We have looked at the positive even numbered dimensions. These are the most suited for pure integral interpretation of reality and are always based on the matching complementarity of opposites. Within the even numbered dimensions 2, 4 and 8 would command a special importance (as suited for integral interpretation of reality).

However so far we have only considered the positive numbered even dimensions!

So our next task is to give a meaning to the corresponding negative numbers.

I have mentioned the philosopher Hegel as proving initially inspirational in terms of formulating the holistic mathematical meaning of 2-dimensional understanding.

However I gradually became disenchanted with Hegel's approach. It seemed to me that that he was led into a fundamental error in elevating the mere formal rational interpretation of his logic above the very spiritual intuition that it really illustrated. In other words though one can formulate in rational terms the basic principle that all phenomenal interactions are based on the dynamic complementarity of opposites, the true understanding of this principle can only be attained in pure intuitive awareness (where the separate identity of either pole is no longer seen to exist).

So I gradually realised that the paradoxical rational formulation itself is properly designed to lead one on to a purer spiritual intuitive realisation of the meaning it contains. And quite literally this requires the negation of any rigid attachment to the intellectual formulation.

I then discovered in St. John of the Cross a person who had addressed these very issues with a profundity that is perhaps unmatched in the Western tradition.

St. John principally in "The Ascent of Mount Carmel" and "The Dark Night" deals in detail with these problems.

Firstly he deals with the problem of the standard linear attachments that arise from normal day to day duties. So to remove such rigidities the "active nights" of sense and spirit are required. This entails the dynamic negation of linear i.e. 1-dimensional, understanding (or more correctly rigid attachment to such understanding). Sense in this context would relate to the more superficial empirical type of understanding while spirit would relate to more deep-rooted concepts and volitional capacity.

All going well in the contemplative life this leads on to the illuminative stages.

Now this would be characterised in our terms by 2-dimensional understanding with a much more spiritually intuitive worldview properly consistent with the both/and logic of the complementarity of opposites.

However St. John warns strongly of the dangers of secondary attachment to such illuminated understanding. In an intellectual context this would imply undue attachment to the rational forms by which 2-dimensional understanding is mediated (without corresponding development with respect to appropriate intuitive recognition).

So this leads on to need for the passive nights of sense and spirit. So once again whereas the active nights relate directly to linear type understanding, the passive nights relate to more refined circular appreciation (of which 2-dimensional understanding represents the earliest kind).

Thus in holistic mathematical terms, the passive nights refer to the negation of "higher" even dimensional understanding. And the goal of all this purgation is, as St. John emphatically stresses, "nada" (or nothing). In other words the purpose of passive purgation is to remove any undue attachment to circular type understanding (that mediates the spiritual light).

Now there is an extremely important - though completely unrecognised - connection here with the Riemann Zeta Function.

In the Riemann Zeta Function the values for all even valued dimensions is defined. And all cases the resulting values are expressed in terms of corresponding powers of pi.

As we have seen in the previous post, in holistic mathematical terms this rational interpretation expresses the relationship as between circular and linear understanding. So the paradoxical nature of such understanding arises from the attempt to convey circular type relationships through - the necessary use of - linear language.

So when undue importance is attached to this new rational logic, it leads to an inevitable reductionism of its true meaning (that is ultimately spiritual) in rational terms. So in the end Hegel tends to elevate philosophy (in the expression of such rational forms) over religion!

However when rigid attachment is negated with respect to these paradoxical rational forms, their true meaning is revealed in a purely intuitive manner (= 0 in phenomenal terms).

And the remarkable finding here is that in the Riemann Zeta Function, when the even dimension is negative, the value of the function is always = 0.

Now these are referred to misleadingly as the trivial zeros of the Function.

However what is entirely missing from present understanding is that these values actually relate to holistic rather than conventional mathematical interpretation.

Now if one for example tries to calculate the value of the Riemann Zeta Function when s (i.e. the value of the dimension) = - 2 in the standard conventional manner i.e. 1 + 4 + 9 + 16 + ..., the series will diverge (i.e. infinite in common expression).

However in the way the Riemann Zeta Function is defined, the value that arises from the Function = 0. So what is not explained - or indeed is even capable of explanation in standard terms - is to give such values an intelligible meaning.

However what is actually happening is that for negative values of dimensions that the Riemann Zeta Function switches from linear interpretation (where values are infinite) to holistic higher-dimensional interpretation (where values are finite).

Once again conventional mathematical interpretation is based on the default dimension of 1 (i.e. where all calculations are reduced in terms of their quantitative values in this dimension).

And for example in the Riemann Zeta Function whenever the value of s (representing the power or dimesnion involved) > 1, a finite value for the series emerges (using the standard default dimension of 1). However remarkably when the Function here is now calculated - not in terms of dimension 1 - but rather in terms of the actual qualitative dimension arising in each case, a finite result emerges.

Thus we have two methods of calculation a) the conventional linear manner (based on a default dimension of 1) and (b) the holistic mathematical calculation based directly on the qualitative dimension actually involved.

Now interestingly when s = 1, the Zeta Function diverges in conventional terms (with no finite value). And because we are here using the Function is already defined in terms of the default dimension of 1, the holistic mathematical calculation is exactly the same. And this is the one point on the Riemann Zeta Function where its value cannot be defined in finite terms (in either the standard or holistic mathematical fashion).

So when the value of s > 1) quantitative interpretation on the RHS (of the real line) can take place for values of the zeta function in the conventional manner. However qualitative interpretation of the holistic mathematical kind is required on the LHS for s < 0).

And indeed this relationship is even enshrined (again without its true significance being recognised) in the Riemann Transformation Formula!

So enhanced integral understanding actually requires two related aspects. In rational terms it requires appreciation of the higher-dimensional forms that I have been illustrating on these blogs. Though the 2-dimensional has indeed been well articulated in various spiritual and philosophical cultures, without holistic mathematical appreciation it is not really possible to formulate higher even dimensions in a satisfactory manner.

The second aspect relates to appropriate intuitive recognition of what these rational forms are supposed to represent.

Quite simply this cannot be attained without journeying through the various stages of contemplative spiritual development. So initially the rational structure of a certain dimension is posited through initial illumination. However sustained attainment of the appropriate intuition associated with this structure requires a very deep purgation or cleansing i.e. negation of the rigid phenomenal aspects of such understanding. And this is the process that spiritual writers such as St. John have already well documented.

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