- QUALITATIVE NUMBERS: TRANSFINITE

Introduction

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 it is quite clear that angels were essentially religious projections of profound archetypal significance providing a convenient figurative device for portraying the vital role of the unconscious mind.

Thus the biblical war as between good and bad angels can be seen as a metaphor for the ongoing struggle between opposing forces in the unconscious mind. Indeed Satan - frequently used to symbolise evil - is portrayed as a "bad" angel.

In the Bible angels were frequently used to deliver important messages of great spiritual significance. Again this relates well to how so often, especially at critical periods in life, intuitively inspired inspirations - springing from the unconscious - mysteriously support people in the taking of key decisions.

The traditional belief that each person has a "guardian angel", can be interpreted in psychological terms as the role of the personal unconscious, which continually attempts to preserve overall balance and health in the personality.

Angels were also entrusted with the protection of nations and cities
which can be translated into the role of the collective unconscious. Finally
the very movement of matter itself in the orderly preservation of the movement
of the planets and stars was attributed to the intervention of the angels.
This has close connections with modern notions in physics e.g. implicate
reality in turn reflecting a more holistic intuitive (i.e. unconscious)
world view.

Thus angels from a psychological perspective symbolise the role of the
unconscious. As transpersonal development entails the pure development
of the unconscious mind, it is interesting to see how this is treated from
the perspective of Angelology. Indeed when properly interpreted, we have
a detailed medieval system which remarkably parallels the various levels
of transpersonal development i.e. circular, point and radial.

To understand this system of Angelology - as outlined by St. Thomas - one has to look at an important problem in the methodology underlining his approach.

Though clearly operating from a considerable level of spiritual insight, the system of St. Thomas belongs very much to the rational linear level. Indeed. It closely resembles the Euclidean system of geometry in that it starts with a set of "self evident" theological axioms, from which all the major Christian teachings are deduced. There is little in it of the deep existential struggle and uncertainty which leads one on the less clearly defined but more authentic quest for meaning.

Because St. Thomas’ approach is so rational (in the linear sense) it is in fact ill suited to treatment of the higher stages of mystical development (where the linear level is superseded). Reason in the manner used by St. Thomas is the supreme expression of the conscious mind. By definition it is not suited to deal with the more subtle intuitively based experience of the unconscious mind (which is qualitatively different).

The use of a linear form of reason 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, even when spiritually motivated it was accepted 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 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 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 knowledge pertaining directly to the unconscious rather than the conscious mind.

Unfortunately because the rational paradigm which they adopted properly related (solely) to the conscious mind, 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.

Angels and the Enneagram

Interpreted in this fashion, Angelology suddenly assumes an unexpected significance and can 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 hierarchies (with three orders in each hierarchy).

I have already drawn attention to the special psycho-mathematical significance of the number 9. In psychological circles a system called the Enneagram has 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 one’s 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 one’s natural personality. Through an arduous process of spiritual transformation one essentially 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 common to all types.

I refer to the transpersonal levels of development as the circular, point and radial respectively. These correspond to the three hierarchies of angels in the Thomistic system.

In turn each level incorporates three sub-levels which I have identified as the physical, mental and spiritual 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.

Thus again 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 God).

The first hierarchy of angels - in ascending order of importance - corresponds to the circular level (i.e. the subtle realm). Now this level though it marks the beginning of true transcendence of the world of human affairs - involving the use of senses and reason -

is still initially tied to it.

The "lowest" order is that of the Angels, traditionally guardians of men and carriers of messages of relatively limited importance from God.

This can be translated in psychological (transpersonal) terms as referring to the role of the personal unconscious and the beginning of supersensory spiritual illumination.

The next order is that of the Archangels. They were entrusted with carrying the more solemn and important messages from God. Thus in the Bible the Archangel Gabriel informs Mary that she is to be the mother of God.

This in turn can be interpreted as the start of that high level intuition (suprarational illumination) marking a more decisive conversion to God.

The third order is that of Principalities. These traditionally were not concerned with particular events but rather entrusted with the more cosmic concern for the welfare of cities and nations.

This again readily translates into the highly transcendent spiritual
focus of the dark night where cosmic rather than ego consciousness now
defines the personality.

The second hierarchy of angels can be related to the point level. Traditionally these were concerned with the universal causes of things. This of course directly parallels the formless consciousness of the point level largely devoid of rigid phenomena.

