Nature of Holistic Mathematics - Introduction

Q. What is your basic starting point?

PC   Mathematics as we know it is based on the rational paradigm and is related to - what I refer to as - the linear level. There are however "higher" levels on the spectrum of consciousness (circular, point and radial in my terminology) culminating in pure simplicity which lead to the emergence of very different types of mathematical understanding.

Q. Can you briefly say something about these levels before proceeding?

PC  I define each of my levels in terms of directions (which represent the true dynamic interpretation of dimensions).

There are three fundamental sets of interactions involved in all relationships (horizontal, vertical and diagonal) which can be translated in dynamic mathematical terms. Referring to psychological reality, the first relates to objective–subjective interactions (positive-negative).The second relates to cognitive-affective interactions (real-imaginary). The third relates to conscious-unconscious interactions (finite-infinite). (The precise reasons for making these connections should unfold as we proceed)

The linear level attempts to freeze interaction between these opposite poles and is based on a rational logic of clear separation. Understanding is formally interpreted in terms of one direction only. For example in mathematics numbers are strictly one-directional (i.e. one-dimensional). Thus though the understanding of a number involves a two-way dynamic interaction of matter and mind (object and subject), it is formally interpreted in terms of the objective direction only. In other words the number is given a static absolute validity which is unaffected through mental interaction with it.

Again the understanding of a number involves a two-way cognitive-affective interaction (reason and sense symbol). Again this dynamic interaction is frozen and given a reduced merely cognitive interpretation.

Finally the understanding of a number involves a two-way conscious-unconscious (i.e. rational-intuitive) interaction. Once more this interaction is frozen in terms of a merely conscious (rational) interpretation.

Thus in terms of each of the three vital interactions we have here a strictly one-directional (one-dimensional) interpretation. Now the line in geometry symbolises one dimension so I refer to this one-directional understanding as the linear level.

The rational paradigm is its supreme achievement with mathematics – as conventionally understood – representing its most specialised development. In more comprehensive terms it can be accurately defined as the positive real rational paradigm. It is positive in that it is based on objective understanding only (i.e. the subjective direction is reduced to the objective) It is real in that it considers the cognitive direction only (the affective is reduced to the cognitive). It is rational in that it recognises (explicitly) conscious understanding only (i.e. it reduces the unconscious direction to the conscious).

This level could be identified as the more advanced part of the gross realm.

Now each of the other levels can be accurately defined in terms of addressing the limitations of the linear approach and thereby allowing for dynamic interaction.

The circular level is based on the explicit recognition of the objective-subjective polarity. Every relationship is understood in dynamic terms as involving two directions (which continually interact). Relationships are now based on the complementarity of "real" positive and negative directions. This clearly has fundamental significance for mathematics but – quite remarkably – has never been comprehensively addressed.

I call this the circular level because the circle involves a line (i.e.diameter) extending from the centre equally in both positive and negative directions. Also, in terms of the linear level, the paradoxical understanding entailed by this level is circular. It would equate with both psychic and subtle realms. (The transition period during which the negative direction is differentiated equates with the psychic. The period during which both directions are combined equates with the subtle).

The point level as well as recognising the two way horizontal interaction (external and internal) also recognises the two way vertical interaction (cognitive and affective). It is thus four directional. Indeed this is an extremely important development which is not properly appreciated in the mystical literature which basically involves a true appreciation - in any context - of the dynamic relationship between whole and part (part and whole).

I term this the point level because again in terms of my geometrical language it involves the intersection of "real" opposites (represented by the horizontal axis) and "imaginary" opposites (represented by the vertical axis) at the central point of both (which is also the centre of the circumscribed circle). We have here in fact a direct correspondence in dynamic psychological terms with the complex number system in mathematics. In other words in dynamic mathematical terms, reality at this level is complex.

This would equate with the causal realm. However the traditional treatment of this level in Eastern spirituality is too intuitive. Thus its highly important rational dynamics are not sufficiently addressed. My own treatment is somewhat different.

