Patterns of Relational Paradigm

Twofold Operators-Sequence/Jump


Jump Digital Discontinuous


Jump (Branch)

  • Western science is based on Newton's calculus which assumes that the world is continuous.
  • The Calculus of Newton does not handle digital.
  • Example: Noise in digital computers requires redundancy checks and reduction of speed by 100,000 times.

         The twofold operator sequence/jump is a pattern which is also represented by the terms analog/digital and continuous/discontinuous. Sequence and jump are two of the four basic processes of computer programming. In a sequence, instructions which physically follow one another are executed in a sequential manner. In a jump, which is often called a branch, process control jumps from one physical location to another location not physically adjacent or contiguous to the first.

          The exhibit above shows a picture of the sequence and jump, and the analog and digital. Continuous and discontinuous are words which accurately describe this phenomenon.

          The existing Western scientific paradigm is based squarely on the continuous. We assume that space and time are continuous. Forms and processes are continuous. The concepts which we use to make sense of the world are based on the continuous.

          Many of our theories of physical phenomena, such as electricity and magnetism, are based on the calculus of Newton and Leibniz. We use the calculus as a principle tool in understanding how things work. A basic axiom of Newton’s calculus is that the world is continuous. The calculus does not handle the jump, the digital or the discontinuous. By limiting our theories to a calculus which assumes the continuous, we build into our explanations the bias that the world is only continuous. We deny the discontinuous.

          A good example of this bias is the digital computer. As the name implies, the digital computer is digital as opposed to analog. It operates on the basis that an individual switch is either on or off, zero or one. In other words, it is discontinuous. However, we use an analog calculus to understand how it works. It is not surprising that there are things about the digital computer which we don’t understand. A phenomenon called “noise” occurs that does not allow us to determine whether the switch is on or off. To handle this problem, we build in redundancy checks and reduce the computer’s speed. Manufacturers of computer chips have stated that if the problem of “noise” were solved, the speed of the chip could be increased by 100,000 times.

          I believe that our inability to understand computer “noise” is a direct result of our lack of axioms and theory to handle the discontinuous. We need a digital calculus.

          Another example of the lack of the discontinuous is in the area of forecasting. All of our forecasting techniques are based on the axiom that past patterns can be continuously projected into the future. Our forecasts, whether of economic, political, social or environmental events, are not capable of predicting the discontinuous.

          Sequence/jump, like all twofold operators, has the same patterns as described for the twofold operator form/process. Without sequence there is no jump, and without jump there is no sequence. They are co-evolutionary concepts. If we can’t identify both the analog and the digital, both the continuous and discontinuous in a phenomenon, then we don’t have a complete map of the territory.

          In the West the identification of the discontinuous has been, almost without exception, associated with the catastrophic. From the scientific community we get the population bomb, ecological crises, and nuclear winter. From the religious community we get the armageddon, the late great planet earth, and other stories of fire, flood, pestilence and plague. The new age mystics speak of the pole shift, the earthquake generation, and other major earth-related catastrophes. From the political community we hear of nuclear war, famine and economic collapse. It’s no wonder that we have avoided the discontinuous.

          The jump, the digital or the discontinuous is itself subject to the twofold nature of our reality. If there is a catastrophe, there is a celebration. If there is war, there is peace. If there’s evil, there’s good. The introduction of the discontinuous into our world view at least gives us the capability to accurately map our reality.

          Most existing world views attribute the discontinuous to an act of a non-comprehensible God by whatever name we use. The continuous we can control; the discontinuous is in the hands of God. An “act of God” is the name we use for catastrophes.

        If there is predestination, there is free will. If there is continuous, there is discontinuous. The axioms of the relational paradigm state that the twofold operator predestination/free will applies to both parts of the twofold operator continuous/discontinuous. We need to be able to know, understand, model and control the discontinuous. We may have a choice in how the jump is experienced.


  • Max Planck, Quantum Theory.
  • Thomas Kuhn, The Structure of Scientific Revolution.
  • Ilya Prigogine, Dissipative Structures.
  • Erich Jantsch, The Self-Organized Universe.
  • George Land, General Systems Theory
  • René Thom, Catastrophe Theory.
  • Club of Rome, World Simulation Model. "Percipitous Decline."
  • Puberty
  • Laughter


Quantum Theory

          Knowledge of the discontinuous has been present in the Western scientific community for many years. In 1900 Max Planck formulated a description of radiation which required a discontinuous process of emission and absorption involving discrete quantities of energy. These discoveries initiated quantum physics. Radiant energy such as light or X-rays is discontinuous. Quantum theory has also been applied in both physics and chemistry to matter of molecular size and smaller.

