(A) Concrete Stages
 

Positive:  The Square Root of 2
 

We have looked at the dilemma of the Pythagoreans in discovering that the Ö2 is irrational. They rightly saw that there was a lack of correspondence between their qualitative paradigm which was rational and the number system (which now extended to irrational quantities). From a psycho-mathematical approach (which requires complementarity), this was unacceptable.

Needless to say, there are much philosophical riches contained in the notion of a square root. This is where the distinction as between the horizontal approach of numbers representing quantities (within a given dimension), and the vertical approach of numbers representing dimensions or qualities (to which a given number is raised) is so valuable.

I have already characterised conscious (rational) understanding as one dimensional and unconscious (intuitive) understanding as two dimensional. The former is based solely on the explicit recognition of one pole which is in the positive direction. The latter - expressed in reduced conscious one dimensional terms - is by contrast two -dimensional, based on poles in both positive and negative directions.
 
 

Now a natural number e.g. 2, when expressed in the first dimension (i.e. x¹ = 2) of course has only one sign (i.e. direction) which is positive.

However if the same natural number results from making x two dimensional (i.e. x² = 2), then the value of x - expressed in reduced one dimensional terms - has two separate poles (i.e. roots), + Ö2 and - Ö2 in positive and negative directions.

Likewise when unconscious two dimensional understanding (i.e. intuition) is expressed in reduced one dimensional terms there are two poles in both positive and negative directions.

Thus again we can see that there is a remarkable and important complementarity here as between psychological and mathematical understanding.

The proper mathematical understanding of Ö2 complements its vertical psychological equivalent (i.e. the attempted translation of intuition into rational terms).
 
 

The next point is, that as the Pythagoreans reluctantly discovered, these reduced mathematical roots (resulting from the conversion of two dimensional rational quantities into one dimensional terms), are themselves irrational.

Furthermore this will always be the case, where this conversion takes place. (I am not considering cases such as the square root of 4 which is rational, as 4 itself is a perfect square (i.e. x² = 2²) and therefore no dimensional conversion is in fact involved).
 
 

The psychological counterpart of this is most interesting. As we have seen when we convert the two dimensional intuitive format (of the circular level) into the one dimensional rational format (of the linear level), it appears - in terms of this reduced rational level - deeply paradoxical. This is because whereas the intuitive format is inherently bipolar - when expressed in reduced conscious form involving the complementarity of positive and negative directions - the rational mode is unipolar (involving the separation of these directions). Thus the attempt to communicate the intuitive bipolar format in terms of the rational unipolar format (which is inherently different), leads inevitably to paradox. As reduced one dimensional statements the intuitive format seems to contradict the very logic on which one dimensional (rational) understanding is based.
 
 

We could say quite accurately therefore, that the (two dimensional) intuitive format interpreted in reduced (one dimensional) rational format is irrational.

So therefore in terms of the one dimensional understanding of the linear level, the two dimensional understanding of the circular level is irrational. So, from this perspective we can refer to all the structures of the circular level as irrational.

Once again, I am at pains to point out the precise complementarity as between the mathematical notion of irrational quantities (or numbers) and the psychological notion of irrational qualities (or structures). This complementarity of both aspects i.e. quantitative and qualitative understanding - as the Pythagoreans realised so well - is necessary for a proper integrated psycho-mathematical understanding.

Thus we are able to say from the mathematical perspective not only that the Ö2 is irrational, but also why it is irrational. Equally we are able to say from the psychological perspective not only that the structures of the circular level are irrational but again also why they are irrational.
 
 

However it is to be said, in our mathematical quantitative case, from the two dimensional perspective (i.e. x² = 2) that 2 is a rational number. (It is only when x is reduced to one dimensional terms, through obtaining the square root, that the result becomes irrational).

Likewise it is equally true, in the psychological qualitative case - from the two dimensional perspective - that understanding of the circular level is also rational. In fact it could well be argued that this understanding on the basis of the complementarity of opposites, constitutes a higher form of reason, than understanding on the basis of the separation of opposites. (Indeed Hegel reserved the word "reason" for this two dimensional "higher" use, using "understanding" to refer to the one dimensional "lower" use).
 
 
 

Reason and Intuition: comparison
 

It would perhaps be profitable to stay with this a little longer, to clear up this difference between one dimensional and two dimensional rationality.

All understanding in fact involves an interaction of both the conscious (rational) and unconscious (intuitive) processes.

At the one dimensional level, where the unconscious process is undifferentiated, though intuition always provides the necessary insight to interpret rational connections, it remains merely implicit, resulting in a reduced form of understanding.

At the two dimensional level, where the unconscious process is now considerably differentiated, intuitive insight is made explicit, which in reduced form language (of the one dimensional level) involves paradox.

