Thursday, December 31, 2009

Fractal Dimensions

One of the great contributions of Fractal Geometry is that it leads to consideration of the corresponding notion of fractal dimensions. For example this is beautifully illustrated with Koch's Snowflake.
See Mathworld.

So to construct this Snowflake we start with an equilateral triangle. Then marking each line into 3 equal divisions we take the middle third and erect another equilateral triangle on each side. Then we continue to proceed in the same manner (constructing a new equilateral triangle on the middle third of each exposed side).

Theoretically, we can continue in this manner an infinite number of times.

The implication of this is that the perimeter boundary of the Snowflake thereby increases without limit. In fact we can easily see that the initial construction of equilateral triangles on the middle third of the original 3 sides of the starting equilateral triangle increases the perimeter length by a factor of 4/3. Thus as we can keep repeating this procedure indefinitely (in each case increasing the perimeter by 4/3) the line can grown without limit even though the area of the entire figure clearly is of a finite magnitude.

In effect the perimeter line (through this process) encloses to a degree the overall 2-dimensional area.

This leads to a new notion of dimension for this line. To acquire it we basically find in this case what root of 4 (= the dimension) gives an answer of 3, the answer which is 1.261859507..

Strangely enough such a dimension (which necessarily falls between 1 and 2) is generally referred to as a fractional dimension.

However this conceals a very important conceptual difficulty.

We commonly think of fractions as rational numbers. However such a dimension (as in this example) will always be of an irrational nature (which thereby cannot be represented as a rational number).

Indeed the only way to get a rational dimension is to attempt to divide the line with irrational number sections!

This is all very interesting from a qualitative perspective.

In proper qualitative terms, finite notions are associated with (linear) reason and infinite notions with (circular) intuition respectively.

Now the irrational notion relates therefore to a situation where the infinite in some ways becomes embedded in the finite.

So corresponding to irrational understanding in qualitative terms is the derivation of an irrational number dimension (reflecting the paradoxical interplay of finite and infinite notions).

The clear implication of all this is that a corresponding qualitative interpretation attaches to the mathematical notion of the irrational.

In psychological terms this becomes equated with the transition from 1 to 2-dimensional understanding. Initially before two clear polar directions of phenomenal understanding crystallise, one goes through a linear phase where rational understanding increasingly interpenetrates with intuitive understanding (with finite phenomena now in some way genuinely embodying spiritual infinite notions).

This also provides an answer to the clarification of dimensions. So now we can look at irrational dimensions as basically representing transition states as between the whole number dimensions. Basically each new higher dimension (as whole number) represents a new more refined form of rational appreciation (with the corresponding number of directions). Therefore the irrational dimensions in between represent various states (whereby reason and intuition are combined in a - as yet - somewhat confused manner).

Once again in quantitative terms a "fractal dimension" between 1 and 2 represents the extent to which the 1-dimensional line can fill in a 2-dimensional area.

In corresponding qualitative terms, a "fractal dimension" (that is indeed of an irrational nature) measures the extent to which one can embody as it were 2-dimensional appreciation coming from a 1-dimensional perspective. This can never be achieved fully. However it can be approximated more closely through a progressively greater degree of intuitive understanding informing rational interpretation.

Return to Chaos

Like so many, my first introduction to the new science of Chaos and Complexity came from the book "Chaos" written by James Gleick back in 1987.

Recently I returned to read this book wondering as to whether the intervening years had changed my perception or sharpened understanding of the issues raised. As on the first occasion of reading, I was left however with a vaguely dissatisfied feeling, rather like the various courses of a meal that promise a great deal in preparation, yet somehow fail to deliver on eating.

So I stayed a little with this feeling so as to get to its roots.

Basically I would summarise the position as follows. Chaos Theory and all the various versions of Complexity arise from the recognition that so much behaviour in nature is of a nonlinear nature. However Conventional Science, through its linear method, tends to approach nature through modelling it in the form of linear equations (frequently adjusted finally in some way to approximate living conditions).

However this greatly limits the range of possible enquiry while in many ways misrepresenting the true nature of natural behaviour that is studied.

Chaos Theory therefore arises from the commendable insight that nonlinear behaviour needs to be accommodated in the true study of nature.

Admittedly impressive advances have been made in some fields from this insight. For example fractal geometry, not alone reveals many new fascinating insights but also lends itself to a new form of computer art (where self similar patterns endlessly repeat without exactly replicating themselves).

However one great limitation of these new studies is that they are still attempting to approach scientific problems from within the conventional scientific approach (which by its very nature is linear).

So we have the attempted study of nonlinear behaviour (within the confines of a decidedly linear paradigm).

What this means in effect is that the very intuition that is required to - literally - see the important holistic relationships in nature is not actually provided through the new approaches. In fact it leads directly to an equivalent form of reductionism. Whereas the "old" standard approach is criticised as attempting to reduce the whole behaviour of systems to (isolated) constituent parts, the "new" approach actually suffers from the opposite form i.e. of attempting to reduce the rich diversity in nature to some general forms of behaviour.

It seems therefore quite obvious to me that the study of nonlinear systems (in quantitative) terms, properly requires a nonlinear manner of approach (from a qualitative perspective). However as of yet there is no real recognition - either among the "old" standard practitioners or the "new" chaos merchants - of such a nonlinear qualitative approach (which would require a radical revision of what is meant by science).

In the end it is only through this new qualitative approach - which I term integral - that the ability to "see" the important holistic connections in nature (in any required context) can actually emerge.

The fact is that physical and psychological reality are dynamically complementary. Nonlinear behaviour in nature above all requires recognition of this simple fact. In the end the same nonlinear behaviour that governs nature, likewise governs mental life. Though proper appreciation of this basic point, each can thereby operate as a mirror to the other's true identity.

Clarifying Dimensions

As one can see, my integral approach to science (and development) is heavily based on holistic mathematical interpretation of the 1st, 2nd, 4th and 8th dimensions respectively.

However a key insight here is that corresponding to every number (in holistic terms) is a unique dimensional interpretation (with application to both physical and psychological reality).

In psychological terms, this entails that all possible understanding entails both conscious and unconscious aspects. So a dimension in this sense corresponds to a unique configuration of these two aspects.

In scientific physical terms, this equally entails that reality entails both specific (manifest) phenomena related to an underlying (hidden) ground of reality. So here, each dimension relates to a unique configuration connecting such phenomena to this underlying ground.

And of course in holistic mathematical terms, a dimension (in either its psychological or physical state) is given as a unique mapping of real and imaginary coordinates (on the complex plane).

What I want to do briefly here however is to extend somewhat the significance of dimensions (beyond 1, 2, 4 and 8) already mentioned.

Firstly it has to be recognised that every number can be given a positive (+) as well as (-) sign. The question then arises as the negative meaning of the dimensions already mentioned.

Quite simply, whereas the positive understanding of a dimension relates to conscious rational interpretation, the corresponding negative interpretation relates to its unconscious intuitive recognition.

In spiritual terms, whereas development of the positive understanding is generally associated with periods of illumination, the corresponding negative understanding is associated with periods of purgation.

Thus development of of the negative first dimension literally relates - in psychological terms - to the active negation of linear understanding (in what St. John of the Cross refers to as the active nights of sense and spirit). Here sense is identified with more superficial perceptual phenomena whereas spirit relates to deeper conceptual and volitional awareness!

There is in fact a very close relationship as between a negative dimension and the next positive number dimension. So in this context what is unconsciously understood at the first dimension (in the negation of - formerly - posited linear understanding) is then consciously appropriated at the next positive dimension (i.e. the 2nd, as combining both positive and negative aspects). Once again there is a remarkable connection here with the Riemann Zeta Function where the corresponding relationship is maintained in terms of the famous functional transformational formula. So properly understood the two sides connected through the functional formula relate to specific (conscious) interpretation on one side (for positive values of s) and to holistic (unconscious) interpretation on the other side (for negative values, 1 - s).

Then development of the negative 2nd dimension once again relates to intuitive (rather than rational understanding) of the complementarity of real opposites in experience. St. John refers to this as the passive night (of both sense and spirit).

In general, purely contemplative (intuitive) understanding of an empty spiritual nature is associated with the negative even dimensions (where always an exact complementary pairing as between opposite terms exists). In like manner in the Riemann Zeta Function for all negative even values of s, the result = 0.

This was to prove a key revelation in my examination of this function as it quickly led to the realisation that the Riemann Zeta Function in fact provides numerical results corresponding to both conventional mathematical interpretation (where s > 1) and holistic mathematical interpretation (where s < 0). Thus one cannot properly interpret the Riemann Zeta Function (and the associated Riemann Hypothesis) without including both conventional and holistic modes of interpretation!

There are strong reasons for believing however that in terms of even numbered dimensions, 2, 4 and 8 are the most important. Remarkably there is a very good mathematical reason (holistically interpreted) why this is in fact the case!

However I have also explored - to a certain extent - the nature of both 16 and 24 dimensional understanding (which I may return to in later contributions).

For each whole (integer) number, we also have a reciprocal number (as fraction). So these fractional dimensions (of even numbers) can also be given a qualitative meaning!

The easiest to understand is the reciprocal of 2 (i.e. 1/2). Whereas the second dimension relates to holistic understanding (as one moves away from specific phenomena in understanding), the reciprocal relates to the corresponding movement back to specific phenomena (in the light of this new understanding). In other words it provides a new enhanced form of analytical understanding where all phenomena can now be viewed within two respective reference frames (that are opposite with respect to each other). Thus in terms of analytic understanding each frame provides - literally - 1/2 of a total perspective).

