The Tetraktys is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself in seedlike form the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the mysteries of the divine realm. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity--Pythagoras.

 

Pythagoras was one of the first Greek philosophers. He was born somewhere around 570 B.C.E. on the island of Samos, the eastern region of the ancient Greek world called Ionia. After almost forty years of studying with the greatest spiritual teachers and philosophers of the ancient word and becoming initiated into nearly every ancient mystery rite, Pythagoras went on to form his own philosophical community. He tested his students by various means before they were allowed to join the community, and once admitted, students had to endure a five-year period of silence before being invited to study directly with the philosopher. Though this may appear extreme, we must remember that in the ancient world wisdom and knowledge were not freely shared with people in the liberal manner that is common today. The gifts of wisdom were restricted to those who proved themselves worthy of these gifts through initiations and purifications.

 

According to his ancient biographer Iamblichus, "what ever is anxiously sought after by the lovers of learning, was brought to light by Pythagoras." It was because of his profound spiritual attainment and great achievements that the following distinction was made: "of rational animals one kind is a God, another man, and another such as Pythagoras." Some Pythagoreans of later generations came to wonder if the sage was the very god Apollo. Pythagoras, however, had a different view of himself. Indeed, when asked whether he was wise [sophos], Pythagoras answered, "No, I am a lover of wisdom [philo-sophos]," and thus the very word and ideal of "philosophy" was born. (For in the philosophical tradition only the gods possess wisdom; humans merely participate in the divine.)

 

Pythagoras also used difficult mathematical theorems and problems to test and prepare his students for ascent to the divine. It is this use of mathematics in the spiritual development of its students that distinguishes the Pythagorean wisdom tradition from other great traditions. Mathematics was seen as a study that purified the soul from its habitual way of looking at the world and its belief that the physical was ultimate reality. Unlike the practical forms of mathematics we are familiar with today, Pythagoras developed ideal systems of mathematics that functioned as intellectual mandalas that guided the soul to a vision of the structure of heaven.

 

These mathematical studies demand a number of spiritual practices. For one, the kind of concentration and discipline required in mathematics develops powers of meditation. In Zen koan study, as in mathematics, students must develop the energy of concentration or one-pointedness. But unlike Zen, in which practitioners seek "sudden illumination," mathematics leads its students to understand its ideas step by step, and this prepares the mind to receive understanding and insight. Pythagoras recognized that all learning (mathemata in Greek) is an awakening of eternal Ideas within the soul and the source of the unity and light that every true form of understanding and insight brings. Patterns of numbers and geometrical forms possess a clarity of truth, a perfect symmetry, and a magnificent beauty which reflect the true nature of the soul. In other words, the study of mathematics leads one to a recognition of the kinship between the objects of the mind and Mind itself--which Plato calls the brilliant light of Being.

 

These mathematical studies are ladders and bridges to the divine because they share a perfection and beauty that is true of the divine but lacking in the physical world. In the Pythagorean scheme of the cosmos, the soul is an intermediary between the mortal and the immortal, and so is mathematics. Although mathematics concerns plurality and objects imagined to be extended in space (characteristics of the physical world), it also is incorporeal, unchanging, and has a truth that is like the divine. Therefore, the meditation upon and comprehension of mathematical Ideas allows the soul to enter into the Ideas as a genuine intellectual mandala, awakening the energy of the soul and preparing it for the vision of true reality.

 

Pythagoras's studies are known as the quadrivium, and they became the foundation of learning through the Renaissance, though not always with the same spiritual purpose. The four studies correspond to the four levels of the Tetraktys, which is the most perfect symbol of Pythagoras's vision of the mind and cosmos. First in the quadrivium is arithmetic, the study of number, and number is nothing but an extension of unity. Moreover, the four levels of the Tetraktys--1 + 2 + 3 + 4 = 10--contain the basic elements of all arithmetic.

 

The second study is music, the ancient name for the mathematical study of ratios. This branch of mathematics provides the foundation for the musical scale and also has important applications in the third and fourth studies, geometry and astronomy respectively. Since ratios describe a relationship between two numbers, music corresponds to the second level in the Tetraktys. The ratios 1:2 (musical octave; geometrical proportion), 2:3 (musical fifth; arithmetic proportion), and 3:4 (musical fourth; harmonic proportion) form the elements of what is known as the Pythagorean music of the spheres.

 

The third study is geometry, the study of the three dimensions: length, width, and height. In the geometric sphere, as David Fideler points out, one represents the point *[this character cannot be represented in ASCII text], two the line *[this character cannot be represented in ASCII text], three the surface *[this character cannot be represented in ASCII text], and four the tetrahedron *[this character cannot be represented in ASCII text], the first three-dimensional form. "Hence in the realm of space, the Tetraktys represents the continuity linking the dimensionless point with the manifestation of the first body."

 

The fourth study in this system is that of astronomy, the study of three-dimensional objects in a fourth dimension, or motion in space-time. Astronomy embraces each of the prior three studies and a fourth (motion in time)--four representing the four primary elements of physical reality: fire, air, water. and earth.