This angelic hierarchy however really only equates with the transcendent aspects of advanced spiritual development. It is a strong contention of mine that - at least in formal terms - complementary immanent aspects are not adequately dealt with in the Western tradition.

The "lowest" order in this hierarchy are the Powers, followed - in ascending fashion by Virtues and Dominations. There was great emphasis in traditional thinking on the interaction of these orders. However it was however strictly one-way with the "higher" angels overseeing and integrating the activities of those of "lower" rank.

This exactly mirrors the belief held in transcendent spirituality - and indeed in much developmental psychology - that integration of the personality proceeds though a hierarchical sequence of stages with the "lower" being eventually superseded and integrated within the higher stages of development. Thus in Christian spirituality the physical sub-level of sense is transcended by the mental sub-level of reason which in turn is eventually transcended by the spiritual sub-level of pure intuition.

However such one-sided emphasis on (merely) forward integration of stages is ultimately mistaken leading to undue conscious control in the personality (with consequent unconscious repression). Equal attention must be given to backward integration and the release of repression through spontaneous response.

Thus the angelic system accurately represents a personified account
of a transcendent approach to mystical growth.

The angels in the third hierarchy in traditional thinking were able to contemplate the intelligible essences within God Himself. In other words they attained directly to union with God. This again closely parallels the radial level (i.e. ultimate reality) with its habitual experience of the divinity.

The first order of this hierarchy - again in ascending order - was that of Thrones. These were able to perform "right judgement" in exercising divine commands, though not yet capable of full beatific vision.

This correlates well with the early phases of the radial level where one learns to act continually in accordance with the will of God - without distraction of sense - though not in full illumination of spirit.

The second order was that of Cherubim who attained to full wisdom of the ways of God, though not yet capable of complete ecstatic vision.

This relates to the high level of intellectual maturity attained during the radial level, where reason operates in full accordance with the spirit in the attainment of true wisdom.

The third and highest order was that of Seraphim who reached the most complete level of spiritual experience in a burning love of God.

This again relates very well to the fullest experience of mystical union
i.e. transforming union, in ecstatic bliss. In this regard it is very interesting
to note how the visions of well known saints (e.g. St. Francis of Assisi
and St. Teresa of Avila) - recorded in such peak moments of mystical union
- involved the intervention of seraphs.

Angels and Complementarity

Thus the "good" angels really personify "high-level" transpersonal development of the spiritually intuitive unconscious mind.

By the same token, in complementary fashion, the "bad" angels personify "low-level" prepersonal development of the physically instinctive unconscious mind. The powerful evidence of this blind and frequently destructive expression of the unconscious - so much alive in individuals and nations - is very problematic for a culture based on the rational paradigm. The common use of such terms as "demonic influences" and "satanic forces" to describe this negative projection of the unconscious bears testimony to its "angelic" roots.

It is the very nature of the rational linear method to separate opposites
which in dynamic terms are really complementary. Thus in the static angelic
system of St. Thomas, only the "good" angels are recognised. This unfortunately
translates into a view of the spiritual life which is based on the domination
of the "bad" (i.e. physical instincts) by the "good" angels (i.e. reason
guided by spirit). This leads to an unduly transcendent view of mystical
development which ultimately lacks wholeness. In dynamic terms the "good"
angels are fully complementary with the "bad" angels. In other words we
can only fully release the potential for good in the unconscious personality
by simultaneously realising - and indeed accepting - its great capacity
for evil. Rather than attempting therefore to overcome the senses through
reason, one seeks to reconcile both, as complementary equals through the
spirit, in a Trinity without hierarchy.

Angelology and the Number System

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. As we have seen 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 (and thereby irrational from a physical perspective) these angels could in certain circumstances assume a human form (thus appearing rational).

It is very similar - in complementary terms - with irrational numbers.

Such a number is by definition irrational when expressed in the 1st
dimension. Thus Ö2 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 it is once again rational i.e. (Ö2)^{
2 }= 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.

Etienne Gilson in his chapter on "The Philosophy of St. Thomas Aquinas" writes

"The angels are creatures whose existence can be proved and, in exceptional cases, observed; their suppression would render the universe, taken as a whole, unintelligible; and lastly the operations of inferior creatures such as man can be perfectly understood only by comparison with, and often by opposition to, those of the angels."

