PC Once again, linear (one-directional) understanding arises from the symmetry breaking of the two complementary poles (positive and negative) in experience. One pole only is recognised which is literally posited in consciousness.
Circular (two directional) understanding attempts to restore this original
symmetry through dynamic negation of the already differentiated positive
pole. When successful, this generates considerable intuitive awareness.
However there is a certain imbalance in this process which only becomes fully apparent on completion of the circular level. This can be stated as follows.
When one pole is posited in conscious experience, the opposite pole is simultaneously activated in reverse fashion in an attempt to negate or block off the unconscious. Unfortunately, this reverse complementary activity by definition is not recognised by the conscious mind and becomes repressed in the unconscious. In the most general sense, repression is an automatic consequence of all conscious activity. Thus when conscious experience is rigid, significant repression will correspondingly take place.
Now in terms of understanding, the cognitive mode is more directly associated
conscious and the affective mode with unconscious activity respectively.
Thus in a rationally based scientific culture such as Western society experience of reality is heavily based on translation through the cognitive mode. This leads in turn to much (unrecognised) repression of the complementary affective mode.
This repression can prove especially problematic during the later stages of the circular level. The emphasis now is on transformation of all conscious experience in a purely spiritually intuitive form of awareness. Thus "higher" stages are based on a subtle type of cognitive control designed to keep the "lower" instinctive body-self at bay. Despite ones best intentions, this unfortunately can lead to a mind-body split in the personality and the build up of a considerable emotional blockage. Much of the deep depression which typifies the decisive "Dark Night of the Soul" is I am convinced - due to this denial of instinctive response. Thus the circular level often culminates in a terrible crisis forcing one to significantly readjust ones attitude to the spiritual life.
Thus the positing of one mode (as consciousness experience) automatically
represses (unconsciously) the complementary mode.
Now the unconscious will strive to rectify this imbalance but it can only do so in an
indirect and often poorly understood manner. It attempts to project this repressed material into consciousness. However the meaning of this - projected and indirect - activity is quite different from rational consciousness.
Basically a projection (e.g. a fantasy) carries a holistic archetypal
meaning expressive of the desire for integration. A projection in fact
is an invitation to address the hitherto neglected role of the unconscious.
However because of the significant dominance in our culture of rational conscious activity, projections are often misinterpreted in reduced conscious fashion.
For example this is very relevant to the philosophy of materialism (which I mentioned
in connection with Holistic Mathematics 1). With materialism, people project the holistic desire for meaning and fulfilment on to specific economic goods (e.g. the "dream" house, car, holiday etc). However to identify holistic (unconscious) meaning with specific (conscious) objects represents a fundamental confusion.
Thus with materialism, increasing possession of objects ultimately leads to emptiness and futility.
Consciousness really comprises two aspects a direct specific and an indirect holistic aspect. This latter aspect though indirectly presented under the guise of phenomenal symbols - really points to a deeper holistic meaning which is unconscious.
Furthermore it is the intervention of this indirect projected consciousness
which provides the very dynamic to switch modes from cognitive to affective
(and affective to cognitive) respectively. When one mode in conscious experience
(e.g. cognitive) is dominant the related affective mode as well described
in Jungian psychology will tend to be "inferior" and "shadow" operating
in a primitive instinctive fashion. Because it cannot express itself adequately
in conscious form it must rely considerably on the indirect manner of projection.
(This helps to explain the phenomenon for example of brilliant intellectuals who can often be quite infantile in terms of emotional relationships).
In a well developed personality cognitive and affective modes will operate in reasonable balance.
Q. So what significance has all this for Holistic Mathematics.
PC When properly translated in mathematical terms it is
enormously important for amongst other things it explains satisfactorily
the age old riddle of the relationship of whole and part (part and whole).
In mathematics we have "real" numbers and "imaginary" numbers. Now in rational scientific terms our very notion of what is "real" relates to what tangibly exists in conscious phenomenal form. This bias is evident in the very use of the word "reality" which so often is interpreted in reduced conscious terms.
The basis for all imaginary number quantities is the square root of 1.
Now this cannot be obtained in "real" terms for in mathematics we cannot obtain the square root of a negative quantity. Thus an imaginary number can only be expressed in an indirect fashion.
What is remarkable is that a projection can be precisely translated in holistic mathematical terms giving us the precise qualitative equivalent of an imaginary number.
Once again a projection springs from the lost repressed direction of consciousness (i.e. the negative direction) which is unconscious. Now this unconscious relative to conscious is two-directional (i.e. two-dimensional). Now for this projection to be expressed (indirectly) in conscious form we must express it one-dimensional format. In other words we must effectively obtain the square root of what is inherently negative. So this is the precise qualitative equivalent of an imaginary number.
So just as in mathematics the comprehensive number system is complex (comprising real and imaginary numbers) we now psychologically can describe reality in precisely similar fashion. We now no longer view reality in "real" terms but in "complex" terms where every holon has both a "real" and "imaginary" aspect.
Q. Could you elaborate further on this highly important point.
Why do we think that we are living in a "real" world if in fact it is "complex"?
PC It is accepted in physics that at an appropriate level reality is indeed "complex". The important equations establishing quantum physics confirm the point.
The reason why we do not appreciate this however, is because of the basic way especially in science that we fundamentally mistranslate the true nature of experience. Science with its emphasis on measurement, strives to focus on the merely actual (quantitative) nature of phenomena (pertaining to the conscious mind). It thereby attempts to screen out their complementary potential (qualitative) nature (pertaining to the unconscious). Now once again when this potential aspect of a phenomenon is indirectly expressed in conscious terms, then in correct holistic mathematical terms it is "imaginary".
Thus every phenomenon has now two aspects: an actual specific aspect which is "real" and a potential holistic aspect (indirectly expressed in phenomenal terms), which is "imaginary".
