The circular level involves additionally the dynamic negation of these "real" phenomena - leading to the substantial generation of spiritual intuition - and is two directional.
The point level involves intense development of "imaginary" consciousness. Here both "high level" spiritual and "low level" physical phenomena are projected from the psyche and carry an (indirect) potential - rather than a (direct) actual - meaning. Experience is now "complex" and four-directional (two "real" and two "imaginary" ).
However a subtle problem still remains. Though "imaginary" symbols require a holistic (qualitative) interpretation, inevitably some degree of confusion is involved through the identification of meaning with the specific (quantitative) phenomena employed.
In other words secondary ego possessiveness - referred to in Christian asceticism as involuntary attachment - persists throughout the point level. Dealing with this problem requires an increasingly naked spiritual attitude in the pure exercise of will. This slowly leads to the death of all phenomena - "real" and "imaginary" in experience. This is what I refer to as the transition from point to radial levels.
Q. Can you say something more about this
PC As you will recall an extreme vertical spit of the personality involving "imaginary" projections from both "high level" spiritual self and "low level" physical self takes place.
The "high level" involves specialised development of the transcendent (masculine) direction of spirituality. The "low level" involves complementary specialised development of the immanent (feminine) physical direction.
As projections from both "selves" gradually cease, one enters a profound void involving both an intimate form of psychological birth and death. From a human perspective this is an existential crisis of the first magnitude where all sense of customary identity is lost. A double void is involved. From the spiritual perspective we now have spiritual death and a form of entombment. From the complementary physical perspective we have a form of en-womb-ment as one goes back deep into the unconscious to psychologically cross the boundary of human birth. So birth and death are now experienced clearly as complementary notions. Indeed this process eventually leads to (coincidental) rebirth (i.e. physical creation is reborn in experience) and resurrection (i.e. one rises spiritually from the dead). However for some considerable time one may have to undergo a slow psychological agony gradually losing all human and spiritual support.
Though receiving no feedback and finding oneís customary duties extremely monotonous the correct attitude during this phase is to carry them out patiently without seeking attention.
So the transition from point to radial level involves the undoing of all remaining structures (regardless of how subtle). It represents therefore a pure (unconscious) process without the generation of (conscious) phenomena (excepting some remaining primitive fears and disturbances).
However subsequently - when conscious activity is renewed at the radial level - one can paradoxically describe this transition phase in precise mathematical terms.
The holistic mathematical structure appropriate to each of the major levels is given by obtaining roots of unity. (I will deal in more detail with the precise rationale for this in "The Theory of Everything").
The linear level is expressed by the one root of unity thus giving one "real" direction (positive) to experience.
The circular level is given by the two roots of unity giving two "real" directions (positive and negative to experience. If we draw a circle of unit radius, the circular level will be represented by the horizontal line diameter (with positive and negative "real" axes drawn from the centre of this circle).
The point level is given by the four roots of unity giving four directions (two "real" and two "imaginary) to experience. On the same circle of unit radius, the point level will be represented by both horizontal and vertical line diameters ("real" and "imaginary" axes) intersecting at the centre of the circle.
The radial level is given by the eight roots of unity giving now eight directions to experience. In addition to horizontal and vertical axes (on the circle of unit radius) we now have in addition two diagonal axes (with positive and negative directions) drawn through the centre of the circle (at an equal distance from horizontal and vertical axes).
The mathematical properties of these diagonal lines are fascinating and can be expressed in a dual fashion. The first of these is directly appropriate in terms of describing the transition from point to radial level. (The second is appropriate in terms of the holistic mathematical translation of the radial level).
Now if we concentrate on just one of the quadrants of our circle (e.g. where horizontal and vertical lines are both positive), the diagonal line - drawn to the circumference from the point of intersection of these lines - will be an equal distance from both axes. Thus if we drop a perpendicular from the circumference point to the horizontal axis we now have a right angled triangle. This comprises the diagonal line (as hypotenuse) and horizontal and vertical lines of equal length.
As the equal horizontal and vertical lines represent "real" and "imaginary" units respectively, when - using the Pythagorean Theorem - we square both of these and add to find the square on the diagonal, the answer is zero. Thus the diagonal line in this quadrant can be represented with a numerical value of zero. In like fashion the diagonal lines in each of the other quadrants will also have a numerical value of zero.
In holistic mathematical terms this precisely represents the void (which characterises the transition from point to radial levels).
This must be interpreted in appropriate dynamic rather than - as in conventional mathematics - static terms. (Clearly this must be so for in standard mathematics a finite line cannot have zero length!)