With an appropriate change of a few key words, this becomes a statement regarding the number system.

"The transcendental numbers are quantities whose existence can be proved
and in exceptional cases observed; their suppression would render the number
system taken as a whole unintelligible; and lastly the operation of these
"inferior" quantities such as rational (numbers) can be perfectly understood
only by comparison with, and often by opposition to those that are transcendental".

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 terms.

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 Cantor’s 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 and 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 one’s 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 infinite 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 in reduced fashion 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 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. It represents an attempt to integrate in uni-modal form 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 reduced rational manner. This is gives rise to a transfinite approach
in both positive and negative directions and real and imaginary modes.

Likewise in relation to the understanding of number there is the emergence of a transfinite approach 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 interpretations.

Let me illustrate this with respect to the concept of number. The number concept can be interpreted in rational terms as the general class to which all actual numbers belong. However the number concept equally has can 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 creation 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 energy 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 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 fueling 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 refer 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 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 something 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 at all, 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 dimension. 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.

The first problem in 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 Cantor’s terminology is 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 has a reduced quantitative interpretation as a kind of filing system for all actual quantities, but this again is its merely 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 in quantitative terms, the concept of number relates, for example,
to the first four natural numbers. Now 1, 2, 3 and 4 are finite actual
quantities. Their determination in this context necessarily requires that
the remaining natural numbers stay undetermined in actual form. No matter
how many finite actual quantities are determined, of necessity a complementary
set of finite quantities must remain unactualised or undetermined.

Thus the intuitive understanding of the number concept as a potential quality represents one aspect of transfinite number (i.e. the transcendent aspect).

One of the fascinating aspects of Cantor’s 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. Aleph_{0
}).

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. Properly understood this corresponds to Aleph_{1}.

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 in that 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. As we have seen even 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 finite realm. We have now gone full circle to discover the same fundamental numbers in the transfinite realm. Just as we can use 0 and 1 to construct the finite number system (which is potentially infinite) we can equally use 0 and 1 to construct the transfinite number system (which is potentially finite). Ultimately finite and transfinite number systems are fully complementary.

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 are different number perceptions we interpret them as belonging to the same number concept. This is why an invariant notion of infinity is so strong. It is vitally necessary to prevent unwanted dynamic interaction among numbers, 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 finite perception changes then in dynamic terms the 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 uninspired.

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.

The immanent aspect of transfinite numbers parallels closely the horizontal system of number. The transcendent aspect parallels closely the vertical system of number.

In finite terms the system of natural number representing direct quantities
is the horizontal system 1^{1}, 2^{1}, 3^{1}, 4^{1}
etc. (Here the number quantity varies while the dimensional quality remains
fixed).

The transfinite system representing indirect qualities (i.e. the intuitive
insight underlying number perception) can be represented in similar fashion
1^{0}, 2^{0}, 3^{0}, 4^{0 }etc. (now representing
the intuition underlying such number perceptions).

In finite terms the system of natural number representing indirect quantities
(i.e. dimensions) is the vertical system 1^{1},^{ }1^{2},
1^{3}, 1^{4} etc. (Here the dimensional quality varies
while the number quantity remains fixed).

The transfinite system representing direct qualities (i.e. the intuitive
insight underlying the number concept) can be represented in similar fashion
as 0^{1},^{ }0^{2}, 0^{3}, 0^{4}
etc. (now representing the direct intuition of number as dimension).

Number Complementarity

*Fundamental Numbers - Finite and Transfinite*

There is a striking complementarity in psychological terms as between the stages of prepersonal and transpersonal development. The earlier stages represent in undifferentiated fashion what emerges and develops later in a mature and complete fashion. As each of these (psychological) stages has an exact mathematical equivalent, this implies that there is also striking complementarity in mathematical terms as between the earlier pre-rational and later post-rational number types.

In this complementary system the "lowest" number type - at the pre-rational stage, exactly matches the "highest" number type at the post-rational stage.

As we have seen we started the number system with the fundamental numbers 0 and 1. The origin of these numbers is in the earliest psychological state where conscious and unconscious minds are totally undifferentiated. Here original unity - in undifferentiated awareness - in fact represents a nothing in terms of actual experience. Thus the complete (vertical) coincidence of 0 and 1 (i.e. unity and nothingness) in psychological terms has a complementary interpretation in the (horizontal) separation of 0 and 1 in mathematical terms. The vertical coincidence of 0 and 1 (i.e. undifferentiated psychological structures) gives way to the horizontal separation of 0 and 1 (i.e. differentiated mathematical quantities). Thus at the earliest stage of development we can see the original unity of psychology and mathematics which provides the very basis for the intuitive (psychological) and rational (mathematical) approaches to experience.

