Visions of Four Notions – Contents

Introduction to a Quadralectic Epistemology

by Marten KUILMAN

Falcon Press, Heemstede, 08072011, ISBN 978-90-814420-2-2

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It may, indeed be not only possible

But likely that life exists on other

Planets circling other suns;

And if it does then the chances

Are very good that it lives

Under blue skies with white clouds

And a yellow sun, with rivers and

Oceans of water nearby in plenty

John GRIBBIN (1982) – Genesis. The Origin of Man & the Universe

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Fig. 1 – The flying and floating island of Laputa was moving in a curious way along the northern periphery of the island Balnibarbi.Voyages du captaine L. Gulliver en divers pays éloigne – Jonathan Swift (1727).

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O Divine Spirit, sustain me on thy wings

(William Blake – Jeruzalem)

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CONTENTS (press the heading for direct access)

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1. Introduction 7

2. Right from the start 15

2.1. The primordial part 21

2.2. Two parts in the universe 23

2.3. The three division creates dynamism 27

2.4. The tetradic sense of stability 30

2.5. There is more to five 34

3. Preliminary movements 35

4. Modern knowledge and new names 58

4.1. The First Quadrant (I) – The ultimate unity 60

4.1.1. Teilhard de Chardin (1881 – 1955) 63

4.1.2. Carl Gustav Jung (1875 – 1961) 66

4.1.3. Giordano Bruno (1546 – 1600) 72

4.1.4. Jan Christiaan Smuts (1870 – 1950) 74

4.2. The Second Quadrant (II) – The first multiplicity 84

4.2.1. A proto-communication (II, 1) 85

4.2.2. The conscious first division (II, 2) 86

4.2.3. The valuation (II, 3) 93

4.2.3.1. The definition of visible visibility 114

4.2.4. The implementation (II, 4) 117

4.2.4.1. In the First Quadrant (I) 124

4.2.4.2. In the Second Quadrant (II) 126

4.2.4.3. In the Third Quadrant (III) 128

4.2.4.4. In the Fourth Quadrant (IV) 130

4.3. The Third Quadrant (III) – A partial unity 134

4.4. The Fourth Quadrant (IV) – The last multiplicity 140

4.5. A Recapitulation 144

5. Comparisons 147

5.1. A cosmic comparison 156

5.2. A terrestrial comparison 166

6. To the boundaries of imagination 170

6.1. Imagination on a human scale: proposal for a new European history 176

6.2. Imagination on a geological scale: the presence of life in time 186

6.2.1. Two groups of extinct animals 188

6.2.1.1. Trilobites 190

6.2.1.2. Ammonites 196

6.2.2. Large animals as time indicators 203

6.2.3. The relation between Man and planet Earth 209

6.3. Imagination on a cosmic scale: existence in space 225

6.3.1. The cosmic history of the earth 226

6.4. The ultimate consciousness: the universe 233

7. Perspectives 247

7.1. The roots of understanding 249

7.2. Inventarisation 253

7.3. Substantiality 255

7.4. Relativity of Meaning 258

7.5. The Four-fold Man 260

8. Glossary (of quadralectic and related terms) 267

9. Bibliography 277

10. Illustrations 289

11. About the author 293

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Cover of the 2009-Edition (5 march 2009) of ‘Visions of Four Notions‘ by Marten Kuilman. Falcon Press, Heemstede. ISBN 978-90-814420-2-2

1. Introduction

This book is about a new and specific form of the tetradic (or four-fold) way of thinking, for which the term quadralectics has been coined. Its main feature is that any reflection is the result of four considerations, and that visibility has a four-fold bond with the observer. This statement might not look very sensational at first sight, but this view may change when the knowledge of its consequence increases. It will be shown that various forms of division thinking are very elementary indeed.

Tetradic thinking is probably as old as mankind itself. However, it was the Greek philosopher Empedocles (ca 492 – 432 BC), who amalgamated the historic material of division thinking into some sort of theory. He suggested four elements or roots as the main constituents of all common experience: fire, air, earth and water. These basic entities were indivisible, eternal and immutable. The different mixtures of the four elements produced a visible world in all its richness.

Other tetradic world views can be encountered in many forms, embedded in many cultures throughout history. The old Egyptian pyramid builders may have had one, translated in the form of their buildings. The sons of Horus, and their place in the funeral culture, did reflect a tetradic setting of the mind. The Meso-American Indians like the Mayas and Incas were fascinated with quadripartite divisions. European history is bestrewed with possible clues, to be found in Celtic ornaments, the buildings of the Roman Emperor Hadrian, the writings of the early Church fathers, features in the French cathedrals: to name only a few.

It is, therefore, at least remarkable, that this body of cultural knowledge was never placed in a philosophical context and regarded as a ‘system of thought’. Any reference to the number four and its dynamic use is systematically described as ‘numerological’, a term with a definite negative meaning within the world of science.

Quadralectic thinking is a new form of the old, archaic four-fold way of thinking. The neologism expresses a distinct frame of mind in which four phases of being are governing a communication. The following cognitive impressions, which have a cyclic connection with each other, are in the center of the new approach:

The acceptance of a distinct area of complete and unlimited incomprehension in the invisible realm;

The knowledge that certain ideas can be developed for reference and mental guidance;

The admission of the existence of an area of limited comprehension in the visible realm;

The understanding that a wider frame of mind can lead to new discoveries.

The journey through life is an adventure, because we don’t know what lies ahead. The comprehension of human existence is an undertaking on its own, with uncertain boundaries and countless possibilities. And it is shaped by a desire to find out what our presence is all about. This book will try to provide an answer to such a question – a daring project that is bound to reach short of its goal. It is, like so many efforts in life, the attempt, which counts.

It can be stated that the most mature achievement within our personal development is the realization of scale. There are layers of magnitude, which can be seen only in the proper setting of observation. Certain features are within reach, are stumbled upon without any effort. They often seem trivial, and will go unnoticed because of their conventionality. However, there are also qualities at the edges of our comprehension, which can be understood with the utmost endeavor or the use of artificial means to enhance the visibility.

A look at the sky at night will confirm that observation. There are millions of stars beaming their light to the earth. Some are bright and clear even in the glare of the city lights, while others are dim under the most favorable circumstances. The amateur can use a pair of binoculars to enhance details and a telescope will further reveal a staggering number of features. Natural and artificial limitations provide a sense of scale (fig. 2).

Fig. 2 – The notion of scale can be demonstrated in the layers of visibility in the nightly sky. This example by Antonin RUKL (1985) shows the constellation of the Little Dipper (Ursa Minor) and the Polar Star (Polaris). The successive enlargement of a particular area of the celestial sphere discloses the visibility of ever more stars.

The stars at night give a first glimpse of what scale means in the process of visibility. And not only from the human point of view, using different sorts of contraptions to enhance vision, but also from the universe itself. At this very moment, there is light traveling towards us from unknown distant stars, which has not reached the planet earth yet. There is, in other words, a cosmic scale of which only the universe is aware.

The same holds for the very earth under our feet: the soil, the debris of plants and rocks, the animals – a whole universe is out there waiting to be discovered. Layer after layer provides evidence of the material abundance of the earth, ending up in DNA-spirals with unknown information or atoms in space. A deep probe into the existence of material things reveals another universe, which may – in the end – be just as complicated and impenetrable as the one behind the stars.

A theory of vision should be based on the acknowledgement of scale and the impossibility for an observer to ‘control’ the whole field of observation. Every attention paid to detail leads to an unperceived space where our measurements reach short: the invisibility of a cosmic presence, which provides the ultimate scale of being, a presence unknown to us.

Any exploration in this territory is guided by intuition and to a lesser extent by feelings: the two basic human instruments to go ahead in an uncharted territory. If, after trial and error, certain trails are established in the vast unknown, they often present themselves as (religious) beliefs, or in a more concrete form as: faith. Mysterious ways create a delicate network of comprehension, which is not based on material experience, but has formed itself on a spiritual level.

This fragile framework of knowledge can have a considerable impact on everyday life, even if it lacks a firm foundation in the empirical world. The conceptual structure built (by the human mind) in the limitless cosmic domain influences the physical understanding and vice versa. The key to understanding the world lies ultimately in the interaction between the faith to draw limits in the unknown and the boundaries set by the observation of the earth. The present inquiry aims at just that interaction and will be a quest in the human mind to establish the foundations of our insight and knowledge and the ideas and feelings which contribute to their formation.

