In yesterday's blog entry, we saw how in truth every phenomenon possesses a dual identity. Thus from the conscious perspective of understanding, it exhibits an actual specific identity (as its real aspect); likewise from the unconscious perspective it exhibits a potential holistic identity (as its imaginary aspect).
And we saw - in a precise qualitative mathematical manner - how the imaginary represents an indirect means by which the potential aspect thereby becomes embodied in the phenomenon (along with its actual identity).
Thus all phenomena in Type 2 mathematical terms are of a complex nature (combining simultaneously both real and imaginary aspects).
Again drawing on insights from Jungian psychology, we saw how the imaginary (holistic) aspect of understanding is frequently not properly recognised in experience (but rather blindly projected on to phenomena).
So in this manner the imaginary aspect represents the unrecognised shadow side of personality.
When however the unconscious side of personality undergoes appropriate development (as for example in a spiritual contemplative context), the imaginary aspect becomes increasingly refined as intuitive insight which can then - literally - illuminate actual understanding in a wonderful holistic manner where every phenomenon potentially serves in some degree as a potential archetype of a more universal meaning.
And because in dynamic terms both the physical and psychological aspects serve as mirrors to each other, this has an intimate bearing on the true nature of physical and mathematical reality (when appropriately understood in a dynamic interactive manner).
In other words in dynamic experiential terms, both the physical and psychological aspects of reality are complementary. Therefore we cannot ultimately divorce the nature of mathematical (and physical) reality from the psychological means through which it is interpreted.
So expressing this in Type 2 terms from the qualitative perspective all mathematical phenomena possess both real and imaginary aspects (corresponding to specific and holistic interpretation respectively).
Now, once again the reason why this is not readily apparent in conventional (Type 1) terms is due to the fact that qualitative type considerations are here necessarily reduced in a mere quantitative type manner.
So, though Type 1 Mathematics does indeed recognise both real and imaginary aspects of numbers as quantities, from a Type 2 qualitatively perspective, it operates within a paradigm that is solely real!
And of course a key consideration of mine in writing a blog such as this is to keep indicating from a variety of different angles what the present unrecognised imaginary aspect of Mathematics might entail!
All this has an intimate bearing on the interpretation of prime numbers, with the full implications being of a truly startling nature.
So each prime number - as is well recognised in Type 1 terms - has a real (i.e. quantitative) identity. So no one will have any difficulty in recognising prime numbers such as 2, 3, 5, 7, ... as real quantities!
However, we are also now saying that each prime number possesses an imaginary (i.e. qualitative) identity. And this is is where the thinking that has dominated our perceptions for several millennia now will have to change!
Now, to some extent those who are familiar with the Riemann Zeta Function will indeed recognise that associated with the primes is a hidden harmonic structure related to the non-trivial zeros of the Function and will even appreciate the imaginary nature of these zeros (in quantitative terms). In other words the non-trivial zeros actually represent measurements made on an imaginary scale.
Furthermore, they might even accept that these zeros operate in a holistic manner thereby helping to connect the individual primes, which seem at first sight randomly distributed without apparent order, to the natural numbers in a collective manner thus revealing an amazing degree of regularity.
However the mistaken notion still persists that the holistic nature of the zeros can be studied in a merely quantitative manner.
And this is to miss the central point that in direct terms, the nature of this holistic connection is of a qualitative - rather than quantitative - nature.
Once more it might help to appreciate this point with reference to the crossroads analogy I have been employing in recent contributions.
So, using isolated frames of reference, in keeping with the conventional mathematical approach, it is true that both the individual primes and their general frequency (among the natural numbers) can indeed be interpreted in a quantitative manner.
However the crucial insight that is required is the recognition that when we attempt to view both of these aspects as interdependent (in two-way relationship to each other) the individual and general nature of the primes are as quantitative to qualitative (and qualitative to quantitative) to each other respectively.
What this further entails is that the hidden dynamic which in a sense constitutes the true nature of the primes is not in fact amenable to Type 1 mathematical interpretation.
Rather it directly corresponds to a qualitative holistic relationship (which in turn can only be directly appreciated in an intuitive - rather than rational - manner).
Now clues as to what this means are already provided through quantum physics, where an individual sub-atomic particle (such as with light) can seemingly maintain a flawless communication with all other particles thereby ensuring perfect synchronicity in behaviour at a collective level.
We need to look at the primes in the same manner! Indeed putting it more emphatically the very communication capabilities displayed by light particles at the quantum level are rooted in a more perfect communication ability at the level of the primes!
So communication in this holistic sense does not operate in a manner amenable to interpretation through sequential linear reason. Rather it operates in a direct simultaneous manner through which regularity at a collective level is maintained.
And in complementary psychological terms it is intuition - rather than reason - that preserves this qualitative holistic function at a human level of organisation.
In all natural and human processes the two aspects of differentiation and integration are at work. And through these two aspects necessarily interact, they are quite distinct. The former serves a quantitative role whereas - by contrast - the latter serves a corresponding qualitative holistic function.
Now, it is the very nature of conventional mathematical procedures to continually reduce the holistic qualitative nature of integration in quantitative terms (as differentiated phenomenal objects).
However what I am saying clearly is that this need not necessarily be the case and that by adopting a more comprehensive mathematical approach (in keeping with the dynamics of mathematical experience) that the discipline can be liberated in ways that currently might seem entirely unimaginable!
Properly understood prime numbers serve as the prototype for all subsequent physical (and psychological) processes in creation.
As we have seen prime numbers combine the two complementary features that govern subsequent created activity (in their extreme form).
Thus, from an individual quantitative perspective, they seem the most independent of all objects. In this respect they perfectly embody the masculine principle. And the current mathematical paradigm likewise embodies the extreme version of this masculine principle!
However - in relative terms - from a holistic qualitative perspective, they appear the most interdependent of all objects (possessing an amazing holistic ability to communicate with other primes). They thereby preserve a perfect collective synchronicity of behaviour and in this respect they embody the extreme version of the feminine principle.
Now it is true that this holistic capacity of the primes is indeed well recognised in some respects. However the crucial limitation of present understanding is the attempt to identify it in a merely quantitative manner.
Thus, once again it is true within isolated frames of reference, that the holistic aspect of the primes is indeed amenable to quantitative analysis. But crucially when we look at the central feature of behaviour, in the two-way interdependence of individual and collective aspects, we must also include a qualitative aspect.
This new framework cannot be interpreted in a conventional mathematical manner. So in any context, when we identify the individual aspect as quantitative, then the holistic aspect is - relatively - qualitative; likewise, in reverse, if we identify the holistic aspect as quantitative, the individual aspect is now - relatively - of a qualitative nature.
And the way in which we seek to objectively understand the primes (in this dynamic relative framework) is complemented in turn by the manner in which psychological interpretation operates, where again we must admit the two-way interaction of both rational and intuitive processes of understanding.
So clearly we are here transcending the limits of the present mathematical paradigm (Type 1) which in formal terms is confined to a 1-dimensional interpretation (as linear reason).
And this is the price that must be paid for a true understanding of the primes which can be seen in dynamic interactive terms as the supreme exponents of two aspects (quantitative and qualitative) the dynamic interaction of which governs all created evolution.
Now, once again the perfect reconciliation of these two tendencies of the primes takes place in an ineffable manner (prior to created activity).
However, when the prime numbers manifest themselves in space and time, they must necessarily operate as living entities through physical phenomena, where to a degree quantitative and qualitative aspects are already separated. So full understanding of the mystery inherent in the primes points to a complementary ineffable spiritual state, where ultimately the quantitative and qualitative aspects of the primes can again be perfectly harmonised.