Thursday, July 22, 2010

More on Nature of Holistic Mathematics

Just as the standard analytic approach to science is ably served by its corresponding mathematical tool i.e. Conventional Mathematics, likewise the (qualitative) integral approach to science is likewise potentially served by its own respective mathematical tool i.e. Holistic Mathematics.

A great barrier however that I have continually faced is the recognition that Holistic Mathematics - though using the same symbols - is radically different from what most people understand as Mathematics. So what certainly is not intended here is the standard use of Mathematics to deal with holistic type problems (which represents the old reductionist approach)! Rather it requires a new interpretation of mathematical symbols that inherently depends on corresponding appropriate level of intuitive insight for correct usage.


To put this in context, one must appreciate that just as there are many bands on the electromagnetic spectrum (with natural light comprising just one), likewise it is true with respect to the potential for human psychological development. Such development comprises many bands where the nature of the intuitive light - that necessarily informs all understanding - varies greatly.

So what informs rational interpretation - dictating conventional approaches to mathematics and science - is a natural light (in the normal intuition that underpins the use of such reason). However potentially there are many other bands on the spectrum with very different forms of intuitive energy so that here "normal" intuition is no longer appropriate.

For example higher bands of intuition unfold through the process of contemplative spiritual development and in the various mystical traditions their associated states have been documented to a considerable level of detail.

However what has been largely missing is the attempt to develop in systematic manner the implication of such states for understanding of science. For quite simply, a radically distinct approach is needed so as to maintain true compatibility with authentic contemplative understanding.


In my initial quest to deal with this issue I realised that a new type of holistic or qualitative science was needed so as to be consistent with the various stages of contemplative development.

I then further recognised that just as the most developed type of intellectual development would combine both reason and contemplative insight in a refined manner that likewise the most comprehensive kind of science - which I term radial - would likewise combine both analytic (quantitative) and holistic (qualitative) aspects in a balanced interactive fashion.


However I have since come to realise with respect to the qualitative holistic aspect that an infinite variety of interpretations potentially exist.
So what this all mean?


Using holistic mathematical language, I always term the basic approach of science (what one might refer to as its metaparadigm) as linear i.e. 1-dimensional.


Now customarily we think of numbers 1, 2, 3 as quantities. But as we know in Mathematics these same numbers can also be used to represent dimensions (as powers or exponents).

However, in re-opening a major misgiving felt even as a child, I realised that that when numbers are used to represent dimensions in Conventional Mathematics they are given but a reduced i.e. quantitative meaning.

For example when we raise the number 2 to the power of 2
i.e. 2 ^ 2, a qualitative as well as quantitative change takes place. So a table with each (straight) side = 2 metres would have an area of 4 square metres. Thus the 2-dimensional area here (in square units) is qualitatively distinct from the 1-dimensional measurement of each side (in linear terms).

However remarkably in Conventional Mathematics, the result of
2 ^ 2 is provided in a merely reduced quantitative manner as 4 i.e.
4 ^ 1. So this illustrates clearly the truly linear nature of Mathematics where the result of numerical calculations is ultimately expressed in a 1-dimensional manner (so that implicitly this result is raised to the power of 1!)


So - quite literally - Conventional Mathematics is confined purely to a 1-dimensional manner of interpretation (with the qualitative dimension thereby reduced in a merely quantitative fashion).

However associated with every number is a unique qualitative interpretation.
Therefore the default standard interpretation - in what we misleadingly call Mathematics - represents just one of a range of qualitative interpretations that is potentially infinite!


You might well ask what possible bearing this can have on scientific investigation!

In fact when appropriately perceived it has very important consequences.


We can illustrate the standard linear manner of interpretation in science very directly.

When one for example observes an object it is - literally - posited in experience as distinct. Now in holistic terms such positing is denoted by the mathematical symbol + (this time with a holistic meaning). Likewise implicit in the recognition of any phenomenon as distinct is its unitary nature (i.e. where it is literally identified as a unit). And here we have the holistic meaning of the symbol 1.


So standard scientific investigation is thereby one-directional which can be defined here in holistic terms as + 1.

