I have been at pains in recent blog entries, to show how distinctive meanings attach to both the base and dimensional aspects of number, which in dynamic interactive terms are quantitative as to qualitative with respect to each other. Put another way, when one manifests itself in a particle like manner (as number independence) the other - relatively - exhibits a wave like structure (as the relationship or interdependence as between numbers).
This distinction is crucial in clarifying the respective nature of addition and multiplication respectively.
When we add two numbers, in a direct manner the independent quantitative nature of number is involved.
So, for example 2 + 3 = (1 + 1) + (1 + 1 + 1) = 5, where in explicit terms all units are viewed in an independent separate manner as quantitative.
Thus expressed more fully (Type 1 terms),
21 + 31 = (11 + 11) + (11 + 11 + 11) = 51.
(However implicitly the corresponding qualitative notion of number interdependence is required to enable recognition of the new number identity here of 5!)
However when we multiply, 2 * 3 both a quantitative and qualitative transformation is involved, entailing both the Type 1 and Type 2 aspects of number.
So expressed more fully (In Type 1 and Type 2 terms),
21 * 31 = 61 * 12.
We can see this in geometrical terms by imagining a rectangle with length 3 units and width 2 units respectively, with the new area of 6 square (i.e. 2-dimensional) units. In other words the area is made up of 6 units (where now each 2-dimensional unit represents a square with side 1 unit).
Then in a directly qualitative manner, if we imagine 2 rows with three units in each row, then 2 * 3 carries a distinctive meaning, whereby one can readily intuit that the 3 units in each row are similar with respect to the 2 rows.
So recognition of such similarity (with respect to the 2 classes) entails a distinctive qualitative notion of number interdependence.
Thus we must indeed initially recognise the 3 units in each row (in an independent quantitative manner).
Indeed, if were to confine ourselves to such quantitative recognition, we would reduce the multiplication of 2 * 3 to the addition of 3 + 3, where the result is given in a merely 1-dimensional format.
However the crucial distinction in moving from such addition to a coherent understanding of multiplication is in the recognition of common similarity (i.e. the qualitative interdependence of the two classes). So though once again, we must initially recognise both the quantitative independence of the 2 rows and the corresponding interdependence of the 3 items in each row, for multiplication to arise, we must then recognise the common similarity of the 2 classes of 3 (with each other) which equally implies the common similarity of the 3 classes of 2 (with each other).
Thus, with multiplication we are combining the individual quantitative independence of each item with the collective qualitative interdependence of all items!
Once again, because Conventional Mathematics allows for sole recognition of the (Type 1) quantitative aspect, it cannot offer a coherent interpretation of the very nature of multiplication.
Using the analogy with physics it is akin - at a much more fundamental level - to an attempt to explain quantum mechanical behaviour in terms of Newtonian Mechanics!
And this crucial limitation cannot be remedied from within Mathematics as now accepted, but rather requires a radical reinterpretation of all its relationships.
Thus in terms of mathematical experience, a ceaseless dynamic interaction in involved with respect to both the quantitative and qualitative aspects of number.
The key here to understanding is the notion of complementarity. So when the base number is quantitative (as independent) the corresponding dimensional number is qualitative (as interdependent); likewise when the dimensional number is now quantitative (as independent) the corresponding base number is - relatively - qualitative as interdependent.
As we have seen, with multiplication both the quantitative and qualitative interpretations arise with respect to number.
Thus the former quantitative aspect e.g. of the number 2 arises when we recognise the number as comprising independent units.
However the latter qualitative aspect arises when we recognise 2 as in common with other classes of 2. In this way we recognise this similarity of 2 with respect to 1, 2, 3, ......, in fact with respect to a potentially infinite number of classes.
Crucially in dynamic terms the understanding of number is merely relative.
In other words notions of number independence (as quantitative) necessarily imply corresponding notions of number interdependence (as qualitative) ; likewise notions of number interdependence (as qualitative) imply corresponding notions of number independence (as quantitative).
Because however of the distorted - merely quantitative - interpretation of Conventional Mathematics, which has now become deeply embedded in our culture, we misleadingly view the natural numbers, - and especially the primes - in an absolute manner. In fact properly, understood the primes and natural numbers have a merely relative identity that - ultimately - perfectly reflect each other in an ineffable manner.
I will deal with this further in the next blog entry.