The nature of the non-trivial zeros initially appears as a major surprise.
So we start by attempting to discover the underlying relationship as between the primes and the natural numbers.
Now despite the problems we experience in understanding this relationship, at least these numbers appear familiar to us in possessing an unambiguous discrete form as integers.
However we have now arrived at the other extreme in unearthing a set of numbers which are vital to the very existence of the prime and natural numbers, which however possess a form that could hardly be more inaccessible to our common sense intuitions of reality.
First of all these non-trivial zeros relate not to numbers as representing base quantities but rather numbers representing dimensions (as exponents or powers).
Secondly these dimensional numbers occur in pairs as complex conjugates with the same real part = ½ and a varying imaginary part with both positive and negative signs occurring.
Thus for example the first pair of the non-trivial zeros (to six decimal places) relates to s = ½ + 14.134725 i and s = ½ – 14.134725 i respectively.
So we can solve ζ(s) = 0 for both of these values. And all zeta zeros (i.e. non-trivial zeros) occur as conjugate pairs.
However whereas the real part of these zeros appears simple enough (as a rational number) the imaginary part relates to transcendental numbers which are the most elusive and inaccessible of all numbers.
So it might seem initially strange that the relationship between prime and natural numbers (with both sets in the form of discrete integers) should depend on complex numbers (representing dimensional values) with an imaginary part that is always of a transcendental nature.
Putting it another way what appears as most accessible in our number system can be thereby seen to depend on what is least accessible!
However it has to be clearly realised that the number system contains two aspects (that are utterly distinct in terms of each other).
So we have a Type 1 aspect relating to the cardinal relationship as between the primes and natural numbers.
We have also a Type 2 aspect relating to the ordinal relationship as between the primes and natural numbers.
And though the relationship between both seems unambiguous and consistent within each isolated frame of reference, when related together they appear as paradoxical.
So we can only properly conceive of the relationship of the primes to the natural numbers in dynamic terms as quantitative as to qualitative, and qualitative as to quantitative respectively.
Thus it is in this dynamic context of the paradoxical relationship as between the two aspects of the number system (Type 1 and Type 2) that the non-trivial zeros acquire their significance.
Therefore the non-trivial zeros can fruitfully be understood as the essential underlying Type 3 aspect of the number system that is required to maintain consistency as between the two other aspects (Type 1 and Type 2).
Put more simply the non-trivial zeros are the essential requirement for maintaining consistency as between the quantitative and qualitative interpretation of number.`
Alternatively we could say that the non-trivial zeros serve as the essential requirement for reconcilation of both the analytic (independent) and holistic (interdependent) aspects of the number system.
Now of course what I am saying here can have little resonance from the conventional mathematical perspective.
Here the number system is defined solely in Type 1 terms (with an absolute rather than relative interpretation). However, the true significance of the non-trivial zeros cannot be appreciated from this static linear perspective.
When we look at the structure of each non-trivial zero, we can see that both linear and circular type components are combined.
The real part (as a rational number) conforms to linear type interpretation; however the imaginary part conforms to circular type appreciation. As we have seen the complementarity of opposites is evident in the fact that positive and negative values with respect to the imaginary part always occur together in pairs.
Also the imaginary notion itself represents the indirect expression of meaning that is of a holistic (circular) nature.
What is also of great significance is the fact that we strictly we can only ever have approximate knowledge of the imaginary part of each non-trivial zero.
For example we have expressed the first zero to its first 6 decimal places. Using the power of modern computers it would be relatively easy to obtain the value correct to 100 or even 1000 decimal places. However it still would represent but an approximate value.
Likewise the set of all non-trivial zeros is of an unlimited nature (or in reduced linear language infinite).
Therefore though many billions of these zeros have now been calculated, we can never have more than a limited finite set of all possible zeros.
This point is of the first magnitude as it clearly demonstrates that the number system in dynamic terms - which represents its authentic nature - is inherently of an approximate nature subject to uncertainty.
