: 26-29). These examples echo a Babylonian myth in which Marduk, Jupiter, the god of time, slays the dragon of chaos, Tiamat. Likewise in ancient Mesopotamia there was a system of seventeen or eighteen constellations, described as 'Gods standing in the path of the moon,' which begins with the Pleiades in ascendant at the time of the vernal equinox. Thus the first point of the vernal equinox under the Pleiades indicates an even older point of genesis, literally, the dawn of time.

This genesis does not mean that there was no prior 'first point.' In the common myth whereby Jupiter slays the dragon of chaos, while that which cuts refers to the Pleiades, the dragon refers to the constellation Draco, which in the shape of a winding serpent, occupied the celestial pole long before the dawn of the vernal equinox under the Pleiades. The pole, not the horizon, offers the most perfect 'first point.' However, due to precession, after 2800 BC Draco moved away (Allen 1963: 202-212; Bakich 1995: 202). Thus, Jupiter, in its various guises as Indian Indra, Babylonian Marduk, Egyptian Horns, Greek Zeus, and Chinese Taisui, in conjunction with the Pleiades became the point of orientation, replacing Draco, which had left the pole in chaos. In this shift, though it is the equinoxes that fix the point on the eastern horizon marking the intersection of the celestial equator and the ecliptic, the year and time itself are defined not by the sun but by the cycle of Jupiter.4 Pinned historically to Draco's straying from the pole and the ascent of the Pleiades as the basis of order over chaos, genesis, as expressed by the `first point of the Pleiades,' is interrelated not so much with the rise of language, common knowledge, or the evolution of man but with the rise of civilization, writing, history, and clocks.

In that genesis depends upon fixing a point of reference, as that point is inherently unstable, the semblance of order it creates does not endure. While conventionally order is epochal, ultimately it lasts but a moment. This mutability of a point poses an inherent dilemma, for the orientation implicit in the order of the constellations serves two contradictory purposes: one, calendrical, the other, genetic. Over time the order of the constellations can reflect only one of these. Though to reflect the change with a new standard shows deference to evolution in nature caused by what would come to be known as the precession of the equinoxes, in so doing the moment of genesis is effaced by a re-ordering of chaos, a new beginning. By maintaining the original order traditions emphasize the genetic function of constellation systems over their calendrical function. In so doing they favor what abides over ephemeral reality.

That eternal truth be favored over ephemeral truth needs be, for a tradition that opts for what is transient sets itself up to be outmoded. But the reason for change is just as obvious. Certainly in the Kalacakratantra the marking of genesis with a first point in Aries instead of the Pleiades by no means favors ephemeral truth over eternal truth, for at the time the Kalacakra is believed to have been written the vernal equinox fell, as it did when the Manual was copied and still does today, not under Aries but Pisces. One may speculate why the Kalacakra is oriented as it is, but to simply observe the difference in its star system is telling enough. The Kalacakra system rationalizes the positions of the constellations which gives it both computational and metaphysical advantages over the true or apparent order of the stars. In this the Kalacakra system embraces a tradition ancient in its dawn right but one that offers worldly advantages apparently deemed necessary. Thus one sees in the Kalacakra system the inherent conflict between continuity and change, progress and tradition, while in the Manual one finds classical purity for its preservation of an earlier genesis at the price of a less sophisticated technology.

In the pull between continuity and change, deference to tradition is born out by the fact that across Eurasia, New Year celebrations commemorate genesis, the moment of order from chaos. In his The Myth of the Eternal Return, M. Eliade discusses in detail the Babylonian New Year celebration, akitu. Though over time it was celebrated variously at the spring equinox as well as at the beginning of fall, on these occasions the creation epic, Enüma delis, was recited in the temple of Marduk recounting how Marduk slew Tiamat, the dragon of chaos (Eliade 1954: 55-58). Remembrances of genesis are found likewise in the New Year celebrations of Judaism, in the ancient Persian Nawruz festival, among the Germanic peoples, in India, and the Tet tradition of the Vietnamese (Steensgaard 1993: 65; Eliade 1954: 55, 64; al-Biruni 1934: 180). In China New Year's Day is still known as yuandan jt H `the first dawn.'