One needs to obey certain rules in order to succeed in an investigation. Henri ELLENBERGER (1970) formulated the following principles of methodology in the introduction of his stimulating book The Discovery of the Unconscious. The publication deals mainly with the history of psychiatry, but also leads into some very interesting side roads:

1. Never take anything for granted;

2. Check everything;

3. Place everything back into its context;

4. Draw a sharp line of distinction between the facts and the interpretation of facts.

This shortlist of mores was composed by Ellenberger to conduct a sound scientific inquiry, but can also be seen as setting forth four distinct theoretical positions, which can be held within any given communication. The various statements represent a microcosm of reasoning, which correspond very well with the quadralectic path proposed in this book. It is worthwhile to look at the different stages more closely.

The first advice – never take anything for granted – is a very general one: it is the hallmark of a skeptical mind. Without it (scientific) research would be impossible. It looks simple, but in the end it isn’t. Because if this advice is doggedly followed to its final consequences, any progress is out of the question. We need some rudimentary substance, need to take at least something for granted, or belief in some sort of truth, to be able to make an advance, to verify and compare. Ultimate skepticism is, in essence, a static affair.

The second suggestion – check everything – is more dynamic. Checking is an arduous activity, there is no doubt about that. It means searching for more, traveling to unknown places, digging in terra incognita. This dynamism poses problems of a different kind. We can check a lot, but not everything. Nature is simply too rich and the universe too big: quantity will never be exhausted. There will always be areas out of our reach. It has to be accepted that a (subjective) decision must be taken long before everything is checked. A continuous search for the last bit of knowledge would stop an investigation and make progress impossible.

The third instruction of Ellenberger – place everything back into its context – is a very important, but again a difficult one: it asks for a limitation of ‘everything’ to its context. First of all, we have seen that ‘everything’ is an unreachable goal in the multitude of the universe. ‘Everything’ will be, at best, a representative selection of facts on which an observer could lay its hands. And secondly: what is a context? Is it an abstract framework of things, which are connected with each other? But what framework and which connection? It seems clear, that the (subjective) boundaries are temporarily taken for granted, to facilitate the ordering of the facts. Apparently, the first and second pieces of advice are postponed, for the time being. A context, it can be concluded, is a fairly static element within a communication. Thomas KUHN (1962/1970) discussed the general function of a (scientific) context in his pioneering book ‘The Structure of Scientific Revolutions’. A ‘paradigm’ is, in his view, a complex set of accepted knowledge, which forms the a-priori departure of scientific research.

The fourth guidance asked for the application of a division between facts and interpretation. This demands oppositional thinking: on one side, there is material (‘facts’) gathered from a ‘neutral’ Nature and on the other side, the results of our handling of the matter under investigation. Ellenberger pointed here, unintentionally, to the great limitation of the scope of modern science. First, we take ‘everything’ into account (which is de facto impossible), and secondly, when it is time to reach conclusions, a rigid duality is applied in a faint effort to keep subjectivity out of the scientific framework.

We have to realize that our scientific edifice, supported by the before-mentioned morals, is built on a shaky surface. It is better to face the facts, then to deny them. Therefore, the catch word in a modern scientific approach is subjectivity, or the importance of human participation, which has to be brought back into the investigation. Not in a sneaky way, through the back door of a false division between facts and interpretations, but free and openly. Be honest, right from the start.

It was only a mere brief item in a newspaper that attracted my attention some years ago because of its philosophical implications. The Russian ex-world-champion chess M.M. Botvinnik developed, around 1982, a chess-program that did not depend on the brute force of the computer, but highlighted the positional aspects of the game. The chess pieces were given, according to their position in relation to other pieces, certain values. A subsequent inventory led to a list of priorities and ultimately to the most favorable move. Botvinnik regarded the game of chess – with its different pieces, moves and rules within a given environment – as an inexact problem which had similarities with other complex organizations like the economy or in management-systems. Van der HERIK (1983) provided an early survey of the world of computer chess and artificial intelligence in his readable Ph.D.-thesis at the TH Delft.

Life itself is, in many ways, also an inexact problem. The advancement of a game of chess into a theory of life is therefore less dramatic than it seems, in particular, with respect to its crude mechanisms and methods. It is possible, in chess as in life, to place the ‘subjective’ associative-positional aspect of the communication in juxtaposition to the ‘objective’ brute force of trial and error.

A conscious decision depends just as much on our insight and experience, gathered mainly in the past, as from the courage to try something completely new in the future, just to widen our experience. There is no observation without an object. Subjectivity and objectivity are inextricably linked to each other. We have to come to terms with the reality that there is no searching without an assumption or understanding without an a-priori.

Botvinnik’s book on chess programming, written some thirty years ago, was a daring attempt to describe the basic problems of transforming the game of chess into a digital model for the computer (BOTVINNIK, 1970; 1982). However, it also encounters, in due course, some essential characteristics of communication-in-general. Botvinnik pointed out, that any system of decision taking should have three objectives:

The collection of information,

The valuation of the information and

The act of taking a decision.

These three faculties cover, by and large, the visible trajectory of any given communication. The approach is more general than the stages in the methodology of Ellenberger, but the crux of the matter is just as revealing. Information is a rough commodity and data mining is an austere business, despite the fact that the tools seem to be getting better every day.

It will be clear in the end – or rather much earlier – that the act of decision-making in any system is embedded in a sequence of minor decisions, which were made along the way. The final, visible choice is just an option from a long row of invisible, preceding choices. If we follow that sequence to its source, it will ultimately be the type of division, which sets the character for the rest of the communication.

Fig. 3 – The decision-tree of chess study by M. Botvinnik and S. Kaminer (white to play and win). The start position is given in the top left-hand corner. The black and white ‘knots’ – 145 in total – are the possible moves of the players (with restrictions).

Botvinnik proposed a decision-tree (fig. 3) as the most appropriate mathematical tool to tackle the problems in relation to the three objectives given above. He regarded the game of chess as an inexact problem, which could be solved in two ways (as had been indicated by the American mathematician Claude Shannon in 1949).

Firstly, there must be a search for all possible moves by the building of a decision-tree. This action is, in theory, of a relative simple nature, but leads, in practice, to unwanted consequences, which stands in the way of a solution. Decision-trees have the habit of growing into huge foliage, even into infinity. The possibilities increase exponentially ‘in depth’ and become a major stumbling block of finding a proper way out of the multitude. This leads directly to the second point.

How should the number of alternatives be limited? Some restraints must be introduced to manage the size of the decision tree. The ‘pruning’ of the tree by excluding all useless possibilities seems a logical step. The so-called minimaxing method validates the individual possibilities one-by-one (in a dualistic way) and rejects certain continuations if a better solution is found elsewhere. The alpha-beta pruning aims at the computation of particular favourable continuations (values) without examining every imaginable possibility. The establishment of a horizon is crucial. If we are able to apply the minimaxing to the full depth of the decision tree (in other words: fix the limits of the horizon), then we continue to have an exact problem, which can give an exact solution.

It all looks so wonderful objective: just apply a dualistic evaluation on a local level and an exact decision could be reached for the whole, pruned system. ‘Turning the machine into a good aid in the solution of inexact problems’, said Botvinnik (1970; p. 6), ‘can be done in one way only – by constructing a mathematically precise program for the solution of inexact problems, or, in the language of the mathematicians, by formalizing the solution.’ An objective algorithm might hold the golden key to success, but when it comes down to the nutty-gritty of (Shannon’s) valuation of pieces, there is obviously a case of subjectivism.

The decision-tree is only the first step in the preparation of data for a binary computer. The selection and valuation of the possibilities are much more difficult and complicated, because the actual parameters for such an operation have to be stated in clear terms. Subjective history and experience enter the (seemingly) objective reality of the decision tree at the very start of the valuation process.

The solution of an inexact problem is a stubborn entity: the introduction of pruning – in the early stage of collecting objective information – is also the introduction of the first subjective measures within the game of chess (or communication in a wider sense). It is better to face this fact right at the beginning of an interchange of information. There is no point in ‘forgetting’ our own choices during further investigations. European scientific research has been for too long dogged by the heritage of Cartesianism, which glorified the ‘objective’ approach to nature, in which the observer had no formal relation to the observed.

Botvinnik pointed to accumulated experience and intuition as the driving forces behind the solution of inexact problems. Those two properties are also highly subjective in nature, born in a pluralistic environment of constant and repetitive attention (experience) and in the bright, daring setting of a unity (intuition).

