Sacred mathematics—the very term seems magical, with its evocation of hidden knowledge and the secrets of long-lost civilizations. Sacred mathematics and sacred geometry speak to us of a wisdom veiled from our own age, of practices that, to use the original meanings of these words, are hermetic, esoteric, and occult.
We picture the Pythagorean brotherhood of ancient Greece; the numerology of the cabala; the twelfth-century Masons who concealed within the construction of the great cathedral of Notre Dame at Chartres a wealth of numerical and geometric symbolism; the preoccupation of the artists of the Renaissance with the connections between nature, mathematics, geometry, and perspective; and Kepler's discovery that the numerical relationships between musical intervals are mirrored within the order of the solar system as the harmonies of the celestial spheres. Today the same sacred mathematics is alive in the Americas where it occupies a key place in the lives and ceremonies of many of the First Peoples of Turtle Island.
Sacred mathematics is a living form, a manifestation of energy, spirit, and mystery. This may at first sight seem strange to us, for within our society numbers have come, in the popular mind, to be associated with static and abstract things. To mathematicians, however, number is a door that opens into the farthest reaches of the human imagination.
For most people the dehumanizing role of number has become overemphasized. We have a Social Security number, a telephone number, a driver's license number, numbers for our bank accounts and credit cards, and other number codes we use to gain electronic access to our accounts. The stories of our lives can be found on educational, tax, banking, employment, and medical records, all identified as a series of numbers on a computer network. Some may feel that we are close to that nightmare world predicted by the science fiction writers. In Eugene Zamiatin's novel We, the citizens of the future have all been reduced to ciphers. In the 1960s cult television series The Prisoner, each episode began with the hero (played by Patrick McGoohan), being addressed as Number 6, and protesting “I am not a number, I am a free man.” This same policy of stripping away the personality by the use of a number is one of the first disorienting shocks a person experiences when being confined to a prison or inducted into the armed forces.
The scientist or mathematician is sometimes portrayed as a wounded schizoid character who flees the world of emotion and insecurity into an abstract universe of number and certainty. Bertrand Russell said in his autobiography that he turned to mathematics as a young man because: “I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere.” In the end Russell was to discover that deep within the foundations of mathematics there can be no certainty, “I set out with a more or less religious belief in a Platonic eternal world, in which mathematics alone shone with a beauty like that of the last cantos of the Paradiso. I came to the conclusion that the eternal world is trivial, and that mathematics is only the art of saying the same thing in different words”.
Within Native America, number is seen in a profoundly different way; not as dry, abstract, and dehumanizing, but as alive, real, and il-nrrtediate. mathematics is a sacred practice related to the dynamics of the whole cosmos. Through the meaning and manipulation of number many Indigenous people maintain balance and harmony within the ever-changing world of flux, energies, and transformation.
For aboriginal peoples all over the world, as well as for many ancient civilizations, number and mathematics have always played a special role. Sometimes its importance is so high that number may not be used for secular or commercial purposes since its symbolic representations are reserved for sacred ceremonies and the great cycles of time and the gods. Numbers are symbols and manifestations of the transformations, dynamics, and cycles within the world of nature and spirit. Numbers are the manifestations of beings.
All over Turtle Island numbers and other symbols can be found inscribed on sacred rocks. Some of these have been photographed and studied by archaeologists, but many others are known only to Native people. Rocks may be deliberately left covered with moss and vegetation and only exposed during special ceremonies or teachings when the rocks are to be read. Tom Porter, a traditional Mohawk, said that he would approach such rocks in a special way by praying and offering sacred tobacco. If the spirit of that rock wished to speak to him and shake hands with him, then that would be good. Thus the voice of the rock heard by a traditional person will be very different from what is read by a non-Native.
Western scientists who study petroglyphs (writings and markings on rocks) have observed, in addition to hieroglyphs and picture writing, series of lines, or holes, sometimes inscribed in rows, that are referred to as “tally marks.” The sequence of marks, they say, is evidence of a numerical representation that may have been used, for example, to count game. Other rocks display what they believe are the records of astronomical observations, such as the number of nights between each new moon.
At one level rock markings refer to number and computation, but their significance may go far beyond records of days and months or the numbers of game that have been killed.
Since they were a sacred subject Tom Porter did not say much about the markings, but when I spoke to him he did convey the sense that these markings can be read as a map, or as a journey though life. As an analogy, he pictured someone driving from New York to Los Angeles. Such a person does not need to know the detailed geography of the United States, simply the particular highways that must be taken for that journey and the changes that must be made at intersections. The tally marks made on stone can also be read in this way; they refer to a journey, but a journey that goes beyond that of a single individual and one that is connected in some way with the land, The People, and the cosmos.
Numbers, tally marks, and writing are also inscribed upon the sticks used by many different peoples, such as the Condolence Cane of the Iroquois or the Walking Stick of the Mic Maq. The Iroquois speak of a stick that is associated with each person at birth. This stick contains the map of a person's life and the number of its days and can only be seen by the creator.
