# Internumeric Relationships by Addition on the Tetractys

It’d be rude and vulgar of me to leave the Tetractys as some simple geometric diagram used for plotting paths or meditations.  I mean, the Tetractys is a meditation tool, yes, but to use it merely for working with the Greek alphabet with in a mathetic framework is to ignore the deeper meaning of the Tetractys.  For the Pythagoreans, especially, the Tetractys was more than a set of ten dots; it was the key to all creation and all cosmos.  There’s no evidence that anybody’s used it to plot paths on like I did, which is probably because this is an innovative use for an already heavily used tool based purely on number.  As we’re all aware by now, the Tetractys is a representation of the Monad, Dyad, Triad, and Tetrad to yield the Decad: 1 + 2 + 3 + 4 = 10.  All these numbers are holy to the Pythagoreans and to Western occultists generally, but there’s so much more to the Tetractys than this.

One of the traditional ways of understanding the mysteries of the Tetractys was to take the different ranks of numbers present and add them together to yield a particular number.  For instance, the Monad plus Tetrad yields the Pentad (1 + 4 = 5), while the Monad, Dyad, and Triad together yield the Hexad (1 + 2 + 3 = 6).  All these numbers have their own meaning, all of which are based ultimately on the Monad and, in succession, the meanings given to the other numbers built upon the Monad.  I’d thought I’d investigate what some of these properties are and see what the Tetractys represents in building the numbers of the Decad together based on these relationships between the ranks of the Tetractys.  Specifically, these relationships are based on the arithmetical operation of addition, the straightforward aggregation of two numbers by combining their distinct magnitudes into a single one.  Other operations exist, but those are for another time.

So, to start off with, we have four basic numbers, starting with the Monad and ending with the Tetrad.  We can say that, with the exception of the Monad, all numbers are just collections of Monads in a particular relationship:

1. Monad = individuation, undifferentiated, undifferentiatable
2. Dyad = two Monads in relation
3. Triad = three Monads in harmony
4. Tetrad = four Monads in form

Note that some of these can be broken down further into simpler groups.  Without repeating any particular number (such as saying that the Dyad is two Monads or the Tetrad is two Dyads), we end up with two extra identities:

1. Triad = Monad + Dyad
2. Tetrad = Monad + Triad

It’s crucially important to note that the Dyad, Triad, and Tetrad are more than just a collection of monads.  Number in the esoteric sense is more than just a magnitude or amount, but also a relationship formed between the individuals in the collection.  The only number in this set that has no relationship is the Monad itself, since it exists as a unity unto itself without anything to relate to.  The Dyad is the first number that has a relationship, but can be said to be relationship itself; without the Dyad, relationship cannot exist.  In a more arithmetic sense that the Pythagoreans preferred, all numbers can be divided into two partially overlapping groups of odd (able to be divided into unequal parts only) and even (able to be divided into two equal and unequal parts).  Four, for instance, is even because it can be split up into groups of 1/3 and 2/2.  Five, however, is odd, because it can be split into 1/4 or 2/3, and neither of those are equal splits.  However, the Monad cannot be split at all into anything, and the Dyad can not be split into unequal parts, so neither the Monad nor Dyad are even nor odd, and are thus not true number, though they are sources of number.

Thus, based on the individuation of the Monad and relation of the Dyad, all other numbers can be made, such as the Triad.  It is because of this that the Triad is considered by the Pythagoreans to be the first true number, since the Monad and Dyad are something rarer and rawer.  All amounts can be formed from the Monad, but it’s the relationship (Dyad) between individual Monads that produce a number.  Thus, as the Triad is the first true number, it is also the first odd number, and the Tetrad is the first even number.

