Intricacies of Addition and Multiplication (4)

We have seen in the last entry how both the quantitative and qualitative nature of the natural numbers is directly related to the operations of addition and multiplication respectively (which are complementary with each other).

Thus in Type 1 terms (where the base is defined in quantitative terms)

by addition,

1 + 1 = 2 (i.e. 1¹ + 1¹ = 2¹).

Then in Type 2 terms (where the dimension is defined in qualitative terms)

by multiplication,

1 * 1 = 2 (i.e. 1¹ * 1¹ = 1²).

Thus we have switched from the quantitative notion of “2” as base number to the qualitative notion of “2” (or twoness)  as dimensional number in this fashion.

Thus with respect to the base (representing specific objects), the quantitative notion of 2 corresponds to cardinal interpretation

So 2 in this context arises from the recognition of homogeneous independent objects (without qualitative distinction)

Then with respect to the dimension (representing general frameworks for objects) the qualitative notion of 2 corresponds directly with ordinal interpretation.

So 2 (as twoness) in this context arises from the recognition of qualitatively distinct  1^st and 2^nd dimensional frameworks (which requires seeing both as qualitatively interdependent with each other). However  as we will indirectly demonstrate later 2 = 1^st + 2^nd lacks any quantitative distinction.

In this way we can see how addition and multiplication are directly related to both the cardinal and ordinal interpretation of the natural numbers respectively.

However Just like the left and right turns at a crossroads are reversed when we approach it from the opposite direction, likewise when we switch the frame of reference (with respect to both quantitative and qualitative) a complementary reverse interpretation results.

So what is addition from a Type 1 perspective, is multiplication from a Type 2 (and vice versa). And this equally applies to both quantitative and qualitative interpretations of base and dimensional values.

So we can equally start with the base number defined as qualitative and the dimensional number as quantitative respectively.

Now addition with respect to the Type 2 aspect implies the quantitative aspect of this dimensional number.

Thus 1 + 1 = 2 (i.e. ₁1 + 1  = ₁2).

Here number representing dimension carries the standard cardinal meaning where 2 = two dimensions.

Then in complementary fashion, multiplication with respect to The Type 1 aspect implies the qualitative aspect the application of this base number.

 

So 1 * 1 = 2 i.e. (₁1 + ₁1   = ₂1).

To distinguish the switch in the meaning (quantitative and qualitative) that numbers now possess, I have likewise reversed the notation, so that base numbers are now represented with subscripts and dimensions as standard size (just as formally, base numbers were represented by normal size and dimensions with superscripts respectively).

Though the meaning associated with the mathematical representation of addition and multiplication is difficult to intuitively grasp (due to the standard identification of number with merely quantitative values) it can be expressed quite simply in psychological terms.

In other words, number perceptions and concepts continually interact in a dynamic manner, whereby both rational (analytic) and intuitive (holistic) aspects are involved.

Through this dynamic interactive process, we are thereby enabled to distinguish the natural numbers in both cardinal and ordinal terms ,where they can represent both (specific) objects and (general) dimensions respectively.

So for example, we are thereby enabled to appreciate 3 as a cardinal number (applying to specific objects); we are also enabled to appreciate 3 in cardinal terms as applying more generally to dimensions i.e. 3 dimensions.

Equally we are enabled to appreciate 3 in ordinal terms with respect to specific objects (as 1^st, 2^nd and 3^rd) and likewise with respect to more generalised dimensions (again as 1^st, 2^nd and 3^rd).

The crucial point to recognise that this crucial capacity - whereby we are enabled to keep switching from cardinal to ordinal (and ordinal to cardinal meaning) - is directly related to the operations of addition and multiplication (that likewise behave in a dynamic interactive manner).

However as long as we attempt to interpret number in a merely quantitative manner, statements regarding the true dynamic nature of addition and multiplication can carry no resonance.