## Tuesday, December 8, 2015

### Deep Implications for Mathematics

The finding that every number - indeed every mathematical - symbol can be given a coherent holistic as well as analytic interpretation is of the very first magnitude.

In fact it reveals a fundamental unrecognised problem that lies at the very heart of Mathematics i.e. that its symbols cannot in fact be coherently understood in a static absolute fashion.

Rather the number system - and indeed all mathematical relationships - should be properly interpreted in an inherently dynamic manner reflecting the interaction of complementary aspects.

Now for some considerable time, I had been aware that we can define the natural number system in this complementary fashion (using Type 1 and Type 2 formulations that are the inverse of each other).

So again the (analytic) Type 1 aspect is defined as,

11, 21, 31, 41,………

Here the base ranges over the natural number sequence (with respect to the default dimensional value of 1).

What this means in fact is that number is viewed in a merely independent quantitative manner (as parts).

Thus 3 for example = 1 + 1 + 1 (with each unit a quantitative part of the total number)

The (holistic) Type 2 aspect by contrast is defined as,

11, 12, 13, 14,………

Here, in inverse fashion, the dimension ranges over the natural numbers (with respect to a default base value of 1).

The essence of this interpretation is that number is now viewed in an interdependent qualitative manner with respect to its relationship with other numbers.

To illustrate let us start with the default base number of 1. Now, we can keep relating this number with itself (without changing the quantitative value) by moving into "higher" dimensions.

So we start with the 1-dimensional line (with a quantitative value 1) Then we move to a 2-dimensional square (again with a quantitative value of 1). And continuing we can next move to a 3-dimensional cube (once more with an unchanged quantitative value of 1).

So - quite literally - while the base unit remains unchanged as 1, we keep moving into higher dimensions with respect to this unit. These dimensions are possible precisely because we keep using 1 in a related ordered fashion, where it can represent the length, breadth and height of the resulting 3 dimensional cube.

Now in conventional physical terms, we cannot go beyond  3 dimensions (as our very notion of space dimensions is greatly limited by conventional quantitative notions of measurement).

However there is a deeper more fundamental meaning of dimension here which can indeed be extended indefinitely.

Imagine I make the statement. "There are 3 cars in the (same) car park".

This can be treated in the conventional Type 1 manner, where each unit is then defined in an independent manner.

So 3 = 1 + 1 + 1.

Imagine I now make the following statement "There is one car in three (different) car parks.

In conventional mathematical terms, this would be treated in exactly the same manner

i.e. 3 = 1 + 1 + 1.

However, in truth a very subtle change has taken place involving the shifting with respect to complementary reference frames.

In other words to add the 3 items in this case (recognising each as a component of a total number of cars) we have to establish the commonality, through relationship, of each of the three cars that lie in different car parks.

Now in the first case, this commonality (or interdependence) is implicitly provided through the recognition that the 3 cars are in the "same" car park. In this way we are then able to view each car in a - relatively - independent manner.

However in the second case,  because the 3 independent units are in separate classes (i.e. car parks) we would not be able to recognise their common membership (of a group) without the holistic recognition of each unit as a whole.

So this is the crucial difference! In the 1st case we looked at each unit as a part. However in the 2nd case we must now look on each unit as an intrinsic whole (in its own right).

Now it is true that once we have established these units as wholes that we can again add each unit to obtain a total of 3.

However properly speaking ,we must now express this in terms of the Type 2 system.

So if we initially arrive at the total of 3 in Type 1 terms through the addition of parts (i.e. 31 = 1+ 111), then in relative terms, we must arrive at the total of 3 (with respect to the 2nd case) in Type 2 terms through the multiplication of  parts.

So 13 = 1111 * 11 .

However this latter expression can equally be shown as the addition of wholes (with respect to the dimensional number)

i.e. 13 = 11 + 1 + 1 .

Therefore what represents addition in Type 1 terms (with respect to parts) equally represents addition in Type 2 terms (with respect to wholes).

However what represents addition in Type 1 terms (with respect to parts) also, equally represents multiplication in Type 2 terms (with respect to parts).

So in truth number keeps switching - depending on context - as between a part and whole status. It is very much akin in this respect to the way in which sub-atomic matter can switch between particles and waves!

But in Conventional Mathematics, a merely quantitative interpretation of the nature of this relationship is possible (where wholes are reduced to parts).
It is not surprising that mathematicians realise that there is something fundamental missing from their appreciation of the relationship as between addition and multiplication.

