Posts Tagged ‘science of alchemy’

About Alchemy and Logic

Thursday, April 28th, 2016


Rebis, Res Bina, Double Matter

Res Bina (REBIS), Double Matter

This blog is about Logic, Alchemy and the relationship between Logic and Alchemy.

Classic Logic is based on the Square of Opposition of Aristotle. This square was discussed for more than 2000 years. Recently scientists detected that the Square is really a Cube. It misses an extra dimension.

The Square describes  the six oppositions (“dual combinations”) of possibility (“some”, Particular Affirmative), necessity (“all”, Universal Affirmative), not-possibility (“not some”, Particular Negative) and not-necessity (“not all”, Universal Negative).

Geometric Logic is a part of Modal Logic which is a part of Logic.

In Geometric Logic the Linguistic Representations of Logic are mapped to the Graphical Representations of Geometry.

Modal Logic works with the notion that propositions can be mapped to sets of possible worlds.

The idea of possible worlds is attributed to the famous mathematician Gottfried Leibniz (1646-1716), who spoke of possible worlds as ideas in the mind of God and argued that our actually created world must be the best of all possible worlds. Possible worlds play an important role in the explanation of Quantum Mechanics.

Alchemy is based on the Square of the Four Elements. The Four Elements are oppositions. The oppositions of Alchemy are related to the oppositions of the Square of Opposition of Aristotle in an orthogonal model. They form an arc of 90 degrees, which makes Alchemy the complex part (imaginary) of Logic.

In his work De Arte Combinatoria Leibniz developed a general theory of science that was based on a fusion between Alchemy and Logic.

The four Elements of Leibniz

The Square of Leibniz

In this blog I will explain Alchemy and Logic and show that Alchemy and Geometric Logic share the same geometry, the Hexagon (the Seal of Solomon), which is a 2D-mapping of the Cube of Space of the Sefer Yetsirah.

Leibniz calculating machine

Drawing of Leibniz’s calculating machine, featured as a folding plate in Miscellanea Berolensia ad incrementum scientiarum (1710), the volume in which he first describes his invention.

About Alchemy

When thou hast made the quadrangle round, 
Then is all the secret found. 
George Ripley (d. 1490).

One of the most complicated ancient architectures is the architecture behind Alchemy. The architecture of Alchemy is full of strange concepts and allegories.

Alchemy comes from the Arabic al-khemia meaning the Black Soil of Egypt (or the ((Egyptian) Black Art). Alchemy is the art of transformation.

Alchemy is an ancient science that was practiced all over the world. The most famous teacher was called Hermes Trismegistus in Greece and Thoth in Egypt. The old science of Hermes came back in the Renaissance.

The essence of Alchemy is called REBIS (Res Bina), Double Matter. REBIS is the end product of the alchemical magnum opus or Great Work. It is the Fusion of Spirit and Matter (the Body). Spirit is the Creative Part of the Human.

Double matter is sometimes described as the divine hermaphrodite, a being of both male and female qualities as indicated by the two heads within a single body.

The Great Work is the chemical and personal quest to create Double Matter, the Philosopher’s Stone, Spirit in Matter.

The Stone is the agent of chemical transmutation, and the key ingredient in the creation of the elixir of life, said to heal all diseases, induce longevity and even immortality.

In current Physics  transmutation is proved at room temperature in a proces called Cold Fusion.

In Physics one explanation for the fusion between Quantum Mechanics and Gravity is called the Wheeler Feynman Absorber Theory. In this theory the Future causes the Past and vice-versa. We are always in the Now, the Middle between Past & Future.

The Philosopher Stone

The Philosopher Stone

The aim of the Great Work is a fusion with the One (the Blazing Star, Saturn, Point) that is divided in the Two (Sun & Moon, Male & Female, Line) and the Four Planets (Square), the 12 (4×4-4) Constellations or more structures based on a power of a power (of a power….)  of 2. The Great Work moves from 2**N, …., Sixteen to Four to Two to One (or from one to two to four) in a Spiraling Spiral Motion.

The Spiraling Spiral in Chartres Cathedral

The Spiraling Spiral in Chartres Cathedral

The One, to which the elements must be reduced, is a little circle in the center of the squared figure (the cross). The cross is the mediator, making peace between the opposition of the planets.