This void (which represents the absence of all actual phenomena) also represents the potential for existence and is a purely creative notion. In psychological terms it refers to spiritual intuition.
In fact in dynamic holistic terms 0 is inseparable from ¥ (infinity). The void as empty is 0; the void as full - in complementary fashion - is ¥ .
Now this (correct) dynamic interpretation of infinity is very different from that in standard mathematics.
Because standard mathematics uses the rational (linear) approach it inevitably reduces infinite notions to finite terms. Thus infinity is misleadingly interpreted in this approach as an extension of the finite. The number axis which starts at 0 is then supposed to stretch "indefinitely" in both directions to reach infinity. Thus here infinity is at the opposite extreme of 0.
However infinite notions strictly speaking are not linear and rational but rather circular and intuitive.
As modern mathematics increasingly demonstrates when one tries to appropriate the infinite through rational (linear) means, one is led increasingly into paradox. (Indeed this problem has a long history, as ably demonstrated by Zenoís paradoxes nearly 2,500 years ago).
By contrast holistic mathematics which starts from the complementary intuitive (circular) direction is especially suited to the unravelling of these paradoxes.
Q. This is a highly important point. Does
what you are saying throw light on Cantorís work on transfinite numbers?
PC Yes, very much so. However I wish to start this area of the discussion - which may seem surprising - on the medieval theology of angels.
Though their relevance has now greatly faded, angels in the past played an extremely important part in religious experience. There are in fact over 300 references in the Bible to angels. Also, it is clear that Angelology was an indispensable component of medieval theology finding its most developed expression in the system of St. Thomas Aquinas.
From a psychological perspective angels were essentially religious projections
of profound archetypal significance providing a figurative device for portraying
the vital role of the unconscious mind.
Indeed when properly interpreted, the medieval system of Angelology remarkably parallels the various levels of transpersonal development i.e. circular, point and radial.
The use of a linear form of reason in medieval theology leads to a strictly hierarchical approach to cosmic development. At the bottom is the world of "inanimate" matter. Further up the scale are the animals with the power to acquire sense knowledge of this world. On a higher level is man with the capacity to draw general form from sense information through universalising conceptual ability. Man is thus essentially defined by the refined conscious power of reason. Though considerably transcending the senses, it was accepted - even when spiritually motivated - that reason could only provide indirect knowledge of the ultimate truth i.e. God.
It was thereby logically postulated that the existence of a higher order
of being - capable of direct knowledge of God - was necessary to complete
the cosmic system.
So next to God, the angels occupied the highest rung on the ladder of this hierarchy. They were capable of a form of pure spiritual intelligence or illumination which - unlike human reason - did not originate in the senses. They acted therefore essentially as intermediaries between God and man.
When we translate this thinking of medieval scholars e.g. St.Thomas, into modern terms, it is clear that they were in fact dealing with the refined intuitive experience typical of transpersonal or mystical type development. In other words they were dealing with spiritual knowledge pertaining directly to the unconscious rather than the conscious mind.
Unfortunately because the rational paradigm which they adopted which pertained solely to consciousness, they dealt with the unconscious in an unsatisfactory indirect manner. It was thereby split off and projected externally on to separate "imaginary" creatures i.e. the angels.
The true position is that the "good angels" symbolise the higher development of the unconscious mind and thereby represent the potential for spiritual development inherent in every personality. (Likewise the "bad angels" i.e. satanic or demonic influences - both at a personal or collective level - symbolise the lower development of the unconscious, in the reverse movement towards disintegration, and deeply symbolise evil in all forms).
Interpreted in this fashion, Angelology suddenly assumes an unexpected significance and can be seen as a guide to transpersonal development. Indeed when viewed from this perspective the system used by St. Thomas gives a very coherent map of such development which remarkably complements the more intuitive mystical approach.
St. Thomas borrowed and completed a system first fully outlined by the pseudo-Dionysius involving in all 9 orders of angels symmetrically made up of three triads (containing three orders in each hierarchy).
In psychological circles a system called the Enneagram has recently become very popular. It is based on 9 distinct personality types. Each person is classified as broadly corresponding to one type. Understanding the typical characteristics of each type can generate much insight regarding oneís own behaviour and that of others.
Using more mathematical terminology, this Enneagram is basically a horizontal profile of personal development with each number (representing a distinct personality type) being considered broadly within the same dimension (i.e. the linear level which characterises our cultural experience).