The use of experience, or subjectivity, is in many ways a much better guide for human decision-making. Botvinnik distinguished four methods of how to make use of previous gained knowledge:

1. The ‘parrot’ method – following the established rules, without much thinking, because the valuation of the moves is more or less historically established. Certain moves have proven their worth in time to be the right ones in a given, known situation. The method has a mechanical nature. Often used at the opening of a game.

2. The information method – or the passive search for a similarity of positions in the memory of a player, based on experience. This pattern matching is the most productive method in the human approach to the game (of chess). Emanuel Lasker, the world champion of chess between 1894 and 1929, was asked one day how many moves he was thinking ahead. His answer was a tribute to pattern recognition: ‘Only one, but it is always the best one’. This method is particular useful in the endgame, but is also applicable to the openings game, when a transformation of moves takes place.

3. The deliberate method – The ambition to create a position in which the information method (2) can work to its full advantage. Now the art of pruning becomes vitally important. Validation turns out to be the name of the game. The method of biased searching is typical of the endgame and probably of the opening as well.

4. The associative method – or the search for ‘position-fragments’, which can be related to ‘library’-positions. This method demands some sort of subjective overview in connection with a body of favourable ‘historic’ positions, known from literature or own experience. This method of pattern matching is, due to its intensive and specific use of the past, the only one suitable for the middle game and complicated endgame. It is clearly Botvinnik’s pet method.

These four approaches reflect, to a certain extent, the stages in a quadralectic communication. Intuition, pattern recognition, validation and quantitative matching are the very characteristics of the conceptual movements in the quadrants.

The final aim of scientific research is – or should be – the ‘samanvaya’ of the Buddhists: the act of reconciling contradictory ideas by carrying them to a level of understanding at which it can be seen that they are not really oppositional. The gaining of this vision will be the ultimate goal in this book. The tool to reach this understanding consist of a widening of the initial fragmentation of a communication into a wider field.

2. Right from the start

Many books have been written on communication in its widest sense. The interaction of people and their environment has been the subject of endless inquiry. The description of ideas and feelings in human relationships provides an infinite source of inspiration. However, the subject is often approached from the outside and deals with the manifestations of the process rather than the process itself. What actually happens when information exchanges? Why do people react in such and such way if they are faced with certain facts?

In the present approach – which is by no means the only way – a communication is seen as a process of interacting divisions. That means: any information exchange is translated into the sharing out of positions within a segmented reality. Two key words control, in my view, the process of communication: division and movement (fig. 4). These essential qualities will first be looked at in a provisional way to get acquainted with some of the mechanisms of data conveyance between people and environment.

Fig. 4 – Division (A) and movement (B) are the two major constituencies of any given communication (C). A. The dynamism of division leads to various sorts of measurements and subsequently to different types of visibility and B. The dynamism of movement plays an important role in the relation between the observer and the observed and is directly related to a definition of visibility.

The first variable (division) belongs to the foursome of basic arithmetical processes: adding, subtracting, multiplying and dividing. The second entity (movement) comes from a dualistic background, where static and dynamic natures are opposed to each other. Any preference of the usage of these subjects in a particular sequence is an arbitrary matter, but not without primary importance and significance.

Ernst BINDEL (1983) pointed to the present cultural preference of addition as the major arithmetical operation. There is no doubt, he said, that everything is geared towards addition: subtraction is just the opposite of it, multiplying is a continuous addition of the same units and division is the inversion of multiplication.

Counting and (digital) computing proved to be the best way to deal with the type of mechanical problems, which have been in the spotlight of our attention for a long time. And because they have worked so well in that particular area, they are gradually applied to a wider range of questions, including those of the human mind itself. Knowledge and thought became saturated with a world view based on the synthetic use of numbers, an addiction to addition.

Behind this preference lies the unwearied human need for the control of things. Everything should be included in our sphere of influence by listing the subjects one after another, like a Noah counting the animals when they were entering the Ark. That accumulation becomes a unity of knowledge and could manifest itself in power.

The ‘encyclopedic’’ movements in history provide instructive material with regards to the importance of addition in the consciousness of the people. The early medieval descriptions of nature, the sixteenth- and seventeenth century collectors, the French Encyclopaedeans and the modern computer-stored databases (like the Internet) have their devotion to addition in common. The celebration of quantities has a long history, which is closely related with a search for knowledge and power. The eternal dream that more is better has motivated generations of people and will continue to do so.

The importance of division is immanent from the outset of any communication, but not always visible in the classical sense of the word. It is only in the further course of communication, when motion is introduced, that the (primal) division provides the equipment to calculate the most crucial element in every communication: the visibility. This calculation – regardless of its nature – links the main components of a communication to a new unity (a value), which we call ‘visibility’.

The division is established in a cerebral process, which is part and parcel of the communication itself. The choice (following the numerical sequence 0, 1, 2, 3, 4, 5 … etc.) is made at the very moment a thought lingers on an object or subject. What is actually ‘seen’ in the mind of the observer is the product of a calculation.

Visibility can be expressed as a (numerical) figure derived from the shifting interaction of divisions.

The impulses produced in the process of seeing and thinking are stored in our brain. Visibility can be objectified as a code, consisting of a combination of numbers. This basic point of view will be elaborated on in a later stage of the investigation. For the time being, it will be sufficient to note, that the arrangement of numbers is highly dependent on the division which was used to store the information in the first place.

The ‘new’ subjectivity, which is proposed here, deals with a conscious notion of the principle of division. It is not the sort of negative subjectivity as indicated by Bertrand RUSSELL (1945, p. XXI) in the introduction of his classic ‘A History of Western Philosophy’: ‘Subjectivity, once let loose, could not be confined within limits until it had run its course’. Instead, it confines the human choices to clearly stated areas. If this is granted, the ominous anarchism is just a pioneering excursion of the mind to the borders of its own choice.

The coding pattern, derived from signals, together with that other, ever-growing reservoir of structured information, which is called memory (as an array of signals), makes up the stuff we think we are: a pattern of configurations, placed in time and based on a certain type of division. The storage of impressions, actions and relations in our mind is of the utmost importance to create our own visibility (identity). It explains why nobody is the same. The possibility to have access to a great variety of division models during our life, at different times and under diverse circumstances, gives an endless multitude of patterns. Nobody’s coding composition is ever the same, although certain congruences do occur in similar division choices. Understanding will be improved in the latter situation.

The idea that visibility – or in a wider sense: the employed stratagem to position our presence in a communication – could be calculated, is not new. It is the logic outcome of the understanding that visibility is a dynamic process, which is measurable. Long time ago the Catalan monk and scholar Raymond Lull (ca. 1232 – 1316) proposed a communication model based on revolving wheels. A particular combination of the wheels gave a sequence of letters, which had a certain meaning.

The Catalan monk and scholar Raymond Lull (ca. 1232 – 1316).

This elaborate system, known as his ‘Ars combinatoria’, was developed after he reached an illumination on Mount Randa, near the city of Palma de Majorca, in 1274. It was described some twenty years later in such books as the ‘Liber nova’ (1303), ‘Art generalis ultima’ (1308) and the ‘Ars brevis’ (1308). The latter book gave four basic figures to represent the relations between the primary (divine) principles: the A-figure, the T-figure, a table of relations and concentric wheels. All these figures tried to express certain dynamic properties of a communication in such a way that they are countable.

The main feature of the system was the employment of concentric circles, which were divided in ‘camerae’ (fig. 5). Each of these divisions was given a letter. A stood for God, B for goodness (bonitas), C for greatness (magnitudo), D for destiny (eternitas), etc. A total of sixteen divine attributes were distinguished. A new truth was created in a particular combination. Insight was, according to Lull, the outcome of a cyclic process and could be expressed in a simple basic principle.

Fig. 5 – The ‘figure A’, representing God, is the first to seven basic ‘figures’ in the Art of Ramon Lull. The sixteen compartments (or ‘camerae’) around the circumference contain letters from B – R (omitting J which was not used in Latin). They stand for the divine attributes. Each combination gives a truth about God.

A fresh impulse of enthusiasm for Lull’s ideas was given by Giordano Bruno, who edited in 1582 in Paris a book called ‘De Architectura Lulliana’ (De Compendiosa Architectura et Complemento Artis Lullii). This was a genuine effort to describe the dynamic system of Lull as faithful as possible. In 1587, he repeated his mission (in Wittenberg) with an edition of ‘De Lampade Combinatoria et de specierum scrutinio’, which gave a survey of the combinatorial possibilities of Lull’s system (TOCCO & VITELLI, 1890/facs. edition 1962).

