Introduction
 
 

We have seen how the circular level is concerned with the development of superstructures which in mathematical terms are irrational. We have irrational qualities (in the form of vertical stages of psychological growth), that exactly complement irrational quantities (in the form of the horizontal reduced mathematical interpretation of such numbers).

Strictly speaking these psychological structures correspond to mathematical quantities that are algebraic irrational.

For example Ö2 is algebraic irrational as a solution to the equation (i.e. x ² = 2). This means that though this number is irrational in the 1st dimension, it is in fact rational when raised to the 2nd dimension.

In like manner, though the superstructures of the circular level are irrational when expressed in terms of the linear 1st dimension (where opposites are separated), they are in fact rational in terms of the 2nd dimension (where opposites are reconciled), representing therefore a higher level intuitive rationality. Indeed in Hegelian philosophy "reason" is used to refer to this "higher" two dimensional cognitive format.
 
 

The point level involves further progress in terms of the emergence of advanced superstructures. Once again these bear striking similarity to the alternative type of irrational quantities (viz. transcendental numbers).

The key characteristic of these advanced structures is that they involve the attempt to reconcile the line and the circle. In other words one now interprets reality, neither in terms of the (formal) linear rational, nor the (formless) holistic circular paradigms, but rather in terms of the relationship between both.
 
 

Symbolic Significance of Pi
 
 

The best known transcendental number in mathematics is p . This represents the highly important relationship between the line and the circle i.e. the ratio of the division of the circumference (of the circle) by the straight line representing its diameter.

The value of p is 3.14 (correct to two decimal places). Therefore rounded to the nearest integer this is 3.

Interestingly, the other transcendental number which is best known is e which is 2.72 (correct to two decimal places). Again, rounded to the nearest integer this is 3.

Indeed just as 1 is the numerical symbol of the linear level, and 2 the numerical symbol of the circular level, 3 is the numerical symbol of this level.

The linear level of the rational structures, involves the positing of phenomena in one direction. In this sense it is a one dimensional form of understanding.

The circular level of the (algebraic) irrational structures, involves both the positing and negating of phenomena (through the formation of complementary opposites), which makes it a two dimensional form of understanding.

The point level of the transcendental structures - where advanced superstructures are concerned - involves the relationship between the one dimensional linear and two dimensional circular levels. It is therefore a three dimensional form of understanding.

(I am of course using dimensions here in terms of their correct mathematical sense rather than the conventional limited scientific interpretation).
 
 

It is very interesting - if accidental - that the two key transcendental numbers both have a numerical value of 3 (when rounded to the nearest integer).

Another interesting empirical relationship exists in relationship to p and e. The value of e falls short of 3 by .28 (correct to two decimal places). The value of p exceeds 3 by .14. Now there is a 2:1 relationship as between the deficit of e and the excess of p respectively in relation to 3. (This relationship in turn mirrors the 2:1 dimensional relationship of circular and linear structures i.e. transcendental structures).
 
 

There are several remarkable ways in which the significance of 3 as a numerical archetype of the point level can be highlighted. I will attempt to develop these at some length.
 
 

I have mentioned before two complementary ways of defining numbers.

The first is the horizontal - and conventional - approach whereby various number quantities are expressed in terms of the fixed 1st dimension.

In this approach, when I mention the number 3 it is tacitly implied that this is 3¹ .

Even when a number is initially expressed in terms of another dimension, its value will be reduced to the 1st dimension for the purposes of numerical operations.

Thus 3² = 9, which implicitly is 9¹.
 
 

The second is the vertical - and largely overlooked - approach whereby, in complementary fashion, the fixed number 1 is defined with respect to various qualities or dimensions. This is an essential aspect not recognised in mathematics that numbers have a qualitative as well as a quantitative significance. Thus, if the number as defined horizontally (i.e. within a fixed dimension), is in direct terms a quantity, then the number defined vertically (i.e. as a dimension or power to which a given number is raised) is in direct terms a quality. Indirectly of course, all number quantities have a (complementary) qualitative interpretation and number qualities a quantitative interpretation. Unfortunately in conventional mathematics, when numbers are used to represent powers or dimensions, they are merely interpreted as reduced quantities

Thus numbers have both complementary quantitative and qualitative significance. This is however only appreciated through psycho-mathematical understanding which is inherently dynamic. Here, every horizontal interpretation of a number that is mathematical, has a corresponding vertical interpretation which is psychological. Of course in reverse fashion every vertical psychological interpretation of a number, has a corresponding mathematical horizontal interpretation.
 
 

Thus when I mention 3 in the alternative vertical approach, I am referring to 3 directly as a dimension (to which the fixed number "1" is raised)

The vertical definition of 3 is therefore 1³.

Now, from a conventional horizontal point of view, the (reduced) value of any such number (regardless of the power) is 1. Therefore it seems of little interest. However if we take the reverse complementary approach and again express this value in the 1st dimension by obtaining the cube root, instead of one result we now obtain three. The corollary of this is that three dimensional understanding of the point level has - in reduced form - three interpretations rather than one.
 
