(A) Concrete Stages
 

Positive Direction

In psychological terms, we have seen how concrete operational ability involves a new transformation in understanding, whereby a child acquires the ability to rationally control physical objects in the environment.

What this in turn involves is a growing clarity in terms of whole and part distinctions. This requires a sufficient degree of conscious specialisation so as to enable (implicit) unconscious operations to be reduced to, and interpreted within the rational linear approach.

The child can now break objects into parts, which preserve a distinct independent identity. These parts can then be related back to the wholes from which they were obtained. Through this reality can be broken up so as to be analysed, and then organised once more in a composite manner.

Breaking any object unit into parts, involves a fascinating psychological feat, which has far reaching consequences.

Again, this period of psychological development of rational ability, has direct parallels with its mathematical counterpart of rational numbers.

A rational number can - by definition - be expressed as a fraction (where numerator and denominator are both whole numbers).

The simplest type of fraction involves reciprocals of whole numbers. Here the number is inverted, so that the numerator is always 1, and the denominator the original integer.

Thus the reciprocal of 2 is simply 1/2. Alternatively this could be written as 2 ^-1.

What happens in obtaining this simple fraction is that we move from a situation, where 2 is defined in the (positive) 1st dimension, to the complementary situation, whereby 2 is defined in the 1st (negative) dimension. The decisive transformation that takes place is therefore a switching in the direction of the (qualitative) dimension involved.
 
 

This insight in turn is remarkably instructive in terms of understanding the dynamics (psychologically) of concrete rational ability.

A perception provides the quantitative understanding of an object. The concept on the other hand - in relative terms - provides the qualitative understanding. The dynamic act of (rationally) understanding objects, always involves relating the quantitative object (i.e. the perception) with the qualitative dimension (i.e. the concept).

The first clear identification of objects involves using both perceptions and concepts in a (solely) positive external sense. This is the earlier of the composite stages.

The next step - in the later composite stage - involves an increased ability in terms of combining both positive and negative directions in terms of perception. This, as I have explained increases the stability of objects giving them an extended existence in space and time.
 
 

Formation of Concepts
 

Now, the next transformation involves the ability to combine both positive and negative directions in terms of concepts. In other words this involves combining - in relative terms - positive and negative dimensions.

The positive dimensional direction comes simply from the ability to form an (external) concept of an object.

This is followed in turn with complementary subjective understanding whereby the child begins to form an (internal) concept of self. This qualitative understanding of (internal) self - in relative terms - is then negative with respect to the qualitative understanding of the (external) object. In other words it is an understanding of dimensional experience as negative.

The true dynamics of psychological experience, remarkably - even at this comparatively early stage - implicitly requires therefore understanding of dimensions as negative.
 
 

Bearing in mind the complementarity of physical and psychological relationships, this means that space and time have a negative as well as positive interpretation.

What this entails is that - in the physical universe - the dimensions (i.e. space and time) are dynamically created through the negation of matter. The negation of matter dynamically involves the positive creation of space and time.

In complementary fashion, matter is dynamically created through the negation of dimensions. Here, the positive creation of matter involves the negation of space and time. Properly understood, both matter and dimensions have, in relative terms - a positive and negative direction.

The static linear approach involves a gross reduction of these basic dynamics. Physical dimensions and matter are largely considered in separate static terms, where both have only positive poles. Here, there is a marked tendency to consider dimensions as pre-existing and given within whose confines all objects are contained.
 
 

It is exactly similar from a psychological perspective. In dynamic terms, understanding of phenomena involves the continual interaction of perceptions (psychic matter) and concepts (psychic dimensions). Here, the concepts are created through the negation of perceptions. The concepts then arise from the fusion of the positive and negative directions of the perceptions. The negation of perceptions, therefore dynamically involves the positive creation of concepts.

In complementary fashion, perceptions are dynamically created through the negation of concepts. The perceptions arise from the fusion of the positive and negative directions of the concepts. Thus the negation of concepts dynamically involves the positive creation of perceptions.
 
 

To illustrate the dynamics of understanding "psychological fractions" let me take a simple example of understanding the division of a cake into three slices.

Here, the cake represents the original whole object and the slices the subsequent parts.

Identifying the cake as a clearly identifiable separate object, involves the dynamic integration of its (implicit) concept with the (explicit) perception of a cake. This in turn requires giving both a positive and negative direction to the perception.

Identifying the slices as identifiable separate objects involves the same dynamic integration of concept with perceptions giving both a positive and negative direction to the perceptions. This gives the fragmented understanding of multiple slices.

Identifying the slices with the original cake (where both maintain their identity), involves a qualitative transformation involving the reverse (explicit) concept with the (implicit) perceptions. This now requires giving both a positive and negative direction to the concepts (i.e. psychic dimensions). This gives the holistic understanding of the whole unit (i.e. the cake) embracing the parts (i.e. the slices).

