Archive for the ‘alchemy’ Category


March 29, 2018

What happens when the fourfold of Noether’s Theorem is spliced together with the fourfold of Pauli-Jung? Both have Space-Time and Matter-Energy. The former has Conservation and Symmetry, and the latter has Causality and Synchronicity.

I had to remind myself that Conservation means consistency (of matter or energy) through time (and space), and Symmetry means consistency (of form) through space (and time), so in some sense they are dualistic.

Combined, one has the three axes of dual concepts, represented above.

Further Reading:



The Six Elements

March 26, 2018

What happens if one combines the four classical Greek (Western) elements with the five classical Chinese (Eastern) elements? Air is in the former and not the latter. Wood and metal are in the latter and not the former.

I’ve left out the Aether (the Fifth Element) of the West because I follow Empedocles, and I see that the five classical Japanese elements have those four and Void as well. But, I’m just looking at material and tangible elements.

One might consider Wood and Metal to be derivatives of Earth, but let’s consider them to be distinct for this exercise. So, maybe six elements!

Further Reading:



The Marriage of Opposites, Part 3

March 2, 2018

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.

— From The Kybalion by The Three Initiates

There are trivial truths and the great truths. The opposite of a trivial truth is plainly false. The opposite of a great truth is also true.

— Niels Bohr

I have mentioned the alchemical notion of the “Marriage of Opposites” several times (here and here). When opposites marry, what happens as a result? Do they cancel one another out, leaving just a boring average as result? Do they explode in a fiery conflagration, like matter and anti-matter releasing energy? Or do they create a new thing, something that is greater than the sum of their parts?

If opposites annihilate each other, what is the result, emptiness or a void? It is often said that nature abhors a vacuum (“horror vacui”), but I think it is far more true that the mind does. In dualistic thinking, everything that is not one thing must be its opposite. Not good is bad, not happy is sad, not black is white.

In classical logic, the Law of the Excluded Middle says that for any proposition “p”, either it is true or its negation “not p” is true. Thus, “p or not p” is necessarily true, a tautology. Similarly, their combination “p and not p”, cannot ever be true, and so is necessarily false. If one can assume “not p” and derive a contradiction, then “p” must be true (reductio ad absurdum).

In intuitionistic logic, one cannot deduce “p” simply from the falsity of “not p” (that is, “not not p”), one must actually prove that “p” is true. So “p or not p” may still be uncertain, if we don’t know how to prove “p”. However, “p and not p” is still false, based on the falsity of “not p”.

In the viewpoint of Dialetheism, it is offered that there are truths whose opposites are also true, called “true contradictions”. Dialetheisms cannot exist in formal logics because if “p and not p” is true, then you can deduce anything you want and your logic breaks down. Nonetheless, much thought through the years has been dedicated to dialetheisms and their ilk. Please see the recent work by philosopher Graham Priest.

When one considers something and its opposite at the same time, how can you reach an agreement between them? In magnetism, opposite charges attract and like charges repel. All too often, opposite viewpoints vigorously repel each other instead of reaching a happy medium. Each viewpoint considers the other “false” and so they push away at each other, instead of meeting halfway in compromise.

If there is empirical evidence supporting one viewpoint and not the other, and both parties can agree to it, then problem solved. But if viewpoints are more like ideologies, and one side shows evidence that the other side dismisses, what then? Are we only left to agree to disagree? That doesn’t seem like a long term solution.

In this blog I have insinuated but not stated explicitly that a marriage of opposites can often be achieved by combining it with another pair of opposites. Rather than meeting in the middle to a void or an annihilation, one can reach the other side by “going around” the danger, by way of intermediates. Much like Winter reaches Summer by passing through Spring and Summer reaches Winter via Fall, this type of structure is found everywhere in human thinking.

In fact, many systems of pluralistic philosophies are built on fourfolds instead of dualities. For example, see the work of Richard McKeon, specifically this paper.

Further Reading:



Seven Sermons to the Dead, P2

February 14, 2018

Here with inadequate description is another fourfold of entities from Seven Sermons to the Dead.

