In the last blog, I explained how the imaginary number i - which is given a merely quantitative interpretation in conventional (Type 1) mathematical understanding, has an extremely important alternative qualitative significance in holistic (Type 2) terms.
Thus once again - 1 (in holistic qualitative terms) represents the dynamic negation of the (linear) conscious, thereby rendering it unconscious. And the meaning pertaining to the unconscious (as intuitive appreciation) represents the complementarity of two opposite poles (positive and negative). So it is thereby 2-dimensional relating to the fusion of two polar directions of understanding.
Therefore to express such understanding indirectly in a linear rational manner (as 1-dimensional) we thereby must obtain its square root.
Therefore in a very precise manner the qualitative notion of the imaginary is related to the square root of - 1 (interpreting such symbols in an appropriate holistic manner).
So in actual experience the unconscious can thereby only express itself through becoming embodied within conscious phenomena (and in a mathematical context within conscious rational symbols).
Thus properly understood, every such symbol has both a real meaning (with respect to the quantitative aspect of conscious interpretation) and an imaginary meaning (as expressive of a more qualitative holistic meaning) that is directly unconscious in origin.
Indeed ordinary language can demonstrate well this dual significance of phenomena.
For example one might speak of buying a "dream" house. So here the house serves an actual conscious significance (as an identifiable object in experience) while the attachment of "dream" signifies a deeper holistic attachment where the house in some way serves as a representation of potential meaning that is unconscious in nature.
However my point here is that all phenomena - by definition - possess both "real" (quantitative) and "imaginary" (qualitative) aspects and that this of course is also true of mathematical symbols.
The connections here with Jungian thought should seem apparent. One of his great disciples Marie Louis Franz once stated that " "Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four ….",
However in this context it was the (circular) holistic significance - rather than the standard analytic interpretation - that Jung was concerned with as for example in exploring his four functions. Though not a mathematician in the accepted sense, his main ideas however lend themselves admirably to holistic mathematical interpretation. So at least implicitly he was mathematical in this holistic sense.
Of course as Jung knew so well, when the unconscious aspect of understanding is not properly recognised, it gets projected on to conscious symbols in a somewhat blind involuntary manner i.e. as the shadow of one's conscious personality.
This obviously has a deep relevance for Mathematics, where the unconscious aspect remains almost totally repressed with respect to formal understanding.
Consequently the unrecognised unconscious aspect is thereby projected in a highly defensive manner that effectively blinds practitioners to anything that does not fit it with accepted principles.
And I mean this in a very literal manner.
The Riemann Hypothesis for example, when appropriately understood, is capable of a remarkably simple resolution (which can be effectively explained in terms of the crossroads analogy). Now, I would strongly suggest that appreciation of the few steps required to understand this analogy, should effectively convince one that the Riemann Hypothesis cannot be proved (or disproved) from a conventional mathematical perspective.
Indeed it clearly demonstrates that the Hypothesis cannot be even properly appreciated from the conventional perspective!
However I would expect extreme reluctance to follow my invitation to approach this crossroads from within the mathematical profession. And the deeper reason lies at an unconscious level. In other words for most practitioners, a serious investment has already been made in an accepted mathematical belief system (which conveys in many ways a remarkable sense of psychological security). So the price that would have to be paid for simple enlightenment regarding the nature of the Riemann Hypothesis, would be an inevitable collapse in this strongly held belief system. Therefore unconscious projections of various kinds (which I am well accustomed to receiving) would inevitably arise, enabling them to defend the status quo and thereby avoid serious examination of assumptions presently held.
So the contributions in a blog such as this are highly unlikely to ever resonate or even be considered by mathematical specialists.
However there will always be others of a more philosophical bent - perhaps with a talent also for mathematical speculation - who will feel challenged to some degree by what is on offer.
And that fact represents to my mind the start of a process that will eventually lead to an unprecedented revolution in our scientific and mathematical thinking.
I will finish this blog entry by returning to the Riemann Zeta Function.
Though significant progress had indeed been made by people such as Euler and Gauss, it took the complex analytic approach employed by Riemann to reveal the (hidden) harmonic structure of the primes.
Complex numbers entail both real and imaginary aspects. And Riemann used both to ingeniously reveal in turn the two intertwined aspects of the primes, with respect to their individual identity and general wave nature (through the non-trivial zeros).
Now the deeper qualitative reason of why this was possible is very revealing.
As we have seen, from a qualitative perspective imaginary numbers represent the holistic (circular) capacity of numbers (indirectly encoded in a rational linear manner).
So not surprisingly when subject to complex analysis the prime numbers reveal both their real (individual) and imaginary (general wave) characteristics. As I have said before such complex analytic investigation, this wave nature of the primes was completely unknown!
And quite literally in this regard, whereas the individual primes are measured on a real scale, the non-trivial zeros (with which their harmonic structure is related) are measured on an imaginary scale!
Likewise when one combines both real (conventional) and imaginary (holistic) understanding, prime numbers likewise reveal a dual complementary nature.
In other words, we now realise clearly that the prime numbers have both real (quantitative) and imaginary (qualitative) characteristics, with the Riemann Hypothesis then obtaining a wonderful central relevance in establishing the condition necessary for reconciliation of both aspects.
It really is that simple and beautiful! For the Riemann Hypothesis further implies that the proper task for Mathematics is likewise to reconcile its quantitative (Type 1) with its unrecognised (Type 2) aspect.
But making the conversion to seeing such "simple" truth ultimately requires embracing a radical new interpretation of what Mathematics represents.
And this is where a profound difficulty still remains!