The Spiral of the Great Work

The Spiral of the Great Work

The winged dragon represents First Matter (Quintessence, The Fifth Element) and suggests ascension, a merging of matter (body) and spirit (Creativity). Creativity (Enthusiasm, Spirit, Inspiration, Imagination) is the engine behind the Spiral Wheel.

The First Matter is the primordial chaos comparable to what we now call the vacuum, the state of lowest energy. This state contains all possibilities.

It can be looked on as an unorganized state of energy  that is the same for all substances and exists in an invisible state between energy and matter.

The First Matter is the One, the Point in the Middle of the Circle. In current physics a point is called a singularity or a black hole.

Alchemy is about Recombination

Alchemy is about re-combination summarized in the Latin expression Solve (break down in separate elementset Coagula (coming back together (coagulating) in a new, higher form).

Solve and Coagula

Solvite Corpora et Coagulata Spiritus (“dissolve the body and coagulate the spirit”).

Earth (Body) becomes Water (Spirit) by dissolving it in some solvent. Water becomes Air (a moist vapor) by boiling, which further heating turns into Fire (a dry vapor).

Finally Fire becomes Earth by allowing the vapors to condense on a solid material (a stone).

The circulation may repeat if it is done in a reflux condenser, such as the Pelican or kerotakis.



The Pelican

The Pelican feeds its children with blood from the Heart. It is a symbol of the Upper Cycle of the Mind.

According to the early alchemists, the four elements—fire, water, air, and earth—come into existence via the combination of specific qualities, recognized as hot, cold, wet, and dry, being impressed on to the Prime Matter. For example, when hot and dry are impressed on the Prime Matter, we have fire.

(Hot and Cold) and (Wet and Dry) are oppositions. Their fusion is the empty set.

(Hot or Cold) and (Wet or Dry) can be combined -> (Hot, Wet), (Hot, Dry), etc.

The Cold allows substances to get together. The Hot is the power of separation.

Wet things tend to be Flexible (Fluent, Movement, Self, Not-Agency) whereas Dry things are fixed and structured (Agency, Resistance, not-Movement).

When the qualities are changed, the elements themselves are changed. Adding Water to Fire substituting wet for dry; hot and dry becomes hot and wet: steam, or Air.As you can see in de picture below Hot & Cold and Wet and Dry are Binary Opposites (“lines”) that are voided by the central cross, the point of the Empty Set, The Void (A and not-A -> Empty).

Hot & Dry = Fire, Hot & Wet = Air, Cold & Wet = Water and Cold & Dry = Earth.

There are four types of Trinities the Passion-trinity (Dry, Wet, Warm), the Structure-trinity (Dry, Wet, Cold), the Resistance-trinity  (Warm, Cold, Dry) and the Movement-trinity (Warm, Cold, Wet).

The Square rotates. The wheel of the Square is driven by the four qualities. Wet on the rising side, Hot on the top, Dry on the descending side, and Cold on the bottom.

The start is powerless (Cold). By becoming more flexible (Wet) success and power comes (Heat).  At the top Rigidity (Dry) undermines the system and it falls back to a powerless state.

The Four Triangles

The Four Elements arise out of the Quintessence, The Fifth Element, The Symbol of The Heart Chakra, the Seal of Solomon

The four Elements all share the Triangle of the Holy Trinity. Every Trinity is seen from another perspective.

The two triangels, The Up-triangel and the Down-Triangel, are Opposites that are created out of two triangels that contain two Opposites that are fused. Humans are opposites in opposition.

Humans are FourFold

Humans are part of the Bilateria,  animals with bilateral symmetry.

Humans can be described as a fusion of two mirrored bilateral triangles, the Up-triangel related to the Top of the Body (Mind) and the Bottom-triangel of the lower Body both connected by the Heart. The two triangels and the separate triangels have to be balanced.

The principle of Balancing the two Triangles of Top & Bottom of the Body is represented by the Egyptian Goddess Ma’at who weights the Heart (the connector of Up and Down) of the Dead with her Feather (Nothing).