The ‘Ars Lulliana’ had Bruno’s special interest because it fitted in his own inquisitiveness in the imagination and the character of knowledge. HENTSCHEL (1988) is opposed to Yates’ theses that Bruno was an ‘out-and-out magician’ and a ‘Hermetist of the deepest dye’ (HENTSCHEL, 1988; YATES, 1964; 1982) who established his hermetic philosophy for religious and political reasons. This criticism does not, in my opinion, full justice to Yates’ balanced view of Giordano Bruno’s position as an explorer of the intellectual world in the sixteenth century. Raymond Lull and Bruno, some three hundred years later, were searching for the same truth hidden in a universal communication model. This truthfulness did not tie in, for various reasons, with the power-obsessed ideas of the Roman Catholic Church in the period between 1200 and 1600.

The French philosopher and mathematician René Descartes (1596 – 1650) wrote in March 1619 to his friend Isaac Beeckman that his system of knowledge, based on analytical geometry, would be a replacement of the Art of Ramon Lull (KUBBINGA, 1989; ADAM & TANNERY (Ed.), 1908; Vol. X, pp. 156/157). This bold statement of Descartes was not far off the mark. The dualistic powers of Cartesianism became to rule the intellectual climate in Europe for a long time to come. Cyclic reasoning and its associated valuation methods were pushed to the background of the (scientific) stage in favor of a linear approach.

The German mathematician Gottfried Wilhelm von Leibniz (1646 – 1716) gave, at the apex of dualistic thinking, a new appraisal of the work of Lull. His ‘Dissertatio de arte combinatoria’ was submitted in 1666 at the University of Altdorf. The principle of division and the interaction of parts had been the major interest in Leibniz’ life. This interest led him to research every bit of uncharted territory, and he invented – simultaneously with Newton – the differential and integral calculus in 1675. He ventured, as one of the first, in the manufacturing of a calculating machine, which could not only add and subtract, but also multiply and divide – and even find the roots of numbers.

Leibniz traveled extensively on diplomatic missions and met the ‘natural philosophers’ Huygens, Malbranche and Arnaud (in Paris), Boyle and Oldenburg (in England) and van Leeuwenhoek and Spinoza (in Holland). The last part of his life was spent in relative obscurity as a librarian, preparing the history of the house of Brunswick. His opposition to the Newtonian cosmology of absolute matter, space and time resulted in the development of a theory based on the one division. He proposed, in his Monadology (1714), so-called monads as the foundation of all reality.

A major presentation of Lull’s work was accomplished by the teacher and printer Ivo Salzinger in Mainz (Germany). He published the ‘Opera omnia’ (Mainz edition) from 1721 – 1742 (YATES, 1954). Lull’s theory, in which he could ‘calculate from the fundamental patterns of nature an Art, which could be applied by analogy to all arts and sciences’ reached an apex of visibility.

The introduction of communication as a process of revolving wheels is, therefore, the continuation of a distinct historical line of thinking. Its main character is the ‘a priori’-departure that every piece of information, which is exchanged in the universe, is envisaged as the product of a contact between cyclic units. The full credit for the idea of revolving wheels must be given to Raymond Lull. None of the Greek philosophers or their Roman imitators had ever come up with such a view, although Empedocles might have been close to the target.

It was rather unfortunate that the mechanics of Lull’s Art were either misunderstood or misused in the ages to come. On top of that, it is unmistakable true that Lull himself contributed to the confusion, because he left little room for the interpretation of the combinatorial facts. He was also not clear on the nature of the initial division of his wheels. There are an eight or sixteen-division (A-, S- and X-figure), five- and fifteen divisions (T-figure) and a fourteen-division (figure V, seven virtues/vices). Furthermore, the number of concentric circles varied. Two or three circles were most common, but the figura universalis consisted of fourteen circles. The number of combinations of such a device is truly mind-boggling.

Lull can be credited to invent the mechanism of the combinatorial art and provide a practical way to calculate the intensio (approach) or remissio (alienation) between the communication partners. These terms, which were originally used by Scholastic philosophers in the thirteenth and fourteenth century to describe a change in quality (DIJKSTERHUIS, 1950/1986), are reintroduced here. They express, in a transparent way, the changes in intensity, which occur during an interchange of information. There is, for those with a historical conscience, a direct reminiscence to the oppositional forces of Love and Strife, which had long been recognized by Empedocles as the energy output of division thinking.

Partitioning is the most basic and authentic constituent of any reasoning. It brings any individual, who is interested in its nature, right in the middle of philosophy and theology. A part can be defined as a compartment of the universe, bounded by the markers of a division and subject to the laws of the universe. The two components of this definition, which are of equal importance, lead to all the problems related to delimitation and the application of rules. The part is the most important and integrated entity of a communication, and cannot escape its mutual bond.

This point was firmly taken by the ancient Greek, who regarded all gods as subordinate to a remote power called Destiny or Fate. The ultimate destiny that was called Moira. This name means ‘part’ or ‘allotted portion’ (CORNFORD, 1912; OTTO, 1954). The Greek historian Homer, living in the eight century BC, used the name, with one exception, always in an impersonal way. He described the gods, in the great epics of the Illias and Odyssee, as if they were human beings. All gods belonged to the earth, and had their share in life as in death. They mingled easily with the great heroes, like Agamemnon and Odysseus.

However, those heroic people also had their misfortunes orchestrated by Destiny or Moira (‘the Part’). They were, in other words, victims of the process of division, and, ultimately, of the whims of communication itself. Demeter’s daughter Persephone was robbed by Pluto while picking flowers and taken to the underworld to be become his wife. It was only after the intervention of the upper god Zeus, that Persephone was allowed one half-year with her mother on earth and the other half with Pluto in the underworld. The hero Ajax was drowned on his return from Troy. Orpheus was a singer, who was killed by Thracian women, because he withdrew himself from the community. Poor Tantalus, who offered his own son as food for the gods, was punished by having a perpetual hunger and thirst. And the wicked Sisyphus was condemned to push a rock up the hill, over and over again.

Many more gods had their human pleasures and pains, caused by the unpredictable nature of Moira. Her remote presence gradually turned from a spatial into a temporal concept. The single Moira (unity) became a threesome, the Three Daughters of Night. The three Fates portrayed a division in time: past, present and future. This plural aspect was first represented by Hesiod – where the three Moirai were daughters of Zeus and Themis – and came later to light in the Orphic theogony. The Fates were given as the children of Uranus and Gaia. The Cretan philosopher Epimenides (who once claimed that ‘all Cretans are liars’, and was credited as the author of the liar’s paradox) described Cronos and Eunyme as the parents of the Moirai, Aphrodite and the Erinyes. Later folk belief had lost all connections with an allotment or an area of primordial unity.

2.1. The primordial part

The primordial part or share (which the Greek called Moira) can be equated with the ultimate unity. The term points to a status of nature, which is incomprehensible for men, a contradictio in terminus. In the ultimate unity or one-division is no transgression of frontiers, because there is no fragmentation to get things going.

The one-division is a division and not-a-division at the same time. This is hard to imagine. Our human mind is so geared toward division, that a thing, which is (visible), cannot not be there (invisible). We are not used to think and comprehend in an undivided environment. An infinity that crosses our intellectual path, either by chance or after a deliberate attempt to reach the boundaries of comprehension, has to be broken down to be understood. The very moment of division also means that infinity is lost.

It seems as if the Unimaginable One has to multiply to be understood. The scholar Kent PALMER (2000) called this level of deeper Being the Enigma and referred to Merleau-Ponty (Flesh, the Wild Being), Deleuze and Guattari (the Rhizome), Arkady Plotnitsky (Complementarity), and Cornelius Castoriadis (the Magma). They all made their own effort to designate the character of a world of pre-duality.

These researchers follow – as many did before them and certainly others will pursue in the future – a road that was ventured by the before-mentioned philosopher Leibniz (1646 – 1716). He ‘invented’ the monad: a foundation of reality with no material existence, no placement in time and space, no velocity or direction and no movement. In short: a one-division. If a (further) fragmentation takes place, then all the ‘parts’ will carry the imprint of the original division in them. As a result, a colony of monads (one-divisions-in-parts) is created. Each monad is a tiny mirroring of the entire cosmos and has the capacity or potential to express the fullness of the universe.

