PC  Understanding – even at the linear level – necessarily involves a dynamic objective-subjective interaction (world and self). When one becomes conscious (externally) of the world, objects are thereby posited in consciousness. This is the very nature of conscious activity, whereby only one pole is posited. This then creates an imbalance in terms of the neutral unconscious which inherently is based on the complementarity of both positive and negative poles. So through interaction with the unconscious, the negative pole is introduced which causes a switch in consciousness. We now become aware of the (internal) subjective aspect of experience which is likewise posited in consciousness. Experience thus keeps switching from world to self (and self to world) which are both posited (i.e. made conscious) in experience. So even at the linear level, conscious and unconscious are dynamically involved. Explicitly one interprets experience in (solely) conscious terms. Implicitly however the unconscious is continually necessary to enable dynamic switching between both poles to take place.

Of course, when the unconscious aspect is not recognised, this switching becomes mechanical so that objects assume a static absolute validity (unaffected through interaction with mind). Equally subjective attitudes - in the form of unquestioned assumptions - become increasingly rigid so that both poles of behaviour reinforce each other.

Conventional mathematics is a fine example of this at work. In experiential terms, all mathematical activity is inherently dynamic requiring the interaction of conscious and unconscious. However in formal interpretations of this activity only the (rational) conscious is explicitly recognised. Thus though the (intuitive) unconscious still remains essential implicitly to enable interaction as between mathematical objects and the mind to take place, explicitly it is not recognised. Thus objects appear - in this worldview - as absolute. This in turn, is supported by a framework of fixed assumptions (regarding the nature of this activity).

What is not properly appreciated is that mathematics – as we know it – results from extreme specialisation of the linear level, where experience – in dynamic terms – is substantially misinterpreted. Putting it more dramatically, it represents a very limited and distorted interpretation of true mathematical activity.

Q Explain how the important transition from linear to the circular level alters this limited interpretation of mathematics.

 
PC  Actually, this transition is well documented in the mystical literature. However the translation required for mathematical purposes is necessarily somewhat different. Though many factors are involved in transpersonal development, I believe that personality characteristics are especially important. Let us go back for a moment to the switching patterns of the linear level. Now with some people, the unconscious is inherently sensitive and plays a more dynamic role in switching as between external and internal poles. Thus through this process a substantial fusing of opposites takes place generating spiritual activity. Against this background of growing psychic energy in the personality positive translations of linear consciousness become increasingly unstable. One’s ego can no longer rest easily in conscious objects. One then becomes increasingly disillusioned and frustrated searching vaguely for a deeper meaning not provided by conventional pursuits. In other words a true existential crisis – usually on the threshold of adulthood – takes place.

What actually happens is very interesting. Ego attachment to the objects comes from an over-identification with the positive (conscious) pole of experience. Thus to be weaned from this attachment one must be increasingly exposed to the complementary negative (unconscious) pole.

In other words attachment to conscious objects is literally – in dynamic terms – negated and undone through developing this negative pole.

This is referred to in Christian mysticism as purgation or alternatively "learned ignorance" or "cloud of unknowing" which – in my translation – is the negative (unconscious) direction of experience.

Thus we are now in a position to begin to appreciate what holistic mathematics is all about.

In conventional mathematics addition and subtraction (positing and negating) are mutually exclusive and treated in absolute terms. When we combine a (positive) number with its negative we get zero (which has a merely static interpretation).

In holistic mathematics addition and subtraction (positing and negating) - which actually represent the interaction of reason and intuition in experience - are mutually complementary and treated in relative terms. Likewise when we combine positive and negative we get zero (which now has a dynamic interpretation as the pure potential for existence). In Christian mysticism this is sometimes referred to as nada (which literally means nothingness). 

In conventional mathematics the two key numbers are the binary digits 1 and 0. (Note that we actually use a straight line and a circle to represent these digits respectively). 0 can be fruitfully represented as +1-1.

In holistic mathematics the two key numbers again are the binary digits 1 and 0. 1 symbolises reason which – as we have seen - is a linear (one dimensional) approach geared to the generation of (separate) units of experience. 0 symbolises intuition which is non dimensional, and when expressed in reduced rational terms is a two dimensional circular approach involving both positive and negative directions (+1 and -1). In other words we generate intuition through dynamically negating the positive rational direction of experience. Thus 0 here represents pure intuition. This is nothing (in rational conscious terms) but yet represents the pure (unconscious) potential for existence.

