Once again we have seen that in conventional terms, the primes and natural numbers are understood as independent number entities (marked by points on a straight line).
Yet at another level we must recognise a unique form of interdependence that connects both types of number.
So from a Type 1 perspective, with respect to the overall number system, the composite natural numbers represent an interdependent relationship among primes so that each composite natural number (other than 1) can be uniquely expressed as the product of primes.
However this interdependence entails both quantitative (analytic) and qualitative (holistic) aspects that cannot be properly expressed in a real independent manner (geared merely to quantitative type understanding).
So therefore when we say that for example that the number 6 is uniquely expressed in cardinal terms as the product of 2 prime numbers i.e. 2 * 3 - as well as the recognised quantitative transformation - a qualitative holistic dimension is involved (which is intimately related to the fundamental nature of the multiplication process).
This indeed indicates the key distinction as between addition and multiplication. In other words, though multiplication can be interpreted in quantitative terms as addition with for example 2 * 3 = 2 + 2 + 2, this essentially reduces its qualitative aspect in a merely quantitative manner!
What is not clearly recognised is that an alternative Type 2 perspective exists, whereby each number (internally) can be viewed as defined by a circle of (ordinal) natural number members.
Once again the primes are central to this approach with this circle of numbers (other than 1) uniquely defined for prime integers (now understanding as representing dimensional rather than base numbers).
So again to illustrate in the expression 1^3, 1 here represents a base number quantity and 3 a dimensional number (which - relatively - is of a qualitative nature).
So again for example the ordinal members (1st, 2nd and 3rd) of the number 3 are defined in quantitative terms by its 3 roots of unity, i.e. 1, – ½ +.866i and – ½ –.866i, respectively. And apart from 1, which is always one of the roots, the remaining roots are uniquely defined for all prime number integers.
Now these 3 roots in quantitative terms bear a complementary relationship with the corresponding ordinal members of 3, considered directly in qualitative terms.
This simply implies that rather than considering members as separate i.e. independent we consider them in holistic terms as interdependent.
Indirectly in quantitative terms, this interdependence can be demonstrated by adding the three roots which always (apart from 1 representing the number dimension) = 0.
So this simply indicates that pure interdependence has no meaning in quantitative terms!
What it also clearly indicates is that the very notion of interdependence can be given no coherent meaning from the conventional mathematical perspective, which is formally defined in a linear (i.e. 1-dimensional) manner.
Thus once again the conventional mathematical approach, that is qualitatively 1-dimensional in nature in formal terms, attempts to understand relationships in a merely quantitative manner.
Therefore, from this perspective the primes and natural numbers are misleadingly viewed in absolute terms as (solely) representing number quantities.
Quite simply the relationship between the primes and the natural numbers (and the natural numbers and the primes) cannot be coherently understood within the conventional mathematical perspective and no amount of abstraction or sophistication with respect to increasingly complex techniques will ever change this situation!
Indeed worse than that the very attempt to approach the key relationship in this manner will only inevitably lead even further away from true appreciation.
So in a nutshell before understanding the two-way relationship of the primes and natural numbers, the unrecognised qualitative dimension of Mathematics must be incorporated in interpretation as a fully equal complementary partner.
And this leads to an inherently dynamic interpretation of all mathematical symbols as representing the interaction of both their quantitative and qualitative aspects.
So once again the composite natural number 6 can be equally expressed as the product of two primes i.e. 2 * 3. However this can be given Type 1 (quantitative) and Type 2 (qualitative) interpretations (which are complementary).
In Type 3 terms we then combine both aspects of interpretation in a simultaneous manner.
So in Type 1 terms we treat number as representing independent entities (where the quantitative pole of understanding is clearly separated from the qualitative).
This approach in qualitative terms is 1-dimensional so that 2 = 2^1 and 3 =3^1
And the resulting product = (2 * 3)^1 = 6^1.
In Type 2 terms, both the quantitative and qualitative aspects of interpretation are considered as dynamically interdependent.
So each number - in reverse manner - relates to an integer as representing a dimensional power (or exponent) defined with respect to a default base quantity of 1.
So 2 = 1^2 and 3 = 1^3 from this perspective.
So both 2 and 3 in this context relate to the (internal) ordinal composition of the whole numbers 2 and 3 respectively
So 2 has a 1st and 2nd member while 3 has a 1st, 2nd and 3rd member.
A unique structure is associated with each dimensional number when prime, which quantitatively is obtained through obtaining its corresponding, roots.
So the unique structure of 2 (through the corresponding 2 roots of unity) in quantitative terms is given by 1 and – 1 and the corresponding unique structure of 3 (through the corresponding 3 roots of unity) in quantitative terms is given by 1, – ½ +.866i and – ½ –.866i.
Then the corresponding qualitative interpretation entails the combination of these ordinal components (represented in a circular manner) as interdependent.