However the implications of facing this are truly enormous, for as we have seen the basic paradigm on which Mathematics has been built for milennia clearly needs to be changed, whereby both analytic and holistic modes of interpretation (of equal importance) are fully recognised, that are clearly understood to dynamically interact in complementary fashion with each other.

## Monday, December 7, 2015

### Psychological Development and the Euler Identity (8)

In yesterday's blog entry, I dealt with the 2nd level (Band 4), which can indeed extend over many years.

Basically it once again relates to the continuous detachment from an over-identification with the "higher" spiritual aspect of personality, so as to eventually allow equal emphasis to be given to the corresponding "lower" physical aspect. For in truth both are fully complementary - and ultimately identical - with each other. So in an equivalent manner, one is here recognising the need to fully harmonise transcendence (as spirit beyond all form) with immanence (as spirit prior to and thereby inherent in all such form).

Now, one might indeed query to what extent distinct levels of development are even appropriate at this stage of development.

It is important to bear in mind that I always use the notion of discrete stages in a relative dynamic manner, where they remain necessarily continuous with all other stages of development.
At this point, the continuous aspect is - relatively - of much greater importance, whereby one ceaselessly integrates in a two-way interactive manner both the "higher" and "lower" levels (especially with respect to Band 3 and Band 1 respectively).

However a certain discrete element is also necessarily in evidence (relating as we have seen to the - still - dominating influence of the "higher" stages). So to this extent, we can still refer to a (relatively) distinct level!

It is only when this process is completed with respect to all the levels of Band 4  that one can finally enter a state which is truly without stages (where "higher and "lower" levels are fully integrated).

However, even this quickly gives way to a new enhanced experience of "radial" reality in a refined phenomenal manner. And this can again be identified with many further stages that have a discrete meaning in a refined relative manner.

When Level 2 (Band 4) has run its course, a remarkable transformation can then unfold in personality.

In the Christian mystical literature this is sometimes referred to as "Deification", which reflects the relatively transcendent aspect (whereby one now fully realises one's individual  as inseparable from the divine personality i.e. God). Perhaps, more commonly it is referred to as "Spiritual Marriage" reflecting the complementary immanent aspect in the more social experience of loving union with the other (recognised now as inseparable from one's self). In psychological terms, this could be well expressed as the union of both masculine and feminine principles within one's psyche.

However I prefer to look on it as the enhancement of reality, whereby the ordinary can be seen as extraordinary and the extraordinary in like manner as ordinary. In other words one has consolidated in experience a habitual state that is now commonplace, whereby phenomenal reality continuously mediates spirit.

In terms of Euler's identity, Level 3 can be expressed as the two-way interaction of both Levels 1 and 2.

Again Level 1 is represented as e2iπ  = 11 and Level 2 as e 2iπ  = 1 – 1 respectively.

Therefore multiplying both expressions,

e2iπ  * e – 2 = 11 * 1 – 1.

Thus by adding indices on both sides,

e= 10.

And this is the appropriate holistic mathematical expression for Level 3.

What this means in effect is that - whereas one initially experienced ultimate reality in transcendent terms as unity beyond all form - one now understands it in a complementary fashion as nothingness (i.e. emptiness). This in turn is associated with the corresponding immanent experience of ultimate reality as the unity prior to and thereby inherent in all form.

In fact we have here the holistic mathematical expression of what is famously expressed in the buddhist sutra,

"Form is nothing but Emptiness
Emptiness in nothing but Form"

In other words, the ultimate unity of  all (phenomenal) form is inseparable from nothingness.

Nothingness is inseparable from the unity of all (phenomenal) form.

Thus nondual ultimate reality is a plenum-void or alternatively a dazzling-darkness (which of course is fully paradoxical from a dualistic perspective).

Now, from the standard analytic perspective, 1 and 0 are absolutely distinguished from each other (as distinct units).

This indeed forms the basis of the binary digital system, whereby all information can be potentially encoded in terms of these two digits.

However, we have now come full circle to the pure holistic perspective, whereby 1 and 0 are intuitively seen as fully interdependent with each other.

And just as the (analytic) binary digital system can be potentially used to encode all information, the corresponding (holistic) binary digital system can likewise be potentially used to encode all transformation processes. So 1 can be seen as the representation of linear logic (the logic of form).
0 in turn can be seen as the (indirect) representation of circular logic (the logic of emptiness).

