As we have seen linear approach to number (Type 1) is based on the natural number system (where each base quantity is raised to the unchanging power i.e. dimensional quality of 1).
i.e. 1^1, 2^1, 3^1, 4^1,…
The corresponding circular approach to number (Type 2) is initially based on a complementary natural number system where – in reverse – the unchanging base quantity of 1 is raised to the natural numbers as varying powers representing qualitative dimensions.
i.e. 1^1, 1^2, 1^3, 1^4,…
So right away through defining the number system with two distinctive aspects in complementary terms (quantitative and qualitative), we pave the way for its understanding in a truly dynamic interactive manner (with a merely relative validity).
Equally this amounts to recognition of the complementary nature of both cardinal and ordinal type understanding of number!
Now the key to the circular nature of the second aspect of the number system comes through associating each dimensional number (D) in qualitative terms, with its corresponding root (1/D) in a quantitative manner.
So for example to represent the qualitative nature of 2 as dimension (1^2), we find the second root of 1 i.e. 1^(1/2) in quantitative terms which is – 1. This can then be geometrically represented as a point on the circle of unit radius (in the complex plane).
Furthermore because of the unique association of each root with a corresponding dimensional number, we get rid of the confusion in Conventional Mathematics whereby a root can be given a multiple number of solutions.
So – 1 as the 2nd root of 1 = 1^(1/2). However + 1, which misleadingly is given as the alternative square root of 1, is now properly explained as the 2nd root of 1^2 (which qualitatively is of a distinct nature).
(Put another way, multiple roots in Conventional Mathematics arise due to the lack of the explicit qualitative notion of a dimension)!
However as well as the Type 1 (quantitative) interpretation, – 1 is now likewise given a complementary qualitative interpretation as the negation of (linear) rational interpretation (that is literally posited in experience in a conscious manner).
And it is through such dynamic negation that holistic intuitive meaning (of an unconscious nature) unfolds.
So in admitting a 2nd as well as 1st dimension in qualitative terms, we likewise need to explicitly admit the interaction of holistic intuitive type understanding (of a qualitative nature) with standard rational interpretation that is quantitative.
Put another way the 1st dimension (of linear rational interpretation) is based on absolute recognition of meaning (based on independent poles as reference frames).
The 2nd dimension (of holistic intuitive appreciation) is based on relative meaning (based on the dynamic interdependence of opposite reference frames).
Thus the very negation of rational meaning (in a manner analogous to the behaviour of anti-matter in physics) creates a fusion through intuitive type energy serving as the basis for understanding of holistic interdependence (with respect to relationships).
So incorporating the very notion of interdependence with respect to interpretation in a meaningful way in Mathematics, requires acceptance of both quantitative and qualitative type appreciation of its symbols.
At a minimum this requires incorporation of the 2nd with the 1st dimension with respect to understanding (where a dimension is now rightly seen in holistic qualitative terms as representing an overall manner of interpreting symbols).
So once we move to the use of more than one dimension (with respect to interpretation), mathematical reality is now necessarily viewed in dynamic interactive terms, with both quantitative and qualitative aspects.
And just as the geometrical representation of the roots of 1, entail both linear and circular aspects, likewise with respect to understanding, the qualitative interpretation of the number dimensions entails both linear and circular aspects of understanding.
Indeed linear understanding can be correctly viewed as the very important special case where qualitative is reduced to quantitative type interpretation. In other words here the holistic circular aspect – in formal terms – disappears, in a merely linear rational type interpretation of mathematical reality.
Now in direct terms, intuition is of an infinite holistic nature that is empty (in phenomenal terms), However indirectly it can be expressed in a circular rational manner that is paradoxical (in terms of the normal use of logic).
So for example + and – in normal linear logic are understood as independent opposites that are separate from each other.
However from a circular holistic perspective + and – are understood as interdependent opposites that are complementary with each other (and ultimately identical). So once again this latter understanding of interdependence, when conveyed indirectly in rational terms, is paradoxical (in terms of the linear use of reason).
Thus to convey directly the intuitive meaning of interdependence, we must negate the indirect rational constructs used to convey its nature.
Thus whereas + 2 as qualitative dimension represents the rational interpretation of the interdependence of two opposite poles in experience, – 2 conveys the direct intuitive appreciation of such interdependence (which is nothing in phenomenal terms).
As we have already seen this plays a big role in appropriate interpretation of the nature of the first of the trivial zeros (for s = – 2) in the Riemann Zeta Function, = 0. Correctly interpreted the zero here (as with all trivial zeros) refers to the qualitative rather than quantitative interpretation of 0 (as a numerical symbol).