The Pythagorean System is one of the embodiments of the Perennial Tradition. 1 Our knowledge of Pythagoras (582-496 BCE) is similar to that concerning Jesus: we have nothing he wrote, only a number of writings about him. In this review of the Pythagorean System I'll present what I consider to be the most reliable account of his ideas and activities. The primary elements of value within the Pythagorean System were its conception of mathematics as a means of discerning Forms, its influence on the life and teachings of Plato, and its discoveries in music.
Pythagoras was born on Samos, an island just off the coast of present-day Turkey. Samos is currently a part of Greece. Pythagoras' father was reputed to be a descendant of Ancaeus, the founder of Samos.
Raised in a family of wealth, Pythagoras excelled in his studies and gained a wide reputation as a fine young man.
A contemporary of Buddha and Lao Tzu, Pythagoras is said to have studied with Anaximander the natural philosopher and Thales of Miletus, an eminent philosopher credited as one of the developers of mathematics.
Thales, then of an advanced age, advised Pythagoras to go to Egypt, to study with the priests of Memphis and Jupiter. Thales indicated that his own study with these priests was the source of his reputation for wisdom.
On his way to Egypt, Pythagoras studied with the local hierophants in Phoenicia. He was initiated into the mysteries of Byblus and Tyre, and in the sacred rites performed in many parts of Syria. After learning all he could of the Phoenician mysteries, he discerned that they had originated from Egyptian
Higher Wisdom. This led Pythagoras to hope that in Egypt itself he might find genuine adepts of the Esoteric Mysteries.
Egypt has enjoyed the reputation of possessing Higher Wisdom, from before the time of Pythagoras to the modern day. One of the contemporary researchers into Egyptian Wisdom was John Anthony West.|
"Here in Egypt he [Pythagoras] frequented all the temples with the greatest diligence, and most studious research, during which time he won the esteem and admiration of all the priests and prophets with whom he associated. Having most solicitously familiarized himself with every detail, he did not, nevertheless, neglect any contemporary celebrity, whether sage renowned for wisdom, or peculiarly performed mystery; he did not fail to visit any place where he thought he might discover something worthwhile. That is how he visited all of the Egyptian priests, acquiring all the wisdom each possessed. He thus passed twenty-two years in the sanctuaries of temples, studying astronomy and geometry, and being initiated in no casual or superficial manner in all the mysteries of the Gods. At length, however, he was taken captive by the soldiers of Cambyses, and carried off to Babylon. Here he was overjoyed to associate with the Magi, who instructed him in their venerable knowledge, and in the most perfect worship of the Gods. Through their assistance, likewise, he studied and completed arithmetic, music, and all the other sciences. After twelve years, about the fifty-sixth year of his age, he returned to Samos." 2
Finding no persons with whom to associate on Samos, Pythagoras established himself at
Croton, in Italy, forming an elite circle of followers who were called Pythagoreans. He opened his school to men and women students alike, setting up very strict rules of conduct. Pythagoras gave a much higher place to women than was common in his day. At that time, women were regarded as property, without rights. In the Pythagorean community, women studied and meditated along with the men, possessing equal rights.
According to Iamblicus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading, and philosophical study. Music featured as an essential organizing factor of communal life, the students singing hymns to Apollo together regularly, with poetry recitations occurring before and after sleep to aid the memory.
The study of mathematics was an important element in the Pythagorean school. Those who joined the inner circle of Pythagoras' society were called the Mathematikoi, living at the school, owning no personal possessions, and required to follow a mainly vegetarian diet. Other than recitation of poetry and other lessons and singing, Pythagoreans followed a strict code of silence.
Pythagoras and his associates were long held in such admiration in Italy, that many cities invited them to consult with them as to their administration. As might be expected, however, they incurred envy, and a conspiracy was ultimately formed against them.
Pythagoras was ninety when Cylon, a rejected candidate for the Pythagorean mystery school, incited a violent revolution. Standing in the courtyard of Crotona, Cylon read aloud from a secret book of Pythagoras, Hieros Logos (Holy Word), distorting and ridiculing the teaching. When Pythagoras and forty of the leading members of the Order were assembled, Cylon set fire to the building and all but two of the council members were killed. As a result, the community was destroyed and much of the original teaching was lost.
Pythagoras and the Metaphysics of Mathematics
"There are therefore three principles: God, the substance of things, and form. God is the artist, the mover; the substance is the matter, the moved ; the essence is what you might call the art, and that to which the substance is brought by the mover. But since the mover contains forces which are self-contrary, those of simple bodies, and as the contraries are in need of a principle harmonizing and unifying them, it must necessarily receive its efficacious virtues and proportions from the numbers, and all that is manifested in numbers and geometric forms; virtues and proportions capable of binding and uniting into form the contraries that exist in the substance of things." |
Archytas of Tarentum, Fragments of Pythagoras, (400 B.C.)