And all development processes can be seen therefore as representing a certain unique configuration with respect to both types of logical systems!

## Sunday, December 6, 2015

### Psychological Development and the Euler Identity (7)

I identified in yesterday's blog entry the (fundamental) Euler Identity as the appropriate holistic mathematical expression of Level 1 (Band 4).

However though this indeed  corresponds to the pure holistic expression of oneness (in transcendent terms) because this in turn relates to the "higher" spiritual aspect of personality that still to a degree dominates the "lower" physical aspect, one is not yet enabled  to equally experience the pure holistic meaning of oneness (in complementary immanent terms).

So a subtle volitional attachment to the "higher" spiritual aspect remains, which results in a compensatory involuntary projection  of - still - repressed primitive elements with respect to the "lower" physical aspect.

Thus the next level (Level 2) relates to the gradual negation of excess attachment to the "higher" spiritual aspect, which in turn enables one to engage more fully with the unrecognised projections with respect to the "lower" physical aspect.

In this respect, I would find my position in conflict with the Christian mystical position as outlined by even such a careful exponent as St. John of the Cross.

St. John maintains that eventual full union (in the dawning of "Spiritual Marriage" entails that the "lower" physical aspect be gradually be brought into complete conformity with the "higher" spiritual aspect.

However I would rather emphasise that "higher" and "lower" are ultimately fully complementary in nature so that one thereby has to abandon all judgement as to the relative superiority of the spiritual aspect. Otherwise a suibtle repression of physical instinctive behaviour is likely to remain which can only hamper full freedom with respect to the personality.

In  other words, even with respect to "Spiritual Marriage" I would see that an unduly transcendent emphasis characterises the position of St. John (with respect to his explicit  treatment of the issue). And this would be consistent with the official Christian emphasis where the psycho sexual aspect of spiritual development has rarely been properly addressed.

Of course this does not necessarily mean that this issue has not been successfully dealt with in practical experiential terms by St. John and other great mystics. However it would entail however that due to the strong influence of the official Church that a certain discrepancy would be in evidence with respect to the written statements of such figures and their actual lived experience!

There is another issue that I would like to address at this point, which relates to the fact that progression through the respective levels of Band 4 is likely to overlap considerably with on-going development with respect to the levels of Band 5. Indeed in a very real sense, this is equally true with respect to three levels of Band 2!

You will remember that for those destined to significantly advance beyond the linear understanding of Band 2 that considerable conflict is likely to be experienced in late teenage and early adult life, whereby they become unable to fully consolidate development with respect to Band 2. Thus when others are settling down to conventionally accepted roles within society, those destined to advance can experience great disillusionment, requiring a radical conversion to a more authentic spiritual commitment (whatever the social cost). So for lengthy periods of time they can become uneasily suspended as between two realities, no longer able to associate with conventional expectations (based on linear notions) and yet not properly established in the alternative spiritual world view (requiring deep holistic vision).

So in a sense both the rational (associated with Band 2) and the contemplative worldviews (associated with Band 4) can only find their proper fulfilment through the more comprehensive radial development (associated with Band 6) where both Band 2 and Band 4 now can at last find their integrated expression.

Thus in a sense, while the various levels of the spiritual descent (Band 5) unfold , one keeps returning to Band 4 on the one hand to establish a  more balanced contemplative outlook, while equally looking forward to Band 6 (where the linear rational worldview can finally become more established in a properly integrated manner).

Now Level 2 (Band 4) is associated with this formulation of the Euler Identity

e – 2iπ  = 1 (i.e. 1 – 1).

From a holistic mathematical perspective, this implies the negation of the holistic dimensional notion of oneness (i.e. the transcendent experience of everything as beyond all form).

Equally this is related to negation of the dimensional symbols i.e. 2iπ (which as we have seen is associated with the notion of "point being").

In other words one must now attempt to become free of any lingering attachment to the subtle notion of a "higher" self (as identified with one's will as the centre of being). Only then can one's personal will as it were be freely identified with the universal cosmic will as manifested through all creation.

This in turn means that one must thereby detach oneself from any rigid transcendent notion of oneness (as beyond all form) so as to fully embrace the complementary immanent notion of oneness (as already deeply inherent in all such form)..

However this does not occur quickly and can require many years of patient attention that considerably overlaps, as I said on-going development with respect to Band 5. Indeed the attempted elimination of these residual rigid aspects of self can resemble the gradual erosion of rock by sea water.