Eliade describes such ceremonies as a sympathetic need of primitive man to regenerate himself by returning to the moment of genesis as often as possible (1954: 73-77). However, this subjective, speculative explanation masks ignorance of a profoundly simple truth. While genesis derives from a fixed point of orientation, order depends on its consistent return. So, traditions define genesis by the return of the stars to their original positions at the end of each year. In particular these traditions mark the return of Jupiter to its original place among the constellations.

However, though the stars ought to return to their original positions at the end of each year, they only do so approximately. The first point of orientation in truth never returns. Thus it becomes necessary to commemorate genesis lest genesis be lost to flux. A world of flux means that the commemoration must be metaphysical, unempirical, and apparently irrational. But what is more, the fact that the stars' return to their original positions is only approximate and not exact means that not only is it appropriate that genesis be remembered with the New Year, but that it must be so, for as the semblance of order is imperfect, each year the foundation of order over chaos must be shored up against the pull of nature's chaos. In this genesis becomes an ongoing process that depends both on the fixing of a point of orientation but also on its preservation through the yearly re-creation of the world anew, a reconstitution manifested through the calendar, such that traditions which yearly commemorate genesis recognize that at the end of each yearly cycle order is lost, the world destroyed, only to be reformed in the coming year.5 In the calendars of these traditions, the day of genesis belongs not only to the cycles of Jupiter and the sun, but to the moon as well. Thus the first month of the year is often also known by a term that marks genesis. In some Indian systems, for instance, the first month takes on the same function as Krttika, the Pleiades. It separates order from chaos. In the Manual, one finds in the ascendant on the first day of the New Year the constellation Kerteg, the Pleiades. What is more, the designation of the first month as Qubi sara 'portion moon' again refers to the point by which time is fixed and order reapportioned.

In these traditions, rather than the irrational or archetypical fancy of pre­scientific peoples, genesis is quintessentially scientific. It reveals at once the way of nature, the fullness of time, and the frailty of order over chaos. Though by fixing a point of orientation time and space are apportioned across the void, because of the inconstant nature of a point, genesis may only be preserved through ritual commemoration. The order it provides must be sustained through yearly installments of time and space. Such an order is not only imperfect but based on arbitrary convention. Nature, itself, allows no such order. Even so, such apportioning of time and space determines the fate of the people and binds them as one. In this, genesis belongs not so much to nature or to man as it does to an intermediary that forms from nature the conventions upon which people must rely in kind. Herein lies the essence of mathematics, for this control over destiny through the distribution of time and space marks the rise of a universal system governed by elite knowledge and the fall of mankind from the primordial state of nature.

According to Plato, mathematics is the subject most near and dear to god (Burkert 1972: 325). Mathematics creates order out of chaos by apportioning time and space across the void. It drives the wheels of causation by fixing the times and places of the gods. In ancient Mesopotamia Nisaba, goddess of grain, writing, and wisdom, is said to measure heaven and earth, to know the secrets of calculation and with Suen, the Moon, to count the days (Koch­Westenholz 1995: 21-32). In the Hebrew Bible God asks Job, do you know when the mountain goats bring forth? Do you observe the calving of the hinds (38.4-7; 39.1-4)? In Psalms it is God who determines the number of the stars and gives them each their names (147.4). Second Isaiah asks who has measured the waters in the hollow of his hand and marked off the heavens with a span, enclosed the dust of the earth in a measure and weighed the mountains in scales and the hills in a balance (Isaiah 40.12-31)? Second Isaiah also says that it was God alone that slew the chaos dragon (51.9-11). Such are forms of knowledge inherently linked to the essence of god and mathematics both.

Thus, mathematics intercedes between the divinity of nature and the common man. As an intermediary between god and man, mathematics, in its functions and practitioners, is that given specifically to divine knowledge. In the apportioning of time and space and driving the wheels of causation mathematics plays a powerful role in determining the fates of individuals. In this mathematics acts as a great demon, one not necessarily evil, but

generally so deemed.

