Do Numbers Evolve? (6)

As we have seen there are two possible extremes in terms of the appreciation of number.

At one extreme we attempt to separate polarities (such as external and internal, quantitative and qualitative) in an absolute independent manner. This leads to the apparent existence of numbers as absolute fixed entities (of phenomenal form).

This in fact represents the abstract analytic approach to number that characterises conventional mathematical interpretation.

At the other extreme we attempt to view such opposite polarities ultimately as totally interdependent with each other  leading to the appreciation of number as pure energy states (ultimately of an ineffable nature).

For simplicity I refer to the first as the analytic aspect of interpretation (identified with linear reason) and the second to the corresponding holistic aspect (identified with pure intuition, that indirectly has a circular paradoxical interpretation in rational terms).

Actual experience of number is implicitly of a relative nature that necessarily falls between the two extremes. So (absolute) analytic interpretation represents just one limiting perspective that can be approached (but never fully achieved).

Likewise (purely relative) holistic interpretation represents the other limiting perspective that can be approached (but again never fully achieved).

So both aspects are in fact controlled by a fundamental uncertainty principle.

So the attempt to achieve analytic understanding (in an absolute manner) therefore tends to blot out recognition of the equally important holistic aspect; equally the attempt to achieve holistic understanding in a purely relative manner, likewise tends to block out corresponding recognition of the analytic aspect.

Conventional Mathematics is however characterised by such an extreme attention on the analytic aspect, that the holistic aspect (which in truth is equally important) is not even formally recognised.

So it must be said – and continually repeated that current Mathematics – despite its admitted great achievements in the quantitative arena is hugely unbalanced and thereby hugely distorted in nature.

Now, properly understood, the zeta zeros (Zeta 1 and Zeta 2) represent the holistic extreme with respect to mathematical interpretation (where it approaches a purely relative state).

Again it might be instructive to illustrate this with respect to the first of the Zeta 2 zeros, indirectly represented by the two roots of 2.

So these two roots, + 1 and – 1, now relate directly to the opposite polarities (such as external and internal) that condition all phenomenal experience.

Now when experience becomes highly refined in an increasingly dynamically interactive manner, one better realises that each pole only has meaning in terms of the other.

So as soon as one posits understanding with respect to one pole e.g. as a number in objective terms, one quickly realises that this has no meaning in the absence of the corresponding perception of number that is opposite and thereby negative. So now one posits the internal perception of number, before again quickly realising (directly through intuition) that is has no meaning independent of its external object.

Thus a ceaseless dynamic interplay takes place in experience as between two opposite poles that momentarily are identified as separate in quantitative terms (in a fixed rational manner). However these poles are then equally experienced as complementary and ultimately identical (in a directly intuitive manner). So through the interplay, the opposite poles continually keep switching as between their positive and negative identities.

Now indirectly this holistic understanding can be represented as + 1 – 1 = 0. And it must be clearly recognised that each pole (external and internal) has both positive and negative states that continually alternate between each other.

So here we combine the momentary quantitative existence of each pole as independent with the combined qualitative existence of both poles as interdependent.

And such quantitative interdependence = 0 (which in holistic terms entails a purely qualitative meaning i.e. without quantitative identity)

  

And if we take any prime number and then express its prime roots, all of these (except 1) will be unique in nature and cannot recur with any other prime.

So in  holistic terms, each prime number is thereby uniquely expressed through its ordinal members indirectly expressed in a quantitative manner by all roots (except 1) .

And the momentary separate identity of each root (as quantitative and independent) is perfectly balanced in each case by the collective identity of all roots (as qualitative and interdependent).

Now the ultimate limit of such understanding approaches a timeless (and spaceless) state where we can no longer distinguish the (separate) quantitative identity of each member from the (collective) qualitative identity of all members. And this represents ineffable reality (of pure emptiness).

So properly understood the evolution of the number system spans the holistic extreme of pure ineffable reality (of emptiness) and the corresponding analytic extreme of absolutely fixed phenomenal reality (of form).

Thus properly understood in experiential terms, both analytic and holistic aspects interact as matter and energy in the ceaseless transformation of number.

Now we have seen in Type 1 terms that all natural numbers are viewed in quantitative terms as the unique product of natural numbers.

So for example 6 is uniquely presented as 2 * 3.

However there is a complementary Type 2 approach to the primes where the natural numbers in ordinal terms are the building blocks of each prime.

Besides prime numbers (as dimensions) we also have natural numbers as dimensions. However the roots of these natural numbers can be directly derived from constituent primes.

So in type 1 terms,

2¹ * 3¹ = 6¹

Equally 1² * ³ = 1⁶

Then when we find the six roots of 1, holistic order is fully preserved in that these roots while preserving a relative quantitative independence can again be collective combined to give a total of zero (representing their qualitative interdependence)

In this way the primes can be seen to be unique in both Type 1 and Type 2 terms (though the order of relationship with the natural numbers is inverted in each case).

In fact both relationships - ultimately expressing the two way interdependence of primes and natural numbers - mutually imply each other.