Yesterday, I briefly attempted to explain the qualitative significance of the Euler Identity which essentially represents a holistic mathematical description of the precise nature of spiritual transformation where emptiness and form (and form and emptiness) are united. Here the contemplative journey - literally - comes full circle with the transcendent goal (beyond all phenomenal form) finally revealed as identical with its immanent source (as already inherent within such form).

Last week I was looking at a fascinating programme on the mapping of the Universe. We are of course accustomed to the mapping of planet Earth and to a lesser extent our planetary system. But ambitious attempts have already been made to provide a map of the entire Milky Way galaxy. And even beyond that considerable progress has been made with respect to the mapping of the visible universe (made up of countless billions of galaxies). Even some are already attempting to know what lies beyond the visible universe, with the current view that it is infinite in extent (or as I would prefer to put finitely unlimited with respect to its size).

Also it is quite apparent that on the grand scale that we have to move well beyond conventional notions of matter and energy. Though only dimly understood the prevailing view now is that most of the Universe is comprised of dark matter and dark energy. Now this seems to complement my own finding that when we look at Mathematics on a grand scale that the unconscious aspect of interpretation thereby assumes great significance!

So there are strong parallels here to my attempts to map both the mathematical and scientific Universe with the prevailing paradigm akin to the mapping of planet Earth. However there is so, so much more out there waiting to be discovered that we have not yet even begun to consider.

It perhaps will provide more perspective on my approach by giving a little more information on the "map of development" that I have now been using for the past few years.

I break this down into seven major bands (with each band comprising three major stages of development).

The first band is well recognised in Developmental Psychology and is concerned with the gradual differentiation of conscious type abilities (in what might be referred as the archaic, magical and mythic stages).

The second band then is concerned with the specialisation of consciousness through (linear) reason. As we have seen such thinking defines the prevailing paradigm and has become long established in - what we know as - Mathematics and Science. This is the gross realm containing both conop (concrete operational), formop (formal operational) and vision logic understanding. The third stage refers to reason that is infused with intuition, and is especially important for creative work of an original kind!

Now the second band - correctly understood - relates merely to specialisation with respect to the quantitative aspect of mathematical (and scientific) understanding i.e Type 1.

The third band is then concerned - from a scientific perspective - with the unfolding of higher dimensional understanding that is directly intuitive in nature, though indirectly can be logically expressed in a circular rational manner. The stages of this band are sometimes referred to as the psychic, subtle and causal realms and we have already looked at the nature of the higher dimensional numbers associated with these realms.

The fourth band (nondual reality) relates to both an extremely refined intuitive and rational type understanding and is concerned with the consolidation of this circular type appreciation.

So the fourth band - correctly interpreted - is concerned with complementary specialisation of the qualitative aspect of both mathematical (and scientific) understanding.

And putting it bluntly this aspect (Type 2) still remains completely unrecognised from the conventional mathematical (and scientific) perspectives.

However I now recognise three further bands on the full Spectrum of Development which I customarily refer to as radial development.

These would entail the growing mature interpenetration of reason (linear and circular) with contemplation.

In relation to both Science and Mathematics it would lead to the increasing dynamic interplay of both the quantitative and qualitative aspects of understanding. Band 5 would relate to its initial unfolding whereas with Band 6, full specialised understanding would commence. In relation to all the number types therefore, one would now seek the combined understanding of both quantitative and qualitative appreciation in what I refer to as Type 3 Mathematics.

This would likewise be associated with a final advance with respect to the qualitative aspect of dimensional understanding.

We have already looked at the distinction as between the real and imaginary notions of number with respect to dimension. However until both transcendent and immanent directions are fully harmonised in experience, these tend to be used separately.

However, with the unfolding of the radial stages, they can now increasingly be used in a simultaneous manner. So this means in effect that the qualitative aspect of dimensional numbers as complex can now be appropriately reflected in experience. This likewise entails a corresponding dynamic appreciation of the nature of time (and space) in likewise complex terms that once again applies in complementary physical and psychological terms.

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