These connections are not accidental. The line and circle in fact have been long used to symbolise conscious and unconscious respectively. In Western religion – where the rational (linear) approach predominates - Supreme Being (God) is seen as One. In Eastern religion – where the intuitive (circular) approach is more established - Supreme Being is often seen more in terms of a void (i.e. nothing in phenomenal terms).

So just as the binary digits of conventional mathematics are so important in computers for encoding all information systems, likewise the binary digits of holistic mathematics are extremely important for encoding all transformation systems (physical and psychological). Transformation - at all levels of reality - results from the interaction of actual (1) and potential reality (0).

 
Indeed a complete fundamental interpretation of reality would be in terms of a double binary system combining the binary system of conventional mathematics with the binary system of holistic mathematics thus enabling both information and transformation processes to be encoded.

 
Q.  Can you briefly outline the stages of this transitional level?
 
 
PC  We are talking about negative stages here. As I have said before these are referred to in mystical literature as the way of purgation and are very painful and distressing. Though the purpose of these stages is to develop the unconscious (through a refined intuitive capacity) initially one is aware only of what is lost. Indeed it often feels as if one's whole world is falling apart. It is only later – when there is a return to consciousness – that the fruits of these stages become apparent.

At first more superficial linear (rational) stages are negated and undone. These would relate to the concrete contents of external cognitive and affective structures (and to a lesser extent the internal contents of these structures). This is the active night of sense (St. John of the Cross).

Some negation of deeper formal linear stages (esp. external) would also take place but in truth this is not likely to significantly happen till a later stage. This is the active night of spirit (St. John of the Cross).

Q.  Can you elaborate further on the (holistic) mathematical implications of this transition?

  
PC  Well, I have stated before that conventional mathematics is misleadingly interpreted as a rational pursuit. The positive direction of experience is solely recognised at the linear level.

However we now have introduced the negative direction of experience which represents – in direct terms – the intuitive or more generally the potential - aspect of experience. So by using both directions we are able to see mathematical activity – and indeed all experience more correctly in dynamic terms as the interaction of both reason and intuition.

This means that all mathematical objects have now two directions.

In conventional mathematics a number has one direction only (i.e. positive).
Thus it appears - to the rational mind – to exist unambiguously in absolute terms.

However a number now has two directions (i.e. positive and negative). Psychologically this allows for both rational and intuitive interpretation of a number.

It now exists (paradoxically) in dynamic relative terms. 

This also applies to mathematical concepts. The number concept also has two directions.
More generally all perceptions and concepts – as in science – are now relative and two directional.

We could also say that all hypotheses and theorems in mathematics and science have two directions.
In other words every theorem has its own anti-theorem.  

In conventional mathematics an absolute number arises from ignoring the sign of the number.

Likewise in mathematics and science an absolute interpretation of a theorem arises from ignoring the sign (i.e. direction) of the theorem.

Thus in static terms the positive and negative directions coincide and therefore the distinction is unnecessary.

However in dynamic terms positive and negative directions actually conflict with each other leading to continual change.

Thus all theorems in mathematics and science (e.g. the Pythagorean Theorem) have now a strictly relative validity. Again all physical objects – in dynamic terms - have now two directions (positive and negative). In other words there is an external and internal aspect to all objects of science.

This is recognised at the sub-atomic levels (where matter and anti-matter particles ceaselessly interact). However it is strictly speaking true of all objects (at this stage).

More surprisingly the dimensions of science have both positive and negative directions.
Each dimension has its own anti-dimension. (We will elaborate further on the profound implications of this later). Indeed one of the great contributions of holistic mathematics is in demonstrating how inadequate is the conventional interpretation of objects and dimensions in science.

There is a fundamental lack of symmetry in the standard treatment of objects and dimensions. Science deals with rational objects (i.e. objects which can be broken into parts) in natural dimensions. There is little appreciation therefore of fractional or negative dimensions. However in holistic mathematics these have a precise interpretation which perfectly complement psychological understanding.

Basically objects in science are complemented by corresponding psychological perceptions.

Dimensions in science are in turn complemented by corresponding psychological concepts.

The (external) world and the (internal) mind are now fused in the one holistic experience. And as these terms are strictly relative the world and mind have external and internal aspects respectively. In other words once again psychological and physical reality both display positive and negative directions which interact in complementary fashion.