 
 
 

Circular Number System
 
 

The horizontal system of numbers - interpreted as real - is conventionally represented as a straight line. When it includes only positive numbers it extends in one direction. When it also includes negative numbers it extends in two directions.

Surprisingly the vertical system of numbers - expressed in reduced one dimensional terms - can be represented as a circle. Thus in fact the vertical approach gives rise to a circular number system which remarkably complements the accepted linear system.

Not surprisingly, this circular system - representing the direct qualitative significance of numbers - has immense psychological significance.
 
 

This circular system is simply derived by the practice of extracting the successive roots of unity.

Thus if the number 1 is initially raised to the 1st dimension, its reduced value is unchanged and is 1 (strictly speaking +1). This exactly represents the psychological nature of the conscious linear level which is based on the solely positive (i.e. static) identification of phenomena.
 
 

When the number 1 is raised to the 2nd dimension (i.e. the power of 2), its reduced root value in the 1st dimension is now Ö 1 or 1^1/2 . Now numerically this results in two real roots i.e. +1 and -1.

Again this perfectly complements the (reduced) conscious interpretation of the circular level. The experience of the circular level is primarily dynamic and intuitive springing from the unconscious. Now expressed in reduced one-dimensional conscious terms, this experience involves the complementarity of opposites in positive and negative directions. Thus in this reduced interpretation, experience appears profoundly paradoxical. Phenomena both have a positive and negative existence. This arises from splitting a dynamic two-way interaction into static one dimensional terms.

In dynamic terms, an object such as a number does not exist independently of mind. Rather it involves a two way interaction involving both object (positive direction) and subject (negative direction). Thus from a static reduced perspective every number has a corresponding anti-number (just as every matter particle has a corresponding anti-matter particle).
 
 

However, it is when we move into higher dimensions that the vertical approach to numbers is especially rewarding. It provides in turn an accurate and precise structure for the higher levels of psychological development.

When I was writing "Transforming Voyage", I found it very difficult to provide a satisfactory classification for the more advanced stages of development, especially at the point and radial levels. For example, I found myself referring at the point level to superstructures even though I had already dealt with these during the circular level. Therefore, at the point level I had to lamely reclassify them as advanced superstructures.

I slowly began to discover, that surprisingly, mathematical notions provide the most precise means of classifying advanced stages of psychological development. In terms of the main levels - linear, circular, point and radial - the roots of unity are particularly relevant. These provide an exact complementary structure - expressed in reduced linear terms - of the higher dimensions of understanding.
 
 

The purpose of the circular level is to transcend directly conscious phenomena in experience so as to arrive at spiritual union.

As I described at length - in writing about this level - this is a very arduous process. Through transcending conscious phenomena, one unconsciously represses - to a considerable extent - affective physical instincts.

The transition from circular to point level is marked one the one hand by the arrival at a (merely) transcendent form of spiritual union. On the other hand it represents the start of a prolonged process by which the unconscious projects itself outwardly in phenomenal form. This is the means by which it relieves psychological repression. However this creates considerable confusion. The projected phenomena of the unconscious are - in mathematical terms - "imaginary" rather than "real". However there is a tendency to temporarily identify with them in rigid "real" terms. This leads to considerably instability in terms of phenomenal experience, whereby one continually is forced to internalise those phenomena that are initially projected outwardly.
 
 

When we reduce the vertical number 3 to its root values in the 1st dimension (i.e. by obtaining the three roots of unity), we obtain a remarkable complementary translation to the psychological situation I have been describing. The three roots of unity are 1,

( -1 + Ö -3 ) / 2, and ( -1 - Ö -3 ) / 2.

The first root complements the spiritual contemplative union achieved (on completion of circular level). the other two roots are in mathematical terms complex, combining real and imaginary components. This in turn complements the psychological state, where projections from the unconscious which are inherently "imaginary", enjoy a finite - if temporary - existence in rigid "real" terms. One thus experiences the phenomenal world in a "complex" fashion, which however is somewhat confused.

Refined "complex" experience requires that cognitive "real" and affective "imaginary" experience be harmonised and equal. The mathematical complement of this is that real and imaginary parts be equal. As we can see, this is not the case in relation to these two roots of unity. (Complete refinement only comes with the 8th dimension which we will return to later).

Furthermore the two latter roots only differ in terms of the sign of the imaginary component. This is also replicated in psychological terms where a high level of mirror structure activity attaches to the phenomenal projections of the point level.

One projects the unconscious outwardly in phenomenal form. When one realises that this represents virtual or "imaginary" rather than "real" experience, one attempts to internalise the phenomenona, so that the fantasy no longer exists objectively, but rather as the expression of one’s inner unconscious.

Thus properly interpreted, the three roots of unity provide a perfectly precise mathematical structure for the experience in the earlier point level, which simultaneously is both very simple and unified (i.e. in the underlying spiritual state) and "complex" (i.e. in the phenomenal translations involved).
 
