Holistic Nature of the Primes

We are now ready to look in more detail at the cardinal nature of the primes and their relation to the natural number system.

Once again the conventional approach is to look at the primes as quantitative “building blocks” of this system.

So here, all composite natural numbers are derived through multiplication of unique configurations of prime nos.

Thus from this perspective, for example, the composite number 6 is uniquely derived from the multiplication of two primes i.e. 2 and 3. So, 6 = 2 * 3.

However, we have seen that a holistic (qualitative) transformation is necessarily involved when two or more numbers are multiplied.


Therefore, again from a dynamic perspective, if we start by viewing the separate prime numbers independently with respect to their (analytic) quantitative, then in relative terms, the resulting products of these primes must be viewed, in an interdependent manner, with respect to their (holistic) qualitative characteristics.

So, we might indeed initially start by viewing a prime such as 2 in an independent fashion. However when this prime subsequently becomes associated through the operation of  multiplication with other primes, then it literally attains a new relative status in this context, where it is no longer absolutely independent, but rather shares the qualitative characteristic of interdependence with these other primes.

Likewise we might initially view the composite natural numbers in a similar absolute quantitative fashion (as points on the same number line as the primes).

However these natural numbers likewise attain a new relative status as factors of larger composite natural numbers.

So, in this sense, a distinct qualitative status can be associated with all the factors of a natural number.

And this holistic (qualitative) nature of cardinal numbers is closely associated with the famed non-trivial zeros of the Zeta 1 (Riemann) function.


So again, this holistic qualitative status is expressed through all divisors, as factors of the natural numbers.

For example if we take the no. 12, its divisors are 2, 3, 4, 6 and 12. Now, one might also include 1, but as this by default is a factor of every natural number, it is best excluded in this context.

So notice once again the complementarity - which can only be appreciated in a dynamic interactive context - that is at work!

Thus the natural numbers as factors - which relate to the dimensional characteristic of number - complement the quantitative (base) aspect of number.

So in this sense, the divisors of each individual natural number in qualitative (dimensional) terms, lie at the other extreme to the primes, as comprising collectively, the quantitative “building blocks” of the natural number system.

And the Zeta 1 (Riemann) zeros bear an intimate relationship with these divisors.

Thus, if one wishes to count the cumulative frequency of such divisors up to a given number n, then the corresponding frequency of the zeros up to t (where n = t/2π) will provide a very good estimate.

Therefore to illustrate, the frequency of zeros - say - up to 1000, provides a good estimate of the cumulative frequency of the factor divisors of all numbers to 1000/2π  = 159 (to the nearest integer).

The frequency of zeros to 1000 = 649 and the corresponding cumulate frequency of factor divisors to 159 = 671. So, one can already see how a good approximation (97% accuracy) is achieved.

And just as with the well known general formulae for calculating the frequency of primes, ultimately when n (and t) are sufficiently large, the relative ratio of the frequency of zeros to factor divisors approaches 1.


In psychological terms, the role of the unconscious is to complement the conscious mind; likewise in mathematical terms, the role of the zeta zeros is to complement the prime numbers (in indirectly expressing their qualitative relationship with the natural numbers).

So in the terms that I customarily employ, just as the Zeta 2 zeros represent the holistic qualitative expression of the ordinal, the Zeta 1 (Riemann) zeros represent the corresponding holistic qualitative expression of the cardinal numbers.

Therefore, from the appropriate dynamic interactive perspective, the number system is composed of twin complementary aspects, where the ordinal numbers (as analytically understood) are complemented in holistic terms by the Zeta 2 zeros and where the cardinal numbers (again as analytically understood) are complemented in holistic terms by the Zeta 1 (Riemann) zeros.    

In fact the deeper psychological realisation here, in this context, is that the Zeta 1 and Zeta 2 zeros represent the hidden unconscious counterpart - now indirectly articulated in a refined conscious rational manner - of our consciously understood number system.

As reported in Constance Reid’s biography, when the great German mathematician Hilbert was once asked what mathematical problem was the most important, he is reputed to have replied “the problem of the zeta zeros, not only in mathematics but absolutely most important.”

I believe that Hilbert was indeed correct, though for reasons that he would have been perhaps loath to consider.

