Wednesday, May 5, 2010

Holistic Nature of Prime Numbers

In the previous contribution, I indicated that each major stage on the spectrum of possible psychological development is closely linked with the qualitative interpretation of a corresponding number type.

We start with the two original numbers 1 and 0. 1 is directly linked in holistic number terms with conscious reality and the rational interpretation of (unitary) form.
0 - by contrast - is directly linked with unconscious reality and the intuitive awareness of spiritual emptiness (i.e. that is empty of phenomenal form).

In turn, linear interpretation - based on the either/or logic of separate polarities, is directly related with the former number (as qualitatively understood); circular interpretation - based on the both/and logic of complementary polarities - serves as the indirect expression of the latter holistic number.

At the commencement of development, both conscious and unconscious remain in an undifferentiated state; equally this represents an - as yet - unintegrated state (as successful integration depends on prior differentiation).

This equally implies that neither linear nor circular understanding can yet emerge in development.

In holistic terms, the emergence of conscious understanding (where phenomena can be differentiated) coincides with the appreciation of duality (in the first separation of opposite polarities). In this way the infant in some measure is able to obtain some initial awareness of a self as distinct from the collective environment.

So in such duality here we have the birth in holistic terms of the first prime number 2 which then serves as the basis for the holistic emergence of the further primes.

Now the very word that is commonly used to suggest the nature of such development i.e. primitive is highly suggestive of its true holistic mathematical nature.

The very essence of primitive development is that both conscious and unconscious still remain entangled with each other to a considerable extent.
Therefore in a primitive state, conscious phenomena cannot be properly separated from an overall undeveloped unconscious state. In other words both conscious and unconscious remain confused with each other.

The nature therefore of primitive instinctive behaviour is that the holistic desire for meaning (properly pertaining to the unconscious) remains embedded with specific phenomena. Initially with infant experience this confusion is so strong that phenomena remain extremely transient (disappearing as soon as they arise). Gradually however as conscious life becomes more differentiated, phenomena can assume a more stable independent existence (free of unconscious confusion).

This holistic nature of prime numbers has equally important implications for understanding of the physical universe. Initially with highly primitive matter i.e. subatomic particles, the existence of distinct phenomena cannot be properly distinguished from their holistic dimensional background. Indeed this is even recognised to a degree in string theory with dimensions in a sense understood to be contained in the strings.

However with the greater differentiation of matter, phenomena gradually attain a more stable distinct existence whereby they can be successfully placed in a dimensional context of space and time.

The key implication of all this for appreciation of the nature of prime numbers is that their inherent nature closely incorporates the operation of two logical systems that are linear and circular with respect to each other.

So just as primitive behaviour - either in psychological or physical terms - entails the entanglement of distinct phenomena with a qualitatively distinct collective context, likewise, properly understood it is the same with prime numbers.

In other words we cannot just attempt to understand prime numbers as the independent building blocks of the natural number system (that befits a linear method of interpretation).

From the overall holistic perspective, prime numbers display a remarkable degree of interdependence with the natural numbers so that their precise location intimately depends on such numbers.

And once again these two aspects i.e. quantitative independence and qualitative interdependence are properly distinct from each other (pertaining to different logical systems).

Now let us see more precisely what actually happens with the treatment of prime numbers in Conventional Mathematics.

Though the inherent nature of prime numbers corresponds to a much earlier stage of development (where qualitative and quantitative characteristics remain embedded with each other) the actual interpretation conventionally used derives from the much later middle stages of development (where linear rational understanding has achieved its specialised expression).

Therefore Conventional Mathematics attempts to understand prime numbers within an exclusively linear (rational) context even though this misrepresents their inherent nature (as properly combining both linear and circular aspects of understanding).

This linear aspect is then clearly manifest in the interpretation of prime numbers as the independent building blocks of the natural numbers.

Now in fairness considerable attention has been given also to the distribution of the prime numbers. However this has essentially been done through further extension of merely linear methods of investigation.

This is a huge problem the implications of which are yet adequately understood. For example in a static context if I walk up a road I can unambiguously identify for example a left turn. Now in a different context (where I now walk down the road) I can again unambiguously identify a left turn.

However though with respect to each independent frame of reference, both turns are unambiguous as left, in relation they each other they are necessarily both left and right.

It is exactly the same with the study of prime numbers. In a static context one can attempt to interpret the individual identity of prime numbers or their overall collective nature from within a linear rational context. However in dynamic interactive terms these two contexts are complementary requiring both linear (quantitative) and circular (qualitative) understanding.

In other words we cannot properly understood the nature of prime numbers in the absence of a holistic mathematical context. Furthermore, as I will show later, coherent appreciation of the Riemann Zeta Function incorporates results (pertaining to both conventional and holistic mathematical interpretation).

From my current perspective, I would find it patently absurd to view prime numbers - as conventionally understood - as the basic building blocks of the natural number system as equally their very location intimately depends on these same natural numbers.

Once one recognises the dual nature of prime numbers i.e. with respect to both their linear and circular characteristics (in turn requiring two distinctive methods of logical interpretation), then a key question arises with respect to maintaining consistent interpretation according to both aspects.

And in is in this reformulation of the inherent nature of a prime number that the Riemann Hypothesis obtains its true context i.e. as the fundamental requirement for guaranteeing such consistency.

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