Thus for example I look at a flower such as a rose, in conventional
scientific terms this represents a "real" object.
In other words its actual specific identity is now recognised.
However in more correct terms the rose is in fact a "complex" object. It is "real" (reflecting its actual identity). However it is also "imaginary" (reflecting its potential identity). Here the rose serves as a symbol expressive of the holistic (unconscious) desire for meaning. Without this latter holistic aspect, it would not be possible to form a relationship with the rose (or indeed any object).
This in fact highlights the important distinction as between the scientific and artistic interpretations of an object. From a scientific perspective an object has meaning in so far as it can be identified as belonging to some common class. Thus correct recognition of a rose consists in identifying it as a common member of the general class of roses.
The artistic meaning of an object however relates to its unique special
quality. Thus a rose has artistic meaning insofar as this individual uniqueness
can be appreciated.
The scientific (common) interpretation though it involves the senses indirectly essentially results from cognitive translation of the experience and is impersonal.
The artistic (unique) interpretation though it involves reason indirectly
essentially results from affective translation of the experience and
However though these two experiences of the object are qualitatively different, when we identify both as the experience of a "real" object, fundamental confusion arises.
Holistic mathematics provides the precise language to correctly identify in dynamic terms both aspects. If we identify the cognitive (impersonal) interpretation of an object as "real" then in relative terms the affective (personal) interpretation is "imaginary". Of course in reverse fashion if the affective interpretation is "real", then again relatively the cognitive interpretation is "imaginary". Thus dynamically speaking all objects are "complex"(with both "real" and "imaginary" aspects).
Though some objects may be especially expressive of the desire for holistic
all potentially embody this tendency. The famous line from Blakes poem "To see a world in a grain of sand" exemplifies this "complex" nature of objects. The grain of sand seems a particularly insignificant (actual) "real" object. Yet Blake could see it as an especially luminous (potential) "imaginary" symbol. Indeed Blakes letters provide a wonderful expression of the "imaginary" world-view.
The closer one comes to the spiritual goal of reality (mystical union), the more important becomes this "imaginary" aspect of objects. Likewise in complementary fashion - in physics the closer a particle comes to its fundamental physical ground the more expressive it is of potential rather than actual meaning. Thus virtual particles are "imaginary" in the precise (dynamic) mathematical meaning of the term.
Q. Would it be correct to say that
the "real" aspect represents the quantitative nature of
a holon and the "imaginary" aspect represents its qualitative nature?
PC Yes, this would be a perfectly correct reading of the situation. However whereas the former quantitative aspect has a direct visible manifestation, the qualitative aspect is highly elusive and only manifests itself indirectly. This is why it is persistently ignored in (conventional) science. This brings us to the critical area of the relationship of wholes to parts (and parts to wholes). The holistic mathematical interpretation of reality in "complex" terms solves this problem.
All objects of experience are in fact composite (which arise from the holistic organisation of "lower level" part elements).
Take for example an object such as a house. Now this object results from the composite organisation of "parts" such as floors, walls, doors, windows, roofs, chimneys etc. Now the movement from these separate "parts" to the recognition of the new "whole" object of the house involves a qualitative transformation which is intuitive in origin. In this mysterious flash of insight a "new" object is born which transcends the "parts".
Thus the crucial transition in recognition from "parts" to "whole" is itself purely qualitative in nature. However in conventional understanding this recognition of the house is translated in reduced linear fashion as another quantitative object. The holistic qualitative nature of the experience is thereby lost and the house itself becomes just a part of a larger set of objects.
Thus in conventional science the whole has no qualitative meaning and
is interpreted in merely quantitative terms as the sum of its collective
In this worldview parts are simply fragments of a larger whole. This approach lacks holistic integration and understandably leads in turn to considerable fragmentation of overall experience.
In other words (conventional) science deals with a world of (solely) "real" objects.
Of course similar crucial dynamics are involved when we move in any context from the "whole" to recognition of its constituent "parts". Again to take our example of the house a crucial qualitative transformation in understanding is involved enabling one to grasp that this "whole" object is in fact comprised of unique distinctive "parts". (Here the original "whole" is dissolved). However once more the "parts" are interpreted in reduced quantitative terms as "real" objects.
Actual understanding is highly dynamic. Thus when one moves in any system
from "part" objects to recognition of the "whole", a qualitative transformation
is involved which cannot be expressed in quantitative "real" terms.
Likewise in reverse fashion when one moves from the "whole" object to recognition of (distinctive) "parts", again a qualitative transformation is involved which cannot be expressed in quantitative "real" terms.
Holistic mathematics which is inherently based on a dynamic interpretation
of mathematical relationships provides a simple answer to this dilemma.
In holistic mathematics if the "parts" in any dynamic system are "real", then the "whole" to which they are related is "imaginary". In complementary fashion if the "whole" is now taken as "real" then its constituent parts in relative terms are "imaginary".
Thus all "wholes" and all "parts" have both "real" (quantitative) and "imaginary" (qualitative) aspects respectively which are related to each other in dynamic complementary fashion. In mathematical terms holons are now "complex".
As we have already seen in Holistic Mathematics 1, holons have "positive"
(external) and "negative" internal aspects. Now in Holistic Mathematics
2, we see that holons also have "real" (part) and "imaginary" (whole) aspects.
(Of course these terms must be understood in appropriate relative fashion).
So we have the perfect dynamic counterpart here to the "complex" number system in (conventional) mathematics.
At its most fundamental all reality is structured in terms of this "complex" system.
Conventional science greatly misrepresents the true dynamic nature of
Basically it adopts an approach that tries to view reality in absolute static terms. So interaction as between (quantitative) objects and (qualitative) dimensions is minimised in this approach.
However in correct terms the quantitative and qualitative ceaselessly interact.
