It is important to distinguish linear and circular, in holistic mathematical terms, from real and imaginary (which in many ways are closely related).
Initially, linear relates - in a scientific context - to understanding according to either/or logic. Here the crucial polar opposites, which condition all phenomenal experience of reality (internal/external, whole/part and form/emptiness) are clearly separated in interpretation. This then enables rational linkages to be made in an unambiguous sequential fashion. So Conventional Science is characterised by linear (1-dimensional) interpretation.
Such linear understanding is directly posited in a real manner. To posit in this dynamic context (which entails the holistic meaning of +) simply means to make conscious. Real in a holistic mathematical context means understanding that is of a direct conscious nature.
So Conventional Science is linear, corresponding to real rational interpretation in a positive manner.
However in dynamic experiential terms one can negate (as well as posit) phenomena. The very significance of such negation (-) is that it thereby opens the door to unconscious (intuitive) type appreciation.
Now, such intuitive appreciation strictly relates to the generation of spiritual energy (necessary to fuel all conscious dynamics). Just as the positive and negative polarities (in the form of matter and anti-matter) annihilate each other on contact, creating physical energy, likewise the positive and negative polarities of phenomenal understanding likewise "annihilate" each other creating spiritual (intuitive) energy.
Intuition is generated directly therefore in a non-rational manner (through negation of phenomenal constructs).
However when one then later rationally attempts to clarify the nature of such understanding it appears as two-dimensional. This new two-dimensional understanding is based on an alternative circular logical system of interpretation relating to the complementarity - and ultimate identity - of the fundamental polar opposites in experience.
Therefore though we separate polar opposites such as internal and external in (rational) linear interpretation, we recognise these same opposites as complementary, and ultimately identical, from an (intuitive) circular perspective.
So linear understanding is 1-dimensional; circular is 2-dimensional. The structure here of the 2nd dimension of understanding corresponds to the 2nd root in quantitative terms of 1 (i.e. 1 raised to the power of 1/2).
So there is an intimate link as between a qualitative dimension (as D) and its corresponding quantitative root (as 1/D).
Now when the dimension is 1, both D and 1/D are identical. This implies that with linear (1-dimensional) interpretation, no distinction is made as between quantitative and qualitative interpretation!
However for all other integer values of D, a clear distinction is preserved as between quantitative and qualitative (implying the combined use in interpretation of both linear and circular appreciation).
So both/and logic is inherently circular in nature. It is also, as we have seen 2-dimensional (when confined to two opposite poles).
The question then arises as to how such understanding might be converted into linear expression.
Thus to reduce the 2nd to the 1st dimension, we obtain the square root of the 2-dimensional expression (- 1). And this represents the precise holistic mathematical equivalent to the notion of an imaginary number in analytic terms.
So imaginary understanding represents circular type intuitive understanding, indirectly expressed in a linear manner. What this means in effect is that such intuitive understanding therby becomes embodied in conscious phenomena (that are understood in a linear manner).
Properly understood, all phenomena are complex in nature (with real and imaginary aspects).
The real aspect relates to direct conscious appreciation of a local kind.
However the imaginary aspect by contrast relates directly to unconscious appreciation of a holistic nature (indirectly mediated through the local object).
For example if one was to purchase a house, both aspects of understanding would be important. Real conscious appreciation would be used to identify and precisely locate the house. However such a purchase could also fulfil unconscious longings of a holistic kind. So one might well - in this context - refer to one's "dream" house - reflecting such unconscious meaning (which is mediated through the same object).
So real and imaginary, implicitly, always interact in understanding (even in the most rigorous scientific interpretation). When the focus is on analysis, the real aspect is of paramount significance. When the focus by contrast is on qualitative synthesis, the imaginary aspect is now critical. Unfortunately - as we have seen - because of the current solely conscious focus of scientific interpretation, this latter "imaginary" aspect is thereby reduced to the real.
