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Peter CollinsPeter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

Clarifying Perspectives 2

Perspectives, Personality Types and Strings

Peter Collins

The well-known personality systems e.g. Jungian and the Enneagram are defined by the dominance of key primary perspectives.
The study of these systems can provide interesting insights therefore with respect to the differentiation and corresponding integration of such perspectives in development.
Also - though perhaps initially surprising - intimate connections can be made as between these personality systems and the world of physical strings (where primary perspectives first emerge).
Furthermore the structure of these perspectives - at the level of strings - is strongly related to the holistic mathematical nature of prime numbers.

Historical Context of Approach

In the first part, I outlined the bones of a dynamic interactive approach to perspectives that is inherently mathematical in a holistic sense. Indeed the version I outlined there illustrates the - potentially - crucial role of the holistic binary system as a scientific means of encoding all developmental relationships.

However as this approach did not arise in a vacuum, perhaps it may help to put it in a more personal context by briefly discussing its evolution.

I first formed the notion of a distinctive integral scientific approach to development - based on an alternative dynamic interpretation of mathematical symbols - when at College in Dublin in the late 60's. At the time I was greatly influenced by Hegelian philosophy which provided invaluable training in the subsequent handling of dynamic relationships at all levels. Also through Hegel, I was able to draw the vitally important link as between the notions of positing and negation (as used in a psychological context) and the fundamental operations of addition and subtraction. In other words - though perhaps not intentionally - Hegel provided me with the firm basis for a holistic interpretation of these mathematical notions.

I was also at this time very much attracted to Taoism which emphasises the complementary nature of polar opposites in experience (and in this respect has marked similarities with Hegelian thought). One particular expression of the general nature of Taoism - that the primal indivisible unity of reality expresses itself in terms of two opposite poles (yin and yang) at the reduced phenomenal level of experience - proved vital in terms of making another key holistic mathematical link.

I could see here that there was an intimate connection with the mathematical notion of a square root (specifically the square root of 1).

Therefore - in analytic terms - when we take the square root of 1 we obtain two equally valid opposite answers i.e. + 1 and - 1. Likewise in holistic terms, when we attempt to express (integral) nondual meaning in reduced (differentiated) terms, two opposite interpretations of form - which are equally valid in a phenomenal context - can always be given.

This is why I consider it so important to clearly distinguish integral notions of meaning (where opposite poles are understood as complementary and ultimately nondual) from their differentiated counterpart (where the same poles are treated in an independent dualistic fashion). Also because the integration of opposite poles of form (+ 1 and - 1) leads to 0, we can thereby encode the essential relationship as between form and emptiness in holistic binary terms.

This connection goes back to Leibniz who not alone invented the binary digital system (as conventionally understood in analytic terms) but also had a clear insight into its complementary holistic aspect (i.e. where 1 = form and 0 = emptiness).

However from my study of Economics at College, I soon realised that there are qualitatively different types of polarities (that cannot be modelled in the same manner). This quickly led me on to the dynamic interactive notion of four polarities, based on the four roots of unity - applicable to all relationships - allowing for two distinct types of polar classification i.e. horizontal and vertical.

The horizontal polarities relate to the interior and exterior aspects, which necessarily condition all phenomenal relationships within existing levels (which I referred to then as inner and outer). The vertical polarities relate to corresponding individual and collective aspects between levels (which I termed particular and general).

For example Economics has both inner and outer aspects (that are positive and negative with respect to each other). However in conventional treatment the inner is largely reduced to the outer so that economic relationships can be modelled in a scientific objective manner. [1] Indeed this attempted objective type approach is referred to as Positive Economics!
So I formed the strong conviction at the time that integral economic understanding requires that equal emphasis be given to both inner and outer and the dynamic interaction as between both aspects.
Likewise Economics has both particular and general aspects (as in the conventional division as between the micro and macro aspects of the discipline). However - given the widespread emphasis on individualism and the free market system in so many economies - the (general) macro is again substantially reduced to (particular) micro notions. [2]
So once again an integral approach would require that equal balance be given to both aspects (and their corresponding interaction) with respect to both the quantitative and qualitative dynamics of economic activity.
As a trained economist a great deal of my early intellectual work related to the kind of dynamic methodology that would be thereby needed in Economics to preserve both the static analytic understanding with respect to individual poles with overall dynamic appreciation (so that these poles could be successfully integrated).

However as the implications of these polarities apply universally to all relationships, my focus shifted substantially away from sole concern with Economics in subsequent years.

Indeed there are marked similarities here with the four-quadrant terminology of Ken Wilber with the inner and outer polarities corresponding with the interior and exterior and the particular and general with individual and collective aspects of holons respectively.

When I first read about Ken Wilber's independent discovery of the four quadrants in the mid 90's I was quite excited as it mirrored in certain respects my own findings.

The great strength about Ken's research based approach is the manner in which he has been able to apply it impressively to a wide range of scientific and developmental data.

However the main weakness - from my standpoint - is that it is not sufficiently grounded in an appropriate dynamic interactive manner of understanding relationships. Consequently there has always been a marked discrepancy as between the manner in which he attempts to treat the overall integral nature of quadrants (where they simultaneously co-exist) and his differentiated treatment, where he typically deals with quadrants in a somewhat fragmented manner (misleadingly identifying them with specific holons).

In other words despite the considerable practical benefits that have flowed from his four-quadrant model, overall it still somewhat inconsistent (due to the lack of dynamic context with respect to its differentiated and integral aspects). And this problem is also evident in his current treatment of perspectives (which I will deal with more fully in the final part)!

