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- QUALITATIVE NUMBERS: IMAGINARY

*Remarkable Transformation*

We have seen earlier how the transition from linear to circular levels
involves a remarkable psychological transformation in the differentiation
of mirror or negative structures. One then realises that actual experience
of reality involves polarised directions of experience in that objective
and subjective experience take place in opposite systems of space and time.
In other words if movement of phenomena in objective terms (for the world)
takes place in a positive or forward direction, then in relative terms,
movement of the same phenomena, in subjective terms (for the self), takes
place in a negative or backward direction. Thus experience physically and
psychologically moves simultaneously both forwards and backwards in space
and time. The conventional understanding therefore that movement of events
takes place solely in a forward direction is due to a reduced interpretation
of reality reflecting a lack of differentiation of these mirror structures.

With the transition from circular to point levels, another remarkable transformation of understanding takes place, leading to the differentiation of virtual or "imaginary" structures, which in psychological qualitative terms are the exact vertical equivalent of "imaginary" quantities in mathematics.

In mathematics Ö-1 or i is defined as an imaginary number. In other words it is the square root of the negative unit quantity. The problem in conventional mathematics is that there is no way of extracting the square root of such a negative quantity. Therefore in the history of mathematics, for a long time, such a number was treated as a useless aberration, undeserving of further treatment. This has now changed and imaginary numbers are understood in their right as being as valid as real numbers and vital for a full understanding of the number system.

However despite the immense strides in the interpretation and application
of imaginary numbers in mathematics, this understanding is still in a reduced
solely quantitative sense, greatly limiting true understanding.

However just as there is a complementary qualitative interpretation in psychological terms of the mathematical notion of negative quantities, likewise there is a complementary qualitative interpretation in psychological terms of the mathematical notion of imaginary quantities. It now begins to unfold at this transition stage of development.

I have already outlined the conscious process as one-dimensional (i.e. where interpretation takes place in a positive direction only), and the unconscious process - expressed in reduced conscious terms - as two-dimensional (i.e. where interpretation takes place in both positive and negative directions). Also switching of experience involves essentially a change of sign or direction. Thus to switch from conscious to unconscious we must negate the conscious (i.e. introduce the mirror structures).

Thus, in relative terms, the unconscious is in a negative direction with respect to the conscious.

Therefore, combining these two features, the unconscious is two dimensional and negative in sign or direction with respect to the conscious. Therefore if we try to express the unconscious in reduced conscious terms (i.e. by conversion to the first dimension) then it is "imaginary". However because both processes are inherently different, there is no satisfactory way of resolving this attempted reduction.

Thus if the conscious is "real" then the reduced unconscious in relative
terms is "imaginary". And as all experience involves the interaction of
conscious and unconscious processes, the world physically and psychologically
is in truth "complex" (i.e. both "real" and "imaginary").

*Dimensions as Imaginary Objects*

Again there is a fascinating correspondence as between psychological and mathematical notions.

The Argand diagram is often used to represent the complex number system. The real numbers are represented along the horizontal axis dividing at the centre into positive and negative axes. The imaginary numbers are represented along the vertical axis again dividing at the centre into positive and negative axes.

Thus the real numbers are represented mathematically in a horizontal fashion. In like fashion the real structures (based on the conscious mind) in psychological terms represent horizontal understanding (i.e. where understanding takes place within a given dimension or paradigm).

The imaginary numbers are represented mathematically in a vertical fashion. In like fashion the imaginary structures (based on direct use of the unconscious mind) in psychological terms represent vertical understanding (i.e. where the paradigms or dimensions in which experience takes place vary).

As we have seen, the circular level, involving the unfolding of the
irrational structures is highly vertical involving the continual movement
through varying subtle paradigms where experience of reality becomes progressively
based on the unconscious. Throughout this time one is trying to reconcile
the two polar directions of experience. With the culmination of the circular
level and the erosion of all one-dimensional conscious experience this
is achieved. One now lives in the pure unconscious where experience is
bi-polar (i.e. both positive and negative simultaneously).

Having reached the pure unconscious, the next stage involves the attempt
by the unconscious to express itself in conscious form through phenomenal
projections. However this inevitably involves an element of confusion.
The differentiation of phenomena depends on the use of the "real" conscious.
Thus just as in mathematics the attempted reduction of a negative two-dimensional
quantity to one dimensional terms is imaginary, likewise psychologically,
the attempted reduction of a negative two-dimensional psychological process
to one-dimensional terms (through projection into conscious phenomena)
is "imaginary".

However there are still more riches to be uncovered through comparing complementary mathematical and psychological notions.

Formerly when discussing horizontal and vertical axes we referred to the horizontal axes as quantities (within a given dimension) and the vertical axes as qualities (i.e. differing dimensions or powers to which a given number can be raised).

Now, we are referring to the vertical axes as representing in fact the imaginary number scale.

This in fact entails that the dimensions and imaginary numbers in unreduced
understanding - are in fact identical.

*Limitations of Scientific Worldview*

There are some fascinating implications of this finding.

Normally, we tend to think of experience both of objects and dimensions belonging to a "real" world. But however at this level of understanding we realise that if objects are "real" then the dimensions to which they relate are in fact "imaginary". Once again reality is now understood as being - mathematically speaking - complex rather than real.

Also just as for each "real" number there is a corresponding "imaginary" number, likewise for each "real" object there is a corresponding "imaginary" object which in fact is the corresponding dimension. Therefore, interpreted in this fashion "real" objects can be looked on as "imaginary" dimensions. In like fashion "real" dimensions can be looked on as "imaginary" objects.