The intuitive approach basically understands unity and nothingness as ultimately identical. In other words any visible phenomenon which is actualised in experience as a finite unit inevitably involves an invisible dimension representing the potential for experience which is nothing in finite terms. In the pure unconscious state of the emerging foetus original unity indeed is nothing.

The rational (mathematical) approach - by contrast - basically understands unity and nothingness as separate. Here the potential dimension of experience is ignored and an absolute distinction drawn as between existence and non-existence. What actually exists (as an independent unit) is in striking contrast to what is nothing - again in actual terms - in any given context. By ignoring the vertical or qualitative dimensional characteristic of experience, one is thereby enabled to treat everything in horizontal (i.e. one dimensional) fashion.

Thus the rational (horizontal) approach to experience begins with the successful differentiation of being and nothingness. The fundamental numbers of 0 and 1 are thereby born in finite terms.

One cannot over-emphasise the importance of these fundamental numbers. The very basis of addition and subtraction in mathematics is inherent in them.

Nothingness - as pure potential - represents the complementarity of unitary opposites i.e. +1 and -1 (i.e. 0 = +1 - 1 ). This represents the basis of the unconscious combining as it does opposite poles. The potential for experience (in the unconscious) dynamically requires the continual actualisation and de-actualisation of finite phenomena.

The binary number system - based on 0 and 1 - is sufficient to build
the entire number system. Indeed this binary system underlines the highly
important development of computers which has revolutionised the transmission
of information in our present age.

Thus the finite emergence of the fundamental numbers - in prepersonal development - represents the earliest system of numbers.

In complementary fashion - as we have seen - the latest and most complete level of development involves the emergence of the fundamental numbers this time in transfinite form.

Whereas the earliest prepersonal stage of development involves the confusion of unity and nothingness (i.e. conscious and unconscious), the latest transpersonal stage involves their mature fusion. In other words conscious structures - at all levels - have by been fully differentiated thus enabling their mature integration (through the unconscious). Experience is now seen in transfinite terms which has two aspects. The transcendent aspect - in the holistic appreciation of what is eternal and universal - represents the realisation of the pure potential of existence. It is nothing in finite terms.

The immanent aspect - in the specific appreciation of what is immediate and intimately present - represents the pure actualisation of existence. Thus each phenomenon is now uniquely something in finite terms.

This psychological realisation translates into mathematical terms in the clear realisation that 0 and 1 as numerical quantities have a transfinite as well as finite significance.

This involves the full appreciation of the (vertical) intuitive dimension of numbers. Thus if the rational understanding of number gives us the (direct) quantitative appreciation, then the intuitive understanding gives the complementary (direct) qualitative appreciation. This qualitative interpretation does not so much offer (direct) information about numbers but rather acts as a dynamic agent of transformation in this process of understanding. Once again we can draw basic though important parallels.

The horizontal quantitative approach is intimately linked to the binary system of logic (based on the rational separation of opposites). This underlines computer technology and the present attempt to come to terms with the world through rapidly increasing generation of information.

The vertical qualitative approach is intimately linked to an alternative binary system of logic (based on the intuitive complementarity of opposites). This underlines the authentic mystical approach and the inward journey which leads to transformation of the self.

All living processes involve complementary processes of information and transformation in what represents a complex computer system based on a double binary system (both quantitative and qualitative).

To put it bluntly there is far too much emphasis in our society on information. There is a much greater need now for complementary transformation in an unreduced appreciation of the intuitive.

The process of understanding numbers is intuitive as much as rational. Therefore a full understanding of numbers is qualitative as much as quantitative. This implies that such an understanding has complementary transfinite and finite aspects which are dynamically interconnected..

Each number interaction thus constitutes a unique relationship so that
specific numbers no longer belong to a fixed static class. Indeed once
again we have striking parallels with the angelic system of the middle
ages. The scholastic theologians had reasoned that each angel constituted
a unique species and did not properly belong therefore to a general class.
We are now seeing the dynamic interpretation of this viewpoint.

