Thursday, August 18, 2011

The Critical Region (2)

We can perhaps illustrate more the nature of qualitative - as opposed to quantitative - type interpretation of numerical values with respect to the Riemann Zeta function for the case where s = 1.

This results again in the well known harmonic series,

1 + 1/2 + 1/3 + 1/4 + ..... which in standard interpretation diverges to infinity.

Now by subtracting the even terms (*2) from the original series we come up with the corresponding Eta series,

1 - 1/2 + 1/3 - 1/4 + ..... which does indeed converge in conventional terms giving the well known result i.e. Ln 2.

However on closer expression what we have subtracted to derive the Eta result seemingly results in the original harmonic series!

So when we multiply the even terms i.e. 1/2 + 1/4 + 1/6 + 1/8 +... etc. by 2 we thereby obtain 1 + 1/2 + 1/3 + 1/4 +... which is the original harmonic series.

Therefore the Eta series which sums to Ln 2 - on this basis - has been derived by subtracting the harmonic series from itself.

So on this logic Ln 2 = 0.

Well what does this actually mean?

Once again, we have obtained this seemingly nonsensical result through mixing 1-dimensional and 2-dimensional notions of terms. When all the terms are of the same sign, linear 1-dimensional notions apply. However when alternating positive and negative terms are involved 2-dimensional notions (in the balancing of successive positive and negative terms) apply.

So on closer inspection the quantitative result of ln 2, corresponding to the Eta series, requires that we follow one set way of ordering terms. For example if we reordered the terms consistently adding two successive even terms before subtracting

i.e. (1/2 + 1/4) - 1/3 + (1/6 + 1/8) - 1/5 +..... we would get a different result!

So when we initially subtracted the even terms (* 2) from the original harmonic series, we were using a linear (1-dimensional) logic. However in interpreting the resulting Eta Series we are using a strict 2-dimensional logic where successive terms are taken as complementary pairings.

Now the very essence of non-dimensional understanding is that it entails the direct confusion of linear and circular notions. In psychological terms we would say that such understanding remains both undifferentiated and non integrated representing mere potential for the subsequent unfolding in experience of both aspects i.e. (linear) differentiation and (circular) integration. So with successful development the first main requirement is the specialised development of linear (1-dimensional) understanding which dominates conventional scientific and mathematical understanding. Then where authentic spiritual contemplative development unfolds higher dimesnions of understanding can thereby take place.

Geometrically we could envisage the 0th dimension as the point at the centre of a circle (which equally is the centre of its line diameter).

So when we say that ln 2 = 0, we are referring to its qualitative dimensional nature (rather than its quantitative value). In other words 0 in dimensional terms relates to the direct embedding of linear (single term) interpretation that is unambiguously of one sign, with complementary (two term) interpretation where values are taken as a pair where they alternate with successive positive and negative values.

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