Friday, October 30, 2015

Psychological Development and the Euler Identity (2)

The Euler Identity is generally recognised as the most beautiful theorem in Mathematics.

 For example the well known "populariser" of Mathematics, Keith Devlin has stated:

 ""like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence".

So when it comes to the Euler Identity, mathematicians somehow realise that they are dealing with a relationship that greatly transcends their rational ability to comprehend its symbols.

Euler's Identity (or Equation) is generally given in the form,

e^iπ + 1 = 0.

e here refers to Euler's number which is the base of the natural logarithms system.

i is the imaginary unit that represents the square root of – 1.

π (or pi) represents the ratio of the circumference of a circle to its line diameter.


So along with 1 and 0, which are also included in the simple Euler equation, these represents the five most important constants that are used in Mathematics.


Now, it will suit our purposes here to rearrange the Euler Identity slightly.

So e^iπ = – 1.

Therefore when we square both sides we obtain,

 e^2iπ = 1.

I refer to this as the fundamental form of the Euler Identity.


Now, one might initially wonder what this strange equation might have to do with the development process, especially with the most obscure stages approximating spiritual union!

However they key to making the appropriate connection is the realisation that when appropriately understood, in a dynamic interactive manner, all mathematical symbols have holistic - as well as analytic - interpretations.

And when begins to appreciate the holistic significance of the mathematical symbols used in the Euler Identity, though still remaining deeply mysterious, it can be seen to intimately relate to the paradoxical nature of spiritual union.

In fact when one accepts - as I now readily do - the deep integral value of the holistic interpretation of mathematical symbols, the Euler Identity can indeed be looked on as the most appropriate scientific explanation possible of the ultimate nature of spiritual transformation.


Now, e and π represent the two best known transcendental numbers.


The transcendental numbers are in fact extremely elusive, for even though the vast majority of real numbers are of a transcendental nature,very few are known.


In "
The Number Paradigms, I had concluded that the most advanced psychological structures (before pure spiritual union) are of a transcendental imaginary nature. And the dimensional number (power, exponent) of e in the fundamental Euler Identity i.e. 2iπ, is of an imaginary transcendental nature!

In analytic mathematical terms, when one raises a rational number to another rational number (as fraction) one gets an irrational (algebraic) number.

In corresponding holistic mathematical terms, I was able to us to show how development would proceed from rational to the more intuitively inspired levels of "higher" consciousness.


Then in analytic mathematical terms, when one raises a rational or (algebraic irrational number) to another irrational number, one now obtains a transcendental number.

Likewise one is able experience the two-way relationship irrational perceptions (associated with the refined intuitively inspired understanding of the subtle level) with corresponding irrational concepts, this leads to a corresponding transformation to the next "higher" level (with a holistic transcendental structure).

 The question then arises as to what sort of transformation is possible, when one then, having further deepened in contemplative awareness, can experience in two-way fashion, the relationship of transcendental perceptions with corresponding transcendental concepts.

And remarkably, just as in analytic mathematical terms there is one special case (with respect to such relationships) that results in 1, likewise in holistic terms there is the corresponding special case, which in terms of psychological development, results in "oneness" (i.e. spiritual unity).


Once again e is a transcendental number, with a very special property in analytic mathematical terms.

 So when e is raised to a power x i.e. ex, when one differentiates the expression y = ex, and then in reverse manner integrates the expression, the result remains the same


.

 Now, the fundamental Euler expression is of the form ex, where x = 2iπ.


Therefore, this remains unchanged where differentiation and integration takes place. And of course when we differentiate the corresponding result on the RHS we get 0.


Equally with respect to development, in holistic psychological terms, we have the two fundamental processes of differentiation and integration respectively.


And in psychological terms, at the transcendental level of contemplative intuitive awareness, one can approach the special situation, where experience now becomes so spiritually rarefied and transparent, that one can no longer distinguish (differentiated) phenomena of form from the continued (integrated) experience of emptiness.


And this concurs when - quite literally - experience of dimensions turns full circle.


As I have repeatedly emphasised, standard analytic interpretation is of a linear (1-dimensional) nature where the dimensional aspect, which - relatively - is of a qualitative nature - is reduced in a merely quantitative manner.


This is especially true with respect to Mathematics, where for example the concept of "number" is reduced in a merely quantitative manner as applying directly to all number perceptions!


So the whole point of contemplative type awareness is to gradually move from this restricted linear interpretation of concepts to a fully circular holistic appreciation (which is completely paradoxical in dualistic terms).

With respect to the unit circle, its circumference is given as 2π. If we were to consider now the "imaginary" equivalent - which corresponds to the pure holistic appreciation of the circle - the circumference would be 2iπ.


So the dimensional value (of the fundamental Euler Identity) relates to the purely holistic appreciation of number (which is completely circular - and paradoxical - in dualistic terms).


In other words it represents the holistic intuitive extreme of awareness (as purely circular) in contrast to the corresponding analytic rational extreme (as purely linear)..