 

Now another fascinating aspect to these three roots of unity is that they lie as equidistant points on the circle (with a radius of 1 unit).

Indeed this also applies to the earlier roots. Clearly a circle can always be drawn through any single point or any two points.

The two point case is particularly interesting. Simultaneously a straight line can be drawn through the two points (representing the diameter), which also are equidistant points on the circumference of the circle. Also - which is highly important - the two dimensional case can be translated in solely "real" terms. Thus, though the circular level is increasingly intuitive, the unconscious is always translated through the refined use of reason. Thus one still views the world in "real" terms, which in reduced terms is paradoxical involving complementary opposites (i.e. positive and negative poles).

This paradox can be illustrated mathematically by using a positive and negative real axis. Thus the section of the diameter to the right of the circle’s centre represents +1 unit. The section to the left represents -1 unit. When we try to combine both sections the real line appears to have zero length.

In dynamic psychological terms this is precisely what actually happens to "real" phenomena during the circular level. They are continually eroded as intuitive consciousness takes over so that one eventually experiences a void in terms of conscious phenomena. From a spiritual perspective however this void represents unity.
 
 
 
 

Roots of Unity
 
 

When we continue to extract higher order roots of unity, they will always be equidistant points on the circle (of radius one unit). Thus the vertical number 4 in reduced one dimensional terms has +1, -1, + i and - i (i = Ö -1) as its four roots respectively.

As can be easily verified these roots are the points on the circle representing the axes of the complex number plane. These also - as we shall see - have great psychological significance.
 
 

I wish to stress once more the remarkable complementarity as between mathematical and psychological notions. Just as we can identify (horizontal) linear, and (vertical) circular levels in psychological terms, we can also identify linear and circular number systems in mathematical terms also.

Psychologically, the linear level can be expressed as the varied understanding of reality within a given dimension i.e. the rational paradigm. As this recognises only the "real" world and is positive in direction, it is - in mathematical terms - the first dimension. This represents well the nature of conventional understanding esp. scientific.

Mathematically, the linear number system involves the understanding of various numbers within a given dimension i.e. the first dimension. This in turn represents the conventional understanding of number. (Where negative as well as positive numbers are represented a merely conscious reductionist interpretation is involved).
 
 

Psychologically, the circular level involves a given unit or phenomenon being understood in terms of varied dimensions of understanding. In this sense it is vertical. We have already examined the highly important two dimensional case. When this is expressed in reduced one dimensional terms we get two opposite directions of understanding.

Mathematically, the circular number system involves a given unit being raised to varied

powers or dimensions which again is a vertical approach. Again when the highly important two dimensional case is expressed in reduced one dimensional terms i.e. by extracting the square root of unity, we get two distinct number poles.
 
 

The points representing these roots are actually on the circle (of radius one unit). If for example we went on to extract the eight roots of unity, we would get eight points on the same circle equidistant from each other. Again there is significant complementarity. In the linear case the number 8 represents the same line measurement of one unit multiplied by 8. In the circular case, the number 8 - i.e. the eight roots of unity - represents the basic circle measurement (i.e. circumference of circle of radius one unit in the complex plane), divided by 8 to create eight curve portions of equal length).
 
 

From a psychological perspective the point level represents the attempt to connect both linear and circular levels. There is an equally fascinating complementary "point" number system which connects the linear and circular number systems. This number system is represented by 0.

If for example - again in a complex plane with circle of radius one unit - we obtain the eight roots of unity, we will get eight equidistant points that lie on its circumference. These represent the circular number system with the circumference - based on radius of one unit - divided into eight equal curve lengths. Joining the centre (of circle) and these points on the circumference are eight lines of equal length. These represent the linear number system of one unit multiplied by eight.

Of course the centre of this circle is 0 and is a point. Thus just as the complementarity of linear and circular levels is achieved through connection to the central point of personality (the self in Jungian terms), the complementarity of linear and circular number systems is obtained through connecting lines to 0 at the central point of the circle.
 
 
 
 

Binary System of Development
 

The basic structure of psychological experience is best represented in binary terms. As we have seen, 0 is the symbol of the (circular) unconscious, whereas 1 is the symbol of the (linear) conscious mind. Just as all mathematical quantities can be represented as a binary system involving the progressive interaction of these two basic symbols, likewise all psychological qualities (i.e. levels of development) can be represented as the progressive interplay of these two symbols (interaction of rational conscious and intuitive unconscious).
 

I have already shown how the number "2" is simply the static attempt to represent the opposite poles of the unconscious in what in dynamic terms constitutes a void or nothing. In other words the two poles of consciousness cancel out in dynamic terms generating pure intuition. This is 0 in phenomenal terms. However in reduced static terms - where directions of experience are separated - this represents two dimensions of experience with positive and negative poles.

In these basic terms the conscious is one dimensional whereas the unconscious is two dimensional.