Thus, what appears to be a very simple psychological feat of breaking a whole unit into parts), in fact is incredibly complex involving - dynamically - positive and negative movement of objects (in relation to dimensions), and also positive and negative movement of dimensions (in relation to objects).This comes through the ability to fuse positive and negative perceptions (relating them to concepts), and then the complementary ability to fuse positive and negative concepts (relating them back to perceptions).
 
 
 
 

Basic Nature of Physical Reality
 

In complementary terms, physical reality dynamically involves the continual fusing of positive and negative matter i.e. sub-atomic matter and anti-matter particles (relating it to space and time dimensions), and then the complementary fusing of positive and negative dimensions (relating them back to matter).

Science needs a positive-negative definition of dimensions to complement its positive -negative definition of matter. In other words, in dynamic terms, we have dimensions and anti-dimensions. This entails that space and time are fully complementary. Space (universally) is posited through the negation of matter (in space); time (universally) in turn is created through the negation of matter (in time);. This basically means that physical reality - at all levels - involves the continual dynamic interaction of matter and dimensions.
 
 

All this interaction, involves the role of the unconscious. However, as the conscious mind increases in specialisation, one increasingly interprets reality in reduced static terms. This in turn means removing the negative poles of understanding.

Thus, instead of objects and dimensions being created in experience, one believes that they already both pre-exist. The understanding process is then interpreted - not as a dynamic instantaneous creative process - but, rather as the registering of a video, as what independently exists.

Now, I am not questioning at all, the validity and indeed necessity of conscious specialisation of understanding in early development. The point that I am making, is that it comprises only one level of understanding (i.e. the linear level). Other levels such as the circular (which involves the specialisation of the unconscious process), and the point (which involves the specialised interaction of both conscious and unconscious processes), need to be subsequently developed to avoid an overall distorted worldview.
 
 

Negative Direction
 

In mathematical terms, we can have negative as well as positive fractions.

Now -1/2 ( = -2 ^-1 ) is an example of a simple fraction, and involves both a negative sign quantitatively (in terms of the horizontal number 2), and qualitatively (in terms of the vertical power or dimension 1).

Likewise, we have the parallel equivalent on the psychological side, involving a negative direction both in terms of quantitative perceptions, and in terms of qualitative concepts.

I have already outlined the dynamics of (positive) psychological fractions which is the breaking of objects into component parts.

The next stage involves the psychological process of (negative) fractions.

Now, in dynamic psychological terms, when the positive pole is identified as objective, the negative refers to the complementary subjective pole of experience.

What this entails is the ability to break the (internal) self into component parts. In other words one learns to split the overall ego into many different aspects. The child has to learn to play a number of different roles which combine to form an overall ego identity.

Positive concepts - in relation to (external) objective phenomena - are formed when there is a fusion of positive and negative perceptions (with the positive external direction of perception dominant).

Negative concepts - in relation to (internal) subjective phenomena - are correspondingly formed, when again there is a fusion of positive and negative perceptions (with the negative internal direction of perception dominant).
 
 

Extroversion and Introversion
 

This indeed gives us a good insight into the dynamics of extroverted and introverted personalities.

An extrovert tends to experience perception in a primarily positive external direction. Therefore when fusion with the negative direction takes place, the positive dominates leading to conceptual formation in a positive external direction. Furthermore, the positive direction of concepts tends to reinforce the positive direction of perceptions giving typical extrovert personality tendencies.

An introvert by contrast, tends to experience - in relative terms - perception in a primarily negative internal direction.

Therefore when fusion with the positive direction takes place, the negative dominates leading to conceptual formation in a negative internal direction. Then the negative direction of self concepts, this time reinforces the negative direction of self perceptions, giving typical introvert personality tendencies.
 
 

This stage represents a more introverted stage, where both negative perceptions and concepts combine, enabling the child to fragment the ego into different parts (through subjective identification with roles seeing them as components of overall identity). The child learns to adopt a number of personae which make up the overall personality.
 
 

Multiple Personality Disorder
 

The personality does not always split or fragment in this way successfully.

A person with multiple personality disorder is - in psychological terms - a fascinating example of how severe problems can arise at this stage.

Such a person is unusual in splitting the ego into a number of highly distinctive personae, but then not being able to integrate them into a single whole.

It is very usual for this same person to have suffered from severe psychological problems in early life (e.g. due to sexual abuse), leading to considerable unconscious overload. Because of this, there is understandably, great (unconscious) resistance to the negative subjective direction of development in personality, which inevitably would be highly painful. Thus, the negative direction of perception is insufficient, leading in turn to insufficient negative development of concepts.

Conceptual development is necessary to organise experience. Thus the person with multiple personality disorder, experiences marked difficulties in organising the various highly distinctive personae exhibited in behaviour, into a unified whole.

