Therefore when a number, such as 3 (representing the base) is defined in an independent quantitative cardinal manner, the corresponding qualitative dimensional notion of 3 is defined in an interdependent ordinal manner incorporating the relationship of 1st, 2nd and 3rd respectively.

This the former notion of 3 (as cardinal) is represented in Type 1 terms as 3

^{1}.

Then the latter notion of 3 (as ordinal) is represented in Type 2 terms, as 1

^{1}, 1

^{2 }and 1

^{3 }respectively.

Therefore to express 1st, 2nd and 3rd in the context of 3 members we obtain the cube root of each

i.e. 1

^{1/3}, 1

^{2/3 }and 1

^{3/3}.

^{ }

We have already seen how the notion of nth in the context of n equates to the cardinal notion of 1

^{n/n}= 1.

So once again 1st (in the context of 1) + 2nd (in the context of 2) + 3rd (in the context of 3) = 1 + 1 + 1 = 3. Thus in this limited case ordinal reduces successfully to direct cardinal interpretation.

However in now considering 1st, 2nd and 3rd, in relation to each other (representing the 3 members of a group), we move from a linear to a circular notion of identity.

Thus 1

^{1/3}, 1

^{2/3 }and 1

^{3/3 }are represented in geometrical terms as 3 equidistant points on the circle of unit radius (in the complex plane).

So 1

^{1/3 }represents the 1st (in the context of 3) in an indirect quantitative manner as the complex number – .5 + .866i

1

^{2/3 }represents the 2nd (in the context of 3) in an indirect quantitative manner as the complex number – .5 – .866i.

And 1

^{3/3 }as we have seen represents the 3rd (in the context of 3) directly in a quantitative manner as 1.

What is fascinating here is that we have now used the Type 2 aspect of the number system - where explicit emphasis is on the dimensional aspect of number - to provide an alternative interpretation of rational numbers as fractions.

In conventional terms 1/3, 2/3 and 3/3 are given a standard quantitative meaning (which however betrays an unrecognised form of qualitative reductionism).

For example if we have a small cake divided into 3 slices we would represent 1 slice as 1/3 of the cake, 2 slices as 2/3 of the cake and 3 slices as 3/3 of the cake i.e. identical to the entire cake as a unit.

Thus here, 1 (in the context of 3) = 1/3 i.e. (1/3)

^{1}.

2 (in the context of 3) = 2/3 i.e. (2/3)

^{1}.

3 (in the context of 3) = 3/3 i.e. (3/3)

^{1 }= 1.

In this way, seemingly we can deal with fractional notions in an unambiguous quantitative manner.

However this conceals a big difficulty.

If we look at the 3 slices in an independent quantitative manner, they represent individual whole units.

And the cake itself represents an individual whole unit.

However to see the slices as a fraction of the whole cake, we need to switch from the notion of whole to part (with respect to slices) in relation to the overall cake which is now seen in complementary terms as whole.

Therefore as the notions of whole and part are quantitative as to qualitative (and qualitative as to quantitative) in relation to each other.

Therefore implicitly involved in this quantitative recognition of fractions is a corresponding qualitative recognition of an ordinal nature.

In fact the qualitative aspect of what is involved in obtaining (quantitative) fractions can be compellingly demonstrated in holistic mathematical terms.

Thus we start with the cardinal number 3 represented in Type 1 terms as 3

^{1}.

However 1/3 = 3

^{– 1}.

Thus moving from a natural number to its reciprocal (as a fraction) entails in holistic mathematical terms, the negation of the 1st dimension i.e. the negation of rational linear type understanding at a conscious level..

This then makes way for the complementary intuitive recognition at an unconscious level of the simultaneous correspondence of quantitative and qualitative notions which then enables quantitative recognition to switch from initial recognition of 3 as whole units to the corresponding recognition of these units as part.

Thus in the example of the cake, the ability to recognise the 3 part slices as equal to the 1 whole cake entails the corresponding ability to switch as between whole and part (and part and whole notions).

^{}

In reverse manner we have shown that through the Type 2 aspect of the number system that the same fractions now explicitly acquire a qualitative meaning as 1st, 2nd and 3rd respectively.

However again this qualitative understanding implicitly requires the corresponding quantitative recognition of 3 .

Thus once more, in terms of the Type 1 interpretation of fractions 1/3, 2/3 and 3/3 respectively represent the relationship between parts and whole (where 3 e.g. 3 part slices = 1 whole cake).

So though here quantitative recognition is explicit, corresponding qualitative recognition (of the two-way relationship between wholes and parts) implicitly is also required.

Then in terms of the Type 2 interpretation, 1/3, 2/3 and 3/3 now respectively represent the explicit qualitative relationship as between the 1st, 2nd and 3rd members of a group. However implicitly this requires the corresponding quantitative recognition of the group as containing 3 members.

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