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
Such interpretation is formally conducted within a solely quantitative frame
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
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
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
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
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
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
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
And if we are to ensure full consistency with respect to both addition and
multiplication, this translation (unlike spoken language) must be of a
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
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
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
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
0 then equates in dynamic terms with the Type 2 aspect of understanding
(carrying an ineffable qualitative significance than cannot be reduced in a
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
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
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
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.
e – 2iπ =
1 – 1.
e2iπ * e – 2iπ = e0
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
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
Indeed the Euler
Identity is intimately identified throughout with the Type 2 aspect of the
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
e2iπ = cos 2π + i
sin 2π =
Thus the second root
of unity (representing the first of the non-trivial zeros) is calculated as,
e2iπ/2 = cos 2π/2 + i
= cos 2π/2 + i
= cos π + i
= cos 1800 + i
sin 1800 = – 1.