Remarkably the angelic system - adopted by St. Thomas can be equally viewed as an Enneagram. However in this instance it relates to a vertical profile of transpersonal development. The very goal of such development can be seen as an attempt to overcome the limitations and rigidities associated with oneís natural personality. Through an arduous process of spiritual transformation one disidentifies with the characteristics of any particular personality type to assume what is common to all. Thus someone who has completed this vertical task of transpersonal development essentially assumes a cosmic or universal personality with characteristics typical of all types.
I refer to the transpersonal levels of development as the circular, point and radial respectively. These correspond to the three triads of angels in the Thomistic system.
In turn each level incorporates three sub-levels which I have identified as the physical, mental and spiritual (i.e. concrete, formal and formless) respectively. These in turn correspond with the three orders in each hierarchy. Taking each sub-level as representing a distinct dimension of experience, we can see that transpersonal development involves the journey of the same personality through nine dimensions.
So we have two Enneagram systems which are fully complementary.
1) The conventional (horizontal) Enneagram of personal development which involves 9 numbers (each representing a distinct personality type) within the same dimension.
2) The "angelic" (vertical) Enneagram of transpersonal development which
also involves 9 numbers (this time representing the differing dimensions)
through which the same personality journeys (before attaining union with
An extremely important - if surprising - application of the angelic hierarchical system is in relation to the number system.
The real number system - in conventional mathematics - is treated very much in similar hierarchical fashion. Indeed the reduced rational approach leading to such a treatment of numbers exactly parallels the same reduced approach in medieval theology leading to the discovery of angel hierarchies.
We have firstly the rational numbers whose interpretation falls within the linear (rational) level.
We then have "higher" classes of numbers i.e. irrational (algebraic), transcendental and transfinite which correspond respectively to the three angelic hierarchies (already outlined).
The (algebraic) irrational numbers correspond with the first hierarchy of angels. These angels formed a bridge as it were as between the one dimensional world of physical affairs (linear level) and a higher dimensional spiritual world. Though essentially spiritual creatures (and thereby irrational from a physical perspective) they could in certain circumstances assume a human form thus appearing rational. (For example in St. Lukeís Gospel, the angel Gabriel informs Mary that she is to be the mother of God).
It is very similar - in complementary terms - with irrational numbers.
Such a number is by definition irrational when expressed in the 1st dimension. Thus the root of 2 is irrational when its value is expressed (in reduced form) in the 1st dimension. However it is in fact rational in certain circumstances. When for example it is raised to the 2nd dimension (i.e. squared) its value is equal to 2.
Thus every (algebraic) irrational number has - as it were - a rational aspect (i.e. is rational when expressed in an appropriate higher dimension or linear combination of dimensions). This rational aspect enabled Cantor to include the (algebraic) irrationals among the set of real countable numbers.
The second hierarchy of angels best exemplify the dilemma of the medieval theologians in classifying angels. On the one hand such angels were a different species of creatures from rational humans and also - as they were not God - not fully infinite.
Transcendental numbers correspond perfectly with this dilemma in mathematical
On the one hand they are irrational (and not capable of reduction to rational terms). On the other hand they are not fully infinite - or in Cantorís terminology transfinite. They therefore fall between two stools as it were with both finite (discrete) and transfinite (continuous) aspects.
It is fascinating the process by which the medieval theologians inferred
the existence of angels by a kind of logical necessity to fill a recognised
gap in the hierarchy of beings.
It is precisely the same process by which Cantor inferred the existence of transcendental numbers again by a kind of logical necessity to fill a recognised gap in the hierarchy of numbers. (Indeed, in this connection it is well known that Cantor was very much influenced by medieval theology).
The third hierarchy of angels were the most spiritual and ultimately capable of the most direct union with God. However this also posed a problem in terms of translation through the rational approach. In union with God there is no separation. However the very basis of the rational approach is to separate so that even at the highest level the angels (i.e. Seraphim) are treated as remaining distinct from God.
This rational bias has continued with the treatment of mystical union, where in the Western tradition the ultimate separation of God from the creature has been vigorously maintained, even though this plainly contradicts the very notion of union.
In the Eastern tradition - by contrast - the approach to spiritual development has been understood in more intuitive terms where ultimately separation has no meaning. Thus union with God means that one now essentially is God. In other words oneís limited ego based human nature is transformed - through spiritual development - into an unlimited cosmic based divine nature. Thus the potential for divinity is inherent in every person. However few ever fully realise this potential.
The experience of unity, is essentially beyond reason altogether and
purely intuitive. Indeed it represents a reality which is infinite.
In like manner the "highest" order of number similarly transcends reason altogether and is purely intuitive. This relates of course in mathematical terms to transfinite numbers.