Likewise much more enhanced analytic perspectives open up at the higher dimensions. For example when we come back to linear analytical investigation (after encountering holistic 4-dimensional understanding) we now can provide four distinct frames of reference for any interpretation. (So each interpretation now represents 1/4 of the total linear dimensional perspective).

The corresponding negative fractional dimensions would then relate to additional active purgation (with respect to such enhanced appreciation). So the task of purgation is never complete (either with respect to active or passive understanding) with each new dimension opening up a more refined type of phenomenal understanding (that can only be purified at that level).

As with the Riemann Zeta Function, the odd dimensions prove to be much more problematic. However, having said this, understanding of the holistic meaning of the Function has greatly enhanced my interpretation of the odd dimensions in psychological development (and corresponding physical reality).

What it amounts to is simply this!. Whereas the even number dimensions relate directly to integration, the corresponding odd number dimensions relate to increasingly refined differentiated appreciation of phenomena.

What actually happens in development, is that a certain amount of intuitive confusion remains (as each new odd number dimension unfolds). So in the corresponding purgation with respect to such dimensions, this confused aspect is removed leaving only the purified rational element.

So whereas the negative even number (integral) dimensions are concerned with the purification of confused rational elements, conversely the negative odd number (differentiated) dimensions are concerned with the purification of confused intuitive (i.e. unconscious) elements. Once again this corresponds remarkably with the Riemann Zeta Function where all negative odd values (for s) yield a rational answer.

Once again the fractional use of such dimensions would occur when one returns to the linear level of experience (from the perspective of the higher odd dimension).

For example if one returns to differentiated linear investigation from the 3rd dimension, 3 independent reference frames will now be available (with each corresponding to 1/3 of the total perspective). One of these will concur with the standard linear approach. However the other two will lead to additional complex interpretations (that are equally valid). Thus comprehensive analytic understanding will now require all three interpretations!

We can also give interpretation to imaginary dimensions. It took me a long time to appreciate this (though it is not really so difficult).

When one remains psychologically in real dimensions, it entails a movement away from form in an increasingly unconscious direction (i.e. spiritual transcendence).

However the corresponding reverse direction in development is the movement from the (empty) unconscious back to form in an increasingly conscious manner (i.e. spiritual immanence).

And when this reverse journey takes place, one in fact is operating out of imaginary (rather than real) dimensions.

Again this has a remarkable correspondent in mathematical terms. If one raises 1 to a real rational value) e.g. 1/4, one obtains the imaginary number i. If now in turn one now raises 1 to this imaginary value i, one obtains a real number!
So this demonstrates the extremely important point that object phenomena and spacetime dimensions are real and imaginary with respect to each other in dynamic interactive terms!

In fact this discovery has proven very illuminating in terms of understanding my own development (which has been unduly transcendent).

Thus - though in principle - I would have subscribed to the view that one should return to the 1st dimension (of linear understanding)in an enhanced manner from higher dimensions, this in fact for me always proved very difficult. I found it in fact easier to keep moving on to ever higher dimensions (in a holistic manner) rather than return to the more specific experience of form! Only recently have I been able to realise that this reflected a lack of sufficient immanent development (which inherently requires the use of imaginary dimensions).

In brief the real dimensions are concerned with obtaining an ever more refined conscious appreciation of reality (inspired through intuition); the imaginary dimensions are concerned with ever more refined unconscious appreciation
(controlled by reason).

We can also give a holistic interpretation to irrational dimensions. However I will tie this in later with a brief discussion on fractals.

Integral Theory of Everything (4)

One way of viewing this integral TOE is in terms of the main contemplative stages required before full union (empty of manifest phenomena) can take place.

Linear understanding (1-dimensional) by its very nature leads to phenomena being given an unwarranted rigidity (by which they have a seemingly independent existence).

So the first task is to remove such rigidity at a conscious level. This is attained by recognising the inherent complementarity - with respect to all phenomena - of both internal and external polarities (2-dimensional understanding). In holistic mathematical terms these are positive (+) and negative (-) with respect to each other in real terms.

In psychological terms this can be attained at either a rational (cognitive) or sense (affective) level without the full integration of either aspect.
In corresponding physical terms this would relate to a failure to fully integrate both personal and impersonal aspects with respect to scientific phenomena.

The second task is then to remove further rigidity at a deeper unconscious level. This manifest itself in experience when phenomena - representing an overall holistic meaning (pertaining to the unconscious) - are given an indirect conscious identity. This is in turn attained by recognising the inherent complementarity of part and whole aspects with respect to all phenomena (4-dimensional understanding). In holistic mathematical terms these are real and imaginary with respect to each other (both having positive and negative directions).

In psychological terms this would now entail substantial integration of both cognitive and affective aspects of experience.
In corresponding physical terms it would entail the true marriage - without reductionism - of specific phenomena with dimensions (combining the interaction of both personal and impersonal aspects of phenomena).

The final task (from the purely contemplative view) is to then simultaneously combine both conscious and unconscious in a refined flexible manner (without secondary rigidity arising). This relates directly to volitional intent which serves in turn as the primary basis for continued integration of both cognitive and affective aspects (8-dimensional understanding). In mathematical terms this would combine the 4 additional complex roots of unity with the 4 already attained).

In psychological terms it means that the balance as between conscious and unconscious (in experience) is now so equal (i.e. where real and imaginary parts are of same magnitude) that phenomena no longer appear to arise. This coincides with experience of spiritual emptiness (i.e. null lines). So here the four "spiritual forces" (in the external and internal manifestations of immanence and transcendence respectively) are unified in the one awareness.

In corresponding physical terms, it means that the four physical forces (where wave and particle aspects remain equal) are also unified as the one superforce (existing in the present moment as mere potential for existence).

Finally, as we have seen both physical and psychological understanding fully coincide here in the same contemplative experience in the continuous present moment (as both the Alpha and Omega of all reality).

Of course paradoxically this Holistic Theory - or rather Experience - of Everything is equally an Analytic Theory of Nothing (in that tells us nothing specifically about reality).

So I would now see this purely contemplative stage as but a stepping tone to more advanced radial engagement with reality.

It might seem initially puzzling why this should be in fact the case! I would use the analogy of the lighting fire to illustrate. Pure contemplation is like a fire - say of wood - that is set alight (without an attempt to replenish its material base). So eventually this fire , having reached its zenith blazing state, is destined to slowly die out. In like manner the state of pure contemplation (without active phenomenal involvement) is not possible to sustain.

However if we keep adding fuel appropriately to the fire, not alone can its blazing state be maintained but its very size and intensity can be increased. Here at any given stage we will have a blazing fire that is consistent with an increasing amount of material that has not yet been fully lit.

Likewise in the radial life as one becomes more active in a sustainable manner, not alone does this entail an increasing level of phenomenal experience (not yet fully transformed in spirit) but equally a more intense and more broadly based spiritual contemplative awareness.

So in the end radial activity is not only necessary in order to transform "the world" (as opposed to the self) but equally necessary to properly sustain and enhance true contemplative experience (of spiritual nothingness).

Wednesday, December 23, 2009

Integral Theory of Everything (3)

I will elaborate further on the significance of the three key polarities (corresponding to the holistic mathematical interpretation of 2, 4 and 8 dimensions respectively).

The first set of horizontal polarities (corresponding to the relationship between internal and external) as we have seen are positive and negative with respect to each other in integral terms.

The significance of this is that all scientific understanding of reality necessarily entails an internal (subjective) observer in relation to what is observed in external (objective) terms.

Now when we view this relationship in a linear manners, both external and internal frames can be given a merely positive identity (that correspond with each other).

For example I can view the "sun" as in objective terms as an external object.

However equally I can view the "sun" in - relative - subjective terms as relating to its internal perception.

Fortunately for Conventional Science - based on a linear approach - both external object and internal constructs are understood to correspond directly with each other. In this way it does not matter if we designate science in conscious (positive) terms as interpretation of the relationship between object phenomena (external) or alternatively between subject constructs (internal).

However when we attempt to view this relationship in a holistic integral manner, these two frames of reference, external and internal - that seemingly correspond with each other (in linear terms) - are diametrically opposite (in a complementary circular manner).

So now both external and internal are understood as positive and negative with respect to each other (in complementary fashion). Though the rational interpretation of this relationship is paradoxical, the actual reconciliation of both aspects takes place in a directly intuitive (unconscious) manner.

So one clear implication of the Integral Approach is that scientific understanding cannot be conducted in (mere) rational linear terms but rather combines both (linear) sequential and (circular) paradoxical understanding (implying the explicit combination of both reason and intuition).

The refined 2-dimensional appreciation of the role of external and internal aspects of understanding (n both linear and circular terms) corresponds to - what I refer to as - the Integral 1 approach.

The second set of polarities relates to the hugely important distinction as between whole and part.

Once again in linear terms (where both are treated as independent) these can be given a rational interpretation in merely conscious terms (i.e. as positive). For example research data would be identified with the part and theory with the whole aspect respectively.

However in circular integral terms (where both are treated as interdependent) the relationship is as real to imaginary. So if we give the part (in any context) a real identity, the corresponding whole aspect is - relatively - imaginary.

So we have the vital insight here that in order to preserve the appropriate qualitative distinction as between part and whole, we must interpret reality in complex rational terms (with both real and imaginary aspects).

This has several important applications (just a few of which I will mention).