So once more, great patience and perseverance is required.

## Saturday, December 5, 2015

### Psychological Development and the Euler Identity (6)

I will elaborate here on the holistic mathematical significance of the symbols 2iπ, which play a central role with respect to the (fundamental) Euler Identity.

Now 2π represents the circumference of the (real) unit circle (i.e. with radius 1). Therefore, 2iπ can be expressed as the circumference of the corresponding "imaginary" circle (with radius i). However what does one holistically meaning by "imaginary" in this context?

Well the holistic meaning of "imaginary" represents the indirect attempt to express the dynamic complementary notion of the union of opposites (i.e. + 1 and  – 1) in the standard rational manner.

What this in effect means is that when the pure intuitive notion (represented by the complementarity of opposites) is fully realised  in experience, any remaining linear notion of + 1 and   – 1 as representing separate notions of distance are dissolved.

This can be easily seen through drawing the unit circle with its line diameter horizontally through its centre. Therefore the distance to the right of the mid-point is + 1 and the distance to the left  – 1 respectively.

Now when the pure fusion of these opposite poles takes place in an intuitive manner, any remaining linear identification of + 1 and  – 1 as being separate is thereby dissolved. So the situation is akin to a circle that now has shrunk to its midpoint (where its circular circumference is indistinguishable from its line diameter).

So quite literally in a precise holistic mathematical manner therefore, 2iπ represents now the important notion of a dimension as a single point (without quantitative characteristics).

Interestingly in Christian spiritual mystical literature this stage of contemplative realisation is likewise often represented by a point (as the point or apex of one's being). In Hindu literature the word "bindu" is used, which likewise carries the same  connotation of "a point" , though now given a more impersonal meaning as the "seed", as it were, that potentially contains all of creation.

Indeed in more modern physical terms, we could refer to this point as a singularity (which likewise carries the potential sense of containing all  creation).

However if the transcendent is later to be united with the corresponding immanent direction, it is important to appreciate this notion of point in both a personal and impersonal manner i.e. as the pure centre of one's personal being and likewise as the pure cosmic centre of the impersonal universe. And of course ultimately both meanings should be united in the same undivided experience!

I will also comment here briefly on the holistic mathematical significance of e (which as base number is raised to the dimensional power of 2iπ).

In conventional mathematical terms, e is unique in the sense that when we raise e to a power x, its differential (and of course corresponding integral) expression) remains unchanged.

So if y = e, dy/dx = e.

The holistic mathematical significance is likewise remarkable and revealing, for it implies the state where differentiation n psychological experience becomes dynamically indistinguishable from corresponding integration. In other words, when contemplative experience becomes highly refined, it no longer becomes possible to distinguish distinct phenomena of form (as differentiated) from the continuous experience of spiritual emptiness (as integrated).

Not surprisingly, this coincides with the arrival at that still point as the centre of being. And when both of these are simultaneously related, we have the pure notion of oneness as the transcendent holistic spiritual experience of the interdependence of everything (which is beyond all form).

However certain limitations with respect to this initial experience of pure transcendence are likely to remain.

The fact is that a point or dot (though without distinct physical characteristics) must still be given a specific location to have meaning.

Likewise in psycho spiritual terms, the arrival at the pure point of being implies that a certain attachment must still necessarily pertain to the volitional aspect of the will (which this point represents).

This means in effect that a certain conscious attachment will still exist at the "higher" aspect of personality with respect to the very desire to do what spiritually is deemed most appropriate in experience.

This likewise entails that a corresponding compensatory attachment will remain at the "lower" physical level of personality, that necessarily remains of an (unrecognised) unconscious nature.

In other words primitive  physical projections of an involuntary nature therefore will still be emitted into consciousness.

In this way during Level 1 (Band 4) the "higher" spiritual self still attempts to exercise - to a degree - undue dominance over the "lower" instinctive self.

So this unfortunate tendency then needs to be addressed at the next stage.

## Friday, December 4, 2015

### Psychological Development and the Euler Identity (5)

I have stated that the (fundamental) Euler Identity can be effectively used to clarify the holistic mathematical nature of the 3 Levels of Band 4.

However, this of course entails that we can appropriately interpret its symbols in a true holistic manner (that is directly dependent on the interaction of nondual intuition with a refined paradoxical appreciation in rational terms).