Thus the experience of a phenomenon such as a house (in our example), represents a "complex" reading of a dynamic interaction pattern that is both "real" and "imaginary" i.e. both object and dimension. So at this level of translation of experience objects cannot be separated from dimensions for all holons have now both "real" and "imaginary" aspects.
Q. This four-directional approach to holons leads to a fundamental reinterpretation of the nature of the universe. Can you elaborate on this?
PC Again one must keep remembering that the conventional scientific approach is linear and one-directional. This then leads to treatment of the universe in objective quantitative terms as the ultimate "real" whole with all physical phenomena as partial "real" fragments of this whole.
This view breaks down with the two-directional circular approach. The universe is now seen in dynamic relative terms as comprising (at all levels) both positive (external) and negative (internal) directions which are complementary. In another words one can no longer separate objective and subjective poles. The universe comprises both physical and psychological aspects at all levels - which are mutually complementary.
Now we have reached the four-directional point level which allows for a much more subtle translation.
In dynamic terms the whole is contained in each part; equally all parts are contained in the whole.
Thus each unit of life (microcosm) be it an atom, a cell, a human organism etc. is actually the universe (i.e. the whole is contained in each part). However the (collective) whole (macrocosm) is also the universe (i.e. the parts are contained in the whole). Now this apparent paradox is simply solved by viewing the universe dynamically in mathematical "complex" terms.
Thus if the individual microcosms comprise "real" universes, then the macrocosm (to which they are related) is an "imaginary" universe. Of course in reverse fashion if the macrocosm is "real", then the microcosms (to which it relates) are "imaginary".
Thus we should not conceive of the universe in absolute static terms as "one". Rather it comprises the ceaseless dynamic interplay of the one and the many (and the many and the one) which are "real" and "imaginary" with respect to each other .
So a complete dynamic interpretation of the nature of the universe is now four-directional and "complex" (i.e. has "real" and "imaginary" aspects with positive and negative directions). In other words it comprises the dynamic interaction of "part" and "whole" aspects with both internal and external directions.
Q. You showed at the circular level how the mathematical operations of addition and subtraction have a dynamic relative interpretation. Do the operations of multiplication and division now likewise have a dynamic interpretation?
PC Yes, they do. There is a problem in the conventional mathematical approach to multiplication and division that is usually overlooked. When one multiplies two or more numbers a qualitative dimensional transformation is involved. Thus 2X2 for example gives a two dimensional (rather than a one-dimensional) result which can be represented geometrically by a rectangle. Thus though the answer quantitatively is 4, a qualitative transformation takes place which is in two rather than one dimensions. (Strictly speaking therefore the result is 4 square units). If we divide this result by 2 again strictly there is a qualitative change of dimension.
However conventionally when dealing with numbers this qualitative distinction is ignored and the answer is interpreted in reduced quantitative terms.
Thus with multiplication (and with the reverse operation of division)
in mathematics a
qualitative as well as a quantitative transformation takes place. However the qualitative transformation is generally ignored in favour of a merely (reduced) quantitative interpret-ation.
The holistic mathematical interpretation of multiplication and division is somewhat different. Multiplication relates to the process by which parts are dynamically combined to form a new whole. Division relates to the complementary process by which a whole is split into separate parts. Now as we saw earlier in our example of the house that when we move from "parts" to the "whole" or from the "whole" to the "parts" both a quantitative and qualitative transformation is involved. "Parts" and "wholes" have quantitative (objective) and qualitative (dimensional) characteristics respectively which are "real" and "imaginary" with respect to each other. Thus multiplication and division are dynamic "complex" operations in holistic mathematics.
Q. Will you now outline briefly the main stages of the transition from circular to point level?
PC Personally, I think this is the most misunderstood phase of the spiritual journey.
Though I could find ample precedents in the literature for the corresponding transition from linear to circular levels, I could find no precedents for this equally important phase. Surprisingly however there is a remarkably close parallel in physics to my approach relating to Stephen Hawkings work on Black Holes. (We will look at this shortly).
The "Dark Night of the Soul" which especially typifies Western Mysticism is the culmination of the transcendent "otherworldly" approach to spirituality. This represents the dominance of "the masculine principle" where spiritual progress is seen in terms of overcoming the "lower" body (affective mode) through the rational discipline of the "higher" mind (cognitive mode). Though considerable spiritual development may take place in this manner when pushed to extremes it inevitably leads to much repression and severe emotional difficulties. Thus the "Dark Night" often culminates in near psychological collapse.
The transition from circular to point level in my translation represents
the slow recovery from this crisis where the emphasis switches from the
transcendent to the immanent approach, from ascent to descent, from the
"higher" mind to the "lower" body.
By now conscious activity has been largely sublimated through dark contemplation. However because of emotional repression these vast reservoirs of spiritual intuitive energy remain trapped within the unconscious.
So recovery takes place through a new form of "lower" imaginary activity whereby the repressed material within, is increasingly projected from the unconscious on to linear objects.
For a while one may feel as if one has returned to "normal" reality. Old skills are relearned. Hopes and ambitions long since forgotten resurface as one gets a foothold in the stream of life once more. However things are now strangely different. Normally the world is seen in "real" terms as objective (out there). However, the world now seems to have become inverted. It increasingly represents the projection of ones subjective psyche with only an indirect existence "out there". In other words the world is literally reborn in "imaginary" fashion (in dynamic mathematical terms).
With "imaginary" structures, positive and negative directions are very
Because we are dealing with projections, the true meaning resides in the unconscious. One is led to continually internally assimilate what has been externally projected.
Thus experience increasingly becomes virtual and transient as one comes to terms with these projections.
In terms of these "imaginary phase we have concrete linear stages in positive and negative directions (which are closely associated). These represented more superficial projections (both cognitive and affective). We then have the formal linear stages representing more deep-rooted projections in both directions.
