More on Dynamic Nature of Number (4)

We have seen how number - when properly understood - entails a dynamic interaction between its quantitative and qualitative aspects (which are complementary with each other).

This raises the key issue of consistency with regard to the two aspects, when used in relation to one another.  

So again in Type 3 (radial) terms, it is now clearly understood that both the quantitative aspect of number (as independent) and the corresponding qualitative aspect (as interdependence) necessarily are of a relative nature.
 

Type 3 understanding represents both Type 1 (cardinal) and Type 2 (ordinal) aspects of number (where both are understood in dynamic terms as complementary).

So  strictly the cardinal notion of number (as explicitly understood) requires implicit understanding of the corresponding ordinal notion.

For example In Type 1 (base) terms, the cardinal notion of number is understood in a quantitative manner.

Therefore, using the number "3" to illustrate, this is explicitly understood as the sum of independent homogeneous units (that lack qualitative distinction).  Therefore 3 = 1 + 1 + 1.

However implicitly, a qualitative relationship must be recognised as existing between these individual units. Otherwise there would be no means of arriving at their composite total = 3.

Therefore explicitly the adding of 1 + 1 + 1 (in an independent quantitative manner) implicitly requires the qualitative ordinal recognition of these units (as 1st, 2nd and 3rd respectively).

Then in a reverse manner in explicit Type 2 (dimensional) terms, the ordinal notion of number as comprised of related ordinal components, implicitly requires the cardinal notion of number (as comprised of independent units).

Therefore we can only explicitly recognise the 1st, 2nd and 3rd ordinal members of a group of 3 members, if we already implicitly recognise 3 in cardinal terms (as composed of single independent units).

So in dynamic interactive terms cardinal and ordinal notions of number, ultimately, mutually imply each other in a holistic synchronous manner.


The huge question then arises as to the consistency of quantitative (cardinal) and qualitative (ordinal) aspects with respect to each other.

This implies for example that an indirect quantitative means is required to convert - as it were - qualitative to quantitative notions (that then can be seen to be related in a consistent manner).

Now again, in conventional mathematical terms, this key issue is avoided through an (unrecognised) form of reductionism whereby qualitative (ordinal) notions are viewed in one limited manner.

If we start with just one item which in cardinal terms is 1, we can unambiguously view this as the 1st (of a group of 1).

If we then move on to two items, which again in cardinal terms is 2, we can once again view this additional member as the 2nd (of a group of 2).

Then if we move on to three items, which now in cardinal terms is 3, we can now view the additional member as the 3rd (of a group of 3).

In this way we can continue to unambiguously identify additional ordinal members, by always considering its as the nth member of a group of n.

We can in fact mathematically show how each additional ordinal member, thereby becomes indistinguishable from corresponding cardinal members


Now quantitative meaning relates directly to linear (i.e. 1-dimensional) rational interpretation (which accurately characterises the conventional mathematical approach).

Dimensional meaning - which relative to base cardinal interpretation is of an ordinal nature - is represented as 1ⁿ.(where n is the dimesnion defined with respect to the default fixed base of 1).

Therefore to reduce from n-dimensional to 1-dimensional format we obtain the solution of
 xⁿ = 1ⁿ

Therefore x¹ = 1^n/n  = 1¹ = 1 (in cardinal terms).

Thus for the 1st unit x¹ = 1; then for the 2nd unit x¹ = 1; likewise for the 3rd unit x¹ = 1.

So the qualitative (ordinal) notion of 1st + 2nd + 3rd  has now been successfully reduced in mere quantitative (cardinal) terms as 1 + 1 + 1.

Therefore, the linear interpretation of ordinal notions is only possible through treating each additional ordinal member as the nth member of the nth group.

In this way ordinal notions seem compatible with absolute cardinal notions of number (in a Type 1 manner).

However when we treat ordinal notions in a truly relative manner, within each group, this comforting position breaks down irretrievably (in effect requiring a greatly enlarged mathematical framework).