Now the starting formulation of the (fundamental) Euler Identity can be directly associated with Band 4 (Level 1).

So, e2iπ  = 1.

However, as explained in the last blog entry to distinguish both the Type 1 and Type 2 aspects of number, each is now presented more fully with respect to both a base and dimensional number.

Therefore e2iπ  = 11.

However, uniquely in the natural number system 11 appears as the first number in both Type 1 and Type 2 aspects.

Therefore we need a more refined way here to distinguish the (true) Type 2  from the corresponding (conventional) Type 1  understanding.

Thus I will here denote the Type 1 as,

e2iπ  = 11 and the Type 2,

as e2iπ  = 11 respectively.

Now in the first (conventional) case the base unit is understood in an explicit manner, with the dimensional unit merely implicit.
What this means in effect is that "1" is understood in the analytic quantitative sense as representing an independent identifiable unit. The dimensional notion of 1 here remains merely implicit indicating in effect that 1 is a number on the real number line (that is - literally - of a 1-dimensional nature).

Now in the second (formally unrecognised) case the dimensional unit is now understood in an explicit manner, with the base unit now merely implicit.

Again what this means in effect is that one now recognises the notion of  1 (in its general 1-dimensional nature as "oneness" as potentially applying to all numbers)

Now remarkably this very notion is inherent in all multiplication (without however being recognised as such).

For example to illustrate imagine that we arrange coins in rows of 3.

Then if we have one row this can be represented as 1 * 3
If we have 2 rows (of 3) it will then be represented as 2 * 3,
Then if we have 3 rows it will be represented as 3 * 3 and so on.

Therefore though the number of coins in each row remains constant as 3, the corresponding multiplication operator can vary here over all the natural numbers (that lie on the real number line).

And the essential point here is that to use this operator we need to recognise the common shared identity of each row (i.e. the "oneness" of each row).

Thus implicit in the very ability to carry out multiplication is this holistic notion of "oneness" where we can identify each class (in this case row) as possessing a shared identity.
And this shared identity can be extended without limit.

So once again the notions of independent identity (in a quantitative sense) and shared identity (in a qualitative manner) are if effect completely confused in conventional mathematical terms both being interpreted in a mere quantitative terms.

So unfortunately in Conventional Mathematics a continual reduction of holistic notions of number takes place in a reduced analytic manner.

Therefore when 1 is used to represent a dimension - rather than this being seen in its true holistic sense as potentially applying to all numbers - it is immediately reduced in an analytic manner as actually applying in all cases. This then inevitably leads to a reduction of the infinite notion itself whereby it is misleadingly understood as a linear extension of finite notions.

Therefore the very basis of the experience of Band 4 (Level 1) is that one now can directly intuit the true potential meaning of 1 (i.e. as oneness) in a manner that transcends all actual notions.

In other words one literally experiences a state of oneness as the pure potential for being. And as we have seen this has deep implications in mathematical terms (which unfortunately has not all been realised by the profession).

However it is in the very nature of human development that this rarefied state cannot be sustained in the absence of proper integration with the world of actual form.

So once again it is rather like the mountaineer that has finally set foot on the peak of Mount Everest.
Certainly triumphant feelings of joy and exultation will be momentarily present. However very quickly one will need to turn one's attention to the considerable task of now safely descending the mountain so that one can once more set one's feet firmly on the plains below.

So in my own case, while I was recovering from a serious ulcer bleed in hospital, I experienced moments of great lightness and joy (where my spirit seemed as it were to detach itself from the body). These then continued for a short period after leaving hospital, before the on-going precariousness of my situation quickly restored normality.

Having concentrated so intensely on the spiritual ascent for so many years, I had now reached a situation where my whole psycho physiological apparatus was undergoing constant severe stress. So now in a somewhat weakened state, attention started to turn to the problem of a necessary rebalancing in the successful negotiation of the spiritual descent.

## Thursday, December 3, 2015

### Psychological Development and the Euler Identity (4)

For some time I remained deeply puzzled by a very interesting facet of the (fundamental) Euler Identity.

So once again, e2iπ = 1.

However, it is well known in conventional mathematical terms that when we raise any number to 0 that the resulting answer = 1.

Therefore e0 = 1.

This then seemed to suggest that as  e2iπ = e0  that therefore 2iπ = 0, which would seem absurd!

Indeed again in conventional terms,  e– 2iπ = 1/e2iπ = 1/1 = 1.

