Intricacies of Addition and Multiplication (5)

So far, we have dealt with numbers in their simplest form i.e. as individual natural numbers, which can be defined with two aspects that are Type 1 (quantitative) and Type 2 (qualitative) with respect to each other.

Thus, when the base number in Type 1 terms - which can vary with respect to the natural numbers - is quantitative, the default dimensional number of 1 is defined in Type 2 terms as qualitative.

Then in reverse terms when the dimensional number from a Type 2 perspective - which now can equally vary with respect to the natural numbers - is qualitative, the default base number of 1 is defined in Type 1 terms as quantitative.

However through the dynamics of interaction reference frames keep switching.
Therefore from this complementary perspective when the base number in Type 2 terms - which can likewise vary with respect to the natural numbers - is qualitative, the default dimensional number is defined in Type 1 terms as quantitative.
Then in reverse terms when the dimensional number in Type 1 terms - which varies with respect to the natural numbers - is quantitative, the default base number is defined in Type 2 terms as qualitative.

Therefore, every natural number, through the dynamics of experiential interaction, acquires both quantitative (analytic) and qualitative (holistic) meanings, with respect to numbers representing (specific) base objects and (general) dimensional categories respectively.

So once again the number 3 for example can be given a quantitative (analytic) meaning with respect to either (specific) objects or (general) dimensions e.g. 3 cars or 3 mathematical dimensions. This indeed coincides with the cardinal interpretation of the (collective) number as comprised of homogeneous independent units (without qualitative distinction) where 3 = 1 + 1 + 1.

However the number 3 can equally be given a qualitative (holistic) meaning with respect to either (specific) objects or (general) dimensions as "threeness" (i.e. quality of 3) with respect to the distinct individual identification of the cars and dimensions in question.
This likewise coincides with the ordinal interpretation of the (individual) numbers, which comprises of unique identity of each unit (without quantitative distinction) where 3 (as threeness) = 1st + 2nd + 3rd.

Strictly speaking we cannot have quantitative appreciation (of a cardinal nature) without corresponding qualitative appreciation (of and ordinal nature) and vice versa.

However it is vital once again to appreciate that both of these aspects correspond to two distinct interpretations of numbers, that are quantitative and qualitative with respect to each other.
Thus the nature of number is inherently dynamic and interactive comprising twin opposite poles that behave in a complementary manner.


We now move on to the more intricate situation of properly explaining the nature of addition and multiplication (from this new dynamic interactive perspective).

The addition of two number quantities (as base numbers) in standard Type 1 terms seems quite simple.

So for example  2¹ + 3¹  = 5¹.

Now the multiplication of the two number (as dimensional numbers) likewise appears simple.

So 1² * 1³  = 1⁵.

However  what appears as multiplication from the Type 1 perspective represents in fact addition from the Type 2. i.e.  1² * 1³  =  1^{2 + 3}.

This simply indicates how addition and multiplication are themselves directly linked to the Type 1 and Type 2 aspects of number respectively. So what represents addition from one perspective, represents multiplication from the other (and vice versa).


Now we will consider the more difficult case where two natural numbers (with base ≠ 1) are multiplied.

Indeed this consideration is of the utmost importance as all natural numbers (≠ 1) can be uniquely expressed as the product of prime factors!

Let us therefore in this context consider the case of 2 * 3 i.e. 2¹ * 3¹.

In conventional mathematical terms a merely reduced quantitative result is given.

So from this perspective, .

2 * 3  = 6 (i.e. 6¹)

However this unsatisfactory from a simple geometrical interpretation will indicate that the result should be represented in square (2-dimensional) rather than linear (1-dimensional) units.

So therefore, properly understood the multiplication of these two numbers requires both Type 1 and Type 2 aspects.

Therefore 2¹ * 3¹ = 6¹ * 1²

Thus in quantitative Type 1 terms, the result is 6 (i.e. 6¹). However in qualitative Type 2 terms the result is 2 i.e. 1² .

The combination of these two aspects reflects the fact that the quantitative result (6) is now expressed in 2-dimensional terms.

However though relative to the base number result (i.e. 6), the dimensional number 2 is qualitative, within its own frame of reference (as applying to dimensional rather than base numbers) it is quantitative in nature.

So the base result from a quantitative perspective is 6 (i.e. 6 unit objects); and  the dimensional result from a quantitative perspective is 2 (i.e. 2-dimensional).


We now equally need to reflect the corresponding qualitative meaning of the symbols.

Again in conventional mathematical terms we strictly must represent 2 * 3 as representing separate
independent objects. For example imagine the 6 objects are circular rings as represented below.

Then we let 3 represent each row (containing 3 objects) and 2 each column (containing 2 objects) as below.

 O O O

 O O O

Now by treating each ring as independent we can con count 3 in each row so that the total result = 3 + 3 = 6.

However to represent 3 + 3 as 2 * 3 , a decisive qualitative transformation is required whereby rather than seeing the objects in each row as independent (in cardinal terms) that they be rather seen in qualitative ordinal terms as composed of 1st, 2nd and 3rd members.

Without such recognition we could not meaningfully identify the two separate rows (which requires a common recognition with respect to the 3 members involved).

So the identification of each row requires the qualitative recognition of the notion of 3 (as "threeness") thereby establishing the common shared identity of the 3 members involved.

Then the key insight by which the operator 2 can be used to multiply the number in each row (i.e. 3) requires the common recognition of the identity of members in both rows.

So each member in the 1st row thereby is seen to share a common identity with each member in the 2nd row and it is this qualitative recognition of interdependence that thereby enables us to use the operator 2 in a multiplicative sense.

So overall the multiplication of 2 * 3 contains quantitative aspects respect to both the base and dimensional nature of the units involved.

So we thereby can identify 6 independent units (in quantitative terms) that are expressed in a 2-dimensional fashion (as square units).

However equally we have the qualitative recognition of 3 (through the ordinal recognition of 1st, 2nd and 3rd members in each row and the qualitative recognition of 2 (through the ordinal recognition of a 1st and 2nd row).

Ultimately therefore the resulting recognition of 6 (through multiplication) entails the quantitative recognition of each member as independent and the qualitative recognition of the six members sharing a common identity through the one to one correspondence of the 3 members in each of the 2 rows.

So in base terms not alone do we establish the quantitative recognition of 6 as cardinal, we now equally establish in qualitative terms ordinal recognition of 1st, 2nd, 3rd, 4th, 5th and 6th members.  

Likewise in dimensional terms, not alone do we establish the quantitative recognition of 2 (i.e. as 2-dimensional units), we equally establish the qualitative recognition of 2 enabling us in this context to distinguish 1st and 2nd dimensions in an ordinal fashion.