First Initiation into Numerics

Pythagorean Number Philosophy is based on the pattern of the first ten Numbers.  These Numbers are the core pattern of philosophical laws for contemplating the mysteries of The One.

Understanding the Number pattern from the Monad to the Decad is the first stage in an alchemical process to harmonize personal consciousness with the flow of universal consciousness.  Through a crystallization process this pattern is hard-wired into our subconscious Net of associations, and becomes a platform for stepping into mystical states where pure knowing is achieved (gnosis via mathesis).

The Pythagoreans gathered around the holy Tetractys as the principal symbol of their philosophy of Numbers.  It displays the progression of the Monad to the Decad as a dot-pattern in the form of a triangle, called a figurate.

The Tetractys, symbol of Pythagorean Number Philosophy

The Tetractys encodes many secrets of the universe, and is to be taken as a cosmological symbol.  There are 4 levels, symbolizing 4 philosophical or metaphysical worlds.  The numerical progression of levels (1:2:3:4) gives the fundamental harmonic proportions used for constructing the Pythagorean Diatonic scale.

In total we have 10 dots, which are viewed as the 10 aspects or qualities of being within The ONE.  This was taken to be the complete map of all that is, since all other numbers could be reduced to one of these.  The use of base-10 counting was seen as a reflection of this philosophical truth within human understanding.

As a figurate, the Tetractys is the 4th Triangular Number.  The Triangular Numbers are given as the sums of consecutive numbers, beginning with 1.  So 1+2=3, 1+2+3=6, and 1+2+3+4=10, making 10 the 4th Triangular.  Taken philosophically, since 10 reduces to 1 (1+0=1) we have that 4 is really 1.  So there is a pattern from 1 to 2 to 3 to 4, but 4 is really 1.  Since we return to 1, we must continue to 2 and 3, but then 2+3=5, 1+2+3=6, 4+3=7, 1+3+4=8, 2+3+4=9, and 1+2+3+4=10.  So by staying within 1 to 4, we can generate all 10.

When seen from modern mathematics, these calculations would seem infantile, but when viewed philosophically, we can see why the Tetractys could merit such veneration.  We may not be surprised, then, to recall that the Divine Names in most ancient mysteries were spelled with 4 letters.  In Greek ThEOS and in Hebrew YHVH.  The latter is known as the Tetragrammaton, or 4-letter Name, which encodes a profound universal pattern of Numbers, which I shall explain in future posts.

The Pythagorean doctrine of the Decad may be responsible for the Qabalistic doctrine of Ten Sephiroth.  The Qabalists also based their metaphysical system on a system of Ten, and while the earliest of Qabalistic texts refering to these Ten Sephiroth, the Sepher Yetzirah, is generally dated by scholars to be at least several hundered years after the time of Pythagoras, but the correspondence between the two is nearly the same.  They can be studied in conjunction, yielding the fruits of cross-fertilization.

The First Stage of Self-Initiation into Pythagorean Number Philosophy is to develop in one's consciousness the ideas and images that follow from contemplating the following series of seed-ideas.

The Meanings of Numbers:

1 - MONAD - Beginning, Location, Unity

2 - DYAD - Duplication, Reflection, Extension

3 - TRIAD - Multiplication, Growth, Expansion

4 - TETRAD - Order, Regularity, Structure

5 - PENTAD - Mediation, Adaptation, Modification

6 - HEXAD - Reciprocation, Symmetry, Harmony

7 - HEPTAD - Equilibrium, Poise, Rest

8 - OCTAD - Rhythm, Vibration, Pulsation

9 - ENNEAD - Perfection, Attainment, Mastery

10 - DECAD - Completion, Result, Wholeness

Once the mathematician-alchemist has distilled and digested this Number pattern, repeated study leads to crystallization and a kind of prototype for the Stone of the philosophers.

The Philosophy begins by establishing the Greek names for the numbers, to remind us that they are to be thought of as philosophical qualities.  Then we try to understand the Numbers as states in a dynamical movement, as indicated by the meanings of the key-words.  These key-words have been chosen very carefully by the sages so that they may help you unlock the secrets which lie hidden within our philosophy.

In future posts, we will explore the link of Number to Geometry.  The geometry associated with the Numbers gives us a way to visualize the pattern in ways that lead to better understanding, but also gives us a Path to follow and a way to Truth.  We will also see how Letters can stand in for Geometry, leading to a sacred script for describing and preserving the mystical experiences of the mathematician-philosophers.

On The Pythagorean Definition of the Quadrivium

The philosophical tradition surrounding the name of Pythagoras derives its mistique from how little we know of its origins, while at the same time can claim the subsequent mathematical development and scientific advancements as verification of its basis in wisdom and truth.