Finally the radial level – as well as horizontal and vertical interactions – also recognises the four way diagonal interaction (conscious and unconscious). It is thus eight directional. This level involves in any context a true appreciation of the dynamic relationship between finite and infinite (form and formlessness).

I term this the radial level. (The diagonal lines are radii of the circle and can be shown to literally represent rays of light). It refers to non-dual reality (the unitive life in Western Mysticism).

My treatment of the radial level is one of continual development whereby dynamic interaction (at the three levels of polarity) are progressively integrated.

What I term pure simplicity refers to an idealised advanced radial stage of total harmony - representing both full differentiation and equally full integration of all eight directions (which can be approximated to but never fully reached). This equates with transforming union in Western mysticism.

To sum up, the linear level involves rational specialisation, the circular level intuitive specialisation, the point level specialisation of the relationship between reason and intuition (i.e. formless spiritual development) and the radial level the progressive integration of all three.

Q. Briefly, what about pre-linear levels? How do you deal with them?

PC  There is a direct correspondence (i.e. vertical complementarity) in my approach as between pre-linear and trans-linear levels.

The earliest level (what I call binary) represents the confusion of all eight directions. The first development represents differentiation of conscious and unconscious directions (separation of actual from potential reality).

The next level (which I call prime) involves confusion of remaining four directions. Consequent development involves differentiation of affective and cognitive directions (separation of personal and impersonal reality).

The final pre-linear level (which I call natural) involves confusion of remaining two directions. Development then involves differentiation of external and internal directions (objective world and subjective self).

Q. So how does this all lead to alternative systems of Mathematics?

PC  Well, lets go back to the (rational) linear level. Using Piaget’s terminology we have the unfolding here of concrete operational and formal operational stages. Now, Mathematics – as we know it – is based on the cognitive structures of these stages.

However, we can equally identify at the each of the "higher" levels more refined concrete operational and formal operational stages leading in each case to the unfolding of different mathematical systems. Now the understanding of the relationships of these systems is inherently dynamic and holistic with – among other uses – direct application to fields such as transpersonal psychology. .

What I have been doing is making explicit the nature of these cognitive structures (at each level) and then clarifying the appropriate type of (dynamic) mathematical understanding associated.

Q. If this work is so important – which it appears to be – why has it not already been attempted?

PC  This is a fascinating question which I have asked myself many times. I would interpret it as a translation problem. There is no doubt that the "higher" stages of transpersonal development have been mapped out in considerable detail in both Eastern and Western traditions. However the treatment has invariably been from a spiritual perspective and clothed in the language and symbolism of religious culture. In other words the accounts we have of the perennial philosophy do not provide the appropriate mapping required.

The (rational) mathematical perspective is even of less value. Mathematicians – with few exceptions – treat mystical spirituality as of little relevance to their own discipline and generally have a closed mind in relation to the fundamental issues that I am addressing.

Q. What precisely started off your own interest in Holistic Mathematics?

PC  I was at University – 30 years ago - studying Mathematics and experiencing a growing distaste for the procedures involved. In particular I had strong reservations regarding the use of infinite notions (which I believed were misleadingly treated as extensions of the finite). This is fundamental and central to the whole notion of mathematical proof. Mathematical activity involves a dynamic interaction which is both rational and intuitive. Now finite notions pertain (directly) to rational understanding whereas infinite notions pertain to intuitive understanding. However because (conventional) mathematics is formally interpreted in rational terms only, then this inevitability leads to this reduction of the infinite to finite terms.

So even then I had a vision of a "new" holistic type of mathematics which would explicitly involve intuition – as well as reason – in its presentation. I formulated a dynamic relative explanation of the number system and of mathematical proof at that time before becoming absorbed in philosophy and mystical spirituality.

