As we have seen 2 in the cardinal number system can be represented as 1 + 1 in quantitative terms.
This can geometrically be represented by marking off two equidistant points representing the units on a straight line (which in qualitative terms is 1-dimensional).
In the corresponding ordinal system, 2 (as dimension) can be represented as 1 – 1 in qualitative terms.
This in turn can be geometrically represented by marking off two equidistant points representing the “qualitative” units on a circular circumference (based quantitatively on a radius of 1 unit in the complex plane).
Now if we attempt to add 1 – 1 (as the ordinal representation of the qualitative interdependence existing among these two number members) the answer is zero. This strictly indicates that the result is 0 from a cardinal perspective. In other words the interdependence of the two numbers - by definition - requires screening out notions of separate independence (on which the cardinal system is based).
Therefore in quantitative terms we can write the complementarity of 2 opposite poles - to which the ordinal notion of 2 as dimension relates - as separate in a cardinal manner. However strictly in qualitative terms, such recognition requires identifying + 1 and – 1 as identical.
We have already illustrated this is the recognition that a turn at a crossroads can be labelled left or right (depending on the polar frame of reference from which it is approached).
So the ordinal notion of 2 in this context relates to the left turn and right turn (which are + 1 and – 1 in relation to each other) as interdependent.
So if we designate the left as 1st with + 1, then right as 2nd is designated as – 1. However if we equally designate the right as 1st with + 1, then the left as 2nd is designated as – 1.
So simultaneous recognition of interdependence involves the two-way recognition of both + 1 and – 1 as identical in qualitative terms.
In the language of Heraclitus such interdependence can be expressed paradoxically using dualistic language as:
a left turn is a right turn; a right turn is a left turn.
Now the dynamic ordinal nature of 2 is the simplest to explain.
However the ordinal nature of any natural number n can be expressed with respect to its corresponding n roots.
So for example with respect to 4 we obtain + 1 , i, - 1, and - i .
As again the ordinal nature of the 4 members 1st, 2nd, 3rd and 4th, relate to a shared interdependence, in quantitative terms this relationship would be expressed as 1 + i – 1 – i = 0.
However the qualitative recognition of such interdependence now requires 4-way intuitive recognition of these 4 unit poles as interdependent.
The importance of this ordinal approach in the present context is that it can be used to show a whole new perspective on the relationship of the primes to the natural numbers (which is the reverse of that demonstrated by the cardinal approach).
Once again in the cardinal Type 1 approach, the natural numbers can be expressed as the unique product of prime number factors.
However the very significance of prime numbers as dimensions, is that they can all be defined in the Type 2 system by a unique set of numbers that represent the natural numbers in ordinal terms!
In other words if n as dimensional number is prime and we then obtain the n roots of 1, we will generate a unique set of roots (as representing its ordinal members).
And it is only when n is prime that this will hold!
So we have now addressed a key problem problem with respect to the cardinal (Type 1) approach where all members – because of their homogeneous quantitative nature – can be given no unique ordinal distinction!
In other words the problem is solved through the Type 2 approach where the prime numbers are now uniquely defined in terms of a set of circular numbers that correspond to the natural numbers in ordinal terms!
We will return to this in the next entry.