The world is flux. Nothing lasts forever. Whenever we have made our plans and laid down the path of our future the trickster will come along and play a trick on us. The celebration of chance, through gambling, becomes important to Native people. But gambling is not simply a “game”—in our sense of the word as entertainment or diversion—but a sacred ceremony that acknowledges the basic metaphysics of the cosmos, including the power of the trickster and other spirits. The Mohawks have the Peach Stone Game that must be played before corn is planted, the Mic Maqs have the game Waltestaqaney. These are played on behalf of the whole society and may, for example, act to renew the compacts and alliances made with the spirits of the universe.
Playing gambling games and creating their rules implies that such people have a knowledge of the laws of probability and are able to compute the odds of various outcomes. A role taken during such games is that of the tally keeper, who must not only carry out rapid mental arithmetic but must have devised a method of recording the tally of points. Tally keepers were also required on the Northwest coast where, during the Potlatch, records of the exchanges of copper and other very valuable gifts were kept.
The sciences of geometry, surveying, computation, and number representation were all involved in the architecture of traditional buildings and in the construction of temples and earth works. The Hopewell people, whose culture flourished from around 1000 B.C. to A.D 500, and who may be the distant ancestors of the present-day Iroquois, must have practiced sophisticated surveying skills to have been able to build their elaborate mounds and earthworks.
The buildings and temples of the lost Anasazi people of the Chaco Canyon were aligned astronomically with the position of the rising sun on certain days of the year. What is particularly striking is that in some locations the horizon cannot be seen because of an intervening hill or building—nevertheless the walls of the building are aligned, not with where the sun happens to be first seen at that location, but with where it would be seen if there was nothing obscuring the view!
In many other cases architecture is tied to cosmology and special directions. The traditional longhouse of the Haudenosaunee is made in the likeness of the cosmos, its curved roof representing the sky and its floor Mother Earth. Likewise, the orientation of the building ensures that its two openings meet the rising and the setting sun at a certain time of the year. The buildings of many other peoples also have doors or windows that open out onto some astronomical event, or that channel light onto a wall or marking. The ability to place an opening so that it will be aligned with the rising sun at the solstice or equinox, or with some other event, clearly implies the ability to predict and calculate the location of these events in the sky. Indeed, the design of a building represents a perfect integration of mathematics, astronomy, surveying, and architecture.
In celebrating these triumphs of Indigenous mathematics one should not forget that the astronomers and mathematicians involved may not necessarily have been men. It is so easy to be blinded by our own history of Western science and technology to assume that the leaders of scientific evolution must always be male. A story told to me by the University of Ottawa ethnomathematician Michael Closs illustrates how easily we can jump to assumptions.
Some time ago balls of string were discovered by an-thropologists on the Northwest Coast. When they were unraveled they were found to contain long knotted strands to which pieces of cloth had been tied at periodic intervals. The anthropologists quickly figured out that the knots represented days and that the regularities of the pieces of cloth at every twenty-eight days represented the appearance of a new moon. Clearly these strings were astronomical records, the “lab books” left by the Indigenous astronomer-mathematicians of the West Coast.
This account was generally accepted until someone noted that these balls of string were usually kept by women. Suddenly the penny dropped—the regular markings were not recording the phases of the moon, but a woman's menstrual cycle. And so one interpretation was replaced by another.
In an Indigenous community a woman's period is described as her “moon.” It is a time of great power, when a woman will be very careful of the medicine and the sacred objects that she touches, for during her moon her own spirit can overwhelm everything with which she comes in contact. In a culture in which everything connects and is related, it is no coincidence that the knots in a string refer both to the cycles of power and fertility within a woman's body and to the waxing and waning of the moon. Indeed, each is a reflection of the other, and it could well be that women first began to study and represent the harmonies and regularities of cosmic and human cycles. For the interrelationships of the cosmos are expressed through the periodicity of heavens, earth, and the human body.
Algonquian, Siouan, and Iroquoian people use a base-ten system as we do, while the ancient Mayan practice of counting by twenties is still preserved in Central America, in parts of California, and by the Inuit people.
The basis of these number systems is reflected within language. In English our use of the base-ten system of counting is shown by the way we reuse the numbers one, two, three, etc., as we reach thirty, forty, fifty, sixty and so on. Thus, after twenty-nine we go to thirty and begin counting again with thirty-one, thirty two and so on. On the other hand, counting by twenties was practiced by the Celtic people of Europe, just as it was by the Mayans and Olmecs. Its remnants still survive in the Biblical-sounding phrase “three score years and ten” for seventy, or in French quafre-vingf (four twenties) for eighty.
Words for numbers indicate the way a people counts. In many cases they show that counting began with the hand. How natural it is to represent numbers as the fingers of one hand as do the Zuni:
Following the five one moves to the other hand with:
6. topalikya “another brought to add with”
and so on to
10. astemthal “all of the fingers”.