So this would now seem to suggest that 2iπ  = – 2iπ, which again would be absurd from this conventional perspective.

In fact, e2kiπ = 1, where k = 1, 2, 3, .......n

Therefore, seemingly,  e2iπ  =  e4iπ  =  e6iπ  = .......= 1, implying that 2iπ = 4iπ = 6iπ etc.

It was only that I began to clearly realise something with truly enormous mathematical implications. in that all numbers have two aspects, which depending on context, keep switching in the dynamics of experience.

Now I refer to these two aspects as analytic (quantitative) and holistic (qualitative) with respect to each other.

This is remarkably similar to quantum mechanical reality, where sub-atomic particles can reveal themselves as particles and waves (also depending on context).

In fact the deeper implications here entail the startling realisation that quantum mechanical behaviour is itself inherent in the very nature of the number system (when appropriately understood in a dynamic interactive manner).

So the way we avoid confusion with respect to the issues raised is to now define the natural number system in terms of two complementary aspects (which I refer to as Type 1 and Type 2 respectively).

The Type 1 aspect can be identified with the conventional analytic interpretation of number (i.e. in a merely quantitative manner).

So here each natural number is defined in terms of a default dimensional value of 1

So the natural numbers in Type 1 terms can be represented as

11, 21, 31, 41,……..

The Type 2 aspect can then be identified with the (unrecognised) holistic interpretation of number (in an intrinsic qualitative manner).

Here the base number remains fixed as 1, whereas the dimensional value now varies over the natural numbers as

11, 12, 13, 14, ......

In (Type 1) quantitative terms, 1 for example, represents an independent quantitative unit (that literally lacks any qualitative distinction).

However in (Type 2) qualitative terms, 1 represents a qualitative unit (with the capacity of relating all independent units). In fact this is conventionally referred to as "oneness" (i.e. the quality of 1).

Now quite remarkably - though there is as yet  no recognition of this crucial fact in conventional mathematical terms - we naturally keep switching between the Type 1 (analytic) and Type 2 (holistic) aspects of number even with respect to the most commonplace number operations!
For example imagine a class (or group) of 2 objects. Because these are already implicitly recognised as belonging to the same group. we can then give "2" the standard (Type 1) quantitative meaning (as comprised of two independent entities).

However now imagine two separate classes containing in each case 1 object!

Here, the identification of 1 (as now common to both classes) properly requires the unrecognised (Type 2) qualitative meaning of 1.

In other words we are able to identify that 1 is common to both classes through the shared qualitative notion of "oneness" in each case.

So we have now moved from the independent (separate) notion of 1 (in Type 1 terms) to the related or interdependent notion of 1 (in a Type 2 manner).

However in conventional mathematical terms, these two complementary aspects of number, which operate in a dynamic relative manner, are reduced in an absolute - merely quantitative - manner.

When one truly grasps this point, then one profoundly realises that the very foundations of the number system - as currently understood and by extension all Mathematics (and its related sciences) - are hugely inadequate.

For millennia now, we have tried to understand Mathematics in merely quantitative terms.

However despite its great advances this has led to a greatly distorted interpretation (even in quantitative terms).
So the clear message is that in future we will slowly come to realise that both quantitative and qualitative aspects are necessarily involved  in even the most trivial mathematical operations and that proper understanding must be based therefore on the dynamic interaction between both poles.

To return to our immediate topic, the amazing discovery that I made is that the Euler Identity in fact directly relates to the (hidden) holistic aspect of the number system!

So to resolve the problems that I raised at the beginning of this entry, we must define number in Type 2 terms.

Therefore the (fundamental) Euler Identity is now expressed as

e2iπ = 11

Then e0 = 10

e– 2iπ = 1– 1

And e2iπ  = 11,  e4iπ  = 12 and e6iπ  =  13 respectively.

So in fact all these expressions which lead to the same value in a Type 1 quantitative manner, are associated with distinctive holistic values from a Type 2 qualitative perspective.

So we will next show more directly how the 3 levels of  Band 4 can be associated with the specialised intuitive understanding of these fundamental numerical values.

## Wednesday, December 2, 2015

### Psychological Development and the Euler Identity (3)

As we have seen, the Euler Identity can be used in a fascinating manner to provide a holistic mathematical interpretation of the nature of contemplative spiritual union.

The deeper implication of this is that - properly understood - Mathematics entails a dynamic interaction between analytic and holistic aspects that lie at opposite extremes of each other.

