Zeta Zeros and the Changing Nature of Number (7)

In the last blog entry, we dealt with the paradoxical nature of the Riemann (Zeta 1) zeros.

Thus again in Type 1 terms the the natural numbers are uniquely derived from prime factors in a quantitative manner.

Then in complementary Type 2 terms, the prime numbers are uniquely derived from the natural number factors (of the composites) in a qualitative manner.

In other words through the interdependence that arises (by multiplication) of the prime numbers with each other, their qualitative nature is thereby expressed.


Once again the very essence of multiplication - as opposed to addition - is that it creates this interdependence (of  a qualitative nature) as between units.

Therefore again in the simplest case when we multiply 1 by 1 i.e. 1 * 1 = 1² , though the (reduced) quantitative value remains unchanged as 1, the dimensional nature of the units is changed (in a qualitative manner).

So here the 1st dimension is necessarily related to the 2nd dimension (so that each must be considered in the context of each other as interdependent). Indeed it is this interdependence that is inherent to the nature of dimensions, that enables ordinal distinctions between numbers to be made!

 By contrast when we add 1 and 1 i.e. 1 + 1 = 2¹, though the qualitative nature of the units remains unchanged as 1, the base units are transformed (in a quantitative manner).

In this way the operations of addition and multiplications are themselves understood as complementary in nature (with both quantitative and qualitative attributes respectively).

Now we can separate the pure nature of addition and multiplication through concentration on the limiting case where the dimensional and base units are fixed at 1 respectively.

However, when we seek to combine the primes, through multiplication, as for example 2 * 3 (where the base quantities are no longer 1) both a quantitative and qualitative transformation is involved.

And we have shown above, the manner in which both the quantitative and qualitative aspects of such number transformation, are expressed through the Type 1 and Type 2 interpretations of the number system respectively.


Now the situation here is very much like our example of the crossroads. When we consider just one direction of approach to the crossroads (either N or S), left and right turns can be given unambiguous meanings.

Likewise when we consider the relationship of the primes to the natural numbers, from either the Type 1 or Type 2 aspects of interpretation, an unambiguous direction to this relationship can be given.

However, as we know, when we try to combine both N and S directions of approach to a crossroads simultaneously, the very notion of left and right turns is rendered paradoxical. So what is left from one perspective is right from the other, and what is right from one perspective is left from the other!

Similarly, when we simultaneously attempt to view the relationship of the primes to the natural numbers from both the Type 1 and Type 2 perspectives (i.e. in Type 3 terms) again we are left with pure paradox. So what is prime from one perspective, is a natural number from the other; and what is a natural number from one perspective is a prime from the other.

So the remarkable nature of Type 3 understanding is the realisation that the primes and natural numbers are ultimately fully interdependent in an ineffable manner.

However such complete interdependence can only be approximated in the phenomenal realm.

So the Riemann (Zeta 1) zeros in effect express the approximation to this state (where the primes and natural numbers are fully interdependent).

Though numbers of a complex nature (with the imaginary part of a transcendental nature) are used to represent these zeros, they truly represent the closest one can approximate in the phenomenal realm to pure energy states.

Thus, properly understood, the Riemann (Zeta 1) zeros lie at the opposite extreme to the conventional understanding of number.

Conventional understanding (including of course the primes and natural numbers) is based on completely rigid notions of form based on the clear separation of opposite polarities such as external/internal and whole/part. In this way we can absolutely separate the primes and (composite) natural numbers with the direction of causation one-way as between the primes and natural numbers in a merely quantitative manner.


However properly understood, the Riemann (Zeta 1) zeros represent the opposite extreme, approaching pure relativity in an ineffable manner, where the opposite polarities are understood in dynamic manner as complementary and ultimately identical with each other.

In this way the very nature of the primes and natural numbers is ultimately understood as fully identical with each other, though this state can only be approximated in phenomenal terms.

So the Riemann (Zeta 1) zeros represent the complementary holistic extreme to the analytic conventional interpretation of the primes and natural numbers.

In psychological terms theses zeros thereby represent the (holistic) unconscious basis of our standard (analytic) conscious interpretation of the cardinal number system.


In similar fashion, the Zeta 2 zeros represent the corresponding holistic extreme to the conventional ordinal interpretation of number. So again in this regard they serve as the (holistic) unconscious basis of the standard (analytic) conscious interpretation of the ordinal number system.