If knowledge is power, that which controls time and space holds ultimate authority as the dominant discourse in society. Though it does not transcend nor create the void, mathematics both reveals and creates the manifestations of the divine. It creates the objects of veneration and the conventions upon which we rely for change. It brings dimensions, qualities, and limitations to that which is otherwise ineffable and forms the matrix of rational order upon which peoples rely in kind. In this it transcends faith, transcends even the word, for it is knowledge that faces the void and so begets the word.

As mathematics controls time and space, it influences all fields of learning and human endeavor. Across Eurasia it was by the favor of heaven that governments ruled. The status of every reign, whether rising or falling, was legitimated by mathematicians. Mathematics was a regular field of study circumscribed within various bureaucracies and institutions and responsible for carrying out assigned functions at certain times. However, mathematics served and continues to serve a dynamic function that comes into play under ad hoc circumstances as well. Coining symbols, making myths, and setting epochs are rather mundane examples of occasions for statecraft that require mathematics. A more dramatic example is taken from the Council of Nicea in 325 when Christian Church initiates, faced with divergent teachings from the likes of Arius (d. ca. 336), formed, out of fundamental mathematical concepts, the doctrine and creed all others would henceforth take on faith. It is on such occasions that the pervasiveness and centrality of mathematics as the dominant discourse become apparent.

Though it transcends the word and reveals and creates the conventional qualities of divinity, mathematics does not transcend god. Because our knowledge is imperfect, what ultimately determines the future reverts back

not simply to nature manifest in random chance and chaos, but to god, that is, the reality of what is, a reality which depends on the intercession of mathematics between nature and man. Manifest as spirit, we know this reality as 'the sign of the times.' Though no man has perfect knowledge to determine absolutely what will be, it is possible for individuals, no matter how low or high born, to possess sufficient knowledge to understand the spirit of the times and so transform the given order, whether for good or ill, according to a prevailing logos that is fixed and irrevocable and against all exigencies excluding chance. The manual refers to this possibility when it says that under a certain star a lowly orphan can rise to seize all the lands of middle earth (27r). Changing the established order is not an easy task, however. Even among those who possess the wherewithal to succeed and the courage to try, few get out unscathed. Not everyone can see the face of god and live.

Because our knowledge is imperfect, even though mathematics is the subject most near and dear to god, the semblance of the physical world created through mathematics in turn creates the illusion that there is no divinity, that the void is imaginary, that things are straight and ordered. Thus mathematics redefines and even effaces god. In bringing order from chaos mathematics shapes the terms of both religion and government. However, the order mathematics provides does not last. Words lose their meaning. Perfect knowledge is merely a matter of rhetoric, not reality. In rhetoric perfect knowledge is capricious, conventional, arbitrary, and exceedingly banal. For those outside its fold the power of knowledge brings no peace but instead causes terror and awe. Though it creates religion and government, mathematics immediately comes into conflict with them. Thus as its power would otherwise be absolute, mathematics must in some way be subordinated.

Throughout history when those who acquire superior knowledge blaspheme their challenge to the given order, they are ridiculed ad hominem for their fallibility. Priests holding divine power are subjugated by emperors holding earthly power. In ancient Babylon though the priests maintained order and were considered to possess divine knowledge, a part of the rhetoric of order over chaos included deriding them for their haughtiness (Langdon 1956: 860-861). In China mathematicians were subordinate to the emperor, son of heaven, a living god, their knowledge diffused through bureaucracy and specialization, so too in Islam (Wittfogel and Feng 1949: 217; Sayili 1960: 47). In Judeo-Christianity when it was known that the stars would not reveal perfect knowledge, they were subordinated to a priori omniscience and time as duration. Tied to the stars, when they fell, mathematics fell with them. In the chasm left from the falling of the stars, heaven was remade. And

so we have the myth of Lucifer, the brightest star in the heavens, who, for his great knowledge, likened himself to God and so was cast out of heaven to rule the underworld as Satan (Isaiah 14.12-16).

Going about the earth weighing and measuring, the devil as a strict determinist symbolizes all of mathematics, for as absolute power corrupts absolutely, it is in mathematics that we find pure evil. While self-interest, pride, and lack of humility all come with the pursuit of supreme knowledge, what is truly insidious about mathematics is the inclination to follow the logical implications to their extremes, that is, to hold that we can either know everything or nothing at all. This inclination is the greatest temptation and something the Buddha warned us to avoid.