Jung's Approach to the Four Quadrants

I soon discovered however that the notion of the four quadrants (in all their varied manifestations) is a very old idea and that others, especially Carl Jung, had already undertaken valuable pioneering work with respect to its universal applicability. Also this was far from an accidental feature of Jung's work but indeed central to his overall contribution.

The idea that four has a special significance in the integration of reality is demonstrated for example in the way we organise space-time dimensions (with 3 of space and 1 of time), in the way that we recognise four fundamental forces of reality, the ancient four elements (fire, air, earth and water) and even the old medical system of the Greeks (with the identification of four "humours").

Indeed there is a definite mathematical justification for this emphasis on "4", as it is invariant with respect to the dual transformation of "2" in horizontal, vertical and diagonal terms. In other words 2 + 2 = 4 (horizontal); 2 X 2 = 4 (vertical); 2 (raised to the power of 2) = 4 (diagonal)."

However in geometric terms there are really two notions of 4.

The first is the standard linear definition where 4 can be represented through successive unit intervals on a (straight) line. However the other - which is deeply relevant in a developmental context - is the circular definition where 4 is represented by equidistant points on a circle (with opposite points joined by horizontal and vertical lines). In this way the circle is literally subdivided into four quadrants. [3] It is this inherently dynamic notion of the four quadrants that informed Jung's work. Thus when Marie Louis von Franz said, "Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four .", it was the circular - rather than the linear - 4 that she had in mind.

Though Jung was not a trained mathematician, implicitly he sought to understand reality in holistic mathematical terms. He understood - at least at some level - the mathematical origin of the four quadrants (and eight sectors) and appreciated their integral significance by drawing attention to their widespread pictorial representation as the most commonly used mandala symbols.

Jung modelled his personality system on four functions, which he conceived in dynamic terms as two pairs of opposite poles (sensation and intuition with thinking, and feeling respectively). He understood clearly the inevitable tension as between the need to consciously differentiate these functions (as separate) and also achieve their unconscious integration (through maintaining equal balance as between opposite aspects).
However it is the very nature of personality that these functions do not initially emerge in a balanced fashion with typically one as dominant (thereby defining personality) with a second and perhaps third operating in a supporting auxiliary fashion. However the inferior function - which is always opposite the most dominant in personality - generally remains somewhat undeveloped and inaccessible through remaining repressed in the unconscious.
So if for example the sensation function is most dominant then the inferior function will be its opposite i.e. intuition.
Using holistic mathematical language, the dominant function is directly posited (as conscious understanding). However because the equal validity of its opposite is thereby not recognised, it is negated in a somewhat rigid manner (whereby it becomes repressed in the unconscious).
In this dynamic complementary manner the superiority of one function (as conscious) is necessarily associated therefore with the corresponding inferiority of its opposite function (as unconscious).

Indeed I found Jung's way of dealing with these functions (where both conscious and unconscious aspects are combined) of immense value, helping me to solve a key holistic mathematical problem that had troubled me for several years.

I found it initially quite easy to provide a satisfactory holistic mathematical explanation for circular duality (based on the square root of unity) i.e. two opposite polarities that are understood in "real" conscious terms.
However the holistic mathematical notion of the four quadrants - based on the four roots of unity - combines two "real" and two "imaginary" results.
Now through Jung's own formulations, I was able to see that all phenomena have both conscious and unconscious aspects and that "imaginary" polarities relate to the indirect expression of what pertains directly to unconscious meaning. [4]

Therefore in dynamic interactive terms, experience not alone switches as between horizontal polarities i.e. exterior and interior (within a given level) and vertical i.e. individual and collective (between levels) but also continually switches as between the "real" (conscious) and "imaginary" (unconscious) expression of these same polarities.

Thus when one function is dominant (in conscious terms) the opposite function remains considerably repressed in the unconscious. It then seeks to indirectly express itself in confused fashion through involuntary projections which enter conscious experience (but cannot be appropriated at that level). Therefore ultimate psychological harmony requires that equal emphasis be given to both the "real" (conscious) and "imaginary" (unconscious) aspects of phenomenal experience. Then as the unconscious aspect is properly integrated, the involuntary nature of projection ceases with both conscious and unconscious now smoothly identified in spiritual terms.

Indeed in this crucial respect - where the complementary nature of both conscious and unconscious is recognised - Jung offers an inherently more integral account of the four quadrants than Ken Wilber (who typically attempts to view them in merely "real" conscious terms).

Then by introducing two attitudes i.e. introversion and extraversion, Jung goes on to delineate, what are in fact primary perspectives, through his eight personality types. [5]

Thus each of Jung's four functions has both an interior (introvert) and exterior (extravert) expression. In this respect it bears marked similarities to Ken Wilber's recent outline of his eight primordial perspectives (where each quadrant designation is given both an "inside" and "outside" interpretation).

Indeed we can see further similarities as between Jung's functions and Wilber's quadrants. For example Wilber would - misleadingly as it happens - designate the Left-Hand Quadrants with personal and the Right-Hand with impersonal "it" understanding. This would roughly equate with Jung's functions of feeling (personal) and thinking (impersonal) respectively. Also sensation and intuition - as Jung defines them - relate to individual and collective aspects. Typically therefore the sense-based individual (in Jungian terms) is most competent dealing with (actual) individual facts; the intuitive by contrast is more readily alive to the (potential) holistic collective pattern inherent in such facts.

When one looks at Wilber's approach to the four quadrants from a Jungian perspective it represents a very marked S (sense) based rather than N (intuitive) understanding. For example Ken's failure to recognise heaps as "potential" holons represents the lack of such intuition. Then when we combine both the S (sense) and N (intuition) aspects we can see both holons and heaps in more dynamic interactive terms as "complex" holons that continually interact as between a "real" (holon) and "imaginary" (onhol) status. [6]

However by the inclusion of two additional attitudes (perception and judgement) the Jungian profile of personality types was subsequently extended to 16 (as used in the well-known Myers-Briggs typology).

