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Pythagoras and the Philosophy of Number (5 of 5)
Part 5: The Golden Ratio, Proof of the Divine Presence
x. Pythagorean Philosophy and the Golden Ratio
This essay is the fifth and concluding part of a five-part series on Pythagorean philosophy. The focus of this section is on the esoteric symbolism and meaning of the Golden Ratio, which is also commonly called the Golden Section or Divine Proportion. Sometimes it is also simply called “Phi”, for reasons that we will see below.
In the sections that follow, I’m going to describe what the Golden Ratio is, how it’s measured, and where in Nature it can be found. Then, having established this foundation, I’m going to discuss why, in Esoteric Philosophy, this particular geometric proportion is revered as such an important symbol.
Here, we will discover why this simple geometric concept was held in such high regard by not only the great Greek sages of Pythagoras and Plato, but also by the Mystery Schools, Master Masons, and Temple Builders of antiquity. This reverence can be found across all cultures, from Asia, the Americas, to Greece, to Egypt. In fact, it was the Egyptian priesthood that was the original source of the Grecians’ knowledge concerning the topic of the Golden Ratio; both Pythagoras and Plato were initiates of the Egyptian Mysteries.
As we proceed in our discussion of the Golden Ratio, I’m going to be referencing back to pervious elements that we’ve already covered in this series on Pythagorean philosophy.
In particular, I’ll be making frequent revisits to the Pythagorean principle of the Triad.
The overall aim here is to highlight how the Golden Ratio shares an intimate and immediate relationship to the core ideas of the Pythagorean School’s perspective on the esoteric significance of the number Three.
The fact that scientists observe the Golden Ratio ubiquitously throughout nature - from the anatomical design of the individual human organism, to the collective design of the Earth ecosystem, up to the interplanetary design of the solar system as a whole - serves as a form of “scientific proof” that both supports and validates the thinking of the ancient Pythagorean sect.
The fact that the Golden Ratio can be ubiquitously found in every aspect of nature that scientists observe is a form of “scientific proof” of Deity’s omnipresence within Creation.
Wherever life, energy, motion, and structure are to be found, the geometry of the Golden Ratio is subtly present; it is one of the key elements of life’s blueprint.
See any decent book on the Golden Ratio (there are many well-illustrated volumes available) for depictions and diagrams of how this sacred geometry is encoded throughout nature, mankind, and the universe as a whole. The main point is that it is well known among scholars and lay people alike how ubiquitous within Nature the Golden Section is.
In Pythagorean philosophy, the three principles of the Golden Ratio are associated with the philosophic principle of the Triad.
When we link these two concepts together, the Triad and the Golden Ratio, we begin to appreciate the simple but profound spiritual implication that the Golden Ratio is indicating to us.
Deity, manifesting its Consciousness in the form of a Triad, leaves a living signature of its presence everywhere in Nature. This living signature is triadic and is expressed as the Golden Ratio, which is a unique three-termed geometric proportion whose measurements are encoded in the material fabric of life at all levels of existence, from the atom to the solar system.
Referencing back to our discussion of systems, the existence of the Golden Ratio appears as a core design feature that all systems innately share in common. To be a system - meaning, to draw upon the “system archetype” - is to embody and express the Golden Ratio as a fundamental design feature in one’s own being. In a Universe of systems-within-systems, it is no wonder the Golden Ratio can be found everywhere.
y. The Golden Section and the Golden Ratio
Note: this section is heavily influenced by (and also cites directly) the work of philosopher Scott Olsen, who is the author of a small book titled “The Golden Section: Nature’s Greatest Secret”, which is one of the finest explorations of this topic I’ve found. I also recommend his 2002 white paper, titled “The Indefinite Dyad”.
As introduced above, the Golden Ratio describes a special and unique type of three-termed geometric proportion whose relationship pattern reveals the “triadic” workings of a single Divine Self.
In particular, the Golden Ratio demonstrates the truth that nature itself is organized according to a triangulated pattern, with two opposing extremes - a thesis and an antithesis - counterbalanced at all times by a third intermediary element, which in the case of the Golden Ratio, is called the “Golden Mean”.
The function of this Golden Mean is to equilibrate and synthesize the two opposing extremes that accompany it in such a way that the Three are made to work harmoniously as One.
The One behind the Three is the Self; the Three within the One are the three worlds of the Soul or Mind that this Self is moving its consciousness through.
In sum, the Triad - which is the Divine Self - inhabits creation through the linking of triads (i.e. through triangulation), with the Golden Ratio being the initial manifest Triad from which this entire chain of being descends.
Let’s now unpack this further.
The fundamental cosmology of esoteric philosophy is premised upon the idea that God, as a Unity, expresses itself creatively by partitioning its Oneness into a Threeness or Triad.
When this Triad emerges, Unity remains always overtop, with the three principles of the Triad or Trinity existing within it. Together, the three elements or principles of this Trinity form an inseparable bond with each other, such that the Three really operate and function as One.
The primary philosophic value and importance of the Golden Ratio is that it depicts the spiritual partitioning of One into Three using the cool, impersonal language of number, geometry, and mathematics.
The fact that scientists observe the presence of this sacred ratio universally throughout nature serves as a type of abstract scientific “proof” of the ancient Pythagorean teachings about the divine nature of the Triad.
But before going further with that discussion, we need to first introduce and explore what the Golden Ratio is, how it’s defined, and what makes it unique within the world of esoteric philosophy and sacred geometry.
Let’s begin by first reviewing some simple definitions for the mathematical terms Ratio and Proportion, as the Golden Ratio is a special form of these two mathematical principles.
At root, the mathematical concepts of “ratio” and “proportion” are about comparing and contrasting two or more numbers, with the aim of locating an underlying pattern between them.
Ratio is, simply put, the relationship of one number to another. For instance, 1:2 or 4:8 are examples of Ratios, describing how one value (1 or 4) is linked to or related to another value (2 or 8).