The implication of this is that higher dimensions such as the 3rd and 4th, cannot be envisaged in purely vertical (unconscious) terms, but rather involve the interaction of both conscious and unconscious. In other words higher levels involve both the interaction of line and circle. This brings us back to the transcendental nature of these higher structures. p which is the most famous transcendental number involves the interaction of line and circle. Likewise, the advanced superstructures of the point level involve the interaction of the linear rational and circular holistic paradigms and are transcendental.
 
 
 

Nature of Transcendental Quantities
 

It would be very valuable to now probe more deeply into the precise nature of both transcendental quantities and qualities.

Rational structures represent real conscious understanding of the one dimensional kind i.e. the linear level.

Irrational structures (algebraic) involve the reduced one dimensional translation in real terms of what is inherently two dimensional and intuitive i.e. the circular level.

Irrational structures (transcendental) are even more subtle and involve the reduced one dimensional translation - again in real terms - of the relationship between one dimensional (formal) linear and two dimensional (formless) circular understanding i.e. the (early) point level.
 
 

However - as we have already seen - the reduced one dimensional translation of this point level raises additional problems in that it generates "complex" as well as "real" solutions.

The unconscious in the form of intuition is the direct source of vertical dimensional understanding. This is inherently of a continuous and infinite nature.

The conscious in the form of reason is the direct source of horizontal understanding (within the given linear dimension). This is inherently of a discrete and finite nature.
 
 

Therefore, when we strive to relate both intuition and reason we are in fact combining qualitatively different notions. Most of the mystery surrounding transcendental structures springs from this problem. One is attempting to give linear conscious expression to a relationship which literally transcends such expression. Thus there is inherent paradox always in relation to such structures.
 
 

It is precisely similar in complementary terms with transcendental numbers. A transcendental quantity represents the relationship between a vertical dimension and a horizontal quantity (within a dimension). It inherently involves therefore a mixture of the continuous infinite and discrete finite notions.

There is tremendous confusion in mathematics in terms of the usage of finite and infinite notions. This is due to the attempt to incorporate both within a merely linear and thereby reductionist framework.

Thus, the straight line used to represent finite quantities is commonly understood to have infinite extension. Also it is understood that any finite portion of a line is capable of infinite division yielding line segments of finite length.

The nature of transcendental numbers highlights many of these difficulties. It results from the dynamic attempt to fuse continuous and discrete notions and combines elements of both.

A transcendental number such as p appears to have a finite value. Correct to four decimal places it is 3.1416. However as is well known its exact value cannot be determined. Thus the uncertainty principle applies to it, as indeed it does to all transcendental numbers. In a certain sense, the value of transcendental numbers is relative and approximate. Determinacy is always against a background of indeterminacy.
 
 

This highlights the inadequacies of the absolute approach to mathematics. We try to make clear distinctions so that a number is either finite or infinite. In truth - properly understood - all numbers inherently involve a relationship between what is both finite and infinite. Though at the level of rational numbers, finiteness appears justifiable, at the level of irrational numbers severe problems arise.

Indeed, this parallels a similar problem in physics. Newtonian physics appears to be valid at the level of conventional reality. However at the subatomic level of quantum particle behaviour, it breaks down as reality becomes increasingly irrational. So there is a highly important common factor at work in terms of both mathematical and physical behaviour.
 
 

To properly understand the nature of number we must correctly appreciate the psychological dynamics through which such knowledge is formed.

Knowledge of number always involves a dynamic relationship as between the perception of a number and the number concept. The perception provides direct knowledge of the actual number which is finite, whereas the concept provides direct knowledge of the potential number which is infinite.

Let us take the number "3" to illustrate. To meaningfully understand this number, we must be able to relate the actual perception of the particular number "3" to the general concept of number to which it relates. In other words, we must appreciate that the particular number "3" is a member of the general class or set of numbers.

Now whereas the particular number perception is finite and determinate, the general number concept is potentially infinite and indeterminate. In other words, the number concept does not - in direct terms - represent any actual number, but rather the potential for all numbers to exist.

When the perception and concept of number fuse we have a paradoxical form of understanding where finite and infinite notions overlap.

The positing of the finite number perception "3", in dynamic terms necessarily involves the negation of all other numbers. In other words to determine the number "3", all other numbers must remain in finite terms undetermined. Thus the finite existence of "3" is relative. Its determinacy is against the background of indeterminacy (of all other actual numbers).

Likewise the positing in infinite terms of the number concept, involves of necessity the negation of any actual number perception. In our illustration this means the negation of the perception "3". Thus the infinite existence of the number concept is relative. Its determinacy is against the background of indeterminacy (of any particular number).
 
 

What this entails is that the act of understanding a number involves a dynamic interaction of intuition and reason. In direct terms the (formless) infinite dimension is provided through intuition whereas the (formal) finite quantity is provided through reason.

When we interpret this inherent dynamic interaction in reduced static terms we are left with a paradoxical situation where determinacy is always in the context of indeterminacy, where the finite is always in the context of the infinite.
 