*Prime and Transcendental Numbers*

In prepersonal terms, the second stage of development involves the emergence of the primitive structures.

In transpersonal terms, the second last stage of development involves the emergence of the point level.

Once again these two stages are complementary. The impulses and instincts which emerge in a blind undifferentiated fashion at the prepersonal (primitive) stage are finally fully understood and integrated within the personality at the later transpersonal (point) stage.

Putting it in more mathematical terms at the primitive stage, the infant is not able to properly distinguish phenomena (within dimensions) from the dimensions themselves. There is still therefore considerable confusion of the conscious (rational) and unconscious (intuitive) processes.

During the linear and circular levels one learns to differentiate the
conscious and unconscious processes respectively. Then at the point level,
one is able to properly integrate these two processes. Thus one no longer
confuses the quantitative with the qualitative aspect of experience.

Again we have precise mathematical parallels in the number system. The prime numbers directly complement the primitive structures whereas the transcendental numbers directly complement the point structures.

A prime number has no factors (other than itself and 1). Composite numbers
involve factors (e.g. 6 = 3 X 2). Such a number is two dimensional and
involves a quantitative value (i.e. 6) and a qualitative value (i.e. no.
of factors = 2). With prime numbers - which have no factors - quantitative
and qualitative aspects are necessarily separated. Thus the complete psychological
fusion of quantities and dimensions with the primitive structures exactly
complements the mathematical separation of quantities and dimensions with
the prime numbers.

The point structures involve the psychological ability to integrate - without reductionism - the (quantitative) linear and (qualitative) circular paradigms.

A transcendental number involves an integration - in reduced terms - of the relationship between the qualitative (or dimensional) and quantitative aspects of number. This - as we have already seen - is perfectly symbolised by the most famous transcendental number p . This represents the quantitative relationship between the circle and line (i.e. its diameter).

Once again the transcendental numbers (mathematically) directly parallel
the point structures (psychologically).

As the prepersonal primitive structures directly complement the later transpersonal point structures, in like manner the prime numbers directly complement the transcendental numbers.

Prime numbers are somewhat elusive. Though there are - in mathematical terms - an infinite set of such numbers, it is very difficult to identify additional prime numbers. Transcendental numbers - which are as dense as prime numbers are sparse - are also notoriously difficult to identify.

Given the complementarity of prime and transcendental numbers, this
suggests that the general behaviour of prime numbers is in fact transcendental.
This is remarkably confirmed by the prime number theorem which provides
a means of predicting - not particular prime numbers - but rather the general
distribution of such primes. Thus the frequency of primes for the natural
numbers up to n is approximated by by n/log_{e}n, the accuracy
of which increases as n increases. This of course is intimately related
to the exponential e which after p is the next most important transcendental
number.

* *

The complex numbers are also of particular value in relation to theorems
on prime numbers. Complex numbers are closely related to transcendental
numbers. Whereas a transcendental number attempts to express the inherent
relationship as between what is real (i.e. number as quantity) and what
is imaginary (i.e. number as quality or dimension), solely in (reduced)
real or imaginary format, a complex number involves the expression of the
same relationship using both formats.

*Natural and irrational numbers*

The third - and final - overall stage of prepersonal psychological development involves the emergence of what I term the natural structures. These structures indeed emerge naturally in the personality with the conscious mind not yet properly differentiated from the unconscious. Whereas increasingly, particular phenomena are now defined clearly in conscious terms, the overall holistic appreciation of the world remains unconscious and confused.

The corresponding third last - which indeed is the first - overall stage of transpersonal psychological development involves the emergence of the structures of the circular level. In contrast to the natural structures - these involve the proper (explicit) differentiation of the unconscious structures. Whereas the natural structures gradually prepare the way for the specialised use of reason, the super structures of the circular level prepare the way for the specialised use of intuition. Indeed - in the context of mature spiritual awareness - they entail a natural stage of transformation.

Once again what is implicit at the earlier prepersonal stage (i.e. the
confused use of the unconscious) is made explicit at the later mature transpersonal
stage.

Again we have in mathematical terms number types which directly parallel these psychological stages.

Corresponding to the natural structures, we have of course the natural numbers. With the natural structures one is not yet able to properly analyse and break whole units into parts. Likewise with the natural numbers, we only deal with the (positive) integers being not yet able to understand part numbers (i.e. fractions).