Q. You have introduced a lot here. Let us now move into the circular level proper. It would be helpful to tell us a little about this level.

 
PC  Again, to recap, the positive direction of experience is developed during the (rational) linear level. The corresponding negative direction of this rational level is developed during the transition from linear to circular level. It represents the radical undoing of conscious rational activity and the corresponding generation – deep within – of intuitive ability. However this remains concealed for some time.

The circular level properly commences when there is a return to conscious activity.

However because of the considerably development of (unconscious) intuition, this new conscious experience undergoes a profound transformation. In mystical terms this is generally referred to as illumination where all objects become archetypes of a transcendent spiritual order.

However what is remarkable is that this transformation can be given a precise holistic mathematical interpretation.

We are now dealing with two different logical systems. Reason as we know is the one directional (conscious) approach where opposite poles are separated (with only the positive pole recognised). Intuition by contrast represents the fusing of opposite conscious poles as pure (psychic) energy and is non-directional (i.e. non-dimensional)

A considerable translation problem exists whereby we now need to adequately express intuition – which is based on a different logical system – in holistic mathematical terms.

Though intuition is dynamically nothing (in actual conscious terms) when we express this experience in the (rational) language of separate opposites, we must refer to it as involving two poles (i.e. positive and negative). In terms of the one-directional logic of the rational paradigm - where positive and negative poles are mutually exclusive - this reduced rational expression is deeply paradoxical. Indeed it is irrational. For example whereas in rational terms a proposition is either true or false, it is now both true and false (i.e. has a solely relative truth value). In terms of our former (rational) logic this assertion is therefore irrational. Putting it another way, rational understanding is based on an either/or logic (where opposite poles are clearly separate); irrational understanding is based on a both/and logic (where opposite poles are now dynamically complementary).

As the Pythagoreans discovered to their horror this problem is exactly replicated in reverse complementary fashion in conventional mathematics. When we try to extract the square root of a number such as two, we get both a positive and negative answer.

Also the result is irrational. The number cannot be expressed rationally (i.e. as a fraction).

In fact an irrational number represents a mixture of finite and infinite aspects. Whereas the value of an irrational number can be approximated in finite rational terms, its true value extends indefinitely (in the continued expansion of its decimal sequence). Strictly speaking all irrational numbers are relative and subject to an uncertainty principle.

This is exactly complemented in holistic mathematics. During the circular levels all objects and dimensions (also perceptions and concepts) have finite and infinite aspects. On the one hand we are dealing with natural symbols (finite). However in the light of pure intuition, these objects radiate an archetypal spiritual quality which is infinite.

Thus these objects now enjoy a dynamic relative existence which can only be approximated in (reduced) finite fashion.

Just as rational understanding – at the linear level – is the holistic qualitative counterpart to the notion of a rational number, we now have the exact holistic counterpart – at the circular level – to the notion of an irrational number.

So the circular level – in holistic mathematical terms – is the qualitative counterpart to (algebraic) irrational numbers.

 
Q.  Can you tell us something about the stages of this level?
 

PC  I believe that my stages correspond closely to the Western mystical tradition (e.g. as in Evylyn Underhill’s classic study), with a somewhat different mapping.

We have – as at the linear level - concrete stages and formal stages. The concrete stages represent objects (within dimensions). The formal stages represent dimensions (to which objects are related). In psychological terms the concrete stages relate to mental perceptions; the formal stages relate to mental concepts.

Whereas the stages at the linear level have a solely positive direction, stages at the circular level have both positive and negative directions.

We have first the concrete irrational structures in a positive direction (The Awakening of the Self). This stage relates to the illuminated experience of concrete sense objects whereby they serve as symbols of a transcendent spiritual order. This inevitably leads to a new form of subtle attachment (i.e. to objects as archetypal representations).

We then have the concrete irrational structures in a negative direction which represent the radical undoing of the former structures (The Purification of the Self). This is referred to by St. John of the Cross as the passive night of the senses.

This leads on to more deeply refined conceptually based structures (formal irrational structures) in a positive direction (The Illumination of the Self).

Again because of secondary ego attachment to these intuitively based concepts we have the formal irrational structures in a negative direction. This occurs through further severe purgation in the radical undoing of all remaining consciousness (The Dark Night of the Soul). St. John refers to this as the passive night of the spirit.

If successfully negotiated this leads close to that nada of pure faith – a "real" transcendent void - where intuition remains concealed in total darkness.