Proportion comes into play as the comparison of one ratio against another. For example, 1:2::4:8. One could read this statement as: 1 is to 2 as 4 is to 8. In technical terms, this is called a “discontinuous geometric proportion”.
Among the various types of proportions studied by mathematicians, there is one particular kind of proportion - called a “three-termed continuous geometric proportion” - that is of crucial importance to our discussion of the Golden Ratio.
While most “discontinuous proportions” are stated in the style above, with one ratio of two numbers compared to another ratio of two numbers (meaning four total numbers are involved in the analysis), “continuous geometric proportions” are unique in that they express proportion using only three numbers.
Here, in this three-termed proportion, we find that the first term (such as 1) is to the second term (such as 3) as the second term (3) is to a third term (in this case, 9). As a mathematical equation, this statement can be expressed as 1:3::3:9, which can then be reduced to this simple three-term statement 1:3:9.
The Golden Ratio describes a special type of “continuous geometric proportion”, one that must be discovered geometrically rather than numerically.
The Golden Ratio is a unique type of three-termed proportion in that its discovery must come from the segmentation of a line, rather from the comparison of ratios of whole numbers. This is because two of the measurements of this three-termed proportion are, like Pi, irregular numbers which can’t be expressed in decimal form because the number value behind the decimal continues indefinitely.
For this reason, the “Golden Ratio” must be derived geometrically. More specifically, it is derived from a particular way of segmenting a line. If you cut a line in two at approximately the 61% mark, then the proportion that results between the lengths of the two resulting line segments, when added to the original length of initial unsegmented line, form three distinct values that fit into a special three-termed continuous geometric proportion relationship, one which philosophers for millennia have been calling the Golden Ratio.
Note: In the above description, I stated that the Golden Cut must be made at “approximately” the 61% point of the line. In reality, to truly hit the Golden Ratio, this Cut must be made at an exact point that is between 61% and 62% (i.e. at the 61.803398874…% mark). Only when the “cut” is made at the exact right point is the Golden Ratio formed.
Just so we don’t miss anything, let’s go over this again:
As a result of the special segmentation of a line (this segmentation being called the Golden Cut or Golden Section), three values are created: a) the original length of the original line; b) the length of the longer of the two line segments formed by the cut; and c) the length of the shorter of the two line segments formed by the cut.
The Greater of these segments (the length of the original line) is termed Φ, the middle segment is termed 1 (the longer of the two segments formed by the Cut), and the lesser is termed φ (the shorter segment that counterposes the longer segment formed by the Cut).
Note that Φ is called “Phi”, is pronounced “ph-eye”, and is capitalized, while φ is called “phi”, id pronounced “ph-ee” and is not capitalized.
The Golden Ratio is mathematically expressed as follows: Φ:1::1:φ. This can then be reduced to the simple three-term equation Φ:1:φ.
The diagram presented at the heading of this section depicts what this Golden Section and Golden Ratio look like in geometric form. (Note: the actual measurements of my hand-drawn diagram are not exact).
Again, it’s important to emphasize that the basis of the Golden Ratio is geometric and not strictly numerical.
This is because the numerical values of two of the three terms of the Golden Ratio (the “Greater” (Φ) and the “Lesser” (φ)) are, like Pi (π), “irrational numbers” that cannot be fully expressed as a simple fraction between two integers. Consequently, its decimal representation never ends and repeats infinitely when you try to write it out in decimal form.
To find the Golden Section geometrically, the challenge is to take a single line (representing Unity) and make a single incision such that three different line segments of three non-repeating lengths are generated as a result. The further challenge is to have these three line measurements relate together as a “continuous geometric proportion.”
Only one type of incision can be made to produce this result: the Golden Section. Based on this special cut, three line segments are generated. The Greater of these segments is termed Φ, the middle segment is termed 1, and the lesser is termed φ.
Before moving on, I want to briefly point out how the Golden Ratio is related to the well-known mathematical pattern known as the Fibonacci Sequence.
One of the ways that scientists observe the presence of the Golden Ratio in their studies of Nature is that they find a certain set of key numerical values in their measurements, ones which correspond to the “Fibonacci Numbers”. These are a special set of numbers which, went presented in ascending sequence, form an ever-increasing (or decreasing) chain of continuous geometric proportions which are entirely keyed to the pattern of the Golden Ratio.
The significance of the Fibonacci Sequence is that the ratio between any two consecutive numbers in the series approximates Phi in ever greater degrees of exactitude.
For example, in the Fibonacci Sequence, any number divided by the previous, smaller number in the series approximates Phi. This holds increasingly true as you go deeper and deeper into the sequence.
Thus, the Fibonacci sequence is, as Scott Olsen puts it, “forever closing in on the Divine Limit,” which is the Golden Ratio.
z. Plato, The Golden Ratio, and the Ancient Mysteries
The foundational teachings on the Golden Ratio - at least the ones publicly available for the layperson to access - come down to us from Plato.
Even without the writings of Plato, however, we can appreciate the special importance that the ancient priesthood and Master Masons put on the Golden Ratio, as this Divine Proportion is encoded in the architecture of every great sacred temple and religious monument going back to the time of the Pyramid Builders.
It was at these ancient sites that the sacred initiation rituals of the ancient Mysteries were performed. Given that the anatomy of the human organism is also keyed to the Golden Ratio, perhaps there is a link between the sacred rituals performed at these sites and the idea introduced earlier that there is a possibility of establishing a bond of resonance between a system or subsystem (i.e. a microcosm) existing on a lower plane of the cosmic ladder existence and a “macrocosmic” supersystem engulfing and embedding that lesser system, as a part to a whole.
It is likely that the initiation rites of the ancient mysteries were designed around the intent of establishing such a resonance between the initiates of the Mystery Schools and the greater Spiritual Self that exists supremely behind physical creation.
