Zeta 1 Zeros - Key Significance (1)

The Zeta 2 zeros represent the complementary qualitative aspect to the the Peano based approach (using the operation of addition) where each natural number is constructed from the addition of 1 to the previous number.

In this context we showed the unique nature of each prime in terms of this complementary qualitative approach which was seen to set a unique set of ordinal numbers up to - but not including - the prime itself.

So once more, to illustrate the prime number 5 can be uniquely defined in terms of its 1st, 2nd, 3rd and 4th ordinal members (indirectly defined in quantitative terms through the 4 (non-trivial) of the 5 roots of 1.

However, as we have seen  the final root i.e. the 5th reduces to 1 (which is then true for the final ordinal member of all the prime numbers).

Therefore paradoxically in this approach, even though each prime is uniquely defined in terms of its first (p – 1) ordinal members, the final pth member is not uniquely defined in this manner.

The problem with the Zeta 2 approach is that we must start with the pre-defined notion of the unique nature of each prime in cardinal quantitative terms, before suggesting - through the Zeta 2 zeros - its complementary holistic qualitative expression! 

So we show in turn how each prime is composed in turn - in Peano-like natural number fashion (using addition) - of units in cardinal (quantitative) and ordinal (qualitative) fashion.

Therefore from this (addition) perspective, the primes are made up of natural numbers.


However to directly demonstrate the corresponding uniqueness of the primes, we need to move in another complementary direction - using this time the operation of multiplication - where each natural number in turn is uniquely composed of primes (as factors).

This is initially achieved in the accepted standard manner through the cardinal (quantitative) composition of each natural number (using its constituent prime factors).

Then - and this point is crucial to appreciate - in a similar manner as shown through the Zeta 2 zeros, we now demonstrate that the relationship of the primes to the natural numbers (through the unique combination of factors) equally can be given a qualitative (holistic) interpretation. 

And the reconciliation of these two complementary perspectives (quantitative and qualitative) is provided by the Zeta 1 (Riemann) zeros, which again act as an essential bridge enabling the consistent interplay of both the analytic and holistic interpretation of the number symbols involved.

Again with the Zeta 2 aspect, one looks internally as it were within each prime to understand the whole/part configuration of its natural number elements, in a truly interactive manner, where both cardinal (quantitative) and ordinal (qualitative) interpretations interact.

In fact the true position is subtler than this for each quantitative aspect has an - initially hidden - qualitative aspect and likewise each qualitative aspects has an - initially hidden - quantitative aspect.

So in dynamic interactive terms we always pair complementary opposites.

So initially if we start in Type 1 terms with the base quantitative aspect (in explicit terms), then the dimensional aspect of number (in Type terms) is - relatively - qualitative in nature.

However if we alternatively start in Type 1 terms, with the base aspect now qualitative (in explicit terms,) then the dimensional aspect of number (in Type 2 terms), is - relatively - quantitative in nature.

So this parallels very closely the situation in quantum physics where waves have a particle-like aspect and equally particles a wave-like aspect, depending on relative context.


From a psychological perspective, this entails that both conscious (rational) and unconscious (intuitive) aspects ceaselessly interplay with each other - in two-way fashion - in our understanding of number.

So the standard mathematical interpretation of number - and indeed by extension all mathematical relationships - greatly distorts their true behaviour, through formally adopting a merely reduced conscious interpretation in an absolute rational manner! 

Thus with the Zeta 1 approach, one now looks externally as it were to the natural number system, as a whole, to again understand the whole/part configuration of its prime factor elements, in a truly inetractive manner, where both cardinal (quantittaive) and ordinal (qualitative) interpretations interact.

And whereas the Zeta 2 (internal) aspect highlights the use of the operation of addition, in complementary fashion, the Zeta 1 (external) aspects highlights the corresponding us of the operation of multiplication.