We have briefly described in the previous blog entry the truly relative nature of the ordinal number system, entailing both quantitative (analytic) and qualitative (holistic) aspects in complementary relationship with each other.
Then the special limiting case of Conventional Mathematics, where number is treated in an absolute manner, represents the reduction of the qualitative aspect in a merely quantitative manner (in what represents 1-dimensional interpretation).
Both the relative and absolute interpretations of the ordinal nature of number are indirectly expressed in a quantitative fashion, through the circular number system (in obtaining successive roots of 1).
And as we have seen in previous entries, when we remove the one absolute case to obtain all the remaining truly relative interpretations, we are left with the Zeta 2 zeros as solutions to the finite equation,
1 + x1 + x2 + x3 + ... xt – 1 = 0 (for different values of t)
So for example when t = 3, we get,
1 + x1 + x2 = 0, so that x = – .5 + .866i and – .5 – .866i respectively.
And these two values then express - in an indirect quantitative manner - the relative qualitative ordinal notions of 1st and 2nd (in the context of a group of 3). As we have seen 3rd (in the context of a group of 3) equates with the absolute case of the ordinal notion, where it is always fixed in meaning as the last member of its respective group.
This recognition of everyday ordinal notions as - literally rooted - in "higher" dimensions of understanding (relating to the unconscious realm) is a product of Band 5 understanding (Level 2).
And these - formerly hidden - unconscious dimensions of qualitative mathematical understanding are uncovered directly in an appropriate holistic intuitive manner. However, they then can be given an indirect rational type explanation in a circular (paradoxical) logical manner.
However the very nature of Band 5 understanding is that an ever closer complementarity is established as between the "higher" super-conscious levels and corresponding "lower" sub-conscious (primitive) levels.
As we have seen with respect to the "higher" levels (in the descent to integration with the middle levels of Band 2), direct appreciation of the truly relative nature of the ordinal number system is revealed. In a complementary fashion, in the attempt to raise up - as it were - in a corresponding ascent, the "lower" levels to integration with the middle levels, direct appreciation of the truly relative nature of the cardinal number system likewise gradually unfolds.
Now, once more in the absolute quantitative manner of Conventional Mathematics, the primes are viewed as the "building blocks" of the natural number system.
However the primes represent (in their individual identities) but a special limiting component of the natural number system.
As we have seen, when the primes are used in conjunction with each other (as the unique factor combinations of composite natural numbers) they thereby attain a truly relative qualitative (holistic) status (through the interdependence of such factors).
Therefore, whereas the primes in their independent identity express the quantitative (analytic) aspect, the primes in their interdependent identity (as constituent factors of the composites) - relatively - express the qualitative (holistic) aspect of number.
Of course, as always reference frames can switch. So each individual prime likewise has an interdependent identity (as a group of ordinal members). And the multiplication of the primes (as factors) has a corresponding quantitative identity. The important point to remember is that complementarity always applies in dynamic terms as quantitative to qualitative (and qualitative to quantitative) respectively.
Thus when we start by viewing the primes in conventional quantitative terms, the collective relationship between primes (expressed through the unique factor composition of the composite natural numbers) is then - relatively - of a qualitative (holistic) nature.
And once again this qualitative relationship - now with respect to the collective relationship of the primes with each other - can indirectly be expressed in a quantitative fashion.
And this is what precisely the famed Zeta 1 (Riemann) zeros represent i.e. the indirect quantitative expression of the qualitative (holistic) nature of the primes (in their collective relationship with the natural numbers).
In fact, using more psychological terminology that would be especially familiar to Jungian followers, the Riemann (non-trivial) zeros represent the perfect shadow number system of the primes.
So when we rationally view the primes in a (conscious) quantitative manner, appreciation of the Riemann zeros should properly operate in a complementary intuitive (unconscious) qualitative manner; however when from the reverse perspective we now look on the relationship between primes in an intuitive (unconscious) manner, then the Riemann zeros should properly operate rationally in a complementary (conscious) quantitative manner.
Therefore in the dynamics of experience, continual switching takes place as between the primes and Zeta 1 (Riemann) zeros with respect, in both cases, to their quantitative (analytic) and qualitative (holistic) aspects.
However because in conventional mathematical understanding the unconscious element is so reduced in mere conscious terms, we thereby remain blind to the necessary dynamic interaction as between these two key aspects of the number system. Though the interaction still necessarily takes place implicitly (without which the number system would have no meaning) the unconscious aspect remains completely hidden (through being reduced in a conscious rational manner).
To put it more simply, if we look on the (cardinal) natural number system as a two-sided coin, if on one side we have the natural number system, then on the other we necessarily have the Zeta 1 zeros (and vice versa). However this can only be appropriately understood in a dynamic interactive manner entailing the balanced interaction of both conscious and unconscious appreciation.
So in order for the assumption to hold that all real numbers lie on a straight line (in a conscious rational manner), it is necessary for all the Zeta 1 zeros to likewise lie on a straight line (as the indirect quantitative expression of the holistic qualitative basis of the number system).
For if we think about it for a moment, it is not only necessary that all numbers (as independent entities) lie on a straight line, but equally that the relationships as between those entities (as interdependent) equally lie on a straight line.
Because of conventional quantitative reductionism we assume that both aspects (number independence and number interdependence respectively) relate to the same straight line.
However when we properly distinguish both quantitative and qualitative aspects, then if the (independent) quantitative entities lie on a real line, then the (interdependent) qualitative relationship between them must lie - relatively - on an imaginary line (as quantitative and qualitative are real and imaginary with respect to each other).
And this indeed is precisely inferred by the Riemann Hypothesis!
However clearly this cannot be proven in conventional mathematical terms. Rather it is a necessary condition that must necessarily hold so that our faith in the subsequent consistency of the entire mathematical enterprise is justified.