Once again we can fruitfully see these 16 personality types as representing unique manifestations of 16 primary (or native) perspectives.

When we draw comparison as between this and Ken Wilber's system it suggests that there are 8 additional primary perspectives that are yet unaccounted for in terms of his understanding of the quadrants.

I will quickly suggest the basis of these missing types here. Just as we have an inside and outside of every quadrant (in Wilberian terms), likewise we have an (actual) individual and (potential) collective interpretation.

So in his delineation of primary perspectives Ken is concentrating on merely "real" actual manifestations (pertaining directly to conscious understanding). However we can equally focus on "imaginary" potential perspectives (relating directly to unconscious appreciation). And in the dynamics of understanding there is a continual interaction as between "real" (actual) and "imaginary" (potential) aspects.
So in Jungian terms, Ken so far has defined his 8 primary perspectives in merely "S" (actual) terms. [7]
However - as yet - he has given no corresponding interpretation in an "N" (potential) intuitive manner.
Though of course no personality type can be reliant on just one perspective (and must necessarily combine all in some measure), however - by definition - one perspective will be especially associated with each personality type.

One weakness of this extended Jungian approach is that it provides little clarity as to the precise manner in which the subsequent integration (of all perspectives) can take place. Jung himself dealt with this issue in merely general terms offering the view that in the "normal" case, one function would be likely to be well developed in early years. Then entering adulthood a second auxiliary function would undergo considerable supporting development and then with the transition of middle age that perhaps a third would also be brought into the "light" of mature conscious use.
However in most cases the fourth (inferior) function would remain largely unconscious and thereby inaccessible.
This would of course imply that typically the primary - and indeed composite perspectives - would remain somewhat lop-sided in normal development with undue concentration on those directly related to the more developed psychological functions in the personality.

Eight "Missing" Personality Types

Because of my dissatisfaction with the "integral" aspects of the Myers-Briggs system, I sought some 15 years ago - using my own "complex" four-quadrant model - to seek a revision.

I realised that the primary personality types (representing in turn the primary perspectives) could be obtained directly from various configurations of the four quadrants (without the need for extraneous variables).

One way of doing this is to view each personality as combining all four-quadrant locations (i.e. interior and exterior and individual and collective) with each permutation or configuration representing relative different strengths of these locations.

Thus when we represent the four quadrants as a, b, c and d respectively then we can obtain 24 distinct permutations (i.e. arrangements) of these four letters. Thus the first letter would represent the most dominant quadrant, the second the supporting auxiliary, the third the weaker auxiliary and the fourth the inferior quadrant. Each of these permutations - representing varying strengths in the manner of configuring the quadrants - defines a distinctive personality type.

When I examined these results more closely, I found that 16 corresponded very well with the 16 types identified in the Myers-Briggs system. However this left 8 additional personality types that had not yet been accounted for. I have described these findings in detail elsewhere so I will only summarise here.

Basically the first 8 represent the horizontal types (corresponding to the S group in the Myers-Briggs) defined by a firm grounding in the actual "real" world. Therefore development for the S type is most likely to plateau with the specialised understanding at a given stage (of the middle level).

The second 8 types represent the vertical types (corresponding to the N group in the Myers-Briggs) defined by a greater affinity with the potential "imaginary" world. Therefore development for the N group is likely to be more problematic with much greater fluctuation as between levels taking place. Intuitives therefore often have less (conscious) control over their lives requiring the right unconscious conditions to be present before they can function effectively in a creative manner.

The third group of 8 represents the diagonal - or mystical - types (with no correspondent in the Myers-Briggs system).

These are defined neither by conscious (S) nor unconscious orientation (N) as such but rather by the need to spiritually reconcile both of these aspects.

The essence therefore of the mystical types is that personality is not primarily defined in (differentiated) either/or but rather (integrated) both/and terms.

Thus in terms of the Myers-Briggs, the first letter defines one as either E (extravert) or I (introvert). However the mystical type is defined by neither of these aspects (as separate) but rather their complementary identity. So mystical types are primarily centrovert (i.e. centred on ultimate nondual reality) though in secondary terms may be defined in either extravert or introvert terms.

The second letter on the Myers-Briggs is defined by either S (sense) or N (intuition). However once again the mystical type is neither sense nor intuitive oriented as such but rather directly spiritual (as the means of combining both).

The third letter on the Myers-Briggs defines one as either a T (thinking) or F (feeling) type. However the mystical is primarily motivated by volition (i.e. will) which again provides the means of harmonising the secondary aspects of thinking and feeling.

Finally the fourth letter on the Myers-Briggs defines one in terms of either P (perception) of J (judgement). The perception type will typically be more pragmatic dealing with issues on their individual merits. The judgmental by contrast will tend to rely on generalised (i.e. collective) rules. However again the mystical personality is primarily defined by a more spiritual form of discernment (as the means of harmonising these aspects).

From my own experience I never could identify with the Jungian view of the decisive crisis in one's life occurring at mid-life. With mystical type personalities the decisive psychological crises (and their may well be several) generally happen much earlier in life e.g. in one's teens and early 20's with mid-life representing - by comparison - a very peaceful form of transition. In other words the mystical type proves especially sensitive from early age to shadow impulses from the unconscious preventing the normal manner of adaptation around the dominant (conscious) function.

