Operations: Multiplication and Division
 

Let us examine these operations firstly from an analytic perspective.

As we saw with the operation of addition
1 + 1 = 2 or more precisely 1¹ + 1¹ = 2¹

So the result here i.e. 2 represents a horizontal (quantitative) transformation that takes place with respect to the same (qualitative) dimension of 1.

Now if we multiply 1 by 1 the dimensional characteristic changes

So 1¹ * 1¹ = 1²
 

In this case a vertical (qualitative) transformation (of the dimensions) takes place with respect to the same number quantity.

The operation of addition alters the horizontal notion of number (as quantity).

The operation of multiplication alters in similar manner the vertical notion of number
(as dimension).

So in this context multiplication represents the vertical counterpart
of addition on the horizontal level.

Again, as we have seen the operation of subtraction alters the horizontal notion of number (as quantity)
1 - 1 = 0 (strictly 1¹ - 1¹ = 0¹)

Again if we divide 1 by 1 again the dimensional characteristic changes

In other words 1¹ / 1¹ = 1⁰

So again (in this context) division represents the vertical counterpart in qualitative (dimensional) terms of the horizontal operation of subtraction in quantitative terms.
 

As always we have an alternative interpretation of these same operations that is dynamic and holistic.

Multiplication and division - in this dynamic holistic sense - are intimately connected with the relationship between whole and part (and part and whole).

So with multiplication we posit in vertical terms (with respect to dimensions).
With division we negate in vertical terms (again with respect to dimensions).
Again in a (linear) differentiated approach, only the positive direction with respect to dimensions takes place.
This leads to a (reduced) dualistic interpretation where - in any context - whole and part are separated (and considered in opposition to each other). So we literally generate multiple facts (parts) and multiple wholes (theories) that are reduced in terms of each other.

Again in an integral approach, a bi-directional interpretation (in vertical terms) is employed leading to a nondual interpretation of dimensions.
In psychological terms this means that there is a dynamic bi-directional relationship as between concepts (wholes) and perceptions (parts).
In corresponding physical terms it means that there is a bi-directional relationship as between dimensions (wholes) and object phenomena (parts).
Thus in dynamic terms the whole has no meaning in isolation from the part; equally the part has no meaning in isolation from the whole.
So when the positive aspect is solely recognised (as with asymmetrical differentiation) parts and wholes are indeed treated in isolation from each other.
We see this very clearly for example in conventional science where empirical research (parts) and theoretical analysis (wholes) are often conducted separately.

When we consider the relationship between horizontal and vertical polarities we must allow for both quantitative and qualitative transformation.
For example if I look at a house I may recognise various constituent parts e.g. doors, walls, windows, roof, chimney etc.
However the actual recognition of the relationship of such parts as a house requires a decisive qualitative transformation, where this relationship assumes a new whole identity (which we recognise as a house). Of course the great problem is that this qualitative transformation is so often overlooked, with the house in turn being interpreted in a largely (reduced) quantitative fashion.
Not surprisingly fragmentation in experience frequently results, whereby wholes (which are qualitatively distinct from parts) are yet interpreted - in reduced quantitative terms - as the sum of the parts.