Indeed, this also explains why the various "personalities" are so distinctive in the first place. Because of the lack of conceptual control, there is far more spontaneity possible, where the personae tend to arise directly out of the impulsive needs of the unconscious. People with this disorder are the supreme actors. Because their egos are simply made up of parts (without organisational control), they are thereby able to get fully into these various parts, identifying literally with the various roles they play.

Indeed, this can be a considerable problem with professional actors also, where, sometimes, over identification with role playing can inhibit true self knowledge.

In normal circumstances, rational conceptual development - externally and internally - tends to have a considerable restraining and indeed repressive influence over instinctive behaviour. Thus, though vitally necessary, it is a two edged sword. Whereas on the one hand it greatly facilitates organisation of one’s environment (externally and internally), it also tends to depersonalise, blocking off contact with one’s basic instincts.
 
 
 

(B) Formal Stages
 

Positive Direction
 

Though the concrete stages involve a considerable increase in (mental) conceptual ability, (sense) perception is still very much in the ascendancy. This indeed, is why experience is so concrete. Concepts, essentially serve a back-up function, bringing order and coherence to the data provided by the senses.
 
 

However, the balance decisively changes during this stage. It is important to appreciate why this is the case.

When experience switches in direction, it tends to negate the phenomena previously generated. Thus, when experience switches from external to internal, during the concrete stages, it dampens down the tendency to generate (positive) perceptions. Thus when experience next switches to a positive direction after the (internal) concrete stage, there is a reduced inclination to generate sense perceptions. This - in relative terms - gives scope for the specialised development of the more holistic conceptual ability of the mind.

This stage - which is referred to by Piaget as formal operational thinking - involves the purer abstract use of reason in an external direction.
 
 

Though normally a considerable level of sense information is still combined with this general rational ability, it is possible to carry it to a high degree of specialisation, where the emphasis in understanding is not on specific content, but rather on general format.

Pure mathematics itself is a fine expression of this specialised use of reason. Highly abstract symbols are used, so that one can more clearly focus not on their specific sense content, but rather on the general mental connections between them.

Indeed, this is the great appeal of mathematics for so many people in the (mistaken) belief that at last here is a haven of absolute truth which does not depend on the unreliable senses.

Strictly speaking, this is not the case. Whereas it is of course true, that the role of the senses is greatly reduced in pure mathematical activity, it can never be fully dispensed with, as there is always a need to use symbols which in the final analysis must be sensibly verified. Indeed, considerable importance attaches to the precise nature of these symbols adopted. The choice, confirmation and economical use of mathematical symbols constitutes an austere but genuine art form with effective communication depending on aesthetic as well as rational criteria.
 
 

Use Of Symbols in Mathematics
 

Indeed, it would be perhaps instructive to comment on the remarkably appropriate psycho-mathematical nature of key symbols commonly used for the basic operations of addition, subtraction, multiplication and division. (I have already commented on the amazing "aptness" of the symbols for unity and nothingness).

Addition is rooted in the basic psychological ability to consciously posit phenomena. Likewise in mathematics adding involves giving numerical quantities a positive sign.

As I have indicated, that though two combined aspects (i.e. a horizontal quantitative and vertical qualitative), are always involved in this understanding, linear understanding always requires interpretation of this relationship in reduced static terms.

The very symbol used to represent addition in mathematics involves the intersection of a horizontal and vertical line (i.e. +).

Subtraction is rooted in the basic psychological ability to negate phenomena. This in turn requires the ability to disentangle the quantitative-qualitative confusion. Thus we now see a dynamic relationship involving differentiated qualitative and quantitative aspects. We have correspondingly + separated into I and -- (The former carries the residual meaning associated with positing phenomena, and the latter now the complementary meaning of negation).

It is therefore again apt that -- has culturally survived as the universal symbol for negation in mathematics.
 
 

I have already referred to multiplication in dynamic terms as being a diagonal process of understanding involving the interaction of (vertical) qualitative transformation of phenomena combined with a reduced (horizontal) quantitative interpretation.

Also there is an inability to clearly differentiate positive and negative directions.

A mathematical symbol commonly employed for multiplication (i.e. X) involves an intersection of two diagonal lines of opposite directions.

Finally, the complicated process of division, requires further differentiation of positive and negative directions. Again this is borne out by the simplest mathematical symbol for division (i.e. /).

It is no accident that these symbols have stood the test of time. In Jungian terms they have profound archetypal significance, suggestive of an ultimate reality where psychological and mathematical processes are identical.

Therefore, appropriate symbols when employed in mathematics - far from being neutral - can act as powerful catalysts in terms of suggesting underlying connecting relationships.
 
 
 
 

 Mathematical Faith
 

Mathematics, which prides itself on reason, is in fact only rational, in a very restricted sense.

