Hermetic Theory of Creation: Understanding Sacred Geometry and the Language of Archetypes

Hermetic Theory of Creation Series

Part 1: Revealing Secrets of Sumerian Theory of Time
Part 2: Hermetic Theory of Creation: God is an Artist Who Crafted Platonic Solids from the Tapestry of Time
Part 3: Sacred Geometry in Egyptian Myths and the Great Pyramid
Part 4: Messages from the Great Pyramid Builders Conceal Quantum Theory of Space-Time
Part 5: Hermetic Theory of Creation: Understanding Sacred Geometry and the Language of Archetypes
 
 

Three Keys to Hermeticism

Space, Time, and Mind serve as the cornerstone principles of Hermetic Philosophy. The piece titled Hermetic Theory of Creation: God is an Artist Who Crafted Platonic Solids from the Tapestry of Time delves into a Hermetic perspective of creation that emanates from Time. In contrast, this article embarks on a different path, proposing a Hermetic theory of creation rooted in pure Mind. Yet, these two theories interlink. The Mind initiates the creation of a foundational structure of unfolding polyads, which can subsequently be perceived as the underlying structure of Time. This presentation is a modern synthesis of philosophical theories of creation that spring from the Mind. The article’s distinctive contribution lies in elucidating the semantics of archetypal proto-forms and the significance of Sacred Geometry as it pertains to the evolution of the Universal Mind. The mathematical theory presented in this article explores the depth of the mystical phrases: In the beginning was the Word and I am the Alpha and the Omega, the first and the last, the beginning and the end

Synthesis of the Three Universal Mind Doctrines

Helena Petrovna Blavatsky (HPB) renowned for her seminal works “Isis Unveiled”  and “The Secret Doctrine,” as well as for co-founding the Theosophical Society, is said to have utilized an ancient tome replete with geometric symbols. Each symbol gracing the pages of this arcane book corresponded to a specific phase in the cosmogenesis orchestrated by the Universal Mind. The principal symbols include a simple circle ⵔ, a circle with a central point ☉, a circle bisected by a horizontal line ⴱ,  circles with various forms of a tau cross, a circle intersected by a cross ⴲ, and the swastika 卍. For practitioners adept in clairvoyance and meditation, contemplation of these symbols purportedly serves as a key to unlocking the Akashic Records, ethereal archives detailing various stages of universal creation. Consequently, these symbols are imbued with profound metaphysical significance.

In stark contrast to HPB’s approach based on understanding symbols and meditation to describe the Universal Mind, Georg Wilhelm Friedrich Hegel presented a rigorously rational and logical framework for grappling with the same concept, often termed the “Absolute Spirit” in his philosophy. In Hegel’s dialectical framework, the Universal Mind unfolds itself through a process of thesis-antithesis-synthesis, gradually manifesting its inherent rational structure within the realms of nature, human history, and individual consciousness. This teleological evolution, as Hegel saw it, moves toward increasing complexity and self-awareness, culminating in the full realization of the Universal Mind understanding itself, thus bringing the process of cosmic and historical development to completion. In this view, the physical universe, as well as human society and thought, are not merely creations but expressions and stages of the Universal Mind’s journey toward self-realization.

In the philosophical and mystical discourse on the Universal Mind, the sequence of natural numbers serves as a symbolic framework for understanding the evolution of cosmic consciousness. This notion finds its roots in the Pythagorean tradition, where numbers were considered the foundational elements of reality. Neo-Platonists like Plotinus further elaborated on this by associating numerical sequences with the emanations from the One, the ultimate unity. Jewish Kabbalists, with their structured Tree of Life featuring the ten “Sefirot,” also employed numerical principles to depict the flow of divine energy into the material world. 

In this article, we introduce a theory of the Universal Mind that integrates elements from the three key approaches previously mentioned: Sacred Geometry, Hegelian dialectics, and numerological sequences. While our framework retains the Hegelian concept of the Universal Mind as a self-reflective entity, we diverge from Hegel’s view of it as a philosopher’s mind. Instead, we adopt a more modern cognitive model. Contrary to Hegel’s model rooted in scholastic philosophical categories, our approach integrates the contemporary theory of metaphors, analogies, and the rigorous language of Abstract Geometry, enhancing its adaptability and strength. We will use the notion of interpretation and symbol as part of the theory.  