The root of this uncertainty relates to the inherent problem of reconciling independent number notions (that enjoy a relatively separate phenomenal existence) with interdependent notions (where no such separation exists).
Putting it more strongly, the apparent independence which numbers (such as the primes and natural) possess represents but an illusion, due to a failure to recognise the consequent interdependent nature of the number system.
And this illusion is solidly placed at the foundations of the conventional approach to Mathematics which is formally based on mere rational type interpretation (suited to such independent notions).
However as we have seen all mathematical understanding experientially entails both rational and intuitive aspects (relating to independence and interdependence respectively).
So the explicit recognition of this necessary interplay of reason and intuition leads to an inherently dynamic interpretation of number that is necessarily relative (of an approximate uncertain nature).
What I have been attempting in these blog entries is simply to show how this dynamic uncertainty necessarily applies to the key relationship as between the primes and the natural numbers (and the natural numbers and the primes).
And such appreciation arises from clear recognition that the relationship as between the primes and the natural numbers (and the natural numbers and the primes) is as quantitative as to qualitative (and qualitative as to quantitative) respectively.
So in dynamic terms we always have a necessary trade-off as between quantitative and qualitative type appreciation of number.
The more we focus on one aspect to achieve greater precision in interpretation, the more blurred as it were becomes the other aspect.
And due to the extreme focus on the merely quantitative aspect of number in Western civilisation, we no longer even recognise any interaction with a qualitative aspect leading therefore to utterly mistaken absolute notions.
So numbers - when appropriately understood - are not absolute unchanging entities with a merely quantitative meaning!
Rather they represent dynamic interaction patterns of a relative nature (as between quantitative and qualitative aspects) that are inherently subject to uncertainty.
Now it is true that in Physics we can at an everyday level use Newtonian notions very successfully in tackling a wide range of problems. However it is now more clearly recognised that these represent but a special case with respect to a more general quantum mechanical interpretation of reality that is of a relative uncertain nature.
Likewise our everyday notions of prime and natural numbers as unambiguous absolute number entities again can be used successfully at a certain level (i.e. of linear interpretation). However these strictly represent but an important special case with respect to a more general dynamic interpretation of number that is relative in nature.
So the number system inherently is uncertain though possessing the appearance of a more absolute nature when interpreted from a particular limited standpoint.
The great tragedy once again as I would see it of our mathematical history is that we have elevated this special limited case (1-dimensional interpretation) as being synonymous with all valid interpretation.
However this view, as I am attempting to demonstrate here, is utterly without foundation.
And the appreciation of the true significance of the non-trivial zeros and their essential role at the very centre of the number system will eventually unmask this great illusion that we have laboured under for several millennia.
There is even a more radical implication to the non-trivial zeros!
Once again because of the misleading absolute type interpretation we have formed of the number system, we thereby view it in abstract terms as inanimate with a merely objective independent identity.
However once we accept the truly dynamic nature of this system, representing the interaction of the fundamental poles of phenomenal existence (i.e. internal and external and whole and part) then we must accept that inherent in the number system from the very onset is a holistic type of intelligence. And the truly amazing nature of such intelligence is that - by definition - it is implicitly able to solve the most inaccessible of all problems from the very beginning in reconciling its quantitative and qualitative aspects.
This is a problem that we will spend the rest of evolution trying to properly appreciate! But again the remarkable fact is that its solution is already inherent in the number system from the moment it attains its phenomenal identity!
So the great mystery is how the most intractable of all problems (viz. the relationship as between quantitative and qualitative meaning), has already been implicitly solved when the number system becomes manifest in existence.
The only coherent answer that can be given is again that a truly wonderful holistic intelligence is already implicit in this system from its onset which later is gradually made more explicit through the process of phenomenal creation. So we will spend the rest of evolution attempting to see into this intelligence (i.e. through greater human understanding of its original nature).
We will later show how this dynamic understanding of number enables us to look at both the physical and psychological aspects of existence in a completely new light.