As we have seen, properly understood every number keeps switching as between whole and part aspects in a dynamic manner.
This reflects the fact that the understanding of Mathematics always entails an interactive experience where number keeps switching as between these two aspects (depending on context).
And again, properly understood Mathematics has no strict meaning apart from its corresponding understanding.
Unfortunately, for millenia now we have become increasingly conditioned to the untenable notion that Mathematics has an independent abstract existence (apart from the enquiring mind).
And this has led to - what I refer to as - the 1-dimensional approach which in fact defines all Conventional Mathematics.
As this is so important, I will briefly clarify what the 1-dimensional approach precisely entails.
All phenomenal experience - including of course mathematical - is conditioned by fundamental polarity pairings.
Two of these pairings are especially important.
The first relates to external (objective) and internal (subjective) polarities.
Fro example when one experiences a number, both of these polarities are necessarily involved. So a number that is viewed as objective, existing in external space, has no strict meaning in the absence of a corresponding mental perception, which - relatively - is internal in nature.
So rather than number having a static absolute existence, in truth number represents a dynamic interaction pattern as between two polar aspects that are - relatively - external and internal with respect to each other.
And this, by extension, applies to all mathematical constructs.
We could truthfully say therefore that in mathematical terms, objective truth has no meaning in the absence of corresponding mental interpretation.
So what happens in Conventional Mathematic is that an attempt is made to totally freeze this interaction as between external and internal, so that mental interpretation is viewed to be in absolute correspondence with the objective situation (which is then given an abstract independent existence).
In this way we can see how conventional mathematical "truth" takes place within just one isolated polar reference frame (i.e. as objective in an absolute manner).
The second key polarity pairing relates to the relationship - which we have been directly looking at - as between whole and part. This could also be referred to as the relationship (in any context) as between general and particular, (or individual and collective), qualitative and quantitative etc.
Again, the experience of number (and indeed all mathematical relationships) entails the dynamic interaction of whole and part aspects, which keep switching, depending on context.
Once more, conventional mathematical interpretation attempts to absolutely freeze this interaction by reducing the qualitative aspect in a mere quantitative manner.
So once again, we can see how such mathematical "truth" takes place within just one isolated polar reference frame.
So 1-dimensional interpretation refers therefore to interpretation that is explicitly conducted in an absolute manner within an isolated polar reference frame.
And the very essence of such interpretation is that dynamic interaction cannot be recognised to take place as between opposite poles (though implicitly some unconscious interaction must necessarily take place).
Many years ago, when I first recognised the all-embracing importance of these two fundamental polar pairings, I slowly began to see that they concurred exactly with a new holistic mathematical manner of interpreting mathematical symbols.
Now, basically when we become conscious with respect to a phenomenon in an independent rational manner, we thereby posit in a conscious manner. So here we have the holistic meaning of the plus sign as used in addition (i.e. +).
Then to switch as between opposite poles e.g. from the objective to the mental recognition of the object, we must implicitly negate the external pole (in an unconscious manner). So here we have the corresponding holistic mathematical meaning of the negative sign as used in subtraction (i.e. – ).
Now the extent to which such unconscious negation is involved, determines the degree to which recognition of the interdependence of opposite polarities takes place.
This occurs directly in an intuitive manner, whereby psychic energy is generated. In fact it parallels very much the manner in which matter and anti-matter particles annihilate each other creating physical energy. So the holistic intuitive realisation of interdependence entails the direct coincidence of both positive (+) and negative (–) poles.
Note here how the holistic interpretation is paradoxical with reference to the corresponding analytic (1-dimensional) interpretation (where poles are separated in an absolute dualistic manner)!
Though the direct intuitive realisation of the interdependence is nondual in nature, indirectly it can be expressed in a circular manner through paradoxical reason.
This contrasts heavily therefore with corresponding analytic interpretation that is expressed in the standard linear manner through the unambiguous use of reason!
Put simply, all analytic interpretation of mathematical symbols is 1-dimensional in nature. Once again this is the only interpretation that is formally recognised within present Mathematics.
However all holistic appreciation (of an authentic nature) entails "higher" dimensional interpretation of which the simplest is 2-dimensional.
This basically relates to appreciation of the interaction of the first pairing of polarities - which I refer to as the horizontal polarities - i.e. internal and external.
And once again in holistic mathematical terms, these are positive (+) and negative (–) with respect to each other.
It took me more time to fully appreciate the holistic mathematical significance of the second set, which relates directly to the dynamic interaction as between whole and part.
Now remarkably, the whole notion (in the authentic appreciation of qualitative interdependence) is imaginary (i) with respect to the corresponding real (i.e. quantitative) interpretation of a unit (1).
Basically the imaginary notion represents the attempt to express the holistic notion of interdependence in an indirect analytic type manner.
Now, as we have here the unconscious appreciation of qualitative interdependence, this entails the negative direction of understanding i.e. whereby positive recognition of the exclusive independence of 1 (i.e. as pole or direction) is thereby to a degree successfully eroded.
Thus, this unconscious appreciation of negation is 2-dimensional (as it dynamically also necessarily includes the positive direction). And to express this in the standard linear (1-dimensional) manner we take the holistic equivalent of a square root (i.e. in expressing what is 2-dimensional in a reduced 1-dimensional manner).
So the relationship of part to whole (quantitative as to qualitative) is as "real" to "imaginary".
And the imaginary - like the real - likewise has two directions that are positive (+) and negative (–) with respect to each other.
Because holistic qualitative notions are inherently of an unconscious intuitive nature, to indirectly recognise their nature, they must necessarily be projected into conscious experience.
So the positive direction arises when we are aware of how an object indirectly conveys a holistic meaning. For example, an athlete might have a dream - say - to one day win Olympic gold. So an Olympic final would not just be understood in a conscious manner, but would likewise serve a powerful holistic unconscious purpose.
Now in similar manner, all mathematical symbols, not only serve a real quantitative, but likewise an imaginary qualitative purpose.
So not alone do "real" and "imaginary" have an important quantitative interpretation (as in Conventional Mathematics) but equally an important holistic interpretation!
So from this qualitative perspective, the great limitation of present Mathematics is that it is conceived solely in "real" terms (i.e. with respect solely to its quantitative aspect).
Therefore from this perspective, I am clearly maintaining that a comprehensive mathematical approach must be complex (i.e. with both real and imaginary components).
So Conventional i.e. Analytic (Type 1) Mathematics, in this qualitative sense represents the "real" component of mathematical understanding.
Holistic (Type 2) Mathematics represents the "imaginary" component and Comprehensive i.e. Radial (Type 3) Mathematics represents "complex" - both "real" and "imaginary" - mathematical understanding.