Q. How does this four directional "complex" approach relate to the interpretation of numbers?
PC We saw at the previous circular level how both perceptions and concepts have - in dynamic terms - both positive and negative directions.
Thus once again if we take the (concrete) number perception "2", the positive direction represents the number "2" (in relation to the perceiving mind). The negative direction represents the perceiving mind (in relation to the number "2").
In like manner the (formal) number concept has also two directions. It is the very fusion of opposite directions - in both cases - that generates the essential intuitive insight to enable perception and concept to dynamically interact in experience.
However if the number "2" represents a quantitative "part" of the number system, then the concept of number represents the qualitative "whole" to which it is related.
Now we have already seen that - in any context - if the (quantitative)
"part" is understood in "real" terms, then the (qualitative) "whole" (to
which it relates) is imaginary.
Likewise if we now view the "whole" in quantitative "real" terms, then a corresponding "part" when viewed in qualitative terms is "imaginary".
Thus in holistic mathematical terms perceptions and concepts are "real" and "imaginary" with respect to each other. Because perceptions and concepts dynamically are qualitatively different - signifying the relationship of "parts" to "wholes"- we can no longer represent both in "real" terms. Rather we must represent them in "complex" terms (with both perceptions and concepts having complementary "real" and "imaginary" aspects).
Thus a number holon has now now four directions. If we take the "part" quantitative number perception of "2" as "real" then the "whole" qualitative number concept (to which "2" is related) is "imaginary". If we take the "whole" number concept as "real", then in dynamic mathematical terms - the number "2" (to which it relates) is "imaginary".
Thus reality is now structured dynamically in "complex" mathematical
terms (which in qualitative holistic terms is the precise equivalent to
the "complex" number system of conventional mathematics.
All holons are now" complex" and four directional.
For example a mathematical theorem has in dynamic terms four directions.
Thus the Pythagorean theorem to illustrate has both a (partial) quantitative and (holistic) qualitative significance which are complementary. Again if the former aspect is "real" then the latter aspect in relative terms is "imaginary".
Holistic mathematics itself with its qualitative bias can be seen as the specialised study of the "imaginary" aspect of "complex" understanding.
A fully comprehensive approach to mathematics (and indeed science) must necessarily combine "real" quantitative analysis with "imaginary" qualitative synthesis.
Q. How does the four-directional "complex" approach apply to dimensions?
PC The application here is of fundamental importance as it gives an entirely different appreciation of the true (dynamic) nature of space and time.
In the linear (one-directional) approach time has only one "real" dimension i.e. the arrow of time moves irreversibly forward.
At the circular (two-directional) level - as we have seen, time has now two "real" dimensions i.e. movements in time are strictly relative with both positive and negative directions.
Now at the point (four-directional) level, time is now four dimensional with both "real" and "imaginary" aspects in positive and negative directions.
What is remarkable is that we can now mathematically convert time into space and space into time. If space is quantitative in nature - then in relative terms time is qualitative. Therefore what is "real" in terms of time is "imaginary" in terms of space. Alternatively what is "imaginary" in terms of time is "real" in terms of space.
Thus our four dimensions of time (two real, two imaginary) can equally be translated as four dimensions of space (two imaginary, two real). We could also translate reality in terms of space and time each having two real dimensions or finally in terms of time and space both having two imaginary dimensions.
Alternatively we could say that every "real" object in dynamic terms represents an "imaginary" dimension. Every "real" dimension in turn represents an "imaginary" object.
Let me illustrate this new dynamic understanding in terms of the understanding of an object (e.g. a rose).
Now we have already distinguished as between the (quantitative) scientific and (qualitative) artistic appreciation of an object which are - in dynamic terms "real" and "imaginary" with respect to each other. Thus the (scientific) cognitive perception of the rose takes place in "real" space. Now this has two directions i.e. the external rose (in relation to internal observer) and internal observer (in relation to external rose). These directions are positive and negative with respect to each other.
Thus the quantitative (scientific) recognition of the object takes place in both positive and negative "real" space.
By contrast the (artistic) affective perception of the rose takes place in "imaginary" (qualitative) space where the rose serves as an (indirect) symbol of holistic (unconscious) meaning - which again has positive and negative aspects. Thus we have now translated this experience accurately in "complex" space dimensions.
However when recognition of an object switches from "real" to imaginary" space, simultaneously recognition switches from "imaginary" to "real" time. (Likewise when recognition switches from "imaginary" to "real" space, time switches from "real" to "imaginary" format. Thus space and time in dynamic terms are vertically complementary.
The psychological implications of this can be expressed as follows: in order to make the cognitive mode conscious, the complementary affective mode must be temporarily screened out and made unconscious. Likewise to make the affective mode conscious, the cognitive mode must be also temporarily made unconscious. In this way the mind is enabled to keep switching as between cognitive and affective modes, between quantitative and qualitative states, between "real" and "imaginary" dimensions.
When we allow for both cognitive and affective recognition of an object (both of which are necessarily involved in dynamic experience), we must translate space and time in "complex" fashion.
Thus the experience of the rose literally takes place in "complex" dimensions.
This of course likewise applies to all processes in nature. In general terms the cognitive mode is a control and the affective mode a response device respectively. In terms of all processes in nature we have equivalent behavior in terms of control and response systems which continually interact. And these take place in "real" and "imaginary" dimensions respectively.
Thus the true relation as between observer and what is observed (which is fundamental to all scientific enquiry) is now even more subtle.
We now have to allow (not alone for the horizontally opposite directions of object and subject), but also for the (vertically) opposite directions of space and time.
In other words we now have a dynamic interaction which moves in both a positive and negative direction in "real" and "imaginary" dimensions.
What is fascinating is how actual understanding of object phenomena is always polarised and cannot take place in the present moment. Thus an object is now experienced as relatively "past" and "future" in both "real" and "imaginary" time. Now these temporal poles in "real" and "imaginary" time are once again connected through the present moment which alone truly exists.