Thus the S ("real") group of personalities is inherently very stable and geared to deal most efficiently with waking reality (within a given level). The N ("imaginary") group is inherently less stable but geared to deal more creatively with dream reality (requiring greater dynamic interaction between levels). The M (mystical) represents equally the most "complex" - yet potentially - the most (spiritually) simple of all the groups. If growth for these types stagnates they can become extremely problematic; however given appropriate development they equally have the capacity to become the most integrated. Indeed the burning desire for high level integration (whatever the cost) is often what sets these apart from other types.

This classification of personality types into three groups therefore enables us to describe in general terms the nature of development required for appropriate spiritual integration to take place. Put another way it provides the framework for integration of primary (and composite) perspectives that is appropriate for each group.

Thus for the "real" (S) waking type, stable spiritual integration can generally be obtained at the centaur. Appropriate radial development for the centaur will thereby tend to be largely centred on the middle stages (with relatively little vertical exposure to other levels).

For the "imaginary" (N) dream type, spiritual integration would generally require - in addition - considerable subtle level development (giving a more distinctive vertical aspect to radial type development).

For the "complex" (mystical) deep sleep type, spiritual integration would also require considerable mastery of the causal level (allowing freedom of attachment from both the direct and indirect conscious expression of phenomena).

Finally the question as to whether a mystical type can fully embrace activity as well as contemplation depends on the relative strength of the two weakest functions. If these differ considerably from the stronger then complete development of radial activity with respect to both its active and contemplative expressions is unlikely.

String Theory and Primary Perspectives

When one accepts that the Jungian personality system (and its extended interpretations) is defined by the relative dominance of particular primary perspectives, this provides an unexpected link with the most basic manifestations of matter (where primary perspectives first originate).

When properly understood this provides a fascinating means for solving the key fundamental riddle of string theory where the successful modelling of reality requires a greater number of dimensions than used in conventional experience.

However there is a relatively simple explanation for this.

As we have seen, when we start with four quadrants, ultimately we can generate a much greater number of primary perspectives. So associated with the four quadrants we have 8 (primary) perspectives in the original Jungian model, 16 in the Myers-Briggs typology and 24 in my further adaptation of this model.

Now if we look on the four quadrants as representing dimensions of space and time, then we can perhaps appreciate that out of these standard four dimensions, we can create a greater number of perspectives (representing unique configurations of space-time).

When we reflect on the nature of space and time that would be appropriate for string reality, it makes no sense to view it as made up of separate dimensions (for by its very nature, differentiated dimensions do not yet exist at this level of reality). This is why indeed string theorists speak of dimensions as being somehow contained in the strings! So the only way to meaningfully speak of dimensions in this primitive state of matter would be in terms of varying configurations of the yet (undifferentiated) four dimensions. In other words a "dimension" here refers to an - as yet - undifferentiated configuration of - what eventually - are viewed as the four (separate) dimensions at the level of observable phenomenal reality. However once again because these dimensions can be configured in various ways we thereby obtain a much greater number than the standard four dimensions.

In earlier versions of string theory, 26 dimensions were required (which was based on 24) with two additional added. This could be explained in holistic terms with reference to the fact that the 24 primitive "dimensions" that are associated with the extended Jungian model of personality types are all based on the original binary "dimensions" of 1 (form) and 0 (emptiness). In this way space-time reality emerges from initial binary conditions (as pure potential for existence).

In the early generalised version there were 10 dimensions which again was based on 8 (with two additional added). So we have here a direct comparison with the earlier Jungian model of personality types. [8]

So once again the "dimensions" as used in string theory can be fruitfully viewed as representing the fundamental primary perspectives of reality at this level (where perspectives have not yet been properly differentiated). In this way they directly complement the "dimensions" that apply in personality terms as representing the key primary perspectives on which experience is based. Therefore at the "highest" level of development, all of these perspectives are once again present in experience (this time in a fully differentiated and properly integral fashion). Thus we can approach perspectives in personal terms through the personality types (that exemplify the key primary perspectives). Equally we can approach perspectives in impersonal terms through the fundamental matter types i.e. strings (that likewise exemplify the key primary perspectives of physical reality). In this way we can draw unexpected integral links as between two previously unconnected areas.

Perspectives and the Enneagram

The most recent model of string theory (M-theory) is based on 11 dimensions (9 + 2). Not accidentally the most popular model of personality is based on 9 distinctive types (i.e. the Enneagram).

So it is to this that we now turn!

The Enneagram is based on another popular organising principle, where we start with two polarities and through dynamic interaction obtain a third. For example in Hegelian terms, we can represent this basic triadic arrangement as thesis, antithesis and synthesis. Indeed the same basic idea lies behind the important Christian Doctrine of the Trinity with Father, Son and Holy Spirit viewed as three expressions of the same Spirit. Then we extend the same triadic notion in vertical as well as horizontal terms we generate 9 (i.e. 3 raised to the power of 2).

The number 9 has indeed strong properties as a symbol of integration possessing some remarkable numerical features (that are perhaps not well known). Relative to the base 10 system (which we conventionally use), 9 is (10) - 1.
In psychological terms, whereas differentiation comes from the positing (+) of phenomena as separate, integration comes from their corresponding negation (-) as complementary.
In base 2 (where we use the binary digits 1 and 0), 9 is represented as 1001.
This number is a palindrome (reads the same in a forward and backward direction).
Alternatively if one combines through subtraction the reverse direction of a number with the forward then the result = 0. So a palindrome in this sense is an important numerical symbol of integration.

Further in psychological terms integration is best achieved through the successful balancing of differentiated extremes. If we write the digits of 1001 in descending order (from the highest to the lowest) we get 1100. If we then order the digits in reverse (from the lowest to highest) we get 0011. Then by combining both results (through subtraction) we once again get 1001.
So the very nature of 1001 with its equal emphasis on form and emptiness in its structure results from the combination of its opposite extreme expressions.