Therefore in holistic terms, the fundamental Euler Identity defines in a truly remarkable and deeply mysterious manner the phenomenon of spiritual union, which however can only be approached - rather than fully attained - in an experiential dynamic interactive manner.

And this union (as the holistic expression of "1" is now indistinguishable from the corresponding experience of nothingness (or emptiness) as the holistic expression of "0".


So once again, one may appreciate how we have come full circle.

In conventional mathematical terms (at the analytic level of interpretation, 1 is clearly separated from 0 (in a quantitative manner).

However now in corresponding holistic terms, 1 is fully interdependent with 0 (in a qualitative manner).

 So the (fundamental) Euler Identity - when properly appreciated in a dynamic interactive manner - represents the mysterious marriage, where both the analytic and holistic interpretation of its symbols ultimately dissolves in an experience that is truly ineffable.

Tuesday, October 27, 2015

Psychological Development and the Euler Identity (1)

Gradually, I realised that Band 4 (as I identify it) would not directly lead to the Spiritual Marriage that I had anticipated. Rather it gave way to a new series of stages that I was to later identify with Band 5.

Of course experience can never be exclusively identified with just one Band (however prominent it may be in experience)!

So rather than expecting the completion of Band 4 before proper development with respect to the radial stages could unfold, I began to accept that it would remain continuous with the "descent" levels of Band 5 (that acted in a complementary fashion to the corresponding "ascent" levels of Band 3).

Whereas the 3 levels of the "ascent" (with respect to Band 3) were associated with the movement away from the firm ground of the linear levels in a transcendent spiritual direction towards emptiness, now the corresponding 3 levels of the "descent" (with respect to Band 5) would be associated with the corresponding movement back in an immanent direction towards the world of form (now seen as freely interpenetrating with spirit).

Thus the contemplative goal of achieving equal balance with respect to both transcendent and immanent directions of spirit could not be thereby completed until contemplation (as formless) itself attained appropriate balance with respect to active engagement (with respect to the world of form).


During this time which effectively represented the first level of Band 5, a certain psychological relaxation took place within, whereby the internal and external polarities now became separated to a degree.

This thereby enabled me to pursue a more refined objective expression of the 3 levels of Band 4. It also encouraged me - for the first time -  towards active engagement with others, with respect to my emerging world view.

It was at this time that I attempted to address a key unresolved issue with respect to my holistic mathematical interpretation of the stages of development.

As I mentioned in previous blog entries I had earlier written an online book "The Number Paradigms" the basic thesis of which was that all the stages of development were of a holistic mathematical nature, which thereby provided the most appropriate approach to development (from an integral scientific perspective).


So I was able to match all the main stages - and indeed sub-stages - of the first 3 Bands to the major number types (now given their holistic mathematical interpretation).

Thus the "lower" levels of Band 1 were associated with the binary, prime, natural numbers and integers respectively.  This then culminated in the specialised linear understanding of Band 2 associated with the rational numbers.

Then the "higher" levels of Band 3 were associated with the irrational (algebraic), transcendental and transfinite numbers (which included positive/negative and real/imaginary directions).

Therefore, I had identified the "highest" stage before pure formless experience (transfinite) as of a transcendental imaginary nature (interpreted in a holistic mathematical fashion).

Transcendental in this context entailed the highly subtle appreciation of the relationship between meaning that is both linear and circular respectively. Put another way it represents neither conscious nor unconscious in isolation, but rather the mid-point or intersection of both!

Imaginary would then represent the indirect conscious expression (as a refined projection) of what directly represents unconscious holistic meaning. So when can immediately intuit such a projection as entailing the both conscious and unconscious domains, any involuntary attachment thereby ceases.

In other words one is then free to experience the pure formless nature of spirit!


However at this point, of spiritual realisation, development itself turns full circle with the realisation of the unity of all form.

So one started as an infant in earliest development with the analytic appreciation of the numbers 1 and 0 (whereby something could be literally distinguished from nothing).

However now development - as I said - turns full circle, with the holistic interpretation of 0 (as spiritual emptiness), which now in turn is identified as interdependent with 1 (as the unity of all form).


The problem however  that I experienced was to show mathematically how this could arise.

Earlier I had achieved this through showing how perceptions and concepts are relatively quantitative as to qualitative with respect to each other.

So I identified 3 main sub-levels with respect to each level.

The first related to sense perceptions which were most closely related quantitative notions of form.

The second then related to mental concepts, which in turn were more closely related to qualitative notions (of a more spiritual nature). For example the perception of "a cat" is readily identifiable in an actual quantitative manner. However the concept of "cat" strictly applies in an infinite manner to all possible perceptions of "cats".

The third sub-level then related to the two-way interaction of perceptions and concepts.

This in turn entails the interaction of quantitative and qualitative notions which enables the transformation to the next level.

Remarkably in holistic number terms, the perceptions can be looked on as base numbers and the concepts - relatively - as dimensional numbers respectively.

Now in conventional analytic mathematical terms, if for example, we raise a prime number to another prime number, we get a transformation to a natural number.


So, for example 23 = 8. Though 2 as base and 3 as dimensional numbers respectively, are both prime, 8 is a new (composite) natural number.