The Structure of Scientific Revolutions

          Thomas Kuhn, a scientific historian, wrote The Structure of Scientific Revolutions in the early 1960’s. Kuhn found that the scientific community at any point in time shares, implicitly and explicitly, a common world view or paradigm. This dominant paradigm provides the basic structure within which all scientific thinking and experimentation are conducted. Consequently, all experiments and the resulting publications tend to support this paradigm. There exists an explicit but universally acknowledged ban; do not ask questions which cannot be answered in the system of accepted structures. Most scientists are not interested in creating a new paradigm.

          However, issues develop which the dominant paradigm cannot answer. Initially these issues are ignored or put on hold until further research can be conducted—of course always within the existing paradigm.

          The new does always emerge, but according to Kuhn, never in a piecemeal or continuous fashion. Someone eventually introduces a new concept that is so far-reaching, so above the battle, that defending the existing paradigm becomes less important than exploring the new. It is a leap to a new level of perception and is accepted without debate.

          Objective logic or debate is never an effective tool for determining the axioms of the paradigm. Logic by definition is based on the axioms, so the use of logic to determine the paradigm is a subjective process, which rather gets in the way of an objective debate.

          Kuhn’s theory, applied to society at large, confirms Arnold Toynbee’s observation that all major civilizations which have existed throughout history perished for exactly the same reason—they could not change the paradigm which brought them to power.

          There is a natural tendency for leaders of any scientific or social system to oppose a change in paradigm. From one view, the leaders are the winners at the old game, the keepers of the rules. They have little incentive to change the rules. From another view, the leaders’ primary role is to ensure the survival of the existing system. Their job is to oppose the new paradigm.

          In any case, the introduction of a new paradigm has always been traumatic. Our whole view of the world, common sense, rules of thumb, and basic behavior patterns are all built on the axioms of our paradigm. If we change the paradigm, we have to reconstruct our world. It can be frightening.

Dissipative Structures

         Ilya Prigogine, a Belgian chemist, won the 1977 Nobel Prize for his theory of dissipative structures. The theory states that living or open systems are involved in a continuous exchange of energy with the environment, non-living or closed systems. Closed or non-living systems all are governed by the second law of thermodynamics which says that in any energy exchange some energy drops to a less ordered, less structured state. It’s called entropy and basically says that the world is running down. According to Prigogine’s theory, living systems operate in an opposite manner. Open systems, or dissipative structures as he calls them, run up rather than down. They increase rather than decrease order.

          Not only do dissipative structures increase order, but in addition, the increase occurs in a discontinuous manner. This transition to a higher order is universally accompanied by turbulence or “perturbation.” Living systems will occasionally go into extreme fluctuation and perturbation, but rather than falling apart, they jump to a higher and more complex order.

          Erich Jantsch in The Self-Organizing Universe has applied Prigogine’s theory of dissipative structures to individual and social evolution.

General Systems Theory

          The General Systems Theory of George Land is similar to Prigogine’s Theory of Dissipative Structures. Land states that all systems go through similar stages of evolution. The first stage is formative and self-oriented. The system defensively gathers everything to itself in an inward directed growth stage; it formulates new patterns. The second stage is replicative in that it tries to endlessly duplicate those things which actually seem to work. The successful patterns are manifested in outward directed growth. The third stage is both integrative and perturbative. The problems and rewards are shared, while at the same time the system gets so good at knowing what works for it that it uses up its environment and goes into a state of disharmony or disorder. This state is really “destructuring” rather than “destructive” and is necessary to shake the system loose from its old paradigm so that the new may appear. At the end of this process the system jumps to a new level and the stages are repeated.

Mathematics and Modeling

          The mathematics of the Western scientific community has had very few tools to deal with the discontinuous. The calculus, our forecasting or time series analysis techniques, and our modeling or simulation techniques all assumed that the world was continuous. Any discontinuity was ignored or handled outside of the mathematical equations.