Unveiling the Universal Mind: Bridging Symbolism and Mathematical Structure

In civilizations like ancient Egypt, India, and Greece, philosophical discourse was deeply embedded in the language of analogies, metaphors, and symbols. HPB’s magnum opus, “The Secret Doctrine,” offers compelling evidence that these early cultures utilized a rich symbolic lexicon to decode both cosmic mysteries and the labyrinthine workings of the human mind. The Hermetic principle “As above, so below” encapsulates this perspective, suggesting that analogical thinking is not merely a rhetorical device but is woven into the cosmic fabric, bridging the gap between the macroscopic and microscopic realms, as well as the corporeal and the transcendental.

The dawn of deductive reasoning, especially as formalized by Aristotle, ushered in a paradigm shift, moving the intellectual focus towards empirical observation and mathematical rigor. Within this frame, analogies and metaphors were increasingly marginalized, deemed useful primarily as pedagogical tools or explanatory aids, but not as foundational elements for scientific postulates or laws.

However, modern cognitive research, notably the pioneering studies of George Lakoff and associates, has reinvigorated the conversation around the integral role of metaphor and analogy in shaping human thought. According to Lakoff, our conceptual landscape is profoundly metaphorical, influencing not just our language but our very perception and interaction with the world. Cognitive science, therefore, is restoring the primacy of metaphors and analogies, acknowledging their fundamental role in shaping both everyday cognition and scientific inquiry.

To construct a 21st-century framework for understanding the Universal Mind, it’s imperative to incorporate both these age-old and cutting-edge perspectives. Metaphors, analogies, and symbols serve as the cognitive scaffolding through which the Universal Mind engages with and constructs reality. This aligns with the ancient tenet “As above, so below,” positing the Principle of Analogies as an elemental force in shaping the universe’s interlinked, multi-layered architecture.

The success of mathematical modeling in capturing the complexities of nature is central to our modern understanding of the Universe. Far from being just a descriptive tool, mathematical language may reflect the Universal Mind’s operating principles. Therefore, integrating mathematics isn’t just an option—it’s a foundational necessity for grasping how the Universal Mind orchestrates complex systems and realities.

Inspiration for the New Theory

In Plato’s theory of Forms, each material object is a reflection of an ideal, non-material Form. However, the “Third Man Argument” presents a dilemma: if there exists an ideal Form representing an object, then there must also be another Form that encompasses both the object and its initial ideal Form. This requirement implies an infinite chain of Forms, creating a challenge to the logical coherence of Plato’s metaphysical system.

In modern mathematics, John von Neumann developed the concept of ordinals to formalize sequences. Using a simple production rule: {X} transforms into {X, {X}}, von Neumann ordinals can be generated. This is a recursive definition, starting with 0 as the empty set ∅. Using this production rule we can generate ordinals as follows:

 

Let us look at this sequence from a symbolic perspective. If we treat the curly braces {} as symbolizing the “idea of” or “reflection of,” each new ordinal incorporates all the previous “reflections” into a new set. The production rule {X} transforms into {X, {X}} says: reflect upon all that is, make it a separate object, and then add that object to what already is. The von Neumann ordinals were created in a specific context to solve specific technical problems of set theory. However, on a purely symbolic level, unrelated to the context of set theory, the sequence of von Newman ordinals looks like a representation of “Third Man Argument.” 

From our neo-Pythagorean and neo-Platonic viewpoint, which posits Universal Mind is guided by mathematical principles, von Neumann’s ordinals could offer a resolution to Plato’s “Third Man Argument.” Instead of considering the infinite series of Forms as a problem, it might be more appropriate to regard it as the foundational structure of the Platonic Universe. In this revised interpretation, each ordinal becomes an essential “node” in the infinitely expansive fabric of Forms. If numbers are the foundation of the universe in Pythagorean thought, then these ordinals could very well be the skeletal framework of a Platonic reality. Each ordinal encompasses and reflects the complexities of the preceding one, offering a mathematical underpinning that not only answers the issue of infinite regress but also enhances our understanding of a universe interconnected through Forms or ideas.