The relationship in physical terms as between object phenomena and dimensions (of space and time) is real to imaginary.
In corresponding psychological terms the relationship between perceptions and concepts is also real to imaginary. Furthermore both of these aspects are fully complementary. This - as we shall see - has huge implications in terms of making sense of many of the findings of string theory which really points to the need for an entirely new understanding of the dimensions of space and time.

The relationship as between cognitive and affective in understanding is also real to imaginary. In other words, in Jungian terms, when the cognitive aspect of understanding is real (conscious), the corresponding aspect is imaginary (as the projection of the holistic unconscious); in like manner when the affective aspect is now real the cognitive is then imaginary. So in the dynamics of experience, we keep switching as between real and imaginary (with the revealed aspect conscious and the alternative hidden aspect unconscious).

Finally, a correct appreciation of such complex interpretation exposes the limitations of conventional mathematical proof (which entails a basic form of reductionism where the whole is confused with the part). So, the general proof that applies holistically to all cases cannot be directly identified with any specific instance (applying to the part).
This leads to need for reformulation of the notion of proof in dynamic relative terms where he it is seen as representing but a special form of social consensus.

The third set of polarities relates to fundamental form and emptiness. Again in linear terms these can be treated in an independent manner. However in corresponding integral terms they require 8-dimensional understanding for appropriate interpretation.

In physical terms this leads to the appropriate holistic mathematical way of understanding the nature of the physical forces; in corresponding terms it provides the corresponding means to translate the volitional psycho spiritual forces; and once again both sets of forces are complementary in experiential terms.

We will look at this again in the next contribution.

Sunday, December 20, 2009

Integral Theory of Everything (2)

The Integral Theory of Everything is based on finding a holistic mathematical acceptable manner of interpreting the dynamic relationship as between the 3 fundamental polarities (that underlie all experience of reality).

Once again these polarities relate to (i) the distinction as between internal and external within a given stage of development (horizontal polarities); (ii) the distinction as between whole and part between different stages of development (vertical polarities); (iii) the ultimate distinction as between form and emptiness both within and between different stages of development (diagonal polarities).

I identify 4 stages of integration corresponding with the qualitative interpretation of 1, 2, 4 and 8 dimensions respectively (relating in turn to the 0th, 1st, 2nd and 3rd powers of 2).

The standard conventional scientific approach - based on linear rational understanding - corresponds with the 1st dimension (Integral 0 approach).

Basically such an approach - which is based on clear separation of the opposite polarities in experience - does nor allow for corresponding complementary understanding. Therefore it necessarily deals with the key polar interactions through a form of reductionism. Thus typically in conventional science, the internal aspect is reduced in terms of the external; the whole in any context is reduced in terms of the part; finally emptiness is reduced in terms of form.

The first truly integral approach (Integral 1) allows for the dynamic complementarity of both internal and external through - what qualitatively represents - 2-dimensional understanding.

Here internal and external (conscious) phenomena are considered in holistic mathematical terms as both positive and negative with respect to each other in real terms. However the Integral 1 approach does not yet allow for appropriate interpretation of the relationship as between whole and part and form and emptiness.

The second approach (Integral 2) allows for additional dynamic complementarity as between whole and part (which are now qualitatively distinguished from each other). In holistic mathematical terms this corresponds with 4-dimensional understanding. Here both real and imaginary distinctions are made with both positive and negative directions. So in any context, if for example the "part" is designated as real (in conscious terms) then relatively the "whole" is designated as imaginary (i.e. pertaining to unconscious meaning).

So the Integral 2 approach allows fully for the necessary interaction of conscious and unconscious in experience; in physical terms this corresponds with the interaction of discrete phenomena with an underlying holistic ground (or field) of matter.

The third approach (Integral 3) then allows for further dynamic complementarity as between form and emptiness which in holistic mathematical terms corresponds with 8-dimensional understanding.

Such understanding is based on the four complex roots (of the 8 roots of 1) where both real and imaginary aspects are equal. The fascinating point here is that these equally constitute null lines (with no magnitude). So in corresponding qualitative terms, these allow for the ultimate unification of form and emptiness (with the complex relating to form and the null designation to emptiness respectively).

In psychological terms this would entail that the interaction as between real (conscious) and imaginary (unconscious) would now be - literally - so equal or harmonious that phenomena (of form) would not even appear to arise in experience thus facilitating union with emptiness.

Friday, December 18, 2009

Integral Theory of Everything

We hear a lot these days in scientific circles regarding the search for a TOE (Theory of Everything). For example some people that developments with respect to String Theory could lead to a TOE (possibly in the near future).

Personally I believe that such optimistic views are quite unfounded and that a TOE within the accepted confines of Conventional Science is simply not tenable.

Current science is still of a highly reduced nature.

Once again I will place this in the context of my own approach. Potentially, we can identify an infinite number of qualitatively different dimensions of understanding. Each corresponds to a unique holistic interpretation of number through which reality can be validly interpreted. However, Conventional Science uses just one of these dimensions i.e. the 1st corresponding to linear rational understanding.

As we have seen this default linear model of interpretation is not able to preserve the qualitative distinctions pertaining to the key polarities that necessarily underline all scientific understanding.

For example all knowledge of reality involves a dynamic relationship between the (subjective) knower and the (objective) known. Conventional science can only accommodate such a relationship either by effectively ignoring the subjective component altogether or alternatively reducing it in merely objective terms.
As we have seen the solution to this problem requires the recognition of complementary - as opposed to merely separate polarities - ultimately leading to the need for an alternative type of circular logic.

Secondly all knowledge of reality involves a dynamic relationship as between whole and part. (The psychological counterpart of this is the need for both perceptions and concepts). Once again Conventional Science can only deal with this relationship through effectively reducing - in any context - the whole to the part. Indeed, strictly speaking this very problem lies at the root of all mathematical proof!
However in truth the whole is qualitatively distinct from the part. Recognition of this fact requires incorporating imaginary as well as real rational notions. In effect this requires that both conscious and unconscious aspects of understanding be formally recognised in interpretation leading to - what I refer to as - the complex rational approach.

Finally all knowledge entails a dynamic relationship - in the most fundamental manner - as between form and emptiness. Again Conventional Science can only deal with such a relationship through effectively reducing - in any context - emptiness to form in the mistaken view that somehow spiritual type awareness can be screened out of scientific interpretation.

Though the spirit cannot be directly interpreted in a phenomenal manner, indirectly more subtle ways of rational interpretation are indeed available to better accommodate the relationship as between form and emptiness. In my own approach this is achieved at the level of 8-dimensional interpretation.

Of course I do not deny the continuing great value of Conventional Science and the enormous contribution that it has made. However I think that it is still perfectly valid to attempt to place it in a significantly wider context. This points clearly to its key strength and limitations while equally opening up entirely new perspectives for valid scientific enquiry. And, make no mistake, this will indeed be more clearly recognised in future.

So an acceptable scientific TOE is not - I believe - possible with the present understanding of science.

However I do believe - that in a certain qualified sense - a TOE is indeed possible within the integral scientific approach (which I am outlining in these contributions).

Teilhard de Chardin

Back in the mid 60's the two greatest intellectual influences on my development were Albert Einstein and Teilhard de Chardin.

What I subsequently found fascinating was the degree to which they were very close contemporaries. De Chardin was born in 1881 (two years after Einstein). However they both died in April 1955, within 8 days of each other, in the US (which had become their new adopted homeland). Indeed even geographically they were very closely united at the end with de Chardin dying in New York and Einstein just 50 miles away in Princeton.

Though in my own person a raging internal debate was going on regarding their respective philosophical standpoints, I am not aware of any substantial contact taking place between the two men during their own lifetimes.

What is similar in the work of both is a great drive towards understanding the unity governing all creation.

However though Einstein was certainly inspired by deep wonder and intuitive insight (of a genuine religious kind) he always sought to understand the laws of nature as if somehow separate from the enquiring mind. This is the underlying principle which drives the classical approach to science. Indeed one could validly argue that - despite the counter intuitive findings from quantum mechanics and elsewhere - that this is still the guiding principle of modern physics in the search for a merely "objective" solution as the Theory of Everything.

However - though a great scientist in his own right - de Chardin's standpoint is quite different. For him there is always an irreducible psychic - as well as physical - aspect to matter. And the subsequent story of creation is the manner in which both interact through evolution. De Chardin believed that increasing complexification of matter reflected the growing influence of the psychic aspect (bringing a capacity for greater organisation). Ultimately with the unfolding of evolution this psychic aspect was set to become ever more prominent leading eventually to an Omega point in creation where all matter would become transformed in spiritual energy.

Now, I would not necessarily agree on the details of de Chardin's perspective. However that is not the key point.

For Einstein the task of unification is to understand the immutable physical laws governing nature. The proposed answer here is in the form of a (detached) rational theory. So, the relationship of the knower to what is known is either effectively ignored or explained in merely reduced physical terms.

However for de Chardin the task of unification is to understand the key relationship as between both the physical and psychic capacities of matter (which are interdependent). Though this approach certainly does not deny the validity of rational understanding - in fact it requires a much more refined use - the proposed answer is ultimately found through authentic contemplative experience. Here the knower (as subject) and what is known (as object) are transformed into their common identity as pure emptiness (in spirit).

Though it is continually ignored in conventional science, the key issue remains of the relationship of the knower to what is known.

Though once again the findings of quantum physics spell the death knell for the merely reductionist view, this still remains the overriding dominant perspective in our culture.

Meanwhile truly vast territories of new scientific understanding with potential riches to significantly transform our whole relationship with the environment remain totally unexplored.

