Clarifying Analytic and Holistic Interpretations of "2" (1)

I return once more to illustrate the crucial distinction as between the analytic (quantitative) and holistic (qualitative) aspect of number.

Once again from the additive approach each number can be expressed in quantitative terms as the combination of individual independent units.

So for example in the simplest case 2 = 1 + 1.

Because these units are defined in a merely homogeneous absolute quantitative manner, they are thereby lacking any qualitative distinction.

However if the units were indeed absolutely independent of each other, then there would be no means by which such numbers could subsequently be related to each other.

So this latter capacity by which (individual) numbers can be related to each other assumes the complementary notion of number interdependence (which is inherently qualitative in nature).

So complementing the quantitative notion of "2" is the corresponding qualitative notion of "twonness"  where the two units, that are assumed independent of each other in quantitative terms, are now understood as interdependent with each other,

Put another way, the quantitative notion of "2" could not exist without the corresponding qualitative notion of "twoness". Equally the qualitative notion of "twoness" implying the interdependence of both units could not exist without the quantitative notion of "2" (where we must start with the independent units).

Therefore the clear implication is that the true notion of number - far from being abstract in an absolute manner - is inherently of a dynamic interactive nature (comprising both quantitative and qualitative aspects).

And the notions of independence (by which numbers assume a separate individual individual identity) and interdependence (by which they are collectively related to each other) are thereby of a merely relative  nature.

Thus the conventional quantitative approach - which is accepted without question - is but a reduced interpretation of number that ultimately greatly distorts the very nature of mathematical activity!

I have expressed before how both aspects of number i.e. quantitative and qualitative can be represented.

With respect to the quantitative aspect, each natural number is defined with respect to the default dimension (i.e. power or exponent of 1).

So 2 from this quantitative perspective is defined as 2¹.

When a number is then raised to a dimensional power (≠ 1), its ultimate value is then given in a reduced quantitative manner.

So with respect for example to 2², we can perhaps readily appreciate that is geometrical terms, this would be represented by a square (with area of 4 square units).
Therefore as well as the quantitative transformation implied by the expression, we also have a qualitative transformation in the nature of units involved.

However with conventional number interpretation, this qualitative aspect is simply ignored with the value of 2²  given in a merely linear quantitative manner as 4 (i.e. 4¹).

With respect to its qualitative aspect, 2 is expressed in a complementary fashion as 1².

Therefore in this case, we are concentrating on the inherent qualitative nature of the number.

Now one could indeed attempt to express 1² as 1^{(1 + 1)}, just as we could equally attempt to express 2¹ 
as (1 + 1)¹,

However the addition of these units refers to two distinctive meanings (which are completely ignored in conventional interpretation)!

In the quantitative case - which I will refer to as the Type 1 aspect - the units involved are assumed to be independent of each other.

So again, 1 + 1,  in Type 1 terms refers to units that are understood independent of each other.

However in the corresponding qualitative case - which I will refer to as the Type 2 aspect - the units are now considered, by contrast, as interdependent with each other.

So now 1 + 1, in Type 2 terms, refers to units that are understood as related i.e. interdependent with each other.

Indeed if one reflects briefly on the nature of a 2-dimensional object, one can quickly appreciate that the 2 dimensions involved  i.e. length and width, cannot be meaningfully independent of each other.

Though the length and width, separately viewed, are both of a 1-dimensional nature, clearly the very appreciation of the two 2-dimensional object, requires that we can view these two dimensions as related to each other in an ordered manner.

Therefore if we tried to maintain the separate independent identity of the two dimensions (as in Type 1 terms), we have the consideration of 2 units objects (that are 1-dimensional in nature) represented as 2^1.

However when we now recognise the relationship of the two dimensions (length and width) as interdependent then we have the consideration of 1 unit object (that is 2-dimensional in nature) represented as  1².

The deeper realisation here is that all mathematical symbols can be viewed in both Type 1 (analytic) and Type 2 (holistic) terms. In the first case, we concentrate on the independent (quantitative) aspect of relationships. Then in the second case, we concentrate on the interdependent  (qualitative) aspect.

And crucially both of these aspects can only be properly understood in a dynamic interactive manner as complements of each other, where both quantitative independence and qualitative interdependence are understood in a relative - rather than absolute - manner!