Q.  How does this "new" logic of the circular level lead to a dynamic relative interpretation of mathematics?

 
PC  Let me illustrate with reference to the number system. There is a central confusion in the conventional rational interpretation of the number system. Numbers are represented as lying on a straight line (which stretches in both directions). Though any section of this line is necessarily finite, the misleading impression is given that through "indefinite" extension it becomes infinite. Thus the infinite is treated in this rational approach misleadingly as an extension of the finite.

At this level of holistic mathematics, every number is dynamic and relative. Let us explain precisely what this means.

Every number has both concrete and formal aspects which are related. The concrete
aspect is provided by the number perception and the formal aspect by the corresponding number concept.

In this dynamic definition of number, every (particular) number has two directions. If
for example the positive direction of "2" represents a rational number perception (in relation to the perceiving mind, the negative direction of "2" represents the perceiving mind (in relation to the rational number perception). It is the fusion of these two directions which creates the essential intuitive insight enabling the switch to the corresponding (holistic) number concept to take place.

The rational number concept in like manner has two directions. Again the fusing
of both directions creates the intuitive insight enabling a switch back to the
rational number perception. In the dynamics of understanding, actual
number existence is provided through reason and potential number existence through
intuition.

Thus in terms of the number system, the finite determination of a number always
implies other numbers (with a potential for existence, but not determined in this
case). Thus the number system – in dynamic terms is strictly relative and open-ended. It is indeed finite but unbounded (necessarily containing numbers which
are both determinate and indeterminate).

Indeed this is equally true of any system in dynamic terms. (Interestingly Stephen
Hawking now speaks of the nature of the universe precisely in this fashion).

Q. You question the conventional notion of mathematical proof. Can you say something
about this?

 
PC Let me illustrate my answer with reference to the Pythagorean Theorem which
is one of the most important in mathematics.

Now in conventional terms an absolute proof exists for this theorem. Indeed
a wide variety of different "proofs" in fact exist. However at the heart of all such "proofs" there is an enormous (unrecognised) difficulty due to the reduction of the infinite to finite terms.

A proof establishes the truth of a proposition for "all" cases. But this – as we have
seen in relation to the number system - cannot be defined absolutely in finite terms.
Rather it can only be defined relatively where "all" cases (necessarily) include both those that are determinate and indeterminate.

Thus the Pythagorean theorem and indeed all "proven" theorems are only true in relative terms. Thus in dynamic terms this theorem (positive direction) continually interacts with its own anti-theorem (negative direction). This leads to the subtle changing of understanding in the ceaseless interplay of (specific) perceptions with corresponding (general) concepts. What is often forgotten is that each person’s understanding of a proposition involves a different configuration of reason and intuition and is unique. No two people’s interpretation is exactly the same. So ultimately a reliance on social consensus – as with all communication – is necessary to establish the validity of a proposition.

The pitfalls that are involved is well illustrated by the problems of verifying a proposition such as "Fermat’s Last Theorem". The "proof" is long and difficult with only a limited
number of trained mathematicians capable of following the steps involved. At the moment the consensus is that the proposition has indeed been proven. However after Andrew Wiles’ initial "proof" a subsequent flaw in his analysis was discovered which was later corrected. However it is certainly possible – if perhaps unlikely – that a further problem with the proof could arise. The longer it survives further scrutiny the stronger the consensus will be that Fermat’s Last Theorem has indeed been proven. However this – as indeed any "proven" proposition in mathematics - has a relative rather than an absolute truth value.

Holistic Mathematics with its two directional approach clearly establishes the precise nature of this relative truth.

 
 
Q.  Can you say a little about the holistic mathematical approach of the Pythagoreans?
 
 
PC  Again this approach is well illustrated by the famous Pythagorean Theorem where the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides.

It was important in the Pythagorean philosophy to establish a correspondence as between the (external) quantitative data of reality and the (internal) qualitative paradigm used to view this reality. Indeed the very fusing of these aspects was seen as the very way to create an archetypal holistic appreciation of mathematical symbols raising the mind to the level of pure contemplation. So the great potential spiritual value of mathematical activity was readily appreciated. Indeed the right angled triangle serves itself as a powerful archetype of this approach combining horizontal, vertical and diagonal lines representing (quantitative) matter, (qualitative) mind and (pure) spirit respectively.