Thursday, December 17, 2009

Surprising Connections

Evelyn Underhill published her classic book "Mysticism" in 1911. This is interesting - for what might initially seem an entirely unconnected reason - in that neatly divides the period as between Einstein's "Special Theory of Relativity" in 1905 and his "General Theory of Relativity" in 1916.

In his earlier contributions Einstein particularly focused on the nature of light (electromagnetic energy) and its intimate connections with spacetime. In the latter he probed the mysterious nature of gravity.

Then for the final 35 years or so of his life he turned his attention to the task of properly integrating these two forces in a unified field theory providing the basis for understanding the whole of nature.

As we have seen it is the nature of the integral approach that it establishes clear complementary connections - ultimately in holistic mathematical terms - as between all key physical concepts and their (unrecognised) psychological complements.

So electromagnetic energy and gravity have their psycho spiritual complementary expression equally as the fundamental motivational (or volitional) forces giving purpose to all phenomenal activity. And as we have seen they have been long recognised in the mystical literature as the immanent and transcendent aspects of spirit respectively.

Now why Underhill's work is of considerable interest in this context, is that she probed deeply into process by which unification of these two forces eventually is attained.

Her thesis was quite simple in that she believed that initially the quest for ultimate unification came through the relative dominance of one of these forces (constituting a distinct personality type). So basically she divided aspirants (to ultimate unity) as corresponding either to devotional (immanent) or metaphysical (transcendent) charcteristics.

Indeed she gives many fascinating portraits of these types. It is also interesting - even though she does not explicitly make the distinction - that they can be easily classified under four force headings (allowing for both external and internal expressions in both cases).

So firstly we have those conforming to the (external) immanent type that would be directly associated with the psycho-spiritual expression of the electromagnetic force.

These are often identified as nature mystics who literally see worldly phenomena bathed in a new spiritual light. One good example, which she gives, is the poet William Blake; another is the American Walt Whitman. A recent more modern example of a cognitive - rather than affective kind - is provided through Teilhard de Chardin (who understood the evolution of creation as its gradual transformation towards pure spiritual energy).

Secondly we have others conforming to the (internal) immanent type that would be directly associated (in psycho spiritual terms) with the electroweak force.
Madame Guyon is an oft quoted - by Underhill - extreme representative of this type. She possessed a very unstable emotional temperament leading to frequent shifting states (very reminiscent of the nature of radioactive decay in matter).

More balanced representatives of this emotional type (also mentioned by Underhill) would include Teresa of Avila, Catherine of Genoa and Henry Suso. Initially this type tends to be unduly preoccupied with interior devotional states (thus posing a barrier to true spiritual integration). However in the process of contemplative purification such a personality limitation is overcome to a significant degree.

Thirdly we have the (external) transcendent type that would be directly associated with the psycho spiritual equivalent of the gravitational force. Now getting clear examples here can be difficult as usually both external and internal aspects exhibit themselves to a marked extent!
However Meister Echhart would fit this category due to his clear understanding of the unconditioned dark lying as the source of all created life.

In a less mystical - and more intellectual - sense Einstein himself and the philosopher Hegel would also conform to this type. In this context the intellectual focus is very much on obtaining a coherent global synthesis of the structure of reality (in the very manner that gravity itself provides such structure in global physical terms).

Finally the fourth (internal) transcendent type is very well exemplified (in extreme form) by St. John of the Cross representing the psycho spiritual equivalent to the strong force. With such types the most severe degree of interior purgation is required before contemplative union can be achieved. This leads to a remarkable degree of faith that - literally - binds the personality strongly together.

However perhaps the most important for our purposes here is the Flemish mystic Ruysbroek (who Underhill especially admired).

Almost uniquely among recognised Christian contemplatives, he combined both immanent and transcendent aspects in a wonderfully inspiring manner (thereby giving perhaps the best indication of the true nature of union).

In physics the state where all forces are united as one is often referred to as the supersymmetry of forces. In similar fashion Ruysbroek refers to the highest stage of the contemplative life (where the spiritual union of forces is obtained) as the Superessential Life. He speaks of this - ultimately - ineffable state using some of the most wonderful, joyful and truly inspiring language ever written.

Perhaps we could briefly attempt to summarise this in paradoxical terms as the dazzling darkness (i.e. the intersection point of immanence and transcendence both internally and externally).

So just as this represents the psychological source (and end) of all life, in complementary fashion the source (and end) of all physical life is represented by the intersection point of the four forces where all are united as pure emptiness (yet potential for all life).

And remarkably it is in the same contemplative experience of union that both sets of forces (physical and psychological) are united in complementary fashion as pure emptiness (and potential for creation of all phenomenal form in both a physical and psychological manner).

Spiritual Forces

As we have seen, it is the every essence of the integral approach to find - in any given context - the complementary notion (with which it is intimately related). So with Integral Physics the task is to find - for any established physical concept - a matching psychological partner (which is mathematically identical in terms of its holistic qualitative nature).

So we are now looking for the matching psycho spiritual counterparts to the four physical forces (already defined in holistic qualitative terms).

In physical terms the forces are essential to explain phenomenal notions of movement.

In corresponding spiritual terms, complementary forces are required to explain phenomenal notions of psychological movement.

This is what we refer to as (fundamental) motivation i.e. that volitional sense of purpose (desire for meaning) which serves as a precondition for all psychological understanding.

Though ultimately there is just one force in this sense, with all subsidiary representations expressive of the same meaning, in phenomenal terms we can identify four that are equivalent in complementary manner with their physical expressions.

We have already talked about spiritual light and spiritual gravitation in other contributions.

In the mystical literature, a distinction is made between the immanent and transcendent expressions of the spirit. These relate in psychological terms to the electromagnetic and gravitational forces respectively.

The very goal of the immanent aspect is to see light as revealed through phenomenal form; the corresponding goal of the transcendent aspect is to go beyond all phenomenal form (in darkness) in pursuit of a pure hidden light.

Thus in terms of the spiritual journey, the illuminative stages would be more associated with the immanent aspect and the purgative stages with the corresponding transcendent aspect.

Once again these two aspects are real and imaginary with respect to each other. When the spiritual light (as the immanent psychological expression of electromagnetic energy) is made conscious as real, the corresponding transcendent expression (as psychological gravity) remains unconscious (as imaginary).

In reverse fashion when the transcendent is made real, the immanent remains imaginary;

Likewise as in the case of the physical forces we treated both the weak and strong as the negative of the electromagnetic and gravitational respectively, likewise we can give both external and internal expressions to immanence and transcendence respectively. In the former case one identifies experience mainly through relationship to the world; in the latter case it is mainly with refernce to the personal self.

The implications of all this is that we are led to see an intimate complementarity as between both the physical forces (motion) and spiritual forces (motivation) each of which can be giving exactly matching holistic mathematical interpretation (in qualitative terms).

So if we for example identify the real particle aspect of spiritual light in a phenomenal context (as immanence) then the corresponding wave aspect remains imaginary (as unconscious); and vice versa.

Likewise there is a null sense to such light in that - from its reference frame as pure spirit - it represents pure actualisation (of phenomenal existence).

And once again we cannot hope to understand its manifestations (immanent and transcendent) with respect to form without equal recognition - in terms of itself - of its purely empty nature.

Ultimately it is in the same contemplative experience of pure union, that both the physical forces and the spiritual forces are together united.

We cannot therefore in direct manner attempt to understand the ultimate integration of either the physical (or spiritual) forces through mere intellectual interpretation. Such integration is experientially realised in pure mystery. However appropriate intellectual appreciation can indeed indirectly act as an important catalyst for this experience.

Wednesday, December 16, 2009

Nature of Light

As suggested in the last contribution, the holistic mathematical nature of each of the physical forces is given by the four complex roots of unity.

If we take the first of these roots it is in the form 1/k(1 + i) where k represents the square root of 2, and i the square root of - 1 respectively.

In holistic terms 1 denotes the real manifestation of (unitary) form and i the corresponding imaginary manifestation.

Now it is the very nature of light that it can reveal itself - phenomenally - in either wave or particle form.
So when for example the wave aspect is manifest in experience as real, then the corresponding particle aspect remains imaginary; likewise when the particle aspect is manifest as real, then the wave aspect is imaginary.
So both wave and particle aspects of light keep switching as between a real and imaginary identity depending on which one manifests itself (according to a chosen observational context).

However what is fascinating about the complex roots of unity is that they can be equally shown (in geometrical terms) to represent null lines with magnitude = 0.

Thus we have an equally important definition of light as pure emptiness (which in dynamic terms represents mere potential for existence).

As is well known in terms of its own frame of reference, light does not travel in finite time and remains continually in the present moment.

Furthermore, these two interpretations of light - both in terms of complex form and simple emptiness respectively - are interdependent. In other words the actualisation of light as form has no meaning in abstraction from equal appreciation of light as mere empty potential for existence.

Finally the presence of the square root of 2 in the root suggests that irrational (i.e. paradoxical) understanding is necessary to appreciate the nature of light.

The square root of 2 symbolises above all else what is irrational (in algebraic terms). And the appreciation of the wave/particle duality of light requires the paradoxical appreciation of its complementary nature.

What is truly remarkable is that we can then in a holistic mathematical manner interpret the qualitative nature of the other physical forces in like manner.

As we have seen the relationship of electromagnetic energy (of which light is a key manifestation) to gravity is that of real to imaginary.

Therefore if 1/k(1 + i) is the formulation given to the electromagnetic force, then
1/k(- 1 + i) is the corresponding formulation - in relative terms - of the gravitational force. So once again it can manifest itself in either wave or particle form with an equally important formless i.e. empty null identity as mere potential for existence of gravity.

