As one may perhaps appreciate, the qualitative use of numbers (as dimensions) plays a key role in holistic mathematical understanding.
So I have spent at this stage more than 40 years in the slow - and often painful - task of unravelling the hidden meaning contained in all numbers (as dimensions).
In the attempt here to explain my present position, I will relate how this understanding actually unfolded in development.
The first key insight was in recognition of the linear nature of the rational paradigm which underpins interpretation of both mathematics and science.
So in qualitative terms, such understanding is defined in holistic mathematical terms by the number 1 (i.e. 1-dimensional interpretation). As we have seen this entails in Conventional Mathematics that all number quantities are ultimately defined in terms of a (default) power of 1. For example 2 ^ 2 = 4 (i.e. 4 ^ 1). In physics it implies for example that object phenomena are unambiguously posited in just one (external) direction.
The next key step was the recognition of the nature of 2-dimensional understanding. This arose through a deep interest for a time in Hegelian philosophy. Rather than a linear use of reason Hegel favoured a more circular form based on the complementarity of opposite poles (popularly represented as thesis and antithesis).
The key insight here was to see in holistic mathematical terms a direct connection between such poles and the square root of a number.
This can be generalised by saying that there is a direct connection between the n roots of the default number 1 in quantitative and the corresponding n dimensions in qualitative terms. However whereas linear either/or logic applies in the former case, circular either/or logic operates in the latter.
For example in quantitative terms the two roots of 1 are given as either + 1 or - 1.
However corresponding qualitative 2-dimensional interpretation is both + 1 and - 1.
What this means is that all understanding now entails a dynamic interaction as between the knower (as internal subject) and what is known (as external object) with a merely relative meaning so that these two poles keep switching in experience.
Though it may not be articulated in a holistic mathematical manner such understanding represents a normal stage with respect to authentic contemplative development. Here the rigid barriers as between the world (as objective) and the self (as subjective) gradually break down with neither having any absolute identity (in isolation).
The importance of this "higher" 2-dimensional understanding is that it is replicated at a corresponding "lower" level by the behaviour of sub-atomic particles. So the provision therefore of a coherent interpretation of quantum behaviour in qualitative holistic terms therefore requires (at a minimum) 2-dimensional understanding. What this means in effect is that in terms of such understanding quantum behaviour intuitively readily resonates with esperience (corresponding to such circular rational interpretation).
2-dimensional understanding represents the first of the truly integral approaches (based on the complementarity of opposites). I usually refer to this as the Integral 1 approach. Once again it is based on the complementarity of opposite real (conscious) poles
4-dimensional understanding is the next truly important integral approach (Integral 2).
Just as the earlier approach establishes the complementarity of internal and external aspects (the knower and what is known) this likewise establishes the complementarity of whole and part which is perhaps the most fundamental of all relationships in physics.
Standard linear interpretation cannot properly maintain the key qualitative distinction of whole and part. Instead it simply reduces (in any context) the whole to the parts in quantitative terms.
Not surprisingly, gross reductionism pervades the conventional approach to science and mathematics (which for the most part is not even recognised as such by its practitioners).
Precisely because the whole - when properly understood - is qualitatively distinct from the parts one cannot maintain such a distinction without incorporating the qualitative dimension of science on an equal basis with the quantitative.
Of course wholes and parts can be given a quantitative interpretation. However in the dynamics of recognition by which the mind switches from wholes to parts (and parts to wholes) an unconscious aspect is necessarily involved.
Properly understood therefore in the recognition of any object phenomenon an unconscious (as well as conscious) aspect is involved. Indeed even in popular language this is often recognised. For example one may speak of a "dream" house. So here the house has a local (conscious) identity; however equally it possesses a holistic (unconscious) identity as the embodiment of a deeper meaning.
However, strictly this necessarily applies to all phenomena. Indeed the very desire of scientists to study certain phenomena often speaks strongly of this holistic (unconscious) meaning (which cannot ultimately be separated from associated recognition of a conscious kind).
The key breakthrough here (which owed much to an interest in Jungian Psychology) was the recognition that the "embodied" unconscious aspect of understanding is "imaginary" in a precise holistic mathematical sense.
