Wednesday, February 12, 2014

Important Role of Euler Identity

We have been concentrating mainly in these blog entries on the intricate issues associated with the relationship of the primes to the natural number system (which directly relates to the fundamental nature of that system).

My strong contention is that this relationship quite simply cannot be portrayed in a coherent manner within the parameters of conventional mathematical interpretation.
Such interpretation is formally conducted within a solely quantitative frame of reference.

However as the fundamental nature of the number system in fact relates to the two-way relationship as between both quantitative and qualitative aspects, this cannot be portrayed - without gross reductionism - in merely quantitative terms.

Now the quantitative aspect of the number system can initially be directly identified with the cardinal interpretation of numbers, where they are viewed in collective terms as whole independent units.

Thus the number 3 for example will be considered in this context as an independent integer. Thus when represented in terms of its individual units i.e. 1 + 1 + 1, these by definition lack any qualitative distinction.

The corresponding qualitative aspect by contrast arises through consideration of the unique individual members of each number group.

Thus in this context 3 is composed of a 1st, 2nd and 3rd member (which are qualitatively distinct).

Then in inverse manner to the cardinal the collective sum of these members lacks any quantitative distinction!

So properly understood, the relationship as between the cardinal and ordinal aspects of number is one of direct complementarity, where both aspects can only be properly understood in a dynamic manner relative to each other.

Thus in my approach I identify the cardinal with the Type 1 and the ordinal with the Type 2 aspects of the number system respectively.

So, the number system should be considered in dynamic terms as the relationship as between its Type 1 and Type 2 aspects respectively.

In this context the relationship as between the primes and the natural numbers is seen in a dramatic new light.

From the Type 1 perspective, each natural number (other than 1) is expressed as a unique combination of prime number factors. Thus the natural number 6 is uniquely expressed as 2 * 3 in cardinal terms.

However, from the Type 2 perspective each prime number is expressed as a unique combination of members (other than 1st) in ordinal terms. So the prime number 3 is expressed by its 1st, 2nd and 3rd members in an ordinal manner. However as one member must always be independently fixed (i.e. the 1st) before the other ordinal relationships can take place, the 1st member is not unique in this sense!

So we now have two diametrically opposed perspectives on the relationship of the primes to the natural numbers. In the Type 1 case, the primes are seen as the independent building blocks of the natural number system.

In the Type 2 case first the natural numbers are seen as interdependent members of each prime number group (where each prime constitutes a unique circle of interdependence).

Therefore in a dynamic context where both aspects interact, the primes and the natural numbers are seen as perfect mirrors of each other (in an ultimately ineffable manner).

The key significance of the relationship as between the primes and the natural numbers (and the natural numbers and the primes) is that this then serves as the means through which the two-way relationship as between the quantitative and qualitative aspects of the number system are transmitted.

However this relationship can only take place indirectly through the zeta zeros.
The deeper reason for this is that - again in a dynamic context - the two key operations of addition and multiplication are quantitative and qualitative with respect to each other.

Thus if we look at number relationships for example from the quantitative (cardinal) perspective, addition and multiplication are in fact initially incompatible with each other.

This is similar to the situation where communication is initially not possible as between two people speaking two different languages (with each unable to understand the other's language).

To facilitate communication therefore it would be necessary to be able to translate each language into the other.
Likewise such a two-way translation is required with respect to the number system.
Thus for addition and multiplication to be reconciled with each other - which is the same as achieving the reconciliation of the quantitative (independent) qualitative (relational) aspects of the number system - the Type 1 aspect must be translated in Type 2 terms and the Type 2 aspect in Type 1 terms.

And if we are to ensure full consistency with respect to both addition and multiplication, this translation (unlike spoken language) must be of a perfect nature.

Remarkably, this translation is achieved through the zeta zeros. And as I have repeatedly stated in these blog entries we have in fact two sets of these zeros associated with the Type 1 and Type 2 aspects of the numbers system respectively.