 

However, each of these studies is preparatory to a fifth and transcendent study: dialectic, the study of the metaphysical realm of Ideas and the bond uniting all the previous studies. In dialectic, one represents God, the first principle and source of all Being, that which is beyond all names but which the ancients called The One Itself. Perhaps the greatest mystery in all philosophy is why The One gives birth to many. From the perspective of the ancients, The One is so perfect that all creation is but the overflow of its perfection. The One is characterized by superabundant potency and creates the world without causing any lack in itself. This overflowing gives rise to the dyad of infinity and limit, qualities which define the class of gods in the Pythagorean theology. (Although Pythagoreanism recognizes many gods, each god is intimately allied to The One Itself.) The third level in this hierarchical unfolding of the cosmos is called Being itself, the triadic source of all Being, Life, and Intelligence (and the paradigm for the soul and its three parts--desire, will, and mind). The tetrad corresponds to the final link in the great chain of being: the physical plane and the four physical elements.

 

The first ten numbers that constitute the tetrad are also recognized as symbols of the unfolding and extension of the unity of the divine Mind. Numbers, therefore, are archetypal principles and divine powers that have causal power which manifests in certain qualities in our world. One is the principle of the monad, the symbol of unity apparent in the circle, the figure enclosed by one line which represents completion and wholeness.

 

Two, the dyad, is expressed in the figure of the vesica piscis and is the power of all duality, contrast, and manyness--for example, male/female, heaven/earth, left/right.

 

Three, the triad, introduces a mean between the two extremes, the power to bring order and harmony to manyness. In the Pythagorean tradition, the soul is the mean that unites the mortal and immortal and binds them into a whole. In geometry, the triangle is born from the vesica piscis as the first plane figure with its three equal lines and angles.

 

Next is the tetrad. In arithmetic, the tetrad represents a stage of completion, for as we have seen in the Tetraktys, the first four numbers make a sum of ten. Geometrically, the tetrad is expressed by the square as well as the four sides of a pyramid. Metaphysically, the tetrad represents matter and the four primary elements.

 

The rest of the decad has essential qualities that we can briefly describe, but a more comprehensive description is beyond our scope. The pentad represents the principle of life, the hexad structure and order, the heptad the completion of cycles, the octad change and renewal. The ennead is the limit and horizon of number, pregnant with the potential birth, and the decad is the completion of the paradigm of the whole.

 

The Tetraktys embraces the multiplicity of these ten principles within the unity of the equilateral triangle. In Greek mathematics, numbers were studied in their different families or kinds. There are even numbers, odd numbers, and irrational numbers (all of which we are familiar with), but the Greeks also recognized triangular numbers, square numbers, pentagonal numbers, and so forth. A triangular number is a series of numbers that can be represented in the figure of an equilateral triangle: 1, 3, 6, 10, 15, 21, 28. ... As you can see, this geometrical form produces as continuous numerical progression expanding the base of our triangle with successive numbers: 1, 2, 3, 4, 5, 6, 7. ...

 

The Pythagoreans identified the particular triangular number ten as profoundly significant for a number of reasons. For one, it embraces the first ten number principles in its geometrical form (ten is not an element in any of the other geometrical number series, i.e., the square, pentagonal, hexagonal, and heptagonal). This is deeply significant because it expresses what the Pythagoreans recognized as the unifying power of the triadic principle. The triad is the key to all ancient metaphysics because it is the structure or form which naturally unites duality into harmonious union. (Indeed, the Pythagoreans considered the triad, in certain respects, the first true number. In Greek the word for number is arithmos, which comes from the root "ar" which means "to join," as in "h-ar-monia," the very function of the triad.) The monad and dyad were said to be prior to number, being the principles of oddness and evenness respectively, and in this respect the very source of number itself. According to Porphyry, the student of Plotinus, "Things that had a beginning, middle and end the Pythagoreans denoted by the number Three, saying that anything that has a middle is triform, which was applied to every perfect thing."

 

The perfection of the triadic form is most evident in the form called the mean (or three-term analogy). An analogy is the comparison of two relationships. For example, a grandfather is to a father, just as a father is to his son. The analogy expresses the comparison between the two sets of relationships (also called ratios), and in this comparison the father is the mean between his own father and son.

 

In Plato's account of the Pythagorean cosmology, The Timaeus, the Pythagorean philosopher says, "But it is impossible for two things alone to cohere together without the intervention of a third; for a certain collective bond is necessary in the middle of the two. And that is the most beautiful of bonds which renders both itself and the natures which are bound remarkably one." The mean analogy therefore is also the basis for the entire Pythagorean and Platonic theology. The great Platonic philosopher Proclus expresses this explicitly in his Elements of Theology: "Every divine order has an internal unity of threefold origin, from its highest, its mean, and its last terms."

 

The profundity of this idea can be recognized if we examine it in the case of self-knowledge (gnosis). The pursuit of self-knowledge embraces three aspects: the knower, the knowing, and the known. The great puzzle of this pursuit is how the extremes of knower and known can be unified. According to Plotinus, this unity is possible because in truth "the knower, the knowing, and the known are one." The philosophers explained this by looking toward the mean analogy in which the power of knowing functions as a mean to unify the extremes.

 

The knower is to the knowing just as the knowing is to the known.

 

Interestingly enough, the mean or three-term analogy, being a logical structure, has a total of four forms in which the basic statement can be expressed. When we draw out this whole analogy, its intrinsic unity, harmony, symmetry, and order become apparent. We watch each of the three terms move through each place in a logical dance that creates an intellectual mandala that expresses the one underlying the many.