If you begin counting on the fingers of the left hand and continue onto the right, it is natural to refer to the number seven as being the index finger of the right hand. Thus, the Yurok refer to seven as tserucek or “pointer,” while eight, identified with the middle finger, is knewetek or “long one”.
At the number ten, one has completed counting on the fingers, and thus the southern Wintun refer to ten as pampa-sempfa or “two hands” and the Maidu refer to it as as ma-tsoko, or “hand doubled”.
Following ten one can begin all over again, or continue, using the toes. Thus, the Unalit people of the Arctic refer to eleven as atkahakhtok, “it goes down,” and sixteen, which they begin to count on the other foot, as qukhtok. Finally, when counting is completed with all the toes and the fingers, some peoples refer to this number twenty as “one person”.
In some languages a particular number does not have a word of its own, but is formed by addition or multiplication. Thus in Crow the number ten is expressed as two (upa) and eight (pirake), i.e., as nupa-pik; while in Kutchin the number six is expressed as neckh-kiethei (which is two (nakhai) multiplied by three (kietheir). Likewise, the number eight is expressed as two times four.
The fact that number words occur in a language means that counting and other mathematical operations must have been used in the culture over a very long period—often these words are quite short, indicating that they have a very ancient use. (In most languages the shortest words generally tend to be the oldest, longer words having been created at a later date in a compound way.)
So far we have only seen the smaller numbers that can be arrived at by counting on fingers and toes. Counting can go on indefinitely and some languages are able to accommodate extremely large numbers. The Ojibwaj people, for example, represent one million as me-das-wac da-sing me-das-wac me datching me-das-wac me-das-wac. It is also referred to as ke-che me-dasWac or the “great thousand.” Indeed, just as we in English can speak of ten million, one hundred million, and one billion (or one thousand million in the British Isles), so, too, the Ojibwaj language is able to vocalize numbers.
What is particularly impressive about having words for such things as one billion is that the human mind, and human language, are referring to what lies outside the realm of direct sensual experience. While it is abstractly possible for the mind to generate the number one billion through mathematical operations it is not possible to appreciate the nature of this number directly. Yet, because so many things within Indigenous science are based upon experience one wonders what use these numbers could have been put to. If they were not experienced through references to things within the physical domain of nature possibly they may have had some cosmological correspondence, or connection to the world of processes, spirits and energy.
Numbers are contained within the languages of Turtle Island as well as being represented as marks, notches, and symbols on rocks, canes, knots on a string, or fragments of shell on a wampum belt. But there is far more to number than its representation. While to us number can be thought of as quantity without quality, to The People each number possesses a quality and a spirit of its own. In this case Indigenous mathematics goes further than the mathematics of the West, for its number celebrates, acknowledges, and mirrors the animating and transformative processes that go on in the world of spirit and nature.
Let us begin this exploration of the deeper nature of number in sacred mathematics with the number four. As far as I know, all over Turtle Island this number expresses, on the one hand, a state of balance and harmony, and, on the other, a dynamic movement of spiritual forces within the cyclica 1 nature of time.
For most Western mathematicians, the heart of their topic lies in abstraction. But in seeking to understand sacred mathematics we should never forget the great principle of Indigenous science, that it is based on both traditional knowledge and direct experience, which includes the whole world of dreams, visions, spirits, and powers. Within sacred mathematics, a number is never abstracted from the animating spirit that gives it life, nor from the concrete situations in which it is used.
The number four is not an abstraction that stands on its own without reference to any process or object in the world of spirit and nature. The number four is the Four Directions and the Four Winds. It is embodied within powerful concrete devices, images, or algorithms—call them what you will—such as the medicine wheel and the sacred hoop. The number four is a spirit and its various representations are manifestations of the dynamics of the cosmos that are animated by that spirit.
The great teachings of the People from all over Turtle Island can be found within the sacred hoop of the number four. In their Thanksgiving Address the Mohawk people acknowledge the Four Winds that come from the Four Directions and bring strength to the people and rain to the earth. Through the activity of the Four Winds the cycles of the seasons are manifest upon the earth. And so the Four Winds are the animating powers or spirits or energies that bring about maturity, continuation, renewal, and refreshment.
The Four Directions are pictured as spokes on the medicine wheel and refer not only to the transformation of the seasons but also to the movement from birth to death; to health and of healing; to the dynamics of the individual psyche; to the concept of justice; to the meaning of the sacred colors; to the history of a group; to the tasks that must be carried out by the different peoples of the earth, and to a host of other teachings. Again and again one sees that, from the perspective of Indigenous science, sacred number is not abstract but concrete and experiential: The spirit of each number unfolds into an interlocking multiplicity of different meanings and teachings.
