We saw yesterday how ordinal notions represent the reduced expression, in a 1-dimensional quantitative manner, of "higher" dimensional qualitative notions of number.
Therefore the notions of 1st and 2nd (with respect to 2 numbers) represent the reduced interpretation of the two related "higher" dimensions (corresponding to 2).
In an indirect quantitative manner, these are expressed as the two roots of 1 (i.e. of 11 and 12 ) i.e. 11/2 and 12/2 which are – 1 and + 1 respectively.
Though these two roots, – 1 and + 1, are clearly separated (at the reduced 1-dimensional level), they are understood intuitively as interdependent with each other at the corresponding "higher" 2-dimensional level of understanding, where the true qualitative nature of "2" (as "twoness") is directly apprehended.
However, indirectly their qualitative nature can be indicated at the 1-dimensional level through obtaining the sum of the two roots i.e. – 1 + 1 = 0.
In other words, the qualitative identity of "twoness" where "both" units are directly understood as interdependent, is strictly without any quantitative identity i.e. = 0!
Therefore, when we adopt this holistic interpretation of the nature of number, the various roots of 1 assume an extraordinary new importance.
So in general terms, the n roots of 1 provide the indirect quantitative means of expressing the notions of 1st, 2nd, 3rd,...., nth (with respect to a group of n numbers).
The "higher" dimensional interpretation here directly refers to the intuitive qualitative realisation of the group interdependence of the n units (which would be understood as clearly independent in an individual manner at the conventional 1-dimensional level of interpretation).
Put another way, at the standard 1-dimensional level of rational quantitative interpretation, each unit has an absolute fixed identity.
However, by contrast, at the n-dimensional level of direct holistic intuitive appreciation, each unit is directly understood in intuitive terms as interdependent with every other unit. This now creates circular paradox in terms of standard analytic interpretation at a rational level, as it implies for example that the 1st unit can equally be the 2nd, 3rd,...., nth unit. Equally each unit can switch identity with each other unit depending on the initial relative perspective that is adopted.
Once again we saw this clearly at work in the case of 2 dimensions with respect to the understanding of a crossroads.
So when we identify the left turn (heading N) as the 1st direction, the right turn (heading S) is the 2nd direction. However when we switch reference frames so as to approach the crossroads (heading S) the right turn is now the 1st and the left turn the 2nd direction respectively.
And in (indirect) quantitative terms, when the left turn (heading N) is + 1, the right turn (heading N) is – 1. However when the right turn (heading S) is + 1, then the left turn (heading S) is – 1.
So 1-dimensional rational interpretation (within isolated N and S polar reference frames) leads to unambiguous notions of fixed identity i.e. as 1st and 2nd in any context.
However 2-dimensional intuitive appreciation (within simultaneous N and S polar reference frames) leads to paradoxical notions of a merely relative identity (where 1st and 2nd switch locations)
In corresponding quantitative terms, 1-dimensional rational interpretation leads to unambiguous notions of separate identity in quantitative terms with respect to + 1 and – 1.
However 2-dimensional intuitive appreciation, leads to paradoxical notions of relative identity in qualitative terms, where + 1 and – 1 are interchangeable.
Now it should be apparent from the crossroads example, that the consideration of 2-dimensional appreciation, necessarily includes 1-dimensional. So we cannot attempt to understand the nature of interdependence at the 2-dimensional level, without first considering independent identity at the 1-dimensional.
And just as 2-dimensional includes 1-dimensional interpretation, likewise n-dimesnional includes 1, 2, 3,...., n – 1 dimesnional interpretation.
Therefore once again, the n roots of 1 holistically relate to the quantitative interpretation of n dimensions in reduced 1-dimensional terms, where the notions of 1st, 2nd, 3rd, ...., nth now can attain a relatively separate independent identity (depending on context).
So one of the n roots of 1 (strictly n roots 11, 12 , 13 ,......, 1n) is always necessarily 1, reflecting the 1st dimension, which defines the nature of conventional mathematical interpretation.
In other words ordinal notions of 1st, 2nd, 3rd etc which strictly have a qualitative - rather than quantitative identity - are simply reduced in conventional mathematical terms to their corresponding cardinal notions.
Therefore, though properly understood in qualitative terms, the notions of 1st, 2nd, 3rd are merely arbitrary (depending on context) they are given an absolutely fixed identity in conventional mathematical terms. So the the notion of 1st is identified unambiguously with the 1st unit (where it = 1). The notion of 2nd is likewise unambiguously identified with the 2nd unit (= 1), the 3rd unambiguously with the 3rd unit (= 1) and so on.
So to return to the simple example of "2"!
In conventional cardinal terms this is expressed as
2 = 1 + 1.
Now if one likewise insists on an ordinal expression,
2 = 1st unit + 2nd unit.
However, as we have seen, by the very nature in which ordinal notions are defined in conventional mathematical terms,
this reduces to 2 = 1 + 1 (i.e. the cardinal definition).
Thus the massive reduction (in every context) of qualitative to quantitative type notions, is directly replicated in conventional mathematical terms by the corresponding reduction of ordinal to cardinal type notions. This reflects the attempt to give number notions a merely fixed independent identity in quantitative terms. However the very ability to relate these numbers with each other requires a subtler interdependent identity (where numbers can continually interchange with each other).
Once again this latter aspect appreciation, requires an entirely new holistic mathematical approach.
Now only does this have an immense unrecognised potential in its own right, but it is also vitally necessary to provide a coherent framework for all quantitative interpretation.