The purpose of this essay is to illustrate a number of mathematical concepts that come up frequently when magicians or mystics turn their attention to math. Thus there is no thesis, the goal is simply to make the reader conversant with these concepts.
Every student of magick or mysticism would benefit from studying math, because what it leaves the student with is a set of lenses through which to view the world, and a set of tools for tinkering with the things these lenses reveal. Moreover, math is an art with no limits - any tool can theoretically be applied to any task, if the task is viewed properly. The methods of math carry quite well into the realms of metaphor, into other types of mental pursuit - mysticism, logic, philosophy - resulting in overall expansion of the student's cognitive ability.
This mirrors the method of the magician, who presents hirself with a never-ending set of new experiences and ideas. These in turn have to be assimilated into some sort of overall framework, whether this be qabalah, mythology, chaos, materialism, or humanism. This framework is never completely stable, and the magician may find hirself at a nexus where the framework is suddenly and severely tested. It must then be rebuilt or replaced - how is this to be done? The student who is well grounded in "solid" cognitive disciplines such as math, logic, psychology, symbolism, or philosophy will have a better chance of surviving these types of "paradigm shifts" unscarred. With magickal maturity, even the framework itself becomes only a tool that can be changed at will, as a serpent sheds its skin.
The history of mysticism is closely tied to the pursuit of mathematical understanding. Pythagoras is perhaps the most well-known example of a mathematician-mystic, but the truth is that math's footprints can be found everywhere in magick: the occurance of the Golden Ratio in Egyptian temples, the pentagram, and the religious paintings of Leonardo da Vinci... numerical symbolism that has changed little over thousands of years... the encoding of astronomical data at sites like Stonehenge... the age-old practice of gematria, assigning numerical value to the letters of the alphabet... the remarkable similarity between modern cosmology and ancient qabalah... and so on. Most of magick's pre-eminent figures, notably John Dee and Aleister Crowley, were well-educated in math, and math played a considerable role in their work.
Despite popular opinion, numbers are not the central currency of math... actually, patterns are. But numbers are as good a starting point as any.
The numbers themselves have acquired a considerable mass of symbolic meaning over the centuries. Much of this is familiar and intuitive; some of it is particular to one system of magick or another. In this place I can only give a small sampling of the considerable volume of mystical or magickal significance that has been given to the numbers.
An important case in point is zero. Zero denotes the "presence of absence," and anthropologists consider use of zero by a culture to be an important cultural advance. (The most intriguing instance of zero, anthropologically speaking, is in the Mayan culture. Though the Mayans did not develop the wheel, or advanced metallurgy, they recognized zero as a number, evidence that mathematical achievement in a culture does not necessarily go hand in hand with technology.)
When nature is closely examined, it does not adhere to the Platonic ideals of straight lines, smooth curves, and whole numbers. Instead, it defies these things and insists on being irrational. Several of the most important numbers in math, which literaly define manifest existence, fall into the subclass of "transcendental" irrationals. (This is actually a technical label, though for us mystics and magicians its use provides no small sense of irony, considering their importance.) Transcendental irrationals cannot be described with finite algebraic statements, as can the square roots, cube roots, and so on. These include pi (3.14159), e (2.7183), and phi (1.6181).
The prime numbers have a special significance in the theory of numbers and in some of the more advanced forms of mathematics. But one does not have to earn a math degree to appreciate the importance of primes. It is enough to understand that they are "indivisible" (in the realm of integers at least) and that this special property has earned them some special attention from mystics and magicians. Every number, for example, can be expressed as a product of a unique set of primes, and a number of experts in gematria, including Aleister Crowley, considered the prime factorization of a number an important clue for understanding its qabalistic significance.
Imaginary numbers (based on the "impossible" notion that -1 has a square root) are a recent invention, and have not yet fully captured the imagination of mystics and magicians. This is due in part to lack of general understanding about them; their behavior is counterintuitive and their use is still quite technical, largely only occuring in engineering problems. However, they offer one way in which a limitation could be used to expand conceptual understanding by toying with the very stuff of conceptual limitation. (Non-Euclidean geometry and transfinite arithmetic might well serve a similar purpose in mystical thought, but exploration along these lines has been almost non-existent.)
The numbers which can't be quantified, which instead are infinite, also acquire significance. The notion of infinity was considered heretical for a long time -- fear of being labeled a heretic by the Church, in fact, caused Isaac Newton to delay the publication of his "theory of fluxions" for over twenty years. Infinity can be a rather hard concept to deal with, and has proven itself an excellent tool for expanding the boundaries of one's conceptual framework.
In magick, two of the most obvious, and perhaps maddening, uses of numbers are numerology and gematria. Numerology follows from the idea that numbers can hold mystical or philosophical importance, going further to suggest that these meanings are inherent, and therefore any occurence of a given number indicates the presence of that number's particular "vibration." For example, a common practice is to assign a number value to each letter of the alphabet, and use these to calculate a 1-digit number from a person's name, by adding them and continuing to add until 1 digit remains. This 1-digit number then serves as an indicator of personality, destiny, etc., in the way the star-signs of astrology are sometimes used.
Gematria looks similar to numerology on the surface, but in practice is quite different. Gematria is a practice used by qabalists to find "hidden connections" between words and phrases. A numerical value is assigned to each letter, and then the "enumerated" values of words and phrases are compared. For example, the Hebrew words for Unity (AChD) and Love (AHBH) each add to 13, while the ineffable name of god (IHVH) adds to 26; therefore God can said to be Unity and Love (13+13 = 26). Traditionally this is reserved for words in Hebrew or Greek, though many magicians now experiment with English gematria.