So, based on the six above identities, we can form the rest of the numbers from the Pentad (5) to the Decad (10).  If we omit the identities from above and reduce all things to a collection of Monads, Dyads, Triads, and Tetrads, we end up with two ways to form the Pentad, and one way each to form the Hexad, Heptad, Octad, Ennead, and Decad:

1. Pentad = (Monad + Tetrad) or (Dyad + Triad)
2. Hexad = Dyad + Tetrad
3. Heptad = Triad + Tetrad
4. Octad = Monad + Triad + Tetrad
5. Ennead = Dyad + Triad + Tetrad
6. Decad = Monad + Dyad + Triad + Tetrad

Yes, this is all basic arithmetic that we’ve been able to do since kindergarten.  Of course, it’s always the simplest things that hide some of the more profound secrets.  I won’t go over all the associations and theologies behind the numbers for that; you can get a copy of the Theology of Arithmetic by Iamblichus for cheap (or even, dare I say it, for free), and you can read about what the Pythagoreans thought about the numbers of the Decad way back when.  What I want to point out is, at a high level, what these additions of the numbers mean based on the four concepts of monadic individuation, dyadic relation, triadic harmony, and tetradic form.

Monad
The Monad is an individual, unchanging, static, and stable.  It is the only thing that exists, and thus cannot be differentiated from anything (since there’s nothing to differentiate it from).  While we can say that it contains all opposites and extremities within itself, it’d be more proper to say that no concept of opposition or extremity exists within the Monad.  While the Monad exists, nothing exists within the Monad; it can become all and any qualities, but it itself has no qualities.  It is the source of all nature, but is itself beyond nature.  It cannot be divided since it is a unit, an atom, the core of existence itself.  The Monad cannot move, as there is nothing within which it can move (which would imply something that is Monad and something that is not-Monad).  The Monad has no shape, consisting only of a single point that indicates both all sizes and all angles but without anything else to connect to.

Dyad
The Dyad is relation and difference.  Between two Monads, we now know of two things that can be compared as equals, but as different equals.  The Dyad is representative of differentiation, distinction, opposition, and motion, all of which can be thought of as different types of relation.  The Dyad represents a line defined by two points, but is still without shape; it can possess direction and magnitude, but is as yet without definition.  The Dyad allows for things to exist within, around, and outside of other things, since it creates space between and among other things.  While the Monad is pure potential for creation (and all other things), the Dyad is the act of creation itself, since it distinguishes a Creator from the Creature, or the Acted from the Actor.  The Dyad is space, change, action, and relativity.

Triad
The Triad is harmony and proportion, formed from a combination of individuation and relation.  It is the first odd number, and the first number that can be added from other distinct numbers.  The Triad gives the first shape of something, as three points can define an enclosed space.  The Triad indicates actuality, the Creature made through Creation (Dyad) from the Creator (Monad).  However, it is also indicates harmony, since two distinct and different things are linked to and joined by a third.  With the Triad, there is real existence as opposed to potential existence or becoming existence.  Quoth Iamblichus, “‘this’ belongs to the Monad, ‘either’ to the Dyad, and ‘each’/’every’ to the Triad”.  With Triad, there is time: beginning, middle, end; there is communication: speaker, listener, message; there is work: actor, action, acted upon. However, like the Monad, the Triad is static, since it provides for space and size but not change, since it is construction and creation that brought a static shape to being.

Tetrad
The Tetrad is the root of form, formed from a combination of individuation and harmony.  With three points we can define a two-dimensional shape, but with four we can define a solid three-dimensional object.  Moreover, the Tetrad is dynamic, since it is even; while the Triad measures static quantity, the Tetrad measures dynamic quantity, since it provides for motion and change while the Tetrad does not.  Further, the Tetrad allows for forms present in relationship to each other; while the Triad offers a two-dimensional form, the Tetrad allows for two-dimensional forms next to each other as the Dyad allows for Monads to be next to each other.  With both individuation and harmony, one can choose to be part of a harmony or break away from it, acting either inside or outside a given group, and allows for distinct existence apart from, aggregated with, or in conjunction with others.