Now initially all seemed well. The Pythagoreans used the rational paradigm in their mathematical approach to view a world where only rational quantities were known to exist.

However in the simple right angled triangle where both horizontal and vertical lines are 1, the length of the hypotenuse (diagonal line) must be expressed as the square root of 2, which is an irrational number.

The real reason why this discovery was so devastating was that it conflicted with the holistic nature of their enquiry. The rational paradigm was generating a quantity which was irrational, so the desired correspondence as between quantitative (physical) and qualitative (mental) reality was broken.

Subsequently in Western mathematics a merely reduced interpretation of numbers has survived. As well as irrational numbers, we now have imaginary, complex, and transfinite numbers all interpreted within the confines of the same (limited) rational paradigm.

Through this mathematical translation of the circular level, we can now explain why irrational number quantities arise and match them with a corresponding irrational paradigm which is qualitative.

The Pythagorean approach – though well intentioned – did not go much beyond the linear level. It was therefore not sufficiently transpersonal to make the necessary translations to solve their problem of irrational numbers.

Q. A key axiom of holistic mathematics is the complementary structure of physical and psychological reality. Can you illustrate this now using your two-directional approach?

 
PC  Let us initially go back to the linear level. When we say that that the linear level is one dimensional we are really referring to its interpretation of time. The most striking characteristic of this level is the view that time is strictly one dimensional moving in a positive direction only. Indeed, the belief that the arrow of time moves irreversibly forward is perhaps the most characteristic feature of linear thinking and indeed conventional science.

The other three space "dimensions" actually relate to the quantitative characteristics of physical phenomena. In this sense, any object that exists is by definition three dimensional (in spatial terms).

Therefore, physically at the linear level we have only one (positive) dimension of time (i.e. time moving in a forward direction). This (qualitative) dimension of time - understood in an absolute sense - is clearly separated from the three (quantitative) "dimensions" of space.

Psychologically, at the linear level we also have one (positive) dimension of time. This (qualitative) dimension of time actually relates to the way concepts are viewed at this level. Thus for example in the linear interpretation, the one concept of object relates to all object perceptions.
The three dimensions of space relate to the way objects in space sensibly appear at this level (i.e. as three dimensional).
However, this conventional linear interpretation of space and time represents a major  distortion whereby dynamic interaction as between objects and dimensions (perceptions and concepts) is frozen.  The holistic interpretation - which is inherently dynamic - is
fundamentally different.

The circular (holistic) level however is two directional with complementary positive and negative poles.

In physical terms this means - at this level - that time has now two directions one positive and one negative. In other words, time is now purely relative.

In dynamic terms, if time is moving forward for object A (with respect to object B), then - relatively speaking - time is moving backwards for B (with respect to A).

For example it takes about 8 minutes for light to travel from the Sun to the Earth.

Thus the Earth is eight minutes forward in time (relative to the Sun). However when this light does reach Earth, eight minutes has already passed on the Sun. Thus the original source of light on the Sun is now eight minutes backwards in time (relative to the Earth).

Notions of linear time are only possible when we fix our frame of reference. Thus when we take the time that light takes to travel from Earth to Sun, or Sun to Earth (separately) the measurement in both cases will be positive in time.

However - taken relatively - measurement for both is of necessity forward and backwards in time.

Therefore from a dynamic perspective, for creation as a whole, time is purely circular with all objects moving simultaneously in complementary fashion, both in a positive and negative direction in time. There is a synchronicity or global coincidence of opposites therefore in the universe, whereby all objects fundamentally are connected through the present moment. There is in fact an a-causal principle of connection working at the circular level. Linear notions of cause and effect only apply when one deals with partial systems. Strictlly speaking therefore they have no meaning in terms of holistic systems. Because of the dynamic complementarity of opposites at the circular level, all objects are simultaneously both cause and effect in terms of the overall system.

Speculation as to the overall age of the universe represents a basic confusion, whereby partial linear notions are applied to the whole system. In dynamic terms the universe did not come into existence in linear time. Rather it continually exists in a central moment or point instant (through which relative notions of time are rooted).

We have at this level of relativity two dimensions of time (i.e. positive and negative). We now also have two complementary dimensions of space (again with positive and negative directions). These space dimensions simply refer to the dynamic two directional nature of all physical objects.

Thus in terms of the holistic mathematics of the circular level there is a fundamental symmetry to dimensions with both space and time having positive and negative directions

This new interpretation of time is reflected through the manner in which conceptual understanding changes.