But what are the Four Directions? At one level they could be compared with the four points of the compass, east, south, west and north, but they are also the directions of the Four Winds, and they are associated with the spirits or energies that can be found in each of these directions. They are also the sacred colors.
Sacred mathematics is not a branch of knowledge or a particular discipline that can be taught separately from other subjects and disciplines. The famous mathematician John von Neumann said that mathematics is the relation of relationships. The whole notion of relationship is central to Indigenous science. While we in the West place emphasis upon objects and categories, the Native mind deals with process and relations of relationship. The Mohawk language, for example, is rich in the various terms it employs for human relationship.
My Mohawk friends tell me that their language contains over 120 terms to express family relationships. To speak Mohawk is to enter into a web of interconnections to family, relatives, and clan that are the Mohawk people. Their language itself stresses the complexity of relationship and this also is the basis of mathematics.
Although we in the West use numbers every day we may never have actually sat down to think what number really means to a mathematician. Writing in James R. Newman's famous four volume collection, The World of Mathematics, the philosopher Bertrand Russell offered the following definition for number: “A number is anything which is the number of some class”.
This may not be of too much help to the philosophically uninitiated, indeed it may sound a bit like a circular definition. Russell offers a further definition: “The number of a class is the class of all those classes that are similar to it”.
Let us not bother too much about the details of this definition; the thing to be taken from it is that numbers are seen as concepts within a worldview whose language is based in nouns; a worldview that deals in the concepts of classes, or collections, of objects. While numbers cannot be held in the hand, they are abstractions that can be grasped in the mind; they are the objects of thought that can be manipulated and moved around in a mental universe.
To the mathematician and scientist these mental objects exist through their interrelationships of addition, subtraction, multiplication, and division. Otherwise they are without quality, and when we speak of them as “having value” we mean this only in the quantitative sense and not as referring to any intrinsic spiritual or moral value.
Within Indigenous science, number is a being—a living entity immersed in flux. To enter the world of numbers and sacred mathematics is not an act of abstraction but a sacred process. To understand the transformations of number, for example, is to seek a relationship with the dynamic processes of energies and spirits. This was once true within our own Western culture as well. Number was sacred to the Pythagorean brotherhood of ancient Greece, it lies close to art and poetry and to the world of enfolded metaphors and numinous symbols, and, fpr some mathematicians, its sacred qualities remain.
An insight into the sacred nature of number and its transformations can be gained by further considering the number four and its representation as the spokes of a medicine wheel, or in the sacred hoop. In many of the world's cultures four is both the symbol of balance and harmony and of process and movement. Early medicine in the West spoke of the four temperaments and four humors, and science pictured matter as an equilibrium of four elements.
The psychologist Carl Jung made a study of symbols whose power transcended the particularities of individual cultures. Again and again he returned to the number four as the intersection of vertical and horizontal, the four-fold division of the sacred hoop.
Jung believed that the human psyche was a dynamic process involving four forces, or psychological functions. One of these was called the thinking function and Jung pictured this as occupying one side of a duality in opposition to the feeling function. Likewise the sensate function of the psyche was seen as being at the opposite end of the spectrum from that of intuition.
Jung imagined the psyche as being structured in a way that is similar to that of the sacred hoop, with the four psychological functions lying along the sacred directions. Ideally the human psyche should occupy a point of balance within the center of the hoop, but most of us live out of balance. A person who approaches different events, including human relationships, only in a rational, logical way can be seen as living from the thinking function, having consigned the feeling function to the shadow world. Eventually this undeveloped and undifferentiated feeling function will begin to exert its subversive power on the personality.
We generally first approach life's decisions through the thinking function, by considering the pros and cons and analyzing the situation, but the final decision is usually based upon the feeling function, that is, upon a general sense, or gut feeling, of what would be the right thing to do. But a person with an inferior feeling function may arrive at a purely logical decision without having an underlying sense of conviction of the essential rightness of his or course of action.
Conversely, a person with an inferior thinking function may, with great conviction, take an important step in life without having a proper understanding of the implications involved. Such a person invests this new phase of life with great energy, only to encounter unforeseen difficulties and practical obstacles.
A person with an overdeveloped psychological function is not centered within the circle of his or her personality and is in danger of being flooded and overwhelmed with the un-differentiated energies of the inferior functions. Clearly the health of the personality depends upon reaching a balance with all four functions. It involves moving to the center of the circle and being at that point of balance within the four functions through what Carl Jung called the process of individuation.
For Jung, the number four was the essence of balance and harmony. In this respect his observations are not far from the traditional Native American teachings of the medicine wheel in which a person or a society seeks to respond to the dynamic movements of the wheel in order to maintain a state of harmony within the flux of the world.
The connection of the four directions to the sacred hoop and to the central point of harmony and balance can be approached through a variety of images. It is the human being standing in the center of the bowl of heaven. It is the central point of the observer within the circle of the horizon. It is the fire in the center of the tepee. It is the center within the council circle.