We aim to construct a model of the Universal Mind using the framework of Abstract Geometry. In this framework, the evolutionary trajectory of the Universal Mind is represented not just as an ordered sequence of natural numbers, but also as a progression of regular polygons converging towards a circle. We introduce the concept of interpretation to explain how shapes from Abstract Geometry can be understood as ideas we encounter in real life. With this notion of interpretation, the evolution of the Universal Mind, as represented by Sacred Geometry symbols, can be seen as an evolution of ideas, in a manner akin to Hegel’s Science of Logic.

Structures and Transformations

Structures offer a means to represent systems, scenes, or situations through the use of objects and predicates. These predicates can describe the properties of the objects or the relationships between them. For instance, the situation “George and Mary are married and have a child named Eliza” can be represented as:

< Mary, George, Eliza; Married(Mary, George), HasParent(Eliza, Mary), HasParent(Eliza,George) >

This structure could be represented as a graph:

Trinity

 

The meaning represented by this structure can be expressed in various ways. To enhance our formalism, we aim to introduce the ability to transform structures using transformation rules. For instance, a structure with the relation HasParent(X, Y) can be transformed into a structure with the relation HasChild(Y, X) using the rule:

<X, Y; HasParent(X, Y) ) > ~> <X, Y; HasChild(Y, X) ) >

Applying this rule to our structure yields a transformation:

< Mary, George, Eliza; Married(Mary, George), HasParent(Eliza, Mary), HasParent(Eliza,George) >

~>

< Mary, George, Eliza; Married(Mary, George), HasChild(Mary,Eliza), HasChild(George, Eliza) >

Abstract Geometry Elementary Forms and Notations

Abstract Geometry involves the representation of geometric objects through the use of structures. By the end of this article, we will have devised an interpretation of Abstract Geometry shapes as archetypes, serving as 21st-century Platonic forms.

We will work with several types of objects: points (P), line segments (L), equilateral triangles (T), squares (Sq), and circles (Cir). We will represent the relation of connectivity with the letter “C.”

For instance, consider a triangle constructed from three interconnected points a, b, and c. The representation of this triangle in the language of Abstract Geometry is as follows:

<a, b, c; P(a), P(b), P(c), C(a, b), C(b, c), C(c, a)>

triangle symmetries

 

A morphism is a mapping of objects within the structure that preserves the relational structure. The reflectional and rotational symmetries of a triangle align with the Abstract Geometry morphisms of the structure into itself. Such a morphism of an object into itself is termed an automorphism.

We will employ set notation. By using pattern variables like {X} followed by one or more predicates containing variable X, we can characterize a set of all ‘X’ meeting the predicate conditions. To account for situations where one set is contained within another or one shape is contained within another, we will introduce a specialized relation, “in”:

in({a, b}, {a, b, c})

in(a, {a, b, c}) 

With set notations, we can describe squares and triangles as follows: 

triangle ans square

<a, b, c; T({a, b, c})> 

<b, c, d, e; Sq({b, c, d, e})> 

<{X}; Sq({X})>

To describe circles and regular polygons that have a center, we will regard an object comprising one object “a” and set {X}. Such an object is portrayed as  (a, {X}), a, {X}, or {a, X}. Using this representation, we can discuss a pattern shared by both circles and regular polygons such as squares:

centroid

<(a, {X}); R(a, X)>

Depending on the cardinality of set {X}, this pattern could represent either a centered polygon or a circle. We will term this pattern an “abstract centroid” with the abstract radius R.

For an object (a, {X}) to be identified as a square, we will need to append another predicate:

<(a, {b, c, d, e}); Sq(a, {b, c, d, e})>

<(a, {X}); R(a, X), Sq((a, {X}))>

We have the flexibility to omit the relation R(a, X) and/or introduce details about the circle’s radius. Comparable notations can be employed for triangles, regular polygons, and circles that designate object “a” as their central point:

<(a, {X}); T(a, {X})> 

<(a, {X}); R(a, X), Cir((a, {X}))>

Using these notations, a square inscribed in a circle can be described as:

<(a, {X}), (a, {Y}); Sq(a, {X}), Cir(a, {Y}), in({X}, {Y})>

Monad, Polyad, and Centered Polyad 

In the realm of Abstract Geometry, Monads and Polyads serve as essential tools for representing the cardinality of structures. These structures can be considered as abstract forms derived from the shapes of Euclidean Geometry. Just as geometric shapes stand as manifestations of Abstract Geometry, Monads and Polyads act as cardinal representations – indicating the number of objects within a given structure.