To understand how Perennialist teachings provide an entree to a supersensual world, we must begin with the most fundamental questions. Why is our physical world not governed by incoherent chaos? Why is this "Cosmos" (as the Greeks termed it) intelligible to humans? What is the essence of these organizing principles: "pattern," "structure," "form," and "order?"
Pythagoras and Plato believed that elements in our empirical world are ordered by supersensual Forms residing in a higher dimension. These Forms were necessary to explain the structure we see in the world around us. The only reason the physical universe is intelligible at all is that objects retain the same form, different things take on the same form, and we are able to communicate with one another about the meaning of these patterns and relationships.
Pythagoras and Plato believed that Forms are neither material objects, aspects of or abstractions from material objects, nor mere concepts in our brains--they exist on their own terms, apart from the physical universe, eternal and immutable. Physical objects are what they are by virtue of their participation in specific Forms.
Aristotle maintained that these Forms are merely "abstractions from" physical objects and have no separate being in a supersensual world. Modern scientists such as Rupert Sheldrake have come up with theories similar to Aristotle's, seeing forms as merely constituents of objects. Sheldrake called the process of retaining the same form "morphic resonance."
Most modern scientists ignore metaphysical questions about the essence of Forms. But we cannot avoid this question if we take seriously what Perennialist teachers have maintained: that humans have the capacity to attain a higher state of consciousness through understanding the supersensual world of Forms or Ideas.
"Within the human consciousness is the unique ability to perceive the transparency between absolute, permanent relationships, contained in the insubstantial forms of a geometric order, and the transitory, changing forms of our actual world. The content of our experience results from an immaterial, abstract, geometric architecture which is composed of harmonic waves of energy, nodes of relationality, melodic forms springing forth from the eternal realm of geometric proportion."
Robert Lawlor, Sacred Geometry
Mathematics and especially geometry were seen by such Perennialist teachers as Pythagoras and Plato as among the most effective means of understanding and entering a spiritual realm composed of eternal, unchanging (invariant) Forms or Ideas. In Plato's Commonwealth, Socrates says that only those versed in geometry will be allowed entrance into the ideal state.
"For Pythagoras, mathematics was a bridge between the visible and invisible worlds. He pursued the study of mathematics not only as a way of understanding and manipulating nature, but also as a means of turning the mind away from the physical world, which he held to be transitory and unreal, and leading it to the contemplation of eternal and truly existing things that never vary. He taught his students that by focusing on the elements of mathematics, they could calm and purify the mind, and ultimately, through disciplined effort, experience true happiness."
John Strohmeier and Peter Westbrook,
The Life and Teachings of Pythagoras
Geometry means "measurement of the earth." When the Nile flooded each year in ancient Egypt, obliterating the property boundaries, priest-mathematicians used geometry to re-establish the markings for specific areas. To the Egyptians and the mystical philosophers,
geometry was regarded as a magical science with the power to reveal to humans the properties of given elements (points, lines, angles, surfaces, and solids) that remain invariant under specified transformations. In general, we use geometry to study spatial order through the dimensions and relationships of forms.
But the study of geometry must be at the highest possible level to enable students to achieve understanding of a higher realm of being. First they must be fully aware of the various levels of being, the characteristics of each level, and the entities peculiar to each level.
State of Being
Forms, Higher Ideas
Sensing, Feeling, Thinking, Reasoning
Trance states, Instinct
The discerning study of simple geometric concepts enables students to gain an understanding of the second, metaphysical, level of being, since geometry deals with non-material concepts and invariable symbols. For example, the study of a simple circle of any size leads to the discovery of what is called a transcendental number, Pi.
Pi - P
Radius (CD) = .5
Diameter (AB) = 1
Circumference = Diameter x pi (3.14159265)
If we take any circle, dividing the circumference by the diameter results in Pi: 3.14159265.
P--Pi is a transcendental number. A transcendental number cannot be the root of any polynomial equation with integer coefficients, meaning that it is not an algebraic number of any degree.
Throughout human history there have been many attempts to calculate this number precisely. One of the oldest approximations appears in the Rhind Papyrus (circa 1650 B.C.E.) from ancient Egypt in which a geometrical construction is given where (16/9)2 = P.