Pentad
Alone among the numbers, the Pentad is the only one that can be formed in two distinct ways: from the Monad and Tetrad (a combination of individuation and form) and from the Dyad and Triad (a combination of relation and harmony).  In a way, it’s fitting; between all the numbers of the Decad, the Pentad is the middle of them.  Consider that any two numbers that add up to 10 have 5 as the mean (9 + 1, 8 + 2, 7 + 3, etc.); the Pentad is halfway to the Decad, and itself is vital to life.  It is the combination of pure potential and discrete aggregation (Monad and Tetrad), as well as of relation and harmony (Dyad and Triad); it is the combination of an even and odd number in either case, and considered to unify opposites in a dynamic way that allows for growth and change as opposed to the static way of the Triad.  If we consider the Pentad as the sum of Monad and Tetrad, we obtain a view of eternality and potentiality combined with and suspended among temporality and discretion (the four changeable elements acting under unchanging Spirit); if we consider the Pentad as the sum of Dyad and Triad, we obtain a view of motion and action mixed with and changing stasis and relationship.  In either case, the Pentad is where life and concrete reality itself begins, since in the Pentad there is balance, reciprocity, distribution, and especially of growth.

Hexad
The Hexad is the combination of relation and form, producing a dynamic harmony.  Unlike the Pentad, which is dynamic growth, the Hexad is a balance between things in motion.  The presence of distinct qualities bestowed by the Tetrad in relation of the Dyad allows for various dynamic forces to exist dynamically, moving with and acting, co-acting, or reacting together without destruction.  As the Tetrad represents a body and the Dyad represents motion, the Hexad represents a body in motion and can move in six ways, or three sets of two ways: up/down, left/right, forward/backward.  Seen the other way, as the Tetrad represents qualities and the Dyad represents opposition, the Hexad represents an ordering and balance of opposites.  Further, as two Tetrads, the Hexad represents what we commonly see as “Merkava stones”, two interlocked tetrahedrons that represent a combination of bodies and opposites that together unite to form a whole.  While the Pentad is the number of life, the Hexad is the number of order.

Heptad
The Heptad is the combination of harmony and form, producing foundation.  This is hard to describe in a single word, but within the Heptad there are all things finally present to create everything, yet is short of actively creating everything; all manifest sources are present in the Heptad (seven planets of astrology, seven vowels of Greek speech, etc.), though they are as yet too unmanifest on their own.  As a combination of Triad and Tetrad, the Heptad represents the four elements and three reagents, or the three processes that transform the four elements so as to create all things.  As an odd number that cannot be divided, the Heptad is similar to the Monad in that it provides for potential creation, but unlike the Monad, the Heptad is a collection of seven entities that provide the foundation of all manifest things, while the Monad is an undifferentiatable source from which all manifest and unmanifest things come.  If the Hexad represents order, then the Heptad are the things that are ordered within the cosmos provided for by the Hexad, the meat to fill out the Hexad’s bones.  The Heptad is that which essentially exists; the Heptad is essence.

Octad
The Octad is the first addition that involves three numbers: the Monad, Triad, and Tetrad.  Thus, the Octad combines individuation, harmony, and form.  As the Heptad is the combination of the Triad and Tetrad, we can say that the Octad is that which results from the essences of creation into which they flow.  However, as we saw with the Pentad, we can also say that the Monad and Heptad combine such that the Heptad is mixed in within the Monad, as the seven planets are within the eighth sphere of the fixed stars, as the four elements are within the Quintessence.  However, we can also say that the Octad is the combination of two Tetrads, allowing for mixtures and combinations of that which otherwise could only relate to each other by processes; although Sulfur combines and transforms Air into Fire and vice versa if we use the Tetrad + Triad view, we end up with dry air or cool fire between Air and Fire if we use the Tetrad + Tetrad view.  The Octad represents solution and combination of qualities, a single entity produced from essences or qualities and their interquality transformations.  The Octad is mixture.

Ennead
The Ennead is the combination of relation, harmony, and form.  Based on how we might conceive of this, we can say that the Ennead combines the Tetrad and Pentad, the Triad and Hexad, the Dyad and Heptad, or the Monad and Octad, but at its root it combines the Dyad, Triad, and Tetrad.  At its core, it lacks the Monad and possesses the Dyad, indicating that the Ennead is an active number related to creating but not as creator or creature.  In the Ennead is all creating of manifest things, combining tetradic body, triadic intermediation, and dyadic motion.  In the number nine are all the other numbers brought together, the final single-digit whole number.  As there were nine Muses who lead to all Art and nine Curetes who watched over the infant Zeus, the Ennead brings things to completion and perfection without itself being perfect.  The Ennead is realization.

