And remember both of these aspects are necessarily involved with respect to a coherent dynamic appreciation of number!
So in effect this requires from one perspective, that a satisfactory means be found to convert the qualitative aspect of number interpretation (indirectly) in a quantitative manner; equally from the other perspective, a satisfactory means is required to convert the quantitative aspect of interpretation (indirectly) in a qualitative manner.
I then began to realise that the ordinal notion of a given number could be (indirectly) converted in a quantitative manner through obtaining the corresponding roots of 1.
Let is illustrate carefully what is unloved here with respect to the simplest case relating to the number "2".
Now the qualitative notion of "2" is expressed through the Type 2 aspect of the number system as 12.
"2" here relates to the interdependent identity of two dimensions (i.e. directions), that strictly in this qualitative state have no (independent) quantitative identity. In fact - just as with the fusion of a matter and anti-matter particle in physical terms - "2" here refers directly to an energy state (from both physical and psychological perspectives).
Therefore when one appreciates the inherently dynamic nature of number, it is no longer feasible to attempt to abstract it in an absolute manner from the natural phenomena of experience. Rather number now represents the encoded nature of such phenomena (with these phenomena in turn representing the decoded nature of number). So in this sense a two way interaction characterises the relationship of number and natural phenomena (in physical and psychological terms).
So just as the Type 1 (quantitative notion of number) has strictly no direct meaning in qualitative terms, likewise in a complementary dynamic manner, the Type 2 (qualitative) notion of number has likewise no direct meaning in a quantitative manner.
So once again in this context, 21 represents the Type 1 (quantitative) expression of "2", whereas 12 represents the corresponding Type 2 (qualitative) expression.
Once again from a Type 2 perspective, neither of the two dimensions (directions of 2) can be separated! In psychological terms this entails an intuitive recognition of "2" as representing the interdependence of opposite external and internal polarities (in psychological terms). In a corresponding physical manner, it represents an energy state (again as the interaction of opposite poles).
However by now attempting to express this (qualitative) 2-dimensional state in the standard 1-dimensional manner, we can, in a sense, now indirectly express these two dimensions in a quantitative type manner.
In effect ,this entails obtaining the two roots of 1 (or more precisely the roots of 12 and 11 respectively).
Remarkably when now expressed in this manner, the two roots give quantitative expression to the qualitative notions of 1st and 2nd in the context of 2 members.
Therefore the intuitive notion of "twoness" (where dimensions cannot be separated), gives way, at the 1-dimensional level (based on the separation of such polarities), to the ordinal notions of 1st and 2nd These then can be indirectly expressed in a quantitative manner as + 1 and – 1 respectively.
Therefore the notion of "twoness" - though again directly of an intuitive nature - is represented at the common rational level of discourse as relating to 1st and 2nd units (which can interchange locations).
We have already had a good example of this in the representation of the directions of a crossroads, where in relation to each both left and right are understood to be paradoxical.
So once again, though the understanding that a turn can be both left and right defies normal rational logic (in a 1-dimensional manner) it can be intuitively grasped (in 2-dimensional terms). Then when we express this in a reduced manner (as 1-dimensional) if the 1st turn is "left", the 2nd turn is thereby - relatively -"right". Likewise if the 1st turn is now "right", the 2nd turn is thereby - relatively - right.
And as we have seen in a quantitative manner, when the 1st turn as "left" is posited as + 1, the 2nd turn as "right" is thereby negated as – 1. However when the 1st turn as "right" is then posited as + 1, the 2nd turn as "left" is now thereby negated as – 1.
So the clear intuitive realisation of "twoness" (at the 2-dimensional) level can only be approached in rational (1-dimensional) terms in a circular manner, through paradox.
Thus once again the intuitive notion of "twoness" is expressed at the rational level in a relative manner as both 1st and 2nd members (where 1st is equally 2nd and 2nd equally 1st).
This leads to the fascinating conclusion that ordinal notions, which are deeply implicit in our handling of all quantitative mathematical relationships at the accepted 1-dimensional level, in fact reflect the reduced conscious understanding of "higher" dimensional relationships that are directly apprehended in an intuitive (unconscious) manner!
Thus, the notions of 1st and 2nd (in the context of 2 units) requires 2-dimensional understanding in qualitative terms.
Then the corresponding notions of 1st, 2nd and 3rd (in the context of 3 units) requires 3-dimensional understanding in qualitative terms. The very meaning of 1st and 2nd now changes in the context of this "higher" dimensional appreciation!
In more general terms, the notions of 1st, 2nd, 3rd,......., nth (in the context of n members) requires n-dimensional understanding, in qualitative terms.
Therefore, our standard ordinal notions cannot be properly understood within the conventional mathematical framework, as it is geared to merely quantitative type appreciation of an analytic nature.
In fact a radically distinct form of Mathematics, which I customarily refer to as "Holistic Mathematics", is required. In other words - rather than the customary quantitative (analytic) interpretation of mathematical symbols - a distinctive qualitative (holistic) appreciation is required.
Thus every mathematical symbol has given both quantitative (analytic) and qualitative (holistic) aspects, both of which necessarily interact in the dynamics of experience.
The quantitative (analytic) aspect directly relates to rational interpretation (corresponding to 1-dimensional understanding). However indirectly this must be implicitly fuelled by intuitive type appreciation (of a qualitative nature).
The qualitative (holistic) aspect, by contrast, relates directly to intuitive appreciations (corresponding to "higher" dimensional insight). However indirectly this is then expressed in a quantitative type manner through circular (paradoxical) rational type relationships.
In conventional mathematical terms, the ordinal type appreciation of number is simply reduced in cardinal type fashion. In other words the qualitative aspect of number understanding is reduced in a quantitative type manner. We will see precisely how this happens in the next blog entry.