Independent and Shared Nature of Primes

We saw how every number can be equally given a holistic (qualitative) as well as analytic (quantitative) meaning.

In direct terms, the latter holistic meaning is equivalent in psycho-spiritual terms to an intuitive energy state.

And as psychological and physical aspects of reality are dynamically complementary, this entails that number can equally be given a holistic meaning in terms of physical energy states.

Because of the close correspondence as between the Riemann zeros and certain data representing the excited energy states of atomic particles, this clearly suggests that the zeros relate directly to the holistic - rather than analytic - aspect of number.

However speaking still somewhat in general terms, a fascinating and crucially important point can be made regarding the holistic nature of number.

As we know modern science, with its strong quantitative bias is rooted in analytic interpretation.

However what is not all yet realized in our culture is that likewise all of the various arts are intrinsically rooted in the holistic interpretation of number.

In other words all the qualitative attributes universally manifest in created phenomena are encoded in the notion of number (when given its appropriate holistic interpretation).

So again every number, as it were, has its own unique holistic signature, which forms the fundamental basis for the qualitative aspects of all phenomena.

Therefore sometime in the future for example it will be readily acknowledged that that true aesthetic appreciation (in all its forms) is rooted in enhanced holistic mathematical appreciation.  

And sadly we still live in an age where the holistic aspect of mathematics is not even formally recognised!

It must be emphatically stress however that holistic mathematical appreciation cannot be considered as an optional add-on to present analytic understanding, as it requires a radically distinct mind-set, where increasing specialisation in a contemplative type vision of reality is required.

In the past, such specialisation in contemplative training was related to advanced spiritual practice (associated with the various religious movements East and West).

However rarely was sustained attention given - with the possible exception of the Pythagoreans - to the implication of advanced contemplative states for mathematical interpretation. So perhaps it is only now that the true need is emerging in our culture for a radically new mathematical approach.
  

Perhaps in making this point I could usefully relate my own experience where holistic understanding naturally emerged while attending University following on profound disillusionment with the conventional mathematical approach.

Then as the holistic aspect underwent considerable development, for many years I suffered a sharp decline in ability to follow the established abstract approach to mathematical problems.

And it is now only in recent years - following several decades of holistic training - that I have been gradually able to look at a fundamental problem such as the Riemann Hypothesis from a dynamic perspective (entailing both analytic and holistic aspects) where it now appears in an entirely different light.    

In some future golden age, I suggest that mathematics by its very nature will entail the balanced integration of both its analytic and holistic aspects. However we are still very far away from that day due to a completely blindness at present to the need for true holistic appreciation.

So to return to the primes, we have seen that each prime can be given both an analytic (quantitative) and holistic (qualitative) interpretation.

Again from the former perspective, each prime is unique it that it has no constituent factors (other than itself and 1). Thus it thereby serves as a quantitative “building block” of the natural number system.

However from the latter perspective, each prime is unique as it is composed in ordinal terms of a group of natural number members, which are themselves (apart from the last) unique.

Thus again for example 5 as a prime is unique from the former quantitative perspective in that it has no factors (other than 5 and 1).

Then from the latter qualitative perspective, 5 is unique in that its 1^st, 2^nd, 3^rd, and 4^th members are distinct. Indirectly this is demonstrated through the corresponding four (of 5) roots of 1, i.e. 1^1/5, 1^2/5, 1^3/5, 1^4/5 which cannot repeat for any other prime.

The final root i.e. 1^5/5, by definition = 1, and this then provides the means by which the ordinal notion is reduced in conventional mathematical terms.

So in conventional usage the ordinal positions are not treated as interchangeable, but rather fixed with the last unit member of each number group.  

Now conventionally, as we know the natural numbers (apart from 1) are obtained through a unique combination of prime factors.

So again from this perspective 6 = 2 * 3.

6 therefore is uniquely composed in quantitative terms of its two component prime “building blocks” i.e. 2 and 3.

So from the analytic perspective 6 is now likewise considered as an independent quantity in an absolute manner.

However though 2 and 3 initially can indeed be considered in isolation as independent “building blocks”, the very fact of combining them creates a new unique holistic interdependent identity. And this is a vitally important point.

Once again in isolation 2 and 3 have an independent identity as prime quantities.

However when combined together as factors of a new composite natural number i.e. 6, they now attain a shared status as prime factors of that number.

Thus there is a clear distinction to be made as between the analytic (quantitative) notion of a prime as an independent “building block” (in isolation from other primes) and the corresponding holistic (qualitative) notion of that same prime as a shared factor of a composite natural number. 

We have already seen in a previous blog entry the distinction as between the analytic and holistic aspects of number intimately relates to the complementary nature of the operations of addition and multiplication respectively.

And if one is to understand the true nature of dual sum over natural numbers and product over primes expressions (which are common to all L-functions) then this distinction is vital.