Monday, August 12, 2013

The Holistic Nature of the Number System (7)

I will demonstrate briefly here the complementarity as between the linear and circular approaches to number.

The quantitative aspect in the linear approach is expressed through addition.

So for example 21 = 11 + 11

However the qualitative aspect - indirectly expressed in a quantitative manner - in the circular approach is expressed through addition.

So  0 = 1st + 2nd i.e. indirectly expressed in quantitative terms as 0 = + 1  – 1.

Then the qualitative aspect in the linear approach is expressed through multiplication.

So for example 13 = 11 * 11 * 11

However when this is expressed in a circular manner - in an indirect quantitative fashion through the 3 roots of 1 - we arrive back at the quantitative aspect of number,

i.e.  1 * (– .5 + .866i) * (1 (– .5 – .866i) = 1

Thus for any prime number p (except 2) the product of its p roots = 1, while the sum of its p roots = 0.

Now the analytic interpretation of this is of course well known. However there is a much deeper corresponding holistic significance in demonstrating the complementary nature of addition and multiplication (when reflected through linear and circular frameworks respectively).

So whereas Conventional Mathematics can indeed analyse circular relationships in quantitative terms, it necessarily does so analytically from within a rational linear framework.

However to properly interpret these relationships from a corresponding holistic perspective, we must do so within a rational circular framework (where paradoxical complementary relationships indirectly reflect holistic understanding that is directly of an intuitive nature).

Once again this holistic aspect is entirely missing from Conventional Mathematics (as formally understood).

So when we appreciate mathematical relationships in an appropriate holistic manner, we are then enabled to seamlessly convert from quantitative to qualitative aspects respectively through continually switching in turn as between linear and circular modes of interpretation.   

Perhaps the most important holistic mathematical appreciation of all relates to the qualitative significance of the imaginary notion.

Imaginary numbers are indeed used mow extensively in quantitative analytic terms within Conventional Mathematics.

However the corresponding qualitative holistic interpretation of imaginary numbers - which is equally important - is entirely missing from conventional mathematical interpretation.

I have found my acquaintance with Jungian notions extremely helpful in arriving at the holistic mathematical meaning of the imaginary.

Again in Jungian terms, when a conscious function such as thinking is unduly dominant, its unconscious (shadow) complement is projected in a somewhat blind manner on to conscious phenomena.

Now in scientific terms, “reality” is defined in a merely conscious rational manner which is properly geared to analytic type interpretation.

However we are all perhaps aware that our actual experience is conditioned to a degree by unconscious type projections (which relate to a distinctive holistic meaning).

So when we experience objects there is indeed a specific local aspect (according with conscious recognition).

Likewise however there is a universal holistic aspect (according with unconscious meaning).

For example one might speak of a “dream house”. Now the house indeed can be consciously verified as an object; however there is also a holistic aspect here in the experience that serves a deeper (unconscious) holistic desire for meaning.

Now the “imaginary” in qualitative terms, simply refers to objects as indirectly representing this holistic aspect of meaning.

In this sense all reality is complex with objects having a - relatively separate - local identity (as real) and a whole relational identity - as imaginary.

As we have seen, this is equally true of all mathematical objects (such as numbers).

Indeed the ordinal nature of number properly relates to its qualitative holistic identity.

In direct terms this relates directly to an unconscious - rather than conscious - recognition that is then indirectly expressed rationally in a circular (paradoxical) fashion. It then can be converted, as it were, into linear type interpretation through giving it an “imaginary” rather than “real” identity.

As we have seen, in quantitative terms,  the imaginary number i is given as the square root of  – 1.

Now, in the context of the Zeta 2 Function, – 1 is all important, serving in qualitative terms as the expression of unconscious meaning. In other words, in experiential terms, the very way in which the unconscious is activated in experience, is through the dynamic negation of conscious type meaning.

So once again, in Jungian terms, if the emphasis on conscious experience is unduly dominant (as it most certainly is in Conventional Mathematics), then the unrecognised unconscious aspect will be blindly projected onto consciously understood phenomena (and then directly confused with them).

And of course this is  precisely what has happened in Mathematics. Though ordinal appreciation is qualitative - inherently relating to the unconscious - this is blindly projected on to the conscious recognition of mathematical objects (as quantitative) and directly confused with them.

So we are still labouring under the huge fallacy that the number system can be understood as quantitative in a merely reduced manner, though in truth we can give no meaningful order to this system without qualitative recognition.

And without such qualitative order in an overall context for relating mathematical objects, the quantitative aspect itself can have no strict meaning!

So again the definition in quantitative terms of i is the square root of – 1; 

In holistic terms, – 1 defines the 2nd dimension. So to indirectly express this in terms of the 1st we again obtain the square root.

What this means in effect is that from a holistic perspective, the imaginary notion relates to the indirect expression (in a linear quantitative manner) of meaning that is properly of a qualitative nature.

If we then apply this to the Zeta 1 zeros, they are all postulated to lie on an imaginary line (through .5).

What this holistically implies, is that all these zeros directly represent a qualitative - rather than quantitative -meaning.   
In other words the very nature of the zeta zeros is that they represent an indirect quantitative way of expressing the qualitative (ordinal) nature of the number system as a whole.  

Now the Zeta 2 zeros express this holistic nature more directly (in a circular manner) thus demonstrating the unique qualitative nature of each prime number group (representing a individual natural number members in an ordinal relationship with each other).

The Zeta 1 zeros express this holistic nature more indirectly (in an imaginary linear manner) and demonstrate the unique qualitative nature of the (composite) natural numbers (representing a unique grouping of prime numbers in ordinal relationship with each other).

So put most simply, the zeta zeros represent, from two distinct perspectives, the (hidden) qualitative nature of our number system.

Though this representation is necessarily given an indirect quantitative expression, as its true nature is holistic, it cannot be successfully interpreted in a conventional mathematical manner (that is exclusively concerned with quantitative meaning).

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