Strictly speaking we are referring to real (i.e. conscious) polarities that are based on the continual switching of external and internal aspects with respect to experience.

Now all experience to a degree implicitly requires the same ability (even with respect to linear type understanding).

When the interaction between polarities is limited, interpretation tends to become dualistic in a somewhat rigid manner. This supports the common sense belief that objects thereby possess an independent validity (not affected by our interaction with them).

However when this interaction becomes especially marked, interpretation moves increasingly from a dualistic to a nondual mode. So one now directly intuits in experience the common identity of both aspects (which is the true basis of holistic appreciation). Indirectly when expressed in a rational manner, such appreciation therefore appears circular i.e. paradoxical in terms of conventional linear reason.

So properly understood in a dynamic interactive manner, with holistic understanding we always have a mix of dual and nondual elements.

Thus the intuition of the common identity of positive and negative poles (i.e. external and internal aspects of experience) provides the nondual appreciation of two-dimensional interpretation.

This could be expressed as + 1 – 1 = 0. So the qualitative nondual appreciation (i.e. direct intuition) literally has no quantitative meaning (= 0).

The dualistic appreciation then comes from the reduced expression of this relationship in a linear (1-dimensional) fashion. This equates with the two roots of 1 = + 1 and – 1, now understood in a relative separate manner. Thus the dynamic holistic experience of 2-dimensional understanding, necessarily entails the interaction as between this direct nondual intuition (of a qualitative nature) and its reduced (circular) rational expression (in quantitative terms).

This in turn leads to a new mode of (algebraic) irrational understanding in a precise holistic mathematical sense.

We have seen how the conventional scientific paradigm is based on the linear levels. So the paradigm is defined correspondingly in a (linear) rational manner.

However, we have now moved to the stages of the circular level (on which a new holistic scientific paradigm is based). And this paradigm is defined in a corresponding (circular) irrational manner.

When in the expression a

^{b}, both a and b are given non-unitary values, we inevitably obtain both a quantitative and qualitative transformation in the variables which is the essence of the holistic meaning of an (algebraic) irrational number!

When we look at an irrational number such as the square root of 2 , it contains both finite and infinite aspects in its very nature. Thus the value of the square root is a finite number which can be approximated to any required value. For example correct to 4 decimal places, its value is 1.4142.

However equally there is an irreducible infinite aspect in that its decimal sequence continues without limit (and cannot be represented as a fraction).

Thus if we were to interpret an irrational number in a comprehensive mathematical manner, we would accept that it represents the interaction of both finite (quantitative) and infinite (qualitative) aspects.

Now the very nature of holistic understanding at this stage, as we have seen is that it indeed combines both dual (quantitative) and nondual (qualitative) aspects in a dynamic manner.

Therefore in a precise holistic mathematical sense, such experience of reality is of an (algebraic) irrational nature.

And just as Conventional Science is built on a (linear) rational paradigm, correctly understood, we can now define a completely new type of science based on a (circular) irrational paradigm.

And the essence of this science is that reality is now inherently understood in a dynamic interactive manner, where both intuitive (qualitative) and circular rational (quantitative) notions are combined.

This is why I was so excited at this time. Though I would not have been capable yet of adequately conveying the true nature of this scientific approach, I could already see clearly in principle that such an approach did exist which could be applied to all scientific disciplines.

So this led in the aftermath to a remarkable flurry of holistic intellectual activity, as I tried to map out the implications of this approach with respect to Philosophy, Economics, Physics and most of all Mathematics.

So we can have (algebraic) irrational perceptions and (algebraic) concepts (again understood in a precise holistic mathematical manner).

It is during the earlier supersensory stage, that irrational perceptions first unfold.

For example if I look at a flower such as a rose (from a conventional scientific perspective) it is defined by the collective group of roses. So its meaning depends on its common identity with all other roses (that collectively are defined by the same class).

In other words the rose here is given but an objective quantitative definition.

However if I now understand in a true holistic manner, each rose will possess a unique identity (in qualitative terms) while yet belonging (in quantitative terms) to the common collective class of all roses.

So the actual experience now explicitly entails the interaction of both finite (quantitative) and infinite (qualitative) aspects.

Therefore it is in the very nature of Conventional Science to continually reduce in every context, the true qualitative aspect of phenomena in mere quantitative terms!

Thus at the supersensory stage, one can already maintain, to a degree, a more balanced recognition of objects. Therefore symbols, while maintaining a local quantitative identity, yet also possess universal attributes in qualitative terms. And the very basis of this qualitative identity is nondual spiritual intuition!

Then at the more advanced suprarational stage, irrational concepts can properly unfold.

So it is this more general stage that one can begin to properly formalise the very nature of science that is built on irrational concepts.

Indeed we are accustomed from the (linear) rational perspective to refer to the physical sciences (where the physical world is clearly separated from its psychological interpretation).

However from the holistic perspective of the suprarational stage, both physical and psychological aspects interact in an explicit dynamic manner.

So we necessarily move therefore from physical to psycho-physical reality (or equally from the alternative perspective physico-psychic reality).

Therefore from this holistic perspective, we cannot meaningfully divorce physics from psychology (on the one hand) or alternatively psychology from physics (on the other).

Indeed the very basis of this new holistic science is the establishing of complementary relationships as between both physical and psychological reality, which becomes the means of properly integrating both spheres in experience.

At an even deeper level, what I am attempting to illustrate here is that the holistic interpretation of number, ultimately provides the best means of mapping out all the general types of science possible for the interpretation of reality.

Once again Conventional Science is defined by the rational paradigm (with its philosophic rationale corresponding therefore to the holistic interpretation of rational numbers).

However what is remarkable is that associated with all the other major number types such as (algebraic) irrational, transcendental, imaginary and complex numbers, we equally have unique new types of science (with their philosophic rationale in turn corresponding to the holistic interpretation of these respective numbers).

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