The four points of the hoop can be thought of as the different poles of two sets of dualities while, at the center, is the resolution of all conflict and duality. Thus the number four is created out of the opposition of dualities and resolved within unity. The dualities themselves can be pictured as the vertical movement of gravity set against the horizon of the world, or as the vertical movement of energy along the spine of the human body and its extension outward into the fingertips. It is the tree of the world that rises vertically out of the earth; it is the cottonwood tree of the Sun Dance; it is the human being who stands at the central point of connection between the Sky World and the world of Mother Earth; and it is the domain of physical experience that extends horizontally outward from this point.
The connection between balance, harmony, and equilibrium contained within the number four is also found within the world of Western physics. This can be illustrated by the example of a stool whose legs serve as an example of the movement toward harmony. It is not difficult for an amateur carpenter to make a serviceable three-legged stool, for as soon as you set it on the floor it will be in balance. Even when its three legs are not exactly the same length the stool will not rock and, even if the floor is uneven, it will be balanced.
To the mathematician this is no surprise, for: Three points define a plane. Mark two points anywhere on a piece of paper and it is always possible to draw a single, unique line through them. As the mathematician would say, “Two points define a line”.
Put down three points in space, and (with the exception of the special case in which they are in line) the third point will not lie on a line. What these three points do is define a plane. One could take a card and maneuver it until it would just touch each of these three points. In this way a flat surface, a twodimensional plane can be uniquely defined by only three points. No other flat surface, angled to the first, will exactly touch these three points.
Back to the stool. The tips of its legs can be thought of as three points; these three tips define a unique plane or flat surface—it is this same flat surface upon which the stool sits! It is a fundamental fact in mathematics that it is always possible to put down a three-legged stool so that the tip of each leg just touches the floor. Moreover, this is a unique result, for no other plane will touch the three tips of the legs. This means that there is only one single orientation of balance, no other orientation is possible. If there were, for example, two different planes, or flat surfaces, then the stool could wobble between them. No matter if the lengths of the legs differ, the stool will always rest on the floor without rocking.
Now add a fourth leg and the whole picture changes; for equilibrium is lost. If you put four points in space it is not generally possible to have a of card touch all of them; one will always be outside the plane. We can arrange the flat paper so that it touches any three of the four points, but not all four at the same time. So it is with our four-legged chair. Any three of the legs will define a perfectly flat plane where the chair is in balance; the fourth leg will either be too long or too short. And this means that there will be different orientations for the chair in which three of the tips touch the floor while the fourth remains in the air. When you sit on the chair you wobble between these different balance points.
This example demonstrates that the very sensitive and difficult process of achieving balance is a fundamental property of space, the universe, and the number four. Since four legs will not, in general, balance, the act of coming to balance means that we must carefully adjust the length of the legs until a unique plane is defined. In the old cartoon, the carpenter shaves a little wood off one of the legs, only to find that another leg has now become too long and so on. Achieving a final balance involves a dynamic process of constant adjustment.
The four-legged stool serves as a simple metaphor for the way in which a series of adjustments must be made in order to achieve balance through the number four. For, while in the case of a three-legged stool whatever wood we shave off from one leg is done quite independently of the other two, when it comes to the four-legged stool any adjustment to the length of one leg must always be made with reference to the lengths of the others.
For a stool, coming to balance is a holistic, integrating process, for any change in the length of one leg must involve all four legs. Likewise, for a person, or a group, or an environment, to come into harmony involves a cooperative working in all the four directions. Coming to balance is a process of renewal in which movement in one direction immediately connects all others. There is no individual who is apart from the group, no group that is independent of nature, for as we attempt to change ourselves in one direction we affect an overall change in the whole.
All this is contained in the number four.
And more.
Number, as defined by Bertrand Russell, is static, an object within the mind. But as we have seen from the teachings of the sacred hoop, the number four involves the movement of “coming to balance.” Four is a dynamic process in which equilibrium must be constantly renewed.
Within Indigenous science the number four is not a thing, it is not a mental abstraction, but a living spirit; likewise the sacred hoop is not a static diagram on a piece of paper, but an unfolding process. The hoop is a movement in which each of the sacred directions gives way to the other, for it is always in rotation. And so within the number four stands each of the sacred directions, each one being also a point of arrival and of departure. Each number, therefore, also contains its neighbor—for it gives birth to it and dies away from it.
Numbers are in a constant state of transformation and becoming. They are the symbolic manifestation of the alliances that exist between the different powerful spirits and energies of nature.
The center is the point of balance within the medicine wheel; it is the person on the mountaintop who stands at the center of the bowl of the earth; it is the fire at the center of the tepee; it is that point that both participates in the rotation of the cosmos, and yet, being above it, also stands still. In the language of Carl Jung and his friend the physicist Wolfgang Pauli, the center is the speculum, the mystical mirror that stands between two worlds and reflects each into the other, yet belongs to neither.