Buddhist Abhidharma

As given in one whole division of its sacred scriptures, the highest or ultimate knowledge in Buddhism is expressed in terms of perception as 'the law of that which is manifest' (S. abidharma, Tib. chos mngon pa; M. iledte nom [see Ligeti 1944: 166-190; Conze 1973]). The accomplishment of this law is understood as 'perfect virtue' or 'that which has reached the utmost limit' (S. paramita Tib. pha rol to phyin pa, M. cinadu kijayar-a kürügsen). Limitation implies a conventional order placed upon otherwise illimitable nature. This law regarding the perfection of wisdom is thus expressly of a tradition, Buddhist, not universal, and so a matter of faith (Conze 1973: 23).

As for the Manual, though it is important to remember that mathematics is a unique tradition of its own and as such often incongruent with Buddhist teaching, the Manual is ultimately a Buddhist text. Its language reflects Buddhist law and culture. Mentioned in the Manual are the dharma (nom), the Greater Vehicle (Yeke kölge), i.e., Mahayana Buddhism, treatises (sastir < S. sastra) and tenets (tayalal) including the 'six aspects of unity' (qamtudquy-yin jiryuyan qubi), the Vinaya (Dülb-a) section of the Ganjur concerning the monks' discipline, the monks' discipline, itself (says-abad < S. sikshapada), a monk's vow (sangvar), the clergy (quvaray), temples (süm-e), monasteries (keyid), and so on.

Given this emphasis on faith, perfect knowledge is held to supersede mathematics. In Buddhist metaphysics mathematics is classified as a minor science and said to belong to the realm of relative or conventional reality (M. inayungki ümen; Tib. kun rdzob bden pa; S. samvrti-satya) and in contrast to ultimate reality or supreme truth (M. ünemleküi ünen; Tib. don dam pa; S. paramartha) which belongs to the realm of transcendent wisdom. The relationship between transcendent wisdom and mathematics is born out in the legend of the origin of mathematics in which, having been unable to teach the ultimate truth in China, Buddha instructed Manjusri to awaken people's minds by means of mathematics (Cornu 1997: 39). The Manual

also makes this point, as follows:

While those whom the aggregate of doctrines on supreme truth frees from conceptualization do not strive in conceptualization to be counted as individuals, those who do not yet understand . . . need conventional truth . . . . Thus, Buddha, the Bodhisattvas, the god Manjuri . . . have taught . . . mathematics and the benefits of reckoning. (1v)

This relegation of mathematics under the rubric 'conventional' does not mean, however, that Buddhist mathematics is to be deemed practical, physical knowledge on par with that possessed by an 'ignorant commoner.' The contradiction is explained by the concept of 'non-duality' in the Middle Way position of the Madhyamika Buddhist School. As described by Liu, knowledge is said to be 'non-dual' in that to merely render a concept into binary opposites is bound to be tentative. For this reason the Middle Way distinguishes between two truths, supreme and mundane, but understands concepts in terms of 'three forms of two truths' (Liu 1994: 39-52). In this analysis of knowledge in terms of 'three forms of two truths,' while in conventional terms mathematics contains both physical and metaphysical aspects, in ultimate terms it is metaphysical. However, political and philosophical reasons dictated that the mathematical conception of knowledge, derived empirically from the study of nature, should not be the `supreme truth.' As Liu describes, after Buddhist teachers had minimized the significance of Prajnaparamita and Madhyamika or Middle Way schools in the late 5th century A.D., philosophers in the middle of the 6th century AD were able to re-assimilate the teachings of these schools by transcending the parameters of mathematical metaphysics (Liu 1994: 82). They did so by assigning 'three forms of two truths' to that which was said to belong to the mundane realm, when it had previously been associated with supreme truth. This created an absolute metaphysic of 1. Emptiness, 2. Non-duality of `existence' and 'emptiness,' 3. Neither 'duality' nor 'non-duality,' and 4. Non-difference of the three forms of two truths out of that which belongs to `Supreme Truth' (Liu 1994: 151). In this way, Buddhist 'four forms of two truths' created a metaphysical realm that was beyond the simultaneous, contradictory coupling of physics and metaphysics, or beyond the physical realm altogether. In so doing Buddhism subordinated mathematics.

