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. 1

^{1 }+ 1^{1 }= 2^{1}).
Then in
Type 2 terms (where the dimension is defined in qualitative terms)

by
multiplication,

1 * 1 = 2
(i.e. 1

^{1 }* 1^{1 }= 1^{2}).
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.

_{1}1 + 1^{ }=_{1}2).
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. (

_{1}1 +_{1}1^{ }=_{2}1).
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.