At the linear level, concepts are strictly one dimensional. All numbers for example are understood to belong to the one concept of number.

However at the circular level, concepts acquire two directions. The positive direction is the objective concept (in relation to mind). The complementary negative direction or anti-concept is the mind (in relation to the concept). In psychological terms concept and anti-concept - in direct terms - involve experience of positive and negative time (relative to each other).

It is the fusion of concept and anti-concept that creates that essential light or insight enabling the switch to particular perceptions in space to materialise. This intuitive light - through which concepts and perceptions are related - represents an immediate present moment continually regenerated in experience. This continual present is primary. Phenomenal translations of reality – in space and time - have only a secondary relative meaning.

This holistic reinterpretation of reality has revolutionary implications for science changing in particular the very way we look at dimensions. There is now a fundamental dynamic equality as between objects and dimensions, so ultimately there are as many dimensions as objects. (We will look at the precise interpretation of this truly multidimensional interpretation of reality later).

Also this holistic two directional approach demonstrates that time – in both physical and psychological terms - necessarily moves backwards as well as forwards.

The explanation of this is surprisingly simple. When I am aware externally of the world (in relation to self) time moves forward for the world. Equally when I am aware internally of the self (in relation to the world) time moves forward for the self. However, the world and self are opposite with respect to each other. Therefore when time moves forward for the world - in relative terms – it moves backwards for the self. Likewise when time moves forward for the self, it moves backwards for the world.

This is a highly important point in that any relationship involving an observer (psychological) and what is observed (physical) necessarily involves space and time moving in both directions. In other words they are entirely relative. Highly spiritual people instinctively realise this and successfully fuse both directions to become deeply centred in the present moment. Phenomenal experience in space and time is then understood as a secondary expression of this underlying spiritual present which is absolute.

Likewise in nature because all objects have external and internal aspects, space and time necessarily move forwards and backwards for all physical processes. Again these serve as relative phenomenal expressions of an underlying absolute present reality.

To sum up this section, physical and psychological reality are understood – in two directional terms - as fully complementary. In the mirror of (horizontal) symmetry they now mutually reflect each other.

In physical terms space and time have two directions; psychologically these are complemented by perceptions and concepts which likewise have two directions.

Indeed it is through the very formation of perceptions and concepts that we are (dynamically) enabled to experience space and time.

 
Q.  I know you believe that this approach is very relevant in terms of understanding the
paranormal. Can you briefly elaborate on this?

 
PC  All reality – as by definition it involves a dynamic mind-matter interaction - is paranormal at the circular level. Of course the reason why conventional science misses this is because it is strictly limited to the one directional thinking of the linear level. However we can now illuminate understanding of those events generally considered as paranormal.

For example some people have a special gift of healing with the capacity to cure illness and physical disabilities. This I believe is directly related to an innate spiritual capacity involving a circular interpretation of time.

In the conventional linear view past and future are clearly divided. The past has already happened; the future has yet to happen. However in the circular view past and present comprise two related poles which are continually connected through the present moment. This in effect means that every "past" event is actually recreated and changed through the present moment. In other words past events have no meaning in themselves but only through interaction with the subjective mind both of which are connected through the present moment. If the event objectively is "past" (in relation to mind), then subjectively the mind is "future" (in relation to this event). Both of these temporal poles are connected through the present moment. Spiritual healers have a strong intuitive realisation of this present connection which can often free sufficient creative power to significantly change "past" physical and emotional events.

In like manner future events are connected through the present moment. Once again if an objective event is "future (relative to the subjective mind), then relatively speaking the subjective mind is "past" (relative to the objective event). The circular approach to time is once again relevant in terms of understanding the phenomenon of precognition. This comes comes from a strong intuitive sense that "future"events are already contained in the present moment giving one a heightened capacity to read "the future".

The paradigm of conventional science is insidious in the way that it entirely ignores the personal aspect of matter. This leads to an essential loss of belonging in a world that appears alien and impersonal. In many ways our increasing ability to control this environment through technology only compounds this underlying problem of alienation.