A Monad (M) signifies the singularity of an object. While a point can be considered the quintessential representation of a monad, any object in abstract geometry with a unique identity can also be labeled as such. The notation for representing a point as a monad is as follows:

 <a; P(a)) > ~><a; M(a)) >

A Polyad, on the other hand, comprises a set of multiple distinct objects. We employ the predicate PN or P(N) to designate a polyad containing N objects. For instance, P4 represents a polyad consisting of four objects.

These Polyads and their assemblies can be likened to Pythagorean figurate numbers. Visualize a Monad as a singular dot, while a Polyad is a sequence of dots. A Pythagorean tetractys could be thought of as a composite of a Monad and three polyads: P2, P3, and P4.

Tetractys

Abstract Geometry shapes like triangles (T) and squares (Sq) correspond to polyads P3 and P4, respectively:

<  a, b, c ;  T({a, b, c})  > ~>< a, b, c; P3({a, b, c})) >


<  {X} ;  Sq({X}) >  ~> < {X}; P4({X})) >

A Centered Polyad (CPN or CP(N)) consists of a set of N objects focused around a central object. A square with a center is a centered polyad CP4.

< (a, {X});  Sq(a, {X}) > ~> < (a, {X});  CP4(a, {X})) >

< (a, {b, c, d, e});  Sq(a, {b, c, d, e}) >  ~> < (a, {b, c, d, e});  CP4(a, {b, c, d, e}) >

A Centered Polyad with a cardinality of N can be transformed into a regular polyad with a cardinality of N+1:

< (a, {b, c, d});  CP3(a, {b, c, d }) > ~> < a, b, c, d ;  P4({a,  b, c, d, e}) >

This transformation rule holds significant implications for the evolution of forms and can be generalized as:
< (a, {X});  CPN(a, {X }) > ~> < {a, X} ;  P(N+1)({a, X }) >

The Essence of Identity and Part-Whole Relations in Mathematical Structures 

Assigning identity to an object is a foundational concept in mathematics. This process encapsulates complex mathematical structures within a single symbol, making it analogous to reflection; much like reflection, identities double the objects they represent. The assignment and subsequent analysis of identities are crucial in constructing infinite sequences, akin to von Neumann ordinals. In this subsection, we will delineate a formal production rule central to our framework for generating infinite sequences.

Structures, such as a line or a triangle, can be assigned identities:

ln =  <a, b; P(a), P(b), C(a, b) >

tr=  <a, b, c; P(a), P(b), P(c), C(a, b), C(b, c), C(c, a) >

Since every object with an identity is a monad, assigning identities to objects implicitly introduces transformations of structures, as demonstrated below:

<a, b; P(a), P(b), C(a, b) > ~> <ln; M(ln)>

<a, b, c; P(a), P(b), P(c), C(a, b), C(b, c), C(c, a) > ~> <tr, M(tr) >

Importantly, the overall structure is not lost when transitioning from a structure to its identity. Commonly, we employ the part-whole relationship, denoted as ‘HasPart’ or ‘HP’ for brevity. To incorporate this fundamental aspect of the HP relation into our structural framework, we must also have transformations that shift objects describing parts to a structure that delineates the identity of the whole. Consequently, our monadic structures describing identity should evolve as follows:

<a, b; P(a), P(b), C(a, b) > ~> <ln, a, b; M(ln), HP(ln, a,), HP(ln, b)….>

<a, b, c; P(a), P(b), P(c), C(a, b),… > ~> <tr, a, b, c,  M(tr),  HP(tr, {a,b,c}),.. >

Observe that we transformed a structure into an abstract centroid, wherein the HP relation serves as an abstract radius of the centroid. Merely assigning an identity to a structure and applying common-sense semantics associated with part-whole relations culminate in a structure that is an abstract centroid.

Furthermore, consider that each structure is a representation of polyads. We can iterate the logical inference above at the level of polyads as follows:

 x= <{X},PN(X)> 

<{X}; PN(X)> ~> <x;M(x)>

<{X};PN(X)> ~> <{x, X},CPN(x, {X}), HP(x, X)>

Assigning an identity to a polyad consequently generates a monad and, subsequently, a centered polyad.