The position of God lies, in Leibniz’ view, outside the realm of monads and potential being. God is full being, characterized as the intellect of the universe. Leibniz used the principle of sufficient reason (which tells us that for every event or fact there is a sufficient reason, even though we may not know what it is) to prove the existence of God. There must be a sufficient reason for the universe and that is its creator.

The quadralectic world view accommodates for the position of God in a similar way, but determined the entity as the principle of division. Leibniz’ assumption as God being the intellect of the universe is transferred to the position of God as the principle of division. This condition refers to the process of division only and not the actual number. There is – within the state of Being – no preference for a particular form of division.

The problem of unity (and one-ness) surfaced again at the beginning of the twentieth century as the set theory. This theory was an attempt to unify all mathematics concepts into one single theory, which contains every theory and idea in mathematics. This was thought to be achieved by regarding every number as a set. Every set contains, in reverse, the same number of elements as the number of the set. This approach (to unity) resembles the position of the ‘division’ within the modern way of thinking: every number is a division, and every division is related back to a number.

Problems did arise in the form of the ‘Paradox of the Biggest Set Ever’. If a set is defined as a ‘collection of elements’ (like a division is a collection of parts) then the question will rise: what is the biggest set there is (or: what is the biggest part there is)? This query assumes knowledge of the biggest number (part), which, unfortunately, cannot be determined. Infinity leads, by definition, to nowhere.

This paradox finds its origin in the definition of a set. A set (and a part), as a unity, needs a plurality to be understood. A ‘set of all sets’ refers to circular thinking. In the end, the truth must be faced that the definition of a set is wrong. Consequently, the set theory lost its acceptation as a theory of ‘everything’. What remained were the seven axioms of the standard set theory which are intended to be sufficient for the deduction of all mathematics (extensionality, subsets, pairing, sum-set, infinity, power set and choice) (BARROW, 1991; p. 36). The set theory did serve, despite its disappointing outcome, the better understanding of unity and plurality.

The logical paradoxes of set theory and the weird properties of infinite sets, which were investigated by the German mathematician Georg Cantor (1845 – 1918), established a climate of uncertainty in which constructivism could flourish. This widely applicable ‘ism’ starts from the assumption that knowledge is in the heads of the persons and constructed from a personal experience. All kinds of cognition are essentially subjective and there is no way to prove that the experience of one person is the same as the attainments from another person. The rigors of deductions, which lead the argument into contradictions, is avoided in a constructivist environment.

The constructivists ideas are particular suitable to side-track the singularity theorems, leading to such creations as the Big Bang theory – the most popular appreciation of the birth of the universe at the moment (WEINBERG, 1977). ‘The important lesson that we learn here’ said Barrow (1991, p. 187), ‘is that the notion of what is ‘true’ about the Universe appears to depend upon our philosophy of mathematics.‘

The (theoretical) world of one-division is born in a logic necessity to create room for further division. People realized this from the early days of man-kind. The Great Unknown or Creating Nothingness got names all over the world. And we have to admit that the labeling of this place as the ‘First Quadrant’, as will be done in the context of this book, is just another name for an unimaginable universe of which we are a part.

2.2. Two parts in the universe

The simplest division of unity (the one-division) is a partition in two pieces. The first proceeding (separation) can result in two equal parts (when severed in the middle), or in two unequal parts. The subsequent act of comparison – in a dynamic approach – differs in both a-priori starting points. The former action (halving) has been called mediation or dimidiation. It is different from a handling of division with multiple (initial) parts. The ancient Egyptians applied the first type of division-by-two as a fundamental calculation method. The method is still used in modern binary arithmetic when a bit-shift is performed that moves the number one place to the right.

The two-division is by far the most important division in a human thinking process. It marks the distinction between unity and multiplicity, expressed in the one and the many. It is the raison d’être of the part, which has distinguished itself from the universe, although it is still part of it. The ‘one’ of the one-division is a complete different entity than the ‘one’ of the two-division. The first unity is a total of itself, while the second is an outlined part of the many (minimal two).

The world of duality is secure: it is ‘either – or’, with nothing in between. This disposition may explain the widespread occurrence of this type of thinking, from the past to the present day, in all forms of biological existence. It is, in short, a natural survival mechanism based on reflexes.

Kent PALMER (2000) pointed, in his article on ‘Intertwining of Duality and Nonduality’, to the difference in use of the terms ‘dualism’ and ‘duality’. He distinguished a philosophical and a mathematical interpretation of the dual. The first (dualism) ‘has to do with the production of nihilistic artificial and extreme opposites’, like the Mind/Body dualism. The second (duality) is a mathematical concept, ‘and would be better stated in terms of complementarity rather than duality’ (What is Life and Living, 2000).

Palmer noted that ‘throughout the history of the Western tradition dualism uses duality as part of the armament by which it builds up dialectically opposite arguments and philosophical positions, which spar with each other.’ He pointed to the Orient, which took – in his view – a different direction of thought by placing an alternative in the middle between dualism and dogmatic monism. Aristotle’s principle of the Excluded Middle (‘every proposition is either true or false’; On Interpretation; Metaphysics, Book 3) and the banning of contradiction was seen by Palmer as the first principle in his metaphysics. He observed that Aristotle ‘forced our (Western) tradition down the road of Dualism.’

LLOYD (1971) described polarity and analogy as types of argumentation in early Greek thought and concluded ‘the fact is that other societies, whether ancient or modern, provide a great deal of evidence concerning dualistic theories and beliefs of different sort.’ The Greek philosopher and physician Alcmaeon of Croton (c. 540 – 500 BC) believed that most human things go in pairs. He was the first to identify the brain as the seat of understanding, which acted as a vital organ for perceptions, thoughts and sensations. Alcmaeon’s dualistic thinking formed the base of his theory of health. He composed a ‘Table of Opposites’,’ which was used by the Pythagoreans.

Alcmaeon saw the human being as made up of opposites: the hot and the cold, the moist and the dry, etc. A disease was seen as a ‘monarchy’ of one of the members of the pair. Health meant the rule of a free government (of the extremes) with equal law. The term ‘isonomy’ expressed the balance of all possible antagonisms (in the human body). The philosopher Parmenides (c. 540 BC) exploited the dualistic theme further in his ‘Way of Seeming’ (the every day perception of reality of the physical world) as a continuation of the opposition of light (day) and darkness (night).

This latter theme had contemporary roots in Zoroastrianism, with its ancestry in the northern part of Persia (Iran). The Iranians have a distinct history of lower division thinking, which originated in a distant past and can be traced to the present day. The tenacity, in which the oppositional way of thinking has been preserved over the years and the historic efforts to escape from it, provides a microcosm of human division thinking. Their history is also a narrative of the development of division thinking in general. A brief outline of its chronicle will be given here to indicate the transition of (division) conceptions within a cultural setting.

Zoroastrianism is the oldest monotheist religion, based on the word of the prophet Zarathustra (or Zoroaster), living somewhere between 1200 and 600 BC. The message of goodness was orally transmitted until the words were put on paper under the Sassanians (226 – 651 AD) in a book called the Avesta. The sacred sections are formed by seventeen great hymns, called the Gathas. Later literature includes the Phalavi Texts, with quotations and paraphrases from lost Avesta texts.

Zoroastrianism was the major religion in the Persian empires from the sixth-century BC until its substitution by the Islam in the seventh century AD. The teachings of Zarathustra had influences on other religions like Judaism, Christianity and Islam. The (dualistic) concepts of a free will, Heaven and Hell, the future resurrection of the body, the Last Judgement, and life everlasting were borrowed from Zoroastrianism (BOYCE, 1979).

The Supreme Being was called Ahura Mazda (Ormuzd, God), meaning ‘Wise Lord’. He was seen as the creator of heaven and earth and visualized as a symbol of light or fire. In this capacity, he reflected the one-division. However, this god of light was permanently fighting with Ahriman, the representative of a destructive spirit, living in darkness. Good (light) and evil (darkness) were in an eternal opposition.

