In the last couple of entries I showed that from a dynamic interactive perspective - which is the appropriate way to view the nature of number - randomness and order are necessarily complementary aspects.
Thus, in cardinal terms, what is random from the individual perspective (i.e. each single prime) is highly ordered from the opposite collective perspective (i.e. the overall relationship pf the primes to the natural numbers).
Then in a reverse ordinal manner, what is highly ordered from the individual perspective (i.e. each natural number member of a prime group) is random from the overall collective perspective (i.e. in the selection of primes).
So when we combine both reference frames (cardinal and ordinal) the notions of randomness and order lead directly to paradox.
Now once again, it is vital to appreciate a fundamental point relating to - what I term - the analytic and holistic aspects of mathematical appreciation.
Analytic appreciation in every context takes place within single isolated poles of reference. So the analytic aspect in this crucial respect is of a rational linear (i.e. 1-dimensional) nature.
Conventional Mathematics is then formally defined in such a linear analytic manner.
This leads to an unambiguous (dualistic) interpretation of relationships in a somewhat static absolute type fashion.
So for example the interpretation of randomness and order are unambiguously separated from this perspective.
What is random therefore is - by definition - not ordered and what is ordered in thereby not random!
However by contrast holistic appreciation takes place simultaneously as between multiple reference frames. So the holistic aspect - at a minimum - requires two opposite poles in - what I term - 2-dimensional appreciation.
By its very nature holistic appreciation is inherently of a dynamic interactive nature leading to a circular (paradoxical) form of interpretation.
Now all understanding - including of course mathematical - inherently entails both analytic (Type 1) and holistic (Type 2) appreciation.
However, quite remarkably, no formal recognition whatsoever is given to the holistic (Type 2) aspect in Conventional Mathematics. Such mathematics therefore, for all its admittedly great achievements, is fundamentally of a reduced - and thereby distorted - nature. Thus it is not suited to appropriate appreciation of the true nature of the number system in the two-way relationships of the primes and natural numbers.
I have illustrated countless times before the precise nature of both analytic and holistic type appreciation respectively in terms of the two turns at a crossroads. If one heads N and encounter a crossroads, a left (L ) or right (R) turn has an unambiguous meaning; if on passing though. one then turns around heading S and once more encounter the crossroad, again either a left (L) or right (R) turn has an unambiguous meaning.
This is because interpretation is takes place in a linear analytic manner (within independent frames of reference.
However if one now considers the interdependence of both N and S, through attempting to view both polar reference frames simultaneously, then deep paradox results with respect to the interpretation of both turns at the crossroads. For what is left (L) approaching in the N direction, is right when approached from the opposite S direction; and what is right (R) when approached from the N, is left (L) from the opposite S direction.
Now implicitly this recognition that L and R turns at a crossroads have a merely relative arbitrary validity, depending on polar context, is of a holistic (2-dimensional) nature.
Again, whereas in direct terms, analytic appreciation is of a rational nature, holistic appreciation is directly intuitive (though indirectly can be expressed in a rational fashion).
Now because Conventional Mathematics has no formal recognition of the holistic - as opposed to the analytic - aspect of interpretation, this inevitably means that it cannot deal with the key notion of interdependence (in any relevant context) except in a reduced manner.
So in the context of the present discussion, randomness and order are akin to the L and R turns at our crossroads.
So what is random from the cardinal perspective is ordered from the corresponding ordinal perspective; and what is ordered from the ordinal perspective is random from the corresponding cardinal perspective.
Thus, if we are to grasp the truly relative nature of randomness and order (and order and randomness) with respect to the behaviour of the primes, we must be able to appreciate in a truly holistic manner (where opposite frames of polar reference can be mutually embraced).
However, once again in formal terms, our very appreciation of number with respect to both its individual and collective aspects is of a strictly analytic nature (viewed within a merely quantitative frame of reference).
So true holistic appreciation requires the balanced recognition that all mathematical understanding entails both quantitative and qualitative type appreciation.
So we have the customary analytic mathematical understanding with respect to the number system (where quantitative is abstracted from qualitative interpretation).
Then we have the greatly unrecognised holistic aspect of mathematical understanding (where both quantitative and qualitative aspects are understood as dynamically interdependent).
The non-trivial zeros (Zeta 1 and Zeta 2) relate directly to this latter unrecognised holistic aspect of the number system (where both quantitative and qualitative aspects of understanding are interdependent).
So, quite obviously, the non-trivial zeros - which are an inseparable component of the number system - cannot be satisfactorily interpreted in the conventional analytic manner.