Mirror Development
 

The development of these structures marks the decisive transformation from the linear to the circular level, where the focus gradually switches from differentiation of the conscious to the unconscious structures.

Each of the crucial transformations between levels is marked by a key distinguishing characteristic.

We have been examining development in terms of process, mode and direction.

The choice in terms of process is conscious v unconscious understanding (finite v infinite in psycho-mathematical terms).

The choice in terms of mode is cognitive v affective understanding (real v imaginary in psycho-mathematical terms).

The choice in terms of direction is external v internal understanding (positive v negative in psycho-mathematical terms).
 
 

In terms of infant development, differentiation of process starts first in the separation of finite from infinite, then differentiation of mode in the separation of real from imaginary and finally differentiation of direction in the separation of positive from negative. (There is however considerable overlap involved).

The development of the linear level leads to a world-view largely based on the rational paradigm. This emphasises the conscious process (over the unconscious), the cognitive mode (over the affective) and the external direction (over the internal).

This tendency is especially pronounced in mathematical understanding, which involves the most specialised use of the rational paradigm. It is heavily biased towards (rational) finite rather then (intuitive) infinite understanding, to logical thinking rather than the senses, and to an objective rather than a subjective orientation.

Therefore, though the unconscious process, the affective mode and internal direction are - necessarily - always implicitly involved in the dynamics of psycho-mathematical understanding (as in all understanding) they are explicitly ignored, with the consequent activity being interpreted in a (reduced) positive real rational fashion.
 
 

Nature of Transitions
 

Each of the major subsequent transitions involves transcending a key element in terms of this reductionism, enabling the shift from a static absolute to a dynamic relative type understanding.

The first of these transitions, from the linear to the circular level, involves directional orientation.

Positive and negative directions are now explicitly recognised as complementary and understood as equally valid components in psycho-mathematical understanding. This in turn leads to the breakdown of any notion of "absolute" relationships. (Indeed, quite literally, in mathematical terms, an absolute number is one where the positive or negative sign is ignored). Truth is no longer understood as (statically) objective, but rather as the dynamic interaction between complementary (relative) objective and subjective poles.
 
 

The second transition, from the circular to the point level, involves modal orientation, where both real and imaginary modes are explicitly recognised as complementary and therefore also as equal components in psycho-mathematical understanding. This in turn leads to the breakdown of any notion of distinct "real" relationships. Truth is no longer understood in (solely) cognitive terms but rather as involving the dynamic interaction of complementary cognitive and affective poles. All relationships are now psychologically understood as complex, involving both "real" and "imaginary" poles.
 
 

The third transition, from the point to the radial level, involves process orientation, where both conscious and unconscious processes are explicitly recognised as complementary and again as equal components in psycho-mathematical understanding. This in turn leads to the breakdown of any notion of distinct "finite" relationships. Truth is no longer understood in (solely) conscious terms but rather as involving the dynamic interaction of complementary "finite" and "transfinite" poles.
 
 

It is the first of these transitions that we are concerned with here.

Whereas at the linear level, mathematical and psychological understanding are clearly separated (and indeed considered as largely unrelated disciplines), this begins to subtly change with this transition. It marks the birth of true psycho-mathematical understanding.
 
 

Let me illustrate this change with reference to the concept of number.

At the (absolute) linear level, understanding is unambiguous. Number is a definite rational objective notion, independent of self. Also there is a direct correspondence as between the concept of number and individual number perceptions. Thus we have "positive" number concepts and "positive" number perceptions which are directly related to each other.
 
 

Psychic Anti-matter
 

The development of the mirror structures dramatically changes this understanding.

I have spoken of mirror structures as a form of psychic anti-matter, directly complementary with physical anti-matter.

Thus every concept has a corresponding anti-concept and every perception has a corresponding anti-perception. These must be understood in dynamic relative terms. If the concept and perception represent the positive direction of understanding respectively, then the anti-concept and anti-perception represent the negative direction.

What this amounts to is a recognition of the true dynamics of understanding, which always involves the relationship between subject and object (observer and what is observed).
 
 

From this perspective numbers are not absolute objective independent entities.

If we take the perception of any number (say 3) we have a two-way relationship.

We have the objective number perception (in relation to the subjective mind). Alternatively, we have the subjective mind (in relation to the objective number).

In dynamic terms, the number does not independently exist, rather is created through the interaction of mind and matter (i.e. internal and external directions of understanding). We are thus always participants in the creation of number perceptions.

If the external direction of a number perception is positive, then the internal direction is - in relative terms - negative. Thus if the former direction represents a perception, then the later direction represents an anti-perception.