The Balance of Ma'at

The Balance of Ma'at

As you can see the Square of the Cycle of the Four Elements is visible in the picture below. The total picture shows a Hexagram with a Center (The Cross).

The points that are missing in the Square are the top- and the bottom of the two triangels which are the Whole (Up, Heaven) and the not-Whole, Emptiness, the Bottom (Hell).

The Seal of Solomon

The Seal of Solomon

A hexagon is a 2D-projection of a Cube.

The hexagon of Solomon can be transformed into the Cube of Space of the Sefer Yetsirah.

Cube of Space

Cube of Space

The Cube of Space can be transformed into the Tree of Life.

The Tree of Life also called the tree of knowledge or world tree  connects heaven and underworld and all forms of creation and is portrayed in various religions and philosophies all over the world.

The Tree of Life shows the levels/scales (three) and possible paths of the Opus Magnum, the alchemical journey.

The Cube of Space can be transformed into the Tree of Life

The Cube of Space can be transformed into the Tree of Life.

The Spiral of Alchemy is about the Chemical Transformation of the Body by the Conceptual Transformation of the Mind.

About Truths for a Fact and Truths for a Reason

People have always believed in the fundamental character of binary oppositions like Hot & Cold and Wet & Dry are used in Alchemy.

In this part we move to the Field of Logic. In this case Hot/Cold and Wet/Dry transform into Necessary /Not-Necessary and Possible/not-Possible.

Chemical Wedding (Union of Opposites)

Chemical Wedding (Marriage of Opposites)

Oppositions can be divided in:

  1. Digital oppositions, contradictories, contain mutually exclusive terms (Gender (Male/Female)).
  2. Analogue oppositions, antonyms  or contraries: contain terms that are ordered on the same dimension (Temperature (Minimum->Cold/maximum->Hot)).
Square of Opposition

Square of Opposition

The doctrine of the square of opposition originated with Aristotle in the fourth century BC and has occurred in logic texts ever since.

The logical square is the result of two questions: Can two things be false together? Can two things be true together?

This gives 4 possibilities: no-no: contradiction (A/O, E/I); no-yes: subcontrariety (I/O); yes-no: contrariety (A/E); yes-yes: subalternation (A/I, E/O) better known as implication (A->B).

The four corners of the square represent the four basic forms of propositions recognized in classical logic:

  1. A propositions, or universal affirmatives take the form: All S are P.
  2. E propositions, or universal negations take the form: No S are P.
  3. I propositions, or particular affirmatives take the form: Some S are P.
  4. O propositions, or particular negations take the form: Some S are not P.

The square of opposition was debated for many reasons for more than two thousand years.

One of the discussions is about the difference between possible and necessary. In the so called Master Argument Diodorus proved that the future is as certain and defined as the past. The essence of logic (necessaty) implies the non-existence of freedom (possibility). X is possible if and only if X is necessary.

The term ‘possible’, in Aristotle’s view, is ambiguous. It has two senses known as one-sided possibility and two-sided possibility (or contingency). Being two-sided possible means being neither impossible nor necessary, and being one-sided possible simply means being not impossible.

Leibniz distinguished between necessary Truths for a Reason, which are true for a reason—i.e, their opposite is a contradiction—and contingent truths, (Truth for a Fact) such as the fact that the president of France is François Hollande. A contingent truth cannot be proved logically or mathematically; it is accidental, or historical (based on facts (events)).

About Geometric Logic and n-dimensional “Squares” of Opposition 

Modal Logic is about the fusion of necessity and possibility, contingency.

Logical Hexagon

Logical Hexagon

In Modal Logic the propositions are modelled in a logical hexagon where:

  1. A is interpreted as necessity: the two propositions must be either simultaneously true or simultaneously false. A is a Proof or Law.
  2. E is interpreted as impossibility. E is an Observation that contradicts the Proof of A.
  3. I is interpreted as possibility:  the truth of the propositions depends on the system of logic being considered . I is an Idea implied by the Proof that Contradicts E.
  4. is interpreted as ‘not necessarily’. O contradicts A.
  5. U is interpreted as non-contingency: neither logically necessary nor logically impossible, its truth or falsity can be established only by sensory observation.
  6. is interpreted as contingency: propositions that are neither true under every possible valuation (i.e. tautologies) nor false under every possible valuation (i.e. contradictions). Their truth depends on the truth of the facts that are part of the proposition. Y is a possible proven theory.