The search for the ultimate particles in physics is an illusion for
all phenomena by definition do not presently exist. Rather they serve as
a temporal relative expression of an underlying ground which is truly fundamental.
Q. Let us now move into the point level proper. Once again can you say something about the nature of this level?
PC Just to recap, the linear (one-directional) level is based on a rational logic. The circular (two-directional) level is based on an irrational logic (which represents a reduced translation of intuitive understanding). The point level is based very subtly on the relationship between these complementary logical systems. It expresses therefore the relationship between the linear and the circular, (the rational and the irrational).
If we imagine a circle divided by horizontal and vertical lines (representing real and imaginary axes respectively) into four equal quadrants, the point at the centre of the circle will also be at the centre of both lines. Thus this midpoint is central to both lines and circle. In like fashion the point level concentrates on what is central to both the (rational) linear and irrational (circular) forms of understanding. All the stages of the point level in a fundamental way are concerned with this central spiritual relationship.
We now have the new problem of expressing the nature of the point level in holistic mathematical terms. Again a fascinating solution is at hand.
There are in mathematics two classes of irrational numbers. The first class is that of algebraic irrational numbers. These can always be expressed as a solution to a polynomial equation. Thus to give an example, x2 = 2 is a simple polynomial equation. The solution is the important square root of 2 (with positive and negative signs). The irrational nature of the answer actually results from trying to express a number from a vertically higher dimension in (reduced) linear (one-dimensional) terms. Indeed all (algebraic) irrational numbers result from this attempted linear translation of numbers expressing higher vertical dimensions.
The second class is that of transcendental numbers which cannot be given as a solution to a polynomial equation. (As we will see presently) holistic mathematics is able to give a convincing reason as to why this in fact is the case).
Now the most famous transcendental number is pi, which expresses the numerical relationship (i.e. ratio) between the circumference of the circle and its line diameter.
In like manner in dynamic qualitative terms the stages of the point level express the relationship as between circular and linear understanding. Thus the stages of the point level in holistic mathematical terms are transcendental.
There is another way of making this qualitative connection which demonstrates the power of holistic mathematics to discover important relationships in the most unlikely circumstances.
Hilbert, - the famous German mathematician at the turn of the century drew up a last of the most important "remaining" unsolved mathematical problems. One of these related to the Hilbert number (2 raised to the power of the square root of 2). The problem was to prove if this number (representing a rational number raised to an irrational power) was transcendental. Though Hilbert was not confident of an early proof, it was solved by Gelfond in the 1920s as part of a more general solution. All such numbers were proven to be transcendental.
This seemingly obscure mathematical result has a precise holistic mathematical equivalent which clarifies in remarkable fashion the nature of transpersonal development.
Let us go back to the linear level. We have here the development of rational perceptions (concrete stages) and rational concepts (formal stages). Now as I have been at pains to demonstrate, in holistic mathematics perceptions represent objects and concepts represent dimensions. Though they are qualitatively different this is confused at the linear level.
However when one relates appropriately to reality one actually experiences perceptions (objects) and concepts (dimensions) in correct dynamic manner - as qualitatively different, so that a subtle transformation takes place whereby experience becomes transpersonal and one moves into the circular level.
In holistic mathematical terms the stages of the circular level are irrational (representing horizontal objects experienced in vertical dimensions). Now, any appropriate rational number (horizontal) raised to a rational power or dimension (vertical) in conventional mathematics is likewise irrational. So we have an important and unexpected connection here as between (conventional) mathematical and "higher" psychological behaviour.
The point level involves the attempt to experience rational objects (linear) in irrational dimensions (circular). This parallels the problem of the Hilbert number (and the raising of a rational number to an irrational power or dimension). In both cases the result is transcendental. Thus an exact correspondence as between (conventional) mathematical and holistic mathematical behaviour once again holds.
In psychological terms qualitative transcendental stages relate to neither (horizontal) objects nor (vertical) dimensions as such but rather the relationship between both.
Likewise a transcendental number represents a hybrid relationship that is part horizontal, part vertical. An algebraic solution involving polynomial equations with vertical "higher" dimensional numbers therefore cannot apply.
Thus to sum up the either/or logic of the linear level is rational; the both/and logic of the circular level is (algebraic) irrational. The logic of the point level expresses very subtly the relationship between rational and irrational logic and in holistic mathematical terms is transcendental.
Q. Can you briefly outline your distinctive mapping of the stages of the point level?
PC Yes, I am very critical of the mapping of this level in existing accounts. In Western mysticism the point level is generally treated as a continuation of the circular level and its distinctive features largely missed. In Eastern traditions though very detailed intuitive mapping of these stages is available, the treatment is invariably too passive providing little integration with important rational dynamics. Also in both traditions the significance of "lower level" physical projections is overlooked. In formal terms though not necessarily in practice both traditions tend to place undue emphasis on the transcendent aspect of mystical development.
My own approach is consistent with holistic mathematics. Again as at the circular level we have (horizontal) positive and negative stages (which are highly interdependent).
In addition we have (vertical) "real" and "imaginary" stages. The personality increasingly splits into a "higher" level (spiritual) self where extremely subtle translations of experience take place and a" lower" level (physical) self from which intense intimate fantasies are projected. Now with respect to each other the stages of the "higher" and "lower" selves are "real" and "imaginary".
In terms of "higher" level (spiritual) development, we can once more identify concrete and formal stages. Again these transcendental stages are very subtle involving a dynamic interplay as between (linear) rational and (circular) intuitive understanding.
During the concrete stages because of close interaction of positive and negative directions a remarkable psycho-physical understanding of reality unfolds. Because physical and psychological reality have now become complementary in experience, one is able to make original and striking connections as between physics and transpersonal psychology.
