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

In yesterday's blog, entry, I distinguished how the notions of independence and interdependence respectively are associated with two complementary aspects of number, dynamically interactive in nature, that are quantitative and qualitative with respect to each other.

So for example whereas we can denote the quantitative (independent) notion of "2" as 2¹, by contrast the corresponding qualitative (interdependent) notion in an inverse complementary manner as 1².

Thus whereas the first aspect (Type 1) relates to two number units of a linear (1-dimensional) nature the latter (Type 2) relates to one number unit that is square (2-dimensional) in nature.

Therefore, once again the key distinction as between both aspects is that the the former (Type 1) relates to units that are independent of each other, whereas the latter (Type 2) relates to units that have and ordered relationship that are thereby interdependent with each other.

Thus properly incorporating the key notions of number independence (as quantitative) and number interdependence (as qualitative) respectively entails again that both must be considered together in a dynamic interactive manner, where independence and interdependence have now a strictly relative - rather than absolute - meaning.


These dynamic relative distinctions are then vital in illustrating the key difference as between the operations of addition and multiplication respectively.

We can now illustrate this simple case of addition i.e.

1 + 1 = 2.

In conventional mathematical terms, both of these units are considered as homogeneous in a merely reduced quantitative manner where they are considered absolutely independent of each other (thereby lacking any qualitative distinction)..

However we can now more accurately relate addition directly with the Type 1 aspect of number.

So now 1¹ + 1¹ = 2¹.

Both units here are still considered explicitly as independent in a quantitative manner. However now the interpretation is strictly of a relative nature (that implicitly allows for the numbers to be related to each other).

There is a basic issue with conventional interpretation that is completely overlooked and in effect treated in a grossly reduced manner. For if the two units are in fact to be treated as independent in an absolute manner, then this begs the question as to to how they can be combined with each other to form the new number identity of "2"!

In other words, we can see clearly here that in conventional mathematical terms, the qualitative notion of number relationship (i.e. number interdependence) is simply reduced in a a merely quantitative (independent) manner.

Because of this basic reductionism, there is no way in conventional terms to properly distinguish the nature of multiplication from addition.

So again in conventional terms 1 * 1 = 1 (with the quantitative value remaining unchanged).

However when we now properly distinguish as between the Type 1 and Type 2 aspects of number,
this operation of multiplication is represented as

1¹ * 1¹ = 1².

In general terms in the number expression a^b, I refer to a as the base and b the dimensional number respectively.

Therefore in this simple case  we can see that with addition, the base number changes while its dimensional counterpart remains unchanged. This is associated explicitly with a quantitative change in the units involved.

Then, by contrast with multiplication, the dimensional number now changes while its base counterpart remains unchanged. This  in turn is explicitly associated with a qualitative change in the units involved. In other words in the multiplication of  1 by 1, we must explicitly recognise that the units are related to each other (i.e. as interdependent) in a qualitative manner!

Therefore by confining ourselves initially to the simplest possible operations representing addition and multiplication respectively, we can clearly distinguish how multiplication differs from addition.

So quite simply, expressing this finding appropriately in a dynamic interactive manner, the operations of addition and multiplication are quantitative and qualitative with respect to each other.

Thus once again whereas addition in our illustration was associated with a quantitative transformation in units, (with the qualitative dimension unchanged), by contrast multiplication was associated with a qualitative transformation in the units (with the quantitative base unchanged).    


The implications here are truly enormous, though apparently completely overlooked in conventional mathematical terms.

Put simply, we cannot coherently understand the simplest use of the operations of additions and multiplication respectively from within the present universally accepted mathematical perspective.

In fact put bluntly, we cannot coherently understand any mathematical relationship from this accepted conventional perspective.

Thus an intellectual revolution much more fundamental than that which has already partly occurred in physics with quantum mechanics is urgently required for the coherent understanding of all mathematical relationships.

For so long we have tried to reduce in every mathematical context qualitative to quantitative type meaning.

This has now created the considerable illusion that Mathematics is all about abstract quantitative type relationships (where qualitative considerations have no relevance).

In truth this is utterly unfounded. Just as we cannot properly reduce the understanding of water solely to its hydrogen atoms. in a much more fundamental manner we cannot properly reduce mathematical understanding to its mere quantitative aspect!