Now if we go back to the definition of the string we can precisely illustrate what is involved.
The conventional (linear) real definition relates to the fact that the patterns of vibrations of the physical 1-dimensional string (representing the fundamental stuff of matter) create all possible matter particles.
However the new (linear) imaginary definition relates to the fact that the patterns of vibrations of the same string (representing the fundamental stuff of psychological interpretation) create all possible dimensions for such interpretation. And as physical and psychological notions are complementary in holistic terms, this equally implies that it is the vibrations of the second aspect of the string that create all possible physical dimensions of reality (i.e. spacetime).
So the string has in fact two intimately related aspects (that are real and imaginary with respect to each other). Indeed we could rightly refer to this as the complex string!
Now if we follow this a little further we will see why differing dimensions of qualitative interpretation are so important.
At the linear 1-dimensional level, only the real positive aspect of the string (as the fundamental "stuff" of all matter) will be recognised.
Then at the 2nd dimension of interpretation the real negative aspect now also becomes apparent as one recognises the role of both conscious (reason) and unconscious (intuition) in the dynamics of understanding. The means - in complementary physical terms - that the string is now given both a specific (local) and a holistic (collective) identity.
Then at the 4th dimension of interpretation, the imaginary aspect of the string unfolds. So at this level of appreciation, one understands the string in complex terms (with both real and imaginary aspects). The imaginary aspect here corresponds to the holisic (circular) identity of the string indirectly expressed in a linear manner.
So as we reach higher dimensions in qualitative terms, actual interpretation with respect to physical reality becomes ever more refined with new aspects of understanding unfolding (which remain hidden at lower dimensions). And it is not just psychological interpretation that changes in this manner but likewise physical reality itself (which directly corresponds to such interpretation).
Thus one could validly say: not alone do we have a uniquely different psychological interpretation of reality corresponding to each new "higher" dimension (the range of which is infinite) but equally we have a uniquely different physical reality!
However, to be even more precise, we should recognise that all these interpretative dimensions (with their corresponding physical realities) are but appearances, or - if one prefers - secondary realities.
True primary reality is non-phenomenal and ineffable, existing continually in the absolute present moment. Phenomenal reality (of matter and spacetime) always serves therefore as a relative expression of what is primary.
So a key insight once more is that the basic relationship as between part (as physical particles) and whole (as the dimensions to which they are relate) are as real to imaginary.
This brings us back to the the crucial point.
To properly investigate physical reality, we need to incorporate a complex rational approach. Such an approach combines real interpretation (suited for analytic interpretation of a quantitative kind) with imaginary interpretation (suited for corresponding holistic interpretation of a qualitative nature).
And because real and imaginary are so closely related to linear and circular, we can also express this complex rational approach in another way (which is highly revealing).
As we know the (analytic) binary system that is so important in IT is based on just two digits 1 and 0 that potentially can be used to encode all information.
However there also exists a corresponding (holistic) binary system that potentially can be used to encode all transformation processes.
So a comprehensive binary system - with potential power to encode with respect to both information and transformation - has both analytic (linear) and holistic (circular) components.
Once more, the fundamental relationship as between whole and part is based on the distinction as between linear and circular understanding (or alternatively real and imaginary).
Put another way the basic relationship as between matter and spacetime requires such a distinction!
The reason why present String Theory is finding it so difficult to properly incorporate spacetime in its approach is because of its inherently reductionist nature. In other ways it is still attempting to explain the whole (dimensions) in terms of the parts (strings) as physical particles. And as long as it persists with this approach it will continually fail to achieve a satisfactory resolution.
So the quest for a coherent TOE along the present reductionist lines is not attainable.
However a much greater prize might yet be won if it leads finally to incorporation of a true holistic aspect in physics based - literally - on recognition of its hidden qualitative dimensions.
With such a realisation, science could then be truly revolutionised in a manner greatly exceeding all previous developments.