 

When only reason is recognised, interpretation of numbers is greatly confused. The actual number is understood in absolute finite terms. Then, it is assumed that the number concept is in direct correspondence with the number perception. This is because of the mistaken understanding that both perception and concept are rational. Thus, the number concept is effectively interpreted - in formal terms - as the set of all actual numbers rather than the qualitatively different (formless) notion of the set of potential numbers.

Because in the conventional interpretation, there is no opposition as between concept and percept, the understanding of number quickly becomes rigid and absolute.

Inspiration goes out of understanding as intuitive capacity becomes dulled. Finally because reason is directly geared to understanding finite notions, the infinite is effectively treated - in reduced terms - as an extension of the finite.

Thus the understanding of number, which is a dynamic interaction necessarily involving both reason and intuition, is interpreted in reduced static terms as involving only reason.
 
 

Let us now attempt to reformulate the more correct understanding which explicitly recognises the dynamic relationship between reason and intuition.

Every number perception is in relative terms determinate and indeterminate. It is determinate in that it can be posited in actual finite terms. It is indeterminate in the sense that the negation or exclusion of all other numbers - so necessary for its existence - involves a potential infinite notion which is qualitatively different.

It is similar in reverse terms where every number concept is again - in relative terms - determinate and indeterminate. It is determinate in that it can be directly posited (intuitively) in potential infinite terms. It is indeterminate in the sense that the exclusion or negation of an actual number involves this time an actual finite notion which again is qualitatively different.

The perception of a number involves a quantity (within a given dimension), The concept of number involves the dimension (in which the number is placed). If the perception is the horizontal number, the concept - in relative terms - is the vertical number. In alternative terms, if the perception is the unfolded explicate number, the concept - relatively - is the enfolded implicate number. The holographic nature of reality is inherent in the very interaction of all our concepts and perceptions.
 
 
 

Dynamic Interpretation of Numbers
 

We will now state this dynamic approach as succinctly as possible.

Every number is relative (i.e. has both positive and negative directions), in both horizontal (quantitative) and vertical (dimensional) terms. If the rational quantitative aspect is "real", then the intuitive dimensional aspect is - in relative terms - "imaginary"

(Of course, this implies that if the vertical dimension is "real" that again - in relative terms - the horizontal quantity is "imaginary").

This psychological interpretation of number can be now seen to directly complement the complex number system geometrically represented by the well known Argand diagram. Here the real axis is horizontal with positive and negative directions. The imaginary axis is vertical again with positive and negative directions. Both axes cut each other at the centre.
 
 

The appropriate way of understanding any (mathematical) number quantity is in terms of the corresponding (psychological) number dimension or quality

Thus a rational number can best be appreciated within the context of the rational linear paradigm. Thus the number "2" for example is a rational number which conforms in its nature to rational notions of enquiry. It has in other words an absolute identity with a definite finite value.
 
 

An irrational (algebraic) number however is best appreciated in terms of the irrational circular paradigm. The number "Ö 2" is an irrational number (in terms of the linear 1st dimension). However when expressed in terms of a higher dimension (viz. The 2nd dimension) it is now rational. Thus when we raise Ö 2 to the power of 2 we obtain 2 which is rational. All irrational (algebraic) numbers are rational in this vertical sense that when raised to the correct dimension (or combination of dimensions) they are thereby transformed into rational format.

However - as the Pythagoreans realised so well - there is something very unsatisfactory about the reduced linear interpretation of irrational numbers such as Ö 2. They then lose their absolute identity becoming relative and partly indeterminate. Thus though the value of Ö 2 can be approximated in finite terms to ever higher degrees of accuracy, its value can never be precisely known. In its non-repeating decimal expansion it is infinite.
 
 
 
 

Uncertainty Principle
 

We have here - as with all irrational numbers - a good example of the uncertainty principle.

We can on the one hand focus solely on the quantitative aspect. We thereby increase the finite accuracy of an irrational number at the expense of losing its infinite nature. Thus when we are satisfied to approximate the finite value of irrational numbers to a required degree of accuracy we are focusing solely on its (reduced) rational quantitative aspect. Eventually, the irrational number becomes rational in the acceptance of an approximate value. Thus to accept 1.4142 as the value of Ö 2 correct to 4 decimal places is effectively to reduce it to a rational number.

We can on the other hand focus on the intuitive aspect. We try and grasp here the precise qualitative aspect of an irrational number. Thus the Pythagoreans were concerned to fit their interpretation of such numbers to their qualitative paradigm which was rational. When they failed to achieve this inherently intuitive qualitative task, they had little interest in the merely quantitative operational aspects of such numbers.

An irrational transcendental number such as p is best understood within the corresponding transcendental paradigm of the point level.

A transcendental number combines elements of both the linear and circular approaches to numbers. Indeed p - its best known member - is deeply symbolic of this approach being literally the ratio of circle and its line diameter.