 

Infinite Sequence of Polyads

The transformation of forms, particularly polyad forms, is conceptualized using rules. Two pertinent rules are as follows:

  1. Assigning identity to a form results in a centered polyad with the same cardinality:

        x= <{X},PN(X)> 

        <{X}; PN(X)> ~> <{x, X}; CPN(x, {X}), HP(x, X)>

  1. A centered polyad with cardinality N can be represented as a polyad with cardinality N+1:

       < (a, {X});  CPN(a, {X }) > ~> <  {a, X}; P(N+1)({a, X }) >

It’s important to underscore that these rules describe a sequence of cognitive acts of the Universal Mind. The first is assigning an identity to an object, and the second is reconciling that identity with a part of the object. Applying these two rules in tandem generates an infinite sequence of expanding polyads and centered polyads. The HasPart (HP) relation is nonsensical when the original form is a monad; therefore, akin to the Pythagoreans, we commence our count from the smallest polyad with cardinality 2.
P2 ~> CP2

CP2 ~> P3

P3 ~> CP3

CP3~> P4 

This sequence mirrors the infinite self-reflection mentioned in the “third man argument” within the context of Abstract Geometry. Essentially, these two rules are an extended version of the von Neumann rule {X} transformed into {X, {X}}. Rather than reflection, we employ the semantically similar concept of assigning an identity to an object and subsequently expanding the object around its identity using the HasPart relation. This method strongly mirrors intellectual reflection.

Morphism of Structures and Hermetic Interpretation of Symbols

In this section, we delineate the novel Hermetic Magic, which transmutes a sequence of unfolding polyads into a series of evolving archetypes, serving as a contemporary iteration of Platonic forms. This transformative process is realized through what is termed a second-order morphism.

Hermetic Symbols 

The usage of symbols forms the bedrock of disciplines like mathematics, physics, and chemistry, yet the philosophical implications of this remarkable ability often go unexamined.  In Greco-Roman culture, we employ symbols that are more concise and compact than the objects they represent. Hermetic symbols in myths, dreams, and alchemical writings are camouflaged as real objects, thereby eluding our understanding. However, deciphering these enigmatic entities may not be as daunting as it seems. To interpret the symbols found in dreams and myths, we must examine the various qualities of objects as signs or symbols rather than focusing on the object itself. The logic employed in the interpretation of myths often mirrors that used in the interpretation of dreams.

Consider one of the most renowned dream interpretations: Joseph’s interpretation of Pharaoh’s dream from Genesis. In Pharaoh’s dream, he stands on the riverbank when seven fat cows emerge from the river, followed by seven lean cows that ate up the seven fat ones yet remain lean. Pharaoh then dreams again, this time seeing seven full and healthy ears of corn growing, followed by seven withered ears growing after them. The withered ears devour the good ears. Joseph interprets the two dreams as one. The seven fat cows and full ears symbolize prosperous years, and the seven lean cows and withered ears symbolize the ensuing years of famine.

Joseph’s interpretation of Pharaoh’s dream serves as a comprehensive guide to interpreting Hermetic symbols, making the entire process transparent. First, there are fundamental common cultural symbols. For instance, a river is a prevalent metaphor for the passage of time. Second, we transition from tangible objects to abstractions – the properties and relations of the objects must be interpreted as symbols. The seven cows must be interpreted as a polyad P7 – seven entities emerging from time. What can emerge from time? That could be weeks, days, or years. The fatness of the cows symbolizes prosperity or abundance, while leannes symbolizes poverty or scarcity. Eating symbolizes the consumption of resources. The art of interpretation lies in crafting a logically consistent narrative using the qualities of objects present in the scene (rather than the objects themselves) as symbols that narrate the story. 

Second Order Morphism of Structures 

The structures of Abstract Geometry aim to represent the shapes of Euclidean Geometry. Structures can be interpreted as geometric shapes. In a Hermetic context, the reverse is also true – geometric shapes represent ideas or semantic archetypes. With the aid of Abstract Geometry and Structures, we can construct a bridge between geometrical shapes and their most profound meanings.