In base 10 if we take any number and subtract its reverse the result will always be divisible by 9. [9] Furthermore this result will display marked palindromic tendencies. For example the number based on the date on which I am writing is 100604. The reverse number (reading the digits backwards) is 406001. When we subtract these two numbers (smaller from larger) we get 305397. Then when we divide by 9 we obtain 33933. So in this case we obtain a full palindrome! [10] Therefore despite its different origins we can best view the Enneagram system as an alternative means of highlighting the main primary perspectives (with each personality type exemplifying the dominance of one perspective). However - by its nature - the Enneagram is perhaps better suited for dealing with the process of integration of these perspectives.

And indeed there is more emphasis in the Enneagram system on the integration of perspectives (based on fascinating numerical sequences that have complementary psychological significance).

So in the Enneagram, each personality type is listed as a specific number. For example, 5 - which is well-represented on sites such as this - represents the Investigator. However each type is also defined by a wing, whereby the features of one of the number types on either side will also be incorporated in personality. Therefore the 5 type (Investigator) as a wing will possess the qualities of either the 4 (Individualist) or 6 (Loyalist). So one is never just a 5, but rather a 5 with a 4 wing or alternatively a 5 with a 6 wing. In this way through inclusion of the wings, 18 personality types are thereby defined. [11] Now it is remarkable enough that such a system of 9 numbers can consistently define personality characteristics in this manner with the wing of one's personality always on either side of one's defining number.

However there are even more remarkable number features defined in the Enneagram with direct implications for the integration of personality - which as we shall see - have far reaching implications for the perspectives of string theory.

So in the Enneagram the numbers 1 - 9 are divided into two sub-systems 396 and 142857, which define the appropriate pattern of subsequent integration for each type.

For example let us continue with the personality primarily defined by the characteristics of the 5. The appropriate pattern of integration for this type is given by the reverse sequence of digits in this (cyclic) number. In other words for integration a 5 should go to 8, then to 2, then to 4, then to 1 and finally to 7. (Interestingly the pattern of disintegration is defined in exactly the reverse manner i.e. where a 5 goes to 7 then 1, then 4, then 2, and then 8!)

142857 is the best known example of a fascinating type of number in mathematics (related to primes). The first 5 prime numbers are 2, 3, 5, 7 and 11. Now 7 is especially interesting here in that it is the first prime number (in base 10) whose reciprocal has a maximum cyclic sequence. For example if you take up a calculator, type in 7 and then press the 1/x button you will see the 6 digits 142857 continually recur. So the unique maximum cyclic sequence here is made up of 6 digits (1 less than the prime number 7 to which it is related). If you arranged these 6 digits around a circle you can take up the cyclic sequence starting with any digit. So if you start with 1 the sequence is 142857, with 2, 285714, with 4, 428571, with 5, 571428 with 7, 714285 and finally with 8, 857142. Remarkably when we multiply 142857 by 1, 2, 3, 4, 5 and 6 respectively we generate the same cyclic sequence of digits in each case!

When we multiply 142857 by 7 we obtain 999999 (remember the Enneagram is based on the number 9!) When we split 142857 into two halves and add we get 999 (i.e. 142 + 857). Just one more connection at this stage! If we subtract the two "halves" (i.e. 857 - 142) we obtain 715. Now multiply this by the sum of "halves" (999) and we obtain 714285 (which is the original cyclic sequence of the 6-digit number).

So we see here the most remarkable connection as between the circular pattern of integration proposed for each of the 6 personality types (in this subsystem) and the properties of the best-known example of the most circular of all numbers (i.e. the unique digit sequence of the reciprocal of the prime number "7"). [12]

However for the 3 types in the other number subsystem (396) integration also takes place in a reverse circular fashion. Thus a 9 for example will integrate by going to 3 and then 6. Disintegration comes from moving to associated number types in a forward circular fashion (i.e. when a 9 goes to 6 and then to 3)!

However though the Enneagram highlights well the circular nature of mature integration (and confused disintegration) in experience, it suffers from a number of limitations.

Firstly it is not really possible to talk about the process of integration while defining personality types in (neutral) horizontal terms. Thus the process of disidentifying with dominant primary perspectives can require the higher stages of development characterised by profound periods of dynamic negation (referred to in the mystical literature as "dark nights" or purgation). Thus, whereas a certain accommodation with the perspectives associated with other personality types can be acquired without substantial spiritual development, appropriate integration is not really possible without such (vertical) transformation.

In this vertical context another fascinating Enneagram system associated with medieval Christianity had long been in existence (without perhaps its true nature being realised). This relates to the angelic system, which largely due to the work of the Pseudo-Dionysius had been arranged in true Enneagram format into three hierarchies each composed of three triads. This then was later embraced by Thomas Aquinas and formed a key component of his overall theology. Angels, as creatures serving as intermediaries between humans and supreme Spirit (i.e. God) might seem somewhat fanciful to the modern mind. However if we read the use of angel terminology in a more metaphorical fashion to represent the higher spiritual possibilities of the human personality, then these mediaeval orders become invested with new meaning describing the various stages of mystical development.

Thus the first hierarchy of angels (Angels, Archangels and Principalities) were considered to have closest links with human beings. Apparitions communicating key spiritual messages (frequently in dreams) typically came in the form of angels or archangels! Now in more realistic fashion we can interpret this as meaning that the earliest stages of "higher" mystical development (i.e. subtle) relate to mature dream reality where archetypal spiritual meaning is unconsciously expressed. The second hierarchy (Powers, Virtues and Dominations) were traditionally associated with the universal causes of things and can thereby be readily identified with the causal level of development. Finally the third hierarchy (Thrones, Cherubim and Seraphim) were associated more directly with Union with God bearing close comparison with what I refer to as radial reality. So the Thrones were associated with Justice, the Cherubim with Wisdom and the Seraphim with supreme Love (representing the culmination of development).