Likewise, in psychological development when we raise prime perceptions (corresponding to the magic stage) to prime concepts, through the two-way interaction of what are now understood in holistic terms, we achieve a transformation to the new level of natural understanding (which would correspond with the mythic level). 

In particular I felt that I was able to offer a very fundamental explanation as to why development plateaus so heavily at Band 2 development in our present culture!

Though the rational perceptions and concepts at this level - are when properly understood - quantitative as to qualitative with respect to each other, a highly reduced interpretation conventionally operates (especially in Mathematics and Science).

Therefore because of this reduced - merely quantitative - emphasis, generally speaking, insufficient interaction as between the two aspects takes place so as to enable substantial transformation to "higher" levels.

Indeed successful transformation at the earlier levels only occurs because conscious understanding has not yet attained full specialisation. So conscious and unconscious continue to interact (as quantitative as to qualitative) though necessarily in a somewhat confused manner.

However as I returned with a view of continuing my holistic mathematical treatment to the further Bands, I began to see that there was an intimate connection here in a holistic manner with the famous Euler Identity.

I will return to this in future blog entries.

Friday, October 16, 2015

Band 4

Just as Band 2 (in my account) relates to the specialisation of (linear) rational type understanding, Band 4 is designed to relate to the corresponding specialisation of (circular) intuitive type appreciation

After Band 3 had culminated in a transitional crisis (in mid 1996), I then looked forward to - what I imagined - would represent a relatively short period of adjustment in the unfolding of Band 4.  I interpreted this band then as acquiring appropriate balance with respect to contemplative experience in terms of both its transcendent and immanent directions.

Then when appropriate adjustment had taken place, I expected that the radial bands would emerge, entailing the gradual interpenetration of both analytic and holistic type understanding (intellectually, emotionally and spiritually).

And just as the earlier bands had been divided into 3 major levels, I could see that this would also be true of Band 4.


So the first level of Band 4 relates to the specialised development of the transcendent type contemplative awareness. This had dominated the ascent to spiritual Mount Everest, and now been significantly attained.

Therefore  my thinking was that as this task was largely consolidated - though necessarily in a relative approximate manner - I could now start to share the many insights that had been so painfully acquired in marked isolation from the world.

However, even in my most extreme moments, I had never seen contemplative solitude as an end in itself. So for me the holy grail was always the attempt to eventually marry both contemplation and activity in a creative productive manner.

Successful re-entry to the world would require therefore that equal emphasis be given to the immanent aspect, through the physical instinctive side of personality acquiring equal balance with its spiritually refined aspect.


Thus the second level would require further withdrawal with respect to the more dominant transcendent aspect, gradually lessening residual super ego control with respect to the physical self.

Finally with this task largely achieved both transcendent and immanent aspects could be integrated in a balanced fashion. This would then represent the long desired spiritual marriage, where both masculine and feminine principles would be united with respect to my personal self, equally now seen in universal terms as my true cosmic self.


So this entry into this new marriage (of both the spiritual and physical aspects of being) would represent the third level of Band 4.

And then after that, I considered I would be ready to creatively and productively engage with the world in a manner that could properly express my true vocation and gifts.


I was gradually to realise however that I had been painting too rosy a picture of my future.

Looking back, the mistake that I made was in believing that the task of balancing the "higher" transcendent and "lower" immanent aspects had been largely completed on attaining the spiritual "summit". However as time went on, it became apparent that this was not in fact the case.

Indeed, if I had paid more attention to my mountain analogy, it would have been apparent that the full descent of its slopes would have to be successfully negotiated before I could hope to place my feet once again safely on the plains below.

Also in many ways, I was in a somewhat weakened state in psycho-physiological terms due to the sustained stress already endured.


So as time elapsed, I was slowly led to change my thinking.

It became apparent to me that just as undue specialisation earlier on the journey, with respect to Band 2 would have greatly inhibited development with respect to the "higher" contemplative levels, equally undue specialisation now with respect to Band 4 would likewise have inhibited a successful descent towards the radial bands.

Therefore I began to understand that a much longer period of preparation was necessary, which would require in reverse fashion journeying through the same three levels of Band 3.

However whereas before the ascent with respect to these bands was mainly in the contemplative direction away from worldly activity, now on the descent, the direction in reverse fashion was back from contemplation towards worldly activity. And this latter experience of descent, I was now to refer to as Band 5.

In fact strictly speaking both the ascent and descent are necessarily involved  with respect to both Bands 3 and 5. However it is very much like a tide where the waves continually come in to and recede from the shore.

However, whereas with Band 3 the overall tide is going out with respect to the shore of worldly engagement, with Band 5 the reverse is happening with the tide now flowing gradually back towards this shore.


Therefore though Band 4 (in the specialisation of contemplative activity) still remained relevant, its attainment was to a degree now continuous with progress through the 3 major levels of Band 5. In fact the gradual consolidation of this task is what enables the contemplative waves eventually to role more successfully back on to the shore of fruitful engagement with the world!