Thus from the conventional analytic (Type 1) extreme, mathematical symbols are understood in terms of absolutely rigid symbols of form, interpreted in a linear (1-dimensional) manner.

However at the opposite holistic (Type 2) extreme, these "same" symbols are now understood in a completely fluid interactive sense, ultimately representing pure energy states in a circular (0-dimensional) fashion.

Thus the paradox inherent in the analytic, can then only be resolved through the corresponding holistic interpretation.

Therefore in order to achieve this, one must move from initial linear rational understanding (of the mathematical symbols in the fundamental Euler Identity) to a fully transformed circular intuitive appreciation (that appears paradoxical at the linear level).

So in the spiritual journey required, understanding - literally - comes full circle (deeply symbolised by the dimensional symbols 2iπ, which trace out the circumference of the "imaginary" i.e. holistic circumference of the unit circle).

However we can go even further here in demonstrating the relevance of the Euler Identity for the 3 levels of Band 4.

Once again, it may be helpful to initially explain - without holistic mathematical terminology - the precise nature of each of these three levels.

Now, again the 1st level (Band 4, Level 1) is characterised by the arrival at a transcendent form of union (representing the culmination of the development that took place during the previous stage (Band 3, Level 3).

This is identified as the attainment of a pure state of spiritual emptiness (i.e. as beyond all phenomenal form) at the "higher" level of personality.

However a certain imbalance is still likely to remain, whereby the "higher" spiritual still unconsciously dominates to an extent the corresponding "lower" physical aspect of personality.

Therefore the arrival at this transcendent state as beyond all physical form, remains to a degree  unconsciously associated with a subtle repression of such form at the "lower" instinctive aspect of personality. This then results in the compensatory activity of the "lower" self to still involuntarily project residual unreformed physical material into consciousness.

Thus the subtle aversion to physical form (at the "higher" level) leads to the involuntary projection of this unrecognised form (with respect to the "lower" aspect of personality).

Therefore, once again we can only meaningfully conceive of such transcendent union in an approximate manner, which depends very much on personality and the intensity of the spiritual drive involved.

So in some "lesser" cases an attenuated degree of transcendent union could be reached with a substantial amount of residual "lower" level imperfection remaining.

However in other cases, especially where a gifted mystic - and perhaps prospective "saint" - is involved, a much greater degree of  cleansing with respect to instinctive impulses would be required.

The second level (Band 4, Level 2) is then concerned with seriously addressing this remaining imbalance, so that the immanent aspect of spirituality (where spirit is seen as deeply inherent in all physical form) can ultimately be properly balanced with the corresponding transcendent aspect (where spirit is likewise seen as beyond all phenomenal form).

Now, I would suggest that especially for male mystics, this residual problem is likely to involve too much super ego control as exercised by the cognitive function.

Because in one's training one, has for long been conditioned to view the physical i.e. instinctive as inferior to the spiritual self (as mediated through one's refined rational capacity), there can be considerable difficulty experienced in sufficiently allowing this primitive emotional self speak for itself. However it is only when one can hear the voice of this self (with all its lustful impulses) freed from moral censorship, that the blind involuntary nature of such instinctive craving can be successfully dissolved.

So the second level here entails yet another withdrawal in the literal negation of such excess identification with one's higher self. This can equally entail a substantial withdrawal from all the social recognition that fosters undue identity with this self.

In my own case, this was to lead to the need for a further lengthy period of "intellectual isolation" free of the need for outside confirmation of my ideas.

The at the third level (Band 4, Level 3) one is then finally enabled to engage in the task of the balanced integration of both one's "higher" and "lower" selves (i.e. the transcendent and immanent aspects of personality).

Though this is commonly referred to in the Western spiritual tradition as the entry into "Spiritual Marriage", I would rather refer to it as the attainment of  an "enhanced" normality where one no longer self consciously separates the spiritual from the material aspects of living. Put another way, the stage has now been reached, where one can begin to properly fulfil one's inherent potential through the actual circumstances of day to day activities.

Put more simply, this is true freedom to finally be oneself! Thus the longing to be (one's true self) finally culminates in a true state of be-longing (where this self is experienced as inseparable from God).

## 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.

## 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!

## Tuesday, August 25, 2015

### Growing Crisis

I do not remember many details of my life from the early to mid 90's. However in general terms it proved to a period of growing stress, that I only began to properly understand much later.