What is this center? It is the process of completion that leads beyond the fourth. It is the fifth direction—the number five. Yet the center is also the first number, the self, the I, the number one. At the center of the medicine wheel a person will find him or herself, for it is here that they come into balance at the one. Within a traditional ceremony a person acknowledges the four directions and this acknowledgment becomes the fifth direction.
The essential processes of sacred mathematics lie beyond our Western operations of addition and subtraction. Number is dynamic, it is a process in which the number five is born out of the acknowledgment of the number four, and is at the same time the origin of that four. Five is contained within four, it is born out of four, yet five is greater than four since it contains the four. And so the four directions are five.
This aspect of sacred mathematics can be seen within the sacred pipe ceremony in which, as the pipe is passed from person to person, its bowl is held at the stationary fifth point while its stem is rotated to acknowledge the four directions. Yet that same static point itself participates in a greater movement as the pipe is passed from person to person around the circle of smokers whose center is the sacred fire—and so the fifth point is in a rotation that returns to its starting point.
In that same ceremony the pipe carrier acknowledges Mother Earth and the sky. The ceremony begins at the center, in the heart, the number one that is also the number five. Out of this five are acknowledged the four directions and through them Mother Earth and the sky. So out of the one and the five, the four directions become six (north, south, east, west, up, and down). Returning to the center this six becomes seven. The dynamics of the sacred numbers are constantly rotating, unfolding one out of the other, folding back into each other. Each number is a starting point and an ending point. Number is no longer an abstract object, it is a process, a dynamic, a moving alliance of spirit and energy. In mathematical terms Indigenous numbers act as an algebra.
In the previous chapter David Bohm's idea of the implicate order was introduced as a metaphor for understanding reality within Indigenous science. Bohm argued that discoveries at the subatomic level suggest that the quantum world is much closer to the implicate order than it is to the explicate. The mathematical formalism used in quantum theory, however, is still locked within the order of explicate forms.
Up to his death Bohm and his colleague Basil Hiley were searching for a mathematics better adapted to this notion of an implicate order and the holomovement. One of the mathematical forms they were exploring is called Grassmann algebra. It was developed by Hermann Grassmann, the famous German mathematician and expert on Sanskrit, in the nineteenth century as an attempt to express the nature of thought. Thoughts, Grassmann argued, do not belong to our explicate world of causality and linear sequences in time; rather, they unfold out of each other and fold back again. One thought follows on the other, they are not distinct objects with clear boundaries; rather, one thought anticipates the next and thereby contains it. The thought that comes afterward contains the memory or trace of the former. Thus, the movement of thought within the mind requires a mathematics of implicate forms.
I believe that this mathematics of enfoldment and unfoldment is close to the spirit of the sacred mathematics of Indigenous science. In order to acknowledge, celebrate, and come into relationships with the flux and movements of the world, and the spirits and energies that lie behind the appearances of things, Indigenous science developed a mathematics in which the numbers are not static objects of thought but dynamic processes and actual living beings.
Something very similar happened four thousand years ago in ancient China. It began during the Shang dynasty and resulted in the I Ching. It is commonly said that the “broken” and “unbroken” lines of the I Ching are a form of binary arithmetic, corresponding to our more familiar 0, 1notation. But this is an oversimplification that misses the I Ching's deeper nature, for each line within the system can be either static or transforming. While the static lines remain the same, the moving line transforms into its dual opposite—a broken line transforms into an unbroken line. Likewise, the moving lines within a pattern of the static lines cause the nature of each of the I Ching characters to change. Each character is created out of a pattern of six lines. Yet, by virtue of its moving lines, as a particular character emerges out of the chance manipulations of the dried stalks of the yarrow plant, it is in the process of transformation and movement within the circle of the I Ching.
Like the thoughts presented by a Grassmann algebra or the implicate order of Bohm and Hiley, the character that emerges out of the permutation of the yarrow stalks contains the other characters, and meanings, within its process of transformation.
The sixty-four characters of the I Ching are arranged in a circle and become part of an underlying movement that transforms the characters and takes them to various points within the circumference of the circle. In this way I believe that there are deep connections between the meaning of the I Ching with its transforming relationship to the flux of the universe and the sacred mathematics and gambling games of Indigenous science.
The meaning of number four was discussed through the medicine wheel and the sacred hoop. We must not forget that, for the Blackfoot, this circle is always left open so that the new may enter. Nothing is permanent, no situation is ever fixed, and no category is ever closed. The trickster is ever present in all aspects of Indigenous science. The trickster may appear as coyote, as raven, or as Napi—the Old Man. The trickster was even metamorphosed into the rabbit, manifested as Br'er Rabbit, when his stories were passed on to black slaves.