Shapes are patterns of an entirely different sort from numbers, though numbers are commonly called upon to describe shapes. A shape of course is any visual figure, though this essay will restrict that to mean the shapes of geometry (as other shapes, such as sigils and letters of alphabets, are somewhat, though not entirely, beyond the present scope).
While today there is a loosely-felt notion that shapes derive from numbers and equations, to the Greeks who developed the primary concepts of western mathematics it was quite arguably the other way around. A shape can be used to communicate relationships, proportions, harmonies, and disharmonies, through its properties -- complexity, number of sides, number of elements, nature of angles (acute or obtuse), curvature or lack thereof, symmetry, and proportion. In fact, it can be quite difficult to express the same concepts in any other fashion with the same subtlety or clarity. For example, if one is trying to communicate something non-linear, a shape is frequently more efficient than an equation, a set of numbers, or a paragraph.
The basic or "primary" shapes have acquired mystical meanings over the millenia, arguably based on the effects their appearance have on the brain. Thus the circle has come to represent unity and continuity, as has the sphere; the square has come to represent order (which is frequently seen as four-fold), as has the cube; and the triangle has come to represent process, as has the tetrahedron. Geometric figures of various types have acquired importance as well, notably the pentagram and the hexagram. Sometimes the shapes that defy reason or easy description have acquired significance: the hypercube, the Moebius strip, the Klein bottle.
In this century there have been two important advances in the field of geometry. Chaos and fractals are the result of a new way of seeing shapes, not as things made of boundaries, but as the organic products of simple, autonomous repeating processes. While fractals are shapes of infinite complexity they can be described with simple equations, and with somewhat rudimentary elements of programming and math (and a powerful computer) anyone can generate them. They are important because they represent a new way of thinking about the manifest universe as a self-organizing system. The second important advance is the synergetic approach to geometry espoused by Bucky Fuller. Fuller assumed nothing about the universe beyond his own perceptions and by seeing with the eyes of a child turned math on its head. His approach is remarkably reminiscent of the alchemical method, and his every conclusion, instead of giving us a cold, hard figure, has immediate philosophical implications.
Proportions are the essence of the Pythagorean philosophy. In the Pythagorean (and Platonic) way of thinking, proportions represent limitless potential, because they are recipes that allow the "stuff" of universe to shape itself into manifest forms. Proportions describe both numbers and shapes, and various other things like musical notes.
It is difficult to overstate the importance of proportion. Proportion defines our sense of aesthetics -- our notions of what is beautiful and sacred. It also describes the processes of life and of nature; growth and decay follow "gnomonic" patterns that can often be described with a numerical ratio. The proportions between musical notes define the difference between music and cacophany, and the differences in proportions between the various musical scales allow endless subtlety. Proportion is at the heart of alchemy, where the ratios at which the various materials are combined has tremendous impact on the results.
The primary notion at work is that of harmony, that simple proportions give rise to artifacts that feel natural, beautiful, and sacred. Thus harmony acquires considerable importance in mysticism, not just in the writing of sacred music but in the design of ritual spaces. The temples of Greece, India, and Egypt, and the cathedrals of Christian Europe, were built using the same small set of principles, proportions that acquire almost archetypal importance. Because of these proportions, one knows that one has stepped into a sacred place.
The most important of these is the Golden Proportion, 1:phi, roughly 1:1.618. This proportion was known to mystics of antiquity and was derived from numerous occurances in nature. Since our eyes are accustomed to seeing things with this proportion, it is not only the basis of sacred architecture but of art and aesthetics as well, lying at the center of what our eyes see as beautiful. Interestingly, the Golden Proportion can be found on the pentagram, which is not only an important figure in magick, it is a figure sacred in many religions around the world.
Finding patterns in our environment is instinctive human behavior. Frequently what comes up is a repetitive cycle, a sequence of events that continually repeats. The motion of the sun and moon across the sky is among the most important of these, as well as the seasons and the motions of the planets. These things became primary concerns with the development of agriculture. The planting and harvesting of crops has to be done in concert with the seasons, or else it will fail and everyone will die. Agriculture in turn made civilization possible. Thus civilization and everything that comes of it are built upon the notions of the cycles of sun and moon.
It is no accident that important mystical and magickal notions follow from the role of the sun, the moon, and the seasons, as well as their cycles. The cycles of the Morning/Evening Star are of importance as well, as is the Saros cycle of lunar and solar eclipses. The complex movements of the planets and of the "fixed" stars have been watched by mystics for many thousands of years, whose mystical conclusions about their significance gave rise to the art of astrology.
More complex patterns, not repeating but endlessly progressing, are important as well. The sequences of nature are closely tied with proportion, since they are frequently derived by a simple ratio. The arithmetic, geometric, and harmonic sequences were assigned mystical importance by the Platonists and Pythagoreans. A sequence of a different type, the Fibonacci sequence, holds great importance. Not only are the individual numbers of this sequence common in nature, but the proportion between one number in this series and the next approaches the Golden Proportion as the sequence progresses.
Copyright notice.
This is an original work by Callisto Radiant (T. Roberti) that has been placed on the Web for public use. Callisto Radiant may be reached at Sabrin1315@aol.com. You may share it, copy it, print it, etc., so long as this copyright notice is shared, copied, printed, etc., along with it.