The Alchemical Paradigm

John S. Uebersax

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1  Introduction

H ere I propose to supply a short mathematical definition for what I elsewhere call the alchemical paradigm. As I use this phrase frequently, it is appropriate that it be given a clear, concise definition.

The idea is simple--extremely so--and one easily and readily understood at an intuitive level. In fact, there's nothing new to the idea at all--it's something manifest in countless ways in life and in the world. The present goal is to give this basic principle a slightly more formal expression.

One benefit is that, because, while the idea itself is common and familiar, it belongs to that vast domain of things we know but are only partly conscious of. Some (e.g., the philosopher Michael Polyani) call this realm tacit knowledge .

An advantage with making tacit knowledge into explicit knowledge (or objective or scientific knowledge; all these terms apply) is that the idea then becomes more fully conscious. We may then relate it to a wider array of concepts, using the full powers of the rational mind. A second advantage, which follows from this, is that an explicit, articulated idea also becomes publicly or collectively conscious -- something we can talk about with other people. This promotes scientific thinking and related practical applications.

In short, until an idea, however obvious it may seem, is defined in a formal logical or mathematical way, it is only partly conscious, and, in that sense, only partly 'real'.

Further, were a phrase, term, or explicit concept like what we here call 'the alchemical paradigm' well known and familiar -- part of the common parlance -- then people would more routinely see in conflict an opportunity for reconciliation, and means to attaining it. In other words, becoming conscious of conflict would lead automatically to the realization: "Ah, we are in conflict. Now let us apply some basic method to resolve it."

While this might seem overly idealistic, at least we may say with greater certainty that the opposite is true: if there is no awareness of means to resolve conflict, then parties are less motivated to become conscious of the conflict, to pause and step outside or above it, and attempt to resolve it.

Note. While the immediate concern is possible application to social sciences, the proposed paradigm and notation may also have uses in the physical sciences. It is the very universality of the paradigm that causes it to be overlooked as a scientific principle.

Basic idea--an intuitive explanation

In few words, the essential idea is this. In life and experience, we frequently encounter complex systems.

Sometimes complex systems conflict with one another. A dramatic example is two countries, each a complex system, at war.

One way to resolve conflict between two complex systems is to accomplish what we might call their union, conjoining, combination, synthesis. An reconciliation, alliance, or union between the aforementioned warring countries would be an example. Here we just present a rudimentary algebraic or notational system to describe this common and basic process.  

2  Definitions

2.1  Definition: complex system

A complex system is simply a system that is complex. The meaning of the term is sufficiently obvious that we may rely on readers intuitive understanding, and a few examples. Common complex systems include the following: a biological organism; an automobile; a molecule; a city, society, nation, or culture; the human mind or psyche; a belief system.

2.2  Notation


A = {a(1), a(2), a(3), ...}. [Equation 1]

denote the complex system, A. A is defined as the set of all objects and relations among objects that characterize the system. Although we could make a logical distinction between objects and relations, for present purposes this is not necessary: both objects and relations may be considered general elements of the set of all things that comprise complex system A.

We may similarly let B = {b(1), b(2), b(3), } denote a second complex system, B.

2.3  Guiding premise

A guiding premise is that typically when two complex systems conflict, they do so because only some, and not all, of their constituent elements are in conflict with some elements of the other system. If the specific conflicting elements can be removed, without harming or destroying the integrity of the original complex systems, then doing so may eliminate the conflict between the two systems.

2.4  The relevance of alchemy

We wish to dispel any negative connotations associated with the word, alchemy. Alchemy, historically, began as the science or proto-science of metallurgy. From this it evolved into two branches: chemistry, and a more vaguely defined 'spiritual' science of esoteric and mystical practices. Contrary to modern popular opinion, it now appears that only a few alchemists were concerned with turning lead into gold. Likely this was more often a metaphor among spiritual alchemists for 'transmuting' or transforming the mind or soul.

In any case, we have good reason to accept alchemy, in the broadest sense, as the historical science or the set of logical principles concerned with the transformation of complex systems. This includes the union or joining of complex systems, such as in the case of making an amalgam or alloy of two metals.

This example is exactly relevant to our present concern. To make an amalgam of two metals, one begins with crude ore. From the crude ore, the metals are extracted. They may then be heated or otherwise refined. Finally, the two refined metals are combined.

An analogous three-stage process can be applied generally to the combining of complex systems, or the resolution of conflict between them. We may call the three steps: (1) separatio (separation); (2) purificatio (purification or refining); and (3) coniunctio (conjoining).