Once such a resonance was established, the initiate was said to be “twice-born”, meaning that the experience of establishing such a resonance is one of ego-death and spiritual rebirth, with the sovereign Divine Self becoming awakened, through resonance, within individual psyche of the initiate.
In this way, the initiate becomes God’s instrument on Earth, with the human organism serving as God’s specially designed vehicle or vessel within Nature. Through the use of this special instrument, God is able to re-awaken its consciousness within its creation and gradually take control over it.
While many authors and commentators on the Golden Ratio have noted, in astounding detail at times, that the mathematics of the Divine Proportion is encoded in many if not all of the great temples and monuments of antiquity, what most have missed is how this sacred geometric relationship is also purposefully referenced, albeit in a cryptic, veiled, and intentionally abstract way, throughout the various writings of Plato.
Plato is the great Athenian philosopher and sage whose relatively small corpus of teachings (only part of his teachings have survived through time) have steadily grown in influence to become, through time, perhaps the single most influential body of philosophical teachings in the history of Western Civilization.
As Scott Olsen has uncovered in his research, there are numerous veiled references to the Golden Section in Plato’s writings. But before proceeding with an analysis of Plato’s writings, let’s pause and thoughtfully consider the question: why would these references to the Golden Section be veiled in Plato’s (and other’s) writings?
The implications of the Golden Ratio (only a couple of which we will touch on here) are vast and profound; many Mysteries are concealed within this simple yet important geometric division of a line.
Consequently, the Golden Ratio was kept as part of the sacred arcana - i.e. the heritage of esoteric knowledge that was revealed to Man at the origin of our present world cycle and which has been protected, in an unbroken lineage, through time via a network of priesthoods, Esoteric Schools, secret societies and initiatic orders since this point of first origin. (For more, see my earlier chapters on the “Cycle of the Arya”, as well as my Atlantis publication).
This institution of sacred knowledge, which the Golden Ratio would have been a part of, was served by an inner caste of priests who ran the Mystery Schools of the ancient world. These Mysteries were enthroned in the great temple complexes of antiquity - these were the sites of both great public religious rituals (i.e. the “Lesser Mysteries”) as well as to highly secretive spiritual rituals and initiatic rites carried out in absolute privacy (the “Greater Mysteries”).
As a foundational aspect of the Esoteric Teachings, the Golden Ratio would have been taught only to initiates. Consequently, only veiled references to it could have been made publicly. Plato, an initiate of this institution, would have been obligated to preserve this tradition and only make veiled reference to it in his public teachings.
As an initiate of both the Greek and Egyptian Mysteries and as a student of Pythagorean Philosophy, Plato was someone who would have earned the right to have been initiated into the deeper spiritual teachings and mysteries concerning the esoteric philosophic significance of the Golden Section and Phi.
As an initiate, Plato was restrained from revealing the fullness of what he knew about the topic in his public writings and lectures; instead, he makes only veiled mention of it.
Scott Olsen elaborates on this: “Plato, as a Pythagorean, was under an inviolable oath not to reveal the deeper truths of the mystical Pythagorean mathematical order. Like Pythagoras, he spent considerable time in Egypt studying the mathematical mysteries with the priesthood there and in his writings intentionally conceals deeper truths under a shroud of mystery. As a teacher and writer, … (Plato) presented anomalous puzzles, problems, and incomplete solutions, along with significant hints both within the Academy and in his dialogues.”
The task for the student of esoteric philosophy - whether of Pythagoras and Plato or of a different tradition - is to work out the solution to abstract spiritual problems, ones that will be presented to them in a deliberately cryptic fashion.
The Mysteries are called that for a reason - the initiatic method of the priesthood was to stimulate and inspire the disciple or initiate to draw upon their own inner resources in order to solve a complex problem or challenge put before them.
In his teachings, Plato is following the characteristic initiatic method of the ancient Mysteries, the ones he was an initiate of. His pedagogical style should therefore be understood as being “initiatic” in design and intent.
Plato deliberately and with great intentionality sought to stimulate his students to grow and evolve by presenting them with “mysteries” in the form of challenging puzzles and deliberately enigmatic teachings. These “mysteries” would be presented to them in order that they may be inspired to accept the challenge being offered and to begin, for themselves, an internal and external quest to bridge the gap between their current state of understanding and what they need to know and become in order to advance within the initiatic program of the Mysteries.
It is through this method - by imitating the archetypal format of the Mysteries - that Plato was able to simultaneously reveal and conceal the deeper mysteries of the Phi and the Golden Ratio in his public teachings.
As Scott Olsen remarks: “In the dialogues, Plato carefully selects several interrelated problems that are very subtly posed. Taken together, they point to the great mystery of the Golden Section.”
He continues: “It is clear from Aristotle and other members of Plato’s Academy that, in unwritten lectures, Plato more openly revealed the deep truths of how the One combined with the Greater and Lesser to produce the Hierarchy of Intelligible Forms, Mathematics, and Sensible Particulars. … And yet, even to the Academy members, the presentation was necessarily enigmatic.”
In his writings, Scott Olsen has highlighted numerous examples of Plato making veiled reference to the sacred division of the Golden Section, a few of which I will share here.
As an example of how Plato enigmatically both revealed and concealed the sacred importance of the Golden Section in his writings, Olsen writes: “In the Parmenides, Plato presents the ‘worst difficulty argument’, which asks: how can the immaterial Intelligible (i.e. archetypal) and the Sensible (i.e. material) worlds have any contact or interaction” with each other?
The answer is not indicated in the same work that presents the problem; rather, a solution is cryptically suggested or hinted at in another of his works, The Timaeus, where, as Olsen writes, “Plato makes it clear that a continuous geometric proportion is the best of all bonds.” This involves there being an “intermediate geometric mean relationship” between the two opposing extremes (the Intelligible and Sensible worlds). This means that a “Golden Mean” must unite them.