However, while transcendent wisdom is held to be unattainable by mere mortal man and only comprehendible by Bodhisattvas who, in an infinite regress of binary oppositions, ultimately do not exist, to the extent that the perfection of virtue imposes limitation and moral order upon the void, its eternal perfection expresses duration in time. Thus, while with respect to Buddhist law, mathematics belongs to the realm of conventional reality, with respect to the logos of nature, transcendent wisdom tends to express a conventional understanding of time. Thus, in the realms of what are ultimate and conventional, the relationship between Buddhist dharma and traditional mathematics is interdependent, symbiotic.

Even though mathematics is but a minor subject in the classification of Buddhist sciences and at once removed from the doctrine of supreme truth, it is by means of mathematics that knowledge is classified into the Five Sciences. Numerology determines the abstract concepts of Buddhist dharma, e.g., the four noble truths, the seven jewels of royal power, the twelve stages of dependent origination, and so on. But most importantly the distinction between instant and duration, the true or apparent nature of time, drives the current of Buddhist rhetoric not forward from the present order towards utopia but back from the present order to the void and so as the dominant discourse shapes the perception of reality. To the extent that Bodhisattvas belong to heaven, their divinity is mathematical. Cosmology legitimizes the government. The almanac fixes the times for specific purposes. Ritual governs and heals the people. The almanac, determining of good and bad results, fixing the circumstances of weddings, and a host of activities have direct influence over the lives of common people. In Modern times when Buddhism was all but annihilated in Mongolia, it was services such as the drawing of natal horoscopes, the almanac, and ritual healing which remained indispensable. As G. Tucci notes concerning the Vaidurya dkar po, as mathematics was ever the handmaid of liturgy, its influence was all pervasive (TPS 137). In these ways mathematics maintains its role in Buddhism as arbiter of ultimate reality such that when one looks at Buddhism in practice, the rhetoric expressed is to a large extent not the moral perfection of transcendent wisdom but the mathematical perfection of divine knowledge.

Although the Manual tends to be a metaphysical representation of the universe, it also describes physical processes. Thus, though contradictory, the Manual empirically accepts both causation and chance. Regarding causal relationships derived from empirical observation, it describes the motion of the sun throughout the course of a year, noting the solstices and the equinoxes and commensurate changes in the amount of daylight and darkness from month to month. It notes changes in the weather and biological processes such as the season certain plants bloom, the time fish mate, when birds migrate, the month the tiger bears its young, and so on. In keeping with its metaphysical modality, however, its métier are chance occurrences, as in the following, "If one builds a home, five people will die (43r)." These seemingly chance occurrences are derived, as befitting the logic of the system, through divination.

As with the term `mathematics,' `divination' pertains to knowledge. The root of the word, deus (god) or divus, indicates the source of the seeker's information (Herodotus 1996: 114). If science is about the prediction of events, from a deterministic point of view, every event is inevitable. As chance can neither be defined nor understood, science is held to be about causes not chance (Prigogine 1997: 4-5). Hence, Cicero in De Divinatione takes on what he considers the Stoic definition of divination as "the foreknowledge and foretelling of things that happen by chance" (Cicero 1979: 220). This, however, is a misstatement of the Stoic argument, for in Stoic determinism chance ought not to exist. The rationale for divination, on the other hand, is from the point of view of an uncertain universe. Here every event is random. There is no such thing as causation. As causation does not exist, something arbitrary is required in order to act. In common experience, there are times when the causal chain is clear and other times when it is not. Divination is simply a way to act when options seem arbitrary. It is common to individuals and assimilated into elite or institutional knowledge systems guided by the true or apparent view of nature.

While divination as a means of knowledge is universal, the forms of divination practiced in elite knowledge systems are not. Though a particular method might arise independently in one place or another, it also happens that traditions of divination methods are passed from generation to generation and culture to culture. Some of the earliest attested examples of divination are found inscribed on bones and tortoise shells in China and on cuneiform tablets in Mesopotamia.'

