So far example 3 * 5 = 15 (in conventional terms).
More accurately this result could be written as 151.
In other words, in conventional mathematical terms, the reduced quantitative value of the multiplication operation (expressed in 1-dimensional terms) is solely considered.
However, it seemed obvious to me, even then that a qualitative transformation is equally involved.
Thus, as with a rectangular field, 3 * 5 would be represented in square (i.e. 2-dimensional) rather than linear (1-dimensional) units.
Therefore, when we multiply numbers both a quantitative and qualitative transformation is involved (with the qualitative aspect relating - relatively - to the change in the dimensional units involved).
The deeper investigation of this issue then led me to the realisation that there are two distinct aspects to number definition - that are quantitative and qualitative with respect to each other - which keep switching in the natural dynamics of experience.
So we have the Type 1 aspect of number that is quantitative in the accepted sense that is defined in (default) 1-dimensional terms.
Thus in quantitative terms, all real numbers lie on the number line.
So, if again we take 3 to illustrate, its Type 1 definition is given as 31.
This of course equates with the cardinal notion of number which can be represented in quantitative terms as composed of independent individual units.
So 3 = 1 + 1 + 1.
It equally applies that because of the homogeneous nature of each unit, they are thereby lacking any qualitative distinction.
Now the Type 2 definition of number is directly related to its dimensional expression (as power or exponent) which is now defined with respect to a default base number of 1.
The idea here is when we raise 1 to a power (> 1) that clearly no quantitative change is involved. However the qualitative nature of the units does indeed change.
So for example if we raise 1 to the power of 3 (i.e. obtain the cube of 1), no quantitative change is involved with the result remaining as 1. However we are now dealing with 3-dimensional - rather than 1-dimensional - units which in this context - relatively - represents a qualitative rather than quantitative notion.
So this latter Type 2 definition of 3 is given as 13.
So the vital point to recognise here again is that number has in fact two distinctive aspects (Type 1 and Type 2) that are quantitative and qualitative with respect to each other.
Thus whereas the quantitative is based on the notion of independent units, the corresponding qualitative notion - by contrast - relates to number interdependence.
When on reflects on the nature of higher dimensions (> 1) this quickly becomes apparent.
Clearly to construct a 2-dimensional square figure the length and width cannot be considered as independent but in fact must be related to each other in a precise manner. So when we represent the unit line as its length, then the corresponding unit representing the width, must be now drawn at the end of the line (representing the length) at right angles to it.
Thus the length and breadth are thereby not independent of each other but rather related to each other in a definite manner.
Then we further go on to represent a cube (as the representation of 13), the units representing length, breadth and height must now be related to each other again in a precise manner.
Thus the crucial distinction as between the quantitative (Type 1) and qualitative (Type 2) aspects of number is that the former relates to the notion of units which are independent, whereas the latter relates to the corresponding notion of units that are interdependent (and thereby related to each other).
Therefore, properly understood, number is a dynamic interactive notion with complementary aspects that are relatively independent and interdependent with each other.
Without the independent aspect, we would not be able to distinguish the very numbers (that are to be related to each other in subsequent operations); without the corresponding interdependent aspect, we would have no means of achieving a common relationship as between these distinct numbers.
So both aspects (i.e. quantitative independence and qualitative interdependence) are vital for all number operations, which can only be properly understood in a dynamic relative manner.
However Conventional Mathematics is riddled through and through with the most massive reductionism, whereby in every context, notions that are properly qualitative in nature, are reduced in a merely quantitative - and thereby greatly limited and distorted manner.
To conclude this entry, I mentioned earlier how the Type 1 (quantitative) notion of 3 can be represented as
3 = 1 + 1 + 1 (where each unit is independent).
However the corresponding Type 2 (qualitative) notion of 3 cannot be represented in this manner (as the various units are related and thereby interdependent with each other).
So the latter representation is given as:
3 = 1st + 2nd + 3rd (where the various units are now related in a qualitative type manner).
So we now see that both the cardinal and ordinal notions of number are relatively distinct in nature (pointing in turn to both its quantitative and qualitative aspects).