The exact quote of Plato’s that Olsen is referencing from The Timaeus is as follows:
“Two things cannot be rightly put together without a third; there must be some bond of union between them. … And the fairest bond is that which makes the most complete fusion of itself and the things which it combines. Proportion (in Greek: analogia) is best adapted to effect such a union. For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it, and again, when the mean is to the first term as the last term is to the mean - then the mean, becoming first and last, and the first and last both becoming means, they will all of them by necessity come to be the same, and having become the same with one another will all be One.” - Plato
Here, Plato is explicitly referencing a three-termed continuous geometric proportion, where you have a set of three numbers in which the middle term serves the function of being intermediary and mediator between the first term (i.e. the Greater) and last term (i.e. the Lesser).
In this quote, Plato also references how this median term (which in the case of the Golden Ratio is One) serves the purpose of offering a "bond of union" between the two “extreme” terms in the set, these being the “Greater” (Phi) and the “Lesser” (phi).
Adding further commentary to this is Alexander (as in “Alexander the Great”), who was a student of Aristotle’s, with Aristotle being one of the prize students of Plato’s Academy, having studied directly under the Master Teacher.
Concerning Plato’s teachings on the Golden Ratio, in his Commentary of the Metaphysics, Alexander writes: “Thinking to prove that the Equal and Unequal are the first Principles of all things … he assigned equality to the Monad, and inequality to Excess and Defect (i.e. the Greater and Lesser); for inequality involves two things, a great and a small, which are excessive and defective” (in relation to the Golden Mean).
Here, Alexander is once again emphasizing Plato’s point that the bond between the Greater (the excessive; Φ) and Lesser (the defective; φ) is provided by the Monad (i.e. the One; the Golden Mean). This confirms the notion that what Plato is making veiled reference to in his writings is indeed the Golden Section.
In another quote, this from The Republic, Plato starts to unpack some of the deeper philosophical meaning behind the dialectic of the Greater and Lesser and the role that the Golden Mean plays in uniting them.
In particular, in one line of this famous work Plato writes: “… suppose you have a line divided into two unequal parts, to represent the Visible and Intelligible orders…”
Here, Plato is equating the Greater (Phi) with the “Intelligible World" of Archetypes and the Lesser (phi) with the “Visible World” of material forms.
Olsen further elaborates on the significance of this quote from The Republic:
“In The Republic, Plato asks us to divide the line unevenly, representing the Intelligible and Sensible Worlds. In effect, he is telling us to generate a continuous geometric proportion between the whole and the parts with the simplest of all cuts, the Golden Cut. When we apply the same ratio, the Golden Cut, to the two segments, we now get the most interesting geometric proportion between the parts: Φ:1:φ.” This geometric proportion is the Golden Ratio.
Olsen continues: “Thus, the worst difficulty argument is answered through continuous geometric proportion. The Intelligible and Visible worlds are interlocked, interlaced, and fused together through the influence of the Golden Mean in its Greater and Lesser relationships with Unity.”
In The Timaeus, Plato again touches on the philosophic meaning of these two extremes (the Greater and the Lesser), while also drawing an association between the Monad (i.e. the Golden Mean) and the World Soul, which, like Atlas, serves to bind together the opposing dimensions of Heaven and Earth, one comparatively excessive with Spirit, the other comparatively deficient.
Plato writes that “The World Soul binds together, into one harmonic resonance, the Intelligible World of forms (i.e. archetypes) and the Visible World of material objects.”
Here we find a restatement of a truth that is foundational to the ancient Esoteric Teachings: that the World Soul (the archetypal Monad) is the intermediary and intercessor through which the energies of Heaven (the Greater) move down into and fertilize the material elements of Earth (the Lesser).
What we call “material reality” is actually the end product of this triadic motion of Consciousness to moving into Mind or Soul to create and inhabit a world of material forms as an end result.
In a final quote, this one from The Statesmen, Plato draws an analogy between the fundamental values of philosophy, which are to serve the Good and release the Beautiful, and the function that the Golden Mean (i.e. the Soul) plays within the Golden Ratio between the two extremes of Greater and Lesser.
Plato writes: “The Greater and the Lesser are to be measured in relation, not only to one another, but also to the establishment of the standard of the mean. ... [T]his mean comprises that which measures them in relation to the moderate, the fitting, the opportune, the needful, and all the other standards that are situated in the mean between the extremes. … It is in this way, when they preserve the standard of the mean, that all their works are good and beautiful.”
What Plato is implying is that the Soul plays the role of intermediary and intercessor (i.e. as the Golden Mean) between the spiritual and material worlds, the former being the Greater and the latter the Lesser. By serving the Soul within, the philosopher brings balance to the spiritual and material aspects of his own nature. Thus, we find in philosophy an emphasis on walking the “Middle Path”.
It is the way of philosophy to walk this Middle Path: i.e. to consider the relationship between two extremes and walk purposefully between them, mediating one by means of the other and vice versa.
The challenge for the initiate of the Mysteries is to keep always to this Middle Path in order that they always serve the Good and release the Beautiful.
By doing so, they gradually bring balance to all extremes, including those existing both within and without. This is the service to God that the philosopher performs: he becomes the alchemist; the balancer of forces; the equalizer of energies.
In this way, as Plato tells us, by keeping to the Middle Path, the philosopher balances the Greater and the Lesser.
And when this happens, the Greater and Lesser both become “means to each other”. Meaning: they become equilibrated together such that each extreme brings balance to its counterpart.
And when that happens “they will all of them of by necessity come to be the same, and having become the same with one another will all be One.”
27. Triangulation and the Philosophy of Equilibrium
Throughout this series on Pythagorean philosophy, we’ve discussed at length the philosophical significance of the number Three - or as the Pythagoreans called it, the Triad.
In our analysis of the Golden Ratio, we find the core elements of the Pythagorean Triad restated in objective geometric and mathematical terms.