The significance of flux and the power of the trickster is present in what anthropologists refer to as gambling games, ways in which The People come into relationship with the deepest processes of the universe. Sa'ke'j Henderson describes the meaning of the Mic Maqs' Waltestaqaney (or Waltes Game) as connecting The People with their ancestors who first made alliances with the spirits and energies of the universe. Waltestaqaney was created by the hero Glooscap and by playing the game one is taken back into that world through the great circle of time.
I have never seen the Waltes Game played but Sa'ke'j Henderson and Marie Battiste gave me photographs and explanations about it that had been written by their people. To begin with I was struck by the implements used to play the game. In a world of categories and abstractions a game is just a game, and it does not much matter from what materials a pack of cards, or a chess board is made. By contrast, Indigenous science deals in direct experience and these things do matter.
As we have already seen, a person who holds a piece of bone senses it as the bone of a particular animal and, moreover, as a particular part of that animal. Since all things are connected, and all things are contained within each part, to hold that bone in the hand is to be connected with the spirit of that animal, and through that to the Keeper of the animals and to all the other animals of that species. Bones and feathers, leather, and fur are direct connections—the apparatus of Indigenous science.
Each piece of the Waltes Game is not a symbol of something but a thing in itself, a direct connection to a world of spirits. The waltestaqan'ogan, or bowl used in the game, is made from the burl of a hardwood tree that has been boiled in salted water for many hours before carving. The six dice (walfestaqank) are made from the bone of caribou or deer. They are flat on one side and curved and marked on the other. In addition there are fifty-one sticks (kitmaqank) that are counted in groups of three; plus three kisikui'skwaq, who are wives of the Old Man and notched on one side; and one kisikuo'p, or Old Man, that is notched on both sides.*
Playing the game is complicated and takes all night to complete. As befits Indigenous science, the mathematics used to keep tally changes during the game's three phases. From the point of view of mathematics, the Waltes Game deals with the laws of probability, of computation, and of tally keeping; in the game transforms itself, for the system of counting used from the perspective of metaphysics it deals in exchanges that must be made within a world of flux. One element of the Waltes Game has entered Western culture, for in their early contacts with white settlers the Mic Maq people adopted an aspect of their game into the game of poker by making certain cards wild—“Indian poker”.
What is striking about the Waltes Game is that a person who has been losing to such an extent that he or she has almost no sticks left can stake everything on a final throw of the dice. By “dancing with the devil” in this way, the losing player stands a chance of winning the game outright. One wins by sacrificing everything and, in a world of flux, reversal of fortune may always be around the corner.
The origin of this game of chance is told in one of the many stories about Waltestaqaney: An old man and an old woman longed for a child and finally a son was given to them. The man loved his son dearly and taught him all that he knew. One day the boy died. The old man grieved and, finally, accompanied by a few brave men, went to the Land of the Souls to ask that his son should be returned to him. At first the Keeper of the Land of Souls was very angry and refused to allow the old man to see his son. But finally he was so impressed by the great love shown by the old man that he agreed to play Waltestaqaney with him. If the man should win, then his son would be returned, but if he lost he must return empty handed.
The game began, and continued day after day. Gradually the old man's strength began to fade, but he prayed that he could continue. The game went on until the old man had lost almost all his sticks. His only chance was to “dance with the devil” and stake everything on one last chance. In his final tossing of the dice the old man won the game and the Keeper had enjoyed the game so much that he allowed the spirit of the son to return to the land of the Mic Maq people; in addition he gave The People the gift of fruit trees and bushes.
The Haudenosaunee also have ceremonies that are related to the drawing up of contracts, and each year the Mohawk people gather in the longhouse to play the Peach Stone Game and determine who will carry out certain of the year's tasks. The Peach Stone Game is played to decide, for example, if men or the women will plant corn for the coming year—the women say that if the men win then they know it is going to be a bad year for corn!
Giving such importance to chance is one of the ways in which the metaphysics of Indigenous science differs from traditional Western science. Within Indigenous science all is process, and the various aspects of nature that we see around us—the sun and moon, winds, rocks, trees, and animals—are manifesta-tions of this underlying flux. But there is an order within this flux, for there are alliances, compacts, and relationships between the energies and spirits of the world. Sacred mathematics, as reflected in number, architecture, symbolic representation, and in the ceremonies of gambling, places The People in touch with this order. It is an order that itself is always in a state of transformation; the world is movement, and movement of that movement, and movement of the movements of movement. Its mathematics consists of relationships, relationships of relationships, and relationships of relationships of relationships.
In the distant past the Elders made contracts with the powers and spirits and energies of the world to enter into relationship with the plants, the animals, the winds, the sun, and the movement of the heavens. In the world that lies between the Sky Kingdom and Mother Earth, The People renew these relationships through their ceremonies that help maintain harmony and balance in the world.
The basic element of this worldview is that balance lies in flux, transformation, and chance. Harmony always requires the presence of the trickster, the one who overturns laws, transcends boundaries, and can win everything when down to the last counter.