The Logical Hexagon has the same geometry as the Seal of Solomon. It contains the Square of Opposition (A,E,I,O).

The Logical Hexagon can be transformed just like the Seal of Solomon to the Cube of Space now called The Cube of Opposition.

The Logical Hexagon is used in many fields of Sciences like Musical Theory or Scientific Discovery.

The Hexagon of Opposition based on Musical Theory

The Hexagon of Opposition is used in Musical Theory

Every opposite can be defined by a string of zero’s and one’s like 1010. An opposite is 0101. This means that there are 2**4/2 = 4×4/2= 16 / 2 = 8 possible opposites. This is called the Octagon of Opposites.

The Octagon contains six reachable and two unreachable points in the Center (0000, 1111), the Empty Set (the Void, the Hole, Contradiction) or its Opposite (The Whole, Tautology).

Logical Geometry

Logical Geometry

The rhombic dodecahedron RDH with the standard bitstring decoration

Octagon of Opposition

It is clear that it must be possible to create higher dimensional models based on n-based-logics.

When n becomes very big the geometry will tend to a n-Dim circle in which every point is in opposition to the point at the other side.

The big problem with n-based logics is language. We don’t have the names to articulate the many grades of opposition that are possible.

The other problem is that a negation is not a symmetric operation (A = not-not-A) but a rotation with an angle 360/n.

The n-opposite-geometries are highly related to Simple Non-Abelian Groups (SNAGs). They play an important role in biology.

The last step is to transform the static n-opposite geometries by making n very large (infinite) to a dynamic “opposition field”.

Related Models

In current Psychology Hot & Cold are called Communion and Moist & Dry, Agency. Together they create the so called Interpersonal Circumplex.

Interpersonal Circumplex

The four stages of Learning by Jean Piaget.

The mathematical model behind the theory of Piaget is called the Klein Four Group or Identity/Negation/Reciprocity/Correlation-model.


INRC-model of Piaget

INRC-model of Piaget

In the Science of Ecology (Panarchy) Agency & Communion are called Connectedness and Potential. The Panarchy model looks like a Mobius Ring.

panarchy model

panarchy model

A model that describes the Four Perspectives on Security:


The Semiotic Square of Greimas:

Semiotic Square

Semiotic Square

The Chinese Sheng-Cycle: The Chinese Five Element Model is comparable to the Western Four Element model. It contains as a Fifth Element, the Whole-Part-relationship (Earth, Observer State). Emptiness (Nothing, the Tao) is represented by the recursive Pentangle inside.

The Sheng Cycle

Tetralemma (Ancient Indian Logic)

Tetralemma (Ancient Indian Logic)

The lesson

When thou hast made the quadrangle round, then is all the secret found“.

The secret: It is impossible to move though the center because the paths that go through the center are a contradiction or a tautology (I am what I am, JHWH).

We have to move with the Cycle (making the rectangle round (a circle)), around the Singularity (the Hole of the Whole), going Up and going Down all the time.

The only way to solve this problem is to join the opposites by accepting only one thruth-value, by:

  1. Accepting everything (Wu Wei),
  2. Denying everything (Stoics) or
  3. Diminishing the amount of dimensions (“becoming Simple”, “like Children”) by moving up in abstraction (Hot & Cold -> Temperature -> Everything is Energy) to the level of zero dimensions, a point, becoming One with the First Matter (Tao, Vacuum, The Kingdom).

“If we then become children, would we thus enter the kingdom?” Jesus said unto them, “When ye make the two one, and when you make the inside like unto the outside and the outside like unto the inside, and that which is above like unto that which is below, and when ye make the male and the female one and the same, so that the male no longer be male nor the female female; then will ye enter into the kingdom.” (Gospel of Thomas, Logion 22).