In predicate logic, the abstract symbols of language are interpreted as objects and relations on structures. The interpretation is a mapping that associates symbols for objects and relations used in formal language with the objects and relations on the structures. Our version of interpretation is a second-order morphism. While a first-order morphism is a mapping of objects that preserves the relational structure, a second-order morphism allows the mapping of relational symbols while preserving the polyadic relations.

Structures and Abstract Geometry, along with second-order morphism, enable us to formalize the act of Hermetic interpretation. As an example, let’s consider Pharaoh’s dream as a collage of related structures:

<r, {X}; P7({X}),  River(r), ComeFrom(r, X ),  Cow(X) , Fat(X)>

<r, {Y};P7({Y}),  River(r),  ComeFrom(r, Y), Cow(Y) , Lean(Y)>

< {Y}, {X}; P7({Y}), P7({X}), Lean(Y), Fat(X), Eat({Y},{X}) >

Consider a second-order morphism that maps predicates as follows:

River ~> Time

Cow  ~> Year 

Fat ~> Good

Lean ~> Poor

Eat ~> ConsumeResourcesOf 

This morphism will transform our collage of structure as follows:

<r, {X}; P7({X}),  Time(r), ComeFrom(r, X), Year(X) , Good(X)>

<r, {Y}; P7({Y}), Time(r),  ComeFrom(r, Y), Years(Y) ,Poor(Y)>

< {Y}, {X}; P7({Y}), P7({X}), Poor(Y), Good(X),  ConsumeResourcesOf({Y},{X}) >

We have translated Pharaoh’s dream into its interpretation by Joseph using a second-degree morphism. Second-degree morphisms afford a vast degree of freedom of interpretation, and obviously, Joseph’s interpretation is not the only one possible. The presence of the second part of Pharaoh’s dream subtly hints at a second, more consequential interpretation of Hermetic parabola depicted as Pharaoh’s dream. Joseph’s interpretation serves merely as a Hermetic key that reveals the entire algorithm of Hermetic interpretation. 

We have uncovered a foundational mathematical structure that underpins the interpretation of dreams and Hermetic symbols. Second-order morphisms provide a considerable degree of freedom in interpreting Abstract Geometry forms, allowing these forms to transform into archetypes—generic constructs from which intricate meanings can be constructed.

This revelation opens up a compelling line of inquiry: Could the elemental building blocks of meaning be a product of a structured evolution of Abstract Geometry forms? The ensuing discussion will delve into this tantalizing notion, probing the possibility that our fundamental sense of meaning may indeed arise from a systematic evolution of geometric structures.

The Dynamics of the Universal Mind

Armed with the idea of the interpretation of form as archetypes, let us interpret the first stages of the Universal Mind’s unfolding.

Unity, Monad, and the Unfolding Mind

In the realm of the Universal Mind, the initial stage, often termed stage 0, is characterized by a pristine, undifferentiated consciousness symbolized by a simple circle ⵔ. At this point, there exists nothing other than the Universal Mind itself.

As the Mind embarks upon its first act of self-reflection, it gives rise to the identity of the singular object within its domain, which is itself—pure “I”. At this juncture, the Mind undergoes a transformation, transitioning from a stage of pure existence to a state of recognition, distinguishing itself as a unique entity. This identification doesn’t imbue the Mind with any attributes other than the fundamental notion of ‘oneness’ or ‘objectness.’ The Universal Mind thus recognizes itself as a distinct object, giving birth to the next stage of its evolution—stage 1. This phase is epitomized by the pure monad, which is symbolically represented as a circle with a central point ☉. 

Within our framework, the Mind at stage 1 is a Monad:

<I, M(I)> 

The first stage serves as an elemental archetype for the concept of “oneness,” functioning as a foundational reference point in our understanding of unity and singularity. Whenever the notion of oneness is invoked—whether in philosophical discourse, religious teachings, or contemplative practices—the archetypal imagery of this initial stage, symbolized by the simple circle  ☉, acts as the implicit or explicit referent. It embodies the quintessential essence of unity, providing a symbolic anchor for all subsequent discussions and explorations of what it means to be “one.”

