A major feature of the holistic mathematical approach is an alternative circular notion of number that gives rise to the various dimensions of development. As each dimension - when expressed in a reduced linear manner - leads to a distinctive direction (drawn from the centre of the unit circle in the complex plane to its circumference), I also refer to these distinctive dimensions as directions.
The deeper meaning of this statement relates to the fact that all phenomenal experience is conditioned by opposite sets of polarities that dynamically interact with each other in experience. This means that opposite poles are clearly separated, with meaning in any context identified with just one pole.
In this regard, I identify 3 key sets of such polarities.
Firstly, we have the horizontal set of external/internal polarities that operate within a given stage of development.
Secondly, we have the vertical set of whole/part (or alternatively collective/individual) polarities that operate between different levels of development.
Thirdly, we have the most fundamental diagonal set of form/emptiness polarities that operate simultaneously within and between all levels of development.
When these three sets operate in a truly dynamically interactive refined manner (without dualistic restrictions) all hierarchical distinctions in development are ultimately clearly seen to have but an arbitrary relative meaning.
So internal is no longer separate (except in a merely relative manner) from external (and vice versa); wholes are no longer separate from parts (and vice versa); and finally form is no longer separate from emptiness (and vice versa).
The big breakthrough that I made - initially in the late 60's - was the realisation that these fundamental polarities possess a unique holistic mathematical numerical structure. In fact they are associated directly with - what I now refer to as - the Type 2 appreciation of number.
The relationship between internal and external was the earliest polarity set to reveal itself in this manner.
Largely under the influence of Hegelian philosophy, at the time I began to realise that whenever an object is made conscious in an external manner in experience, that it is thereby posited in that manner (where positing now represents the holistic mathematical notion of addition i.e. + 1).
Then in order to switch to corresponding internal recognition, the object must be dynamically negated - at least to some degree - in an unconscious manner, before again being consciously posited. So here we have the corresponding holistic mathematical notion of subtraction (as negation) i.e. – 1.
In reverse fashion, to switch back from conscious internal recognition (as mental construct) to its external manifestation as object, once again we must dynamically negate in some measure (at an unconscious level).
Therefore like left and right turns at a crossroads, when properly understood in an interactive manner, the external and internal polarities of experience are understood as positive and negative with respect to each other Expressed in an equivalent manner, external and internal are dynamic complements of each other!
This then directly expresses the Type 2 understanding of the number 2.
In conventional Type 1 terms, 2 is understood - literally in 1-dimensional manner - where it is more fully expressed as 21.
However in holistic Type 2 terms, 2 is now understood directly in terms of the dimension (power, or exponent) of 2 that is expressed in an inverse manner as 12.
Then to indirectly express in the accepted linear manner this notion of 2, we obtain the 2 roots of 1, to obtain + 1 and – 1 respectively.
In other words when we attempt to express the nature of 2-dimensional understanding in the accepted linear (1-dimensional) fashion, it appears as deeply paradoxical (or circular).
And this is literally because number is now represented in a circular fashion, where the two roots of 1 are equidistant points on the circumference (of the unit circle in the complex plane).
Mathematical interpretation especially is solely defined in a limited 1-dimensional manner.
For example if we take the notion of any number - e.g. 2 - we can posit it externally as a unitary number object in conscious terms (+ 1).
However equally we can posit it internally in conscious terms as the corresponding mental perception of number (+ 1).
So in 1-dimensional interpretation - where the conscious aspects of understanding are solely recognised - both external and internal poles are assumed to directly correspond with each other in an absolute fashion.
However in 2-dimensional interpretation (in holistic terms) external and internal are now clearly seen as opposites of each other (from a dynamic interactive perspective). In other words such understanding crucially incorporates the unconscious aspect of understanding which is vitally necessary to enable switching as between opposite polarities to take place.
Therefore, the deeper realisation that accompanied this breakthrough in terms of understanding the holistic basis of polarity interaction, is that Mathematics itself can no longer be understood in a merely conscious rational fashion, but must also explicitly incorporate its (unrecognised) unconscious aspect.
And when one properly reflects on the matter, this finding should be seen as indisputable, for the very nature of all mathematical understanding necessarily entails the dynamic interaction of objects with mental constructs (that are dynamically positive and negative with respect to each other).
We can easily appreciate, for example in the context of left and right turns at a crossroads, is that when the direction of approach switches as between N and S directions, that these turns then have a purely relative meaning.
However in the dynamics of mathematical experience, polarities (such as external and internal) are continually switching. Therefore, properly understood from this perspective, all mathematical truth is rendered strictly relative. However once again the full realisation of this will require the explicit incorporation of the unconscious aspect in all interpretation!