Wednesday, October 5, 2011

Mathematical Dimensions and Psychological Development (1)

I have already pointed to the fact that - properly understood - every number expression represents a dynamic interaction as between a a base quantity and a dimensional number (that is relatively of a qualitative nature).

So what we might refer to in conventional (Type 1) terms as the number quantity 2, more accurately is expressed as 2^1 (where 2 is quantitative and 1 - relatively - of a qualitative holistic nature).

However because the very nature of linear (1-dimensional) understanding is to reduce qualitative to quantitative type meaning, from a Type 1 perspective, interpretation of numbers is invariably reduced in a mere quantitative manner.

However when correctly appreciated in holistic Type 2 terms, number expressions have an intimate bearing on the interpretation of all the main stages on the spectrum of psychological development.

As we know number is the best means we have for quantitative ordering in experience. In fact such ordering is synonymous with number appreciation. And seen from this Type 1 perspective we have various number types such as prime, natural, rational, irrational etc.

What is not recognised is that the same number types have a profound relevance for the qualitative interpretation of development.
In fact the holistic structure of each of the main levels of psychological development can be precisely matched to expressions entailing the main number types. Furthermore we can subdivide each "number" level into three coherent stages showing how the transformation into the next level on the spectrum takes place.

So, three main stages are involved with respect to unfolding of each level.

Firstly we have the unfolding of specific perceptions (characteristic of the level in question). In the second stage this gives way to unfolding of the more generalised conceptual understanding (again characterising the level). Now the relationship of such perceptions and concepts is as quantitative and qualitative with respect to each other!
Then in the third stage increasing dynamic interaction takes place as between perceptions and concepts causing a transformation to a new level (whose structure in turn corresponds with a new distinctive number type).

Development commences from a completely undifferentiated state where form is indistinguishable from emptiness. This relates in qualitative terms to - what I refer to as - the original numbers. So unity (1) is initially indistinguishable from nothingness (0). Then when development commences the first initial differentiation takes place leading to the birth of duality (2) in experience.

Though we still have great confusion, in a qualified sense we can identify three stages (1) where duality takes place with respect to incipient perception (2) when duality now takes place with respect to the incipient formation of concepts and finally where some level of interaction of both takes place. This can be identified with the archaic level.

All going well, the dynamic interaction of understanding that is quantitative and qualitative with respect to each other, leads to a transformation to the next level.

This level is then identified in terms of the holistic interpretation of the prime numbers that entails an intimate relationship of both conscious and unconscious aspects of experience. Indeed we can accurately use the word primitive to refer to such experience where holistic collective notions (pertaining to the unconscious) are continually confused with distinct specific notions (pertaining to the conscious). We can identify this with the magical level.

Again in a qualified manner we have the unfolding of three stages firstly with respect to specific perceptions of a primitive kind, then more general concepts (of a primitive nature) and finally the growing dynamic interaction of both perceptions and concepts.

When we raise a prime number to a prime number dimension in quantitative terms, a transformation takes place in that we derive a new (composite) natural number.

Likewise in holistic psychological terms when we relate specific perceptions of a primitive quantitative nature to their general concepts (that relatively are qualitative in nature) a transformation to a new level of understanding takes place which correlates with the holistic nature of natural numbers.

One of the key characteristics of natural numbers is that numbers are whole (and not yet divided into fractional components).
Likewise at this level (which relates to mythic development) wholes - especially with respect to concepts - cannot be properly broken into parts. Put another way, abstract ability is not yet sufficiently developed so that objects are still given a (whole) personality. Only later when the conscious aspect is further differentiated from the unconscious does true abstract ability of an impersonal nature properly unfold.

One fascinating feature of this level is that one learns to - temporarily - negate both perceptions and concepts (while holding them in memory). This is made possible through the greater level of phenomenal constancy characteristic of this level.
This literally means therefore that such dynamic negation of concepts in experience relates to negative rather than positive dimensions (in number terms).