Then the electroweak force will be expressed - relatively - as the negative of the electromagnetic i.e. 1/k(-1 - i) and the strong as the negative of the gravitational i.e. 1/k(1 - i).

So the holistic mathematical formulation clarifies the exact nature of the forces in qualitative terms (and their corresponding relationship to each other).

It also strongly suggests how they are unified (through their null identities).

However more about this anon!

Fundamental Forces - physical and psychological

When Einstein was seeking his Unified Field Theory only two forces were involved i.e. the electromagnetic and gravitational.

Subsequently two additional forces have been recognised i.e. the electroweak and the (strong) nuclear.

When one looks at it, these appear in holistic mathematical terms very much as internal counterparts to the other two forces (which operate extensively throughout nature).

Indeed the weak force (which for example can be used to explain the radioactive disintegration of certain atoms) offers itself as the internal (negative) counterpart to the electromagnetic; likewise the nuclear force readily suggests itself as the internal (negative) counterpart to the gravitational.

I have already drawn attention to the natural complementary relationship as between the electromagnetic and gravitational forces. Whereas we would - literally - see the first as "light" the latter by contract would be "heavy".

In fact the holistic mathematical relationship as between electromagnetic and gravitational is as real to imaginary. Now it is postulated that both forces can phenomenally reveal themselves as waves or particles (which in turn are real and imaginary with respect to each other). In other words when one aspect - say wave - reveals itself as real the other remains imaginary (and vice versa). This would further suggest that when the real aspect of one force is manifest that the alternative aspect of the other force is likewise made phenomenally manifest. So if the wave aspect of the electromagnetic force is real (i.e. manifest) then the particle aspect of the gravitational is likewise made real. (And both are deeply interdependent in this manner!)

Now it is fascinating that the very words popularly used to describe the two new i.e. weak and strong also suggest a complementary pairing. Indeed women have long been described in the cultural stereotype of the feminine principle as the "weak" sex (relating to intuition and the unconscious); men then by contrast are viewed as "strong" (relating - apart from physical attributes - to reason and the conscious mind). So the Western cultural stereotype values conscious reason (as "real") above unconscious intuition (as "imaginary").

So in fact the relationship of the two new forces weak and strong is likewise as real and imaginary with respect to each other (with both in turn representing the negative expression of the electromagnetic and gravitational forces respectively).

In this way we can provide a wonderfully simple mapping of the holistic mathematical relationship of the four forces to each other through the 4 complex roots of the 8 roots of unity.

Once again in holistic mathematical terms it is easy to see the reasons for Einstein's difficulties in unifying the gravitational and electromagnetic forces and indeed Quantum Mechanics with General Relativity.

Due to the reduced philosophical nature of conventional scientific interpretation, the whole - in any suitable context - cannot be properly distinguished from the part. So effectively wholes are treated simply as "aggregate" parts (without any proper qualitative distinction being maintained).

So the relationship of Quantum Mechanics (dealing with minute parts) to General Relativity (dealing with the whole cosmos) is as real to imaginary.

However in relative independence from each other, both Quantum Mechanics and General Relativity can be given a real scientific interpretation (in the conventional sense).

However in relation to each other this breaks down. So if Quantum Mechanics is treated as "real" then General Relativity is "imaginary"; likewise if General Relativity is "real" then Quantum Mechanics is "imaginary".

What this actually implies that we cannot properly incorporate both aspects without including holistic (qualitative) as well as analytic (quantitative) understanding.

In other words, comprehensive scientific interpretation requires, in qualitative terms, a complex rational approach that can successfully combine both real (analytic) and imaginary (holistic) elements.

Tuesday, December 15, 2009

Integral Science - holistic mathematical nature

In Conventional Mathematics both real and imaginary numbers are used with respect to their (merely) quantitative interpretation.

However the key starting point of Holistic Mathematics is the realisation that every mathematical symbol can also be given a corresponding qualitative meaning.

The limitation therefore of Conventional Mathematics is that it is confined in qualitative terms to merely real understanding (corresponding to default one-dimensional interpretation).

The role of Holistic Mathematics is to provide corresponding imaginary interpretation in qualitative terms.

Whereas real interpretation corresponds directly with conscious, imaginary interpretation - by contrast - corresponds directly with unconscious understanding. Once again the real aspect relates to linear logic (where opposite polarities in experience are clearly separated) whereas the imaginary aspect relates in turn to circular logic (where such opposite polarities are treated as complementary).

Thus when we separate two polarities (to concentrate on one as independent) we have linear understanding; however when we combine these two polarities as complementary (to concentrate on both as interdependent) we have (two-dimensional) circular understanding. Though in direct terms such circular appreciation is of an intuitive nature, indirectly it can be given a reduced (linear) rational expression (corresponding to the qualitative square root of interdependent unitary form). And this is imaginary understanding.

So in this way the qualitative nature of imaginary understanding corresponds closely with its quantitative counterpart.

In like manner whereas Conventional Science provides the real aspect of understanding , Holistic (i.e. Integral) Science provides the corresponding imaginary component.

So - from a qualitative holistic mathematical perspective - in these contributions on Integral Physics, I am attempting to provide the vital imaginary aspect of qualitative understanding (that is ignored entirely in the conventional scientific approach).
Such an approach requires that one establishes - for example in any physical context - a complementary psychological interpretation. Through realising such complementarity in experience, one can intuitively unite both aspects (in a true contemplative manner).
Thus if we go back to the previous contribution, it deals with the complementarity - in qualitative terms - of both the physical notion of the "black hole" and the corresponding psycho spiritual notion of the "dark night". Now it is important to stress that such a relationship cannot be - literally - seen without adopting a distinctive type of understanding (that is not provided through conventional scientific appreciation!)

From a true integral scientific perspective the ultimate goal of understanding is - not to attempt to know about reality but rather - to experientially become one with reality in an ineffable manner. So reality is truly utterly mysterious with any theoretical understanding - necessarily - of a limited and partial nature.

One advantage of the integral approach is the manner in which it enlarges one's philosophical perspective to clearly see that the quest for any final TOE (as with string theory) is futile. Of course exciting new discoveries can indeed be made in this way, and we should always rightly strive to know more; however the quest for any final (merely objective) reduced explanation of reality is not possible.

Of course a truly comprehensive approach to science would employ - in qualitative terms - a complex rational approach combining both real (analytic) and imaginary (holistic) understanding.
However once again in these contributions I am confining myself to the integral (i.e. imaginary) aspect which is so greatly neglected in present scientific understanding.

Sunday, December 13, 2009

Black Holes and the Dark Night of the Soul

As we have seen, from a psychological perspective a sudden acceleration in the customary speed of interaction between polarities of experience (such as internal and external) leads to a dramatic increase in psychological gravity.

In common language this represents an experience of grief which in the spiritual contemplative life is identified as a purgative period for the soul.

The higher stages of development (defined by their unique number dimensions) therefore represent progressively greater interaction i.e. psychological velocity as between opposite polarities (associated in turn with purer experience of the spiritual light). The transitions as between such stages thereby requires an acceleration with respect to previous interaction (associated again with purgative phases of development).

The most profound of such purgative phases is often referred to as the Dark Night of the Soul. In truth there can be many "dark nights" on the spiritual journey. However the most dramatic - which is referred to as "the passive night of spirit" by St. John of the Cross - would generally bridge the transition between 2-dimensional and 4-dimensional understanding.

St. John provides a particularly intense account of such a "night" in his writings (no doubt based directly on his own experience).

It would perhaps help initially to clarify what the term "passive night of spirit" actually entails.

First of all St. John distinguishes as between "active" and "passive" nights. Now active in this context relates to phenomena of a directly conscious (i.e. linear) nature.

However passive more properly relates to indirect circular paradoxical phenomena that tend to be very refined and fleeting which mediate the pure intuitive light. Now during the first major period of illumination (corresponding to 2-dimensional understanding) this refined type of understanding would be especially prevalent.

However the problem that arises in the spiritual life is that attachment can build up with respect to such phenomena (leading to a certain rigidity in their use) thereby dimming the true quality of the light.

So for pure contemplative union writers such as St. John would prescribe an intense period of further purgation whereby remaining attachment with respect to even these subtle phenomena (as mediators of spiritual light) must be largely eradicated. Therefore there is need for a "passive night of spirit". "Night" of course symbolises darkness rather than "light" which in a psycho spiritual context implies deep immersion in the unconscious regions of personality.

Therefore because such purgation relates to the removing of remaining linear attachments with respect - to what is inherently - circular type appreciation, it leads to consequent extreme curvature in the psychological experience of space and time. Here for a considerable period of time no light can thereby escape! Also there is intense psychological suffering involved largely relating to a profound degree of psychological gravitation (i.e. grief). During such a time it seems as if the whole world - as it were - becomes drawn inside one's internal psyche causing an immensely suffocating feeling.

What we have here therefore in this experience of the "dark night" is the psychological counterpart to the black hole. The black hole likewise is associated with extreme curvature in space and time due to the enormous effects of physical gravity brought about by a burnt out star. So just as the most intense form of the spiritual "dark night" is reserved for dynamic personalities who have already shone as spiritual "stars" (through extraordinary spiritual illumination), likewise black holes are associated with particularly dynamic and immense forms of physical stars.

Likewise, just as with the "dark night" where all psychic matter in one's immediate experience gets sucked inwards in experience, likewise with the black hole where matter in its vicinity likewise gets sucked in towards the hole's centre.