For the real message of the zeta zeros is that there is an unconscious basis to all mathematical understanding, which must be incorporated with conscious appreciation in a coherent integrated manner, for appropriate understanding of number and other mathematical relationships - and ultimately the entirety of all created phenomena - to take place.

In other words, analytic (quantitative) understanding of a conscious rational kind must be fully complemented with holistic (qualitative) appreciation, which is directly of an intuitive unconscious nature (though indirectly expressed in a refined rational manner).

And when with respect to mathematical understanding, we properly incorporate this holistic unconscious aspect (now brought fully to conscious recognition), its very scope is expanded immeasurably, so that we are then perhaps enabled to see that universally in nature, all phenomena - whether physical or psychological - are ultimately encoded in number.


I must say that it puzzles me how the conventional emphasis still remains so centred on the misleading notion of the primes as absolute “building blocks” of the natural number system in a merely quantitative manner. For the very metaphor that is commonly used to describe their synchronous relationship with the natural numbers i.e. “music of the primes”, points directly to a significant qualitative aspect!

And when one reflects on the matter, it becomes readily apparent that the very notion of the primes implicitly implies the natural numbers and vice versa.

So we may indeed start by attempting to consider the primes as pre-given independent entities in a quantitative manner. However on further reflection, it should then become apparent that the unique spacing as between each prime depends on the natural numbers (through the qualitative relationship of interdependence which the primes collectively have with the overall natural number system).

Thus clearly we cannot give meaning to the value of any prime in quantitative terms, without knowledge of the collective interdependence of the primes with the natural numbers in a qualitative manner (and vice versa).

In political discussions nowadays, we frequently hear the expression “Nothing is agreed until all is agreed!” Well it is somewhat similar with respect to the primes. Therefore we cannot pre-define the independent identity of any individual prime until the collective interdependent relationship of all primes to the natural number system is decided and vice versa.

Thus from a dynamic interactive perspective, the primes and natural numbers mutually depend on each other in a bi-directional synchronous manner that is ultimately ineffable.

This crucial point can also be made from another related perspective.

So often in conventional mathematical terms, the emphasis is placed externally on the distribution of the primes with respect to the overall natural number system.

However there is an equally important internal distribution as between the primes and natural numbers - the significance of which is greatly overlooked - which relates to the factors within each individual number.

So, as we have seen, each individual number is composed of a unique configuration of prime factors.

For example, 12 = 2 * 2 * 3. However because 2 occurs here twice, we would say that 12 is composed of 2 distinct prime factors (2 and 3).

However 12 can also be viewed with respect to its (natural number) divisors, which (excluding 1) are 2, 3, 4, 6 and 12.



Just as there is an important distribution externally, connecting the primes (as base numbers) with the natural numbers (with respect to the collective system), rightly understood, there is an equally important internal distribution connecting the (distinct) prime with the natural number factors (of each individual number).

And once again in dynamic interactive terms, these two distributions are analytic (quantitative) and holistic (qualitative) in two-way fashion with respect to each other.


As is well known, n/log n provides an estimate externally of the frequency of the primes (with respect to the number system as a whole).

And log n/loglog n provides a corresponding estimate internally (within each number) of the ratio of natural to (distinct) prime factors.

So if we let n₁ = log n, then this latter ratio is given as n₁/log n₁. Thus the form of the internal individual mirrors that of the external collective distribution (and vice versa).

Now, both of these estimates (where reference frames are treated separately) are conventionally understood in a merely quantitative manner.

However, when we consider these prime relationships with the natural numbers (both external and internal) - like in our crossroad example - in an appropriate dynamic interactive manner, they are then clearly seen as complementary with each other.

This entails therefore, that if the external relationship is considered in a quantitative manner, then the internal - relatively - is thereby of a qualitative nature; likewise if the internal is considered in a quantitative manner, then the external is thereby - relatively - of a qualitative nature.

In other words, both the external distribution of the primes (collectively with respect to the natural number system) and the internal distribution of prime to natural number factors (individually with respect to each natural number) possess both quantitative aspects of relative independence and qualitative aspects of relative interdependence with respect to each other.


And both external and internal distributions are mutually intertwined.

So we cannot truly consider the external distribution in the absence of the internal, nor the internal in the absence of the external.

Thus once again, the distribution in each case between primes and natural numbers (and prime and natural number factors) is mutually co-determined in a dynamic holistic synchronous manner that is ultimately ineffable.