As we have seen number can be defined in two ways (which dynamically interact in mathematical experience).
Now to see this clearly, numbers must be defined with respect to base and dimensional numbers (which properly in a dynamic manner, are quantitative and qualitative with respect to each other).
So again to take the simplest (non-trivial) case of the number 2, this can be defined in two distinct ways.
In the conventional quantitative approach - related to (pure) addition
2 = 1 + 1
i.e. expressed more fully
21 = 11 + 11
So the essence of this quantitative relationship is that default dimension to which the (base) number quantity (2) is expressed remains fixed as 1.
And as I have repeatedly stated, Conventional Mathematics, geared exclusively to the quantitative interpretation of relationships is precisely defined, from a qualitative perspective, in terms of its linear (i.e. 1-dimensional) mode of interpretation.
So in Conventional Mathematics with respect to multiplication a merely reduced (i.e. quantitative) interpretation is possible.
So in this context if we multiply 2 by 2 i.e. 22 , the answer will be given as 4 i.e. 41.
Thus the answer in reduced quantitative terms is expressed with respect to the default dimensional value of 1 (which again illustrates well the 1-dimensional nature of mathematical interpretation).
However if we think about it for a moment, clearly a dimensional transformation (of a qualitative nature) is likewise involved.
Thus in simple geometrical terms, 2 * 2 would be represented by a square (with side 2 units).
Therefore the area of this square is properly expressed in square (i.e. 2-dimensional units).
Thus a dimensional change of a qualitative nature is clearly involved through this simple multiplication process.
However in conventional mathematical terms the qualitative nature of this transformation is simply edited completely out of the process with the result expressed thereby in a reduced i.e. merely quantitative manner.
And such reductionism universally characterises the nature of multiplication from a conventional mathematical perspective.
It is hardly surprising therefore that mathematicians eventually realise that there is something fundamentally missing from their appreciation of the relationship as between addition and multiplication (as multiplication is inherently interpreted in a reduced merely quantitative manner).
This issue struck me so strongly at the age of 9 or 10, that I already realised then that there was - literally - a fundamental dimension with respect to conventional mathematical understanding that effectively was overlooked. In other words in the mere quantitative interpretation of number, its qualitative dimensional aspect is simply ignored.
So the conventional quantitative approach to number - where number is defined in a cardinal manner - I refer to as the Type 1 aspect of the number system.
So again in Type 1 terms,
21 = 11 + 11
However in the corresponding Type 2 aspect, the number 2 is expressed in terms of a (pure) multiplication process.
So in Type 2 terms,
2 = 1 * 1
In other words,
12 = 11 * 11
Now 2 in our definition of number has been inverted, with 2 representing the dimensional qualitative nature of number, which is defined with respect to the default (base) quantity of 1.
So again if we think of this in geometrical terms, when we square 1, we obtain a 2-dimensional figure (with area 1 sq. units)
Therefore, though in quantitative terms nothing has changed (with the 2-dimensional area the same as its 1-dimesnional side), clearly a qualitative change in the nature of units has been involved.
Thus, the very purpose of the Type 2 approach is to isolate the qualitative aspect of number transformation, which is related directly to the pure nature of multiplication.
As I have repeatedly stated in my blog entries, the mystery of the relationship of addition and multiplication is the same mystery as the relationship of the quantitative and qualitative aspects of number.
Once again there are insuperable difficulties in attempting to understand this relationship in a mere quantitative manner!
However having isolated the qualitative aspect of number transformation (through its Type 2 aspect), the next problem is in attempting to give expression in an indirect quantitative manner to this aspect.
The deeper reason for this is that when one accepts that there are necessarily two distinctive aspects to number (which are quantitative and qualitative) with respect to each other, then the ultimate consistency of both aspects becomes the indisputable key mathematical requirement.
So demonstrating such consistency - which clearly cannot be proved within the reduced conventional mathematical perspective - requires the ability to (i) indirectly convert from the qualitative to the quantitative aspect and (ii) the ability in reverse manner to convert from the quantitative to the qualitative aspect, while establishing complete harmony between both aspects.
Put another way it requires establishing the interdependence of both quantitative and qualitative aspects (from two complementary perspectives).
Now once again, by its very nature, this is not strictly possible within the conventional mathematical perspective (defined as it is in a mere quantitative manner).
Such interdependence relates directly to a holistic - rather than analytic - type appreciation of mathematical relationships.
It is central to appreciation of the true nature of the number system and yet in formal mathematical terms currently does not even exist.
To sum up this blog entry, properly understood in dynamic interactive terms, there are two distinctive ways of interpreting every number.
The Type 1 relates to its quantitative aspect, directly associated with the (pure) notion of addition.
The Type 2 relates to its qualitative aspect directly associated with the (pure) nature of multiplication.
Both of these continually interact in a - necessarily - relative manner in experience and are directly associated with the cardinal and ordinal aspects of number interpretation respectively.
Conventional mathematical interpretation therefore represents but a highly reduced - and greatly confused - interpretation of the number system.
We will have more to say about the - much misunderstood - ordinal nature of number in the next entry.