However, as the formal method of translation in mathematics is decidedly rational, this creates enormous problems in relation to their understanding, so that paradox abounds at every turn. This is simply due to the translation of what is inherently intuitive through the opposite rational format (which is based on an entirely different logic).
I will now elaborate on this important point.
As we have seen the understanding of number goes through many different stages as transpersonal development unfolds.
At the linear level, one adopts the interpretation of numbers as objective independent entities enjoying an absolute existence. This indeed is the conventional rational interpretation of numbers. More correctly it is a positive rational real approach. It is positive in recognising (solely) the objective direction to numbers; the negative subjective direction is thereby formally ignored. It is rational in recognising (solely) the cognitive mode to numbers; the affective mode of the senses is thereby formally ignored. Finally it is real in recognising (solely) the conscious process to numbers; here the unconscious process of intuition again is formally ignored. We have therefore at this level a highly reduced understanding of numbers.
The transition from linear to circular level involves the first modification in understanding where the negative direction to number is now explicitly recognised. Understanding is now considerably transformed. Perceptions and concepts lose their absolute independent interpretation to be replaced by a relative one, where all meaning involves a dynamic two-way dialogue between the (internal) knower and what is (externally) known. There are thus two directions involved in understanding which are fully complementary.
This of course applies directly to the understanding of number. If the external number perception (i.e. the number in relation to the mind) is positive, then the internal number perception (i.e. the mind in relation to the number) is negative. Like matter and anti-matter particles, all number perceptions have identical counterparts of opposite direction. In this dynamic sense, if the positive direction relates to numbers, then the negative direction relates to anti-numbers.
The same thinking applies to the number concept which in dynamic terms has both a positive and negative direction.
The circular level initially involves the attempt to translate this dynamic relation between the (complementary) positive and negative poles of experience in a reduced static fashion. Thus, what is inherently intuitive is translated through the objective rational mode based on a different logic. In terms of this rational mode, such understanding is therefore paradoxical. Indeed it is irrational. This takes place firstly in relation to concrete perceptions of reality, later followed by a deeper formal conceptual understanding.
In terms of number experience we now have the development of a positive irrational real approach. Again it is positive because one translates understanding of number in an objective manner. It is irrational because such translation is based on the inherently dynamic fusion of opposites generating number experience. It is real in that it is - indirectly through translation - still a conscious interpretation of number.
The next development - also occurring at the circular level - is to explicitly interpret this subtle irrational understanding in a subjective direction also, rendering a complementary negative irrational interpretation of number.
The transition from the circular to the point level involves a new psychological transformation where - in addition to the two complementary directions - the two complementary modes of experience (i.e. cognitive and affective) are also explicitly differentiated. One clearly understands that if cognitive (mental) experience of reality is real, then in relative terms affective (sense) experience is imaginary. Experience of reality is in truth of a complex rather than a real world with both "real" and "imaginary" aspects.
We have therefore - in psychological terms - "imaginary" as well as "real" approaches to the understanding of number. During this lengthy transition phase we have the emergence of such "imaginary" understanding firstly in a positive and then a negative direction.
The point level now follows. Just as during the circular level there was an indirect attempt to translate the two-directional nature of experience in a subtle and indirect uni-directional form (through irrational understanding) there is now a similar attempt during the point level to indirectly translate the bi-modal nature of experience in an even more subtle and indirect uni-modal fashion (i.e. either "real" or "imaginary" format). It represents an attempt to integrate the dynamic relationship as between the former linear and circular levels of understanding.
In relation to number, this involves an attempt to express in uni-modal format an understanding which represents the dynamic relationship between both rational and irrational understanding. This is transcendental understanding.
At the "higher" level of personality one translates in "real" terms. Thus we have the emergence of both a positive and negative transcendental approach to the understanding of numbers. At the "lower" level of personality one translates reality in "imaginary" terms. Here we have the emergence of a complementary positive and closely associated negative transcendental approach to the understanding of numbers.
These translations of the point level are extremely subtle and indirect
depending on a high level of intuition for their generation.
Ultimately, such phenomenal translations must be abandoned altogether. This happens during the transition from point to radial level. All attachment to phenomena of either a direct conscious or indirect unconscious kind ceases as one enters a void which represents the pure creative potential for existence. So now, experience has a (potential) infinite rather than an (actual) finite aspect.
The radial level then involves an attempt to express this key relationship as between the potential (infinite) and actual (finite) processes again in a satisfactory manner.
In relation to the understanding of number a transfinite approach emerges which I will now attempt to explain.
