I have defined this first composite stage as that time, where conscious phenomena achieve a degree of constancy and can therefore be clearly identified.
A remarkable transformation in psychological understanding is necessary to achieve this, which involves the ability to separate quantitative from qualitative characteristics (i.e. to separate partial object phenomena from holistic dimensional experience). In other words for the first time the child is able to clearly locate objects in an environment of space and time.
This crucial change represents a degree of differentiation of conscious structures, enabling more creative interaction with the unconscious. The child now is able to posit distinct object perceptions through the conscious mind and also through interaction with unconscious - at a superficial level - general classifying concepts.
In psychological experience, perceptions represent the quantitative physical data, whereas concepts represent the qualitative mental dimensions. In other words we place our perceptions in an environment of space and time, through classifying and ordering this data mentally through the use of concepts.
In a direct sense the physical data come through the affective mode of response, and the mental concepts through the cognitive mode of control.
The existence of objects and dimensions psychologically are therefore purely relative and the result of dynamic interaction.
Accepting that psychological and physical reality are complementary,
this implies that the universe itself is a dynamically interacting process
involving - at all levels - complementary patterns of response and control,
which generate matter and dimensions (of space and time).
Experience of separate phenomena is always composite in nature. Of necessity, to identify an object we must be able to relate the (quantitative) perception, with a (qualitative) concept. If I for example identify a window, this involves a dynamic interaction of the (quantitative) perception of " a window", with the (qualitative) concept of "window". Neither the perception or concept - in isolation - can provide me with experience of the window.
Thus experience of natural objects arises from this composite interactive
understanding, whereby prime perceptions and concepts are brought into
a meaningful pattern.
It is exactly complementary in relation to numbers. Composite numbers are formed from combining, quantitatively and qualitatively, prime numbers. When this is done for all possible combinations, we generate the natural number system.
This is all tied up with the mathematical process of multiplication, which involves both a (vertical) qualitative as well as (horizontal) quantitative transformation.
Again, if 2 is multiplied by 3, there is a vertical transformation, in that there is a move from one to two dimensions. (This can be simply represented in geometrical terms with 2 and 3 representing adjacent sides of a rectangle). The horizontal transformation is in the (reduced) one dimensional quantitative value, which is 6.
Though this qualitative aspect is necessarily involved, conventionally
it is ignored and mathematical operations involving numbers are interpreted
in terms of their (reduced) quantitative values.
This exactly mirrors the psychological experience of objects. Object experience always involves composite groupings of prime elements, whose integration results in the experience of the object.
The experience of a house for example, involves putting together a grouping of other objects - which themselves are composite - into a pattern which when recognised is identified as "a house". Thus component elements could be separately identified as walls, windows, doors, a chimney, a roof etc. However just combining these together does not give a house. A qualitative holistic dimension - related to the unconscious - is required, leading to creative insight which enables one to interpret the new pattern. However, though this intuitive insight is continually required to "read" the object patterns, it remains implicit, and the objects formed are reduced to mere horizontal interpretation. Thus, conventionally, one interprets that a composite object such as a house has a self contained independent existence, which is consciously observed.
In truth, it is an arbitrary reading of a dynamic interaction pattern
involving both conscious (horizontal) and unconscious (vertical) understanding.
Thus, understanding of new composite objects, involves what is really a psychological multiplication process, where the combination of different elements is associated with a qualitative transformation, rendering a new object. Psychological addition by contrast would merely involve recognising more of the same object, where the qualitative transforming insight is not required.
Quite, literally, when the child acquires this (psychological) multiplication ability, new objects quickly appear without limit in experience, creating the natural world. These are the natural structures.
We could also speak of experience of natural dimensions, which is a kind of free flowing unbounded linear experience of space and time.
Again, this complements the natural number system which of course though
linear and finite is unbounded horizontally and vertically.
QUALITATIVE NUMBERS: INTEGERS
As I have stated elsewhere, a child initially believes that objects only exist, when immediately present in experience. Thus though able to recognise objects, they still enjoy a somewhat fleeting existence. They are quickly forgotten when not in the present field of experience.
The next advance in understanding, involves an ability to separate the two directional poles, whereby the child is more easily able to distinguish the (subjective) self from the (objective) world. What this in turn entails, is the implicit recognition that these poles are opposite to each other.
Switching from the external world to the internal self in experience,
requires an ability to sustain the temporary loss of objects. With external
experience, the ego becomes identified with, and attached to the objects
thus created. To switch to conscious internal experience of the self, requires
an ability to let go. In this painful experience, the self becomes to some
extent separate and independent from the objects experienced. It is this
very consciousness of separateness which gives objects a more solid independent
existence. Of course, the internal ego also begins to acquire an independent
existence. Independence implies, as it were, a self contained existence,
where both the object and self enjoy a separate (whole) identity. Indeed
it implies that both enjoy a distinct integral existence. This enables
their continued existence, in an extended environment of space and time.
Again the same pattern is replicated in the number system. The integral number system involves all positive and negative integers with 0 at the centre of the system.
However, once more, the mathematical approach to negative numbers involves a strictly reduced and static interpretation.
In psychological dynamic terms, negation always involves the ability to switch directions in experience, thus tending to cancel out and undo mere positive (i.e. conscious) activity. Thus nothing (i.e. 0) in dynamic terms is the highly creative balanced fusion of positive and negative experience. In other words, conscious experience is fully cancelled and undone leading to a creative fusion in intuitive spiritual awareness (the "void" of Eastern thought).
Thus the very ability of a child to comprehend negative numbers, and
to carry out operations using them, requires a switching in experience
from an external (objective) to an internal (subjective) direction. In
other words, the act of subtracting numbers involves a dynamic objective-subjective
interaction. This dynamic interaction in turn is the source of intuitive
ability, which is highly necessary as a lubricant of the thinking process.
However, once more the contribution of the unconscious remains merely implicit, and subsequent interpretation of the mathematical process is carried out in reduced directional terms. In other words, instead of treating both positive and negative numbers as dynamic interactions, they are treated as static one-directional independent objective entities.
So just as the vertical (qualitative) aspect in understanding is reduced to a merely horizontal (quantitative), interpretation, likewise the negative (subjective) aspect is reduced to a merely positive (objective) interpretation and is solely understood in horizontal rather than vertical fashion.
Now, I am not questioning the necessity of this reductionism at an early
stage of child development. Specialisation of conscious understanding requires
it. However the interpretation of mathematics rarely seems to go beyond
the linear level. I am for the moment simply raising problems that must
be addressed at higher levels of psychological development.