Thus form a holistic qualitative perspective the key structural characteristics of the (psychological) dark night and the (physical) black hole are complementary.

Once again this implies that for appropriate qualitative understanding of physical reality that both a linear and circular aspect must be employed.

This realisation in fact has far reaching implications. Spiritual writers acknowledge that the darkness experienced during the spiritual night in fact relates to a hidden form of light (that remains unseen).

Ultimately pure darkness (relating to pure psychological gravity) is of an inherently empty nature (without phenomenal characteristics).

In complementary fashion this entails that the holistic nature of physical gravity is likewise empty (as is similar with electromagnetic energy).

In psychological terms we identify pure darkness in unconscious terms with gravity (the unseen light) and then the seen light with its pure conscious manifestation.

Now the relationship of conscious to unconscious is of real to imaginary in holistic mathematical terms.

Therefore the corresponding relationship of electromagnetic energy and gravity (in both its physical and psychological manifestations) is also of real to imaginary.

The important consequence of this is that the unification of these two forces requires a complex rational approach combining both real (analytic) and imaginary (holistic) components.

Therefore such unification cannot be properly attained using conventional scientific method (which is qualitatively geared solely to real interpretation).

This perhaps explains why Einstein's subsequent attempts to obtain a unified field theory (consistently combining both the electromagnetic energy and gravitational forces) was ultimately doomed to failure. We shall return to this in a later contribution.

Saturday, December 12, 2009

Curved Spacetime

The next significant breakthrough that Einstein was to make was the realisation that space and time becomes curved in the presence of gravity.

As however gravity is an especially weak force in normal circumstances this curvature is so small as to be undetectable. However in the presence of matter with a substantial degree of mass the gravity force can exercise a significant degree of influence in warping surrounding space time. Einstein also postulated that gravity would cause light to bend in the vicinity of such mass. Indeed Rutherford's experimental verification of this in 1919 was accepted as proof of Einstein's General Theory thus paving the way for universal acclaim.

Once again there is a fascinating holistic correspondent to the curving of spacetime. Not only is this of interest in its own right but ultimately it throws considerable light on the true relationship of the electromagnetic to the gravitational force (demonstrating why a fully unified field theory is untenable in conventional scientific terms).

From the accepted physical perspective gravity arises in the presence of matter (with the actual force exercised depending on the degree of mass involved).

Corresponding appreciation of the holistic nature of curved spacetime arises from the complementary experiential loss of matter (with again the actual force experienced depending on the psychological weight i.e. mass involved).

As we have seen grief is synonymous with the psychological notion of gravity. And grief is experienced in human terms through the loss of what - literally - matters most to us e.g a loved one, a job, health etc.

The question then arises as to how the profound experience of grief can substantially curve psychological spacetime!

The key here is to recognise that the customary linear understanding of spacetime (resulting in absolute notions) is based directly on a rational interpretation (corresponding to conscious understanding).

However as we know from a psychological perspective, the conscious is in continual interaction with the unconscious mind. And whereas the conscious corresponds directly with analytic type appreciation of reality, the unconscious corresponds with holistic appreciation.
Furthermore just as conscious (analytic) understanding is of a direct linear nature (where the polar opposites of experience are separated) unconscious (holistic) understanding is of a direct circular nature (where the polar opposites are experienced as complementary).

So for example following the death of a loved one, grief is the typical reaction. This intensification in psychological gravity arises as the direct consequence of the personal loss experienced. This in turn leads to consequent loss with respect to a truly important set of phenomenal relationships (that thereby carry special weight). So we have therefore a considerable death with respect to psychic matter. This changes the focus inward in an unconscious search for holistic meaning.

And it is this holistic quest (relating to circular understanding) that therefore causes curvature with respect to the psychological experience of space and time.

Though difficult to bear, grief therefore can play an important role in the authentic development of holistic meaning as opposed to specific understanding of a conscious kind.

However because of the complementarity of the physical and psychological aspects of experience this explanation of the psychological nature of curved spacetime (entailing, both conscious and unconscious aspects) has necessarily a physical counterpart.

We cannot properly understand physical reality without incorporating in a qualitative manner both linear and circular type understanding.

The problem once again with the conventional explanation of curved spacetime in Einstein's approach is that it is confined in formal terms to a rational (linear) interpretation.

In other words though Einstein clearly realises the curvature of spacetime in quantitative terms, he does not recognise that this equally applies to the very mental constructs we use to interpret this reality.

Once again the standard default interpretation (i.e. 1-dimensional) that Einstein employed to the very end of his life strictly does not allow for any curvature of spacetime in a qualitative sense. Such curvature only begins to occur at higher dimensions (e.g. 2-dimensional) that we have looked at in previous contributions.

Indeed proper appreciation of the true nature of curved spacetime (in qualitative terms) provides the key to understanding why Einstein's subsequent search for a unified field theory was ultimately doomed to failure. We shall return to this important issue later!

Thursday, December 10, 2009

General Relativity - holistic equivalence principle

Though path breaking in several respects, Special Relativity was limited in scope in that it did not incorporate the effects of gravity.

So Special Relativity in quantitative physical terms is based on relative comparisons of phenomena travelling at a constant velocity.

In corresponding qualitative psychological terms, Special Relativity is based on relative comparisons of phenomena that are interpreted within given stages of development (defined - as we have seen - by their dimensional numbers).

Einstein made an important breakthrough in his attempts to incorporate gravity in a more general theory when he realised in a moment of clear insight that - what we call - gravity is inseparable from the effect experienced through accelerated movement.

So for example when a car suddenly accelerates in speed one can feel a force pinning one to the seat that is identical to that of gravity.

This unlocked an important key for Einstein. He saw that he could include gravity in a General Theory by extending his findings regarding the relative nature of space and time in the Special Theory to now include accelerated - as well as constant - notions of velocity.

Once again a corresponding holistic qualitative extension exists that has important implications.

It took me some time to unlock this alternative explanation to my satisfaction. However close assocation with the work of the Spanish mystic, St. John of the Cross was eventually to prove rewarding.

St. John in his works deals with the higher stages of development specifically from a spiritual perspective.

For one following an authentic contemplative path, stages of illumination (when one is bathed in a new spiritual light) inevitably tend to be associated with corresponding stages of purgation (where one can become plunged in deep spiritual darkness).

What is surprising about St. John's approach however is that he places little emphasis on the value of the illuminative stages for authentic progress but rather on the dreaded "dark nights" that must be so painfully endured.

Indeed he is rather scathing of the faults of "proficients" who would already have reached a relatively advanced stage of development for - what he considers - an often mistaken attraction to the spiritual light breeding all kinds of spiritual imperfections.

Concentrating further on the precise reason for this problem can give a clear insight into the corresponding holistic counterpart of Einstein's equivalence principle!

Usually a new stage of illumination follows a previous stage of trial and darkness. During this difficult preceding period the spiritual light - while remaining hidden - incubates perhaps for a lengthy period in the unconscious. Then when the time is right - often in a dramatic new moment of conversion - it then becomes released in a conscious manner through new intuitive illumination. However as one dwells in this new light, it gradually interacts with phenomena. And inevitably as phenomenal life is restored (initially in a wonderful manner), secondary attachment sets in. This growing rigidity then leads to a gradual reduction in the dynamic interaction of polarities in experience. This can be accurately described as a decrease in the psychological velocity of understanding which is directly equivalent to a reduction in (psychological) gravity.

So the important point to register here is that just as - in holistic terms - (physical) light has a complement in (psycho-spiritual) light, likewise (physical) gravity has its corresponding counterpart in (psycho-spiritual) gravity.

As we know the force of (physical) gravity is much lower on the Moon than on Earth. So for someone on the Moon's surface, it is very difficult - literally - to keep one's feet on the ground.

Understood in appropriate fashion an exactly similar psychological explanation likewise exists (of which St. John of the Cross was keenly aware).

As phenomenal attachment increases (during the illuminative stages), psycho-spiritual gravity itself starts to decrease. When we think of it, this is even recognised in popular language usage. For example we might refer to a person as having "gravitas" which implies a degree of interior depth. However when the spiritual light shines, the focus of attention tends to become somewhat externalised. One then tends to develop more superficial interests thus lessening this sense of gravitas. This is even applicable to whole nations. For example Ireland spent a period of 15 years during the famed "Celtic Tiger" bathed as it were in a seemingly unending economic light. Unfortunately this subsequently led to a dramatic increase in the superficiality of people's concerns through growing materialism and selfishness that now - in deep recession - is proving very difficult to address.

The key problem that St. John points to - in the exalted context of spiritual "proficients" - is the danger of pride surfacing during the illuminative stages. Now pride is the direct opposite of humility which - literally - refers to a deep grounding in (true) reality. So when pride takes over, one loses the ability to keep one's feet (psychologically) on the ground thereby becoming prone to all kinds of illusions. Once again using the Celtic Tiger analogy, as a people we - quite literally - got "carried away" (due to a considerable reduction in psychological gravity) during the boom years with consumer behaviour reaching manic proportions.

So just as St. John would associate the illuminative stages with a decrease in psychological gravity (leading to the danger of pride) he would see the corresponding purgative stages as vital for restoring necessary balance. Here one can become properly grounded again in reality through a spirit of true humility. (Likewise in economic terms a severy recession may be necessary to cure the manic psychological behaviour of the preceding boom!)

Indeed this helps to explain why the onset of the purgative stages can be so disconcerting. Because of the increased speed of interaction with respect to opposite polarities in experience (representing a corresponding increase in spiritual gravity) former attachments are revealed as key impediments to progress. And as these attachments are painfully eroded, one experiences a deep sense of loss.

