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:

  1. The acceptance of a distinct area of complete and unlimited incomprehension in the invisible realm;
  1. The knowledge that certain ideas can be developed for reference and mental guidance;
  1. The admission of the existence of an area of limited comprehension in the visible realm;
  1. 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).

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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.

  1. 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.
  1. 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.
  1. 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.
  1. 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:

  1. The collection of information,
  1. The valuation of the information and
  1. 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.


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