The dualistic aspect is – on an intellectual level – extended to the three-division in Time. The sacred number three found an expression in a ‘limited time’, consisting of a period of Creation, Mixture and Separation (each lasting a thousand years). Zoroaster was born towards the end of the third millennium, or the first half of the ‘world year’ of six thousand years. The subsequent periods in the second half (of the world year) all had a Saviour (or Saoshyant, one who brings benefit). The first one, Ukhshyat-ereta was ‘He who makes righteousness grow’. The second, in the year 5000, was his brother Ukhshyat-nemah, or ‘He who makes reverence grow’. Finally, there was the greatest of the Saoshyants, Astvat-erata. The latter brings the final renovation of the world, the Frashokereti.

The last stage in the development of division thinking was shaped in one of the few deviations of the basic Zoroastrian doctrine known as the Zurvanite heresy. The Zurvanites found their scriptural justification in the explanation of a Gathic verse (Y 30.3), in which the two primal spirits (good and bad) are depicted as a twin. When there are twins, there must be a father and so Zurvan (or Time), took that place and assumed power over Ahura Mazda and Anra Mainyu. Zurvan became the Lord of the Three Times. This separation of the ‘means’ as a ruler (of a trinity) let to a new quaternity. The number four was, in due course, a prominent element in the cult of the Zurvanites (BOYCE, 1979; p. 69).

The conceptions of Ahura Mazda changed over time and an orientation away from the dualism of early Zoroastrianism could also lead (back) to monotheism. The main agent in this process was the position of ‘power’, being either subscribed to a single creator of the world or in a divided pantheon.

Fig. 6 – The Faravahar or Farohar, was the winged symbol of the Zoroastrians. It was an adapted form of the motif of the ‘spread eagle’. This form, which features a flying bird shown from below, with its wings, tail and legs outstretched, has a wide cultural distribution in time and place.

The faravahar is the symbol of our human spirit, which already existed before we were born and which will continue to exist after death (fig. 6). All events of our world are based on cause and effect. In the center of the figure is a circle, which represents the soul of the individual. The cyclicity points to immortality, because there is no beginning or end. Each wing has five layers of feathers, which might represent the Five Divine Songs (Gathas), the five divisions of the day (Gehs) and the five senses of the human body.

The meaning of the Persian Faravahar symbol is ambivalent. Some scholars see it as a symbolic image of Ahura Mazda, others as an aim to reach the One God, the ‘Wise Lord’. Irach Taraporewala identified, in 1928, the Winged Disk as the khavarenah or royal glory.

The fight between the sepanta meynu (the good force) and the ankara meynu (force of the bad/evil) are essential for the faravahar (spirit) to reach maturity. Only by strengthening the good force in us and suppress the evil force can we reach the final aim: a higher level of future existence. The key to life is vohu mana (divine wisdom), which means the attainment of a good mind, wisdom and good thinking. The divine fire (mainyu athra) is empowered by truth, a key word in the Zoroastrian religion. Alternatively, like it is expressed in Yasna 34.4:

‘Now, we wish Thy fire, Lord, which possesses strength through truth and which is the swiftest, forceful thing, to be of clear help to Thy supporter but of visible harm, with the powers in its hands, to Thy enemy, Wise One’.

The motif of the eagle was used in Egypt as early as the second millennium BC (notably by pharaoh Tut-ank-amoun), and found its way to the Hittites of the ancient Near East. In Syria, it was shown on a seal from the Mitanni civilization (c. 1450 – 1360 BC). Assyrian art had their own, independent, version as a winged disc associated with divinity and divine protection. The design that would become the Faravahar flourished during the Achaemenid kings, from about 600 – 330 BC. It is, however, absent in the art of the Sassanian period, from AD 250 – 650. In fact, it were the European antiquarians of the early twentieth century that gave the symbol a new lease of life. A renewed awareness among Zoroastrians worked in favor of its popularity.

The same ‘heresy’ as the Zurvanites with regards to a (holy) trinity is known in Christian-dominated Europe. This development started with the Arian heresy in the early fourth-century AD. The Cyrenaicean priest Arian, living in Alexandria, shook the young Christianity with the claim, that God deserved a separate place, outside the Trinity. By doing so, he created a quaternity and challenged the basics of division thinking (and consequently, the position of power).

The controversy flared up again between Joachim of Fiore and Petrus Lombardus, which took place at the end of the twelfth century. ‘Joachim accused Peter Lombard of Sabellianism and Arianism, of overemphasizing the unity of God at the expense of His threeness to such an extent so as to make a quarternity of persons by separating the ‘deitas’ or ‘essentia’ of God too distinctly from the persons’ (BLOOMFIELD, 1957). Joachim characterized Petrus Lombardus as a ‘quaternator’, who made God into a quaternity (DANIEL, 1980). And he was completely right: the year 1200 was – from the perspective of a modern observer – a historic dividing line between the old (quaternarian) and the new (trinitarian) way of thinking in the cultural history of Europe.

The world of the two-division is, in nature, a simple one, but becomes complex as soon as dynamism is introduced. The cause of the complications is found in the definition of the communication itself. Its main components – division and movement (fig. 4) – should both be present to fulfill the requirements of a ‘real interaction’. The individual components, operating on their own, do not provide the complete essence of human understanding.

2.3. The three-division creates dynamism

A division of (a ‘linear’) infinity into a grouping of three (triad) requires two points of partitioning. The three-division is in many ways a more dynamic and versatile entity than the two-division (dyad). The latter is direct and elementary. The three-division gives, in contrast, more space to maneuver. The three terms – regardless of their interpretation – can be juggled and a middle term can absorb the sharp edges of opposition. This capacity has challenged the intellectual minds of mankind and given them a vast and near boundless space to create a new reality.

Neoplatonism and Christianity are most noticeable for the development of a system based on the triple division. Both spiritual currents experienced their infancy in the first centuries AD. This crucial time in European and Middle Eastern history was characterized by the hectic activity of different types of division thinking, translated in power struggles and religious currents.

The reason to embrace the trias (as a division system) at that particular time, can be interpreted as a reaction to the growing quadripartite tendencies in the ‘years of power’ of the Roman Empire. The celebrated poet Vergilius (70 – 19 BC) created in his ‘Georgica’ a tetradic epos, in which Empedocles’ cycle of love and strife sounded as a distinct echo. The arcadic opening was followed by a phase of dynamic growth and development, characterized by strife. Than the advancement crystallized again in a static reality, where the identity was born in love. The highest/dynamic understanding was finally found in the last section, but not without strife.

The Roman poet Ovidius (43 BC – c.17 AD) added more material to the acceptance of a four-fold cognitive framework. His creation story of the earth was a tetradic model in a nutshell. He distinguished four ages: the Golden Age (of unbound happiness), the Age of Silver (with the institution of the four seasons and agricultural labour), the Age of Bronze (with a fierce character) and finally, the Age of Iron, where modesty, truth and loyalty fled. Tetradic affinities were found in the architecture of the palaces, bathes and other structures of the emperors like Trajan (AD 98 – 117), Hadrian (117 – 138), Caracalla (212 – 216) and Diocletian (284 – 305). The latter emperor even concentrated (twice) his political organization on the four-division, known as the tetrarchy of Diocletian.

The mythological Roma quadrata (VON GERKAN, 1959; MÜLLER, 1961) was recreated in the Italian Renaissance. The first centuries AD were inspired by the ideas of Neoplatonism, which influenced the persons in power. Plotinus (AD 204 – 270) was a Greek philosopher from Egypt, who gained many followers after his settlement in Rome in 244. He got permission from Emperor Galienus to design a city based on the rules set by Plato, but this project never materialized. Plotinus’ book ‘The Enneads’, compiled by Porphyrios, is still a cryptic compilation of knowledge in which the triadic organization was a guideline. Stephen Mackenna’s translation of the ‘Enneads’ (in English, Larsen Publications, 1992) reached for the essence of Plotinus’ writing. However, a quadralectic mind will be necessary to understand it.

Another reason to embrace the trias can be determined as an action of dynamism derived from the One (monas). The emerging Christian faith followed the spirit of division thinking by taking a direct step from the monas to the trias. God – in the unity of a one-division – was fragmented in a Father, Son and Holy Spirit. The pitfalls of the dualistic system were surpassed and the soul, in its journey through the stages of being, came to rest in a heavenly trias. The motion was a world-fulfilling act of man in reaching the Unity. This transition from a monadic into a triadic system was, among others, recorded by the Greek philosopher Porphyrios of Tyre (ca. AD 232 – 304) in his explanation of the Chaldaeic oracles.