For those who can understand in terms of the second dimension, paradox is no problem. They are readily able to intuit (explicitly) its unreduced meaning and in this sense it is a very valuable "higher" rational means of communication.

For those however, who can only understand in terms of the first dimension, paradox will be largely unintelligible. They are only able to intuit (implicitly) and will try to interpret it explicitly solely in terms of reduced one dimensional understanding.

Needless to say, the quality of paradoxical communication - as with rational - can vary greatly. Basically, effective two dimensional communication, in any appropriate context, will succeed in effectively matching appropriate complementary poles. This will convey a knowledge of their holistic interdependence, that otherwise may not have been realised. By contrast, effective one dimensional communication will succeed in effectively separating opposite poles, facilitating in an appropriate context, analysis of a particular situation.

The two dimensional (intuitive) approach is thus essentially holistic and synthetic (establishing the interdependence of whole and parts).

The one dimensional (rational) approach is by contrast essentially fragmentary and analytical (establishing the separation of whole and parts).

As we have seen the understanding of this concrete supersensory (i.e. irrational) stage is relative. It is two-edged; on the one hand natural symbols are used; on the other hand their true meaning is archetypal transcending the merely finite. We have in other words a dynamic conjunction of finite and infinite realms involved.

Exactly the same pattern is also evident with irrational numbers. From one point of view an irrational number has a finite value. However from another point of view - as its true value can never be precisely ascertained - its value transcends the merely finite, so that again we have a conjunction of finite and infinite realms involved.

Indeed, properly understood an irrational number is relative, constituting a dynamic number relationship which is expressed in reduced form. Because it is relative, its true value can only be approximated. There is a definite element of paradox involved, in that though its value can be approximated, its true value can never be precisely known (i.e. in one dimensional terms).

If we takeÖ2, we can approximate its value (say to 4 decimal places), as 1.4142. However this is not its true value, which in fact can never be determined (in one dimensional terms). We can keep improve the accuracy of our approximation through extending the decimal digit sequence, but this process can never be exhausted.
 
 
 
 

Number System Redefined
 

This is a very important point and leads to a new definition of the real number system (i.e. from a relative two dimensional perspective).

Conventional one dimensional understanding always implicitly involves a confusion as between finite and infinite notions. The number system is expressed as a straight line which can be extended indefinitely. This is usually represented as involving infinite extension. In other words though individual members of the number class are finite, the overall class is defined as infinite. However, this is just a reduction of the infinite - which is qualitatively different - to the finite.

However, we can successfully get around this difficulty in the two dimensional definition. Here the number system is understood in dynamic two-directional terms, where every number has complementary positive and negative poles. When one pole is posited or made determinate, the other pole in relative terms is negated or made indeterminate. Thus every real number from this two dimensional perspective is both finitely determinate and finitely indeterminate. Its positing as an "actual" defined number involves the simultaneous negation (i.e. exclusion) of all other "potential" or as yet undefined numbers.

Thus determinacy and indeterminacy are always linked in relative terms.

All finite number are then - in strict terms - only relative, implying that their determinacy always involves other finite numbers which remain undetermined.

Now, we can keep increasing this finite number "set" at will in the determination of additional numbers. However, the process is inexhaustible, There will still always be a set of finitely indeterminate numbers remaining.

From this relative point of view the set of real numbers is finite. However it is like the universe itself, finite in an unbounded sense. It thus involves the dynamic relationship of finitely determinate and finitely indeterminate members. Indeed, we can therefore quite accurately conclude that there is an essential mystery implicit in the very nature of number and the number system.
 
 
 
 

Illuminated Vision
 

This mathematical finding in turn helps to clarify the worldview typical of the illuminated vision of the circular level.

At the linear level, there is a tendency to believe that the world exists out there (in its entirety) independent of mind (i.e. that it has an objective existence).

However, at the circular level there is always an essential mystery present in experience.

The world now has no meaningful existence independent of mind. Rather it is brought into creation through a dynamic two-way mind-matter interaction. The positing of actual (defined) phenomena in experience always involves the negation of potential (undefined) phenomena. Experience thus is always open-ended, where growth in understanding is accompanied by a sense of awe and reverence and ultimate mystery. The more one actually discovers, by the same process, the more one realises, how much more potentially remains undiscovered.

This is a long way from that closed reduced one dimensional linear view of modern science for example, where many entertain the futile belief of finally unlocking nature's secrets.
 
 
 

Negative
 

Fractional Dimensions
 

This corresponds to the second of our supersensory stages, where the direction of understanding - in relative terms - is negative. Therefore the mathematical counterpart of this would be the (concrete) negative irrational numbers.

