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.