  • The Pleroma: The spiritual universe as the abode of gods and of the totality of the divine powers and emanations.
  • The Creatura: The living world, subject to perceptual difference, distinction, and information
  • Abraxas: The supreme power of being transcending all divinities and demons and uniting all opposites into one
  • Philemon: Jung’s spiritual guide, the narrator

Further Reading:


Seven Sermons to the Dead

February 11, 2018

Four is the number of the principal gods, as four is the number of the world’s measurements.

One is the beginning, the god-sun.

Two is Eros; for he bindeth twain together and outspreadeth himself in brightness.

Three is the Tree of Life, for it filleth space with bodily forms.

Four is the devil, for he openeth all that is closed. All that is formed of bodily nature doth he dissolve; he is the destroyer in whom everything is brought to nothing.

— Carl Jung, from Seven Sermons to the Dead

Further Reading:



The Four Mythological Beasts

January 20, 2018

Another item for my fourfold menagerie is the four symbolic mythological beasts of China. They are associated with the Four Cardinal Directions and the Four Seasons.

  • The Vermilion Bird (South, Summer, named Zhūquè or Ling Guang)
  • The Azure Dragon (East, Spring, named Qīnglóng or Meng Zhang)
  • The Black Turtle (North, Winter, named Xuánwǔ or Zhi Ming)
  • The White Tiger (West, Fall, named Báihǔ or Jian Bing)

They are also part of Japanese folklore and I show the Japanese names of these four god beasts to the right, mostly because I am more familiar with them: Suzaku, Seiryu, Genbu, and Byakko.

Further Reading:


It was nice to see the Four Beasts in the opening ceremony at the 2018 Winter Olympics in South Korea.

[*8.104, *10.64]


The Marriage of Opposites, Part 2

February 21, 2017

sq_menageMy last post made me realize that I had written about six-fold things several times before. The first time was about Richard McKeon’s Aspects of Knowing, the second was about Vaughn Pratt’s Duality of Information and Time, and now we have Edward de Bono’s Six Thinking Hats.

For each of these schema, three pairs of opposites can be shown on the edges of a tetrahedron. I have previously written about the Alchemical Marriage of Opposites, where I imagined two pairs of opposites being in a fourfold. With this new common design, I see that three pairs of opposites can label the vertices of a tetrahedron. In fact, this may be at least as common as double dualities, and I have found several triple dualities to write about in the near future.

In algebraic notation this triple marriage of opposites yields:

(A + A’)(B + B’)(C + C’) = (ABC + A’B’C’) + (AB’C’ + A’BC) + (A’BC’ + AB’C) + (A’B’C + ABC’)

I might even call this diagram a “Ménage of Opposites”, but ménage of course merely means household. Appropriate, nonetheless.


This diagram also represents four pairs of opposites.



The Marriage of Opposites

August 29, 2015

sq_marriageThe road up and the road down are the same thing.

— Heraclitus

The Marriage of Opposites, or Chemical Wedding, or Coniunctio Oppositorum, is a term from alchemy that means combining two opposite substances, or essences, or even ideas into a unity greater than the sum of its parts. This term has also meaning in Jungian psychology as Jung showed many parallels between the processes of self-understanding and alchemy. For example, the marriage of opposites is symbolized by the union of the Animus and Anima.

Opposites are everywhere in our everyday lives and language. They are also prevalent in our social institutions such as religion, politics, philosophy, and science. There are long lists of opposites of life and language and institutions: spatial, temporal, relative, linguistic, mathematical, social, normative, philosophical, and mythological.

A + A’ = ?

What does it mean to combine two opposites into one? Does it mean that the two things are no longer extant and only the combination remains, or that the thing is its own opposite (e.g. 1 + -1 = 1)? Does it mean that a new, third thing is now created that incorporates both of the originals but they still exist as well, like Hegel’s thesis, anti-thesis, synthesis (e.g. 1 + -1 = 3)? Or does it mean that the originals annihilate each other, and nothing remains (e.g. 1 + -1 = 0)?