Some years ago, I decided to write a book outlining – from a Western perspective – the "higher" stages of the psychological spectrum. To my great surprise my earlier vision was realised providing a totally unexpected holistic mathematical framework for all levels and stages of the spectrum. Of course what was happening all along was that I was translating my own transpersonal development implicitly in (holistic) mathematical terms.

I now realise that the true nature of all mathematical quantities, operations and relationships is inherently dynamic with direct application to both physical and psychological reality. Unfortunately the dynamic holistic interpretation has become largely lost because of the (reduced) absolute nature of conventional mathematics. I see my task therefore as to recover this true original understanding.

Q. Are there any precedents for what you are attempting?

PC  I am not a historian of ideas so impressions here are merely personal. The dynamic approach where mathematical symbols have a holistic archetypal significance is common to many cultures. In F. David Peat's book "Blackfoot Physics" - dealing with the indigenous science of North American Indians - there is a fascinating chapter on "Sacred Mathematics". There appears to be a fundamental similarity here with some of the points I make. However I would contend that the understanding of mathematics in these cultures is largely prerational and mythical and not properly differentiated. So I would be claiming to offer a proper transrational interpretation of many of the same ideas, ultimately integrating them with conventional mathematical notions.

Also one aspect of the Tantra tradition views reality as emerging from a central point (bindu) and expressing itself in geometrical forms (yantras) which are then charged with a holistic spiritual meaning. In both Buddhism and Hinduism – and indeed other cultures - mandalas (which are usually geometrical patterns inscribed inside a circle) have been used as aids to meditation. These patterns are often arranged in symmetric patterns of fours and signify integration of the self. Now what I find fascinating is that the most common four-fold form (e.g. the cross within a circle), can be given a precise mathematical interpretation in terms of the complex number system. This in turn is simply a geometrical expression of a coherent circular number system (largely unrecognised) with immense holistic significance.

In the Western tradition, the Pythagorean school is especially relevant  in that its mathematics was inherently based on the integration of analytical and holistic approaches. As we will see later the famed Pythagorean Theorem uniquely combines both aspects. However since the break up of this school 2,500 years ago, the holistic aspect of Mathematics has been progressively lost with a (reduced) analytical interpretation remaining. Indeed this deeply ingrained reductionist approach is itself the main barrier to the acceptance of holistic mathematical notions. Since the Pythagoreans, in the West, mathematics and mystical spirituality have developed largely in opposite directions, with very few people displaying either the interest or the capacity to see them as complementary.

One notable exception was Nicholas of Cusa, a man of many parts and surprisingly modern in his outlook. He believed that the Earth revolved around the Sun (100 years before Copernicus) and that there were other stars in the Heavens with planetary systems which were inhabited (as on Earth). He also combined mathematical training with mystical insight and had an unusually clear understanding of intuition as based on the coincidence of opposites (which would be separated at the rational level). The combination of reason and insight in turn lead to a dynamic open-ended view of truth (where knowledge is necessarily combined with ignorance) and true to this spirit he had a relative understanding of the nature of space and time. Pretty impressive!

Leibniz is another especially interesting figure combining diverse intellectual talents with the insight of a genius. His view of the Universe as being made up of individual monads (or centres of energy) reflecting in various ways an overall macocosm represents a significant attempt to get to grips with the dynamic nature of whole and part (which is of central concern in Holistic Mathematics). He also formulated the binary digital system – where 1 and 0 are separate and distinct - so useful) in this computer age - and considered the idea of an alternative binary system (where 1 and 0 are identical). In the most fundamental sense – in holistic mathematics - reality at all levels can be viewed in terms of the interaction of these two systems (the double binary system) encoding both information and transformation processes simultaneously.

Hegel is also worth a mention. Though his approach – as usual – is somewhat convoluted – he at least attempted to use his dialectic to formulate mathematical notions in dynamic relative fashion.

In modern times Jung, though not producing a formal system showed a deep appreciation of the holistic aspect of mathematics (esp. number). Not surprisingly many Jungian notions (e.g. his treatment of personality types) lend themselves easily to translation in holistic mathematical terms.

