In yesterday's blog entry, I attempted to outline the true nature of the two sets of zeta zeros (Zeta 1 and Zeta 2).
As both of these sets are ultimately fully complementary with each other, they can only be properly understood therefore in a dynamic interactive manner, where they are seen to play a truly key role ensuring consistency with respect to all subsequent number operations.
So properly understood number - and indeed all mathematical relationships - entail both quantitative and qualitative aspects.
Whereas the quantitative relates to the notion of independence, the qualitative - by contrast - relates to the complementary notion of interdependence (i.e. where numbers are defined in relation to each other).
Thus number is necessarily of a dynamic relative nature entailing the interaction of both quantitative and qualitative aspects.
The relationship between quantitative and qualitative in turn leads to the corresponding interaction as between the cardinal and ordinal aspects of number, which are mutually distinct and cannot be successfully reduced in terms of each other.
So the key overriding issue with respect to the number system - and again by extension all mathematical relationships - is the prior need to ensure complete consistency in the relationship of both cardinal and ordinal meaning (reflecting its quantitative and qualitative aspects).
Here the relationship between the primes and the natural numbers is of the utmost importance, for it is through this relationship, entailing cardinals and ordinals, that both aspects (quantitative and qualitative) are mediated in a bi-directional manner.
And at the heart of this relationship between the primes and natural numbers lies a direct paradox!.
Whereas from the cardinal perspective, the primes appear as the independent building blocks of the natural number system (in quantitative terms), from the corresponding ordinal perspective, each prime is already necessarily defined by its natural number members (in a qualitative manner).
Thus, when one allows for recognition of both quantitative and qualitative aspects (which in truth are equally important) the primes are seen in dynamic terms to inherently combine two extreme complementary tendencies (with respect to independence and interdependence respectively).
Thus from one perspective, the primes appear as the most independent of numbers (in an absolute unchanging manner).
Then from an equally valid alternative perspective, the primes are understood as the most interdependent of all numbers (in a purely relative fashion approaching complete ineffability).
Therefore we must properly view the relationship between the primes and natural numbers (and natural numbers and primes) in a bi-directional interactive manner, whereby their mutual identity continually changes as between quantitative and qualitative (and qualitative and quantitative) aspects.
And remarkably this is what happens in actual experience, where appreciation of the cardinal and ordinal aspects of number keep interchanging!
However the consistent interplay of both aspects implicitly requires another two key sets of numbers, which very much mirror both cardinal and ordinal aspects i.e. Zeta 1 and Zeta 2 zeros.
Indeed in a very important psycho spiritual sense, these two sets represent the unconscious shadow counterparts to the consciously recognised cardinal and ordinal aspects of number.
Thus the role of these two sets of zeros is to enable conversion, in a fully consistent manner, as between both quantitative and qualitative (and qualitative and quantitative) aspects.
So, as we saw yesterday, the Zeta 1 (Riemann) zeros provide an (indirect) means of conversion (via the primes) from the recognised quantitative notion of cardinal numbers to their (unrecognised) qualitative counterpart notion.
Likewise from the opposite perspective, the Zeta 2 zeros again provide an (indirect) means of conversion (via the natural numbers) from the inherent qualitative nature of the ordinal numbers to their (unrecognised) quantitative counterpart notion. Now, as we have seen, this entails the simple task of obtaining roots of 1, which of course is well known in conventional mathematical terms. However the deeper appreciation, that these can in fact represent the important quantitative conversion of ordinal type notions, is completely missing!
Thus once again the key role of the two sets of zeros (Zeta 1 and Zeta 2) is to enable the perfect conversion in two-way fashion from the quantitative to the qualitative (and qualitative to quantitative) aspects of number.
Without belief in the guarantee of such consistency we would have no reason to belief in the subsequent consistency of any mathematical operation and the whole edifice would be thereby built on sand.
However this consistency, attained through the mediation of the zeta zeros, cannot be proved or disproved in a conventional mathematical manner, as its very acceptance is already implicit in the use of standard mathematical axioms.
So ultimately a massive act of faith underlies the whole mathematical enterprise.
However in moving towards a true knowledge of the number system, one must be prepared for a radical change in customary beliefs.
Indeed the most profound fact about our number system is that ultimately it operates in a totally synchronous manner, which can only be properly approached through holistic - rather than analytic - interpretation.
Associated with this is an appreciation of the extraordinary status of the dynamic interactive nature of primes as representing the purest relative form of knowledge in the phenomenal universe.
G.H. Hardy must now be surely turning in his grave!