The Zeta 1 zeros (i.e. the Riemann zeros) enable translation of the Type 1 aspect in Type 2 terms.
In effect this implies the conversion of the set of natural numbers in a base (cardinal) manner to a corresponding set of numbers representing dimensional values.

The unrecognised Zeta 2 zeros - which in fact are much simpler to understand - enable translation of the Type 2 aspect in Type 1 terms. In a reverse manner this enables the conversion of the set of natural numbers,  representing dimensional powers, to a corresponding set of numbers representing base quantitative values.

Again from a dynamic perspective it is easy to understand the zeta zeros (both sets) as representing the holistic extreme to our conventional fixed understanding of number in analytic terms.

Thus when understood in pure holistic terms, which concurs psychologically with highly refined intuitive awareness, both sets of zeros are understood as representing pure energy states (with both physical and psychological interpretations).

From a Jungian psychological perspective this implies that the zeta zeros represent the perfect unconscious shadow to our conventional rigid conscious notions of number.

The deeper implication of this realisation is that these numbers cannot be properly understood within the existing conventional framework of Mathematics (based formally on merely conscious analytic notions).

Thus the coherent synthesis of both the analytic and holistic aspects of number will require that Mathematics be radically amended to explicitly include both conscious and unconscious type understanding in a balanced manner. Equally this implies giving equal emphasis to both the quantitative and qualitative aspects of interpretation.

Of course existing Mathematics will still survive (and indeed be greatly enhanced). However it will then be seen as representing just one especially important case of an altogether more comprehensive paradigm.

I have frequently outlined my basic template for this new Mathematics.

1. Standard i.e. present formally accepted Mathematics (Type 1) based on a reduced quantitative frame of interpretation.

2. Holistic Mathematics (Type 2) - which is all but totally unrecognised in present culture where all mathematical symbols acquire alternative holistic meanings that directly facilitates their use in a wide variety of qualitative contexts.

In my own mathematical development, I have largely concentrated on this neglected aspect of Mathematics and have come to realise at least some of its enormous potential e.g. in terms of a precise scientific mapping of all psychological structures on the spectrum of development.

3. Radial Mathematics (Type 3) which entails the dynamic interaction of both Type 1 (analytic) and Type 2 (holistic) aspects. This open up enormous possibilities - which presently cannot even be imagined - for an entirely new type of mathematical understanding that without undue reductionism can embrace the entirety of all phenomenal investigation (in both quantitative and qualitative terms).

Though still necessarily representing but the barest introduction, the above description of the number system is representative of the most perfunctory appreciation of a Type 3 kind.

However my real intention here is to open people's minds in some small way to the extraordinary presently unrecognised potential of Mathematics.

I have stated before however that this dynamic description of the nature of the number system is in itself somewhat incomplete.

You see the relationship of the primes to the natural numbers in both Type 1 and Type 2 terms already presupposes the existence of - what I refer to as - the original numbers 1 and 0.

Now again properly understood in dynamic terms 1 directly corresponds with the Type 1 aspect of understanding (serving as the fundamental basis for all quantitative measurement).

0 then equates in dynamic terms with the Type 2 aspect of understanding (carrying an ineffable qualitative significance than cannot be reduced in a quantitative manner).
Now again initially these are incompatible with each other.

However there is an important conversion.

So 1 – 1 = 0.

Now this result might seem trivial from the standard analytic perspective.

However there is a much deeper holistic significance of immense importance.

Now 1, i.e. + 1 serves as the basis in holistic terms of 1 dimensional understanding.
This implies understanding that formally is conducted within independent frames of polar reference i.e. where objective is clearly abstracted from subjective meaning and where likewise quantitative is clearly abstracted from qualitative interpretation.

Conventional Mathematics as we know is entirely conducted from a formal perspective in 1-dimensional terms.

By contrast, in holistic terms, 1 – 1 serves as the basis for 2-dimensional understanding (often referred to as the dynamic logic of the complementarity of opposites.

I have often used the example of a crossroads to show how this is directly related to the qualitative notion of interdependence.

