Yesterday we looked at the qualitative nature of 2 as a dimension.
When we consider the importance of 1 in dimensional terms, which - literally - defines the linear logical nature of conventional mathematical understanding, we perhaps might realise that 2 likewise carries a similar importance (of a distinctive nature).
In fact 2 in this qualitative dimensional sense - as the dynamic interaction of opposite (unit) polarities that are positive (+) and negative (-) with respect to each other - has been widely used in variety of contexts.
For example it has been heavily employed in the mystical expression of the Eastern mystical religions e.g. Hinduism and Buddhism. The most famous extract from the best known Buddhist heart sutra is:
"Form is not other than Void;
Void is not Other than Form."
We could equally say in a dynamic interactive mathematical context,
"The quantitative aspect (of understanding) is not other than the qualitative;
The qualitative aspect is not other than the quantitative."
We can also add add that this important statement can have no meaning in the 1-dimensional context of present Mathematics (where symbols are defined solely in quantitative terms).
Indeed Taoism gives explicit expression to the complementarity of opposites in the notion that that the original indivisible Tao splits into two opposite aspects yin and yang with respect to all phenomenal processes.
A particularly striking earlier expression in Western thought is given by Heraclitus
in the paradoxical statement
"The way up is the way down;
the way down is the way up."
So once again we have a statement of the bi-directional nature of opposites when considered in a dynamic interactive context.
Indeed in a qualitative mathematical manner we would translate that statement
(where pure interdependence exists) as:
+ 1 = -1; - 1 = + 1
I have illustrated the crucial importance of this many times before in the context of the relationship of the primes to the natural numbers (and natural numbers to the primes).
Now in this mathematical context, the two polarities are quantitative and qualitative (which are + and - in relation to each other).
Imagine one walking in a South to North direction approaching a crossroads. We can give an unambiguous meaning (for example) to a left turn in this context.
Now imagine further that having continued "up" the road in a northerly direction one decides to turn back heading once more towards the crossroad in a southerly direction.
Again when one reaches the crossroads one can unambiguously identify a left turn from this direction. So within isolated (1-dimensional) poles of reference, an unambiguous meaning can be given to a left turn in each case.
However if we now look at this from a 2-dimensional perspective (where left and right have a merely arbitrary identity) deep paradox arises.
Though in 1-dimensional terms, as independent, both turns at the crossroads are designated left, clearly in simultaneous relationship to each other as interdependent, the turns are left and right (and right and left).
So when a unit pole (which can, depending on context, be either left or right) is designated as + 1, the opposite pole is - 1. However when we switch the polar frame of reference (in this case identified with movement "up" or "down" the road), the designation thereby changes.
Thus the very ability to recognise that turns at a crossroads must be left and right in relationship to each other, implicitly requires 2-dimensional understanding of the nature of interdependence.
If now instead of left and right as complementary opposites, we use quantitative and qualitative, let us see how this applies to conventional mathematical interpretation!
Because in formal terms such Mathematics solely recognises the quantitative aspect, this approach can be likened to someone who can only identify a left turn!
So when a mathematician studies the individual nature of prime number behaviour this can be likened to the "up" direction in our crossroads example.
Therefore, unambiguous quantitative results can arise from such interpretation.
Then once again when one now switches to the opposite reference frame in the general interpretation of prime number behaviour, again unambiguous quantitative results emerge.
However when one now simultaneously tries to understand the interdependence as between individual and general prime number behaviour (with respect to the natural numbers) deep paradox arises. For just as with our crossroads example, these two aspects now are understood as quantitative and qualitative (and qualitative and qualitative) with respect to each other.
This poses an immediate dilemma for the conventional mathematical approach which allows no formal recognition for the qualitative aspect of interpretation.
In other words, Conventional Mathematics lacks any genuine holistic method of interpretation. Whereas analytic type understanding is indeed appropriate in terms of quantitative notions of independent relationships (and then strictly only in a relative context) genuine holistic understanding is required for the authentic appreciation of the interdependence of such relationships.
So the present problem with Conventional Mathematics is the most fundamental imaginable in that is has no adequate means (within its own set of interpretations) of dealing with the key notion of interdependence (except in a grossly reduced manner that distorts its very nature).
And the very reason for this that the notion of interdependence relates directly to the qualitative aspect of understanding!
And as the non-trivial zeros relate directly to this key notion of interdependence with respect to reconciling the primes with the natural numbers (and the natural numbers with the primes) the obvious implication is that their true nature cannot be approached from within the present (1-dimensional) framework of Mathematics.
As I say, Heraclitus was one of the earliest Greek thinkers to clearly express the paradoxical nature of interdependence (which in holistic mathematical terms represents the qualitative nature of 2 as dimensional number).
Other well-known Western thinkers, who also explicitly employed such thinking, include Nicholas of Cusa with his coincidence of opposites. (Though not ranked among the very greatest of thinkers his contribution is perhaps especially interesting in that he tried to appreciate such notions in a mathematical context).
Other noteworthies include Hegel (with his dynamic interaction of thesis and antithesis) and Carl Jung (with his understanding of conscious and unconscious as dynamic interacting poles of experience).
The notion has likewise found its way into Quantum Physics (e.g. the complementary nature of light as waves and particles). However the deeper qualitative implications of such understanding have yet to be properly addressed in this context (which would require nothing less than a greatly enlarged new paradigm for science).
So the interpretation of 2 as qualitative dimension is hugely important in terms of providing the most accessible appreciation of the dynamic interactive nature of both quantitative and qualitative aspects in Mathematics.
However it represents just one of a potentially unlimited set of such interpretations (as every number in dimensional terms carries a holistic significance applicable to all mathematical relationships).