There is an underlying spiritual basis to all experience. However in
terms of mathematics this essential aspect is screened out of all formal
Let me illustrate this with respect to the concept of number. The number concept can be interpreted in (reduced) rational terms as the general class to which all actual numbers belong. However the number concept can more correctly be interpreted in intuitive terms as the general class to which all potential numbers belong. Whereas the former properly relates to finite numbers, the latter relates to infinite numbers.
All this can only be understood in dynamic terms. The number concept is formed in relation to corresponding number perceptions. If I become aware of a number e.g. "2", I form a positive perception of that particular number and its objective existence (i.e. the number in relation to the mind). I then form the anti-number "2" in the corresponding negative perception of that number. I thereby become aware of its subjective existence (the mind in relation to the number).
The fusion of number and anti-number perceptions creates a flash of psychic energy in the generation of intuitive insight. In other words one thereby moves from the perception of "2" as a finite number to the general spiritual intuition of number as potentially infinite. In other words one moves from the finite perception of "2" as a particular number to the infinite concept of number. This infinite concept is essentially potential and intuitive (applying to all numbers in general) and none in particular.
This intuitive insight of number quickly "collapses" again in the conscious separation of opposites to a reduced rational interpretation of number as applying to all actual numbers.
There is thus an intuitive or transfinite aspect to the understanding of the concept of number complementing the rational or finite aspect. Conventionally in mathematics the formal interpretation of the number concept is solely in reduced rational terms.
Equally there is a directly intuitive or transfinite aspect to the understanding of any number perception. The formation of a number perception once again dynamically involves the number concept.
We now have formed the reduced concept of number (as relating to any particular number). This is firstly with respect to the positive direction (i.e. the number concept as objectively existing in relation to the mind). One then forms the complementary anti-concept in the negative subjective direction (i.e. the mind existing in relation to the number concept).
Once again there is a fusion of both directions of the number concept generating psychic gravity in the form of intuitive insight. This insight - which again is infinite - provides the potential for existence of a particular number. This then "collapses" once more to the actual existence of the particular number (in this instance "2").
Once more the intuitive or transfinite perception of a number as potentially existing is screened out of formal understanding to be interpreted solely in reduced finite actual terms. (I am using the term "transfinite" to refer to the infinite in dynamic relationship with the finite realm).
The transfinite notion therefore properly relates to the intuitive - as opposed to the rational - aspect inherent in the understanding of all numbers.
This transfinite aspect of experience is well recognised in the mystical literature. It has two complementary aspects viz. the contemplation of immanence and the contemplation of transcendence respectively. In conventional experience, though intuition is always present fuelling the dynamics of understanding, it is continually reduced in favour of (solely) rational interpretations.
However with the mature mystical experience of the radial level, intuition and reason co-exist and interact as equal partners. Intuition provides a direct gateway to infinite reality (and indirectly to finite). Reason in complementary fashion provides a direct gateway to finite reality (and indirectly to infinite). Potential unconscious reality (in the unity of complementary opposites) continually collapses, as it were, to be actualised consciously (in the separation of phenomenal opposites). In like manner actual reality in reverse fashion undergoes a ceaseless state of spiritual transformation by which it once more realises its potential infinite destiny.
Finite and infinite reality are therefore continually mediated and reflected through each other. (I am referring to the infinite in this transitional state as transfinite reality).
Transfinite reality - as we have seen - has two complementary aspects i.e. transcendent and immanent. The transcendent aspect is highly general. All particular finite phenomena are screened out of awareness to reveal their general infinite ground. The immanent aspect on the other hand is highly specific revealing the unique and infinite quality of each facet of nature.
Now all of this is of direct relevance to understanding properly the nature of transfinite numbers.
I mentioned before that the angel hierarchies essentially deal - in reduced form - with the transcendent aspect of spiritual development. The highest of these hierarchies relates to transfinite reality. Likewise the mathematical system of numbers - which closely mirrors the logic of the angelic system - also leads to transfinite numbers as its highest "hierarchy" dealing - again in reduced form - with the transcendent aspect of transfinite numbers.
In dealing with angels I suggested that they represented essentially (unconscious) projections. Thus what is inherently intuitive and subjective could be appropriated in objective rational format. This particularly applies to the highest hierarchy of angels.
It is precisely the same in relation to transfinite numbers. They represent most clearly that aspect of number which is most intuitive and subjective projected outwardly so as to be appropriated in rational fashion.
As we have seen angels are not really separate beings, but rather represent the infinite potential inherent in all personalities.