 

Emergence of Duality: In the Beginning Was the Word

When a geometer assigns a name or identity to a geometric form, that act establishes a conceptual link between the form and the geometer’s own cognitive process. Now imagine if this geometer is not just any mind but the Universal Mind—often referred to as the Mind of God. In this context, the identity assigned to a form transcends mere labels; it becomes a Sacred Name that weaves the form into the fabric of divine cognition. The most primal of these identities is self-identity, encapsulated in the first utterance the Mind makes to signify its own essence:

W= <I, M(I)>

“W” is the first Divine Word. This is the moment of existential affirmation, a self-declaration akin to the Hegelian concept of Being, where the mind proclaims “I AM.” 

In the beginning was the Word, and the Word was with God, and the Word was ‘God.’ 

In general, the act of naming a structure introduces a polyad with a center, due to the inclusion of the ‘HasPart’ relation. However, the monad presents an exception, for it is a structure that lacks subdivisions or parts. To capture this unique property, we introduce a specialized naming relation, denoted by R, that connects a symbol with the structure it represents: R(W,I). This yields an Abstract Geometry structure with two objects.

<W, I;  R(W,I), P2({W,I})>

 

Duade

 

The Archetypal Nature of Duality

Within the Hermetic framework of second-order morphism, the flexibility to define the nature of relations is vast.  In our example, <W, I; R(W, I)>, the relation ‘R’ signifies reflection. Yet in Hermetic interpretation, it can be extrapolated to symbolize any binary relation between two objects. This flexibility allows us to explore the concept of duality as an archetypal paradigm fundamental to human cognition and cosmic order.

The notion of duality serves as a bedrock for our understanding of the world and ourselves. It pervades our categorization of reality: true or false, good or evil, black or white. Even our spatial orientation—left and right, up and down, forward and backward—exemplifies this dichotomy. These dualistic concepts not only help us navigate physical and conceptual spaces but also serve as metaphors that shape our worldviews and moral compasses.

In the Hermetic text, the Kybalion, the principle of duality is articulated with depth: “Everything is dual; everything has poles; everything has its pair of opposites; like and unlike are the same; opposites are identical in nature, but different in degree; extremes meet; all truths are but half-truths; all paradoxes may be reconciled.” By recognizing duality as a foundational archetype, we can better comprehend the complex interplay of forces that construct reality and foster a holistic approach to understanding the operations of the Universal Mind. 

Archetype of Trinity

The concept of duality, often seen as the tension between opposites finds its reconciliation and transcendence in the archetype of the Trinity—a concept that captures a richer, threefold complexity. Rather than seeing opposites as isolated polarities, ancient wisdom and philosophical traditions, such as Hegelian dialectics, present them as connected endpoints of a spectrum. In this spectrum, the third element emerges as a synthesis or resolution, embodying the evolutionary impulse of the Universal Mind. It’s not just about the clash of thesis and antithesis but the birth of a synthesis that transcends and includes both.

In our formal model, this transcendence from duality to trinity is described by the emergence of a centered polyad (CP2) from an original polyad (P2). The formal rule governing this transformation is:

<{X};PN(X)> ~> <{x, X}, CPN(x, {X}), HP(x, {X})>

Starting with a polyad representing duality, such as:

<W, I; R(W, I, P2({W,I}))>

Applying the rule yields a centered polyad:

<c, {W, I}; CP2N(x, {X}),  R(W, I), HP(c, {W, I})> 

This transformation marks the emergence of a synthesis between dual opposites, often symbolically depicted as the birth of a divine child, among other interpretations.

 The second rule for converting a centered polyad into a higher-order polyad is:

< (a, {X});  CPN(a, {X }) > ~> < {a, X} ;  P(N+1)({a, X }) >

When applied to our existing structure:

<c, {W, I}; CP2N(c, {W, I}),  R(W, I), HP(c, {W, I})>

It transforms into:

<{c, W, I}; P3( {c, W, I}),  R(W, I), HP(c, {W, I})>

Thus, we have mathematically generated an archetype of the Trinity:

<c, W, I;  R(W, I), HP(c, W ), HP(c,I)>

triade

The archetype of the Trinity is a profound and ubiquitous symbol, transcending cultural and religious boundaries, and embodying a unified whole expressed in three distinct aspects. Across various religious and philosophical traditions, the concept of the Trinity encapsulates the intricate interrelationships between the different facets of a singular entity. For example, in Christian theology, the Holy Trinity—comprising the Father, the Son, and the Holy Spirit—is each distinct yet fundamentally interconnected. Similarly, in Hinduism, the triumvirate of Brahma, Vishnu, and Shiva represents the cyclical processes of creation, preservation, and destruction, respectively. Ancient Egyptian deities also often appear in trinities, such as the triad of Osiris, Isis, and Horus, representing father, mother, and son, and echoing the patterns observed in nature.