So put another way in Angelology - when suitably interpreted - we have an alternative Enneagram system that is directly related to the integration of primary perspectives.

Coming back however to the conventional Enneagram, another weakness is that in its (attempted) portrayal of integration, it ignores the wing of each personality. However, realistically - especially where the wing approaches the same strength as the dominant mode of personality - integration would take place with respect to both aspects. However this could raise a new set of issues not dealt with in the standard treatment. For example take a 5 with a 6 wing. Whereas the integration process with respect to the first aspect would take place within the larger number subsystem (142857) the integration of the second would be within the smaller subsystem (396). However the relationship as between these two subsystems is far from clear. Also in other cases - say the 5 with a 4 wing - integration for both aspects would be entirely confined to the one number subsystem (142857) which then begs the question as to how integration with the remaining number types 3, 9 and 6 takes place!

Finally its relationship with the earlier Jungian system (and its variants) is not obvious.

Some years ago when I studied this closely, I came to the conclusion that a more compelling system could be built around a Septagram (in base 8) where 4 of the number types would equate well with the four quadrants (in horizontal terms) and the other 3 with (vertical) development of these quadrants through higher stages. In this way we could show how development takes place with respect to (a) the differentiation of primary perspectives and (b) the corresponding higher level integration of these perspectives, relating to distinct personality types.

The Enneagram and Strings

However just as we were able to use the Jungian system of personality types to provide an intuitively satisfying explanation for the nature of dimensions in string theory, equally we can use the main number subsystem in the Enneagram to throw fascinating light on other key aspects of string reality.

Remember that this number subsystem (142857) is intimately related to the prime number "7" as the unique sequence of digits of its reciprocal (i.e. 1/7). Now prime numbers are the most independent of numbers (which cannot be broken into further integer factor combinations). So whereas we can represent 6 in geometrical terms as two-dimensional (i.e. 2 X 3) a prime number such as 7 is strictly one-dimensional (i.e. linear).

However when we take the reciprocal of a prime number (which literally entails the negation of this linear dimension i.e. 1/7 = 7 raised to the power of -1), then we get the most circular of all numbers (excluding in base 10, 2 and 5). So, as we have seen, the unique digit sequence of 1/7 has amazing cyclic (i.e. circular) properties

Now this number behaviour of primes and their reciprocals provides a fascinating clue as to the holistic meaning of prime numbers (which then can be used to scientifically describe the nature of string reality).

So in holistic mathematical terms, prime phenomena relate to a situation of (confused) circular integration whereby the differentiation of separate independent phenomena (in linear terms) is not yet possible. The very identification of material phenomena - even at the minute sub-atomic levels of reality - implies that they have already obtained sufficient independence (as differentiated) so that they can be viewed against a background of space and time.

However by its very nature this is not possible with string reality, where distinct phenomena cannot be yet distinguished from background space and time dimensions. In other words, both these aspects are still entangled with each other (though their indirect effects - as with black holes - can perhaps be observed).

So in precise holistic mathematical terms, strings are defined by prime number notions, where - by definition - (linear) object phenomena cannot be yet distinguished from their (circular) dimensional background.

We can in fact appreciate this by offering a more intuitively satisfying explanation of some ideas used by string theorists. For example, strings are defined as one-dimensional objects (pointing to their prime origins). As observable phenomena this makes little sense. However if we now define such strings as having the potential for independent linear (i.e. one-dimensional) existence, then this manner of expression conveys much greater meaning. Likewise theorists speak of the tension of strings and how the dimensions in a sense are somehow contained in them. Then their vibration leads to the existence of unique matter particles. So in dynamic terms the actual vibration of strings is the means by which differentiation (as independent phenomena) takes place against a background of (separated) space-time dimensions. So the observable phenomena of nature - like the majority of natural numbers in mathematics - represent composite arrangements of prime constituents arranged in a coherent manner. Also the very notion of a string conveys that which has the capacity for linear extension through being drawn out and circular extension (through forming loops). However once again these ideas do not have literal phenomenal meaning at the level of string reality. Rather they express the potential for differentiated (linear) independence as separate phenomena and integral (circular) dimensional interdependence with other phenomena.

Indeed the very word "prime" in holistic mathematical terms has definite connections with the words "primary", "primordial", primitive etc. [13] Therefore, the first emergence of the primary (or primordial) perspectives can be identified with what we know as string reality. However here distinct perspectives cannot be yet clearly identified as they are still entangled with each other in a confused dimensional setting. There is a close correlation with the emergence of perspectives in human psychological terms in the first infant manifestations of primitive instinctive behaviour. Remarkably there are close structural connections here with the nature of string reality (both of which are defined - in holistic mathematical terms - by prime dimensions). The very essence of purely instinctive behaviour is that conscious phenomena cannot be distinguished from the involuntary (unconscious) context in which they arise. In other words with primitive instinctive behaviour, "object" phenomena are directly entangled with their corresponding dimensions of space and time in a purely immediate experience (so that neither aspect can be separately distinguished).

In this context therefore we could fruitfully look on string reality as the purely instinctive nature of matter.


We have seen briefly how there are remarkable connections to be drawn as between the nature of perspectives, personality types and the fundamental strings of physical reality (where primary perspectives first emerge). This contribution has been used to demonstrate therefore the benefits of an integral scientific approach (enabling us to make deep connections as between areas that previously seemed entirely separate).