          In recent years, several areas have emerged which begin to deal directly with the discontinuous. The first of which I’m aware is an area of topology—the study of shapes—called Catastrophe Theory, initially formulated by the French mathematician Rene Thom. This field of mathematics studies the discontinuities which occur on different types of geometric figures. It provides a much needed view, but Thom could have called it Eureka Theory.

          The Fractal Geometry of Nature by Benoit Mandelbrot is another recently formulated area of mathematics which has something to say about discontinuity. Fractal geometry represents naturally occurring phenomena, such as a coast line or stock market prices, through a pattern which can be endlessly repeated at finer levels. I’m not yet sure of its implications but feel that it adds a view previously unavailable.

          The mathematics of modeling or simulation are used in many areas in an attempt to understand our world. Economic forecasts are always based on some explicitly or implicitly stated model. Western science has historically lacked the theory and techniques to effectively model a system which was discontinuous.

          In the early 1970’s a group called “The Club of Rome” developed a simulation model of the world. They used a modeling technique originally developed by Jay Forrester called Dynamo. This modeling technique differed from most others in that it utilized feedback loops. It modeled the network rather than just the hierarchy. The Club of Rome found that regardless of the growth rates or basic assumptions they used in their world model, the results of the simulation always showed a “precipitous decline.” They ran around for several years yelling, “The sky is falling.” And indeed it does. They had discovered the discontinuous.

          The best summaries of these emerging areas of scientific thought that I have found are contained in the first few issues of The Tarrytown Letter published by the Tarrytown Group, and the book The Aquarian Conspiracy by Marilyn Ferguson. But there are many examples and references to be found in every area of our society, both scientific and religious, both West and East, both liberal and conservative.


          The idea of discontinuity certainly is not limited to intimidating and obscure scientific theories. Life is discontinuous. We deal with it as a natural process of living, of birth and death.

          Everything about puberty seems discontinuous. You are either male or female, regardless of your sexual preference or dress. You either can or you cannot reproduce. You are either a virgin or you are not. The male either did it or he did not. The female is either pregnant or she is not. It is all discontinuous, and continuously continues.

          Unfortunately, we have misplaced many of the techniques used by societies to handle these discontinuities. Ritual is a process by which societies consciously deal with the discontinuous, and Western society no longer consciously practices many of its rituals. The rituals are still being conducted; however, without a conscious knowledge or control of these events, the results are often debilitating—war, crime, drug abuse, etc.—rather than uplifting.

          Humor is one process which we still use for handling the discontinuous. The essence of laughter is discontinuity.


  • G. Spencer-Brown in Laws of Form has developed a digital calculus. He calls it calculus of indications.
  • Two axioms of digital calculus
    1. The value of a a call made again is the value of the call.
    2. The value of a crossing made again is not the value of the crossing.

Another View of the Two Axioms

        Perhaps the most important emergent mathematical concept is what I call a digital calculus. The analog calculus of Newton is so prevalent in the Western world view that the recognition of a balancing technique is imperative if we are to understand our reality. G. Spencer-Brown in Laws of Form has developed the axioms and initial theorems of a digital calculus. He calls it a calculus of indications.

        I personally had to spend a substantial amount of time with Laws of Form before I was able to understand and work with the concept. I unsuccessfully tried to find an instructor in several university math departments who could explain the concepts, but the book is clearly not of the existing dominant paradigm and hence by definition is heresy.

          Basically, what Spencer-Brown has done is to provide a calculus for Boolean algebra—the area of mathematics which deals with logic, with the concepts of true and false. He has formulated two basic axioms and a nomenclature which allow the development of an arithmetic for logic, for the digital or discontinuous. Many of the concepts of Laws of Form can be found in other conventional techniques, but Spencer-Brown’s concepts provide a clarity and power not found elsewhere.

          He uses only two symbols: the marked state which is designated by a right corner, and the unmarked state designated by an empty space. Unlike conventional mathematical techniques, he uses the same symbol to designate both form and process. The position of the symbol determines what process is to be executed.

          I believe that Laws of Form provides a missing link in the Western world view. It does need an interpreter who can provide examples of its use and translate it into a more easily understood form.

Form/Process • Hierarchy/Network • Sequence/JumpMeasure/CountKnown/RandomBalance/RelationshipSummary of Emergent Axioms




Patterns of Relational Paradigm


Twofold Operators







Summary of Emergent Axioms

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