With the Golden Ratio, we also find the three core principles of the Triad - the Greater, the Lesser, and the Golden Mean - expressed in a form that can be scientifically studied and observed throughout Nature.
In this light, we can appreciate how the Golden Ratio forms a link between the abstract philosophical teachings of the Pythagoreans and contemporary research being done by scientists and mathematicians today. Thus, the Golden Ratio is a bridge uniting ancient wisdom and modern knowledge.
In our earlier discussion of the philosophical meaning of the Pythagorean Triad, we associated it with the emergence of the Divine Self - the unified being whose supreme consciousness is the single sovereign power and source of all individual beings and life forms that are made manifest within the Universe of material creation.
This Divine Self, born as the Child of the Father-Mother principles of the Godhead, adopts a three-fold constitution in the ontological design of its being: it possesses a Spirit (inherited from its Father), a Material Form (inherited from its Mother), and between them it possesses a Soul, which serves as intermediary between the two.
In the Golden Ratio, we find this teaching re-expressed: the world of Spirit is associated with the Greater (Φ), the world of material form is associated with the Lesser (φ), and the Golden Mean uniting them is associated with the World Soul, whose form takes the shape of a Grand Man.
Plato tells us this directly when he writes: “The World Soul (i.e. the Monad; the Golden Mean) binds together, into one harmonic resonance, the Intelligible World of forms and the Visible World of material objects.”
The means by which the Soul binds them together is by establishing equilibrium between them. This is the function that the World Soul plays in relation to Heaven and Earth. And, in relation to Man’s existence, which is a microcosm of the World, his own soul serves this same function: it unites his greater spiritual aspect with his lesser material nature.
As we discussed above, this passage of Plato's is essentially advocating for the philosopher to walk the “Middle Path” between two extremes.
In this way, from the vantage point of the Middle, the philosopher can work to balance one extreme with other, the end goal being to ultimately bring the two extremes into equilibrium with each other.
In this way, through philosophy (i.e. the walking of the middle path) the extremes of life “will all of them of necessity come to be the same, and having become the same with one another will be all be One.”
Here, we discover the underling reason behind Philosophy’s use of the triangulation method: the philosopher breaks all problems into Three so that they may locate the extremes while also pinpointing the Middle Path existing subtly between them.
Manly P. Hall, the 20th century’s greatest authority on Esoteric Philosophy, reveals to us the simple but powerful adage that “Three is God in Philosophy.” He then concludes that, in order to think with God, the philosopher must: “in all aspects of life, triangulate; find three essential points.”
What he’s implying by these simple statements is that God manifests its unified being as a Trinity or Triad. This Triad is necessary for the unbound conscious of Unity to become focused (by Mind) into an experience of Self-consciousness.
Being products of the Creation that this triadic Divine Self has fashioned, we must learn to think with our God and come to appreciate the fact that God works and builds within Creation in terms of Threes.
By means of triangulation, the philosopher seeks to locate Divinity’s presence in all things - a presence that always expresses itself within Creation in terms of a Triad.
Therefore, by means of triangulation, God’s presence is revealed to Man’s faculties of intellect and reason. Once awakened to this knowledge, the philosopher can locate all extremes and walk the middle path between them.
28. Yang, Yin, and Te
As we just discussed, the Triad of the Divine Self reveals itself as a triangulation between three principles that together form a set.
The Golden Ratio presents the mathematical archetype for how the three terms of this set relate to each other.
In the Golden Ratio, we find the statement: Φ:1:φ. Here we find two opposing extremes (the Greater and the Lesser, or Φ and φ) balanced by a third, synthesizing factor: the Golden Mean.
In the ancient Chinese philosophy of Taoism, we find a similar Triad of principles enthroned at the foundation of their teachings: that of Yin, Yang, and Te.
While Pythagorean philosophy is founded on the concept of absolute Unity, ancient Chinese philosophy is founded upon the ultimate principle of the “Tao".
In the Taoist system, the term Tao essentially covers the same core ideas as the Pythagorean principle of Unity. In the Pythagorean system, Unity becomes a Triad when Oneness partitions itself into an inseparably interwoven trinity of three interconnected principles. In the language of the Golden Section, these three principles are: the Greater, the Lesser, and the Golden Mean, which unites and binds the two opposing polarities of Greater and Lesser together.
In the Chinese system, we find a similar patter of divine unfoldment, with the Tao, like Unity, manifesting its Universal Consciousness by partitioning itself into a Triad of three interconnected principles.
The Taoists termed these three principles “Yang”, “Yin”, and “Te”. Here, Yang and Yin play the role of two divergent “extremes.” They are two opposites, one “excessive” (Yang) and one “deficient” (Yin). As opposites, they eternally counterpose each other. They are opposites, but neither can exist without the counterweight of its counterpart.
In between this dialectic exists a third, subtle principle - what the Taoists termed “Te”. This third, synthesizing principle is connected to both Yin and Yang and serves to bring balance and equilibrium to their duality.
In the Chinese system, “Te” functions somewhat like an “ambassador” of the Tao (i.e. Unity) within Creation (which is the world of diversity). Much like the “Monad” or “Golden Mean” of the Pythagorean system, the Te principle works among, within, and between the Yin-Yang dialectic. Its function is to balance and equilibrate Yang and Yin such that, in time, at the conclusion of the cycle, “all will again be one” (as Plato puts puts) and Universal Creation will be come to completion.
At the conclusion of this cycle of creation, the three elements of the Triad - all having become One - dissolve their boundaries that separate one from the other. Their task complete, these three principles of the Triad are then restored to the eternal condition of Absolute Unity (the Tao). Here these principles will “rest” until their eventual reactivation in a future cycle of existence.
29. Thesis, Antithesis, Synthesis
Regarding our current discussion of philosophy’s use of the triangulation method, the work on 17th century philosopher Friedrich Hegel offers us a valuable framework of insights that we can use to unravel and disambiguate the underlying relationship dynamic that underpins the Pythagorean principle of the Triad.