In like manner transfinite numbers are not really separate numbers at all, but rather represent the infinite potential inherent in all numbers. All numbers therefore have in dynamic terms both a finite and transfinite aspect.
Though it is customary to interpret numbers rationally in reduced finite form, understanding of numbers - as we have seen - always involves an intuitive infinite aspect. Though the finite and infinite aspects necessarily coexist, when one tries to understand numbers from a solely rational approach, they become separated.
As is well known, Cantor managed to build up a whole separate set of
transfinite, in opposition to finite quantities.
One problem with this approach is that he treats the infinite as an indefinite extension of the finite. Thus for example the set of natural numbers 1, 2, 3, 4 ..... is considered to be infinite or in Cantorís terminology a transfinite number.
This in fact is a very confused form of thinking by which the infinite is treated simply as a linear extension of the finite. Properly speaking the infinite is a circular intuitive notion relating to the very potential for number existence. What is then actualised in rational terms is always finite.
From a dynamic relative perspective the set of natural numbers is finite with both determinate and indeterminate aspects. What this simply means is that the positing of any number or set of numbers e.g. natural numbers, is always against a background of other numbers which remain - in finite terms - undetermined. In other words the set of natural numbers is unbounded and open-ended.
In this context the unlimited potential for number existence is provided by the number concept (intuitively understood). It is this intuitive understanding of the general quality of number - rather than any number quantities - which is truly transfinite.
This qualitative transcendent understanding of the general concept of
number is therefore transfinite. Actual number quantities are always finite.
Now it is true that the number concept can serve as a kind of filing system for all actual quantities, but this again is its (reduced) finite interpretation.
In dynamic terms this quantitative interpretation is open-ended, relating both to all finite quantities which are determined in a specific context, and those finite quantities which remain undetermined.
Thus the intuitive understanding of the number concept as a potential quality represents just one aspect of transfinite number (i.e. the transcendent aspect).
One of the fascinating aspects of Cantorís work is that he seemingly
proved the existence of an infinite number of transfinite numbers. This
finding, in a mathematical community accustomed to the notion of only one
global infinite number, caused much controversy.
Cantor matched differing "infinite" sets by establishing a one to one correspondence as between successive terms in their respective series. Thus for example the set of natural numbers and the set of squares of the natural numbers can be placed in one to one correspondence, with 1 being matched by 1(i.e. the first term of each series), 2 with 4, 3 with 9, 4 with 16 and so on. As long as all the terms of such series could be counted, then this one to one correspondence could be established. Cantor found that this was true of the set of real rational and indeed (algebraic) irrational numbers. These series therefore had the same transfinite number (viz. Aleph0 ).
However Cantor established that the set of all real numbers (which includes the transcendental numbers) could not in fact be counted. This set being immeasurably more dense could not be placed therefore in one to one correspondence with the other series. He therefore concluded that this set had a different transfinite number viz. c (the continuum).
Now what Cantor was really discovering here was the fact that there
are two fundamental processes involved in number experience.
Numbers which can be counted are in a definite sense discrete. (This is even true of irrational numbers - because as they can be expressed as solutions to polynomial equations - they become discrete when raised to an appropriate power or combination of powers). They therefore conform to the rational understanding of number. (Rational understanding corresponds to the conscious linear level. Irrational understanding, on the other hand, corresponds to the "higher level" conscious understanding of the circular level).
Numbers which cannot be counted are by contrast continuous. They therefore conform to the intuitive understanding of number. As we have already seen transcendental understanding corresponds to the (unconscious) point level. Not surprisingly therefore the set of real numbers (which includes the transcendentals) cannot be counted and Cantor had to conclude that this continuous set was fundamentally different from the earlier discrete sets.
What he was really discovering were the two fundamental numbers 0 and 1, this time in a transfinite sense. As we saw earlier, the fundamental building blocks of the finite number system are 0 and 1. They are sufficient to construct the entire number system (i.e. the binary system), and underline the miracles of our computer age. The very symbols used conventionally to represent these fundamental numbers (a circle and a line), are intimately associated with the circular level of intuition and the linear level of reason respectively.
We started - in our construction of the number system - with the fundamental numbers of 0 and 1 in the (quantitative) finite realm. We have now gone full circle to discover the same fundamental numbers in the (qualitative) transfinite realm. Just as we can use 0 and 1 to construct the finite number system, we can equally use 0 and 1 to construct the transfinite number system. Ultimately finite and transfinite number systems are fully complementary. All experience simultaneously involves (quantitative) information and (qualitative) transformation. Just as all quantitative information can be ultimately translated through the finite binary system, all qualitative transformation can be equally translated through the (complementary) transfinite binary system. As the one and the many applies to the finite realm, equally the one and the many also applies to the transfinite realm. It is the recognition of an intuitive (vertical) as well as a rational (horizontal) aspect to all experience that leads to this realisation.