1 For example this merely objective stance is evident in the manner in which the fundamental notion of scarcity in Economics is treated. Conventional attempts to alleviate scarcity focus on the provision of more material goods and services (exterior). However dealing with scarcity is equally a matter of mind whereby people can train themselves to want less. So traditionally for example, monks living in monastic communities - often with very limited economic resources - proved that they could find true happiness in this manner. When there is continual emphasis on possessing more - as in our materialistic affluent societies - spiritual contentment can prove ever more elusive. This is largely due to the fact that we are looking at our essential relationship with goods (and services) in an unbalanced manner (where the inner aspect of the interactive relationship is neglected).

2. Though economic decisions always involve moral choices and values (sometimes of a crucially important nature) the free market - as conventionally understood - provides a very distorted model of economic relationships. Because the misleading attempt is made to view markets in merely objective terms, this in turn leads to the abstraction of market forces from any relationship with underlying moral behaviour. For example decisions with respect to large transactions in financial markets have important moral consequences. However this is greatly neglected in our interpretation of such decisions, where indeed we transfer subjective behaviour directly on to the markets themselves (thereby personifying them in human terms). So we speak of markets being "nervous", "having a good day", "rallying" etc. as if they directly embodied human characteristics (in abstraction from the people actually taking decisions that have important moral consequences). Another way in which the moral consequences of important decisions are avoided is by treating market activity increasingly as a "game" in which key participants are referred to as "players".

3. "All numbers in fact have a dimensional characteristic and with the linear interpretation of number it is always 1 (which literally defines its linear nature). Thus 4 in linear terms is strictly 4 (raised to the power of 1).

However with circular numbers the process is reversed, with the dimensional characteristic changing while the base quantity remains fixed as 1. So 4 in circular terms is strictly 1 (raised to the power of 4).

However to give a (reduced) linear interpretation of this circular number (i.e. by expressing with respect to the power of 1 as the 1st dimension), we must extract the four roots.

Thus if x (raised to the power of 4) = 1 (raised to the power of 4), then x (i.e. x raised to the power of 1) = + 1, - 1, + i and - i .

We can geometrically represent these four points in the complex plane (i.e. where the horizontal x-axis represents real and the vertical y-axis represents imaginary numbers respectively). The four points then lie in an equidistant manner on a circle (of unit radius) divided by both horizontal and vertical line diameters thereby creating the four quadrants of the circle.

4. I will briefly explain here the precise holistic mathematical rationale of an imaginary number (more specifically i = the square root of -1).

The conscious recognition of phenomena - whereby opposite poles separate - is associated directly with (dynamic) positing (+). The corresponding unconscious recognition - whereby opposite poles are understood as complementary - is associated by contrast directly with (dynamic) negating (-).

However unconscious understanding cannot be directly viewed in phenomenal terms (which is of a linear one-dimensional nature where opposite poles are separated). Therefore to express what is inherently two-dimensional (as unconscious), indirectly in linear conscious terms (as one-dimensional), we must take the square root of the negative direction of form (which dynamically generates such unconscious understanding). Therefore in holistic mathematical terms, imaginary understanding represents the (linear) phenomenal reduced interpretation of what is inherently (circular) two-dimensional understanding. When the unconscious is not properly developed in experience, the "imaginary" aspect expresses itself in a necessarily involuntary - and thereby confused - manner through projections. However with appropriate spiritual development - where attachment to the dominant conscious aspect of experience is eroded - the "imaginary" aspect can express itself in a very refined manner as the direct expression of Spirit. It thereby serves as the vital process though which dynamic switching between the opposite poles (and perspectives) of experience can take place.

The "imaginary" notion equally has a physical as well as psychological interpretation (as these aspects are complementary).

For example virtual particles in physics are a fine example of the "imaginary" expression of matter. Therefore though virtual particles have a transient phenomenal existence, in direct terms they point to the deeper fundamental ground of matter (which complements the unconscious) from which they emerge.

5. Jung's eight personality types are: introvert intuition, extravert intuition, introvert thinking, extravert thinking, introvert feeling, extravert feeling, introvert sensation and extravert sensation.

With respect to his functions Jung classified two (thinking and feeling) as rational and two (intuition and sensation) as irrational. In fact Jung comes very close here to the complex mathematical formulation whereby two are - relatively - "real" and two "imaginary". Indeed Jung employed another notion "the transcendental function" - in terms of the spiritual harmonisation of conscious and unconscious - that is frequently likened directly to the complex interpretation of numbers as real and imaginary. However it is important to stress that Jung defines feeling in an unusual manner (which does not correlate with conventional usage). So for Jung feeling is not so much an affective aspect of personality but more a cognitive evaluation function (resembling the interior use of reason). Thus - though strongly identifying with the basic thrust of his four-quadrant approach (with its inherent dynamic emphasis) - I have always had reservations regarding the precise terminology he uses to define his functions.

6. Let me be clear here! I certainly do believe that Ken Wilber possesses strong analytic and intuitive capabilities (both S and N personality characteristics in Jungian terms). However what I have always consistently maintained is that there is a marked discontinuity in the manner in which Ken employs both of these capabilities.

Putting it simply, Ken's intuitive energy is largely identified with spiritual states and affective type poetic communication of such states. By contrast his rational energy is largely identified with the structural analysis of development (in a somewhat linear mechanical fashion). Therefore at the level of states in Jungian terms, Wilber acts as a strong N. However at the level of development structures he acts as a strong - and indeed sometimes extreme - S. So once again there is a marked discontinuity evident - certainly from my perspective - in the manner he uses both aspects.