The pattern we’ve been working with so far concerns the existence of a fundamental, archetypal Triad of principles. Within this set of three principles, two oppose each other as inverse extremes, while a third element exists between them and works to balance and equilibrate them.
In Hegel’s philosophy, the basic semantic pattern involved with this three-principle set is described in terms of the three elements of “thesis”, “antithesis”, and “synthesis”. Here we find two opposite polarities (thesis and antithesis) interacting through a third, intermediary agent - the “synthesis”.
This third agent is an integrating, synthesizing principle, one which, as Manly P. Hall writes, lies at “the equilibrium between two points, evaluating two dissimilars from the vantage of the middle.”
What I’ve been calling “triangulation” is really a specialized philosophic discipline in which the philosopher learns to always frame their analysis of the World according to the core pattern of this “thesis-antithesis-synthesis” methodology.
In this way, the philosophic method of “triangulation” describes a technique for establishing a fixed pattern of value between three variables bound together within a set.
Here, each of the three variables in the set is given a specific name, position, and function in relation to the other two. This designation specifies the distinct role that each variable in the set will play in relation to its two other counterparts.
The three role categories that are contained within the “triangulated set” are: a) the thesis; b) the antithesis and c) the synthesis.
Within each set of three concepts that one is triangulating, each individual term will be slotted into one of these three archetypal role positions. Meaning, one of the three terms must play the role of “thesis”, another the role of “antithesis”, and the third the role of “synthesis”.
The two concepts occupying the positions of “thesis” and “antithesis” form a precise pattern of relationship with each other that is “dialectical”.
This means that the two terms oppose each other as inverse opposites. Simple examples of this include: “light” and “dark”; “good” and “bad”, “day” and “night”.
The method here is to first specify a “thesis” and then you attempt to locate a dialectically opposite concept - an “antithesis”. So, for example, if “light” is taken as the thesis, then by necessity “dark” must play the role of antithesis.
In this way, a fixed pattern of semantic value is established between the pair: each is given a value and purpose in relation to the other; each plays a specific “role” in relation to its counterpart.
Once this thesis-antithesis pairing has been established, one then moves to specify the third position in the triad: the “synthesis”.
Thesis and antithesis, being opposites, do not directly interact. They are like oil and water: they naturally oppose and separate from one another. Imagine them like the positive and negative poles of a battery: they fit together, but they also oppose each other.
To overcome this position, a third, integrating factor must be introduced, one that can integrate, balance, and synthesize the two counterposed extremes together.
In this way, through the introduction of this third element of “synthesis", the two dialectic opposites of “thesis" and “antithesis” are brought into an increasing state of functional interaction.
In our battery example, this synthesis represents the concept of the magnetic field that emerges in space and time in order to direct the flow of energy moving from the positive pole to the negative pole. Through this magnetic field, the two opposites of positive and negative are brought into a state of mutual, constructive interaction.
The mission of the synthesis is thus to gradually bring the variance between the two extremes into a state of relative equilibrium with each other.
Left alone, opposing things tend to come into conflict. But with the aid a third element that synthesizes, the pair can be brought into a state of productive and harmonious relationship.
In total, the role of “synthesis” performs two key functions in relation to the two extremes its exists alongside of: first, it functions as a catalyst for their mutual interaction; and second, it serves as a balancing, mediating, and equilibrating agent whose purpose is to bring the their opposing energies into a state of integrated and unified functioning.
Keeping this discussion of the “Hegelian dialectic”, let’s return to the topic of the Golden Ratio to see how its three elements fit in with Hegel's scheme.
In the case of the Golden Ratio, the roles of thesis and antithesis are played by Φ and φ, the former being the “Greater", the latter the “Lesser”. The role of “synthesis”, meanwhile, is played by the Monad, which, as the “Golden Mean”, symbolizes the rulership of Oneness over Duality and of Unity over Diversity.
The interplay between these two opposing principles (Yang and Yin) is the foundation of Creation - without them, a world of individualized life forms, each undergoing its own unique experience of Self-Consciousness, could never be had.
This world of duality exists for a reason, plan, and purpose: it serves as a “platform of experience” for the Divine Self to vivify and inhabit with its consciousness.
But while the Duad (Yang and Yin; the Greater and Lesser) makes an experience of diversity possible, the subtle presence of a third factor - the Golden Mean - ensures that this experience of diversity will end in the re-discovery of Unity as the foundation and destiny of all things.
By walking the “Middle Path”, the Philosopher identifies with Unity over Diversity and thereby becomes an agent for the restoration of Unity within the world of created things.
30. The Philosophical Significance of the Golden Ratio
In a previous discussion of the philosophic meaning of the Triad (covered in Part 2 of this series on Pythagorean Philosophy), I concluded that: “Three represents the rulership of Unity over diversity.” I also wrote that “duality can only exist within the superior context of Unity. Therefore, there is always Unity overtop, within, and throughout diversity.”
As I hope you can now see, the Golden Ratio is the ideal mathematical representation of this profound spiritual Truth: that behind, within, and throughout this universe of individualized self-conscious experience is one Self, one Consciousness, and one Mind.
This underlying Oneness is revealed through the Golden Mean, which symbolizes the triune nature of the World Soul. This Word Soul is the macrocosm - the “Grand Man of the Zohar”; also called “Atlas” - with his head in Heaven (the Greater World) and his feet upon the Earth (the Lesser World).
As the Golden Section informs us, before the triadic nature of this World Soul was first brought into formation, there was Unity.
In the geometry of the Golden Ratio, we find that, before the “Golden Cut” was first made - the single incision that establishes the three-part proportion that we call the Golden Ratio - there was, at the beginning, a single, unified line. Meaning: in the beginning, there was Unity.