So therefore the dynamic understanding of the numbers “2” and “3” from a Type 2 perspective entails the ability to recognise in both cases their separate ordinal elements in a relatively independent quantitative manner (represented by circular numbers) while equally combining these members in a qualitative holistic manner as interdependent.
In the dynamics of understanding this explicitly requires both rational (analytic) and intuitive (holistic) elements of interpretation!
So in Type 2 terms 6 as a composite dimension arises from the product 2 and 3
i.e. 1^(2 * 3) = 1^6.
So this would then generate six ordinal members (represented in circular quantitative terms) as 1, – ½ +.866i, – ½ –.866i, – 1, ½ –.866i and ½ –.866i in a relatively independent manner combined with the holistic qualitative interdependence of all members as interdependent.
Now as the prime numbers representing dimensions become very large, the dynamic interactive nature of understanding must necessarily increase so that ultimately the independent nature of each member quantitatively can no longer be explicitly distinguished from the overall interdependence of all members in a qualitative manner.
Thus this opposite extreme in understanding, of the pure interdependence of number, relates to the situation where p becomes unlimited in size (where in understanding number as material form cannot be distinguished from number as representing intuitive (psycho-spiritual) energy. The corresponding physical complement of this entails that number as material form cannot be distinguished from number as representing physical energy states!
The non-trivial zeros in Type 2 terms therefore provide the ready solution for this interdependence (with respect to quantitative and qualitative aspects) for each number (as internally conceived in ordinal terms) in a manner where its rationale can be explained in an intuitively accessible manner.
In other words this very notion of interdependence in Type 2 terms is inherently based on a dynamic appreciation of the relationship as between the quantitative and qualitative aspects of number.
By contrast the (recognised) non-trivial zeros in Type 1 terms likewise provide the ready solution for this interdependence (of quantitative and qualitative) with respect to the overall number system externally considered in a cardinal manner.
However as explained in the last blog entry, appreciation of the nature of such interdependence is not directly accessible from a Type 1 quantitative perspective, which is directly based on independent quantitative notions.
Therefore it requires the holistic appreciation (arising from the Type 2 approach) to convey in an indirect manner, how the non-trivial zeros in this case relate to interdependent - rather than independent - number notions.
Now I attempted to convey the nature of such indirect appreciation in the last blog entry.
In addition it should be stated that an inevitable uncertainty attaches to the whole process. Though in principle the set of all non-trivial zeros is of an infinite nature, clearly in phenomenal terms, we can only approximate through the understanding of a finite number of this set.
Thus the very relationship of the primes to the natural numbers is thereby of a relative - rather than absolute - nature.
Likewise, the non-trivial zeros can be only approximated in value. Thus, strictly each non-trivial zero (though given an individual identity) only has meaning in the context of the holistic collection of all zeros (which remains ultimately unknowable).
Likewise the whole set only has meaning in the context of each individual zero in quantitative terms (which likewise ultimately remains unknowable).
Therefore, the Type 3 appreciation simultaneously seeks to combine appreciation of the non-trivial zeros in both Type 1 and Type 2 terms.
This leads to the paradox that the primes and natural numbers are ultimately related to each other in an absolute ineffable manner (which cannot be identified in phenomenal terms).
Thus the two-way relationship between the primes and natural numbers (arising from both Type 1 and Type 2 appreciation) strictly represents relative approximations with respect to their ultimate identity (which is ineffable).
Such ineffable reality is variously referred to in the mystical traditions as an ultimate unity (1) that is equally a void or nothingness (0).
In like manner the very number notions of 1 and 0 are already deeply implicit in the subsequent relationship of the primes to the natural numbers (and the natural numbers to the primes).
So the numbers 1 and 0, are already implicit in all phenomena (as their potential for existence). Then bridging the phenomenal world with ineffable reality, both the primes and (other) natural numbers arise (in relative time and space) as the first dynamic interaction of both the quantitative and qualitative (and internal and external) poles which subsequently govern the course of all created evolution.
Such numbers do not therefore exist as abstract entities but rather as the inherent encoding of created phenomena (in both physical and psychological terms). And the consistent two-way relationship of the primes with the natural numbers is a necessary condition for the subsequent unfolding of phenomenal evolution (with respect to both its quantitative and qualitative aspects).
So ultimately there is a great mystery to the nature of number underlying our - seemingly - most obvious intuitions.
For example, the manner in which we readily see a one-to-one correspondence as between the natural numbers in cardinal and ordinal terms (with 1 being paired with 1st, 2 with 2nd, 3 with 3rd,...) would not be possible but for the two-way interdependence of both the primes and the natural numbers.
So the most inaccessible relationship of all is thereby necessarily already implicit in the - seemingly - most simple!
Ultimately therefore, final knowledge of the nature of number (and indeed of phenomenal reality) can only be obtained through pure experience of its unfathomable mystery.