About the Geometry of Negation

About the Logic of Music

About Wu Wei

About Modal Logic

About the Cube of Opposition

About Logical Geometry

About the Square of Religion

Semiotics for Beginners

About Holonomic Alchemy

About Paths of Change

About the Sheng Cycle

How to use the Square of Opposition as a Research Tool

About the Cube of Space of  Sefer Yetzirah

About Truth of Reason and Truth of Fact

Why Humans are Toolmakers

About Chemical Transformation (Cold Fusion)

About the Ars Generalis Ultima of Ramon Lull (1305)

About the Six Domains of  the Polynomic System of Value

About the Logic of Creation

About Leibniz Calculating Machine

About Zero Dimension 

Why Innovation is Re-Combination

How to Balance the Seal of Solomon

About Interpersonal Theory 

About Panarchy

About the Hexagon of Opposition

About n-based logics

About Flow-Systems

Anti-Fragility and the Square of Opposites

An Introduction to Hexagonal Geometry

About Alchemy

Wednesday, December 10th, 2008

alchemyWhen I studied Chemistry at the University of Leiden I wanted to learn more about “Ancient Chemistry“.

At that time I really did not understand the Symbolism behind Alchemy. This changed a few years later when I encountered Jung. Suddenly Alchemy moved to the Social Sciences.

At this moment I am convinced modern Mathematics and Physics are exploring the same issue Jung and many others Alchemists were exploring.

The Material World and the Spiritual World are both based on the same principles. They only operate on different Levels.

When you understand one level it is not so difficult to map this level to another level. The real big problem is to map the simplicity of the parts to the complexity of the Whole.

Human understanding is based on our Imagination. When you are not capable to visualize the Parts and the Whole you are lost. The old Alchemist understood this problem and they produced many visual metaphors to help the explorer of the Spiritual Domain.

ouroborosThe most important unifying concept in Alchemy is represented by the Snake Ouroboros. The Snake is “Eating Itself“.

When you try to imagine what is happening you will understand that the Snake will be spiraling inwards until it simply vanishes.

You also could imagine that the Snake appears again in another “space” where it starts to spiral outwards until it moves back to our reality.

By “eating itself” the Snake is “compressed” into what is now called a “singularity” (A Black Hole).

klein bottleIn the other (Imaginary) world the Snake expands and spirals through another hole (A White Hole) back into Our Reality.

The Imaginary Universe is the Twin Universe of Our Universe.

If the Imaginary Universe is represented by the Complex or Imaginary Numbers there is also an In-Between or Joint Universe.

This Universe existst in a higher dimension. We are able to experience the Real Part of the Numbers. Our Twins only see the Imaginary parts.

The aim of the process of Alchemy is to move through the singularity to another Space/Time. In this Space/Time you are “immortal”. This Space/Time is the Imaginary World of Mathematics, the Complex Plane.

holy weddingTo Reach the World of Immortality You have to Unite the Opposites. Alchemy uses many Metaphors to show how this is accomplished. One of them is the Holy Wedding. In the Holy Wedding the Male and the Female Unite.

It is very important to understand what the Alchemists meant with the Male and the Female. Of course the first interpretation is simply the Human Male and the Human Female.

In this case the Holy Wedding is a sexual metaphor. The Male and the Female Unite when they have Sexual Intercourse.

In Sacred Sexual Intercourse, Tantra, something called the Kundalini, the Serpent Power, is activated.

Awakening the Kundalini without a good Preperation (Psychological Cleaning) and a Good Guidance is very dangerous.

The Rising of the Kundalini will generate tremendous energies (Extreme Heating) and intense Psychological Disturbances, the Kundalini-Syndrome.

Although Sacred Sexual Intercourse is a very powerful tool for Enlightment it is still not the Right Metaphor.

The Male and the Female have to be seen as complementary functions. The Male represents Order/Control and the Female Chaos/Desire. The complementary functions are independent of the sexes. A Female Human is perfectly able to act with a (Dominant) Male Pattern.

Uniting Male and Female is accomplished when you Live on the Edge (in the Middle) of Control/Order and Chaos/Desire (Wu-Wei). When Chaos and Order are in Balance the Union of Both Forces, The Void (also called the Tao), takes over and Synchronicity starts to Happen. The Human Action is Guided by a Higher Force, the Holy Spirit.

pentagleThe Human can also be viewed as a FiveFold Being (The Pentagram). In this case the Human is a combination of Male/Male (Control), Female/Female (Desire) AND Male/Female and Female/Male.

