Ramanujan's Letter (7)

On reflection, I think perhaps that the holistic "conversion" that I provided yesterday for ζ(– 2), could be modified a little.

Initially, I suggested that the "converted" ζ(– 2),

= (e^2iπ – 1)/1²  +  (e^4iπ – 1)/2² +  (e^6iπ – 1)/3² +  (e^8iπ – 1)/4² + ...,  

However, more accurately it should be,

(e^2iπ – 1)/1^{– 2}  +  (e^4iπ – 1)/2^{– 2}   +  (e^6iπ – 1)/3^{– 2} +  (e^8iπ – 1)/4^{– 2}  + ...,  

In fact the standard expression for ζ(– 2) is given as,

1/1^{– 2}  +  1/2^{– 2}   + 1/3^{– 2} +  1/4^{– 2}  + ...,  

This could be then be expressed as,

e^2iπ/1^{– 2}  + e^4iπ/2^{– 2}  +  e^6iπ/3^{– 2} +  e^8iπ/4^{– 2}  + ...,  where e^2iπ = e^4iπ = e^6iπ = e^8iπ  + ...  = 1 (in a reduced quantitative manner). 

 

So in a certain sense the holistic "conversion" for  ζ(– 2) can be expressed as,

= (1 – 1)/1^{– 2} +  (1 – 1)/2^{– 2}   + (1 – 1)/3^{– 2} +  (1 – 1)/4^{– 2}  + ...     = 0.

When we interpret this result in a true holistic manner, it becomes clear what is really involved.

Now in customary analytic terms 1 – 1 = 0 carries a static absolute interpretation.

However in corresponding holistic terms 1 – 1 = 0, carries a dynamic relative meaning.

For example in physics, when a particle (+ 1) interacts with with its anti-particle (– 1), there is a fusion resulting in a pure energy state.  So we are at the other extreme here to the absolute notion of unchanging form.

Likewise in psycho spiritual terms, when the external aspect of number (as a number "object") is fully negated in an unconscious manner through a complete fusion with its corresponding internal aspect (as number "perception") again this results in an energy state representing pure intuition.

So, again we are now at the opposite holistic extreme (in this purely relative experience of number),  from the corresponding analytic extreme (where number is viewed in absolute rigid terms as an unchanging object).

Thus we have the analytic extreme where number is interpreted in a rational absolute manner.
Then we have the corresponding holistic extreme, where number is experienced directly in an intuitive manner (as the full interdependence of opposite poles).  

Now again, 2 in analytic terms, represents the quantitative understanding of this number (where its component units are interpreted in an independent manner).

However 2 in holistic terms represents the qualitative understanding of this number as "twoness" (where its component units are understood in an interdependent manner i.e. where both units are complementary in a relative fashion as + 1 and – 1, respectively.

Now positive 2- dimensional understanding (+ 2) entails the indirect rational interpretation of the complementarity of opposites (which are identified initially as separate from each other). 

However negative 2-dimensional understanding (– 2) entails the direct intuitive experience of such opposite poles as fully interdependent with each other, where objective "truth" and subjective (mental) interpretation fully merge with each other in experience.

And this is precisely what the Riemann zeta function for s = – 2 entails. In other words it represents the direct intuitive realisation of all numbers as combining both positive and negative polarities in a dynamic relative fashion.

So in this relative sense, + 1 from an equally valid opposite perspective is also – 1, + 2 = – 2, + 3 = – 3 and so on.
Thus by combining both directions for each number, it is easy to see why the Riemann function (for s = – 2) = 0. 
And by extension, as all roots of the even integers involve complementary pairings, it is easy to see why the Riemann function for s = – 2, – 4, – 6, ...  = 0.

However it is important to recognise that the symbol "0" in this context carries a true holistic meaning (i.e. as nothing in a phenomenal number manner). However it does represent a highly important psycho spiritual energy state (as the potential for number transformation).

However the clear implications of what I am saying here are utterly devastating for conventional mathematical interpretation.

As we have seen all mathematical symbols carry both an analytic (quantitative) and holistic (qualitative) interpretation, relating to the two aspects of number independence and number interdependence respectively.

Using language from quantum mechanics, all numbers carry both particle and wave aspects which keep interchanging in the dynamics of experience.
However Conventional Mathematics is build on the greatly reduced assumption that numbers can be coherently understood in an abstract absolute manner (with respect to their merely quantitative attributes).

Strictly speaking, this is utterly fallacious with the Riemann zeta function crying out for a dynamic interpretation to make coherent sense of the relative interconnections as between number.