Once again we can identify three stages 1) where natural perceptions (and their corresponding temporary negation) takes place 2) natural concepts (and their temporary negation occurs and finally (3) the growing interaction of both perceptions and concepts in both a positive and negative sense.
So during this level the natural numbers give way to the integers (where numbers can be both positive and negative) in holistic terms. And the scientific structure of the level is based on such understanding.

When we look at the simple number expression 4^ (- 1) we have an integer that is raised to the negative of 1 (as dimensional power). And this number expression in Type 1 terms leads to the generation of a fraction i.e. 1/4.

Likewise in development, when whole perceptions are related to concepts (that can be temporarily negated in experience) we generate analytic understanding of a fractional kind.

This leads to an important transformation in development whereby the mythic gives rise to the rational stages (that are so relevant in conventional scientific understanding).

What is remarkable here is that appropriate holistic mathematical interpretation implies that even the simplest rational task implies the ability to temporarily negate the linear (1-dimensional) mode of understanding.

Let us illustrate with respect to the task of cutting a cake into four slices.

Now, the cake obviously represents a whole and each slice another unique whole object. So the ability to recognise each slice also as a part requires the temporary negation of its whole status. This then enables the link of the slice to the whole status of the cake whereby it is now recognised as a part of that whole. So, initially each slice is posited as a whole perception and then negated (with respect to this status) enabling it to be thereby understood as a part of the greater whole of the cake.
So here we establish the link between whole and part perceptions which implicitly requires the ability to temporarily negate their respective identities while still holding them in memory. And the same ability also implicitly develops with respect to concepts. For example one can initially recognises a specific number such as 2 as a perception. However the recognition that this number belongs to the universal number class i.e concept of number requires the temporary ability to negate its specific status in recognition of its general identity. Then the reverse procedure of once again recognising 2 as a specific perception in turn requires the temporary negation of its number concept. Therefore - though still operating at a merely implicit level the continual dynamic positing and negating of both perceptions (as quantities) and concepts (in qualitative terms) takes place. Once again the (linear) rational level - which dominates conventional understanding of Mathematics and Science - has 3 stages.

The first corresponds to the rational appreciation of perceptions in the ability to break object perceptions into smaller parts and rearranging them again in composite wholes. This is generally referred to in Piagetian terms as conop.

The second stage corresponds to the ability to break general universal concepts into sub-categories and then synthesise them with respect to the original concepts. In Piagetian terms this is formop.

Finally the third stage entails the growing interaction of rational perceptions and concepts. When suitably refined - though this is not generally the case - this leads to a dynamic intuitive based form of rational understanding i.e. vision-logic.

Another remarkable transformation occurs in Type 1 quantitative terms when a rational number is raised to a fractional dimension (as power) in that an irrational number results.

For example in the well known case when 2 is raised to 1/2 we obtain the square root of 2 = 1.4142... which is an irrational number.

However, in Western culture there is very little evidence of the sustained growth of further more advanced stages of understanding (beyond the rational level).

One important reason for this is quite revealing. The source of the dynamic transformation that takes place in the emergence of each new psychological level results from the interaction of perceptions and concepts, that are implicitly experienced as quantitative and qualitative with respect to each other. This enables therefore a fruitful interaction of conscious and unconscious to emerge.

In formal terms the interpretations of Mathematics and Science directly reduce the qualitative in quantitative terms. As this leads to emphasis solely on the conscious aspect of experience, a considerable diminution in true holistic type qualitative understanding can result. For example Holistic (Type 2) Mathematics, which I am outlining in these contributions, is totally unrecognised at present by the mathematical profession!

In experiential terms therefore this can reduce the unconscious intuitive aspect of understanding (within which true qualitative appreciation is embodied) to such a significant degree that sustained progress beyond the rational level is not possible.
In other words the specialisation of rational understanding, which is so characteristic of Western culture can greatly reduce the role of the unconscious in experience with the consequence that dynamic transformation beyond the rational level is significantly impeded.

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