At the Center (In the Middle, At the Cross) of the Four Combinations is always the One, The Void. Most of the time the “new” combinations are called Soul and Spirit.

To Understand this process a little bit better we will change the Metaphor of the Sexes to the Metaphor of the Movements (Up & Down, Inside & Outside). The Opposite Movements are a Tension, that is Voided in the Middle.

Inside Outside
Up Up/Inside (Spirit) Up/Outside (Control)
Down Down/Inside (Soul) Down/Outside (Desire)

When there are Four Movements in Action, there are Six Ways to Integrate the Two Opposites and there are many more ways to combine the Six Ways into a Four Fold Integration. One of the most effectives ways to do this is to travel the Matrix in a Circular Motion(○) With or Against the Clock or by Circulating the Cross following the Pattern of the Moebius Ring (∞) also With or Against the Clock.

To move from the Mortal to the Immortal, Timeless, Domain we have to apply the Ouroboros-pattern. This is accomplished by connecting the Circulations of the Circle or the Moebius Ring at the Singularity, The Middle of the Cross. The circulations move against or with the Clock.

We are now moving from a 2-dimensional Matrix to a 3-dimensional Cube. To avoid confusion we have to define a third terminology for the two opposite Movements. In many cultures the directions of the Winds are used (East, West, North, South) combined with Up and Down but it is up to You to find other Terms.

It is still possible to visualize this structure in 3-dimensional space. This structure is called by the Topologists, The Klein Botlle. The Klein Bottle is a connection of two complementary Moebius Rings with the Void in the Middle.

There are of course other metaphors available to explain what is “happening”. The most interesting ia a Physical Metaphor. It is called the Holographic Universe.

In the Holographic Universe the Hologram is used as a methaphor. A Hologram is a 3-dimensional picture stored in a rotating 2-dimensional structure. A “grain of sand” could contain the complete image of the Universe.

In the Holographic Universe Time is not Linear or Cyclic. The Now expands into the Past and the Future. We are able to change the Past and the Future with our Imagination. Time Travel is possible if we are able to find the same point in the Multi-Universe we started our Travel. The Humans also exists in many Space/Times at the same Moment, The Now.

We are able to see with our (Higher) Conscioussnes the Complete Network of Time Loops that are enfolding in the weird structure that has no Inside and Outside.

Many people still live in the Hierarchical Word, where the Movements are always going Up. They believe in a beginning and an end and believe that every Cause has an Effect. Some people, The Holists (New Age), moved from the Outside to the Inside. To them everything is Energy and Reality is a Circle. They are the people of Soul and Spirit. The Holists believe in the Wheel of Karma or the Wheel of Destiny.

The new Age of Aquarius will be a completely different world. We, the Humans, are not moving around in the Merry Go Round that is created by the Gods. We will be the Co-Creators of the Creative Force that is in the Center of the Wheel.

We are the Whole AND the Part. We are the Cycle AND we are Not the Cycle. We will take part in a new enfolding of the Self-Reference, that is Whole and Empty.

Hierarchical Cyclic Holographic
Movements are always going Up Movements go Up and Down Self-Reference
Triangle/Pyramid/Cube/Tube Circle/Torus Moebius Ring/Klein Bottle
Causal/Linear Cyclic/Spiraling Closed in Itself
Creation Compression/Expansion Enfolding
Evolution Growth and Decay Continuous Creation
Outside Inside and Outside No Inside and Outside
Part/Whole Wave Fractal
Everything is Different Everything is the Same (Make a) Difference
Individual Collective Whole
Control/Desire Soul/ Spirit Center



About the Kundalini Syndrome

About the Klein Bottle and Alchemy

About our Imaginary Twin Universe

About the Relationship between Alchemy, Tantra and The Gnostics

About the Holy Wedding

All You Want to Know About Alchemy

About Alchemy

The Gnostic Science of Alchemy

Dimension of Apeiron by Steven M. Rosen