Friday, August 19, 2011

Differentiation and Integration

Differentiation and Integration are of vital importance in both physical and psychological terms.

Once again these essentially relate to two different systems of logic that are linear (1) and circular (0) with respect to each other.

In psychological terms - which is complemented in physical terms - differentiation entails conscious type understanding of reality based on the separation of opposite polarities (such as internal and external). This corresponds in turn with the linear use of either/or logic (which defines conventional mathematical understanding).

However properly speaking integration - which is again replicated in physical processes - entails unconscious understanding based on the complementarity and ultimate identity - of these same polarities. Though directly of an intuitive infinite nature, indirectly this corresponds with the circular paradoxical use of both/and logic, which in formal terms is completely ignored in conventional mathematical interpretation.

Differentiation relates directly to the quantitative means by which we can analytically interpret reality; integration by contrast relates to the corresponding qualitative means by which we can holistically interpret that same reality.

Obviously the attempt to use just one means of understanding i.e. the linear logical system, for both quantitative and qualitative type interpretation in Mathematics, leads to considerable reductionism and essentially this is the present position with Mathematics. Basically it leads to a confusion of infinite with finite type notions.

Now, differentiation and integration are likewise of considerable importance in Mathematics and perhaps surprisingly there are very close (unrecognised) links with corresponding psychological usage.

When we look at the (simplest) holistic notion of integration in psychological terms, it entails the harmonisation and interdependence in experience of opposite polarities (such as external and internal). Now the complementarity of such opposite poles (+ and -) corresponds directly with 2-dimensional understanding. So 2 here a dimension is pointing to the fact that in this qualitative context these two poles form a complementary pairing.

Then when we differentiate in experience we move from 2-dimensional (where poles are interdependent) to 1-dimensional appreciation (where they are separated). So we now move from the holistic dimension (which is qualitative) to the ability to differentiate objects as separate (in dualistic terms). So the very meaning of 2 switches from the holistic qualitative interpretation, where 2 represents a complementary pairing, to the analytical quantitative interpretation where both poles are now separated (thus enabling dualistic understanding).

Now, remarkably this is replicated in mathematical terms.

If we start with the simple expression y = x^2, 2 here as number properly represents a (qualitative) dimension. However when we differentiate y with respect to x, dy/dx = 2x. So the dimension has now been reduced to 1 (as befits differentiation) with the 2 now representing a base quantity!

In the same manner as sub-atomic phenomena in physics - where all particles likewise have a wave aspect and all waves a particle aspect - likewise properly understood in mathematics all numbers as qualitative terms representing dimensions likewise have quantitative aspects; and all numbers as quantities likewise have quantitative effects.

So it is true that numbers (as dimensions) also have a legitimate quantitative aspect - a fact that is widely exploited in conventional interpretation!

However this obscures the important truth that in relation to each other, numbers as base values (raised to a particular dimension) and numbers as dimensions (to which a particular number is raised) are properly quantitative as to qualitative (and alternatively qualitative as to quantitative) in relation to each other.

This is of vital significance for example in understanding the true nature of prime numbers!