I was aware however that my development had taken a pronounced vertical character, with the continual unfolding of more advanced spiritual states and structures.

However as I ascended "higher" in a transcendent direction ever closer to pure spiritual intuition (without phenomenal structure), this was complemented by the opposite descent with respect to sensation into the physical realms of the unconscious (repressed from earliest childhood).

Thus increasingly my energy was given over to this dynamic interaction as between "higher" and "lower" (unconscious) bands, as attempted integration of the levels of these bands continued.

I was becoming aware however of a growing problem in that the (conscious) middle levels were being continually bypassed in the process.

And as the more horizontal extensive nature of the middle levels is properly suited for conscious day to day responsibilities, this began to create a considerable amount of stress with respect to the conduct of such activity.

I felt - quite literally - as if  my very body was being increasingly stretched in a vertical manner, with less and less oxygen, as it were, available for daily work and social engagements.

Once again the comparisons with a steep mountain climb such as Mount Everest are very apt.

As a climber approaches the summit of Everest, available oxygen in the physical atmosphere becomes steadily depleted. Therefore one can only proceed ever more painfully, as one strives to minimise on undue physical exertion.

Equally as one approaches the transcendent peak in spiritual terms, the psychological oxygen, enabling one to carry out normal conscious activity can likewise become greatly depleted. Therefore the carrying out of day to day activities can prove very burdensome, where no relief can be experienced. One is thereby forced to continually economise on undue conscious effort, which requires that any attachment (positive or negative) be relinquished.

In fact one's breathing in physiological terms becomes almost entirely suspended.
This is compatible with a deep sleep state, where unfolding events leave remarkably little trace in memory. However in the hidden depths, all sorts of physical fantasies of a primitive kind can emerge threatening to disturb this peace.

When I look back now a distinctive form of stress was associated with each of the three major levels (of Band 3).

Level 1 was associated directly with the thinking mental function (connected largely to the expression of my male identity).

The main problem here in terms of stress was associated directly with what would be identified as mental illness.
So as I have stated  clinical depression (e.g. endogenous) - even psychosis - is quite likely as one attempts to survive a lengthy "dark night of the soul" experience.

This was associated in my own case with a severe shedding of a masculine ego identity.
It is in the very nature of things that this problem tends to be reinforced through outward events.
So as stated before, at this time my father (who had always provided unconditional affirmation) died. and I also ran into serious problems with respect to employment.

Level 2 was then associated with the affective feeling function (in the hidden expression of a female identity),
The stress that I experienced now was more directly of an emotional kind in the fading of a significant emotional relationship and several social activities that for a while had given me support. Finally and most significantly, my mother died.
So I interpreted this turmoil as a testing of a newly acquired feminine identity. In other words I was forced to develop the feminine within myself to compensate for its outside loss.

Level 3 was associated directly with the volitional function (in the attempt to reconcile both masculine and feminine identities)

However, complementing the pure (empty) activity of will of the "higher" self were the still involuntary promptings of the "lower" physical self  (where cognitive and affective functions still remained confused).

So stress now started to manifest itself increasingly in a psycho physiological manner.

In other words, psychological stress - which previously manifested itself in mental and emotional terms - now directly affected the body.

However because I had enjoyed remarkably good health in physical terms prior to this, I was not yet attuned to recognise the new problem.

Then in mid 1996, one night, I began to feel somewhat dizzy and lightheaded. When I developed what seemed like a routine virus the following day, I was relieved as seemingly this was the explanation for the previous night's problem.

Though the virus cleared up, I still seemed unusually weak at times and even passed out one night. Eventually I discovered that I had an internal bleed (which was due to a painless ulcer).

By the time I got to hospital the blood count was dangerously low.
So here was a new crisis, where my own life for a while seemed to be hanging in the balance.

Just as the earlier levels had culminated with the death of my father and mother respectively, this was associated with the threat of death with respect to my personal self,  now spiritually experienced in androgynous terms (as both masculine and feminine). Only through such an event could the fear of death be faced in such a way, as to enable the transcendent task of that period of development to be realised.

The person who finally climbs on to the summit of Everest though understandably highly elated, is likely to have already suffered a lot and perhaps be in a fragile physical condition.

In like manner, achieving the transcendent peak of my own journey was associated for a while (in hospital and later outside) with several outpourings of  mystical joy. However in many ways with the real descent now beckoning, I was still potentially in a fragile physical state and subject to continuing psychological stress.