Where Science and Art Coincide (16)

The way to properly understand the Zeta 1 and Zeta 2 zeros is to see them - as they inherently are - as complementary pairings.

Now complementary in this context means that when we establish the relationship for one, the corresponding relationship for the other will be the inverse of the former.

And ultimately this points to the total interdependence in dynamic interactive terms of both sets of zeros.

The Zeta 2 are in fact somewhat easier to grasp and that is why I tend to start with them.

Once again higher dimensional interpretation implies the increasing interdependence of the (fundamental) polar opposites in experience. The notion of a dimension then relates to a specific dynamic configuration of these poles. So in the simplest case, 2-dimensional entails that we dynamically configure experience in terms of two real poles that are complementary opposites of each other.

In spiritual contemplative traditions, the higher dimensions are associated with an increasing refinement in the nature of intuitive awareness. And it is such intuitive awareness that provides the direct basis for the holistic (qualitative) appreciation of mathematical symbols.

However in these traditions, perhaps too much attention is placed on such dimensions as representing intuitive states (arising form the increasing nondual appreciation of interdependence).

There is also an important related aspect involved in terms of highly refined circular rational structures of a paradoxical nature, that likewise unfold (and are indeed necessary to properly consolidate the intuitive states).

In fact this structural aspect arises from the attempt to relate these "higher" dimensions  to customary experience of reality (i.e. at a 1-dimensional level).

The remarkable significance of all this is that this latter form of understanding provides the holistic mathematical manner of successfully converting ordinal appreciation of number (which is of a relative qualitative nature) indirectly in a quantitative (circular) manner.

As I have repeatedly pointed out the ordinal notion of number continually changes depending on context.

So 2nd (in the context of 2) for example is distinct from 2nd (in the context of 3).

Now the Zeta 2 zeros of the finite equation,

 ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime) = 0,

provide the solution to this dilemma enabling us to give an indirect quantitative expression of ordinal meaning (in any context).

For example to mathematically interpret 2nd (in the context of 2) we obtain the 2nd non-trivial root of 1 i.e. 1 + s¹  =  0, so that s   =  – 1.

Then to interpret 2nd (in the context of 3) we obtain the 2nd (of the 3 roots of 1) which represents the first solution of 1 + s¹  + s²   = 0 = – 1/2 + .866i.

Now all the non-trivial solutions (i.e. excluding 1) are unique where prime roots are concerned.

And once again this is the very significance of a prime in this Type 2 mathematical context, where each is uniquely defined (except 1) by its natural number members in ordinal terms.


Returning to the simplest case of 2nd (in the case of 2), the direct holistic (qualitative) interpretation of –  1, must then be conducted in a dynamic interactive context.

So what this entails is that the notion of 2nd (in the context of 2) entails the (temporary) negation of 1 (which equally implies 1st). Thus the very process of recognition of 1st and 2nd (in the context of 2) implicitly implies a continual positing and negating with respect to the two number members involved.

The deeper implication of this is that the very understanding of 1st and 2nd - which in conventional terms is mistakenly believed to be directly implied by cardinal appreciation of 1 and 2 - in fact implicitly requires an unconscious holistic aspect (of a qualitative nature) to be meaningful.

And the key significance of the Zeta 2 zeros is that they provide the means of making explicit this (unrecognised) qualitative dimension of mathematical experience.


However the implications of this could not be greater, for it truly establishes that customary understanding (in a merely quantitative manner) is quite untenable in terms of the inherent dynamics of experience.

So again put simply, again the importance of the Zeta 2 zeros is that they represent the (unrecognised) qualitative holistic aspect of the natural number system in ordinal terms.


Therefore, properly understood, the ordinal nature of number is not merely quantitative in the standard reduced manner of conventional analytic interpretation, but rather represents the dynamic interaction of two aspects that are quantitative (analytic) and qualitative (holistic) in relation to each other.


It is then easy - though not so easy to intuitively appreciate - that the Zeta 1 (Riemann) zeros represent the corresponding qualitative (holistic) aspect of the natural number system in cardinal terms.

Now, as always, the direction of interpretation changes here in an inverse manner. 

In the case of the Zeta 2 we had the problem of attempting to indirectly express the holistic appreciation of the qualitative nature of "higher" dimensions indirectly in a (1-dimensional) quantitative manner.

And we have seen that this is achieved through obtaining the corresponding roots of 1, which then provides an indirect quantitative means of individually defining the ordinal nature of number in any context.