There are two ways to classify divination methods according to a psychological point of view. Autoscopic or direct methods depend simply on some change in the consciousness of the practitioner. With heteroscopic or indirect methods the diviner looks beyond himself for guidance. Examples of autoscopic divination include methods based on sensory perceptions such as gazing into a crystal or listening to a shell, motor automatisms, such as in the use of a divining rod, or mental impressions derived from activities such as looking at the palm of a hand or at cards. For heteroscopic divination the process depends on inference from external facts. Examples of heteroscopic divination include sortilege 'the casting of lots,' such as dice or the tarsus bones of sheep; haruspication 'the inspection of entrails;' scapulimancy `divination by shoulder-blade;' and so on. While the practitioner must actively perform these methods, there are also passive types of divination, such as taking an omen from the weather or from the behavior or cries of birds .

Divination in the Manual is expressed in four interrelated forms: 1. complementary relations, yin/yang; 2. rational methods, such as the systems of elements, trigrams and color schemes, the results of which are derived through logic and appear random against the matrix of the calendar; 3. irrational methods, including omina, the results of which take on a statistical significance, and magical formulae/medical treatments; and 4. patterns in the matrix of the calendar, either orderly or random, the composition of which we have previously discussed.

These aspects of divination fall under two main classifications in the Sino-Buddhist scheme adopted by the Mongols through Uygur and Tibetan traditions: 1. Chinese or Black mathematics (Tib. nag rtsis), distinguished principally by the system of elemental divination (Tib. `byung rtsis), based on the relationships formed between the five elements.

This translation of a relatively common textbook shows that much of the stories we assume about discreet continuities of technical knowledge is not demonstrated by traditional sciences and that pedigree of these works are much more complex, diffuse and wildly and intractably dispersed than our common histories say. As such the extensive introductory matter explodes many of our isolated assumptions about traditional knowledge. Baumann has very much seen the universe through a common textbook, and it is a universe more diverse than expected.


Spiritual Merchants: Religion, Magic, and Commerce by Carolyn Morrow Long (University of Tennessee Press) provides a major account of the folklore and commerce of hoodoo in USA cities. They can be found along the side streets of many American cities: herb or candle shops catering to practitioners of Voodoo, hoodoo, Santera, and similar beliefs. Here one can purchase ritual items and raw materials for the fabrication of traditional charms, plus a variety of soaps, powders, and aromatic goods known in the trade as "spiritual products." For those seeking health or success, love or protection, these potions offer the power of the saints and the authority of the African gods.

Spiritual Merchants provides an inside look at the followers of African-based belief systems and the retailers and manufacturers who supply them. Traveling from New Orleans to New York, from Charleston to Los Angeles, she takes readers on a tour of these shops, examines the origins of the products, and profiles the merchants who sell them.

Long describes the principles by which charms are thought to operate, how ingredients are chosen, and the uses to which they are put. She then explores the commodification of traditional charms and the evolution of the spiritual products industry--from small-scale mail order "doctors" and hoodoo drugstores to major manufacturers who market their products worldwide. She also offers an eye-opening look at how merchants who are not members of the culture entered the business through the manufacture of other goods such as toiletries, incense, and pharmaceuticals. Her narrative includes previously unpublished information on legendary Voodoo queens and hoodoo workers, as well as a case study of John the Conqueror root and its metamorphosis from spirit-embodying charm to commercial spiritual product.

No other book deals in such detail with both the history and current practices of African-based belief systems in the United States and the evolution of the spiritual products industry. For students of folklore or anyone intrigued by the world of charms and candle shops, Spiritual Merchants examines the confluence of African and European religion in the Americas and provides a colorful introduction to a vibrant aspect of contemporary culture.

LISTENING TO THE ORACLE:  The Ancient Art of Finding Guidance in the Signs and Symbols All Around Us by Dianne Skafte ($23.00, hardcover, 288 pages, Harper San Francisco; ISBN: 006251444X)

The ancient oracles haven’t stopped speaking we’ve just forgotten how to listen. Now, noted psychotherapist and author Dianne Skafte shows how we can reclaim this innate ability to receive guidance, illumination, and direction from mysterious sources beyond the self.