We have just seen that the affective sense mode in terms of symbols is so important. Equally the central mode of will is also vital to its activity. Mathematics depends greatly on "blind" faith, - in the unquestioned acceptance of key assumptions - to preserve its "rational" status. Indeed, in a very real sense mathematics constitutes a powerful from of secular religion. However, its assumptions are invariably the product of a reduced linear interpretation of reality. When experience goes beyond the linear level, they become increasingly untenable and much of mathematical activity appears quite irrational.

Let me illustrate this point with reference to the whole notion of "proof".

The faith is nowhere stronger in mathematics than as to what constitutes a "valid" proof. A high level of consensus is involved.

However if we look at what underlines such a proof, fundamental difficulties are encountered.

For example, the Pythagorean Theorem - which perhaps is the most famous in mathematics - asserts that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares of the other two sides. Now the conventional wisdom is that this theorem has been proved satisfactorily beyond any doubt. Indeed a large number of alternative "valid" proofs are available.

The assumption is that because this theorem is true for the general case (i.e. "any" triangle), that therefore it is true in a particular case.

However, this seemingly logical conclusion conceals what is in fact an enormous difficulty. If the general case is true, then it is true for a potentially infinite set of triangles. However the truth in any particular case refers to a single actual finite example.

So what is involved in all "proofs" is the seemingly safe logical deduction that what is true of the general "infinite" case is thereby true for the particular "finite" case. In other words what is true for the "potential" case is thereby true also for the "actual" case.

However this deduction is wholly unwarranted. It amounts to considering the infinite as simply an extension of the finite, whereas in truth it is qualitatively an entirely different notion.
 
 

The mathematical assumption of the infinite as being a (horizontal) extension of the finite - which is implicit in any mathematical "proof" is just another classical example of reductionism. The finite, which involves discrete knowledge is amenable directly to rational understanding. However the infinite, which by contrast involves continuous knowledge is directly amenable to intuitive understanding. However because mathematics does not explicitly recognise the dynamic rational-intuitive interaction involved in understanding, it always attempts to reduce the intuitive to rational explanation. Nowhere is this more important than in the attempt to reduce infinite notions to finite, where the infinite - in effect - is simply treated as a quantitative extension of the finite.
 
 

Thus the conclusion - which underlines all mathematical "proofs" - that what can be proved for the general (infinite) case, is thereby true for the particular (finite) case, rather than being a logical interpretation, is in fact a logical misinterpretation of the true situation.

Reformulating - in an acceptable fashion - the relationship between finite and infinite notions, involves explicitly recognising the dynamics of rational-intuitive interactions.

We will return to this critical matter later, and show that all mathematical proofs, in dynamic terms, can only have a relative validity.
 
 

Thus to summarise, at this stage there is the development, in an external (positive) direction of abstract rational ability, which is the essence of pure logic and all algebraic operations. Once more. I am not questioning the appropriateness of specialisation of this highly valuable skill. Rather, I am questioning the tendency to subsequently reduce all other levels to the rational, leading to considerable distortion in understanding.
 
 
 

Negative Direction
 

This is the internal direction of formal understanding.

It is developed through negation of (positive) formal understanding, where external understanding is once again cancelled out. This is the means by which one is enabled to obtain an abstract conceptual understanding of self (i.e. a means to rationally understand the self).

This ability is often demonstrated in such areas as philosophy and theology (esp. moral) and underlines the discussion of many contemporary social issues where one is dealing with formal abstract principles of subjective conduct.

The complement in mathematical terms would therefore be negative rational numbers (e.g. -1/a).
 
 

There is a very consistent pattern of evolution in terms of the development of the conscious structures.

The affective sense mode comes first developing in the positive external direction. Specific superficial knowledge is initially generated.

The direction of understanding then switches giving complementary affective knowledge of a negative subjective kind.

This in turn - due to the lessening of force of the positive affective mode - frees the cognitive mental mode to develop, firstly in a positive and then in a negative direction.

Due to the cancelling out of specific information involved in these dynamics, when the direction of understanding turns back to the positive, there is the development of more generalised understanding again firstly in the positive (objective) direction and finally in the negative (subjective) direction.

Of course, though there is a definite stage associated with the specialised development of each of these structures, in practice - following specialisation - they tend to be used in a more hybrid manner in conjunction with each other.

Thus there is never a totally definite mode or direction (or indeed process) evident in understanding.

As we have seen, mathematics (esp. pure) might be characterised as involving the cognitive rational mode in a positive (objective) direction, involving the conscious process.

However, as we have seen, in dynamic terms, the affective sense mode, the negative (subjective) direction and the unconscious (intuitive) process are always implicitly involved, exercising a considerable influence - even if often unacknowledged - on the outcome.