What is transmitted through geometry is so subtle that it is exceptionally easy to miss it. When we come upon an entity such as Pi, we have arrived at INVARIANCE. No matter how large a circle, dividing its circumference by its diameter invariably gives us Pi. We are able to understand that the cosmos is put together using specific formulae (such as Pi)--we are discovering the very STRUCTURE of the universe!
Along with the discovery of the ordering principles through which our universe is constructed, in geometry we enter a fascinating world of CERTAINTY. In the ordinary empirical world of shoes and ships and sealing wax, we experience constant uncertainty, having to settle for approximations at best. Students of geometry can feel the difference in worlds as they enter the domain of mathematics; they are suddenly in a world of certainty; they will arrive at conclusions that are not just probable but undeniable.
The world of mathematics is one where all entities are defined and if we follow its principles then our conclusions are unequivocal. We say of it that it is a "closed" world--because most of its elements are defined, but we must remember that the world of mathematics also contains symbols which point beyond it: irrational numbers, transcendental numbers, infinity, point, zero, etc. Students of geometry learn to breathe in a world of certainty and understand that in the higher world of Forms, this sense of certainty is unlimited: all in that world is eternal, unchanging, unified.
As Plato points out, through the higher study of mathematics we can achieve the ability to think non-concretely--attaining independence from sensible objects in our contemplation. The freeing of thought from dependence upon the sensible image is an accomplishment of the very greatest magnitude. Until thought has achieved this power, it cannot penetrate into the realm of imageless consciousness.
Through following the specific exercises prescribed by the Perennialist teacher, initiates gain an awareness of the second level of being, the metaphysical-conceptual domain. Having understood the second, metaphysical, level of being, the seeker is able to move upward to an understanding of the highest world of being: the dimension of Forms.
"The true use of [mathematics] is simply to draw the soul towards being. . .
"Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. . .
"The knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient. . .
"Geometry will draw the soul toward truth, and create the spirit of philosophy."
Plato, The Commonwealth, Book VII
The Magic of Number
The Pythagorean Theorem|
Pythagoras discovered the magical relationship
between the lengths of the sides of right-angle triangles.
The longest side of a right angle triangle is called the hypotenuse. This is always opposite to the right angle.
The side opposite a chosen angle is called the opposite side.
The side next to the chosen angle is called the adjacent side.
Pythagoras discovered that the square of the hypotenuse (length times length) is equal to the sum of the squares of the other two sides: hypotenuse² = opposite² + adjacent².
This is known as Pythagoras's Theorem and it works for all right-angled triangles. Using this theorem, we can discover the length of any of the sides of any right triangle if we know the length of the other two sides.
We can take a triangle with the lengths of its two shortest sides known: 3 cm and 4 cm. Applying Pythagoras's theorem, we can find the length of the hypotenuse:
hypotenuse² = one-side² + second-side²
h² = 3² + 4²
h² = 9 + 16 = 25
h = 5 cm
In studying mathematics, we discover that there are magical, esoteric qualities to numbers. Applying this and other discoveries, we're able to understand that the cosmos is constructed by the use of specific formulae such as Pi and Pythagoras' theorem. By using these organizing principles we discover the mathematical/geometric structure of the universe!
"The philosopher, because he has to rise out of the sea of change and lay hold of true being . . . must be an arithmetician."
Plato, The Commonwealth, Book VII
Pythagoras and the Mathematics of Music
The western tradition of tonal harmony developed from the systemization of Pythagoras' approach to the mathematics of music. Pythagoras is associated with mathematical discoveries involving the musical intervals of the octave, fourth and fifth - the simple ratios of the lengths of stretched strings and the pitch of their vibration. Intervals could be expressed in the numbers from 1 to 4; ... 1:2 the sound of an octave; 2:3 the fifth; 3:4 the fourth.
Pythagoras defined the "consonant" acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension (regardless of their material: gut, steel, rope, etc.) of proportional lengths create tones of proportional frequencies when plucked. For example, a string that is two (2) feet long will vibrate x times per second (Hertz). While a string that is one (1) foot long (x/2) will vibrate twice as fast: 2x. And furthermore, those two frequencies create a perfect octave.
The Greek scale had a mere five notes. Pythagoras pointed out that each note was a fraction of a string tuned to a certain pitch. If we have a string that plays an A, the next note is 4/5 the length (or 5/4 the frequency) which is approximately a C. The rest of the octave has the fractions 3/4 (approximately D), 2/3 (approximately E), and 3/5 (approximately F), before you run into 1/2 which is the octave A.
- x Hertz -- Fundamental Pitch
- 2x Hertz -- An Octave Higher
To hear the five notes of the Greek scale click on the start button below.