Corresponding to the super structures of the circular level, we have the (algebraic) irrational numbers. The super structures involve the expression of the two-dimensional unconscious in reduced conscious terms with complementary positive and negative poles. In terms of the logic of the linear level (which is one dimensional and based on the clear separation of opposites), such understanding is indeed irrational. A vertical qualitative dimension (viz. the two dimensional unconscious) when translated in reduced horizontal terms is thus irrational.

This applies exactly in complementary manner to irrational numbers. This is symbolised by the best known irrational number i.e. Ö2. This involves the attempt to convert the number 2 - from one dimension to a fractional dimension (viz. 1/2). This is done through extracting the square root. Again we get two poles (one positive and one negative) which are separated. Thus the square of 2 has two distinct roots +Ö2 and -Ö2. Furthermore the quantitative value associated with these roots is irrational in that it is not strictly finite. Though we can approximate the value to a high degree of accuracy an irreducible infinite dimension remains involved. This time the problem arises due to the attempt to switch from a horizontal (i.e. one dimensional) to a vertical (i.e. fraction dimensional) format. Properly speaking the irrational in psychological terms always involves the switching from two to one dimensions (or alternatively one to two dimensions). In mathematical terms the irrational properly speaking relates to the complementary switching from 1 to 1/2 dimension in obtaining square roots (or alternatively from 1/2 to 1 dimension).

When we move into higher dimensions in psychological terms (e.g. the
3rd) more complex relationships are involved (transcending the irrational).
When we move into lower (fractional) dimensions in mathematical terms (e.g.
the 1/3 or cube root), complex - as well as irrational roots - are inevitably
involved. This point indeed is central to the true understanding of the
significance of Fermat’s Last Theorem.

Natural numbers and irrational numbers are therefore themselves complementary.
This is illustrated by a very interesting connection. In mathematics the
solution to any polynomial equation (of one variable) will yield a real
valued irrational (if not rational) solution. This requires the coefficient
of the varied terms in the equation (representing different powers) to
be integers. Thus ignoring the sign, these coefficients themselves are
all natural numbers.

*Rational Numbers*

Psychologically intermediate between the prepersonal and transpersonal stages of development lie the rational structures which represent the culmination of the linear level. Whereas each stage of prepersonal development can be matched in complementary terms with a corresponding stage of transpersonal development, the rational structures of personal development - which lie in the middle - can only be matched with themselves. Thus the dynamic tension as between opposites disappears at this level with in turn the psychological vertical dimension being screened out altogether from formal mathematical understanding. Thus at the linear level, mathematical understanding is deemed to be horizontal and rational and entirely separate from (transpersonal) psychology which is by contrast vertical and intuitive in scope.

The rational paradigm, which is the defining characteristic of conventional personal development, like natural light represents just one small band on the overall spectrum of understanding where reason and intuition become divorced to an extreme degree. At all the other bands on the spectrum there is a considerable degree of interaction as between conscious and unconscious (i.e. rational and intuitive approaches). With prepersonal development this interaction is somewhat confused and undifferentiated. With transpersonal development it becomes refined and harmoniously integrated within experience.

Rational understanding involves a basic fallacy, which is so ingrained in out thinking as to become a fundamental axiom. This involves the confusion of whole and part. With rational analytical thinking, parts are simply fragments of the whole. In other words the whole is not contained in each of its constituent parts.

This is exemplified in the very nature of a rational number. If I divide 1 by 2, I get a rational number or fraction (i.e. 1/2). In like manner in concrete practical terms, if I divide any object (e.g. a cake) in two, I get two separate parts, each of which is half of the original whole. The wholeness of the original object is literally lost by this division, leaving us with separate parts. Not surprisingly in a culture largely attuned to such rational analysis, considerable fragmentation of experience results. The wholeness of life is quite simply lost by this process. With so many parts created that are separated from their original wholeness, the uniqueness of these parts vanishes and experience of their authentic meaning is lost. The inner self in like manner becomes progressively divided and fragmented breeding alienation and futility.

It is intuition which provides the ability to properly link parts with the whole (and in reverse manner the whole with the parts). This is what I have referred to as the transfinite aspect of experience. One is then enabled to see the whole contained in each part, and likewise all parts contained within the whole.

As I have been stressing this intimately affects our very experience
and interpretation of number interactions.