When we look at number from a correct dynamic perspective this complementarity is easy to appreciate. One of the unfortunate results of a merely rational approach to number is to consider that that the number concept is itself invariant with respect to different number perceptions. For example though 1, 2, 3 and 4 for example are different number perceptions we interpret them as belonging to the same number concept. This is why the invariant notion of infinity is so strong. It is vitally necessary to prevent unwanted dynamic interaction among numbers, thus allowing an absolute understanding to be established. However in dynamic interaction the variety of finite experience is exactly matched by transfinite experience. Thus for every finite number there is a matching transfinite number.
This transfinite dimension has itself two aspects. When we focus directly on the intuitive number concept (in relation to differing number perceptions) we have the transcendent or wave aspect. To rephrase and reverse the famous line from Blake, we can see each grain (i.e. number perception) in a world (i.e. the number concept). In this sense when each number perception changes the dynamic relationship with the number concept also changes. Thus the concept yields the transfinite number "1" when we relate the finite perception "1" to it. In like manner the concept yields the transfinite numbers "2","3", "4", when related to the finite number perceptions "2", "3" and "4" respectively. Putting it simply if the (rational) finite perception changes then in dynamic terms the (intuitive) transfinite concept must of necessity also change. Due to this changing intuitive contribution, experience thereby is permanently enlightened and inspired. When the transfinite dimension is not properly recognised, intuition itself becomes fixed thus serving only to reconfirm existing patterns of experience which not surprisingly become rigid and lifeless.
When we focus directly on the intuitive number perception (in relation to the number concept), we have the particle or immanent aspect of transfinite experience. Here we see a world (i.e. the number concept) in a grain of number (i.e. each number perception). Once more, each different finite number will reflect the number concept in a unique manner. So once more we have a potential transfinite number in dynamic terms (i.e. the unique intuitive insight) associated with each actual finite number.
This can be given a simple - yet compelling - mathematical interpretation.
As we have seen the (holistic) finite interpretation of number is the
unit (as quantity or dimension).
Implicit in the (horizontal) definition of a number is that it is raised to the power of 1. Likewise implicit in the (vertical) definition is that the (invariant) one is raised to differing powers (or dimensions).
The (holistic) transfinite interpretation of a number is can be expressed though using 0 rather than 1(either as quantity or dimension).
Thus in the (horizontal) definition a number is raised to the power of 0 (i.e. it is non-dimensional).
Now any number raised to the power of zero is equal to one. Thus though we can have differing number quantities in this transfinite system, when raised to 0, they are all equal to unity. This transfinite aspect of number reflects the contemplation of immanence, where differing objects reflect the same unified vision.
In the (vertical) definition, 0 is raised to differing dimensions.
Thus though we can have such various number qualities (or dimensions) in this transfinite system, again when 0 is raised to these dimensions, the result is always zero.
This transfinite aspect reflects the contemplation of transcendence, where differing dimensions are reflections of emptiness (i.e. the void).
Thus once again in holistic transfinite terms numbers can be represented in binary terms as unity and nothing (1 and 0). So starting with the linear (finite) binary system, we now have come full circle to discover the complementary (transfinite) binary system.
Q. Is there any support within physics for
your dynamic transfinite interpretation of nothing.
PC Yes, there is though the full implications of this evidence have been resisted by the scientific community.
The diagonal lines I have mentioned (with zero magnitude) are sometimes referred to within physics as null lines and play an important part in relativity physics and more recently twistor theory.
Indeed these null lines can be accurately used to represent the true
nature of light.
As we know space and time measurements are relative to the speed of light. Thus light itself does not travel in finite space and time. Now this in itself is a startling observation for it points to the utterly mysterious nature of light. (What we know of light is always indirect through interaction with phenomena). However in a primary sense the true nature of light is transfinite and mysterious. Put another way light will travel an infinite distance in zero time. So here we have the two dynamic polarities of zero and infinity naturally linked.
Now light (as electromagnetic waves) represents just one of the four physical forces.
Indeed the same kind of thinking can be applied to all four forces. The essential insight is that a physical force does not represent objects (quantities) or dimensions (qualities) as such but rather the dynamic intersection of both. Therefore these diagonal or null lines essentially serve as holistic interpretations of the fundamental nature of physical forces representing an essential interaction of matter (quantitative) and dimension (qualitative).