Because a transcendental number contains within itself, both the horizontal and vertical notions of number it cannot be properly expressed - in real terms - in terms of either.

If we treat horizontal quantities as "real", then, in relative terms, vertical qualities are "imaginary". If in reverse terms we treat vertical dimensions as "real", then again in relative terms horizontal quantities are "imaginary". We cannot maintain both as "real" simultaneously.

This explains why the value of a transcendental number is irrational, both horizontally in terms of the linear 1st dimension, and also in terms of any attempted vertical transformation into higher dimensions (or combination of dimensions). In other words, the solution to an algebraic equation (of rational coefficients containing these higher powers or dimensions) cannot be a transcendental number.
 
 

Though transcendental numbers seems so abstruse they are exceptionally common. We are accustomed to physical measurements being precise rational numbers. This in fact represents a static idealised approach. In practice, in an important sense, all real measurements of distance are transcendental. If I say that the length of a curtain is 72 ins, this is just a convenient rational approximation. In fact it can never be exactly 72 ins. If we tried to measure it precisely we could better approximate its true length without ever determining it exactly. Indeed, reaching a high level of detail at the sub-atomic level, it would become increasingly difficult to proceed as the measuring device would inevitably influence what we are attempting to measure.

What actually happens is that at this level of reality, particles have no meaning independent of the dimensions they inhabit. The world indeed - in real terms - at this level is mathematically transcendental, involving the dynamic interaction of quantitative and dimensional characteristics of existence. Though we are normally not aware of this at the rational macro level of experience, strictly speaking, everything in creation exhibits a unique element of indeterminacy.

Because of its rational bias the methods of science thus are highly reductionist.

In dynamic terms every object is unique manifesting a quantitative finite aspect and qualitative infinite aspect. Thus though an object may in many respects be similar to other objects, it always maintains an aspect giving it uniqueness. Science however is primarily interested in the merely quantitative aspects of objects. It does this by freezing their interaction with dimensions. In this view objects literally lose their unique transcendental quality and only have meaning in terms of common membership of a given class. Thus objects arising from a dynamic understanding, where reason and intuition interact, are reduced to a merely static understanding which is (solely) rational.
 
 
 

Diagonal Understanding
 

I have already defined the linear and circular systems of understanding both in psychological and mathematical terms. Rational notions correspond to linear and irrational to circular notions. Thus whereas 2¹ is the horizontal number "2" (i.e. the quantity "2" in the fixed 1st dimension), 1² is its corresponding vertical number (i.e. the quality or dimension "2", to which the given number 1 is raised).

When we combine both linear and circular definitions we obtain diagonal numbers.

Conventionally this will simply involve a number - other than 1 - raised to a particular power.

Thus for example 3² is a diagonal number. (Even in conventional mathematical notation it is represented in a diagonal fashion).

Strictly speaking 3² - in terms of our definitions - should be written (3¹) ^{(1 2)} . In this format we can clearly see that a number representing a horizontal linear quantity, is being raised to a number representing - in relative terms - a vertical circular quality. Thus in correct psychological terms if the horizontal quantity is interpreted rationally, the vertical quality - in relative terms - must be interpreted irrationally. Thus strictly speaking raising a horizontal (rational) quantity to a vertical (irrational) power represents the fundamental relationship between line and circle, which characterises from a psychological perspective our definition of transcendental structures.
 
 

There is a remarkable complementary pattern of activity replicated in mathematical terms.

Now if we take 2 as a rational (linear) number and Ö 2 as the corresponding irrational (circular) number.

Now 2 ^{Ö 2} is a diagonal number quantity which can be defined as a rational number raised to its irrational counterpart i.e. a linear number raised to a circular dimension. This is the famous Hilbert number which mathematically has been proven to be transcendental.

The interesting fact is that it again directly involves a relationship as between the line and the circle (i.e. horizontal rational and vertical irrational notions).

Indeed this is just a special case of a more general theorem i.e. Gelfond"s Theorem which implies that any rational (or irrational number) raised to an irrational power is transcendental. The inclusion of irrational quantities does not invalidate this analogy with the line and circle. As we have seen irrational quantities are treated in mathematics in a reduced linear horizontal fashion. Thus raising an irrational quantity to an irrational power still involves the relationship of line and circle.
 
 

There is a fascinating unrecognised parallel as between such mathematical results and psychological development.

Psychological development - properly understood - progresses through concrete stages which are horizontal and formal stages which are vertical. Thus the concrete stages provide the empirical sense data (to be placed within concepts) and the formal stages provide the concepts or dimensions (within which the data are placed)

Thus, for example in Piaget’s well known categories - referring to linear rational development - the concrete operational is followed by the formal operational stage of development. Here, a physical quantitative stage (horizontal) is followed by a mental qualitative stage (vertical).

Because of the high level of conventional reductionism (i.e. where vertical is reduced to horizontal development), experience rarely goes beyond the rational linear level.