This also highlights how limited is the conventional understanding of space and time where we think in terms of a fixed number of "real" dimensions (3 of space and 1 of time).

This fixing of dimensions is a direct reflection of attempted reduction
of experience to solely conscious terms. Understanding the complementary
role of the unconscious changes all this. We now see that objects and dimensions
are completely interchangeable. Indeed there as many dimensions as objects.
Indeed, each object exists in a unique dimension and yet interpenetrates
with every other object and dimension. Once again, dimensions of experience
- physically and psychologically - are not pre-existing empty containers
within which to place objects. They only come into being in the moment
of creation, with the mutual existence of both objects and dimensions.
There is a dynamic switching process involved whereby "real" objects momentarily
exist against an "imaginary" background of dimensions. Then the dimensions
themselves momentarily become "real" against a background of "imaginary"
objects. Thus, in this pulsating display, objects and dimensions are dynamically
"complex" (i.e. both "real" and "imaginary"). This simply represents in
turn the dynamic interplay of "real" conscious and "imaginary" unconscious
processes. However when the activity of the unconscious is sufficiently
dulled - as in conventional understanding - the dynamic interpenetration
is greatly reduced and we become aware solely of a static "real" world
of unchanging objects and dimensions. Unfortunately conventional scientific
understanding of reality, greatly reinforces this uninspiring and distorted
"real" viewpoint.

As a child, I was always fascinated by imaginary numbers and naively dreamed of someday unlocking a "real" valued solution. Even now, with calculators, I will instinctively test them out to see what response they will give to the attempted extraction of a square root from a negative number!

However, now there is a surprising solution to this mathematical problem.

Mathematical logic is heavily based on clear differentiation of polar
directions (which is the whole basis of linear rational thinking). Thus
+1 is clearly differentiated from -1. An attempted solution of an equation
can be either +1 or -1 (as with x^{2} = 1). However by this logic
it cannot be +1 and -1 simultaneously.

Intuitive unconscious thinking, however operates by an entirely different logic. Here the essential basis is that both directions do coincide simultaneously.

Now imaginary numbers in representing the unconscious process correspond closely to this alternative intuitive logic.

Thus Ö-1 can be accurately expressed
as +1 and -1 (simultaneously). Of course this does not carry much meaning
in conventional terms. However this simply reflects that conventionally
mathematics is solely based on the logic of the rational conscious mind
(thereby excluding the logic of the intuitive unconscious).

Linear Imaginary Structures

Just as real structures can be differentiated at differing levels, likewise is it with the imaginary structures.

During the transition from circular to point levels, we have the emergence of what can be called the linear imaginary structures.

In other words, the unconscious starts to project itself, becoming embodied in somewhat rigid linear terms.

In some respects this resembles the earlier "real" linear stage where rational perceptions and concepts are developed in both external and internal directions. One tends to become involved again in the normal concerns of life, tending to see what had gone before in the circular level as a strange "time-out" from reality. Many of the evade skills are restored and new interests developed.

However there are crucial differences. Though there is some easing in one’s psychological state, the mind remains substantially in darkness (i.e. immersed in the unconscious). As conscious activity only operates at the very surface of the personality, one derives no substantial satisfaction from it.

Also there is very little permanence or stability possible in terms of achievements or relationships. Because one operates so much out of the unconscious, one becomes extremely sensitive to the direction of experience. Thus for example when one momentarily puts one’s faith externally in some goal or person (thereby becoming possessively attached), one quickly becomes aware also of the internal direction reflecting the unconscious desire underlying the projected phenomena. This is turn creates conflict and unrest as the one-directional unconscious cannot properly appropriate the needs of the two-directional unconscious. One has to detach oneself from from such phenomena in returning to the unconscious. Thus one lives in a constant state of flux, being unable to hold on possessively to anything for long.

This means that as by definition there is very little separation of positive and negative directions of experience in terms of imaginary structures so that they are by their nature transient and short-lived. Projected phenomena are posited (i.e. emitted from the unconscious). There is a temporary confusion when one interprets them in a real conscious fashion. However the experience of inner conflict and unrest in the realisation of their true origin quickly activates mirror structure activity leading to their erosion.

However it is still possible to identify various stages of development. Initially one will feel attracted to externally projected phenomena of perception. Mirror structure development then changes one in an internal direction to become aware of projected feelings. Later at a deeper level one becomes attracted to externally projected conceptual phenomena and again through mirror structure development internally projected phenomena in the form of values and judgements.

Thus we can summarise these four sub-stages of the linear imaginary structures:

1) Externally projected linear perceptions. This corresponds to positive
(rational) imaginary numbers in the 1st dimension (i.e. in mathematical
terms +( Ö-1)^{1}).

2) Internally projected linear perceptions. This correspond to negative
(rational) imaginary numbers in the 1st dimension (i.e. in mathematical
terms -( Ö-1)^{1}).

3) Externally projected linear concepts. This corresponds to positive
(rational) imaginary dimensions to which the number 1 is raised (i.e. in
mathematical terms 1^{Ö-1 }).

4) Internally projected linear corresponds. This corresponds to negative (rational) imaginary dimensions to which the number 1 is raised (i.e. in mathematical terms

1 ^{- Ö-1 }).

Experience during this period moves from imaginary experience of objects
(in both positive external and negative internal directions) to imaginary
experience of dimensions themselves (again in positive and negative directions).