I have already introduced two number systems as the basis for Type 1 (quantitative) and Type 2 (qualitative) appreciation respectively.
Initially both Type 1 and Type 2 interpretation can take place in comparative isolation from each other. However ultimately from the Type 3 perspective, their truly relative nature becomes readily apparent so that the Type 1 also - necessarily - has a qualitative and Type 2 a quantitative aspect respectively.
And the reason why this seems reminiscent of the wave/particle nature of particles at the quantum level, is because the deeper root of such complementarity lies in the very foundations of Mathematics.
It has to be stated from the outset that I am already defining Type 1 Mathematics in a - relatively - rather than absolute independent manner (which is the way it appears from the the more comprehensive Type 3 approach). Because no such perspective exists in Conventional Mathematics, it is misleadingly defined in an absolute type manner (which does not lend itself therefore to subsequent integration with the other approaches).
So bearing this important point in mind, the Type 1 is defined in terms of the natural numbers that can vary as quantities and a default number number dimension of 1 (that does not vary). And from this relatively independent perspective, the dimensional number is qualitative, with respect to the base number quantity. So once again, from a qualitative perspective Type 1 Mathematics, is properly defined by its linear (1-dimensional) nature.
Thus in Type 1 terms the natural number system is represented as follows
1^1, 2^1, 3^1, 4^1,.....
Now Conventional Mathematics attempts to come to grips with the primes through sole reference to this number system (with the dimensional number system ignored).
So in this context we have the individual prime numbers 2, 3, 5, 7,... (which are looked on as the basic building blocks, or atoms, of the natural number system).
Then, we also have the general relationship of the primes to the natural numbers where each (natural) number is expressed as a unique combination of prime factors.
So we have the individual uniqueness of the primes (as separate from the natural number system) and also - what we might call their relational uniqueness (in that all natural numbers entail unique combinations as a product of prime factors).
And both of these aspects are understood - merely - with respect to their quantitative aspect.
However in truth the true picture requires much greater subtlety.
Now, prime numbers, as quantitative, have a cardinal meaning. For example, we could look at the prime number 7 for as representing a set of objects. Thus, the very notion of 7 necessarily implies the prior notion of 1, 2, 3, 4, 5 and 6 respectively.
Now, from a cardinal sense the relationship between these natural numbers (within the primes as it were) have no relational significance. So each natural number can be built up by the addition of 1). So 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1 and so on.
The important reason for the lack of relational significance, arises directly from the fact that addition can be carried out from - merely - the 1-dimensional perspective.
So 2^1 = 1^1 + 1^1; 3^1 = 2^1 + 1^1; 4^1 = 3^1 + 1^1 and so on.
Now this clearly suggests that true relational capacity in fact entails the dimensional notion of number (when it varies from 1). And as Conventional Mathematics is qualitatively defined by the 1-dimensional approach, it is thereby ill suited to get to grips with the relational aspect of the primes.
This perhaps becomes clearer when we now look at the relational aspect of the primes (as understood in Type 1 terms).
So the natural numbers are understood in terms of the unique expression of a combination of prime factors.
Therefore for example, 6 = 2 * 3 is uniquely expressed as the product of these two prime numbers (as factors).
Now just as the individual aspect of the primes is understood merely in quantitative terms, likewise this is now also the case with respect to their collective relational aspect.
However when we include the default dimensional number with the primes we get a significant clue, that something vital is missing from the conventional perspective.
Earlier we expressed the two unique factors of 6 as 2 and 3.
However more fully expressed, the product of these two numbers = (2^1) * (3^1). However if we think of this in geometrical terms, we would generate a 2-dimensional rectangle with sides 2 and 3 respectively. So in fact, in saying that 6 is the natural number that is uniquely expressed by the product of its prime factors, 2 and 3, we are in fact giving but a reduced quantitative interpretation (that misses out entirely on the key qualitative significance of their relational aspect).
To see this more clearly (in qualitative terms) we replace the base number quantities by 1 (thereby switching to the Type 2 approach).