To those destined to go on to the more intuitive circular level, there will be a period (e.g. in early adulthood) when they attempt to relate (rational) concrete perceptions to (rational) formal concepts within the linear level. However, because of insufficient recognition of intuition in this worldview, severe psychological conflict and indeed an existential crisis ensues. As they learn to transcend this reductionist approach, experience gradually becomes suprarational or intuitive (which expressed in rational terms is irrational or paradoxical).

Thus rational experience in rational dimensions leads to a psychological transformation through which experience becomes irrational.

Mathematically there is a perfect complementary pattern whereby any rational number - which itself can not be expressed as a power of another number - raised to a rational power (or dimension which is not an integer) is irrational.

We have here once again a fascinating example of where a mathematical (quantitative) result has an important (qualitative) psychological significance.
 
 

There are again concrete (horizontal) and formal (vertical) stages of development at the circular level. Here one experiences - separately - irrational or supersensory perceptions (to be placed in dimensions) and irrational or suprarational concepts (representing the dimensions within which perceptions are to be placed).

When one however tries to experience these intuitively inspired perceptions within intuitively inspired concepts a fresh conflict emerges. Again one realises that qualitatively horizontal perceptions contrast with vertical conceptions (i.e. they are linear and circular with respect to each other). Thus a new transformation takes place in the point level where understanding is seen as the relationship between the linear and the circular, or between the formal and the formless (i.e. between the rational and the irrational), which as we have seen represents what is mathematically transcendental.

We have the perfect complement in mathematics with Gelfond’s theorem telling us that any rational (or irrational) number raised to an irrational power is always transcendental. Again a mathematical result has a highly important complementary psychological significance.
 
 

There is yet another way of highlighting this fundamental nature of transcendental numbers i.e. as the relationship between (discrete) linear and (continuous) circular notions.

1 which is a straight line, symbolises the linear conscious level. This deals directly with discrete finite reality.

0 which is a circle, symbolises the (circular) unconscious level. This deals directly with continuous infinite reality.

. which is a point, symbolises the point level involving the (mathematically) transcendental connection between conscious and unconscious. This deals directly with the relationship between the discrete finite and continuous infinite reality.

I have already shown the beautiful way in which the linear and circular number systems are connected through a central point.

The very format of a transcendental number bears testimony to this relationship.

By definition a rational number can be expressed in discrete finite terms as a fraction involving the ratio of two integers.

An irrational number, cannot be expressed as a rational number in the 1st dimension, but can in terms of a higher dimension (or combination of dimensions).

A transcendental number cannot be expressed in rational form, either in the 1st or higher order dimensions. Thus a transcendental number such as p (i.e. 3.14159 ... ) inevitably involves two components. On the left hand side we have a finite discrete number corresponding to (rational) linear notions. On the right hand side, we have by contrast a continuous number - with infinite extension - corresponding to (irrational) circular notions. Separating both number components we have literally a (decimal) point.

So the very structure of a transcendental number bears testimony as to its inherent nature.
 
 

There is yet another fascinating connection here. I have already defined the three basic personality types. The horizontal (linear) type operates largely out of the conscious mind adopting the rational paradigm for interpreting reality. The vertical (circular) type operates largely out of the unconscious adopting the irrational (i.e. holistic intuitive) paradigm. The diagonal type is mainly concerned with the interaction of conscious and unconscious and thereby combines both line and circle (i.e. rational and irrational paradigms).

As is well known Cantor used a diagonal line argument for proving the existence of

transcendental numbers. Likewise, one who experiences reality at the level of transcendental structures of the point level is the diagonal personality type.
 
 

One unexpected connection with the above relates to Fermat’s Last Theorem. It has now been proven mathematically that this famous conjecture in fact is true. Thus if a, b and c are positive integers (> 0), then if n is an integer > 2

then aⁿ + bⁿ ¹ cⁿ.

Clearly where n = 1, this relationship will always hold. Two integers can always be added to give a third integer.

When n = 2, this will hold for a wide range of integers. Thus for example if a = 3 and

b = 4, then c = 5. However for an even greater range of integers for a and b, c will be irrational.

Psychologically this can be explained by saying that the 2nd dimension though intuitive and irrational (in terms of the 1st dimension) is yet rational in higher level dimensions. Thus Ö 2 which is irrational (in 1st dimension) and is rational in the 2nd is the mathematical corollary.

Mathematically, the square root of any number can be represented in real terms.

However the 3rd dimension and all higher dimensions are problematic in that they involve the interaction of both real and imaginary notions and cannot be solely interpreted in terms of real rational notions. In mathematical terms when we extract the 3rd root or any higher root of an integer, one or at most two roots will be rational (with all other complex). In complementary psychological terms the "real" roots - as I have illustrated in the case of the 3rd dimension - have a direct spiritual interpretation (i.e. the transcendent and/or immanent poles of spiritual union). The "complex" solutions relate to interpretation of phenomenal reality.

