This latest contribution in many ways entails a summary of the points that I have made in recent blog entries and in discussion with Anthony Judge in relation to “Transforming the Art of Conversation - conversing as the transformative science of development” at his hugely impressive “Laetus in Praesens” site.
I have been long fascinated by the fact that the two binary digits (1 and 0)
when used in a quantitative manner can potentially encode all
I am therefore of the opinion that the same two digits when used in an
appropriate qualitative manner can likewise potentially encode all transformation processes.
So transformation itself (in all its manifestations) is basically encoded in number when appreciated in a qualitative manner.
Now as geometrical symbols, 1 can be identified with the straight line and 0 with a circular circumference. So the relationship of 1 and 0 in qualitative terms implies the relationship between (rational) linear and (intuitive) circular understanding. (In this context circular refers to the indirect rational attempt through paradox to portray the nature of intuitive understanding).
From a physical perspective this would imply that all transformation processes entail the interaction of a visible phenomenal aspect together with an equally important invisible holistic dimension.
At a deeper level this circular aspect relates to the manner in which the
fundamental polarities - which necessarily underlie all phenomenal
relationships - are configured.
For convenience, I would see that two key sets here are essential to all dynamic relationships i.e. external and internal and whole and part. In dynamic terms, external always implies internal (and internal external). Likewise wholes imply parts (and parts wholes). All phenomenal creation necessarily entails the two-way interaction of both sets of polarities.
Conventional Science and Mathematics are decidedly linear (i.e. 1-dimensional) in the manner that these polarities are treated with isolated independent reference frames employed. So the external (objective) is abstracted from the internal (subjective) aspect; likewise wholes are typically viewed as composed of parts in a mere quantitative manner. Not surprisingly this leads to a highly reduced interpretation of truth!
However an unlimited number of higher dimensional perspectives are possible which all entail an authentic dynamic interaction as between polarities.
The nature of each number, as qualitative dimension, is structurally related to the corresponding notion of quantitative roots of unity.
So the nature of 2-dimensional understanding bears a close relationship
therefore with the two roots of 1, i.e. + 1 and - 1. However whereas with
standard quantitative appreciation, these two values are separated, in
holistic qualitative terms they are interdependent. Thus 2-dimensional
understanding can be therefore expressed as the complementarity of (real)
opposites in the dynamic interaction of poles which are positive and
negative with respect to each other.
These dimensions can be given a geometrical representation (though we must remember that the interpretation is now of a holistic nature).
For example 3-dimensional understanding can be geometrically represented in terms of the well-known Mercedes-Benz logo (which equally is a geometrical representation of the 3 roots of 1).
So in short, each number as dimension, relates to a unique manner of
configuring the dynamic interaction of the two fundamental sets of
polarities. So rather like with a compass with four starting coordinates, we
can obtain ever more detailed notions of direction, likewise starting with
the two fundamental polarity sets we can give ever more refined expression
to the dynamic interaction between opposite coordinates through moving to
higher dimensional numbers! So once again each number in this qualitative
sense represents a unique manner of configuring the dynamic interaction as
between the essential polarities that necessarily underlie phenomenal experience.
This key issue is avoided completely in conventional scientific (and
mathematical) terms through sole concentration on the special limiting case
where understanding in formal terms is 1-dimensional.
Now, I believe that this qualitative holistic notion of dimension intimately
applies to the true nature of space and time. So if we were to map
space-time reality, we could validly say that it is truly multi-dimensional
where the ceaseless interweaving of these qualitative numbers are involved. Going even further, the distinctive qualitative features that phenomena possess, thereby represent multi-dimensional configurations with respect to space and time that are ultimately rooted in the qualitative notion of number.
I would go even further. In dynamic relative terms, phenomena represent but appearances (in continual transformation) of an ultimate reality that is ineffable.
In fact, from this perspective, we can say that such phenomena (which
possess no ultimate substance) fundamentally represent but the dynamic
configurations of number (with respect to both their quantitative and
From a geometrical perspective the quantitative shape of all phenomena can
be understood in terms of the interplay of both linear and circular
properties in varying dimensions.
The corresponding qualitative "shape" of these phenomena in their uniquely
distinctive features can likewise be understood in terms of both specific
and holistic features again with respect to the combined interplay of
multiple dimensional numbers (which again represent a distinctive manner in which the fundamental polarities are dynamically configured).
In a direct sense I would see the quantitative aspect of understanding as
relating to form, with the qualitative relating to the mysterious
transformation of this form.
So if we are to isolate what is common to all patterns of transformation, it
is the intersection of this holistic qualitative aspect with established
quantitative notions of form.
However when one accepts that the very nature of the standard paradigm of
science and mathematics is to attempt to reduce this interaction in a merely quantitative manner, then one can perhaps appreciate why it is inimical to transformation.
It is not that science as such is opposed to such transformation, but rather the present limited version that is wrongly accepted as solely synonymous with valid scientific interpretation!
Now there is much greater freedom for both the development and expression of the qualitative aspect within the arts.
So in the quest to transform present conversation - even scientific conversation - it would be helpful to informally dialogue with artistic metaphors.
Of course acceptance of the (neglected) qualitative aspect of science (and mathematics) would eventually pave the way towards better integration with the arts (with both seen as complementary expressions of the same truth).
My key point again is this!
There is not just one Mathematics (that is qualitatively 1-dimensional in nature) but potentially an infinite set, with each interpretation as the complex expression of a number dimension possessing a partial relative validity. And as phenomenal reality can be expressed as the dynamic interplay of all these dimensional systems in complex space and time (with quantitative and qualitative aspects), ultimately it is vital that we abandon the present total adherence to just one! Even with the best intentions, it therefore continually leads to a reduced form of understanding that eventually can serve as the enemy of true transformation.