Usually the casual causal combination of opposites leads to an averaging of the two. For example if hot and cold liquids are combined, soon you will just get a tepid mixture. This is entropy, where the difference between the hot and the cold material is soon eliminated by the blending of the parts, and in this case actually fueled by the energy of the hot molecules.

The usual meaning of this Chemical Wedding is rather esoteric or mysterious. I propose that what the term “marriage of opposites” really means is quite different. Each “opposite” is an opposing pair itself, so that the marriage is between two pairs instead of two things. Instead of a unity, I suggest that this is the root of multiplicity. Instead of simplicity, there obtains organized complexity. Two pairs of two things yields four things in many ways: 2 + 2 = 2 * 2 = 2 ^ 2 = 4.

Let A and A’ be a pair of opposites, as well as B and B’. Then we can consider a union between those pairs to be:

(A + A’)(B + B’) = AB + AB’ + A’B + A’B’

Our individual genetic material is really a marriage of opposites in that half of it comes from each of our parents. Like Mendel’s peas, the sexual union generates the possibility of four versions of each feature. Similarly, the combination between two pairs of opposites generates a fourfold of possibilities.

How should one arrange such a union on a diagram? In Aristotle’s Square of Opposition, the true opposites are at opposite corners of a square, the “contradictories” (one is true and one is false). There are also “contraries” (cannot both be true), “subcontraries” (cannot both be false) and “subaltern” (only one implies the other) relations between corners. But the square is built from the logic of quantifiers and properties, different from this fourfold.

They can be arranged as the cycle AB + AB’ + A’B’ + A’B: One can naturally consider AB and A’B’ to be opposites, as well as AB’ and A’B. One can start with the two opposite pairs crossed, and this cycle sequence arises simply between them. Also, one nice feature is that only one of A or B needs to be changed to its opposite as we move around the cycle, even as we return to the beginning of the sequence. This is called a “gray code” in terms of binary numerals.sq_marriage2

They can be arranged as the grid AB + AB’+ A’B + A’B’: Here the opposites fall across a diagonal that runs southwest to northeast. This doesn’t have the nice properties of the cycle above, but I have used it for many of my diagrams. Instead of a cyclic symmetry, we have a dynamic symmetry about this diagonal that runs from an imagined origin of separation towards greater mixing and combination. This is the usual binary numerical sequence.



Interestingly, while thinking about the next stage in the 2 + 2, 2 * 2, 2 ^ 2, … series, I found that the superexponential or exponential tower operation is called “tetration”. In fact, Tetration( 2, 2) = 4 as well.

Also, consider “Mirage of Opposites”!



The Fourfold Democritus

May 4, 2015

sq_fourfold_democritusOld Democritus under a tree,
Sittes on a stone with booke on knee;
About him hang there many features,
Of Cattes Dogges, and such-like creatures,
Of which he makes Anatomy,
The seat of black choler to see.
Over his head appears the skye,
And Saturne Lord of Melancholy.

— From Robert Burton’s Anatomy of Melancholy

Truth lies at the bottom of a well.

— Democritus


Christoph Luthy / The Fourfold Democritus on the Stage of Early Modern Science
Isis, Vol. 91, No. 3 (Sep., 2000), pp. 443-479
The University of Chicago Press on behalf of The History of Science Society



A Four-fold Riddle

February 19, 2015

sq_roses_dogs_stars_stonesA rose I give to you
This rose so fresh with fragrance rare,
Its petals bringing joy to you
The fairest of the fair.
Oh roses are like memories
They fade and pass above
But you dear heart will e’er remain
My fading flower of forgotten love.

Fading Flower of Forgotten Love by Agnes Ellicott Strong

John Crowley (author of the AEgypt Tetralogy) has mentioned several times in his books a curious list: dogs, stones, stars, and roses. What can he mean by this?

I propose this is a metaphor (Meta-four?) for the four colors of the Magnum Opus: yellow dogs, black stones, white stars, and red roses.

At least until I find out otherwise!


[*8.13, *9.116]


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