Then again the qualitative interdependent nature of these roots is demonstrated through their sum = 0.
In other words when we can truly appreciate the relative collective nature of the ordinal members of any group, we attain an appreciation of pure interdependence  (which is nothing in quantitative terms).


In the case of the Zeta 1 zeros, we have the opposite problem of attempting to indirectly express the holistic appreciation of quantitative (base) numbers indirectly in a "higher" dimensional qualitative manner.

The basic idea here is not too difficult to express.

All composite natural numbers are uniquely expressed as the product of prime numbers.

So for example 6 is uniquely expressed as 2 * 3.

However just as a rectangular field with length 2 and width of 3 units is measured in 2-dimensional (square) units, likewise the product of these two primes leads to a qualitative (dimensional) - as well as quantitative - change in the resulting expression.

Now as always Conventional Mathematics reduces this transformation in a merely quantitative manner.
However we can begin to appreciate the true significance of the Zeta 1 zeros when we understand them as the indirect means of representing this qualitative change with respect to the cardinal number factors involved.

Now, as I have consistently stated in these blogs, dimensional numbers, in dynamic interactive terms, are qualitative with respect to given base numbers that are - relatively - quantitative.

Thus the Zeta 1 zeros (in reverse fashion to the Zeta 2) relate indirectly to dimensional numbers (of a qualitative nature).


Now qualitative always implies interdependence. So we can point here to a basic difference as between Zeta 2 and Zeta 1 zeros!

Though the meaning of 2nd for example can vary without limit (depending on its group context) it  can be given a relatively independent meaning in each case.

However with the product of primes, the resulting meaning relates always to the combination of prime factors involved.

For example 2 can be used without limit as a prime factor (in obtaining composite natural numbers). However it is always necessarily used (except where it is used itself repeatedly) in conjunction with one or more other prime numbers.

Thus the corresponding meaning of the Zeta 1 zeros relates to the qualitative interdependence of such groupings.

It is even easy to suggest why the frequency of (non-trivial) Zeta 1 zeros continually increases as we ascend (and descend) the imaginary number scale. The reason for this is related to the corresponding fact that the interdependence with respect to prime number factors (comprising composite natural numbers) likewise steadily increases as we ascend the real number scale.

Now the reason why the scale is imaginary - rather than real - is that we are using numbers indirectly to represent qualitative - rather than quantitative - notions. And the correct way of doing this is to use an imaginary - rather than - real number scale!

I have explained before the important difference as between the transcendent and immanent perspectives.

From the transcendent perspective, interdependence is seen as going beyond the individual towards a more universal collective meaning. So in this case interdependence is identified with a collective grouping (rather than each individual member). And we have seen that in terms of ordinal notions of number this is exactly the case. The interdependent nature of meaning here relates to the collective interaction of all members.

However from the immanent perspective, interdependence is seen as going within towards a more unique individual meaning (as reflective of the whole). For example this immanent aspect is well expressed through Blake's famous line "To see a  world in a grain of sand".

Now it is similar here in a mathematical context, where the overall holistic nature of the number system is reflected uniquely through each Zeta 1 zero. This indeed is why each zero represents a pure energy state in psycho spiritual terms. In other words, if we could correctly understand the Zeta 1 zeros in a highly refined intuitive manner (which requires an extremely interactive dynamic type of appreciation), they would thereby lose all phenomenal rigidity and reflect their inherent light in a pure transparent qualitative manner  

And again just as with the Zeta 2 we move from quantitative to qualitative appreciation through summation, in reverse manner we move from qualitative to quantitative appreciation through summation of all the Zeta 1 zeros.
This indeed is why the summation of Zeta 1 zeros can be used to eliminate remaining deviations in the general prediction of prime frequency (up to a given number).


Though it may not be possible yet to grasp the full implications of what is involved here, the key insight to absorb is that proper interpretation of the zeta zeros (Zeta 1 and Zeta 2) requires an entirely new mathematical approach that entails the dynamic interaction of Type 1 (analytic) and Type 2 (holistic) aspects. And remembers there is still no formal recognition whatsoever of the Type 2 (holistic) aspect of Mathematics! 

And the full blooming of such interactive understanding represent mathematical understanding - as it should be - in a comprehensive manner (i.e. Type 3). And again without recognition of the Type 2 aspect, Type 3 is inevitably reduced in merely Type 1 terms!   

So these blog entries are intended as serving as the most preliminary introduction to Type 3 mathematical understanding.

Truly, when these basic insights are grasped, it will be realised that we are thereby at the very beginning of the most important revolution yet - not alone in mathematical - but in our intellectual history.