Every number has now - in dynamic terms - a two-way definition. Thus the number "3" has a two-way direction. 3 is a number (in terms of its positive direction). 3 is also an anti-number (in terms of its negative direction).

Indeed it is the dynamic fusion of these two directions that generates insight which is so vital for understanding. When positive and negative directions fuse this creates a switch from the particular to the general understanding of number. In other words there is a switch from 3 as a particular number perception to the general concept of number (which embraces all particular perceptions). Insight (i.e. energy) in understanding involves this essential ability of making the vital connection relating particular perceptions to general concepts.
 
 

Now the concept of number (following the fusion of perceptions) is made conscious and in turn - in dynamic terms - assumes both a positive and negative direction.

Thus we have again, in relative terms, the concept of number balanced by the corresponding anti-concept. The former represents the objective number concept (in relation to subjective mind); the latter represents the subjective mind (in relation to the number concept).

Again it is the fusion of the two directions of understanding that causes a switch in understanding back to particular perception.
 
 

Thus we can see how understanding is inherently a dynamic interactive process involving psychic matter and anti-matter. The positive and negative directions of particular perceptions (perceptions and anti-perceptions) fuse to cause a switch to general concepts. The positive and negative directions of concepts (concepts and anti-concepts) in turn fuse causing a switch in turn to particular perceptions.

In relative terms, the perceptions constitute the quantitative phenomena and the concepts the qualitative phenomena which provide the dimensional environment for perception. Needless to say, neither phenomena nor dimensions pre-exist but rather are created through dynamic interaction. All of this applies equally to mathematical as well other areas of experience.

The dynamics of directional interaction, I have outlined, are inherent in understanding at the linear level. However the interpretation is in a reduced absolute fashion. Now this reduced interpretation itself has a profound effect on the manner in which interaction takes place.

When there is an explicit awareness of interaction present, the level of fusion rapidly increases both in relation to perception and concepts. What this amounts to is a considerable increase in intuitive capacity, greatly facilitating understanding.

When this explicit awareness is not present, the level of fusion is greatly reduced. This leads to the tendency towards rigid mechanical interaction as between concepts and perceptions.

Understanding becomes largely stunted, where spontaneity, originality and freshness of insight is considerably eroded. One’s experience tends then to simply reconfirm pre-existing rigid notions.

However, as well as the interaction of positive and negative directions, the precise balance is very important. People with a strong extrovert personality tendency often have difficulties balancing perception with organising concepts. The switch from perception to concept formation presupposes that external perception of the world is counterbalanced by internal perception of self. For the extrovert this can be difficult leading relatively to too much perception in an unduly positive direction.
 
 

Likewise the pure scientist or mathematician deals mainly with positive concepts in an external direction and from an absolute perspective (which is well suited to the introvert personality). Here, there is the danger of insufficient counterbalancing negative concept formation leading to little switching to perceptive experience. When this happens we get someone who can easily lose touch with the real world.
 
 

Now, what happens during the transition stage from linear to circular level, is that one is led to profoundly question the absolute approach to truth (which ignores distinctions as between positive and negative directions).

For example, someone with an interest in mathematics, will, during the linear level, try to adopt the conventional absolutist approach (e.g. considering numbers as objective independent entities). However this interpretation now comes sharply into conflict with actual experience. One begins to realise that there is a significant subjective dimension to understanding mathematical relationships which should be explicitly recognised.

Initially because of the strong positive direction of (former) understanding one is led very much in the opposite direction. This is what leads to immersion in the unconscious and the development of mirror structures (anti-structures). During this intensely psychological period, there can be a significant reduction in enthusiasm and indeed ability for (conventional) mathematical activity.
 
 

Anti-structure activity is thus, at this stage, mainly confined to the external level of understanding. There is the start of a process by which mathematical understanding becomes increasingly psycho-mathematical. In other words the conventional absolutist approach (based on separation of objective and subjective poles) gives way to a new dynamic relative approach (which considers objective and subjective poles of understanding as complementary).

Because mirror structure development does not yet greatly affect the internal direction of understanding, in terms of psychological experience, psycho-mathematical understanding does not yet apply. In other words there is not a clear understanding yet of psychology having a mathematical dimension.
 
 

Thus, to summarise, there is during this transition period, explicit differentiation of the negative direction of experience. This has to be contrasted with the merely implicit differentiation of the negative direction during the linear level.

Whereas implicit differentiation of negative direction enables conscious experience to switch from objective to subjective poles, explicit differentiation of negative direction is necessary to enable significant unconscious (i.e. intuitive) fusion to take place in experience.