By contrast, the Newtonian worldview pictured the cosmos as material bodies moving under fixed laws against the backdrop of space. Newton's universe was perfect order and predictable mechanical motion and held no room for the trickster. As the quantum physicist Wolfgang Pauli put it: There is no place for the irrational within the world of classical physics.
Within such a classical worldview there is no room for the lone Blackfoot who, early each morning, stands on the hilltop and prays for the renewal of the sun. Within a Newtonian cosmology the sun will rise tomorrow no matter what we do; nature is dead and our contracts with its supposed spirits are illusions. The deterministic universe of Newton and Laplace is impervious to all that we want or desire and has no need for our ceremonies of renewal.
In fact it is only recently that Western science has begun to entertain a picture that, on the face of it at least, appears closer to that of Indigenous science. Chaos theory (or, more generally, the science of nonlinear systems) explores the different consequences that randomness, chance, and probability can play in our world. It demonstrates that our universe is far from being simplistic clockwork because chance plays a guiding role in a vast number of processes, including weather, fast-flowing rivers, the shock waves from supersonic aircraft, the growth of materials, and the fluctuations of insect populations.
One tool in the description of chaotic systems are fractals, which are capable of modeling the details of extremely complex objects. Fractals are geometrical patterns of infinite complexity. Magnify a tiny portion of one of these fractals and a new world of detail will be revealed. They have been used to portray fractures in metals, the shapes of trees, river deltas, lungs, the circulatory system, clouds, coastlines, mountainsides, the electrical activity of the brain, and the beating of a healthy heart. Fractals are related to both order and chaos. Thus, the heart of nature, it appears, is more concerned with this chance than with simple order.
At one time order was considered to be the norm, with chaos pictured as the breakdown or disintegration of this order—thus chaos was equated with lawlessness. Now scientists are discovering, as Indigenous science has long taught, that stability and harmony can also result from chance. The chemist Ilya Prigogine has studied a host of what he calls “dissipative systems,” which range from living cells and periodic chemical reactions, to the growth of cities and the flow of traffic. In each case regularity and order emerges out of a world of chance fluctuations.
In so many ways scientists are beginning to perceive the underlying duality of chaos and order, the one emerging out of the other. Order and chaos are like the two brothers in the stories told by the Iroquois people. When one of the brothers produce something, the other creates its opposite; when one of the brothers produces order, the other will turn this order upside down. Non-Natives interpret these brothers as “good” and “evil” and see an image of the Western picture of the battle between God and the Devil in their eternal confrontations. Yet to Native people this loses the deeper meaning of the stories, for both brothers are necessary, and each must be acknowledged. To have one brother without the other would be to create disharmony in the world, for order cannot exist without chaos, nor chaos without order.
Acknowledging the power of chaos and chance is one way in which Western and Indigenous sciences are bridging their worldviews. But a distance remains between them in the way they interpret physical law. Western science seeks ever more fundamental laws, expressed in mathematically elegant ways and applying to a wider and wider range of phenomena.
In a sense, these laws lie beyond the realm of matter. As an example, one of the most fundamental laws about elementary particles, the grand unified theory, applies to processes occurring at incredibly high energies comparable to those that occurred during the big bang creation of the universe. Indeed, physicists believe that once this ultimate law is discovered it will describe the big bang origin itself. In other words, such a law describes the coming into existence of matter and energy, space and time. The law has an existence and a rule before the appearance of time and the universe itself; it existed prior to all existence!
Western physics attempts to discover the most basic laws that lie beyond all contexts, and all exceptions. These laws somehow exist outside matter and are imposed upon it. By contrast, the laws, or the alliances, relationships, and compacts of Indigenous science can never be abstracted from the spirits, energies, and forms to which they refer. These laws are like the relationships between members of a family. And if that family were to cease to exist then the law would have no meaning. Rather than law being an abstraction, it is an expression of the rich activity of the world.
But, in other ways, maybe an Indigenous perception of “laws of nature” would not be that far from the ideas that are now emerging within Western science. In my book The Philosopher's Stone I explored the idea that nature in its deepest essence is an inexhaustible and indescribable inscape. The laws of science are expressions of the regularity of nature, but are always limited and prescribed according to some particular context. Since no context is ever closed and complete, but always open to something that lies beyond it, so too, no law of nature can ever be exhaustive. What we see as the laws of nature are not impositions from some abstract domain that lies above matter and energy, space and time; rather they are expressions of the regularities within the world; they are the forms and patterns of process created out of the inner relationships of matter and energy.
Indigenous science expresses the harmony and balance of nature by sacred numbers. These numbers are never static, they cannot be exhausted through abstraction and logical analysis. Each sacred number is an inscape, an animating spirit and a moving process. Within the heart of Indigenous harmony there is always room for the trickster. The circle contains a gap to allow the unexpected to enter.
* As recently as the mid-1950s, police raids were made on Mic Maq homes and Waltes Games confiscated.