During the formal stages an even more remarkable psycho-mathematical interpretation of reality unfolds. Understanding of holistic mathematical relationships greatly accelerates at this time so that mathematics and transpersonal psychology are now seen in dynamic terms - as one.
(Much of my own work is the product of this stage).
In terms of "lower" level stages we can also identify concrete and formal stages (though the sequence of development is reversed).
The formal stages involve extensive projections (intellectual and emotional) with a degree of super-ego control still in evidence.
The concrete stages involve the intense projection of highly intimate fantasies, now largely freed of rational interference.
So simultaneously one moves towards two extremes the "higher" spiritual self (free of "lower" level projections) and the "lower" physical self (free of "higher" level rational control).
Of course the ability to sustain the tension of these two extremes comes from the spiritual refinement of the will which lies at the centre of the personality.
This level culminates in the simultaneous arrival at a double void, a "real" transcendent void and an "imaginary" immanent void. In other words all projections of both the "higher" level (spiritual) self and the "lower" level (physical) self) cease. One reaches simultaneously the ultimate superfinite ground (the Omega of creation) which is also the underlying subfinite ground (the Alpha of creation). Though fulfilled and at peace, one suffers - in existential terms - a profound emptiness. Freed from all secondary phenomenal distractions the fundamental questions of identity and meaning confront one in the starkest manner.
Q. So once more what implications do these stages have for holistic mathematics?
PC Though considerable mathematical appreciation of the overall qualitative nature of reality takes place at the circular level, it tends to become too intuitive, and divorced from the rational quantitative understanding of the linear level.
However because the very purpose of the point level is to reconcile both circular and linear understanding, a closer relationship as between the quantitative and qualitative aspects is now possible.
The implications indeed are quite remarkable. Put simply, every mathematical quantity, operation or relationship that exists in (conventional) mathematics, has a qualitative dynamic meaning (in holistic mathematics) which applies directly to both physical and psychological reality.
Now just reflect on this for a moment. A great deal of the conventional mathematics is highly abstract and carried out for its own sake (without any belief that it applies to the "real" world). This in turn is largely due to the inadequate translation of reality provided by conventional science.
However we are now saying in contrast that all (conventional) mathematical relationships have coherent holistic equivalents, that are qualitative and dynamic; also, that in principle all these holistic relationships apply directly to physical and psychological reality, with mutually complementary interpretations.
This really turns mathematics on its head; for now holistic mathematics - because it is inherently based on a true dynamic translation of reality - becomes primary. (Conventional) mathematics , because it is based on the rational philosophical paradigm which distorts reality, is now of secondary value.
When one then considers - despite its enormous potential importance - the widespread neglect of holistic - as opposed to conventional - mathematics, then one can perhaps appreciate my desire to spread the "good news" of this exciting new mathematics.
Q. Can you justify your claims for holistic mathematics from your own work?
PC Yes. I set out to tackle what I believed to be the most important - and most difficult - problem. This was to translate the full Spectrum of Consciousness in holistic mathematical terms.
I found to me delight that despite all the variety and complexity of multiple levels and stages of development that a remarkably coherent pattern existed that could be fully translated in terms of the qualitative number system.
Now the qualitative number system is dynamic equivalent in holistic mathematics to the quantitative number system of (conventional) mathematics.
A full listing of (quantitative) number types would be as follows - binary (1 and 0), prime, natural, integers (including negative numbers), rational (including fractions), irrational (algebraic and transcendental), imaginary, complex and transfinite.
Now all the levels and stages of the Spectrum of Consciousness can be mapped in terms of corresponding holistic mathematical interpretations of these number types. A perfect correspondence in fact exists. There is no psychological structure that cannot be matched by an appropriate number type; equally there is no number type that cannot be matched by an appropriate psychological structure.
The levels of my Spectrum are obtained from a simple procedure obtaining successive roots of unity. (I will explain in greater detail the precise rationale for this in a later posting)
Thus the linear level (one-directional) relates to a qualitative interpretation of the one root of unity (positive).
The circular level (two-directional) relates to a qualitative interpretation of the two roots of unity (positive and negative).
The point level (four-directional) relates to a qualitative interpretation of the four roots of unity (real and imaginary in positive and negative format).
The radial level (eight-directional) relates to a qualitative interpretation of the eight roots of unity (real and imaginary in positive and negative format and four additional "special" complex roots).
Now the general stages of levels e.g. linear, circular and point relate to number types - not derived directly from obtaining appropriate roots - which are the rational, (algebraic) irrational and transcendental respectively. (Indeed I have already demonstrated in my postings how such qualitative holistic mathematical interpretations are derived).
Finally we have the pre-linear levels. There is vertical symmetry in my approach in that each of the "higher" levels is complemented by a corresponding "lower" level. I call these "lower" levels the binary, prime and natural respectively. Just as the "higher" levels are based on the psychological fusion of directions (two at the circular, four at the point, eight at the radial) the corresponding "lower" levels are based on a corresponding confusion of directions (two at the natural, four at the prime and eight at the binary). Again there is nothing arbitrary about the choice of binary, prime and natural for the pre-linear levels. In each case they are based on a precise dynamic qualitative interpretation of their (conventional) mathematical equivalents.
Q. So the entire Spectrum of Consciousness can be viewed as the qualitative number system?
PC Yes, that is correct. In one way there should not be too surprising. In the quantitative realm, number is synonymous with order. Therefore when one accepts the holistic aspect of mathematical activity, then likewise in the qualitative realm number is also synonymous with order. The number spectrum simply demonstrates this in the field of consciousness. I want to stress that there is nothing artificial about this mathematical mapping. In many ways the number spectrum is more comprehensive than existing accounts. Also - despite formal translation - it is intimately based on personal experience.
Q. You believe that your mapping of the Spectrum has major implications for the philosophy of science. Can you elaborate on this?
