Friday, January 13, 2017

Zeta 3 Zeros - Key Significance (1)

I use the term Zeta 3 Zeros to refer to the combined interactive recognition of both Zeta 1 and Zeta 2 zeros.

Now if we go back to the Zeta 2 zeros, we will recall that we they were initially defined with respect to each prime (representing a unique group of individual members) .

So one again using the prime number "3" to illustrate, the conventional quantitative definition treats the individual sub-units in a homogeneous independent manner (as without qualitative distinction).

Thus in simple additive terms, 3 = 1 + 1 + 1 (where each of the independent units are indistinguishable from each other).

And in more detailed fashion, this quantitative approach - that corresponds with cardinal interpretation - represents the Type 1 aspect of number i.e. where number is defined with respect to a default dimensional value of 1 (which typifies the interpretation of such numbers as points on the number line as - literally - 1-dimensional in nature).

Thus in Type 1 terms 3= 1+ 1+ 11.

The Zeta 2 zeros then in complementary fashion - which can only be properly understood in a dynamic interactive context - represent the corresponding attempt to understand "3" in qualitative fashion.

Just as the assumed independence of the 3 units forms the basis for the quantitative interpretation, the corresponding interdependence of these units forms the basis for qualitative interpretation.

Now this qualitative interpretation of "3" (as "threeness") is directly understood in an intuitive manner (just as the quantitative interpretation is directly understood in a rational manner).

This is then expressed in a Type 2 terms as 1= 0 (i.e. where the qualitative notion of "threeness" - literally - lacks any quantitative distinction.

However, if we were to leave it at this, there would be no means for coherently relating quantitative and qualitative notions of number.

So the key importance of the Zeta 2 zeros is this context is that they provide an indirect quantitative means of expressing the qualitative notion of number in a rational manner.

Seen from the opposite perspective, they provide an indirect qualitative means of expressing the quantitative notion of number in an intuitive manner

Therefore by taking the 3 roots of 1 we can break up as it were this notion of "threeness" in a relatively independent manner (as three individual components) while then relating these again in a relatively interdependent manner. So each individual root has a relatively independent status (as a quantitative value) while the collective sum of roots expresses their relatively interdependent nature (which in quantitative terms = 0).
And through this conversion, we are thereby enabled in two-way fashion, to relate the cardinal notion of number (as quantitative) with the ordinal notion of number (as qualitative) respectively.

So the key importance of the Zeta 2 zeros  is that they provide thereby a means of expressing ordinal notions of the relationship between numbers (that are qualitative in nature) indirectly in a quantitative manner.

Thus again with respect to "3", 1st, 2nd and 3rd as ordinal notions are qualitative in nature. However indirectly these can be "converted" in quantitative terms using the 3 roots of 1.

Likewise in reverse 1, 1 and 1 as the sub-units of the cardinal notion of "3" are quantitative in nature.

However, indirectly these can be likewise expressed in a qualitative manner. So 1 as the - necessary - last unit of 1) is deemed the 1st. Then 2 (as the last unit of 2) is deemed the 2nd, 3 as the last unit of 3 is deemed the 3rd and so on.

In this way the three sub-units of  the cardinal notion of 3 can be converted in an ordinal manner as 1st, 2nd and 3rd units respectively.

And this is how conventionally ordinal and cardinal notions are combined in conventional mathematical terms.

However this implies a merely analytic type interpretation (where the qualitative aspect is reduced in terms of the quantitative in a fixed manner).

So once again 1st (in this context) always refers to the last unit of 1; 2nd refers to the next additional unit i.e. as the last unit of 2; 3rd refers then to the next additional unit (as the last unit of 3) and so on.

Thus from this perspective 1 + 1 + 1 = 1st + 2nd + 3rd

However a paradox arises here. For if the 3 units are in fact indistinguishable, then 1st 2nd and 3rd can equally be identified with any unit.

However this requires a new holistic way of looking at the relationship which now implies the interdependence (rather than independence) of each unit.

This means in effect that any of the 3 units can be potentially deemed as 1st (in a ranking context).
Then when one particular unit is then actually fixed as 1st,  either of the 2 remaining units can potentially be picked as 2nd . Then when one is actually picked as 2nd, then the 1 remaining potential unit becomes synonymous with the 3rd actual unit.

Thus with respect to the last unit, potential (holistic) automatically reduces to the actual (analytic) interpretation. And this again is how in conventional mathematical terms, ordinal notions of number - which are properly of a qualitative nature - are successfully reduced in a quantitative manner.

So again here 1st has the fixed actual meaning of 1st (of 1); 2nd then has the fixed actual meaning of 2nd (as the additional last unit of 2); 3rd has the fixed actual meaning of 3rd (as the last additional unit of 3) and so on.

This equates in mathematical terms as identifying each ordinal number in Type 2 terms with the t th of the t roots of 1 i.e. 1t/t = 11 (in a Type 1 manner).

However the other t – 1 roots of 1, represent in an indirect quantitative manner, the true qualitative holistic nature of the ordinal numbers.

So again with reference to 3, the 3 roots of 1 are 11/3, 12/3 and 13/3, = – .5 +.866i– .5 .866and 1 respectively.

These represent 1st, 2nd and 3rd respectively (in the context of 3 members). However the true potential i.e. holistic meaning,where a necessary interdependence exists as to what  might be chosen as 1st and 2nd, only applies to the first 2 positions; for by definition what potentially is 3rd,when the first two positions have been filled in this circumstance, is now fixed to what actually is 3rd!  

So the Zeta 2 zeros are given in general terms by,

1 + xx+.... xt 1 = 0 (where t is prime).

Thus when t = 3, the two solutions are given by 

 x2 x+ 1 = 0.

One now realises that the ordinal notions of 1st, 2nd and 3rd, .... t th, have a purely relative meaning (depending on holistic context). So for example, 1st (of 2) is distinct from 1st (of 3) which again is distinct  from 1st (of 4) and so on.

Thus the Zeta 2 zeros from this important perspective, express - indirectly in a quantitative manner - the holistic ordinal nature of each prime number.

So again the Zeta 2 solutions for the two non-trivial roots of 3, express - in quantitative terms - the holistic natural number ordinal meaning of 1st and 2nd (in the context of 3 members).

However the paradox here is that the prime number "3" itself is not defined by these solutions. Though 1st and 2nd (of 3 members) are - indirectly - defined by the Zeta 2 zeros, as we have seen the notion of 3rd (as the last of 3 members) reduces down to its analytic meaning, where it is identified in fixed terms as 1 unit. 

And then, this is paradox is true for each of the cardinal primes, where its ordinal counterpart remains holistically undefined by the Zeta 2 zeros.

This then switches attention to the Zeta 1 (Riemann) zeros, where this key problem is addressed.

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