Thursday, April 27, 2017

Number and Development (1)

Even as a small child I was fascinated with the nature of number and the fundamental mathematical operations of addition and multiplication.

This led me on at the age of about 5 to the discovery of my own system of logs (based on the power of 2) whereby multiplication of numbers could be conveniently "converted" into a simple operation involving addition.

Then about 5 years later, I made a much more important and life-changing discovery, when I clearly came to realise that every act of multiplication inevitably involves a qualitative as well as quantitative transformation. However the qualitative aspect is then completely edited out of conventional mathematical interpretation.

This initially arose from consideration of the the area of rectangular fields. So if a field was of length 80 metres and width 60 metres, the area, 80 * 60 = 4800 sq metres. So though the initial measurement of length and width are both 1-dimensional, the resulting area now represents 2-dimensional units. So both a quantitative transformation (i.e. 4800) and qualitative transformation (2-dimensional units) takes place.

However conventionally, when in arithmetic terms we multiply 80 * 60, the result of 4800 is expressed in an unchanged 1-dimensional quantitative manner (which thereby can be represented as a point on the real number line). Therefore I realised  - though I lacked the means then to resolve the issue - that the conventional interpretation of multiplication was simply not fit for purpose. 

So this represented an early "peak" experience into a far more advanced form of mathematical understanding, where both analytic (quantitative) and holistic (qualitative) aspects would be necessarily involved.

Furthermore this moment which crystalised as a very sober realisation was not so much the peaking of a "higher" state - though it certainly contained an important illuminative element  - but rather of a "higher" structure of understanding. This experience then in many ways  served as the crucial driving force over the next 50 years or so towards a greatly enhanced form of mathematical understanding, which I have only recently been able to finally articulate (to my own satisfaction).

Expressed simply, the key issue relates to the notions of number independence and number interdependence respectively, which are always reduced in terms of each other in conventional mathematical terms.

For example, if we take a cardinal number such as 3, it is represented in quantitative terms as the sum of independent component sub-units i.e. 1 + 1 + 1. So the units here of a homogeneous nature, which - literally - lack any qualitative distinction.

Then we add for example two numbers - say 2 + 3 - the result i.e. 5, then seemingly is likewise composed of merely independent units.

However things become subtly different where multiplication is involved.

So let's now consider the multiplication of 2 * 3 .

To make it concrete let us imagine 2 rows with 3 coins in each row!

Now the key to using the operator (i.e. 2) in this case is the recognition that the 3 coins in each row are similar (as duplicates of each other) so that we can make a one to one identification of each coin in the 1st row with each corresponding coin in the 2nd row.

However in the recognition of this shared similarity, we have now moved from the quantitative notion of number independence to the qualitative notion of number interdependence.

However in conventional mathematical terms, this crucial distinction is completely overlooked.

So putting it bluntly, the standard accepted interpretation of multiplication suffers from a fundamental confusion in the failure to properly distinguish the analytic notion of number independence (which is quantitative)  from the corresponding holistic notion of number interdependence (which is of a distinctive qualitative nature).

Now it is recognised that the Riemann Hypothesis is deeply connected with the relationship as between addition and multiplication. However this issue clearly cannot be properly understood within a mathematical paradigm that insists on mere quantitative type interpretation of its numerical symbols!

So on the long road to unravelling the true nature of addition and multiplication, I realised that the first major requirement was the development of the hidden aspect of Holistic Mathematics (where every symbol can be given a coherent qualitative - as opposed to quantitative - interpretation).

And the unfolding of this new holistic mathematical understanding formed a very important - though not exclusive - component of my cognitive development through the various levels of Band 3.

So associated with each of the "higher" levels of Band 3 i.e. Level 1, Level 2 and Level 3 respectively are both refined states (representing ever purer forms of spiritual intuition) and refined structures (representing increasingly dynamic forms that indirectly are defined in a circular paradoxical manner).

And these states and structures relate to cognitive, affective and volitional understanding respectively.