In qualitative terms (conventional) science is based on the rational paradigm. More correctly, it is based on the positive real rational paradigm. (It is positive in that it deals with objective - rather than subjective - reality; it is real in that it deals with conscious - rather than (projected) unconscious - reality; it is rational in that it is based on the either/or logic of the separation of polar opposites, rather than the both/and logic of the complementarity of opposites).
However the positive real rational numbers represent a very limited subset of the overall set of number quantities. Likewise in holistic mathematical terms - the positive real rational paradigm represents a very limited example of the overall set of possible qualitative paradigms.
Thus we can have negative as well as positive paradigms. We can have imaginary as well as real paradigms. We can have irrational as well rational paradigms.
Now all of these alternative paradigms - in the appropriate context - have a valid use in terms of the philosophical translation of reality. However no coherent investigation of the precise nature or scientific uses of these paradigms has taken place. So the conventional rational paradigm of science continues to operate as a considerable straight jacket blinding us to alternative valuable translations of reality.
Q. Can you say how you use horizontal and vertical symmetry in terms of your mapping of the spectrum?
PC With the exception of the linear, complementarity operates at all other levels, both pre-linear and post-linear.
Horizontal symmetry relates to the complementarity of positive (external) and negative (internal) aspects of experience. Thus on the transpersonal journey (post-linear), each psychological stage is complemented by a corresponding physical stage.
Thus one starts out to transform the self, and gradually finds that all creation becomes transformed in the process. In other words in terms of dynamic relationships - changing the self (in relation to the world) automatically implies changing the world (in relation to self). Until the self and world are experienced as identical, imbalance remains. Thus the ceaseless interaction of both positive and negative stages takes place until union is achieved. Then the ego-based personality of the self is fully replaced by a cosmic-based personality whereby one becomes a mediator of the entire life of creation. Thus the "higher" stages of the self, equally represent "higher" stages in terms of physical nature. In other words evolution of the self is inseparable from evolution of the world. Transpersonal psychology correctly understood is inseparable from transpersonal physics. In mystical union they both reach their Omega point whereby they are identical.
During the pre-linear stages the complementarity of opposites leads to an experience where the world is confused with the self. Thus dynamic change continues until the world and self are fully differentiated (at the linear level).
Vertical symmetry relates to the complementarity of the "higher" levels with the "lower" levels. This needs some explanation.
Evolution can be taken two ways. We can look at it in the conventional fashion as the upward movement of creation from the purely physical to the purely spiritual i.e. from the lowest states of matter to the most spiritualised psychological states. However we can equally view evolution in the reverse direction as the movement from the spiritual to the physical. In this alternative approach (what might be termed involution) the most advanced state is the purely physical. Ultimately the pure void of superfinite spiritual reality is identical with the pure void of subfinite physical reality.
Now this is all highly revealing for it means that holistic mathematical structures can be used equally in terms of the "higher" psychological spectrum or the "lower" physical spectrum.
In other words there is a direct (vertical) complementarity in structural terms - as between the levels (and indeed stages) of "higher" psychological development and corresponding levels and stages of "lower" physical nature. The mathematical interpretation of the "highest" (the superfinite void) is equally the interpretation of the "lowest" (subfinite). Indeed this is central to "The Theory of Everything" which I will deal with in a later posting.
Also the interpretation of next "highest" (psychological) level is the interpretation of the next "lowest" (physical) level etc.
What is so often forgotten in physics is that what we see in nature is a reflection of the psychological constructs we use to view this reality. Thus for example, to properly interpret the "lowest" level of physical nature we need the psychological understanding of the "highest" level.
So I am now using my number spectrum which was initially designed to interpret "higher" psychological reality in complementary vertical fashion - to interpret the "lower" levels of physical reality. The number spectrum is now serving as a highly comprehensive mapping for holistic physics. Many remarkable insights relevant to contemporary physics have resulted from this approach.
Q. This sounds very interesting. Can you
give any examples of what you have mean?.
PC One fascinating area relates to the complementarity of the "dark night" mystical phenomenon and the "black hole" of contemporary physics. Indeed in "Transforming Voyage" I devoted a considerable amount of space to the remarkable connections between both phenomena. In terms of my approach vertical complementary exists as between the "higher" level circular stage relating to the "dark night" (on the psychological spectrum) and the corresponding "lower" level natural stage relating to the "black hole" (on the physical spectrum).
As I have stated before the most understood part of the Spectrum of Consciousness relates to the transition from circular to point level where the emphasis switches from "real" higher level spiritual to "imaginary" lower level physical understanding. This happens through the intense projection of (repressed) material in the unconscious. The release of energy gradually relieves darkness and congestion in the psyche, so that the intuition trapped within can express itself externally as pure consciousness. In this way the darkness of the "night" slowly recedes and turns to light.
Now in a quite remarkable way Stephen Hawkings work on black hole radiation - which seemed so revolutionary in the 1970s - parallels closely in physical terms the same structural process.
Hawling explains this black hole radiation through the formation of virtual particles at the boundary of the black hole. Now virtual particles are associated with extremely close complementarity as between parings of matter and anti-matter particles and are transient and elusive. Indeed they are more "imaginary" than "real". Now in Hawkings theory one pole of matter can separate escaping the black hole thus leading to the radiation effect (the other is returned to the black hole). When this happens to a sufficient extent the "black" hole radiates and becomes a "white" hole.
I explain dark night radiation in precisely similar terms. Corresponding to virtual particles we have intimate fantasies. These relate in a direct manner to the unconscious mind with complementary poles thus closely associated. Again they are more "imaginary" than "real". Now through projection, one of these poles (i.e. the positive) escapes the internal psyche (the boundary of the dark night) leading to the release of energy which is materialised. Now just as conscious activity leads to repression (of the unconscious) this indirect conscious activity works in reverse fashion leading to the lifting of repression. This easing in repression leads in turn to the intensification of the degree of projected material. When all repressed material has been released full union can be achieved (i.e. "night" turns to "day").