Combining two integers (raised to the power of 3 or higher) and reducing result to 1st dimension (i.e. obtaining its root), is the mathematical equivalent to forming perceptions that are inherently dynamic and unstable at the point level (or higher) - and then attempting to express results in solely "real" and "rational" terms. It cannot be done. As we have seen once experience enters the 3rd dimension (i.e. the point level) or higher, reality is interpreted in "complex" rather than "real" terms.

By concentrating solely on the real positive valued roots of integers, in my opinion, mathematicians have consistently missed the essential factor explaining Fermat’s Last Theorem. This is that that roots of numbers of degree three or higher inevitably involve complex solutions.
 
 

Self Generating Numbers
 

I will complete this section with an interesting demonstration using numbers which again highlights the archetypal significance of the number "3" in relation to transcendental structures (i.e. involving the relationship between horizontal line and vertical circular notions).

Self generating numbers can be either linear or circular (i.e. cyclic).

The linear format comprises two sub types which I call hierarchical and non-hierarchical respectively.

495 is a well known example of a hierarchical self generating number (in base 10). This means that when we arrange its digits in descending and ascending order of magnitude respectively and obtain the difference, we obtain the original number.

So 954 - 459 = 495

275 is an example of a non-hierarchical self generating number (in base 8). This time when we obtain the difference of a number and its reverse (without ordering), we again obtain the original number

So 572 - 275 = 275.

The most basic example of this type involves 01 (in base 2). When we subtract this from its reverse (i.e. 10) we obtain 01. 10 in binary terms represents the unconscious and 01 the conscious. We can literally see how in binary terms how conscious and unconscious are mirrors of each other. The binary system is therefore the most appropriate means of modelling this psychological reality.

There do not appear to be any linear self generating numbers of this second non-hierarchical type in the denary system.
 
 

All linear self generating numbers as they involve the original numbers and their reverses complement the dynamic interaction psychologically, of structures and their mirror structures.

We also have self generating numbers of a cyclic or circular kind. Again we can distinguish two types involving differing and similar orders respectively.

142857 (in base 10) is an example of the 1st type. If we subtract this from the number (starting with 7) following the cyclic sequence of digits, we obtain a number with a different number with however the same cyclic order of digits.

So 714285 - 142857 = 571428 (i.e. we have generated a different number with same cyclic order of digits).

The above cyclic number is intimately related to the reciprocal of the prime number 7 (i.e. 1/7) and comprises its unique sequence of recurring digits.

Such cyclic numbers are directly associated with prime number reciprocals. They represent an interesting form of mathematical complementarity. A prime number such as 7¹ (with no factors other than itself and 1), represents the most "masculine" and linear of numbers. Its reciprocal which is 7^-1 (which involves the changing of sign or direction of its dimension) represents by contrast the other extreme of the most "feminine" and circular of numbers.

1463 (in base 8) is an example of the second type of cyclic self generating number.

When we subtract 1463 from its "cyclic" reverse (3146) we obtain 1463 which is the same number.
 
 

A particularly interesting form of self generating number involves just two digits. This eliminates linear hierarchical and cyclical order distinctions as between numbers. Also by definition if a two digit number is a linear self generating number it is also a cyclic self generating number.

We have already seen the archetypal example of this number in the binary system.

Thus 01 is both a linear and cyclic self generating number. When subtracted from its reverse number - where digits are ordered either in linear or cyclic form - the same original number results (i.e. 10 - 01 = 01).

Thus this interaction of conscious and unconscious (i.e. linear and circular approaches) is perfectly complemented in terms of its corresponding quantitative symbols.

Further self generating numbers comprising two digits arise in number bases that increase in steps of 3 (i.e. in bases 5, 8, 11, 14, 17 20 etc.) More formally if x represents a number base divisible by 3, then a two digit self generating number exists in base x - 1. The 2nd digit will be (x + 1)/2. If y = (x + 1)/2, then the 1st digit will be

(y - 1)/2.

Thus, for example 9 is divisible by 3. Therefore a two digit self generating number exists in base 8. The 2nd digit is (9 + 1)/2 = 5. The 1st digit is then (5 - 1)/2 = 2.

Therefore the two digit self generating number in base 8 is 25. When we subtract this from 52 we once again generate 25.

The self generating numbers (which by definition are both linear and circular), in first 6 bases 5, 8, 11, 14, 17 and 20 are therefore 13, 25, 37, 49, 5B and 6D respectively.

In all these bases - and indeed all other bases where the numbers occur - the numbers concerned are intimately associated with the number 3. In all these bases 3 is a cyclic prime and the unique two digit sequence of its reciprocal is the self generating number in question. (25 for example is the unique digit sequence relating to the cyclic prime 3 in base 8).
 

Self generating numbers while being interesting in their own right, have considerable psychological significance serving as archetypes of integration. By being combined with their reverses, they resemble the dynamic psychic interaction of structures with mirror structures. Like a healthy personality they still maintain their original identity when subjected to such change.