Now when one experiences loss it leads to the feelings of grief. And here in the word "grief" we have the direct psychological counterpart to the physical notion of gravity.

So becoming established in higher dimensional stages of development requires a preceding period of purgation (through which dynamic interaction i.e. velocity as between opposite polarities in experience greatly accelerates). Not surprisingly such purgative periods are intimately associated with deep feelings of grief (in the experience of psychological gravity).

Thus each higher stage of development is thereby more dynamic i.e. characterised by a greater speed of psychological interaction than its predecessor. In the limit where such speed approximates its maximum, phenomena no longer even appear to arise in experience with the pure light of Spirit now solely remaining. And such a state of pure union is the ultimate goal of the contemplative life!

Monday, December 7, 2009

E = MC2 (Holistic Interpretation)

Einstein's famous equation demonstrating the equivalence of mass and energy arose out of his reflections on Special Relativity.

So physical energy and matter are intimately related to each other.

From an integral perspective, the question then immediately arises as to the relationship between spiritual energy and (our cognitive experience of) phenomenal form.

As we can see from the physical formula, the square of the speed of light is directly involved, with the speed expressed in two-dimensional terms.

Now if we examine this equation from a holistic perspective, two-dimensional understanding entails the structure corresponding to the two roots of unity i.e. + 1 and - 1 respectively.

The implication here is that for physical energy to be fully released from matter its opposite polarities must cancel each other totally. In physical terms, this can be expressed as the situation where matter (+) and anti-matter (-) annihilate each other.

There is a remarkable complementary explanation available in psychological terms.

Here the actual existence of phenomena requires that they be - literally - posited temporarily in experience (without corresponding negation taking place).

When experience is especially rigid, little spiritual energy i.e. intuition is generated. Put another away little dynamic negation (relating to the holistic contribution of the unconscious) is involved.

As however experience becomes more spiritually refined (through contemplative development), successful dynamic negation occurs to a considerable extent. So as soon as a phenomenon is posited, a rapid switch as between internal and external polarities takes place in experience. So, two linear reference frames are now entailed in the cognition of phenomena i.e. (external) object in relation to - relatively - (internal) perception and vice versa. Then from a dynamic circular perspective these frames are understood in holistic terms as paradoxical, the resolution of which generates spiritual energy. And such energy i.e. intuition is necessary to fuel all cognitive dynamics.

In pure contemplative experience, the dynamic negation of phenomena is so rapid that they do not even seem to arise in experience. Put another way it approximates closely to the total annihilation of psychic matter through interaction with psychic anti-matter.

Fascinatingly this also implies a clear recognition that the very notion of the speed of light itself becomes redundant (in the context of light itself).

In other words the unambiguous notion of light having a speed, only applies in the context where (rigid) phenomena are related to light. Though external and internal polarities necessarily interact, registered speed takes place with respect to each (as independent). However clearly in holistic terms, where both directions are understood as interdependent, the speed of light becomes totally paradoxical. Thus in terms of itself, light does not move (and this is why time does not pass for light).

In complementary fashion when one enters into the spiritual light (through pure contemplation) one is then - by definition - fully at rest (without movement).

So the very notion of movement - ultimately with respect to light itself - is but an illusion (due to a failure to recognise its real polar frames of reference as fully complementary)!

So just as in physical terms, from a psychological perspective phenomenal mass and spiritual energy are but two sides of the same coin.

Once again with especially rigid understanding (where the linear aspect greatly predominates) little spiritual energy will be in evidence.

At the other extreme where pure contemplation is entailed, the linear aspect becomes almost entirely negated with respect to the circular aspect (which in turn requires the continual dynamic negation of the linear).

One can perhaps thereby appreciate how unbalanced conventional science is from the psychological viewpoint.

In formal terms it is based purely on the recognition of the rational (linear) aspect of understanding.

As I have outlined repeatedly however, both linear (rational) and circular (intuitive) aspects are necessarily involved in all experience and the true task is to show how these interact in scientific understanding.

There is another interesting aspect.

Einstein's formula tells us that a great deal of energy can be created from a small amount of matter.

In like manner a great deal of spiritual energy can be created from a small amount of phenomenal matter (in experience).

This perhaps explains the traditional monastic approach to the contemplative life, where severe restrictions were generally placed on the range of phenomenal experience open to participants. Thus by continually using restricted phenomena as a focus for spiritual concentration, radical transformation (with respect to their qualitative nature) could be more easily obtained.

However there is a great danger that the pure contemplative life could thereby become unduly narrow and largely detached from popular concerns.

This is why I would see contemplation as representing but a particular specialisation (with respect to intuitive appreciation) which ultimately should be fully incorporated in a balanced manner with the world of phenomenal form (i.e. through radial activity).

Sunday, December 6, 2009

More on Special Relativity

There is a famous quote by Albert Einstein on Relativity.

"Put you hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour and it seems like a minute. THAT'S Relativity."

I have always found this an interesting quote - not so much for its instantly accessible message that can be readily appreciated by anyone - but rather for the hidden problem that it raises (which Einstein does not deal with in his work).

Put briefly, this is a statement about Relativity - indeed about Special Relativity - which relates specifically however to its (qualitative) psychological rather than its (quantitative) physical meaning.

And as Einstein's Theory solely relates to the physical meaning, his clever quickfire illustration here in fact relates to a much deeper issue (which he does not address).

In other words Relativity - in this context Special Relativity - can be given both a (quantitative) physical and (qualitative) psychological interpretation.

However, crucially, whereas the former can be seemingly be explained through Conventional Science, the latter requires a quite distinct holistic (integral) appreciation.

As we have seen the basis of Integral Science is that it incorporates both linear and circular notions (corresponding to partial and true holistic meaning respectively). The fundamental limitation with Conventional Science is - because it is based on purely linear use of logic - that it necessarily reduces in any context true holistic to a reduced partial understanding.

Einstein's illustration can in fact be used to highlight the very limitations of the scientific approach that he pursued so resolutely.

Meaning in a psychological context largely entails the relationship of specific phenomenal events to an overall holistic perspective.

In this context we will explain briefly why in fact time passes so quickly in a psychological context when - as Einstein implies - a man sits with a pretty girl.

What is to the fore here is the holistic (archetypal) aspect of meaning that corresponds directly with the nature of spiritual light. In other words in such an encounter a man however briefly may feel in some measure that his holistic desire for fulfilment is being realised. Indeed one might truthfully say as a consequence that he - literally - will light up inside.

And just as time does not pass for spiritual light (which exists in the continual present moment) likewise with such an encounter it will approximate to a continual present moment (where time does not seem to pass). In the other case, where one places one's hand on a hot stove it is quite the opposite. Here the actual phenomenal event - far from mediating the holistic light of meaning - comes sharply into conflict with it. So the focus of attention switches dramatically from holistic appreciation to the narrowly restricted attention of the immediate phenomenal event. Now with so little light in evidence, experience of time will slow down considerably. Thus in the extreme (of excruciating pain) a single moment will feel like an eternity.

So actual psychological experience of time is based on the relationship of specific events (linear) to an overall (circular) holistic notion of meaning.

Thus we cannot properly interpret this psychological notion of relativity without incorporating both linear (analytic) and circular (holistic) aspects.

What Einstein failed to realise is that - strictly speaking - we cannot likewise properly interpret physical relativity without again incorporating both linear and circular notions.

Once again the explanation that Einstein provides of Special Relativity is but the default linear (1-dimensional) interpretation (where holistic notions are inevitably reduced). Thus Einstein never provides a true holistic explanation of light (the nature of which is utterly mysterious). As we have seen without such an explanation (which ultimately requires an 8-dimensional approach), understanding of fundamental issues such as the origin of the Universe and the unification of the four forces remains very limited.

Generally speaking when one is en-joy-ing an event time will tend to pass quickly. However such an experience may be be very superficial and short-lived. True (and lasting) joy is associated directly with a more permanent grounding in the pure light of Spirit which depends less and less on specific phenomenal events (to act as a catalyst) as it deepens.

My interest in Relativity initially arose out of a desire to understand how the stages of contemplative development alter one's psychological experience of space and time.

And because physical and psychological understanding are in dynamic terms complementary, this led me in turn to a deep interest in the nature of physical relativity.

Ultimately a comprehensive scientific interpretation of Relativity (and indeed any phenomenal behaviour) must be radial (incorporating both quantitative and qualitative aspects of understanding).

However because the qualitative (i.e. integral) aspect is at present almost entirely missing from present scientific understanding, I am especially concentrating on this issue in my contributions.

Saturday, December 5, 2009

Special Relativity - higher dimensional appreciation

In my last contribution, I distinguished as between the 1-dimensional and 2-dimensional qualitative interpretations of Special Relativity. This highlighted the distinction as between the conventional scientific approach (1-dimensional) where effectively holistic circular (paradoxical) notions of meaning are reduced in linear (unambiguous) terms and the more refined 2-dimensional interpretation where both types of meaning can be clearly distinguished.

Essentially all other higher dimensional interpretations entail the same basic distinction as between (analytic) linear and (holistic) circular notions of meaning in an increasingly refined manner.

However some further distinctions can be made (for which my radial mathematical investigation of the Riemann Hypothesis has proved invaluable).

Properly speaking, purely integral understanding i.e. that ultimately is realised in an intuitive manner, is confined to the even dimensions while increasingly refined rational understanding is associated with the odd dimensions. The counterpart in terms of the Riemann Zeta Function is that the value of the function for all (negative) even values = 0, whereas the value for all (negative) odd numbers is a rational number!