Thus - very simply - holistic mathematics points to the true hidden symmetric nature of forces interpreting them - in each case - in true transfinite terms as the intersection of "real" objects and "imaginary" dimensions.
Now when we approach these forces from the standpoint of the world of phenomena, confusion inevitably enters with the forces becoming separated and asymmetrical. Physicists believe that at high enough energies that this original symmetry is restored. However, in the standard mathematical approach to supersymmetry this leads to the intrusion of "infinities" into equations. Essentially this problem is dealt with through the trick of renormalisation (where infinities are conveniently cancelled out). However this is unconvincing and unsatisfactory. Though at the deepest level, nature has both finite and transfinite aspects, standard mathematics (by its very logic) cannot deal with this. However with Holistic Mathematics 3, finite (actual) and transfinite (potential) aspects - intrinsic to every holon - can be very simply interpreted.
Q. Have these "forces" a counterpart at the psychological level?
PC Yes, they have and I have found this complementary approach to spiritual "forces" extremely useful.
The easiest parallel is that between (physical) and (spiritual) light. Just as (physical) light can be represented by a null (diagonal) line, likewise (spiritual) light can be represented in similar fashion. In a primary sense spiritual light is purely transfinite. Again it has secondary (finite) manifestations through interaction with phenomena.
Indeed the correct interpretation of spiritual (light) is as the intersection of quantitative perceptions (objects) and qualitative concepts (dimensions). This is as we have seen results in that intuitive flash (transfinite) so essential to the fuelling of the dynamics of experience.
As we have seen mystics distinguish as between the contemplation of immanence and the contemplation of transcendence. The former - which is directly responsible for the illumination of specific phenomena in nature - is directly comparable with (physical) light and the electromagnetic force.
The latter - which leads to a generalised global understanding of (external) reality - is directly comparable with the gravitational force. (The supreme expression of this force in spiritual terms is the "dark night". This bears direct (vertical) comparison with the "black hole" which of course is the complementary expression of physical gravity).
Now in physical terms the electromagnetic and gravitational forces can be understood in a global macro fashion in nature. The remaining weak and strong forces have a more local restricted interpretation.. Again these are balanced on the psychological side by inner spiritual "forces". The strong force is represented by deep faith (inner transcendence), which literally holds the personality together. The weak force (inner immanence) then represents an outward expression of this (inner) spiritual belief. Thus we have two external physical forces electromagnetic and gravitational in nature complemented internally by the weak and strong forces respectively. Likewise we have two external spiritual forces, immanent and transcendent equally complemented by corresponding internal aspects.
Once again in a primary sense the four spiritual "forces" are transfinite and represent the essential intersection in experience of (quantitative) perceptions and (qualitative) concepts.
These are complementary (diagonally) with the four physical "forces" which - in a primary sense - are likewise transfinite representing the essential intersection of (quantitative) phenomena and (qualitative) dimensions.
Q. Can you now briefly summarise the eight directional approach of the radial level.
PC I have already dealt with the four finite directions. Put briefly, the specific perception of any number has both positive (quantitative object in relation to perceiving mind) and negative directions (mind in relation to quantitative object perceived). Likewise the general concept of number has both positive (quantitative dimension in relation to conceiving mind) and negative directions (conceiving mind in relation to quantitative dimension).
Relatively speaking perception and concept are "real" and "imaginary" with respect to each other.
Now each of these finite directions - representing actual form - is complemented by a corresponding transfinite direction - representing emptiness which is the pure potential (for form). In other words in the dynamics of understanding rational form is always manifest against a background of intuitive formlessness. In other words finite and transfinite reality mutually reflect each other.
Of course this eight directional approach now equally applies to all holons.
Thus a theorem for example - such as the Pythagorean Theorem - has now both positive and negative interpretations. It also has real and imaginary interpretations. These directions in turn have now in addition both finite and transfinite interpretations.
Thus the ideal of the Pythagoreans to use mathematics as a means of spiritual contemplation can now be given adequate holistic scientific expression. Ultimately all the symbols and operations used in mathematics are truly mysterious (transfinite) and a source of deep wonder. Unfortunately the reduced standard mathematical approach blinds us to this reality.
Physical reality is now likewise modelled in eight directional terms. All phenomena have positive (external) and negative (internal) aspects. Likewise dimensions and object phenomena are "real" and "imaginary" with respect to each other. Finally the intersection of both through physical forces are finite and transfinite with respect to each other.