The Holistic Nature of the Number System (10)

In this blog entry, we will look directly at the holistic nature of the famed Riemann (Zeta 1) non-trivial zeros.

Now once again in terms of the Type 1 aspect of the number system, each (composite) natural number is viewed quantitatively in cardinal terms as comprising a unique combination of prime  factors.

So for example 6 (in cardinal terms) can be expressed uniquely as the product of  the primes, 2 and 3.

Therefore 6 = 2 * 3.

Then expressed more fully (with respect to the default dimensional value of 1),

6¹ = 2¹ * 3¹

However as we have seen before, whenever two numbers are multiplied together that a qualitative (dimensional) transformation is likewise involved.

So for example, if we were to represent the product of 2 and 3 in geometrical terms we would get a rectangle with units measured in square (2-dimensional) rather than linear (1-dimensional) units.

Now we can represent this latter qualitative aspect through the Type 2 aspect of the number system.

Therefore 1⁶ = 1^{(2 * 3)}

So just as the adding of two indices implies multiplication, the multiplication of two indices implies exponentiation.

                                             ₃

So therefore 1^{(2 * 3)}  = (1²)

Now as we have seen we indirectly express this in a circular quantitative manner by obtaining the corresponding roots.

So the sixth root of 1 i.e. 1^1/6 = 1^{(1/2) * (1/3)}

Therefore, just as the cardinal notion of 6 is derived from a combination of prime factors, the ordinal notion of 6 (i.e. 6^th) is derived from a similar combination of prime roots.

Thus the 6^th root of 1 (that indirectly expresses in a quantitative manner the ordinal notion of 6) is itself derived from a combination of the 2^nd and 3^rd roots of 1.

However, calculating this 6^th root involves a process of exponentiation, where we initially obtain the 2^nd root (1^1/2) and then raise the resulting value to 1/3.

So once again when we consider the multiplication of prime factors, two distinctive processes are at work.

1)      From the quantitative perspective, the (composite) natural number in cardinal terms that results, represents a unique combination of prime factors. This reflects the Type 1 aspect of the number system.

 So for example, again from the quantitative perspective, the (composite) natural number 6 in cardinal terms represents the product of the two primes (i.e. 2 and 3) 

2)      From the qualitative perspective, the (composite) natural number in ordinal terms that results, indirectly expressed in a circular quantitative manner, likewise represent a unique combination of these prime roots (of 1).

So now from the corresponding qualitative perspective the (composite) natural number 6 in ordinal terms (6^th) represents the product (of indices) of the two prime roots.   

So when we look at this process of prime number multiplication, two complementary aspects are always necessarily involved in a dynamic interactive manner, which are quantitative and qualitative with respect to each other.

From one quantitative perspective, the unique identity of each (composite) natural number in cardinal terms is achieved; from the complementary qualitative perspective, the unique identity of each (composite) natural number in ordinal terms is achieved.

So again for example, we cannot divorce the unique composition of 6 in cardinal from the corresponding unique composition of 6^th in ordinal terms.

And both of these involve distinct mathematical processes that cannot be reduced in terms of one another.

So from this more comprehensive dynamic perspective, the conventional mathematical attempt to explain the nature of the number system in a mere quantitative manner is ultimately seen as quite futile!

Thus when one accepts the complementary dynamic interaction of two distinct processes (with respect to both the cardinal and ordinal aspects respectively), then the key issue relates to the ultimate reconciliation of both aspects within the number system.

In other words this entails (i) the manner in which cardinal and ordinal aspects can ultimately be unified as mutually identical with each other and (ii) the expression of this mutual identity in an appropriate numerical fashion?

We already saw how this identity is achieved with respect to the Zeta 2 (non-trivial) zeros.

Once again the task here is to reconcile the overall (qualitative) interdependence of  the ordinal  natural number grouping comprising each prime number, with the separate quantitative independence of each individual member.

So in the simplest case, the two roots of 1 indirectly express in a quantitative manner the ordinal nature of 1^st and 2^nd (in the context of 2 members).

Each of these ordinal members enjoys a (relatively) separate identity in quantitative terms.

However the overall qualitative interdependence of the group is expressed through their collective sum = 0.

So we can perhaps see here how this unification of cardinal and ordinal aspects is achieved with respect to each prime number (and how this relationship is numerically expressed).

Now the Zeta 1 zeros achieve a somewhat similar task with respect to the number system as a whole.

When one understands the relationships as between ordinal and cardinal appropriately, a dynamic interdependence of both is necessarily involved.

In other words the very reason why each (composite) natural number can be expressed as the product of prime number factors in a cardinal (Type 1 ) manner is dependent on the corresponding fact, that each (composite) natural number can equally be (indirectly) expressed  as the product of prime number roots in ordinal (Type 2) terms.

And of course this also works in reverse with the Type 2 relationship intimately dependent on its Type 1 counterpart.

So the zeta zeros represent the simultaneous interdependence of both sets of relationships, where both quantitative (cardinal) and qualitative (ordinal) aspects for the composite natural numbers are directly reconciled with each other.

Thus the very nature of the zeta zeros is of a holistic nature.

Remember the nature of analytic interpretation (especially as used in Conventional Mathematics) is to attempt to absolutely separate, in a static fixed manner, quantitative and qualitative aspects (and then directly reduce the qualitative aspect in quantitative terms)!

However the nature of holistic is quite the opposite where the quantitative and qualitative aspects are always seen in a necessary dynamic relationship with each other and where one thereby seeks the harmonious identity of both in a relative approximate manner.

So once again the very nature of the zeta zeros is holistic and therefore not strictly amenable to conventional mathematical interpretation.

This time in the case of the Zeta 1, we have the separate identity (indirectly) of - relatively - independent zeros, combined with the overall relative interdependence of the zeros for the number system as a whole. Thus expressed through these zeros, we have the reconciliation for the composite numbers as a whole, of  both its cardinal (quantitative) and ordinal (qualitative) features.

Then also as we have earlier seen in the case of the Zeta 2, we have again the separate identity of - relatively - independent zeros (indirectly representing the ordinal nature of the natural number members of each prime group), combined with the relative interdependence of these ordinal members (within each prime number).

And once again, in dynamic interactive terms the solutions for the two sets of zeros is obtained in a simultaneous manner (with both intimately dependent on each other).


Thus the very nature of the holistic approach is to establish such complementarity (and ultimate identity) in a dynamic interactive manner as between relationships occurring in polar opposite reference frames.

And for their appropriate understanding, the zeta zeros - par excellence - require the extreme specialisation of this approach i.e. where one can simultaneously "see" relationships in terms of both complementary perspectives.

When one properly grasps this point, it is thereby again futile attempting to grasp the nature of the zeros in the standard fixed analytic manner (based on mere quantitative interpretation)