In my own work, I have always attached particular importance to the even numbered dimensions 2, 4 and 8 as being of special significance from an integral viewpoint.

Indeed I then found striking evidence for this conviction in the way that that the rational values for (negative) numbers in the Zeta function up to 8 of an especially low magnitude. This would strongly suggest that the analytic understanding associated with dimensions 3, 5 and 7 is thus of a very diminished nature (as befits the movement towards a pure state of contemplative awareness which culminates in my work with the 8th dimension).

However this is not the end of the story for the next task is then to successfully recombine active involvement in affairs with mature contemplative awareness (in what I call radial reality).

Remarkably again the Riemann Zeta Function would suggest that this readjustment as between contemplation and activity would coincide with the dimensions between 8 and 16 (in what I associate with Band 4 of specialised contemplative development).
When we look at the Riemann Zeta Function, the numerical rational values for the (negative) odd dimensions become larger as between the 9th and 15th dimensions before exceeding 1 (and then rapidly further increase). So crossing the threshold of 1 can be associated here with the true emergence of the radial stages (where mature immersion with form can once again take place).

However my concern here is limited directly to integral (and not radial) scientific understanding for which dimensions 2, 4 and 8 are especially significant.

Once again in my work I associate 2-dimensional with Integral 1, 4-dimensional with Integral 2 and 8-dimensional with Integral 3 interpretation respectively.

As we have seen 2-dimensional appreciation involves recognition of the twin polarities (external and internal) in understanding. In a partial independent sense, understanding with respect these polarities can be treated as separate in an unambiguous (linear) fashion; however equally in a holistic interdependent sense, understanding must be considered as complementary - and ultimately identical - in a paradoxical circular manner.

In holistic mathematical terms, these represent the real polarities of experience (that are positive and negative with respect to each other).

4-dimensional appreciation involves additional recognition of the true distinction as between part and whole notions (that are real and imaginary with respect to each other).

For example, from a direct physical perspective, part notions would be associated with object phenomena and whole notions with corresponding dimensions (of space and time).

Thus at the 4-dimensional level of interpretation, reality is now mathematically complex (in qualitative terms) with both real and imaginary aspects of meaning.

The corresponding distinction at a psychological level is that between perceptions and (corresponding) concepts which again are real and imaginary with respect to each other. So once again in integral scientific terms we preserve matching meanings for both the physical and psychological aspects of scientific interpretation.

This in 4-dimensional terms an "object" can be given four directions of movement in space and time. Two of these - as we have already seen - are real (and positive and negative with respect to each other). However the two additional directions are now - relatively - imaginary (again with positive and negative directions).
Such 4-dimensional understanding thereby corresponds with the Integral 2 approach.

However we can further distinguish in an enlarged 8-dimensional approach the most fundamental polarities of form and emptiness. In holistic mathematical terms these correspond directly with the structure of the four additional roots (of the 8 roots of 1).

In other words emptiness (whether in physical or psycho spiritual terms) corresponds with nothingness (= 0 in holistic mathematical terms). In like manner the diagonal lines representing the geometric expression of these 4 additional roots are null lines = 0.

However in reduced rational terms, these lines (representing the fundamental polarities) can be given a complex mathematical interpretation (where both real and imaginary aspects are of the same magnitude).

What this entails is that - literally in formal terms - such emptiness can be characterised qualitatively in a holistic mathematical sense as both real and imaginary (in equal manner).

The key significance of this formulation is that it characterises directly with the root nature of - what we call - the four (physical) forces.

For example the true nature of physical light (as a manifestation of the electromagnetic force) is utterly mysterious. From one valid perspective it is - at light speed - empty i.e. devoid of any phenomenal characteristics. This indeed explains why light "travels" in the present moment.

However from an equally valid perspective, it can be characterised as made up of equal components that are real and imaginary with respect to each other. In other words light manifests itself in real phenomenal terms as either waves or particles. So when the wave aspect is manifested as real the particle aspect remains relatively - imaginary. However when in turn the particle aspect is manifest as real, then the wave aspect remains imaginary.

Once again we have a matching correspondent on the psychological side, where spiritual light manifests itself likewise in - relatively - real and imaginary fashion i.e. as transcendent and immanent respectively.

I remember receiving a wonderful insight while attending the U2 concern in Dublin during the Summer when it suddenly became clear to me that the so-called unification of the four forces is continually present in the present moment as the very ground of existence (that underlies all phenomenal events in space and time). And this is necessarily the case as the root nature of the forces (i.e. what they are in themselves) is inherently of an empty nature (corresponding directly to Integral 3 interpretation).

Far from interfering with the concert, such an intuition - freely registered in a moment of intuitive certitude - greatly enhanced my enjoyment of the evening.

Friday, December 4, 2009

Special Relativity - illustration of qualitative significance

I will outline briefly the basic holistic mathematical rationale by which the qualitative approach to measurement with respect to Special Relativity can occur.

Conventional Science is based qualitatively on a merely linear (i.e. 1-dimensional) interpretation of mathematical symbols. However corresponding to every number (as dimension) is a unique qualitative interpretation. So theoretically an infinite number of such interpretations is possible.

All interpretations (other than the default linear) entail a unique configuration of both linear and circular aspects of understanding.

In psychological terms this entails a complex mix of both conscious and unconscious (corresponding to rational and intuitive type appreciation).

Phenomenal quantitative measurements of space and time correspond merely to linear type understanding.

However when both linear and circular aspects are incorporated such linear measurements of space and time are seen as merely relative expressions of an underlying holistic basis of reality that exists simply as the continual present moment.

In holistic mathematical terms the precise configuration corresponding to each dimension (with respect to both linear and circular aspects of understanding) is obtained with reference to the corresponding roots of that number.
So for example to establish the qualitative nature of 2-dimensional understanding, we translate its two roots in a corresponding qualitative manner!

In this illustration we will confine ourselves to a comparison as between the (default) 1-dimensional and 2-dimensional interpretations respectively.

The key feature of 1-dimensional (linear) understanding is that it is unambiguous (relating to one positive direction of movement). So from this perspective objects move forward i.e. a positive distance in space; likewise time moves in a solely forward (positive) direction.

However 2-dimensional interpretation is much subtler in nature. Its holistic mathematical nature is obtained with reference to the two roots of unity i.e. + 1 and - 1. So from a 2-dimensional qualitative perspective, there is a valid sense in which space and time move in both a positive and negative direction.

Correct psychological appreciation of this requires recognising that actual phenomena do not exist independently (as merely external objects) but actually entail a dynamic interaction of subjective and objective aspects of understanding that are - relatively - external and internal with respect to each other.

Now if we try to treat these aspects as independent, unambiguous linear interpretation will be possible. So if we identify understanding with the external aspect of understanding (as "objects) one ambiguous direction of movement will result. Likewise if we understand understanding with the internal aspect of understanding (as "mental perceptions") an alternative ambiguous direction will result.

However if we now treat these opposite reference frames as interdependent (as befits integral appreciation) then they will move in both positive and negative directions with respect to each other. Furthermore the understanding of what is positive and negative is purely arbitrary depending on the initial reference frame chosen.

Fro example what is a left or right turn on a road depends crucially on what direction I take (i.e. "up" or "down") on the road.

Thus the analytic linear aspect of (2-dimensional) understanding corresponds to the treatment of either reference frame i.e. external or internal as independent .

The holistic circular aspect corresponds in turn to the treatment of both frames as interdependent (where space and time movements in rational terms are understood as paradoxical). Then in direct intuitive terms, this corresponds to a spiritual awareness of the continual present moment.

For example standard linear interpretation would tell us that the it takes about 8 seconds for the light of the Sun to reach Earth (travelling some 93 ml. miles in the process). So both of these measurements (with respect to space and time) are considered as positive.

However 2-dimensional interpretation would yield a more refined interpretation giving rise to two frames of reference. Now within each frame (treated as separate) both measurements will again appear as positive. However relative to each other (treated as complementary) movement is now paradoxical with both positive and negative directions of movement.
Then the intuitive realisation of such paradox is an awareness of the continual present moment as the primary basis from which the relative secondary phenomenal measurements of space and time have but an arbitrary relative meaning.

Thus the 2-dimensional interpretation is of a qualitatively distinct nature compared to the standard linear (1-dimensional) version.

This has far reaching consequences.

For example - largely as a result of Einstein's work - it is now conventionally understood that the Universe began with a Big Bang some 13 bl. years ago.

From the two-dimensional perspective this is strictly meaningless. The basic confusion here is a gross reductionism whereby interpretation with respect to the whole is not properly distinguished from interpretation with respect to the part.

Linear measurements of space and time are only possible by separating opposite polarities (such as external and internal). However clearly for the whole Universe this is not possible. We seem to forget that such linear measurements themselves crucially depend on the assumption that we can somehow separate ourselves as external psychological observers of the Universe from its original nature (which clearly is not possible).

Thus the correct starting point for a 2-dimensional interpretation is the continual present moment from which all phenomenal interpretations with respect to space and time have but a secondary - merely relative - expression.

In a primary sense the Universe is simply always now. Entering into such realisation we truly experience both its Alpha and Omega (i.e. as the present moment).

In this illustration I have dealt with the simplest possible case of integral understanding (i.e. conforming to the qualitative number dimension "2"). However all other dimensional interpretations (though much more intricate and refined) entail the same basic relationship whereby varying directions of movement (with respect to space and time) are seen as but relative phenomenal expressions of the present moment.