So 1^1 * 1^1 = 1^2.
In other words, the true significance of such a relational capacity, with respect to the primes, is directly due to the manner in which the qualitative dimension, associated with the primes, changes through multiplication.
And again in the conventional mathematical approach, this key subtle distinction - which is vital for comprehension of the relational aspect - can be given no meaning (as it is qualitatively defined within a merely 1-dimensional perspective!)
However to get a fuller picture, we now need to interpret such prime number relationships from within the alternative Type 2 perspective.
So once again the number system (in Type 2 terms) is defined as;
1^1, 1^2, 1^3, 1^4,....
This represents the complete inverse of Type 1, where the base quantity is now fixed as 1, and where the natural numbers - relatively - representing qualitative dimensions, can now vary over the natural numbers.
The key to realising what is happening here is that dimensional numbers are now defined with respect to a (qualitative) ordinal - rather than (quantitative) cardinal meaning.
So 7 as a prime number in this context is written more fully as 1^7 and refers to the qualitative notion of 7 as a dimension (i.e. the 7th dimension).
To unlock the significance of this alternative formulation we must switch to a circular rather than linear representation.
The key to this in turn is to recognise that the notion of a dimension in qualitative terms and its corresponding root in a quantitative manner, are inversely related (with the same identical structure).
So 1^7 therefore has its counterpart in 1^(1/7) as the seventh root of 1.
And once again - but now in an ordinal manner - the notion of 7 (as a prime root of 1) likewise implies 6, 5, 4, 3, 2, and 1 as roots of 1 also.
However unlike the previous case where the relationship between the natural numbers in this grouping, allowed for no qualitative distinction, it is quite the opposite here, with all the natural numbers up to 7 i.e. as the 1st, 2nd, 3rd, 4th, 5th, 6th and 7th roots of 1,respectively, possessing a unique qualitative identity. And the reason for this is that the corresponding ordinal numbers as dimensions i.e. 1st, 2nd, 3rd, 4th, 5th, 6th and 7th, all likewise possess a unique qualitative identity (serving as a means of holistic interpretation).
So this illustrates in truly remarkable fashion, how each number, can in truth be given both a quantitative and qualitative interpretation. And the key to this realisation is the fact that the cardinal and ordinal nature of numbers, properly relates to their quantitative and qualitative characteristics respectively.
And just as the the individual primes have a unique identity in Type 1, it is the reverse in Type 2 terms. So the strong circular type relationships that bind the various roots of a prime number and corresponding dimensional interpretations) together, imply that these ordinal roots lack such identity.
We also then can attempt replicate in Type 2 terms the relational aspect of the primes.
So 2 * 3 = 6 from this perspective is represented as 1^(2 * 3) = 1^6
However again a very subtle point arises that is illuminating.
In Type 1 terms when we multiply two numbers, it does not matter in what order we do so.
So 2 * 3 = 3 * 2.
However this is not true with respect to Type 2.
For 1^(2 * 3) = (1^2)^3 and 1^(3 * 2) = (1^3)^2
Now when we invert this in Type 1 terms, the former is:
(2^3)^1 = 8^1 and the latter (3^2)^1 = 9^1
So when we try to represent the relational aspect of the primes (which as a qualitative aspect properly is explained in Type 2 terms), quantitative uniqueness, with respect to generation of the natural numbers, breaks down. So what represents multiplication in the Type 2 case, represents exponentiation with respect to Type 1 (just as addition with respect to Type 2, represents multiplication with respect to Type 1).
This is just another way of saying - complementary to our earlier observation - that this relational capacity of the primes is directly associated with the qualitative nature of number (an is appropriate to Type 2 interpretation).
So let us return once more to our crossroads analogy, though this time in an enlarged manner.
In previous entries, I used this to indicate the crucial difference as between linear frames of reference (using isolated poles) and circular frames (where complementary opposite poles are taken). I did so initially, where the isolated frames related solely to the "real" aspect of quantitative type interpretation.