So the evolution in the physical journey of the "black hole" takes place through its gravitational collapse and eventual radiation (via virtual particle activity) as pure energy. This is exactly paralleled - in my approach - by the evolution of the "dark night" through intense psychological grief, and then radiation (though projected fantasies) as pure light.
So as well as clarifying a highly important aspect of the mystical journey - which is ignored or mistranslated - in other accounts, we establish a remarkably close linkage to contemporary physics.
This is simply an application of the vertical complementarity aspect of holistic mathematics.
Thus the transition stage from circular to point level (on the psychological spectrum) is vertically complementary with the transition from the natural to the prime level of reality (on the physical spectrum).
I have also been able to use my number spectrum to clarify the difficulty within physics in terms of reconciling the theory of Relativity with Quantum Mechanics. This in turn leads to a proposed resolution which can only be accurately translated in holistic mathematical terms. I will deal with this in "The Theory of Everything".
Q. Can you briefly indicate how the four-directional approach would apply to economics.
PC The economic system - like all others - is intimately concerned with the relationship as between "parts" and "wholes". Indeed the standard breakdown of the discipline is in terms of Microeconomics (parts) and Macroeconomics (whole). However because of the Newtonian paradigm of science employed the interaction between these levels is viewed in mechanical quantitative terms.
Now a comprehensive approach necessarily would be far more subtle.
Basically we need to employ a "complex" paradigm (with "real" and "imaginary" aspects).
The "real" aspect relates to either the whole or parts taken separately and can have positive or negative directions. Thus we could for example start with a scientific explanation of the macroeconomic system (the whole). However as well as a merely positive (objective) explanation of the system we would also require a negative (subjective) explanation. In other words we would evaluate the system in ethical terms (examining issues such as justice, freedom, responsibility etc.)
Equally we could in "real" terms start with a scientific explanation of the part (e.g. the working of a particular market). However again as well as a merely positive (objective) understanding we would also require an ethical evaluation in negative (subjective) terms (e.g. how peoples attitudes are affected by the behaviour of this market).
Now the basis for the "imaginary" approach is the dynamic (qualitative) interaction as between the "wholes" and parts" of the system. Again the imaginary approach can be taken in positive and negative directions. In positive terms we could obtain a global integrated understanding of the evolution of the "whole" system (in relation to the "parts"). In other words we would evaluate the long-term effect in qualitative terms of miscellaneous factors such as social, political and cultural influences, the environment, institutions etc. on the economic system. Alternatively we could obtain a corresponding qualitative evolutionary study of "part" of the system - such as an individual market- (in relation to the "whole").
Again these "imaginary" aspects will have negative directions relating to the overall evolution of social and personal consciousness respectively.
Thus a comprehensive overall paradigm is four directional and "complex".
The "real" (quantitative) aspect is appropriate for static short-term analysis either of the whole system (or parts of the system) considered separately (in both positive and negative directions)
The "imaginary" (qualitative) aspect is appropriate for dynamic long-term synthesis of the "whole" system (in relation to the "parts") or "parts" of the system (in relation to the "whole"). Again this involves both positive and negative directions.
Of course ultimately both (static) short-term "real" analysis and (dynamic)
long term "imaginary" synthesis should be properly coordinated.
Because of the concentration on "real" analysis one of the key problems of (conventional) economics - apart from its neglect of subjective ethical issues - is the lack
of a suitable dynamic context (which comes from the "imaginary" aspect).
This leads even in quantitative terms to much inefficency in economic management
with a tendency for economic doctrines to be still in vogue even when no longer valid.
In other words there is a continual lag in adopting economic theories which have true contemporary relevance.
Q. Once more we have come a long way.
Could you now summarise your findings on Holistic Mathematics 2?
PC As at the other levels, there is an important mathematical system associated with the point level.
The mathematics of the linear level involves the specialisation of the quantitative analytical approach; the mathematics of the circular level involves the specialisation of the qualitative holistic approach; the mathematics of the point level more subtly involves the specialisation of the relationship between the quantitative and qualitative approaches. It provides a satisfactory solution for the relationship between "parts" and "wholes" (in any system) leading in dynamic holistic terms to a "complex" view of reality which greatly extends the scientific integrative power of mathematics.
Once again the logic of linear mathematics is "rational" (based on the separation of opposite poles of understanding. The logic of circular mathematics is "irrational" (based on the dynamic complementarity of separate opposites). The logic of point mathematics is "transcendental" (based on the subtle relationship between "rational" and "irrational" logic).
This new logic is both highly rational and highly intuitive at the same time. Thus there is strong emphasis on the precise rational translation of complex dynamic relationships; yet there is also a clear understanding that the meaning of such relationships transcends reason and must be intuitively grasped.
It simultaneously is static (in its precise rational translations) and dynamic (in terms of the intuitive meaning of these translations).
Again a key feature of Holistic Mathematics 2 is the complementarity of psychological and physical reality. However this complementarity is now understood in both horizontal and vertical fashion. This new vertical aspect of complementarity is extremely useful in terms of translating the structure of physical reality from a corresponding understanding of the structure of psychological reality. (Alternatively one can translate the structure of psychological reality from a corresponding understanding of physical reality). Reality is now seen at all levels dynamically in psycho-physical terms.
Once again however Holistic Mathematics 2 has its limitations. Though it deals successfully with two of the fundamental symmetries of reality (positive-negative and real-imaginary), it cannot properly deal with the final symmetry (finite-transfinite).
Also, though offering a mathematically satisfying holistic view of the overall nature of systems, one is not yet able to satisfactorily link this holistic understanding with earlier analytical understanding. These remaining problems are addressed with the mathematics of the radial level (Holistic Mathematics 3) which I will return to in a later posting.