Wednesday, October 14, 2015

Clarifying Analytic and Holistic Interpretations of "2" (3)

As we have seen, properly understood, number should be defined in a dynamic interactive manner entailing both base and dimensional aspects, which are - relatively - quantitative and qualitative with respect to each other.

Therefore the number "2" is dynamically defined as representing the interaction of its two aspects i.e.  21 and 12. Therefore from the perspective, where the base number "2" is considered in an quantitative  (independent) manner, then the corresponding dimensional number "2" must be then - relatively - considered as of a qualitative (interdependent) nature.

So we have the interaction here therefore of the quantitative notion of "2" with the corresponding qualitative notion of "2" (i.e. as twoness).

One must remember that all phenomenal experience - including of course mathematical - is universally governed by fundamental opposite polarities. Chief among these are external and internal poles, so that no number can be be objectively experienced (as external) in the absence of a corresponding mental perception (that is - relatively - internal). So ultimately what is objectively known cannot be divorced from corresponding subjective mental interpretation!
Thus any absolute notion of truth within Mathematics is untenable.

Likewise we have whole and part polarities, which directly entail complementary quantitative and qualitative aspects with respect to phenomena.

So once again we cannot experience a number in a quantitative manner without a corresponding qualitative counterpart recognition.

So once again number recognition is strictly relative with respect to recognition of both its quantitative and qualitative aspects.

As I have repeatedly demonstrated, the crossroads analogy can be very helpful here!

If standing at a crossroads one denotes a direction as left, then in relative fashion, the opposite direction is right. If however approaching the crossroads from the opposite direction, one now denotes this first turn as right, then the opposite is thereby left.

So depending on the context from which direction (N or S) the crossroads is approached, each turn can be both left and right (and right and left) respectively.

So it is quite similar in mathematical terms, which is governed likewise by twin opposite polarities that keep switching in direction.

Therefore in this context, if we denote the base  as quantitative, then the dimensional number is - relatively - qualitative in nature.

However, when we now switch the frame of reference - which repeatedly occurs in experience - the base number is qualitative, with the corresponding dimension assuming a quantitative meaning.

Now because of the reduced nature of standard mathematical interpretation, these critical distinctions are completely missed with number - whether representing the base or dimension - interpreted in a merely quantitative manner.

This in fact is tantamount to one who insists that both turns at a crossroads represent  the same directiont. And this arises from the solely linear (1-dimensional) interpretation that is employed, which necessarily operates within single isolated polar reference frames.

Thus operating in this manner with the single direction of movement as heading North towards the crossroads, one can indeed unambiguously identify, for example, a left turn at the crossroads.

Then when we again fix a single direction of movement, this time heading South towards the crossroads, one can unambiguously identify a left turn.

However when we now try to combine both of these isolated frames of reference, both turns at the crossroads are identified as left!

Thus to understand that the turns must be relatively left and right with respect to each other, one must implicitly move from a 1-dimensional (where N and S are considered separately) to a 2-dimensional manner of interpretation, where they are simultaneously considered.

Intuitively we can do this automatically in relation to understanding the turns at a crossroads.

However, when it comes to mathematical understanding, we are rigidly locked within a merely linear (1-dimensional) framework because of the massive reductionism involved. So the internal pole of recognition is not distinguished from the external; equally the qualitative pole of recognition is not distinguished from the quantitative.

So effectively this is like attempting to understand the directions at a crossroads where only the N direction with respect to (vertical) movement is recognised and solely the left direction with respect to (horizontal) movement!

Therefore with respect to our understanding of "2", we can switch frames of reference so that the base number is now qualitative, and the dimension - relatively - of a quantitative nature.

In fact when one reflects on it carefully this qualitative aspect of recognition is necessarily involved in all multiplication processes.

Let us take for example the simple example of 2 * 3.

Now imagine we have 2 rows with 3 coins in each row, this could then be represented as 2 * 3.

In the standard reduced manner of conventional mathematical interpretation, such multiplication could be represented merely in terms of addition where 2 * 3 = 3 + 3.

So from this perspective, the multiplication operation results in 6, interpreted in a merely quantitative (1-dimensional)  manner as representing independent units.

However, clearly there is something missing from this reduced interpretation for as we know in geometrical terms when we multiply 2 * 3, the units change from a linear (1-dimensional) to a square (2-dimensional) format. So a qualitative change in the nature of units occurs (as well as the recognised quantitative transformation).

And even to represent the two rows of coins (with 3 coins in each row) requires adopting a 2-dimensional format!

The crucial "missing" element is that the very ability to match off the the two rows, requires that we recognise their common identity. And this common identity relates to the qualitative notion of number interdependence (i.e. the common recognition of "threeness").

Therefore we initially must indeed recognise the independent identity of each of the three coins with respect to each row. This relates to the quantitative nature of the multiplication operation.

However the very ability to then match the two rows requires the common recognition of similarity with respect to each row (as now interdependent with each other). This then relates to the qualitative nature of the multiplication process.

Now in the pure case of simple multiplication, where we multiply individual units (recognised as independent), a mere qualitative transformation is involved in dimensional terms.  

However in the more general case of compound multiplication where we multiply numbers (≠ 1), both a quantitative and qualitative transformation is involved.

Thus to conclude this blog entry!

Both the base and dimensional aspects of number can be given a quantitative (as independent)  and qualitative interpretation (as interdependent) respectively.

However just like the turns at a crossroads, they always bear a complementary relationship with each other.

Therefore if the base is defined in a quantitative manner, then the dimensional number is - relatively - qualitative - in nature.

If however the base number is defined in a qualitative manner, then the dimension is now - relatively - quantitative - in nature.

Thus with respect to the number "2", it keeps switching in experience as between both its quantitative and qualitative aspects - with respect to the independent notion of "2" and interdependent notion of  "twoness" - in both base and dimensional terms.

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