More recently, Ken Wilbers's four quadrant approach to holons can be given a compelling  holistic mathematical interpretation.

However despite these somewhat sketchy precedents what is truly remarkable – when placed in the context of the considerable development of conventional mathematics - is the extent to which Holistic Mathematics has been almost entirely neglected.

Q. Before proceeding any further perhaps you should say something about conventional mathematics?

PC  What is mathematics is itself a fascinating question. Formerly it used to be known as the science of number and quantity. However it also involves the study of the logical operations connecting abstract symbols which has strong philosophical connotations. And of course its applications are used in many different fields. So one cannot precisely define the limits of mathematics..

We can however accurately characterise (conventional) mathematics - irrespective of difficulties in definition - by its one directional approach.

As we will see again and again there are three fundamental dynamic interactions involved in all understanding and the one directional approach represents an especially limiting interpretation of these dynamics.

In holistic terms mathematics represents a positive (objective) real (cognitive) rational (conscious) approach. This qualitative mathematical paradigm exactly complements the quantitative set of positive real rational numbers.

Just as this quantitative set represents a very limited subset of all numbers, the (conventional) mathematical paradigm represents a very limited sample of the full range of mathematical truth.

Q. In the terms you put it, this would seem to lead to the view that (conventional) mathematical understanding represents a fundamental distortion of truth. Is this what you mean?.

PC  Yes, when looked at from a holistic mathematical perspective this is true. Conventional mathematics is of course superb for the task of precise analysis of nature. However its very success at this level has greatly blinded us to the realisation of an alternative type of mathematics (holistic mathematics) - very different in approach - suited to the complementary task of (precise) dynamic integration of reality. This mathematics is not meant in any way to replace existing mathematics. Rather it complements it and when understood properly could greatly enrich existing mathematical research.

Q. And the starting point for holistic mathematics is a precise outline of the psychological dynamics of understanding.

PC  Yes, that’s right. Strangely, I could find no satisfactory answers in psychology , philosophy or mystical spirituality to three fundamental questions as to how understanding switches as between (a) objective and subjective poles (b) cognitive and affective poles and most fundamentally (c) conscious and unconscious poles.

The more I looked into it the more I realised that these interactions could be precisely explained in dynamic mathematical terms. Furthermore each of the major levels of transpersonal development could be fundamentally viewed in terms of these switching patterns.

Thus the circular level represents the specialised development of objective and subjective switching (positive and negative) and is two directional. Confusion still remains in relation to the other forms of switching.

The point level represents the additional specialised development of cognitive and affective switching (real and imaginary) and is four directional.

The radial level represents finally further specialised development of conscious and unconscious switching (finite and infinite) and is eight directional.

Now this provides a dynamic set of conditions which determine the type of mathematical activity that arises. As we have seen one-directional switching leads to the (conventional) mathematics of the linear level. However two-directional, four directional and eight-directional switching leads to three new types of holistic mathematics which can be identified with the circular, point and radial levels respectively.

Q. So there is not just one holistic mathematics?

PC  No, I have identified three distinct types. The purpose of this discussion is to clarify the precise nature of each type.

Q. So these switching patterns you talk about are like initial axioms that determine the nature of mathematical activity that unfolds?

PC That’s right. Freeze all interactions and you get one type (conventional mathematics)

Open up the positive-negative interaction (others remaining closed) and you get Holistic Mathematics (Level 1).

Open up in addition the real-imaginary interaction and you get Holistic Mathematics (Level 2).

Open up all the remaining (finite-infinite) interaction and you get Holistic Mathematics (Level 3).

Q. This reminds me of Euclid’s axioms. Change one of the axioms and you get a new Geometry?

PC  Yes there is some similarity. However the basic approach to Mathematics remained unaltered by changing this axiom. In other words it remained one directional. What I am suggesting is in fact more revolutionary and requires a fundamental shift in what we mean by Mathematics.