Then, in a single creative act, a Cut is made and a Trinity is established. But this Trinity only exists in relation to the original uncut line, which always remains as the foundation. It represents the background context of Unity without which the Triad’s existence would be impossible.
Thus we find Unity as the ruler of creation - not just above creation in the form of the original line but also within creation through the presence of the Golden Mean, where, as the Monad, it subtly rules over the world of diversity it has fashioned - a world whose basis is the interplay of a fundamental duality: Yang and Yin; the Greater and the Lesser.
In truth, by pointing us toward the Golden Ratio, Plato is simply teaching us to walk the “Middle Path”.
Manly P. Hall makes this same point in his various teachings, where he informs us that the opposite of Evil is not Good. Rather, evil represents one form of extreme that will always be counterbalanced by another form of extreme (i.e. by another evil). So, if evil counterposes itself, then “Good” must lie at the middle, synthesizing point between them.
Good does not diametrically oppose Evil; rather, the Good resides at the Golden Mean between two relative extremes or two relative evils. Good is that which works between them to balance and equilibrate all extremes together. In this way, “evil” is overcome, and only Good remains.
Simply put, the philosophical significance of the Golden Section is that it provides an objective, scientific justification for philosophy’s hallmark teaching of the “Middle Way”. The teachings of this Way are found in Taoism, Buddhism, and in all other great systems of Philosophy that follow in the footsteps of the original Aryan revelations.
The way of life taught by the great philosophers of old is the way of equilibrium.
This is the way of the Golden Mean, which is the way of becoming an equilibrating agent who works between two extremes in order to gradually establish a relationship of perfect balance between them. Then, to again quote Plato, “having become the same with one another, all will be One.”
31. Philosophy is the Way of Equilibrium
In sum, the fulfillment of the Plan for Creation is accomplished through the attainment of equilibrium between all imbalanced parts and forces.
Three is the symbol of equilibrium in Space. Through equilibrium, three principles unite and act as-one.
Things therefore grow toward unity by becoming equilibrated; through equilibrium, total being gradually restores the fact of Unity on the plane of diversified existence.
Equilibrium, applied to motion, results in the gradual suspension of motion.
What this means is that, in the fullness of time, the one Divine Self gradually exhausts its own desire to maintain itself as an individuality.
By overcoming its desire to participate in the cycle of conflict and separation required to sustain its own individuality, the consciousness within the Self again returns itself to its primordial state of Unity, where Self and Not-Self are re-united as One.
The process of Unity restating its sovereign authority over duality is accomplished through an adherence to the principle of Three. As Manly Hall stated, in the worldview of Philosophy, “God is Three”.
Through the equilibrium of Three, all imbalanced forces inevitably are returned to an unconditioned state of Absolute Unity. In this state, all things find perfect equilibrium.
By thinking, acting, feeling, and perceiving in terms of Three, the philosopher learns to walk the Middle Path and keep to the Golden Mean. In this way, through Philosophy, one learns to serve God and the Divine Plan with every aspect of their being.
The ubiquitous presence of the Golden Ratio within the natural world is testament to the ubiquitous presence of God behind and within manifest Creation. It serves as a type of “proof of concept” for the abstract thinking of the Pythagoreans and other sects of esoteric philosophy.
The Golden Ratio is an objective expression of the value and importance of equilibrium in the growth and evolution of life. Equilibrium is a divine attribute, one that the Pythagoreans have traditionally associated with their sacred principle of the Triad.
By working to find and establish equilibrium in ourselves, in society, and in Nature as a whole, we become ambassadors of the Golden Mean - i.e. we become instruments for equilibrium’s “release” within a world of Creation that is fraught with imbalances.
Looking within ourselves - at the fundamental design blueprint of human anatomy - we find evidence of this Truth everywhere: that Man is designed according to the measure of God in order to be a perfect servant of God.
The anatomical design of the human being is infused with the sacred geometry and mathematics of the Golden Ratio.
Each of us is, at root, an expression of a divine design. Being expressions of it, we must learn to become like it: to follow its example. For it is the archetype upon which our very existence depends.
It is therefore our destiny to learn the law and obey the law. And this law is one of equilibrium.
When Man learns to understand, value, and imitate equilibrium, then we become conscious participants in the fulfillment of the Divine Plan.
Only by dedicating oneself toward living according to this way of life - the way of the Middle Path; the Golden Mean - can one rightly call themselves a Philosopher.
Pythagoras invented the term; this is what he intended by it. Anything that is not in alignment with this teaching should not rightly be called Philosophy. To resurrect the institution, we must re-establish its dignity by taking away everything else that falls short of its standard.
When we follow the Way that Pythagoras and Plato have pointed us toward - meaning, when we embody the Golden Mean in our thoughts and actions - we assist God and Nature in their grand project of re-establishing perfect equilibrium between Heaven and Earth.
Mankind is designed as an instrument or agent of transformation in Nature. We are collectively the chosen vehicle through which the grand alchemical transmutation of Nature is destined to take place.
First the alchemical transformation must take place within Man himself. Then, through a regenerated and spiritually awakened human, mankind accomplishes the larger transformation within Nature itself. This means that all forces within the macrocosm or World as a whole are balanced and brought into equilibrium, with Heaven and Earth re-united as One. When this happens, the Cycle comes to completion and Man’s great labor is finished.
32. The Golden Mean and the Destiny of Man
The Pythagoreans regarded the pentagram or five-pointed star as the symbol of the Soul - the human soul (the microcosm), but also the World Soul as a whole (the macrocosm).
For this reason, Five represents the divine power of Soul, both within Man and within Nature.
The Power of the Pentad resides within the World Soul itself. Man, a microcosm of this World Soul, channels this power and serves as a gateway or vessel through which it is made manifest within the world of Creation.
The power of Five within Man is the fifth element or “quintessence” lording over the lower four elements of the body.