So we concentrating initially here on the cognitive mode with respect to the unfolding of new logico-mathematical understanding of a holistic nature.

In particular I formed then the holistic understanding of the important circular number system that - in quantitative terms - is defined by the various roots of 1 (in the complex plane).

So for example the 2 roots of 1 are + 1 and – 1 respectively, which in the standard analytic manner are separated in a static either/or fashion.

However these two numbers can equally be given a holistic meaning where + 1 and – 1 are now understood as interdependent in a dynamic interactive manner.

So from a holistic perspective, + 1  implies the dimension (i.e. direction) of experience whereby  phenomenal meaning is posited in a directly conscious manner. – 1 then implies the corresponding dimension whereby such meaning is then negated in a directly unconscious fashion.

Now there are two poles with respect to all phenomenal reality that are - relatively - external and internal with respect to each other. In holistic mathematical terms these poles are designated as + 1 and – 1 respectively.

So when the external direction is posited e.g. as a conscious object, this is designated (within its own independent frame of reference) as + 1.
Likewise when the internal direction is posited e.g. as a mental construct, this is likewise designated (within its own independent frame of reference) as + 1.

However crucially the dynamic switching as between both poles requires the unconscious direction which - literally - serves to negate the phenomenon already posited, resulting in a psycho spiritual fusion of intuitive energy, which then activates recognition of the opposite (unrecognised) pole..

Therefore though phenomena can indeed be consciously posited within their own independent frames of references, when these frames are related as interdependent, like matter and anti-matter particles in physics, a negative fusion with the positive, resulting in intuitive appreciation necessarily takes.

So therefore relative to each other, external and internal poles are positive and negative, which continually keep switching in the dynamics of experience.

I realised clearly at this point that conventional mathematical interpretation is defined in a holistic mathematical manner as 1-dimensional i.e. where formal  interpretation is merely of a conscious nature (taking place within isolated independent frames of reference).

However I had now defined for myself a new 2-dimensional framework whereby both conscious (posited) and unconscious (negated) directions of experience, for all mathematical symbols, are recognised in a dynamic interactive manner.

Therefore it is somewhat meaningless to attempt to define mathematical truth externally in an absolute objective manner as such truth necessarily entails subjective mental interpretation which is relatively of an internal nature.
So properly understood, all mathematical truth (from this 2-dimensional perspective) is of a dynamic relative nature, entailing external and internal poles (as complementary opposites).

So what is formally accepted as mathematical truth (i.e. 1-dimensional interpretation) - where interpretation is viewed in absolute corresponence with objective reality - represents but one special limiting case that is of a highly reduced nature. 

And of course the 2-dimensional interpretation in principle can likewise apply to all intellectual discourse.  So Hegel in many ways applied it to philosophy while Jung showed its relevance for psychology. Likewise it forms a  a staple part of expression in many mystical traditions such as Taoism, while more recently it has been seen to be deeply relevant in physics with respect to quantum mechanical behaviour.

So I had could see how a number such as 2, can be given two distinctive interpretations, which I term Type 1 and Type 2 respectively.

In the first case (Type 1) - which represents the standard analytical mathematical interpretation - 2 is more properly defined as 21. Here the number is defined in the default 1-dimensional quantitative manner.
So 21 = 11 +11 where the units are defined as homogeneous and independent.

In the second case (Type 2) - which represent the new holistic mathematical interpretation - 2 is more properly defined as 12, where the default base number 1 is raised to the power of 2 (i.e. in a 2-dimensional manner).

Here the two units of 2 are considered as interdependent with each other in a qualitative manner. Now such interdependence can only be directly appreciated in an intuitive fashion. However indirectly, one can attempt to translate such interdependence, in the standard 1-dimensional manner by taking the square root, resulting in the two answers + 1 and – 1.
Though these two results are necessarily separated in 1-dimensional terms (either + 1 or – 1) , from the 2-dimensional perspective they are considered as directly complementary (i.e. + 1 and – 1 simultaneously).        

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