Decad
At long last, we finally reach the Decad, the combination of the Monad, Dyad, Triad, and Tetrad; of individuation, relation, harmony, and form.  In the Decad are all the basic numbers of the Tetractys, and there are many ways to add to the Decad using the lesser numbers, but at its core it is the number formed from 1, 2, 3, and 4 summed together.  Just as in the Ennead there is the process of realization and completion but without something to realize or complete, the Decad augments this with the Monad, allowing for something to be filled with the Ennead.  The Decad represents a discrete entity (Monad) that is distinct from other things (Dyad) that is stable unto itself (Triad) given physical a body (Tetrad).  Moreover, it is also something that can grow (Pentad) while maintaining itself in an order (Hexad) that combines all ethereal essences (Heptad) and concrete mixtures (Octad) being brought together (Ennead).  Without any other number preceding it, the entity represented by the Decad would be lacking and could not be fully realized.  Whether it is the universe we live in or the individual people we live as, we are all representative of the Decad and the journey it has taken to get here.  The Decad is the Whole.

I think it goes without saying that this Pythagorean analysis of the ten numbers of the Decad can easily be mapped onto the Tree of Life in Jewish kabbalah or Hermetic qabbalah, and indeed, I recall seeing many of these things present in the explanations given in works like Alan Moore’s Promethea series.  It makes sense, too, since Pythagoreanism is one of the fundamental philosophies underlying Western occult thought, deep enough to not clearly be distinguished as Pythagorean but also profound enough to affect everything that’s built upon it.  While numerology has never quite been my strong suit, this little exploration of the basic numbers has considerably helped.

# Towards a Greek Kabbalah: Tetractys as Cosmic Framework

Alright, so now we understand the Greek letters as symbols of many things: zodiac signs, planets, elements, numbers, body parts, and any number of gods, images, and other concepts.  All this is in addition to their use as instruments of written language as glyphs and of spoken language as sounds and names.  While we’ve come a long way, we’re still only setting out our basic tools for further analysis.  We know of the Greek letters as units, single entities representing a single set of symbols.  To use them, we have to start seeing these letters as relationships, transferring and communicating information and power between other units.  It’s like a single word, such as “my” or “the”, being understood, but without meaning until it’s used in a sentence when it indicates relationships and distance between and among other concepts.  Within a word, letters act as relationships between the letters before and after it, but is there anything else bigger that we might have the letters act as a relationship between?

This is where we start thinking about things cosmically and from an emanationist perspective, and, to be honest, I have some catching up to do on Hermetic and Neoplatonic philosophical language to describe it accurately.  Suffice to say that, like Jewish kabbalah and Hermetic qabbalah, Greek kampala also describes the creation of the cosmos in a series of stages, but not necessarily in a clearly-ordered and sequential manner like the sephiroth do.  Rather, it is through providing multiple levels of understanding of the cosmos that we can better understand it.  The Greeks loved to divvy up stuff into smaller stuff, the foundation of the problem-solving technique “divide and conquer”, and that continues here.  With that, I propose we use the geometric diagram of the tetractys, the holy diagram of Pythagoras:   The tetractys is a diagram consisting of four rows of points, each row having an increasing number of points starting at one and ending at four, arranged in an equilateral triangle.  This was considered to be the foundational blueprint for all of creation according to that mathematical and mystic hero of Greek thought, and was mathematically significant: it combined the Monad (unity), Dyad (two), Triad (three), and Tetrad (four) into a single unit.  These four numbers, representing the One, Power or Differentiation, Harmony, and Cosmos, compose all the things of the universe, as 1 + 2 + 3 + 4 = 10, the unity of a higher order, or the Decad.  All of these numbers were holy to the Pythagoreans, but ten most of all; given its fourfold structure, this diagram was called the Tetractys, and sometimes referred to by the Greek letter Delta (Δ).  The Pythagoreans saw it as so holy that they prayed to it and glorified it:

Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.