The very essence of a dynamic interactive approach is that both spiritual states and rational structures become increasingly interdependent at higher levels (with the relationship between both thereby continually transformed leading to new increasingly refined types of paradoxical understanding). I say it again, this vital aspect of understanding is greatly missing from Ken Wilber's work and has thereby distorted his interpretation of the inherently dynamic nature of Jungian notions e.g. in the manner he criticises Jung of the pre/trans fallacy.

7. If we identify the 8 personality types in the S Group of the Myers-Briggs typology with Wilber's delineation of primary perspectives the matching would be as follows:

ESFP - UL (Outside)
ISFP - UL (Inside)
ESFJ - LL (Outside) ISFJ - LL (Inside)
ESTP - UR (Outside) ISTP - UR (Inside)
ESTJ - LR (Outside) ISTJ - LR (Inside)

Perception (P) tends to be based on an (individual) case by case by case basis whereas Judgement (J) tends be more generally based on (collective) rules.

In the first part I developed an eight-sector model (leading to the outlining of 64 primary perspectives). The question then arises as to whether the Myers-Briggs (which caters well for differentiated primary perspectives in the four-quadrant model) could be enlarged to accommodate these 64 perspectives. The answer is yes! Each of the 16 types in the Myers-Briggs can be further subdivided (with respect to form and emptiness) with profiles reflecting spiritual (religious) and worldly (secular) experience respectively. So if we take one of the 16 in the Myers-Briggs to illustrate (e.g. INFJ) this can be given both spiritual (transcendent) and worldly (immanent) interpretations. Likewise we could subdivide on the basis of the masculine and feminine principles. Though of course we cannot directly identify the masculine principle with men and the feminine with women, it would however be valid in general terms to distinguish men's experience from women's. For example we could explore the extent to which the "typical" spiritual experience of the male INFJ differs from the female INFJ! Also with respect to this final classification we could attempt to distinguish the experience of straight and gay members (with respect to both sexes).

8. The connections here are far from accidental. For example the work of the Indian mathematician Ramanujan with respect to modular forms were very important in terms of the original formulation of the 26 - and in generalised from - 10 dimensions of string theory. Modular forms are defined with respect to the complex number system (with real and imaginary components). Likewise the revision of the Myers-Briggs system was based on the complex number system (on which my four-quadrant system was based). So I was led to the view that we could produce 24 unique arrangements of space-time from the four co-ordinates of the complex number system.

In fairness to Jung he did not solely conceive of his four functions as applying only to psychological reality but suspected that there were deep connections here with the dimensions of space and time. He was right and I am sure if he were alive today would greatly appreciate how his (extended) model can be successfully applied to explain the dimensions of string theory.

9. There are equally remarkable features of the number system which hold when we divide by n + 1 (i.e. add 1 to the number base).

If the number of digits is odd, when we get the difference of it and its reverse, it will always be divisible by 11. (This implies that the difference of all odd digit numbers and their reverses are divisible by 99!) If a number however has an even number of digits when we now add it to its reverse it will be divisible by 11.

So + 1 is associated with the conscious polarised aspect of experience. Likewise it is fascinating how number behaviour is polarised with respect to division by 11 (n + 1). Once again the difference of all odd digit numbers and their reverses is divisible by 11 (in all number bases). However for even digit numbers the sum of all numbers and their reverses is divisible by 11.

10. Number behaviour here is universal with respect to all bases. Therefore in base 8 system for example, when we get the difference of a number and its reverse it will always be divisible by 7 (i.e. n - 1 where n represents the number base). Likewise the result will again display marked palindromic tendencies. One interesting consequence of this is that in the base 2 system (where we employ just 1 and 0), palindromic tendencies will result directly from obtaining the difference of any number and its reverse.

11. Fairly close connections can be made as between the 8 personality types in the Jungian system and 8 of the corresponding number personalities in the Enneagram. The exception relates to no. 3 in the Enneagram (the Motivator).

Now by its very nature this highly adaptable personality bears comparison with the mystical personality types (with one big difference).

Though the motivator (or status seeker) appears to have the capacity for flexible ready integration with the environment, it generally operates more at the level of appearance than authentic reality. Therefore one fascinating possibility with the mystical types is that where dominant functions have not been substantially differentiated, the facility for apparent integration (without substantial spiritual development) might well be possible. However when this is the case - despite appearances - personality development will remain inauthentic. For example this is often evident in great actors who apparently can assume with appropriate gravitas any persona they wish but in real life do not really know who they are. Thus this type is defined largely by appearance through ready identification with pleasing roles, which in a superficial society brings more acclaim that authentic performance.

12. The next prime number whose reciprocal has a full cyclic sequence of digits (i.e. n - 1, where n is the prime number) in base 10 is 17. The unique digits (which continually repeat) are 0588235294117647. All the fascinating features mentioned with respect to 142857 likewise apply here.

13. Number provides the best means we have for order. Though this is accepted without question in quantitative terms, there is little recognition yet that the holistic notion of number - especially with regard to the major number types - is the best means we have for scientifically ordering development.

So we start with the binary numbers - in holistic terms - as the precondition for structuring development. Then with the emergence of the primary perspectives (though not yet identifiable in phenomenal terms) the holistic notion of prime numbers assumes great importance. Later with the emergence of natural phenomena (e.g. quantum reality) the natural numbers assume a significant role (in holistic terms). Then at the ordinary level of macro reality the rational numbers become increasingly important. Indeed science as we know is governed by the rational paradigm.

However when we go into the higher spiritual levels the holistic meaning of other numbers e.g. negative, irrational, imaginary, complex and transfinite assume corresponding significance. Indeed the most scientific way of ordering the various stages of development is in terms of the holistic equivalent of the various number types.


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