So I highlighted the fact that the primes can be given a quantitative type interpretation with respect to both their individual and collective attributes (in Type 1 terms) using isolated frames of reference. However when it comes to establishing their two-way interdependence, we need necessarily include the complementary qualitative aspect.
However I did not explicitly elaborate on the direct nature of this qualitative aspect at the time. So I was illustrating the paradox of direction at a crossroads when one can only identify - as it were - a left turn.
However one could equally start from the complementary qualitative direction, attempting to understand, both their individual and collective natures, solely from this perspective. So, this is analgous to being able to identify solely a right turn, which would create the same paradox of direction (when two opposite directions need to be simultaneously recognised at the crossroads).
Using Jungian terminology the qualitative aspect of Mathematics (that I identify with the Type 2 approach) in a very true sense represents the unrecognised shadow of present mathematical activity. Because of extreme specialisation with respect to merely quantitative type interpretation, the Type 2 aspect - though equally necessary for a balanced mathematical understanding - is entirely overlooked.
And because of this, the true nature of the the amazing relational capacity of the primes - which is of a qualitative nature - is continually missed.
Now the shadow of course suggests the unconscious and indeed the holistic capacity to properly appreciate the relational capacity of the primes is directly asssociated with the unconscious in intuitive type terms. Then when the holistic insights emanating from this source are appropriately channeled in an (indirect) rational manner, we are then dealing with the "imaginary" aspect of interpretation (in a precise qualitative manner).
So proper understanding of the nature of the primes requires - as a starting basis - both "real" and "imaginary" aspects of interpretation corresponding to Type 1 and Type 2 Mathematics respectively.
Initially, when both aspects are dealt with in relative isolation, the "real" aspect is directly associated with the (cardinal number) quantitative aspect; by contrast the "imaginary" aspect is associated directly with the (ordinal number) qualitative aspect.
So from this perspective we have seen, that whereas the individual nature of the primes is directly assocaited with the quantitative aspect (as the building blocks of the prime number system), the corresponding general relational nature of the primes is directly associated with the qualitative aspect (where all natural numbers are expressed in terms of a unique combination of prime factors).
Then in the dynamic interactive approach that constitutes the Type 3 mathematical approach, both aspects (Type 1 and Type 2) are united through an interlocking series of complementary type relationships.
And from this perspective, the very distinction of quantitative and qualitative begins to break down, as the quantitative acquires a qualitative aspect and the qualitative in turn a quantitative aspect.
So to illustrate this again in terms of our crossroads analogy! We start off in a relatively unambiguous manner, identifying a left turn (within isolated frames) approaching it in turn from two different directions (north and south) i.e. Type 1. We then in an unambiguous manner, identify right turns (within isolated frames) coming from different directions i.e. Type 2.
Finally we attempt to understand both left and right as interdependent with each other, in a two-way complementary manner.
This leads to a continual shifting of isolated reference frames, so that what was left becomes right, and what was right left, in quick succession i.e. Type 3.
So when we give the two axes in the complex plane, which do indeed intersect at a
central point like a crossroads, a Type 3 interpretation, both real and imaginary axes (representing Type 1 and Type 2 mathematical understanding) keep shifting so as to approximate ever closer to the mutual ultimate identity of both the quantitative and qualitative aspects of the primes.
So once again just as real and imaginary can be given a quantitative interpretation with respect to the Riemann Zeta Function Function, equally they can - and of course should - be equally given a corresponding qualitative interpretation.
So once again, I have arrivedly at the same conclusion - reflected from yet another angle - that the key issue, points to the condition for the ultimate identity of both quantitaive and qualitative aspects of the primes (which lies at the very centre). And this of course is what the Riemann Hypothesis is all about. At the centre of the crossroads we are neither turning left or right (from a linear perspective) or equally from the other circular perspective, turning both left and right (and right and left) simultaneously.
So here both line and circle (and linear and circular) are finally reconciled.