Now the negative direction (by which phenomena are literally negated in experience) represents the unconscious direction of understanding.Inherently this leads to a 2-dimensional form (combining positive and negative polarities). So to express such understanding in a (reduced) linear form we take the square root!
Thus "imaginary" interpretation is the every means through which the qualitative - as opposed to quantitative - aspect of understanding is expressed in a rational manner.
Mathematics recognises that numbers (as quantities) incorporate both real and imaginary members.
The holistic mathematical corollary of this is that science contains both "real" and "imaginary" aspects (i.e. as quantitative and qualitative aspects respectively).
So this whole blog on "Integral Science" is thereby designed to elaborate in various ways the nature of the unrecognised "imaginary" aspect of science.
And a fully comprehensive approach to science - which I term radial - would be "complex" combining both "real" and "imaginary" aspects (as equal partners).
As we have seen in the previous post, one important application of this relationship as between "real" and "imaginary" relates to object phenomena and dimensions (which are "real" and "imaginary" with respect to each other. Thus a clear implication of this is that we cannot hope to properly understand String Theory without incorporating a corresponding holistic qualitative aspect. And as I was mentioning all the important concepts in String Theory can be given corresponding holistic interpretations with a comprehensive understanding then relating to the relationship as between both aspects.
The next truly fundamental integral approach - which I term Integral 3 - relates to 8-dimensional approach. As well as the 4 dimensions (of Integral 2) this opens up 4 new dimensions (of a complex kind). However in geometric terms the diagonal lines representing the corresponding additional roots (on the circle of unit radius in the complex plane) can be represented as null lines.
The psychological implication of this is that pure spiritual attainment (that is nothing in phenomenal terms) is approximated when both conscious (real) and unconscious (imaginary) aspects are fully harmonised.
Remarkably the corresponding physical implication relates to the very nature of forces. For example light is often in Relativity Theory represented as a null line. So the deeper implication is that light (in itself) represents the complete harmonisation of phenomenal and dimensional characteristics.
Even in the Biblical account in Genesis, the World is created out of light. So the phenomenal and dimensional characteristics of the created Universe can be seen to emerge from a common origin!
For many years 8-dimensional interpretation represented the limit of what I felt could be achieved in integral terms.
Indeed I formulated A Holistic Theory of Everything based on such understanding!
Then shortly after that when studying the holistic mathematical nature of Jungian Personality Types I saw how to extend 4 to 24-dimensional interpretation through obtaining all possible permutations of the existing 4. This - as I have related elsewhere - then led to an extension of the Myers-Briggs system (to include 8 missing types). The realisation that each of these Personality Types represented a unique circular dimension (in the characteristic manner of configuring space and time) led me a similar notion within physical reality (which I now see as the means of interpreting dimensions in String Theory).
So just as with String Theory, I was left with 5 main integral models for holistic qualitative interpretation of reality.
It is only in more recent years - largely through association with the Riemann Hypothesis - that I have been able to significantly extend this understanding.
After some time I realised that 16-dimensional interpretation (and all subsequent powers of 2) could be explained like a compass. So as with the compass we have 4 main coordinates (applying to 4-dimensional interpretation. Then with higher powers of 2 we are able to define our directions ever more accurately. Likewise it is similar with higher dimensional understanding where 2 is raised to 0, 1, 2, 3, 4,...etc.
Then all other even numbered dimensions likewise have a direct integral significance.
The basis of integral understanding is the complementarity of opposites in understanding (with again 2-dimensional interpretation offering the simplest example).
However all other even based dimensions can likewise be represented by the complementarity of opposites (replicated in quantitative manner by the format of the roots of every even number).
Indeed there is an intimate connection here with the Riemann Zeta Function.
As is well known for all even numbered integer dimensions (powers) of the Function, an expression involving pi is involved.
Now pi represents in quantitative terms the relationship i.e. ratio, as between the circular circumference and its line diameter.
In like manner in qualitative terms all the even based dimensions (as interpretative models) represent the direct relationship as between circular and linear understanding!