Now if we take the 1st root of 1 (which is the same as raising to the power of 1) the answer remains 1.
More precisely the answer is + 1. So again in 1-dimensional terms, no distinction can be made as between quantitative and qualitative interpretation.


However when we obtain the two roots of 1 something strange happens in that the answer splits quantitatively in two possible directions i.e. + 1 and - 1.

The corresponding implication in qualitative terms is that when we raise the number 1 to the power of 2 that the answer now splits again in two directions + 1 and - 1.



As one would expect - because of its 1-dimensional bias - there is no recognition of this in Conventional Mathematics (which merely reduces the result in quantitative terms). So 1 ^ 2 is treated in precisely the same manner as 1 ^ 1 (= 1).
However in Holistic Mathematics when we raise 1 to the power of 2, it corresponds in qualitative terms to the taking of two roots.

So very importantly raising 1 qualitatively to D (as dimension) structurally corresponds to raising 1 to (1/D) in quantitative terms. However the qualitative inetrpretation is based on circular (both/and) rather than linear (either/or) logic!


So to make a long story short, 2-dimensional - as opposed to standard 1-dimensional interpretation - in a scientific observational context entails giving every phenomenon two directions that are + (positive) and - (negative) with respect to each other.

This in fact refers to the inescapable fact what all experience of reality (in any possible context) inevitably entails a dynamic interaction as between the knower (as subject) and what is known (as object). So from a 2-dimensional perspective an "object" phenomenon has no strict meaning outside this two-way dynamic interactive context.

So we now can only speak relatively of every phenomenon with two directions involved. Thus if we associate + with the external (objective) aspect, then we must associate - with the corresponding internal (subjective) aspect; however if we equally in any arbitrary polar context fix + with the internal, then in relative terms we must then associate - with the external. So in dynamic interactive terms positive and negative signs keep switching in a relative manner.

This type of circular understanding based on the complementarity of opposite poles, thereby enables one to deal more subtly and accurately with the actual dynamics of scientific experience.

Considerably more refined experiences are possible (and in the appropriate scientific contexts extremely important). For example with 4-dimensional understanding (which is especially significant) every object phenomenon would be given a 4-directional interpretation with both two real and two imaginary aspects that are each positive and negative with each other. This again would correspond, from a structural perspective, with the 4 roots of 1 (in quantitative terms).

What this means in effect is that as well as recognising the dynamic nature of the external/internal dialogue in experience, we also recognise the equally important interaction of whole and part (which inevitably defines all possible experience).

This leads on to the key insight that the imaginary symbol i corresponds directly with the unconscious recognition of an object (in holistic terms). Thus switching from whole to part (and part to whole) in experience always entails the interaction of conscious and unconscious. And in holistic mathematical terms whereas conscious understanding is "real" unconscious - by contrast - is "imaginary".

So the qualitative implication of the complex number system is the recognition that both rational (conscious) and intuitive (unconscious) aspects must be formally recognised and that correctly understood in 4-dimensional terms in any appropriate context the whole is always - relatively - "imaginary" with respect to the parts that are "real". So when we fail to recognise this fact (as in standard 1-dimensional interpretation which is solely "real") the whole is inevitably reduced to the parts.


So again we can characterise the limitation of the present accepted approach as one that formally recognises solely the "real" aspect of understanding.

Thus what I am attempting to address here is the need for recognition of the equally important "imaginary" aspect of mathematical and scientific understanding (i.e. Holistic Mathematics and Integral Science respectively).

Then the radial approach to both - combining them as both equal in importance - can thereby constitute in qualitative terms the complex rational approach. And when both aspects are equal they can be represented as null lines, likewise such an approach also represents the most simple (based on pure spiritual intuition).


We have not time here to deal with further dimensional interpretations (which are ever more demanding in terms of appropriate intuitive recognition).

However we can hopefully generalise what is involved in a meaningful manner.

Just as all the possible roots of unity can be represented by sets of complex numbers on the circle (in the complex plane), likewise all possible dimensions of understanding can be understood in circular logical terms as representing various configurations of both "real" (conscious rational) and "imaginary" (unconscious intuitive) understanding. (Once again what is "imaginary" in this context strictly relates to an appropiate indirect rational means of expressing holistic intuition!)

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