Saturday, December 17, 2016

Zeta 2 Zeros - Key Significance (4)

Let is now as an illustration of the three preceding entries look at the following fractions, 1/5, 2/5, 3/5, 4/5 and 5/5 respectively.

In the Type 1 aspect of the number system, these would be represented as (1/5)1, (2/5)1, (3/5)1, (4/5)and (5/5)respectively.

In the Type 2 aspect of the number system, they would be represented as 11/5, 12/5, 13/5, 14/5 and 15/5 respectively.

Now this latter (Type 2) aspect relates to the 5 roots of 1 (with the first 4 representing the non-trivial zeros of the Zeta 2 function).

There is then a very close relationship as between the Zeta 2 zeros and the famed Euler Identity.

So eiπ   =  – 1. Therefore in - what I refer to as - the fundamental Euler Identity, e2iπ   =  1.  

And e2iπ   cos 2π + i sin 2π.

Then we can use this to calculate the 4 non-trivial zeros in this case.

These are cos 2π/5 + i sin 2π/5, cos 4π/5 + i sin 4π/5, cos 6π/5 + i sin 6π/5 and cos 8π/5 + i sin 8π/5,  i.e. .3090 + .9511 i, – 8090 +.5878 i, – .8090 – .5878 i and .3090 – .9511 i respectively.

So in effect the non-trivial Zeta 2 zeros represent the conversion of the Type 2 aspect of number - that is inherently of a qualitative nature - indirectly in a Type 1 (quantitative)  manner. 

And because of this different nature, it now appears as circular in contrast to the standard linear representation of the Type 1 aspect. Likewise in terms of logical interpretation it leads to dualistic paradox in contrast to the standard unambiguous notions  associated with the Type 1 aspect.

Now if we were to now to take a simple concrete example entailing these fractions, both the Type 1 and Type 2 aspects would apply in a complementary dynamic manner.

Imagine a  cake divided into 5 (equal) slices!

The Type 1 aspect then represents the quantitative relationship between the (part) slices and (whole) cake in this case.

So 1 slice represents 1/5, 2 slices 2/5, 3 slices 3/5, 4 slices 4/5 and 5 slices 5/5 of the whole cake.

However when interpreted in the standard manner this entails a gross form of reductionism, where the whole is defined merely in terms of its constituent parts. This is especially evident in the final case where the 5 slices (as parts) = 1 cake (as whole). 

This form of quantitative reductionism (i.e. where the qualitative aspect in any context is reduced to the quantitative) is so pervasive in accepted Mathematics, that is not even recognised as an issue!

Now when we look at this example from the Type 2 aspect, we have the complementary qualitative relationship between the (part) slices and the (whole) cake.

So we now have 1st, 2nd 3rd , 4th and 5th slices that are defined in relation to (whole) cake and thereby in relationship to each other. So the very essence of this qualitative definition is that the whole cake now assumes a distinct qualitative identity, thereby enabling these unique ordinal relationships to take place.

And, as we have seen, the first four represent the holistic options, whereby a degree of freedom exists with respect to the possible locations 1st, 2nd, 3rd and 4th (in any given context) can take. However - by definition - no degree of freedom exists in relation to the last position i.e. the 5th. So this is why - due to its necessarily fixed location in actual terms, that it thereby becomes equated with the cardinal notion of 1

And as this 5th position can be identified, though different classification criteria, with every possible option this means that each of the various ordinal positions become reduced to representing 1 unit in an actual quantitative context (given in this case by the 5th root of 1). However associated with these are the other 4 options that holistically represent the alternative ordinal possibilities involved, so that potentially each location can be identified with all ordinal positions!

So initially, the Type 1 and Type 2 aspects seem at opposite extremes to each other.

The Type 1 aspect represents the absolute extreme, whereby qualitative notions become reduced to quantitative in a pure analytic manner. Numbers then become identified in an independent fashion as representing rigid unchanging forms, corresponding to a strictly rational interpretation..

By contrast, the Type 2 aspect represents the opposite extreme of pure relativity, whereby quantitative notions become transformed in a pure holistic manner. Numbers then become identified in an interdependent fashion as representing true energy states corresponding to a strictly intuitive appreciation.
Now clearly, when seen in a dynamic interactive manner, both of these aspects are complementary with each other in b-directional terms. 

So we can neither maintain the extreme absolute position (of quantitative independence) nor the extreme relative position (of qualitative interdependence)  as both can only arise through their mutual dependence on each other.

Rather numbers now represent  and intermediate state (Type 3) with aspects that are both relatively independent and relatively interdependent (in both quantitative and qualitative terms).

And this is where  the Zeta 2 zeros are so important as they provide that essential bridge that connects both aspects with each other.

Thus from the quantitative perspective, the circular nature of the zeros (as represented indirectly in a quantitative manner) points to their hidden qualitative aspect in holistic terms.

From the corresponding qualitative perspective, the indirect quantitative representation of ordinal positions indicates how they are necessarily grounded in quantitative notions.  

So the Zeta 2 zeros are thereby once again the bridge where analytic meets holistic, and holistic in turn meets analytic understanding.

Needless t say therefore their true role cannot be grasped in the reduced manner of conventional mathematical understanding (which emphasises solely the quantitative analytic aspect of understanding).

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