The apparently simple operation of multiplication reveals in a glaring fashion the stark inadequacies of conventional mathematical interpretation. In fact the true nature of multiplication cannot be properly explained from this perspective!
The starting point in rectifying this problem is the clear recognition that - far from being static absolute identities - all numbers are inherently dynamic in nature entailing the interaction of twin complementary aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.
For example the number 2 is a cardinal number with a recognised independent quantitative meaning.
This implies that it is defined in a manner that renders it as without any meaningful qualitative (i.e. relational) context.
If numbers were indeed independent in an absolute manner, then by definition, no means would exist for establishing their interdependence (i.e. relationship) with other numbers!
So what we have in conventional mathematical terms is but a reduced interpretation of number. Thus the critical issue of establishing a relationship with other numbers - which is inherently of a qualitative nature - is effectively ignored!
So what is completely overlooked in formal conventional terms, is that every number - which I am illustrating here in the specific case of 2 - has strictly but a relative meaning with both quantitative and qualitative aspects that dynamically interact with each other in a complementary (opposite) manner.
So once again the number "2" has a quantitative (analytic) meaning. So for example, if I look out my front window and see two cars in the driveway, I am using "2" in its accepted quantitative sense.
However "2" has an equally important meaning that is of a qualitative (holistic) nature which can be referred to as "twoness" (i.e. the quality of "2").
Now whereas the former aspect relates to the independent nature of number recognition in a quantitative manner, the latter aspect - by contrast - relates to the corresponding interdependent nature of number which thereby provides for number its crucial relationship context.
In this way through the interaction of both quantitative and qualitative aspects, we are enabled to recognise number relationships in relative fashion, entailing both independence (from other numbers as distinct entities) and interdependence (through common relationship with these other numbers) respectively.
Now when we define the component "building blocks" of "2" in a quantitative manner, we do so in a manner that strictly lacks any qualitative distinction. To facilitate further exposition, I will refer to this as the Type 1 aspect of number!
So from the Type 1 perspective, 2 = 1 + 1, Here each unit is defined in a strictly homogeneous manner (where neither can be meaningfully distinguished from each other).
This begs the very fundamental question is how one is enabled to form number recognition of "2", given that its unit components are defined in an independent manner! So once again the all important qualitative aspect,whereby meaningful relationship as between separate units can be established,is entirely missing from conventional mathematical interpretation.
When we look at the latter aspect of number (i.e. Type 2), we arrive at a complementary appreciation of number that is qualitative (and thereby strictly lacking any quantitative distinction).
Then from the Type 2 perspective 2 (now reflecting its qualitative nature as "twoness") enables us to clearly distinguish the unique nature of both its units.
So here, 2 = 1st + 2nd, where each unit is now uniquely defined in an ordinal manner.
It is vitally important to realise in this context that both the cardinal and ordinal nature of number are directly identified with the two aspects of number (Type 1 and Type 2) respectively, which are crucially distinct from each other. This once again highlights the significantly reduced accepted interpretation, where both cardinal and ordinal aspects are absolutely treated in a merely (Type 1)) quantitative manner.
We are now ready to make the all important leap, whereby the true distinctive nature of addition and multiplication can be clearly revealed.
Because we are now recognising two aspects to number (that dynamically interact in experiential understanding), all numbers are defined in a twin manner containing both a base and a dimensional aspect.
Furthermore to move from Type 1 to Type 2 recognition we invert both aspects in a complementary manner.
So in Type 1 terms the number "2" is more comprehensively defined as 21. So the base here is 2 and the dimensional aspect 1.
The conventional treatment of natural numbers effectively views numbers solely in Type 1 terms (where they are given an absolute interpretation).
So 1, 2, 3, 4, ... can be more fully defined as 11, 21, 31, 41...
However because the reduced quantitative value in each case remains unchanged (when the dimensional number is 1), the implied default dimension (i.e. 1) is omitted altogether.
Furthermore whenever number expressions entail dimensional values (powers or exponents) other than 1, the ultimate value is given in a reduced quantitative manner (defined in terms of 1 as dimension).
So for example, in conventional mathematical terms 22 = 4 (i.e.41). So though in geometrical terms, we can easily see that this would represent 4 square (i.e. 2-dimensional) units - rather than 4 linear (1-dimensional) units, this qualitative change in the nature of the units is simply ignored, with the resulting value i.e. 4 given in a merely reduced quantitative manner (i.e. in 1-dimensional terms).
In Type 2 terms however the number "2" is defined in a complementary manner as 12 (where base and dimensional aspects are switched).
Thus the dimensional number here now varies (with respect to a fixed base number of 1) directly indicating the true qualitative nature of the number "2". Now once again this relates to the qualitative recognition of 2 as a number group whereby both 1st and 2nd members can be uniquely distinguished in an ordinal fashion.
However from a conventional perspective, the Type 2 aspect of number seems pointless, as the reduced quantitative value of each number = 1 (i.e. 11).
Finally in this entry, I wish to highlight the dynamic nature of interpretation that is now required (when we attempt to reconcile both Type 1 and Type 2 aspects with each other.
Expressing it simply, the numbers representing base and dimensional values respectively are always opposite to each other.
Therefore if - as we have seen with the Type 1 aspect - the base number is defined in a quantitative (analytic) manner, the corresponding (default) dimensional number is now - relatively - of a qualitative (holistic) nature.
However if by contrast - as we have seen with the Type 2 aspect - the dimensional number is defined in a qualitative (holistic) manner, the corresponding (default) base number (i.e. 1) is then of a quantitative (analytic) nature.
However, just as in the manner that the directions of left and right turns at a crossroads are reversed when we approach it from an opposite direction, likewise, when the polar frame of reference switches with respect to number the base number can now take on a qualitative (holistic) meaning, while the dimensional number - now relatively - is of a quantitative (analytic) nature.
We will elaborate further in the next entry!