Master Pythagoras

The Pythagoreans divided their teaching into a four-fold system called the Quadrivium, consisting of Numerics, Harmonics, Geometry, Cosmology. Nicomachus of Gerasa, well known as being the greatest Neo-Pythagorean of his time, can help us understand the meaning of this division of their teaching:

"Things, then, both those properly so called and those that simply have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called "magnitudes"; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called "multitudes", a flock, for instance, a people, a heap, a chorus, and the like.

"Wisdom, then, must be considered to be knowledge of these two forms.  Since, however, all multitude and magnitude are by their own nature of necessity infinite-- for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end, but proceeds therefore to infinity-- and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing either with magnitude, per se, or with multitude, per se, could never be formulated, for each of them is limitless in itself, multitude in the direction of the more, and magnitude in the direction of the less.  A science, however, would arise to deal with something separated from each of them, with quantity, set of from multitude, and size, set off from magnitude.

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as "even", "odd", "perfect", and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like "double", "greater", "smaller", [etc], it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

"And once more, inasmuch as part of "size" is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of "size", geometry the part that abides and is at rest, astronomy that which moves and revolves."    

 -- Nicomachus of Gerasa: Introduction to Arithmetic I - ch. 2-3

Another clear description of the way the Pythagoreans divided learning into the Quadrivium is given by the late Neo-Platonist Proclus:

"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold.  A quantity can be considered in regard to its character by itself or in its relation to another quantity; magnitudes as either stationary or in motion.  Arithmetic, then, studies quantity as such; music the relations between quantities; geometry [studies] magnitude at rest, spherics [studies] magnitude inherently moving.  The Pythagoreans consider quantity and magnitude not in their generality, however, but only as finite in each case.  For they say that the sciences study the finite in abstraction from infinite quantities and magnitudes, since it is impossible to comprehend infinity in either of them.  Since this assertion is made by men who have reached the summit of wisdom, it is not for us to demand that we be taught about quantity in sense objects or magnitude that appears in bodies.  To examine these matters is, I think, the province of the science of nature, not that of mathematics itself."   

 -- Proclus: A Commentary on the First Book of Euclid's Elements - Prologue I ch. 7

 The study of arithmetic was given before anything else, so fundamental was the doctrine of Number for their philosophical school.  Nicomachus can help us see why arithmetic must be studied first:

"Which then of these four methods must we first learn?  Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others.  And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order to their proper ends; but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them.

"So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one.  For how can "triple" exist, or be spoken of, unless, the number 3 exists beforehand, or "eightfold", without 8?  But on the contrary 3, 4, and the rest might be without the figures existing to which they give names.

"Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.

"And once more is this true in the case of music; not only because the absolute is prior to the relative, as "great" to "greater" and "rich" to "richer" and "man" to "father", but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similiarly all of their harmonic ratios are arithmetical ones, for the diatessaron is the ratio 4:3, the diapente that of 3:2, and the diapason the double ratio; and the most perfect, the didiapason, is the quadruple ratio.

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin-- for motion naturally comes after rest-- nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantites.

"So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and as it were, mother and nurse of the rest."    

 -- Nicomachaus of Gerasa: Introduction to Arithmetic I - ch. 4-5

It is recommended that a deeper understanding of the philosophy of the Quadrivium is gained by contemplation of these quotations from the mathematician-philosophers.

We shall be exploring the Quadrivium and related areas in future posts.

Proclus on the Mathematical Imagination

Proclus' Commentary on the First Book of Euclid's Elements offers penetrating insight on the nature of mathematical being, showing how the Platonic Ideas of the mathematical concepts are received by the mind (nous) and projected onto the imagination, which serves as a mirror for the reflections of the soul.  Geometrical figures are therefore rightly understood as doorways into the Ideal World.  Contemplation of the relation between the figures and the Ideas can lead us into the Via Mathesis, that is, the spiritual path of self-knowledge gained through mathematical insight.

Proclus (410 - 485 CE)

The following text acts like a seed, that when planted in consciousness and grown through meditative techniques yields a plentiful harvest of seed-bearing fruit (emphasis mine):

"Therefore just as nature stands creatively above the visible figures, so the soul, exercising her capacity to know, projects on the imagination, as on a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions of the ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself.

"It is as if a man looking at himself in a mirror and marveling at the power of nature and at his own appearance should wish to look upon himself directly and possess such a power as would enable him to become at the same time the seer and the seen.

"In the same way, when the soul is looking outside herself at the imagination, seeing the figures depicted there and being struck by their beauty and orderedness, she is admiring her own ideas from which they are derived; and though she adores their beauty, she dismisses it as something reflected and seeks her own beauty.  She wants to penetrate within herself to see the circle and the triangle there, all things without parts and all in one another, to become one with what she sees and enfold their plurality, to behold the secret and ineffable figures in the inaccessible places and shrines of the gods, to uncover the unadorned divine beauty and see the circle more partless than any center, the triangle without extension, and every other object of knowledge that has regained unity."