Wednesday, April 20, 2011

The Trivial Zeros (2)

I have mentioned before how at University a fresh source of disillusionment with respect to conventional mathematical procedures arose with respect to treatment of the infinite notion which once again is given a reduced meaning (robbing it of its true qualitative significance).

I then developed a growing interest in philosophy culminating with a strong interest especially in the Hegelian system.

When I then realised how key notions with respect to this system could be suitably expressed in a qualitative mathematical fashion, this led to the development of that missing aspect I was earlier searching for, in what I term Holistic Mathematics.

Hegelian logic - in contrast to standard Aristotelian logic - is circular and paradoxical in nature. It is commonly expressed in terms of the view that every thesis has a corresponding antithesis. Then in evolution the dynamic interaction of these opposites in experience (and in a wider context historical evolution) leads to a new synthesis, that in turn becomes the thesis for the next stage.

This circular paradoxical logic can also be expressed as the complementarity of opposite poles (such as objective and subjective). In contrast to standard linear logic which is based on clear either/or distinctions as between such opposites, this new circular logic is based on simultaneous both/and inclusion of both poles.

Both types of logic are necessarily involved in all experience enabling us to both analytically differentiate (in quantitative terms) while also holistically integrating (in a qualitative manner).

I have illustrated this repeatedly with a simple example. According to linear logic a turn on a straight road is either left or right. However according to circular logic, it is both left and right.

Now one can either travel up or down this straight road. So if we separate possible reference frames to identify movement as either up or down the road, then we can unambiguously identify a turn as left or right. However if we try to simultaneously combine reference frames, then a turn will be both left and right.

So linear reason - which is the direct product of conscious understanding - is suited to this sequential recognition (according to independent polar frames of reference). However circular reason - as the indirect expression of unconscious intuition - is suited by contrast to simultaneous recognition of interdependent polar frames.

Now remember that the square root of 1 has in quantitative terms two roots + 1 and – 1. The important inverse corollary of this is that the square of 1 (where we express the number with respect to the 2nd dimension) combines opposite polarities of form (in a qualitative both/and manner).

Therefore associated with the number 2 in qualitative dimensional terms is a uniquely distinctive (both/and) circular logic of understanding that is the indirect rational expression of intuitive type recognition.

By extension associated with every number (as dimension) is a uniquely distinctive logical system of interpretation.

So remarkably, in formal terms, Conventional Mathematics is conducted entirely through a default 1-dimensional logical mode. Though this does indeed constitute an enormously important special case, it is in the very nature of this system that the qualitative is reduced to mere quantitative interpretation.

However in every other potential logical system, mathematical reality is interpreted according a distinctive configuration of both quantitative and qualitative type meaning. And all of these systems have an unrecognised potential relevance with respect to mathematical interpretation!

So to properly understand the square root of 1 (in both quantitative and qualitative terms), we must combine both 1-dimensional and 2-dimensional interpretation.

With 1-dimensional interpretation opposite polarities are clearly separated. So according to this logic, the root is either + 1 or – 1 (in reduced quantitative terms).

However according to 2-dimensional interpretation, these opposite polarities are combined as complementary i.e. both + 1 and – 1 simultaneously. This actually points directly to holistic intuitive recognition (which is empty and thereby formless). Thus when we attempt to express this in a (reduced) 1-dimensional manner, the two poles that are inherently combined (in 2-dimensional terms) are clearly separated in an either/or manner.

Thus strictly speaking corresponding to every root (in quantitative terms) is a corresponding unique form of dimensional logic (from a qualitative inverse perspective). Now the structural form of both will be similar. However whereas the roots will be interpreted in reduced quantitative manner according to either/or logic as separate, in the corresponding qualitative appreciation they will be combined in an intuitive both/and manner as simultaneous.

So to sum up:

Every number - and by extension every mathematical symbol - can be given both a quantitative and qualitative interpretation.
Remarkably the qualitative interpretation of number corresponds to a unique dimension of logical interpretation (through which all mathematical symbols can in turn be expressed).
Therefore we have an inexhaustible number of possible logical interpretations of mathematical symbols and relationships (with the linear logic of Conventional Mathematics corresponding to the default dimensional interpretation of 1).
Needless to say however there is a truly enormous amount of meaning waiting to be uncovered through appreciation of the other dimensional interpretations (of which 2 is the most accessible).

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