In this entry, I will interpret the nature of these levels more specifically from a holistic mathematical perspective.
In general terms we can identify a repeating pattern with respect to development.
Initially we have the unfolding of more superficially based sensory phenomena; then with appropriate consolidation - depending on the stage of development reached - the unfolding of more general conceptual structures occurs.
Quite literally, from a scientific perspective, the earlier sense perceptions constitute the quantitative data of experience with the later concepts providing the general dimensional framework (of space and time) in which organisation of perceptions can take place.
Now there is a remarkable correlation here with the nature of number which can be used in a (base) quantitative and (dimensional) qualitative sense.
So remarkably, when appropriately understood in a dynamic interactive sense - which is in the very nature of experience - perceptions and concepts are relatively quantitative and qualitative with respect to each other.
Put another way, without the qualitative dimensional framework provided through conceptual understanding we would remain completely unable to organise the quantitative data (provided through specific perceptions).
Indeed this explains why in early infant life, phenomena enjoy but a fleeting transitory existence, as quite simply conceptual development is still too weak to enable any permanent organisation with respect to such phenomena!
So again, what happens - quite literally - at this time is that the infant directly confuses (specific) phenomena with the (holistic) dimensional nature of experience so that no perspective therefore exists for the phenomena. So the objects and dimensions (of space and time) collapse quickly in terms of each other with no permanency possible.
Now when we look at number appropriately in an experiential fashion, again a continual dynamic interaction exists as between its (base) quantitative and (dimensional) qualitative aspects. So just like the stages of development, we have a continual transformation with respect to the nature of number (when base and dimensional aspects are related in this manner).
For example 2, is the simplest example of a prime number.
Now, in the expression 22, in dynamic interactive terms, 2 represents the (base) quantitative, whereas the exponent i.e. 2, represents the (dimensional) qualitative value.
In conventional terms we know that the (reduced) quantitative result represents a (composite) natural number. So the very nature of the number type has been transformed through this interaction of quantitative and qualitative aspects. And this holistic dimension of understanding is totally missing from conventional mathematical understanding, which attempts to deal with relationships in a solely reduced quantitative manner!
So in early experience, the young infant gradually moves from largely primitive behaviour (based on blind instinct), where no permanency exists to a more natural type where - literally - objects start to naturally exist in experience.
So there is a deep holistic connection here as between primitive i.e. prime and natural experience in psychological terms. This conforms exactly to the true holistic interpretation of the quantitative and qualitative interaction (as the numbers representing base and dimensional values respectively).
When natural object permanency is consolidated in experience, another remarkable psychological transition occurs. This means that objects can be temporarily negated in experience without their existence being lost. In other words the child can hold in memory objects not immediately present in experience.
Now this ability to temporarily negate objects brings a new more integral aspect to experience (just like the negative and positive natural numbers constitute the integers).
This leads to a remarkable new transformation with respect to experience in an increasing analytic ability (whereby whole objects can now be broken into their constituent parts while still capable of being related back to their wholes).
Now again this can be seen to have its explanation in holistic mathematical terms.
For example if we again take the simple expression,
4 – 1, we know that the quantitative result = ¼.
So the holistic mathematical expression of analytic ability (which defines conventional scientific ability) is directly related to the ability to experience perceptions (in negative dimensions).
In other words the very ability to relate parts to wholes in experience requires the ability to temporarily negate the whole dimensional notion in experience.
So implicitly, analytic ability implies the increasing interaction between quantitative (conscious) and qualitative (unconscious) notions, though explicitly this is interpreted in a merely conscious rational manner.
What is remarkable about the middle stages is that quantitative and qualitative are directly reduced to each other.
Thus the rational paradigm (in qualitative analytic terms) concurs directly with the notion of rational quantities (in fractional terms).
Therefore Conventional Science is directly based on the reduction of qualitative to quantitative notions.
So with a developed scientific ability, one achieves a specialised analytic ability with respect to both (quantitative) perceptions and (qualitative) concepts (with however, from a scientific perspective, the concepts interpreted in a reduced quantitative manner).
Now this should lead to a new transformation in experience, which again can be simply demonstrated in holistic mathematical terms.
For example if we once more take the simple expression
21/2 we know that this results in the square root of 2 which is an irrational number.
This implies that in the dynamics of development that increasing interaction as between the analytic use of perceptions and concepts respectively, should lead to a transformation to an irrational status (in holistic mathematical terms).
This means in effect that experience should becomes paradoxical (i.e. irrational) in terms of conventional dualistic understanding.
This even occurs to a degree in normal everyday experience without however its significance being realised.
For example in linear dualistic terms, a left or right turn at a crossroads has an unambiguous meaning (which implies just one polar reference frame).
So if we approach the crossroads from just one direction e.g. travelling North, then interpretation is unambiguous.
However if consider two opposing reference frames simultaneously, (i.e. North and South) the left and right at the crossroads have a merely paradoxical relative identity. So what is designated left from one direction is designated right from the other direction, and vice versa.
So irrational (paradoxical) understanding therefore arises from considering more than one polar reference frame simultaneously.
Now in the relationship of wholes to parts and parts to wholes, two interacting polar reference frames are indeed simultaneously involved. So this should rightly lead to the growth of irrational (paradoxical) understanding which can correctly be explained as an increased appreciation of the notion of interdependence.
However in our culture, human development - especially in scientific and mathematical terms - rarely significantly proceeds beyond the middle (rational) levels.
This highlight the greatly reduced nature of such understanding - rooted in our notion of number - which is given a mere quantitative interpretation.
However properly understood, we always have two notions of number (cardinal and ordinal) which are quantitative and qualitative with respect to each other.
So the very process of enabling the freeing up development, so that it can move significantly past the 2nd band (which acts as a major plateau in conventional terms) in many ways depends on a correct interpretation of our number system (in a dynamic interactive manner).