Wednesday, January 28, 2015

Intricacies of Addition and Multiplication (3)

There are obvious parallels as between the dynamic approach to number and quantum mechanics.

Indeed, I am confident that it will be ultimately understood that this dynamic approach to number serves as the starting basis for quantum mechanical understanding and in that sense is more fundamental. So properly understood, quantum mechanics is rooted in the true dynamic nature of number.

As is well-known all sub-atomic particles manifest themselves in a complementary fashion as both particles and waves.
Likewise, as we have seen, all numbers manifest themselves in a complementary manner through both quantitative (analytic) and qualitative (holistic) aspects.

It might be initially helpful to identify the particle with the quantitative aspect, and the wave with the qualitative aspect respectively.

However just as it is recognised in quantum mechanics that the particle has also a wave, and the wave a particle aspect, likewise we have seen, that when reference frames are switched with respect to number, that the quantitative aspect has a qualitative and the qualitative also a quantitative aspect respectively.

Indeed the parallels go further.

Again as is well-known the uncertainty principle apples with respect to the behaviour of sub-atomic particles. So, for instance, one cannot hope to precisely predict the position and momentum of a particle simultaneously. Rather a trade-off is involved whereby greater precision with respect to one aspect entails increasing imprecision with respect to the other.

This is equally true of number behaviour, whereby an uncertainty principle equally applies to the dynamic behaviour of number. So here for example a trade-off is also involved with respect to simultaneous knowledge of both quantitative (analytic) and qualitative (holistic) aspects.

So greater precision with respect to the quantitative aspect thereby implies greater imprecision with respect to its corresponding qualitative aspect.

This is greatly exemplified by the very nature of Conventional Mathematics. So increasing focus on the merely quantitative aspect of number behaviour has become so extreme that the qualitative aspect is not even recognised in formal interpretation.

So the misleading view that numbers have an absolute quantitative identity (without reference to their qualitative nature) has long become entrenched in accepted understanding.

In truth however numbers enjoy a merely relative identity (based on the dynamic interaction of twin complementary aspects of behaviour).

However the very appreciation of this point will require - as I continually repeat - a radical new paradigm of what Mathematics really represents. Again this "conversion" will I believe signal the greatest revolution yet in our intellectual history!

Before returning directly back to the nature of addition and multiplication, I wish to address a key feature of conventional mathematical understanding that is not properly appreciated.

This relates to its 1-dimensional nature (based on the qualitative holistic meaning 1).

So though dimensions (i.e. powers or exponents) other than 1, are of course recognised in a quantitative (analytic) manner, these are all interpreted within the standard 1-dimensional context (in qualitative terms).

As this is so important, I will comment further here on what it entails.

The 1st dimension is unique is that that it is the only dimension where qualitative and quantitative meaning are reduced in an absolute manner.

So, 1 in the Type 1 aspect of the number system, is represented as 11.

However, 1 in the Type 2 aspect of the number system, is likewise represented as 11 .

So with respect to 1, no distinction can be made as between its quantitative (Type 1) and qualitative (Type 2) interpretations. This is why a linear (i.e. 1-dimensional) rational approach to interpretation entails the reduction of qualitative to quantitative meaning in an absolute manner.

However what is not at all clearly recognised is that every number (other than 1) can equally serve as a valid means of interpretation of mathematical symbols.

The key distinction then in all these other approaches is that a dynamic relative means of appreciation ensues (entailing the interaction of both analytic and holistic aspects).

So the rigid preoccupation with 1-dimensional interpretation in Mathematics has completely blinded us to the existence of unlimited further terrains of potential meaning (where numbers ≠ 1,serve as the holistic means of interpretation).

We can easily illustrate once more the 1-dimensional approach with respect to number interpretation.

For example if we multiply two numbers, say 2 * 3, the resulting answer is given as 6.

However, if represent this expression in geometrical terms, we can quickly appreciate that a qualitative as well as quantitative conversion takes place. So in quantitative terms the result is indeed 6. However this now relates to 2-dimensional (square) rather than 1-dimensional (linear) units.

However in conventional mathematical terms, this qualitative transformation in the nature of the units is simply ignored with the result expressed in a linear (1-dimensional) manner.

Thus from this perspective 2 * 3 = 6 (i.e. 61) . So if you want to appreciate the key problem with respect to reconciling multiplication with addition, it is right here!

Therefore whereas addition (in linear terms) leads solely to a quantitative transformation in the nature of the units, multiplication - by contrast - likewise entails a qualitative transformation.

Indeed from the conventional mathematical perspective, multiplication serves as but a short-hand way of representing addition.

So 2 * 3 (from this perspective) = 2 + 2 + 2. Therefore 2 * 3 expresses the fact that we are adding 2 three times!

However this misses the point completely - as is inevitable from the conventional mathematical perspective - that multiplication essentially, relative to addition, entails a qualitative, rather than quantitative transformation.

The deeper philosophical basis of the 1-dimensional approach is the use of single polar frames of reference in the interpretation of mathematical relationships.

I frequently illustrate this with respect to the interpretation of left and right turns at a crossroads.

I one is heading N, a left turn can be unambiguously identified at the crossroads. So here a single polar frame of reference (i.e. the N direction) is used.

Now having passed through the crossroads one switched direction and then travels S, once again a left turn can be unambiguously identified (using this single pole of reference).

So both turns at the crossroads are now unambiguously identified as left (using separate poles of reference). However we know that the turns must be left and right with respect to each other!

This recognition that identification of turns is merely relative, requires the ability to appreciate N and S as complementary opposite poles (which requires simultaneously linking N and S directions).

The key insight enabling such simultaneous recognition is of a holistic intuitive nature and leads to (circular) paradoxical understanding in terms of unambiguous rational appreciation of an analytic nature.

So absolute unambiguous understanding is based on the use just one polar direction as reference frame. And this is the fundamental nature of linear (1-dimensional) interpretation.

However relative paradoxical understanding always entails simultaneous use of more than one polar reference frame (which in the simplest case entails two).  And this likewise is the fundamental nature of circular (higher dimensional) interpretation.

So rather just one polar reference frame for number (i.e. analytic) we have now extended appreciation to include two such directions i.e. analytic and holistic, in dynamic relationship with each other.

However just as with the interpretation of a crossroads we keep alternating as between absolute understanding (where each turn is unambiguously defined) and relative understanding (where both turns have a merely arbitrary relative interpretation depending on context) likewise we must do the same with dynamic mathematical understanding as we continually alternate as between analytic (linear) and holistic (circular) type appreciation.

Having dealt with these important background issues let us return to the central issue of addition and multiplication.

The remarkable fact that now presents itself, is that once we define both Type 1 and Type 2 aspects of the number system, we can simply distinguish the nature of addition from multiplication.

Thus when we interpret the Type 1 aspect from the standard quantitative perspective, the natural numbers can be derived (as base values) through the continual addition of the starting unit.

So  starting with 11, 21  = 1+ 11, 3= 1+ 1+ 11, and so on.

Thus addition is here directly associated with the quantitative transformation of number.

However equally when we interpret the Type 2 aspect from the (unrecognised) qualitative perspective, the natural numbers can be derived (as dimensional values) through the continual multiplication of the starting unit.

So again starting with 11,  1 = 1* 11, 1= 1* 1* 11, and so on.

Thus multiplication is here directly associated with the qualitative transformation of number.

Therefore when correctly understood, addition and multiplication represent two fundamental operations that are quantitative (analytic) and qualitative (holistic) with respect to each other.

However this key distinction cannot be appreciated from a conventional mathematical perspective (where the quantitative aspect is solely recognised).

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