The fourth-century grammarian, rhetorician, philosopher, and theologian Marius Victorinus (ca. AD 280 – 365) used two sequences of the Trinity. Firstly, he employed the Father – Holy Ghost – Son succession. And secondly, the traditional (Neoplatonic) order was mentioned as Father – Son – Holy Ghost. The latter was favored by Plotinus in the form of One, Nus and Soul. Marius Victorinus expressed his devotion to the three-division in the following verses (v. 210 – 213) of his Hymnus 1:

I praise thy, Unity

I praise thy, Trinity

One will be as three

And three will be as one.

This hymn expresses the three-fold way of thinking in miniature. It displays, in a direct way, the ‘jump’ from the unity (of the one-division) to the multitude (in a three-division). The last sentence closes the division circle again.

The synthesis between the (Neo) Platonian trias and the Christian trinity was brought about some century later by Synesios of Cyrene, who was bishop of Ptolemais around AD 410. This colorful character studied in Alexandria, served the army, visited Constantinople (and Athens) on a diplomatic mission and wrote many books and letters. He died about AD 414. The subjects of his interest were wide and varied, ranging from the breeding of dogs (Cynogetics, not extant) to a treatise on dreams (De insomniis). His election of bishop and his subsequent duties were uncongenial to him. One hundred and fifty-five epistles and ten hymns from his hand are known. Petavius in Paris published the only complete edition of Synesius’ writing in 1612.

The concept of the trias has been extremely powerful throughout the ages, mainly because of its versatility. The three options of positioning surpass the dual cause with its two probabilities. It withdraws itself from elementary language of survival based on reflexes. It gives the act of thinking a human flexibility, which lifts it above the basic instincts of the animal world. It provides, in short, mankind with some sort of dignity.

The adaptable content of the trinity was attributed to God when the times of struggle for spiritual supremacy flared up in the early Christian era. The advantage was that the principle of power (related to oppositional thinking) was still close at hand and the lines of decision taking remained short. The Trinity of Father, Son and Holy Ghost became the representative of God in this world (fig. 7).

Fig. 7 – The trinity of Father, Son and Holy Ghost (Pater – Filius – Spiritus Sanctus) as given in a woodcut from 1524. This tripartition was used in the Christian faith to capture the three-fold way of thinking in a comprehensive way. This picture also shows the position of God (Deo) in the middle, i.e. inside the trinity. Above the triangle is his triple face, as a ruler of the division. The counting of God as an extra entity can easily extend the meaning of the trinity into a quaternary, but the Catholic Faith is explicitly against this move. The symbol of the four evangelists in the four corners (Saint John as an eagle, Matthew as a man, Marc as a lion and Luke as a bull) nevertheless strengthen the idea of a higher division.

The tripartite psychological setting can be associated – in a quadralectic interpretation – with a Nietzsche’s ‘will to power’ (der Wille zur Macht). The two-division is too obvious and simple a tool to decide over the right or wrong (for oneself and for other people). The three-division offers, on the other hand, a suitable frame to manipulate the mind. Different abstract positions can be taken. A position in one (of the three) segments is still near the ‘save’ world of duality. A retreat, if necessary, is a simple move.

2.4. The tetradic sense of stability

A division of (a ‘linear’) infinity into four sections requires three points of partitioning. Three pegs can stake out four regions. The four-division is more stable than the three division. The tetradic frame of mind keeps static and dynamic forces at bay. It is broad, balanced and accommodates the lower kinds of division thinking (with monad, dyad and triad as coagulating agents) within a structured context.

The compartments in a (universal) four-division are called quadrants. They are the most important units within the quadralectic frame of mind, because they are the designated areas of different forms of visibility. Each quadrant has its own type of visibility (which will be discussed later). The definition of the basic element in the four-fold division is as follows:

A quadrant is an autonomous part created in the four-division of a unity.

This qualification does not have a reference to size or shape, because the (ultimate) unity is not defined. A quadrant is a theoretical unit, a reflective compartment, which acts as a storehouse for different forms of visibility and division thinking.

The definition of a quadrant needs some further explanation. It has – in the spatial realm – a reference to the fundamental region, which is a term, derived from symmetry. LAUWERIER (1988) called the area a ‘primitive cell’ in his book on symmetry. Symmetry is, essentially, a division of a given space according to fixed laws. STEVENS (1980) distinguished four types of movements to obtain the multiplication of a pattern:

Stevens gave, in his study on regular patterns, the following definition of a fundamental region:

A fundamental region is the region of a minimum area that can be repeated without gaps of overlaps to make a complete pattern.

This definition of a particular, limited area of space has an affinity with the characteristics of the quadrants (in a quadralectic communication), which make up a unity (or cycle). The quadrants-as-a-whole are valued as a fundamental region, which is defined as an autonomous part and is limited by specific boundaries. The tetradic pattern can extend into infinity, just like a primitive cell in a symmetrical pattern expands in all directions.

Each quadrant is, in a quadralectic setting, related a particular form of (dualistic) visibility (fig. 8). The information starts with a primary pair (visible versus invisible). A further interpretation of the different combinations is derived from a theoretical shift of two four division along each other. The actual procedure and the creation of a universal communication sequence will be dealt with later in this book (p. 90 onwards).

Fig. 8 – The quadrant is a compartment in a universal four-division, which is characterized by a particular form of visibility. The types of visibilities as given here are an approximation from the world of duality. The number of the quadrant does not indicate a hierarchy since the units are placed in a cyclic setting.

It is possible to imagine a ‘growth’ of division thinking (from one to four). An immovable observer (position W) will encounter in the various quadrants a cyclic movement of different types of divisions (fig. 9). The dynamism will increase in the process. The various ‘values’ (of the individual quadrants), which have to be taken into account expand accordingly. The (stable) observer will become, in a later stage, also a fourfold, cyclic entity, which partakes in a dynamic interchange of information.

Fig. 9 – The development of the four-division from a one-division and the valuation of a contact point between the cyclic movement of the division and a fixed (static) observer (position W).

A communication established in the four-division is a most practical way of understanding, because its harmony covers the whole field of human possibilities, ranching from the complete unknown to a subjective interpretation of one’s own, personal choice. All stages of comprehension are ordered in a logical way, allowing for every participant to choose a comfortable niche as a means of operation.

Lower forms of division thinking can be spotted from the higher advantage point of four-fold thinking. The valuation of data exchange, or the quality of a communication, has to be rated accordingly. There is hardly any point to ‘fight’ the oppositional thinker, since that the outcome is not more than a (brief) episode in the conscience of a higher division thinker.

A hind of structuralism seems to be incorporated in the emphasis on division. A multi-divisional communication could be seen as a structure or system, operating independently of its contributors. This impression is, categorically, wrong. A subjective choice at an early stage is necessary, if understanding will ever be possible. That choice is, however, not limited to oppositional pairs – like the binaries in the deconstruction ideas of the French philosopher Jacques Derrida – but encompasses four positions, with the whole scale of possibilities enclosed in each of them. The ‘ultimate’ choice is not restricted to oppositional pairs, but comprises a motivated registration of the position on the communication cycle.

Fourfold thinking is often disguised as a form of wisdom, practiced by elderly people and spiritual leaders. They have reached, by experience or intuition, a state of knowledge, which includes the (four) possible positions in a communication. Any advice offered by the sagacious men and women throughout the ages to the ignorant parties (in a conflict) consisted of a suggestion to broaden their horizons and open their minds to ‘the things that have always been’ (fig. 10).

Fig. 10 – An Ojibway medicine lodge parchment with the ‘things that have always been’. The Mugwa (bear) traveled from east (E) to west (right to left) through the four compartment (lodges) in the earth shelf. Collected by the Canadian Ethnology Service of the National Museum of Man (in: DICKASON, 1984).

2.5. There is more to five

Thinking in a five division is, in a practical sense, the ultimate extension of the human mind. Any higher division – although possible on an intellectual level – is too far away from the material world to be of any use. The distinction between pure division thinking and the numerological approach should be clear at this point. The former relates to a corpus of interactions in which every compartment contributes to the operation of the whole. The latter is only a distinction in the number of contributors, without any internal structural relation to the whole, except in certain features.

The Pythagoreans (described by Plato in the Timaeus) were attracted by the five regular geometrical solids, as found in certain crystals, and regarded them as fundamental building stones (of a tangible world).