Unambiguous notions of left and right at a crossroads also have meaning in the context of independent polar frames of reference.

Thus if I approach a crossroads from a southerly direction, I can unambiguously define a left turn in this context.

Then If I alternatively approach the crossroads from a northerly direction again I can unambiguously assign a left turn.

However when we consider both north and south directions as interdependent, then the notion of direction becomes paradoxical. Here a turn can be both left and right.

Thus through considering both directions simultaneously, any definite knowledge of direction is thereby cancelled out. So if we designate left as + 1, then when we consider north and south as two poles simultaneously, a left turn (+ 1) is likewise not a left turn (– 1).
So the result of the turn being both left and right (in independent dualistic terms) is just a paradoxical way of expressing the qualitative notion of interdependence (which is 0 in quantitative terms).

Now the startling conclusion here is that because Mathematics is formally conducted within (isolated) independent frames of reference, it has no means of coherently accommodating the qualitative notion of interdependence (except in a grossly reduced manner).

Thus enshrined in the original numbers 1 and 0 (when considered in a dynamic interactive manner) are the two fundamental notions of  (quantitative) independence and (qualitative) interdependence respectively.
There are intimate connections here with the fundamental Euler Identity which I define as:

e2iπ  =  1.

Now more completely this can be defined as:

e2iπ  = 11.
I have used the holistic interpretation of the Euler Identity to clarify the basic meaning of 1, – 1 and 0 in the Type 2 number system.

So  e2iπ  =  11.


e – 2iπ   =  1 – 1.


e2iπ  *  e – 2iπ   =  e0  =  10.

I have identified these three holistic relationships as representing the crucial turning point where in fact form and emptiness are identified as equal in mystical terms.

So in the first case though one is largely at the peak of contemplative type unity there is still some  linear attachment to the notion of dimension as representing a point. This can be equated with a slight imbalance of the transcendent over the immanent direction of spirituality. In terms of the 10 famous ox-herding pictures used to express the stages towards achieving spiritual enlightenment, this would equate with no. 7 i.e. the bull transcended. This would equate with the proper realisation of the transcendent aspect (i.e. as emptiness beyond all phenomenal form) without yet full corresponding realisation with respect to the immanent aspect (i.e. as emptiness the source of all such form). 

The second stage then relates to the gradual erosion of this remaining attachment.

Then in the final stage both immanent and transcendent directions can be perfectly balanced. Thus the experience of dimensional nothingness equates with pure spiritual unity of phenomenal form (i.e. 10).

This would then correspond with the 8th picture commonly translated as both bull and self transcended.

So this would now equate with the balanced integration of both transcendent and immanent aspects where the ordinary is transcended in the (spiritual) extraordinary and the extraordinary equally made fully immanent in ordinary phenomena.

Interestingly an empty circle is often used to depict this stage which  replicates well the very symbol for nothingness (i.e. zero) as 0.  So quite literally this experience represents the plenum-void where 1 (as the holistic unity i.e. interdependence of all form is experienced as a pure spiritual emptiness (i.e.0) in dimensional terms.

True holistic appreciation therefore of the Euler Identity is inseparable from the experience of true spiritual enlightenment (which provides the intuitive capacity to see its relationships in the appropriate light).  

This mastery of the true holistic meaning of 1, – 1 and 0 then serves as the prerequisite for the holistic appreciation of the two-way relationship as between the primes and the natural numbers.

Indeed the Euler Identity is intimately identified throughout with the Type 2 aspect of the number system.

Thus the very means through which the Zeta 2 zeros are calculated (which directly equate with the various roots of 1) is through the Euler Identity using the equivalent expression,

e2iπ  = cos 2π + i sin 2π = 11 .

Thus the second root of unity (representing the first of the non-trivial zeros)  is calculated as,

e2iπ/2  = cos 2π/2 + i sin 2π/2 = 11/2

            =  cos 2π/2 + i sin 2π/2 = cos π + i sin π 

        =  cos 1800 + i sin 1800   = – 1.