Thursday, April 21, 2011

The Trivial Zeros (5)

As we have seen in the case of the 1st trivial zero where s (the dimensional number)
= – 2, the value of the Riemann Zeta Function = 0.

However what is vital to appreciate is that this value does not relate to a standard quantitative notion of number but rather to its alternative qualitative aspect (which is unrecognised in conventional mathematical interpretation).

Once again the deeper implications of this finding go back to the very nature of mathematical experience which combines both rational and intuitive processes of understanding in dynamic interaction with each other. However though informally mathematicians may recognise the importance of intuitive insight (especially for creative work), in formal terms Conventional Mathematics is misleadingly presented as a solely rational body of truths. And as we have seen the type of rationality allowed here is confined purely to linear type understanding based on an unambiguous either/or logic.

So the qualitative notion of number arises when one attempts to reflect its holistic (intuitive) rather that (analytic) meaning.

Now to get a clue as to what this 1st trivial zero might mean, we need to enlarge our perspective to recognise that just as there is a wide spectrum for electromagnetic energy (of which natural light comprises one small band) likewise there is a wide spectrum of potential psychological energy states (in human development) of which the rational approach that informs conventional mathematical interpretation likewise comprises just one small band.

Traditionally the higher psychological energy states - that inform pure intuitive understanding - have been largely confined to a spiritual type treatment of development (devoid of any coherent mathematical implications).

As my own evolution in thinking owes a great deal to the Spanish mystic St. John of the Cross it is worth dealing briefly with his approach here.

St. John aimed at a very pure form of spiritually intuitive awareness (free of any lingering phenomenal attachment). This required putting a high degree of emphasis on the importance of negation of such attachments.

Initially St. John emphasises what he refers to “active nights” with respect to both sense and spirit. This could be referred to the context of mathematical understanding as the dynamic negation of linear perceptions and concepts (i.e. 1-dimensional understanding).

He later goes on to more deeply emphasise the importance of the “passive nights” with respect to both sense and spirit. Again in the context of mathematical understanding this would refer to the dynamic negation of circular type perceptions and concepts (of which 2-dimensional appreciation would be the earliest representative).

So these “passive nights” would therefore include - literally - the dynamic negation of 2-dimensional interpretation (to which the first of the trivial zeros relates).
Interestingly St. John himself frequently refers to the goal of this purgative process of negation as “nada” which means nothing.

When I was at school in Ireland the term zero was not used to refer to 0 but rather nought or nothing. And so “nada” as used by St. John in fact accurately refers to the qualitative meaning of 0 (which is actually implied by the trivial zeros).

What of course St. John means by this term is a purely intuitive experience of meaning (that is free of any secondary phenomenal attachment such as rational).

So we can have two extremes with respect to mathematical meaning;
1. rational quantitative interpretation (free of holistic intuitive connotations)
2. intuitive qualitative interpretation (free of partial rational considerations)

When properly appreciated the Riemann Zeta Function incorporates both extremes.

Where s > 1, values of the function correspond to rational quantitative interpretation (i.e. as conventionally understood).

Where s < 0, values of the function correspond to holistic qualitative interpretation (that is not formally recognised in present Mathematics).

Interestingly for values of s between 0 and 1 - to which the Riemann Hypothesis directly relates - both types of meaning are involved.

The huge problem with conventional mathematical interpretation is that it cannot give a coherent explanation for values of the Function where s < 1. Obviously it will be admitted that these values arise through the recognised process of analytic continuation and clearly have considerable importance. However it has no qualitative means of explaining such importance. Rather it attempts misleadingly to reduce all values generated - even though this leads to obvious contradictions - to mere quantitative interpretation.

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