I have dealt at length in the previous entries with the unfolding appreciation of the nature of number, when viewed in appropriate dynamic interactive manner from complementary Band 3 and Band 5 perspectives.

However this development - in my approach - is intimately tied in turn to the unfolding psychological development of the various levels of these Bands.

So the deeper realisation slowly revealed to me during this time was that all physical and psychological development is precisely encoded in a dynamic interactive mathematical manner. And as I was to eventually discover, this encoding applies equally from this new mathematical perspective to every aspect of development (of a cognitive, affective and volitional nature).

Now it is important to keep emphasising - lest there be any misunderstanding on this matter - that the ultimate nature of reality is ineffable (and therefore not capable of direct representation in phenomenal terms). However human experience always necessarily entails the interaction of this ultimate spiritual reality of an empty nature and more accessible phenomena of form, which then indirectly can serve to mediate this reality.

So what I was discovering is that appropriate interpretation of mathematical symbols - that allows for the balanced interplay of both quantitative (analytic) and qualitative (holistic) aspects - provides the purest possible means of mediating as between these two fundamental domains i.e. spiritual emptiness and phenomenal form.

So to put it more succinctly, phenomenal reality is fundamentally encoded in a mathematical manner (that is ultimately ineffable); equally from the alternative perspective, phenomenal reality represents the corresponding decoding of its inherent mathematical nature.

And once again this does not just apply in a more limited manner to what we would recognise as the scientific (cognitive) nature of reality, but equally for example to artistic (affective) and religious (volitional) domains.

However once again it is vital to emphasise that I am not referring to Mathematics here in the merely reduced conventional quantitative manner, but with respect to an utterly distinctive vision (which is directly compatible with the unfolding of these more advanced bands of understanding).

So conventional mathematical understanding is strictly a product of Band 2 type understanding (where rational conscious type interpretation is separated to an extreme degree from the unconscious).

Therefore appreciation of this new mathematical vision requires firstly the recovery of the hidden unconscious basis of Mathematics in direct appreciation of the qualitative (holistic) nature of all its symbols, before eventually then achieving the balanced integration of both quantitative and qualitative aspects (in what I refer to as Radial Mathematics).

From this perspective, the 3 major levels of Band 3 relate to development of the qualitative (holistic) aspect of Mathematics.

Then then just as Band 2 represents the specialised development of the quantitative (analytic) aspect, Band 4 can likewise be seen as representing the corresponding specialised development of the qualitative (holistic) aspect.

It has to be stated clearly that at the present stage of human evolution, remarkably little sustained development has yet taken place beyond Band 2.

However over the past few millennia, a very small minority of gifted human beings have indeed managed to traverse these bands and in some cases have left detailed accounts of the nature of their experience.

However even here, this has generally taken place within specific religious traditions, with major emphasis placed on the more advanced intuitive contemplative states corresponding to such development.

And in a primary sense these states indeed constitute a key feature of such development.

However, properly understand, associated with the "higher" levels of the more advanced bands are corresponding cognitive, affective and volitional structures (of an increasingly dynamic interactive nature). These then provide the basis for new distinctive types of appreciation with respect to science, the arts and morality respectively (with ultimately these three domains seen as truly integrated in a complementary manner).

In particular in this context, I have been at pains to explore the profound implications of such development for our understanding of the very nature of Mathematics.

And this quite literally constitutes a new radical dimension with respect to the more advanced bands, which has not been explored in any sustained manner to this point.

Furthermore, I believe that this new type of understanding will prove vital in coming years in helping our cultures adapt to ill-understood major social transformations indirectly resulting from a slavish adherence to the present reduced model of science.

So I would simply characterise this present model that so dominates accepted notions of Mathematics and Science as 1-dimensional.

This is manifest for example in the attempt to view observed objects as existing in an abstract manner (as external to a passive observer).

An object therefore is thereby given just one unambiguous external direction i.e. dimension.

This is especially true with respect to the conventional way of interpreting numbers that are viewed as abstract quantitative objects (that exist independent of the inquiring mind).

Therefore any consideration of how the (internal) mind interacts with the (external) objects or perhaps even more crucially how the quantitative (independent) objects can thereby be related in a qualitative (interdependent) manner, is thereby avoided.

So all real numbers are rationally represented as points on a (1-dimensional) line.

However with Band 3 development, one gradually begins to realise - rather than just one absolute interpretation (i.e. 1-dimensional) of mathematical symbols - that an unlimited number of other possible dimensional interpretations exist (all with a certain limited relative validity).

The simplest of these is 2-dimensional, where - using standard dualistic language - all interpretation has both positive and negative aspects (that are dynamically complementary).

So an object for example such as number has both external (objective) and internal (subjective) aspects which interact in dynamic fashion.

However every number also has potentially a dynamic dimensional significance.

Therefore for example, with the highly important 4-dimensional case, all objects now have both real and imaginary aspects (with positive and negative poles). So both quantitative (real) and qualitative (imaginary) aspects are dynamically interdependent in relative fashion, with both having interdependent twin poles (as objective "reality" and subjective interpretation respectively).

So this increasingly relative understanding of the nature of mathematical truth is a product of Band 3 development. Here dualistic understanding of a rational kind continually breaks down as one moves to an increasingly holistic intuitive appreciation of the nature of phenomena.

More precisely it is a product of super conscious development, which is associated with the firm belief that one is now traversing "higher" levels of development.

And in my own case this was likewise directly associated with continual refinement of the cognitive mode. So once again rigid rational notions were eroded in the unfolding of a contemplative intuitive worldview, where I sought to directly experience the interdependence of all reality (in a transcendent spiritual fashion).

This in turn led to a dramatic change in my appreciation of the nature of number.

At the conventional levels of Band 2, number is interpreted in a static rigid manner.

Indeed one of the great attractions of Mathematics for so many is the belief in the absolute nature of its symbols. This leads to the notion for example of a prime number as representing an unchanging universal form, frozen as it were in time and space. Though we may have reluctantly conceded with the advent of quantum physics that the apparent rigid nature of physical forms is but an illusion, we cling to the mistaken belief that we can still safely take refuge in the absolute nature of mathematical forms!.

However Band 3 development leads to the gradual erosion of this belief as one discovers the truly relative nature of all mathematical symbols.

This then culminates in the appreciation of number as representing a pure energy state (in contrast to earlier understanding as a rigid absolute form).

So again its is all a matter of interpretation! If one approaches number from a limited linear rational perspective (as at Band 2) it will of course appear to possess an absolute form.

If however one now approaches number from a refined intuitive perspective (in keeping with the most advanced level of Band 3) it will now equally appear to represent a purely relative existence as an energy state.

So these represent the two extreme positions of quantitative (analytic) in the former and qualitative (holistic) interpretation in the latter case, respectively.

And though the holistic understanding is directly intuitive - representing a psycho spiritual state - this properly is associated with an increasingly refined rational appreciation of a circular i.e. paradoxical nature.

However the problem with Band 3 development is that it too can become quite unbalanced. Therefore as I attempted to gain an increasingly specialised appreciation with respect to the true qualitative nature of mathematical symbols, I started to lose touch with former quantitative understanding.

So as well as the ascent to "higher" super conscious levels at Band 3, one must also undergo - for balanced development - a corresponding "lower" descent into the subconscious regions of personality, to eventually discover that these too unexpectedly possess an important mathematical significance.

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