The ‘Platonic’ solids or regular polyhedra are polygons whose vertices and faces are all of the same type. In three dimensions there are only five regular polyhedra: the tetrahedron (number of faces 4), the cube (6), the octahedron (8), the dodecahedron (12) and the icosahedron (20).
Plato (in the Timaeus) associated the cube with the four-squareness of the earth, the tetrahedron with fire, the icosahedron with water and the octahedron with air. The dodecahedron (12) was, according to Plato, for ‘embroidering the constellations’, pointing to the zodiac and the cosmos.

The Chinese culture also ventured in the terrain of the five-division. The way of classifying things by fives was already given in the famous Tao Te Ching and the Shu Ching (the Book of History), both of uncertain date and authorship. The actual theory of the five elements was provided by Tsou Yen (c. 350 – 270 BC), of the Ying-Yang or ‘Cosmologist’ School. He was seen as the founder of Chinese scientific thinking, the latter so thoroughly described by Joseph NEEDHAM (1954/1988; RONAN, 1978/1986).

The four cosmological elements (wood, fire, metal and water) are jointed in the center by a pivotal unit, the earth (fig. 11). This setting leaves a flexible, but uneven configuration of (sub) units. The pentagonal division has this unbalanced character in common with the three-division, but the wider symmetry pattern around the centre compensates for the small scale of triple thinking. The deep and rich world of the five division will, at this point, not further elaborated.

Fig. 11 – The five elements in the Chinese culture, as established by Tsou Yen in the third century B.C. offer the theoretical foundation to introduce the five-fold way of thinking. The elements are related to colours, symbols seasons and directions.

3. Preliminary movements

Division and movement were marked as the most important constituencies of a communication. The division has been briefly indicated as a rational act of slicing up a unity into parts. More parts (in a primary division) mean more opportunities to distinguish detail, but their presence also complicates the understanding.

Personal communication has, for that very reason, always been an act of compromise. Certain situations, mostly in the field of biological survival, ask for decisive and immediate action. Two-fold thinking is most effective in this way. This decisiveness by lower division thinking, however, also operates in complicated situations or crisis in which no rational solution is at hand. When nobody knows what to do, people tend to drift to elementary answers. Dictators are masters in using these tendencies in the human mind. They use the arsenal of oppositional devises, aiming at some sort of existential fear to continue their power.

Fortunately, there are in history also times in which the primary survival techniques are not of direct interest. Then the range of division can be widened, leaving room for nuances and attention to detail. It is granted that the decision speed has to suffer in this process, but the depth of investigation is enhanced. At first, a third term is added, as a go-between, to the ‘either – or’ of dualistic thinking. The triple division system offers more dynamic possibilities, but is, in essence, unbalanced. This fundamental inequality gives the chance to create new value systems, but has also been used to the advantage of those who were interested in power play and hierarchical predominance.

Later, generally as a mature step in the practical application of division thinking, a fourth term can be introduced. The communication system is back as equilibrium. The hierarchy is not the main characteristic of the quadripartite order, and the gradations will be confined to specific places within the subsections. Lower division arrangements also get their own place within the boundaries of the four-fold system.

All these theoretical accomplishments within the evolutionary range of division thinking are only possible when a second conceptual act is performed: the movement of the (hypothetical) parts along each other. An observer can only compare, validate and make visible if a shift in the division is performed.

A division without movement is a static affair, situated and received on an abstract level. Mathematicians can play with it, just like Mandelbrot did with his fractals, born in the highly theoretical environment of broken dimensions (MANDELBROT, 1982). A division becomes ‘visible’ when the parts are involved in a dynamic process of comparison with the observer. When I see an object (a part), it is only because I notice boundaries between the object and its environment. Observation is a continuous process of to-and from or in short: movement. A comparison (leading to a form of ‘visibility’) can only take place as a movement takes place.

The four major means of transportation (or movement) of the initial material in a communication can be listed as follows:

These four types are related to each other in a circular way. The signal gives rise to a symbol, which is understood as a sign and used in a language. The opposite movement is also possible in which a language produces signs, which are interpreted as symbols, functioning as signals. A graphic representation of the arrangement is given in fig. 12:

Fig. 12 – This scheme indicates the means of transportation in a quadripartite communication. The sequence must be seen in a cyclic setting whereby each element is related to, and included in, the other elements. The sequence of quadrants does not indicate a hierarchy of importance.

The four major components of every communication will be discussed briefly, because they reach into the heart of the quadralectic issue. The four constituents are the ‘faces’ of the quadrants. Their character can be read in their expression.

3.1. The signal

The signal is the primary feature of a communication. It stands at the very base of understanding. Without a signal there would be no vestige of a communication at all. The signal as a singularity is a rare phenomenon, if not impossible to imagine. Its definition encounters the same problems as the one, which was experienced in the concept of unity. The very moment a definition (limitation) is given, the spell of its existence is broken. The undivided world cannot be caught in definitions without breaking its authenticity. The same characteristic is applicable to the signal, as a voice from the deep unknown. Its initial unity is broken the very moment it is noticed by an onlooker.

The observer deals more often with an array: a multitude of signals of the same nature. Radar and sonar are examples of array signals, which are used to build up a new kind of visibility. Acoustic and vibration signals are illustrations of the practical application of signals, for instance, in seismic prospecting. The present age of computerization is highly indebted to communication signals, which are sent by modem, cable or wireless. The human body itself creates physiological signals, including clinical and speech signals, which can be registered and analyzed. The world is full of noises of a natural or synthetic character. And last, but not least, there are signals from outer space, which possess the challenging prospect of a contact with other creations in the universe.

The Croatian genius Nikola Tesla (1856 – 1943) was one of the first experts to see the potential of the signal as a source of communication. His experiments with electricity and the wireless transmission of energy (including the possible cause of the Tunguska catastrophe in 1908) are a specimen of the incredible forces, which could be unleashed when the potentiality of the unseen is tapped. Tesla followed in the footsteps of Michael Faraday (1791 – 1867), the blacksmith’ son, who pointed the way – some sixty years earlier in 1831 – to the production of electricity by magnetism (if it was accompanied by motion).

Tesla’s inventions included the AC-power (both 2-phase and 3-phase), broadcast power (radio wave propagation), microwaves and radar. The Tesla Coil is a transformer, which generate very high voltages at high frequencies. A most common form of the transformer is used in neon signs. It is interesting to note here that a signal (high voltage) is used to power a sign.

Tesla’s vision included the transmission of electrical energy through the earth to be picked up wherever it was needed. The globe, even with its great size, responded to electrical currents just like a metal ball. He expressed his original idea in 1911 as follows: ‘The entire apparatus for lightning the average country dwelling will contain no moving parts whatever, and could be readily carried about in a small valise.’

The research in signal processing has taken a huge flight since then and continues to do so. The signal is the messenger of the invisible invisibility of the First Quadrant, who brings never-ending opportunities to mankind. The present interest focuses on the domains of influence, which form a direct link to a division-based communication. In particular, the geometric representation of tessellations could be a practical tool in the interpretation of signals used in human relationships. The verb ‘to tessellate’ points to the arrangement of squares in a mosaic pattern. The Greek word ‘tessares’ means ‘four’, because initially square tiles were used.

The so-called Voronoi tessellation (or its name equivalents like the Dirichlet domain, Wigner-Seitz cell, Thiessen polygon or Brillouin zone) is applied for analyzing cell-like structures and the division of space into regions (fig. 13). The Voronoi diagram can be used in a wide field ‘from archeology to zoology’ (DRYSDALE, 1993). The representation of lattices is useful in the identification of clusters (in astronomy), the location of areas of growth (in biology), the modeling of sphere packing in chemistry, the pattern of settlement (in geography), the estimation of mineral reserves (in geology), data analyzing in marketing and much more. A further investigation into its use in quadralectic thinking might be a worthwhile undertaking.

Fig. 13 – The Wigner-Seitz (WS) cell is a volume made up of space which is closer to a given lattice point than to any other point. The construction starts from a lattice point and the drawing of lines to its neighbours. Next is the drawing of perpendicular bisecting planes to these lines, half way along. The smallest volume enclosed within these planes is the WS cell. The example gives a rhombic dodecahedron for a face-centered cubic lattice.

Visions of Four Notions

Introduction to a Quadralectic Epistomology

Geen categorie

Visions of Four Notions – Contents

1. Introduction

2. Right from the start

2.1. The primordial part

2.2. Two parts in the universe

2.3. The three-division creates dynamism

2.4. The tetradic sense of stability

2.5. There is more to five

3. Preliminary movements

3.1. The signal