Oracles-wise messengers that can help us navigate our lives can be found everywhere: in patterns in nature, chance words overheard in a crowd, images in a dream, a right-on-target tarot reading. During ancient times, people actively sought out these messengers with hopes of gaining greater awareness of past, present, or future events. Today, we need a little help in opening our eyes and minds to the fascinating possibilities of a world that is alive and brimming with consciousness and helpful guidance. LISTENING TO THE ORACLE offers that help. The book is both a spellbinding history of oracles and a hands-on guide to reawakening this intuitive power and putting it to work for guidance and profound psychological growth.

Tracing oracular communications from their ancient beginnings to modern, times, Skafte, a Jungian psychotherapist and professor of psychology, illustrates the powerful influence these revelatory experiences have had on numerous civilizations and cultures throughout history. With the exception of indigenous cultures that still practice oracular ways of knowing, she explains, most people today have removed themselves from the interactive world of oracles because they no longer remember how to receive or trust them. Drawing on her extensive research as well as her own direct experiences, Skafte explores the many ways we can regain the ability to communicate with oracles, beginning by adopting an attitude of warm expectation" in everyday life.

Ranging from oracles of the earth (birds, bees, and oaks) to Nepalese shaman divinations, card readings, and a fascinating experiment with a computer and word divination, Listening to the Oracle blends ancient legends with psychological insight, reflection, advice, and specific instructions on preparing for and receiving an oracle. It is an inspired and highly instructive work that will awaken you to the world’s rich spiritual terrain and teach you how to tap into your own hidden powers of perception.

DIANNE SKAFTE, Ph.D., is academic vice president at Pacifica Graduate Institute where she teaches depth psychology and mythological studies. A professional psychotherapist, she has published numerous journal articles on oracular practices in antiquity.

THE ENLIGHTENMENT PACK: Identify Your Personal Goals, Improve Your Life, Your Work, Your Relationships by Chuck Spezzano, illustrated by Alison Jay ($19.95, box, contains 48 original color cards and booklet, Bulfinch, 0-8212-2357-7)

Why do we feel the way we do? What are our inner strengths? Our weaknesses? And how can we best realize our potential and achieve our goals? This new deck of cards — along with the comprehensive paperback guide provided — makes it fun to embark on a journey of inner discovery, either alone or with friends.

Drawing on his "psychology of vision," a synthesis of Western and Eastern precepts, Chuck Spezzano has created a deck of forty-eight cards divided into six suites — Victim (guilt), Unconscious (independence), Relationship (fear), Healing (communication), Gift (beauty), and Grace (innocence). To use the deck people take cards from each suite and lay them out in any of several patterns. Then, drawing on the easy-to-understand explanations in the book, they analyze their reactions to the cards and their juxtapositions — and open the door to inspiration and enlightenment. Combining the insights of psychology with the interactivity of tarot, this unprecedented package will be a revelation for spiritual seekers everywhere.

ABOUT THE AUTHOR: Chuck Spezzano, Ph.D., has more than twenty-five years of experience as a practicing psychologist and teacher. Now based in Hawaii, he has inspired thousands of people through his seminars and workshops in the United States, Canada, Japan, Europe, Australia, and New Zealand. he has also written Awaken the Gods: Aphorisms to Remember the Way Home


Headline 3

Berakhyah Ben Natronai ha-Nakdan, Sefer Ko’ah Ha-Avanim (On the Virtue of the Stones). Hebrew Text and English Translation. With a Lexicological Analysis of the Romance Terminology and Source Study by Gerrit Bos and Julia Zwink (Brill Academic)  The lore of the supposed magic and medical virtue of stones goes back to the Babylonians and peaks out in the lapidary literature of the Middle Ages. The famous work of Marbode of Rennes, which made lapidaries a very popular type of medieval scientific literature, was translated into numerous vernacular languages. The Jewish tradition, missing a particular lapidary literature of its own, absorbed non-Jewish works like that of Marbode. Several Anglo-Norman Marbode translations could be identified as the main source of the present edited Hebrew lapidary Ko’ah Ha-Avanim, written by Berakhyah Ben Natronai ha-Nakdan around 1300. The edition is accompanied by an English translation, a source study, and a linguistic analysis of the Romance, mostly Anglo-Norman, terms featuring within the text in Hebrew spelling. More