In Pythagorean philosophy, Five is regarded as the number of the human Soul. The perfected human Soul is expressed through the archetypal figure of the Hierophant or High Priest of the Mysteries - an enlightened, god-like human who exists as a foreshadowing of what we are all destined to one day become: fully realized beings (i.e. “Bodhisattvas”) who have molded themselves into the likeness of their own divinely designed Archetype.
These Divine Humans - the Adepts or Hierophants; the “enlightened ones” - have awakened Divinity within themselves and exist to further this awakening within their fellow Man.
In relation to the Pythagorean school, Pythagoras was this hierophant. It is not a coincidence, then, that this sect used the Pentagram as the chosen emblem of their society - it symbolized not only the sage who founded the school but also the Divine Power that his teachings and school were intended to serve and channel.
Esoterically, the symbol of the Pentad also encodes important elements of the Pythagorean teachings, including those regarding the mystery of the Golden Ratio.
For example, the Five-Pointed Star or Pentagram, which is the archetypal symbol that the Pythagoreans chose for both their school as a whole as well as to their numerical principle of the Pentad, encodes the Golden Section throughout its dimensions. The five lines that form the pentagram intersect at points that mark a golden ratio division. Thus, there are two golden section “cuts” made to each of the five lines that comprise the pentagram.
The numeric disambiguation of Five also gives further clues that unlock important aspects of the Pythagorean doctrine.
For example, mathematically, Five can be formed two ways: as the sum of 3 and 2 or as the sum 4 and 1.
As the sum of 4 and 1, the Pentad wields the power of the Ether, the Fifth Element or Quintessence which integrates and synthesizes the four lower elements together.
As the sum of 3 and 2, the Soul of Man is composed of the equilibrium of Spirit (pertaining to the Triad or 3) and the duality of Mind (pertaining to the Duad or 2).
These two principles (the Triad and Duad) together represent the Solar and Lunar aspects of Creation: the Father-Mother principles that together give birth to the Macrocosm and Microcosm alike.
These two opposing principles (representing the Yang and Yin principles of Creation) are the fundamental elements that the initiate is challenged to bring into equilibrium within his own life.
Only once the initiate has attained this for himself will he then be entrusted with the power to accomplish this alchemical act within the larger world environment.
In this way, the Pentad - the power of the human soul - is the agent of restoration for equilibrium within the world of form. This he accomplishes by becoming the Alchemist and harnessing the power of the Ether to transmute all base metals into Gold.
The Universe in its natural state is in equilibrium: there is balance so that nothing dominates. To become the Alchemist, Man must re-discover and discipline himself according to Nature’s perfect, innate demonstration of this equilibrium.
By following Law, Man becomes like Nature: a balancer of forces and a bringer of equilibrium.
Whether we discover the Truth of this Law or not, it will nonetheless continue to act on us in the form of “karma” or “consequence.” As Manly Hall informs us, “equilibrium is actually indestructible, but man is afforded the capacity to manipulate this balance.” Therefore, man has the power to create or destroy equilibrium within himself and others, but never within Nature as a whole. Rather, Nature as a whole has the ultimate power of Law over him.
All other creations of God naturally obey Law; only Man has the power to disobey in order to pursue inner, ego-oriented urges and desires. Disease inevitably results from man’s fall into such a state of imbalance, with the Alchemist serving as the “Golden Mean” who restores balance to these forces, thereby returning life to a state of health.
As the Alchemist, Man’s purpose is to bring about a balance in nature’s elemental forces. In this way, he imitates the Way of God, which is the Way of Equilibrium.
As Manly Hall explains, ”The human is the master of nature. Man’s role in nature is to ensure equilibrium in the environment in which he exists.”
He later elaborates, writing that “Man himself exists only through a balance of forces (5), which he must preserve. He can only achieve this by attaining wisdom (3) through the development of the intellect (4).”
Unpacking this statement further, we can say that Man, as the Alchemist, embodies, in his pentagonal soul design, the divine quality of equilibrium, which is innate to God (the Triad).
By harnessing the power of equilibrium, the Human Soul becomes a channel for the release of God’s innate quality within the world of creation.
Man, as the Alchemist, is the bringer of equilibrium. This he accomplishes by becoming a balancer of all opposing forces, elements, and principles in nature.
This is the specific role in the grand Hierarchy of creation we are designed to play. It is therefore written in our soul’s archetype that we must each evolve to become the Alchemist. Only in so doing will we gain the wisdom, intelligence, and insight necessary to fulfill the divine purpose we have been designed and created to accomplish.
As the Alchemist, Man has the power to unite the four lower elements within himself and thereby awaken his consciousness to a fifth, enlightened state of understanding associated with the element Ether.
This awakening is one in which the Universal Self awakens itself within the individual self of a human personality. Once awakened, this Universal Self becomes the guiding light of the individual human soul’s actions and directives here on Earth.
The Alchemist is the awakened human who has accomplished the Great Work within themselves. Only after having attained this feat within is the Alchemist then granted the knowledge and opportunity to pursue the accomplishment of this divine work within the greater environment of the World Soul itself.
The Alchemist is actually just another name for the philosopher. There is only one Way, and those who embody this Way serve God and their fellow man alike.
The philosopher or alchemist is properly born through the initiations of the Mysteries - the Schoolhouse of the Cosmic Mother. Once born, they work on behalf of God to bring the World Soul’s greater cycle of existence toward its pre-destined state of ultimate conclusion.
This “Omega” state - the pre-destined end state that all of evolution is being pulled toward - is one of a great return of individual existence to “ultimate existence” - i.e. of diversity back into the primordial state of Absolute Unity; the Tao; the Mahaparanirvana.
This is the service - that of becoming awakened, dedicated servants to the fulfillment of the Divine Plan - that we, as humans, have the privilege of offering to God. This is the way of philosophy; the way of the Tao; the way of the Bodhisattva. There are many names, but there has only ever been one Way: the Way of Equilibrium.
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