As explained in the segment concerning ‘Symbols as Representations of Forms and Polyads,’ the interpretation of this structure through second-order morphisms opens the door to multiple layers of meaning. Utilizing second-order morphism as a tool for infusing semantic value transforms these archetypes into objects and situations.  For example, structure describing Mary, George and Eliza could be obtained from our polyad and a second-order morphism. We can map symbols as follows:

I ~> George 

W ~> Mary 

C ~> Eliza

R ~> Married  

HP ~> HasParent

This second-order morphism would transform the structure of Trinity into a structure describing Mary, George, and Eliza  as follows:

Mapping

<c, W, I;  R(W, I), HP(c, W ), HP(c,I)> ~>

< Eliza, Mary, George; Married(Mary, George), HasParent(Eliza, Mary), HasParent(Eliza,George) >

This transformation shows how an unfolding sequence of polyads can serve as the most basic building blocks of meaning, from which more concrete structures in the world emerge. We are constructing a 21st-century Hegelian Dialectic, where the process of the Universal Mind’s unfolding is governed by mathematical rules. The semantics of these unfolding forms are described by second-order morphisms of structure, a concept that mirrors the formal definitions of interpretation used in symbolic logic.

The Trajectory of Universal Mind Evolution

In our framework, the evolution of the Universal Mind is portrayed through a sequence of Abstract Geometry forms—specifically polyads and centered polyads, delineated as P1, P2, CP2, P3, CP3, and P4. When we isolate these sequences into distinct categories, they correspond to two seminal Hermetic symbols that offer profound insights into the evolution of the Universal Mind.

The sequence of polyads, when visualized, forms an expanding triangle, illustrating a progressive unfolding.  

triangular numbers

On the other hand, the sequence of centered polyads manifests as a series of regular polygons that gradually converge toward a perfect circle. 

polygons

These geometric transformations are more than just shapes; they are potent symbols deeply embedded in the Hermetic interpretations of Sacred Geometry. Unfolding polygons provides significant insights. The first insight centers on the increasing number of symmetries, a phenomenon we have labeled ‘Harmony’. The circle, embodying an infinite number of symmetries, is thus a natural symbol of perfection. The second insight concerns the cyclical journey of the Mind’s evolution, ending up astonishingly close to its starting point. This observation resonates with the mystical assertion, I am the Alpha and the Omega, the first and the last, the beginning and the end. Interpreted through this geometric lens, the statement takes the form of a circle with a center ☉ – also the symbol of the Sun – representing both the origin and the ultimate state of perfection. This interpretation not only underscores the foundational principles of Hermetic thought but also provides a visual representation that deepens our understanding of ancient wisdom in the modern world.

Within these geometric constructs, the center of each polygon represents the evolving identity of the form it embodies. As the polygons evolve, converging toward a circle, the center signifies an identity on a journey toward perfection. The radius of each polygon, and ultimately of the circle, serves as a metaphorical link between this evolving identity and the elements that make up the whole. This radius effectively encapsulates the dynamic relationship between identity and the components that collectively contribute to its ultimate form.

This research establishes a foundation for understanding Time as an archetypal structure, evolving due to the conscious progression of the Universal Mind reflecting upon its inherent essence. Nonetheless, a detailed exploration of this concept merits its own dedicated article.

Conclusion

In this exploration, we have constructed a model of the Universal Mind grounded in the theoretical framework of Abstract Geometry. Leveraging the concept of second-order morphism as a mechanism for semantic interpretation, we’ve built a crucial bridge between geometric shapes and the Abstract Geometry archetypes that emerge as evolutionary processes unfold, as well as structures that symbolize real-life situations.

Our focus on natural numbers as the cornerstone of this evolutionary framework aligns our theory with various mystical traditions like Theosophy, Neo-Platonism, and Kabbalism, all of which propose theories of emanation. However, our model distinguishes itself by elucidating numerical evolution as a rational process of infinite self-reflection, drawing parallels with Hegelian dialectics. 

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