And, further, the Pythagoreans swore by the tetractys itself:

By that pure, holy, four lettered name on high,nature’s eternal fountain and supply,the parent of all souls that living be,by him, with faith find oath, I swear to thee.

The tetractys was considered to represent the fourfold nature of creation in eleven different ways, according to Iamblichus in his “Life of Pythagoras”.  Essentially, a single Monad (God, the Good, etc.) created a Duality of Two, which then created a Harmony of Three, which then created an Ordering of Four; by the interaction of these different forces, both within their own groups and across other groups, all other things are made.  In fact, by interpreting each row (with its difference) in different ways, we obtain something resembling Cornelius Agrippa’s Scale of Four (book II, chapter 7):

1. According to the composition of numbers: 1, 2, 3, 4
2. According to the multiplication of numbers: a point (0-dimensions), a side (1-dimension), a square (2-dimensions), a cube (3-dimensions)
3. According to magnitude: a point (0D), a line (1D), a plane (2D), a solid (3D)
4. According to simple bodies: fire, air, water, earth
5. According to figures: pyramid, octahedron, icosahedron, cube
6. According to things rising into existence through vegetative life: seed, length (shoot), breadth (leaf), depth (trunk)
7. According to communities to form nations: individual, household, street, city
8. According to judicial power: intellect, science, opinion, sense
9. According to parts of the animal: rational, irascible, epithymetic (that which desires good), physical body
10. According to the seasons of the year: spring, summer, autumn, winter
11. According to the ages of man: infancy, youth, adulthood, senescence

The tetractys even helped to guide the Pythagorean musical system by taking ratios of the rows of dots:

1. Rows 4 and 3, 4:3, perfect fourth
2. Rows 3 and 2, 3:2, perfect fifth
3. Rows 2 and 1, 2:1, octave
4. Rows 1 and 1, 1:1, unison

In fact, so influential was the use of the Tetractys in Greek thought that it even influenced philosophical schools hundreds of years afterwards, even Jewish kabbalah and Hermeticism.  This can even be seen in one representation of the Tetragrammaton, the four-lettered name of God, represented in a tetractys-like form: Plus, having ten units inside, parallels can be drawn between the tetractys and the Tree of Life, or ten sephiroth.  Dion Fortune in her “The Mystical Qabalah” even drew a comparison between the fourfold Tetractys and the first four sephiroth on the Tree of Life, especially with regards to the tetractys as demonstrator of physical space:

The point is assigned to Kether; the line to Chokmah; the two-dimensional plane to Binah; consequently the three-dimensional solid naturally falls to Chesed.

In terms of the four parts of the body, it might be better to restate tetractys #9 above (according to the parts of the human) in Agrippan terms as the Mind, Spirit, Soul, and Body.  These four parts of the human are that which links us to the divine (Mind, row 1), that which allows us to reason and intellectually understand the world (Spirit, row 2), that which feels and moves (Soul, row 3), and that which is moved and is felt (Body, row 4).  It is by the unification and purification of these four parts of the body do we practice ascension through and beyond ourselves back to the One, but that’s another topic for another day.  Suffice it to say that, through the cultivation and increasing of virtues in the four parts of ourselves, we ascend the Tetractys and the multiple parts of the world we find ourselves connected to.  If you want, give the excellent Summary of Pythagorean Theology by Apollonius Sophistes a read in the meantime, since that’ll be a huge thing for us later on.

The study and meditation of the Tetractys will become bigger and bigger later on, especially once we view it as a kampalic cosmic map much in the way that the Tree of Life functions for kabbalah and qabbalah.  As yet, we’ll leave the individual units of the tetractys unnumbered and unmarked, letting the structure speak for itself.  To let it do so, meditate on the structure and form of the tetractys, offering it the same devotion and glory the old Pythagoreans themselves would have done so.  Hold the image in your mind, and relate all the parts of the cosmos to its structure: the fourfold nature of things resulting from a threefold harmony resulting from a twofold differentiation resulting from a single Source.  To say much about the tetractys at this point would be premature, so I’ll leave it as an introduction on its own as a symbolic representation of what’s to come: first a seed, then a shoot, then a leaf, then a trunk.