I have been discussing the true significance of the Riemann Hypothesis as establishing an intimate correspondence as between the quantitative and qualitative interpretation of mathematical symbols.
The question then arises as to why this should be especially relevant in the context of prime numbers!
Once again we identified the quantitative (analytic) aspect of interpretation with the linear use of logic (pertaining directly to reason); then we defined the qualitative (holistic) aspect of interpretation indirectly with the circular use of logic (pertaining directly to intuition).
Now prime numbers are especially relevant in this context as they combine extremes with reference to both systems. So once we can establish a correspondence as between quantitative and qualitative interpretation in such circumstances (with respect to prime numbers) we can then easily extend this correspondence to all other numbers.
The very definition of a prime number is that it has no factors (other than itself and 1). In this way prime numbers are the most linear (and independent) of numbers. Not surprisingly from this perspective, prime numbers are thereby seen as the basic building blocks for the entire natural number system.
However what is not properly realised is that prime numbers, when used as a qualitative means of interpretation, are also the most uniquely circular of all numbers (with a structural configuration that cannot be derived from other combinations).
So if we are to establish this unique correspondence as between the quantitative (analytic) and (holistic) qualitative use of mathematical symbols, we must first establish its application with respect to the prime numbers.
As I have stated the linear quantitative interpretation of prime numbers is well established in Conventional Mathematics, where once again they are viewed as the atoms or building blocks of the natural numbers.
However the circular holistic aspect relates to their opposite characteristics (en bloc) as being intimately dependent on the natural numbers (for their precise location).
Therefore though no clear pattern is evident with respect to the individual sequence of prime numbers, an amazing regularity of behaviour characterises their general distribution with respect to the natural numbers.
So there are two opposite tendencies at work (in extreme fashion) with respect to (linear) independence of individual prime numbers and (circular) interdependence with respect to the collective behaviour of primes.
Once again though Conventional Mathematics investigates both of these aspects in considerable detail, because of its linear bias it can only do so within a quantitative approach to interpretation.
However as the very key to appreciation of primes entails maintaining correspondence as between both quantitative and qualitative aspects (pertaining to two distinct logical systems) once again their true nature is overlooked.
We can actually learn a great deal about what is involved here by looking at the dynamic nature of prime (i.e. primitive) instincts with respect to human behaviour.
In earliest infant behaviour both conscious and unconscious remain strongly embedded with each other. Indeed one can say that human life begins from the point where they are totally confused with each other. So using psychological language, neither conscious (linear) nor unconscious (circular) activity can yet be distinguished. So here we have the perfect correspondence (in undifferentiated confusion) of both the quantitative and qualitative aspects of prime behaviour.
The very essence of primitive instincts is that holistic meaning (qualitatively pertaining to the unconscious) is identified with specific phenomena (quantitatively pertaining to conscious understanding). So in this sense primitive behaviour represents the confused correspondence of quantitative and qualitative meaning.
Now from a human development problem the ultimate solution to such behaviour requires the differentiation of conscious from unconscious paving the way for integral union in spiritual terms. Thus in this mature state we have the perfect correspondence of quantitative and qualitative meaning (equally entailing the perfect correspondence of reason and intuition) .
Thus the mystery of the primes relates to an initial state (where both the quantitative and qualitative aspects of understanding exist in perfect correspondence as mere immanent potential for existence combined with their pure actualisation in existence as realized transcendent experience of this correspondence.
This means that the secret governing the behaviour of the prime numbers (in time and space) is already encoded as a perfect correspondence as between two logical systems (prior to their experiential manifestation). However equally the full realisation of this secret entails the spiritual transcendence of all lesser phenomenal understanding in space and time (where this perfect correspondence is broken).
So Hilbert was right! When properly understood Riemann's Hypothesis is pointing to a truth that is not only central to the nature of mathematics but to life itself.
In the phenomenal realm of experience, we can never fully reconcile the quantitative with the qualitative, the discrete with the continuous, order with chaos, reason with intuition etc.
It is only in pure spiritual realisation of our destiny that we can approximate a state where these opposites can at last be truly bridged.
Understanding the nature of prime numbers entails exactly the same issues. Both in the beginning and in the end a perfect correspondence exists as between the quantitative nature of prime numbers and their qualitative nature (which is inseparable from psychological development). So resolving the dynamic issues in relation to the former is inseparable from resolving the development issues in relation to the latter. So only at last when the prime problem in qualitative terms is resolved through conscious (quantitative) and unconscious (qualitative) aspects of interpretation now operating as one, can the discrete and continuous aspects with respect to the quantitative behaviour of the primes be also fully merged (transcending phenomenal experience).
In the final contribution in this series I will deal a little more with the precise significance of the Riemann Hypothesis (with respect all non-trivial zeros lying on the line with real part = 1/2)