2.5  Why use Latin terms?

Latin terms here offer the same advantages we see in other sciences. As Latin is a 'dead language', it limits unnecessary variation and embellishment of terms by individual users. This promotes precise and consistent definition.  

3  The Phases

We now consider each of the aforementioned stages.

3.1  Separatio

This stage is necessary to separate the two component systems prior to their purification and union. It can take any of several forms. We distinguish three cases here.

Case 1. In the most common case, analogous to the separation of metals from ores, a complex system is removed from a surrounding matrix. Removal from the surrounding matrix may be necessary in cases where elements of the matrix interact with the complex system in ways that inhibit or complicate the changing of the latter.

Case 2. In situations where an external, intelligent agent (the 'alchemist') is responsible for performing or overseeing the alchemical process, separation may enable this agent to attend more closely to the system and its elements.

Case 3. A third case occurs when two subsystems of a complex system are are in conflict. Then it may beneficial to temporarily separate them, purify them individually, and then re-unite them to form a new, less or un-conflicted system.

Comment. In alchemical metallurgy, separatio may be accompanied by a process like heating. Or, sometimes, a separate new substance, e.g., mercury, is used temporarily to extract metal from ore. Both of these principles may apply more generally to conjoining two complex systems. Concerning this, we shall here leave details implicit.

Definition: prima materia

The term prima materia refers to an original, untreated, and unseparated combination of complex system and its surrounding matrix, or, as in Case 3, an original, untreated system with its conflicting subsystems.

Notation. No new notation is needed for the separatio phase.

3.2  Purificatio

The purificatio phase removes from a complex system those elements which are responsible for conflict, or which are inimical to its joining with the other complex system.


A' = the purified version of A. [Equation 2]


A' is a purified version of A if and only if:

  1. A' is a subset of A (all elements of A' are also in A);
  2. There is at least one element in A that is not in A'.


Comment 1: We may again consider the example of the heating of metals, and how this may, by analogy, apply to purification of other complex systems. Heating may correspond to a temporary quickening or energizing of the processes of a complex system, the breaking of static bonds, and the supplying of energy for transformation.

Comment 2: Purificatio may also be effected with the help of a mediating complex system, C, as in the case of mercury. That is, A' may be produced by exposing A to C. The idea is that C will connect with or bond only with the targeted elements of A. After the purification, A' and C may then be separated.

3.3  Ennobling

While it is not essential to the present discussion, we may take the opportunity to mention a complementary or converse operation to purificatio, which we term ennobling. Whereas purificatio prepares a complex system by removing elements, ennobling prepares the system by adding new elements.


A* = an ennobled version of A; [Equation 3]


A* is an ennobled version of A if and only if:

  1. A is a subset of A* (all elements of A are also in A*).
  2. There is at least one element in A* that is not in A.

It is clear that complex systems may be both purified and ennobled in anticipation of their conjoining. We may denote a system A that has been first purified, then ennobled, by A'*, etc.

3.4  Coniunctio

The coniunctio phase joins the two suitably separated and prepared complex systems. Synonyms for coniunctio include conjoining, joining, conjunction, alchemical marriage, etc.


(AB) = the complex system produced by conjoining of complex
systems A and B.
[Equation 4]

4  The Result

The net result of our alchemical paradigm -- the three-stage process of separatio-purificatio-coniunctio -- we denote as below.


(A'B') = the new complex system produced by the conjoining of
the two purified complex systems, A' and B'.
[Equation 5]

As in the case of algebra or other mathematical notation systems, from these basic principles many other results follow.

5  An Example

We consider again the example of two countries, A and B, in conflict.

For our purposes, we consider country A to be a complex system with three elements:

We consider country B to also contain three relevant elements:

Now, however much one might wish to do so, it is neither realistic, just, nor feasible to simply remove the friction-causing parties, a(2) and b(2), from their respective countries. However, we may justly intervene to eliminate the relational elements a(3) and b(3). Applied to this problem, our alchemical paradigm is as follows:

Separatio. For example, this may consist of distinguishing countries A and B from their respective neighboring countries or their broader cultures (e.g., if country A is the United States, to recognize that one is considering the United States, and not 'The West' in general.)

Purificatio. To use appropriate means (e.g., education) to politically restructure A and B to remove the structural elements a(3) and b(3).

Coniunctio. The sources of conflict now having been removed, to institute economic trade, cultural exchange, etc., between the purified systems A' and B'.



To Cite this Page

Uebersax, John S. (2007). "The Alchemical Paradigm". Online article. Retrieved from on mmm dd, yyyy.

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John Uebersax PhD

7 Feb 2007 (first draft)