In truth the self and its environment are as equal friends sharing an intimate dialogue. The two directional approach (where mind and matter are complementary) is the correct starting point for restoring this inherently equal partnership.
Deeply spiritual people intuitively relate in practice to reality in this manner. Thus physical and psychological events become  - to a considerable degree - complementary. Because of this two directional approach, meaningful synchronous events (i.e. coincidences) occur with great regularity in the lives of such people. Conventional science - as it is based on a one-directional logic (which separates the objective and subjective) cannot explain this. However - as we have seen - this acausal principle of synchronicity is simply explained by the two-directional logic of holistic mathematics.   
 
 
 

Q. As an economist, how do you apply this two directional approach?

 
PC  I am especially concerned with the inadequate paradigm used in Economics which is still based on the Newtonian model of physical science. This has led to a very distorted interpretation of the true nature of economic activity. This linear one-directional approach leads to a preoccupation with "Positive Economics". Thus the subjective "negative" aspect of behaviour – though vitally important – is either entirely ignored or misleadingly interpreted in (reduced) objective terms.

Let me illustrate this with relation to the fundamental economic concept of scarcity.

In the standard approach economic resources are finite whereas human wants are unlimited. Thus scarcity is inevitable. The solution to the economic problem is ultimately seen – in solely objective terms – as the provision of more of these "scarce" resources. This leads in turn to a preoccupation with economic growth as a means of promoting human welfare.

However this is all highly questionable. Economic goods and services strictly speaking have no meaning in themselves. Rather it is the dynamic relationship between (objective) goods and (subjective) attitudes that gives them meaning. To emphasise the objective pole only is totally unbalanced and leads to a philosophy of materialism.

There are in fact two complementary ways of tackling scarcity. The first – which we have already looked at – emphasises the objective pole in making more economic goods available.

The second emphasises the subjective pole  in a downward adjustment of attitudes so that one learns to be free from possessive attachment and content with less. This latter pole is often emphasised in spiritual communities with true happiness requiring little dependence on material possessions.

Thus economic welfare essentially relates to the dynamic relationship we have with material goods. When there is a distorted emphasis on merely the objective pole, then happiness is measured in terms of accumulation of wealth, and scarcity will always exist. Material goods cannot satisfy deeper personality needs, so paradoxically – as evident in affluent societies – when people accumulate possessions, they tend to want more which only increases the level of perceived scarcity.

Sufficient emphasis on the complementary subjective pole by contrast fosters a spiritual attitude of detachment , and the consequent realisation of the abundance of nature.

Thus a dynamic approach to Economics requires the harmonious balance of both poles so that a healthy relationship with material goods can be successfully achieved through the satisfaction of moderate wants.

The distortions of this one directional approach carry over into every area of Economics.

For example nature is viewed in impersonal terms as an asset to be exploited for material resources. Despite a growing emphasis on environmental issues this highly damaging attitude still dominates economic activity. Also the one-directional (conventional) interpretation of the "free" market – which excludes moral considerations such as justice and responsibility - is highly misleading and the antithesis of freedom in any meaningful sense. The application of the two-directional approach would therefore lead to a fundamental reinterpretation of all economic notions providing the basis for a truly balanced scientific approach.

Q.  We have covered a lot of ground. Can you briefly some up your main findings on this type of Holistic Mathematics?
 

PC   Conventional mathematics is based on the either/or logic of the linear level where opposite poles are clearly separated (object and subject), the holistic mathematics of the circular level is based on a both/and logic where opposite poles are now mutually complementary.

In qualitative terms the mathematical paradigm of the linear level is rational; the corresponding paradigm of the circular level – in relative terms – is irrational.

Mathematics at the linear level has an analytical quantitative bias; the mathematics of the circular level by contrast has a holistic qualitative bias.

Mathematical truth at the linear level has an absolute static meaning; mathematics at the circular level is inherently dynamic and relative. It converts the important mathematical operations of addition and subtraction (positing and negating) into dynamic relative terms.

An essential feature of Holistic Mathematics 1 is the inherent complementarity of all physical and psychological processes. Thus the mathematics of this level applies equally to psychological as well as physical structures. Indeed the application of this mathematics fundamentally changes the very way we view objects and dimensions in science.

However Holistic Mathematics 1 has its limitations.

It is only able to deal successfully with horizontal symmetry (external and internal aspects of holons). It is not able to properly explain vertical symmetry (whole and part aspects) and diagonal symmetry (finite and infinite aspects).

Also in its specialised development it leads to a purely qualitative holistic understanding
(which is intuitively based) that becomes divorced from complementary quantitative analytical understanding (which is rationally based). Higher levels of understanding are needed to address these deficiencies.