Equally, in terms of our number terminology, we could refer to this concrete supersensory stage (which is characterised by subjective paradox) as the negative irrational stage.

There is another very important point that is worth developing.

The square root of a number (e.g. 2) can be expressed as 2^1/2.

Also, the negative direction of the root can be expressed as - 2^1/2.

In other words obtaining a square root involves raising a number to a fractional dimension.

The psychological counterpart of this, is that the process of reductionism (i.e. from two dimensional intuitive, to one dimensional rational format also involves experience of fractional dimensions).
 
 

We can perhaps appreciate now, how limited is the conventional understanding of dimensions. Here there are considered to be four dimensions (3 of space and 1 of time). There are two observations I will make on this view.

a) The dimensions are considered to be positive (i.e. movement taking place in one direction only). with integer values (i.e. we do not think of experience as taking place in fractional dimensions).

b) There is a lack of symmetry in the view which allocates three dimensions to space and only one to time).

The very fact that this conventional view seems so obvious and in keeping with normal experience, really only indicates how deeply established is this reductionist perspective.
 
 

By carefully outlining the (vertical) psychological structures of understanding and showing in each case the (horizontal) mathematical equivalent (in the form of number definitions), we can appreciate that the true nature of dimensional experience is far more subtle and complex.

Thus we have positive dimensions and negative dimensions (even in the context of the linear level of understanding). I have already outlined how the (formal) rational stages implicitly involve experience of negative dimensions.

Now, these (concrete) supersensory states of the circular level, involve experience of fractional dimensions. The fraction involved is 1/2. What this simply means is that the unit of experience appropriate to the two dimensional intuitive perspective of the circular level is expressed in the reduced terms of the one dimensional rational perspective.

(Intuition by its own terms forms a dynamic unity. It is only two dimensional in the rational language of the linear level. So what is involved is splitting the dynamic dimensional unit into two halves, which gives us fractional dimensions).
 
 
 
 

Scientific Significance

What I am saying is of direct scientific as well as psychological relevance. This illustrates, how matter (i.e. phenomena) and dimensions are created out of continual directional (i.e. objective and subjective) and process (i.e. conscious and unconscious) interaction. It is exactly complementary in the physical world with directional (i.e. external and internal) and process interaction (i.e. independence and interdependence).

Thus an (unreduced) scientific world view should include - as in mathematics - notions of negative as well as positive dimensions, and fractional as well as whole dimensions. And this, as we shall see, is just the beginning of a comprehensive worldview.
 
 
 

Sequence of Development
 

By using the mathematical language of number types to describe the various stages, an extremely interesting pattern emerges, which can be expressed as follows.

The horizontal understanding of one stage grouping in development becomes the vertical understanding of the next stage grouping.

Thus going back to the linear level, we have positive development of phenomena

(i.e. natural perceptions), then positive development of dimensions (i.e. natural concepts), then (fractional) development of phenomena (i.e. rational perception) and (fractional) development of dimensions (i.e. rational concepts). Now this is followed by irrational development of phenomena (i.e. supersensory perception) which - in rational terms - is paradoxical. This suggests that the next grouping of stages of the circular level (i.e. suprarational) involves conceptual experience in irrational dimensions.

I will return to this later showing how it is intimately linked with a famous mathematical proposition.

There is a most important point which needs addressing at this stage. Though in theory - from the mathematical perspective - we can use higher dimensions (than 2), and consequently obtain higher order roots, such operations are qualitatively different than the two dimensional case, and cannot be given a precise psychological significance at this stage.
 
 
 

Significance of Three or More Dimensions
 

For example, if we obtain the cube root of any positive real number, three roots will be involved. However only one of these is real (the other two being complex).

In other words we cannot successfully reduce a real number from three dimensions to one dimension in solely real terms.

(There is a remarkable psychological counterpart to the three dimensional case. The circular level which we are now dealing with, is inherently two dimensional).

Also if we deal with real numbers in positive integral dimensions of size n ( where n is greater than 3, and attempt to reduce to the 1st dimension (i.e. obtain the nth root), once more the result will not be expressible in solely real terms. In other words, whereas there will be one or two real roots, depending on whether the root is odd or even respectively, the remaining roots will be complex involving imaginary values.

Thus only first and second roots of real numbers can be expressed solely in real terms.

Third and all higher roots always involve complex values.

There is an important link here with Fermat’s Last Theorem, which I will return to, where I will suggest a sound psycho-mathematical reason for its validity.
 
 
 
 

(B) Formal Stages
 

Positive
 
 

Intuitively Inspired Concepts
 

This circular stage of high intuitive illumination (i.e. the positive suprarational) corresponds with the linear stage (i.e. the positive rational) insofar as conceptual development is concerned. However whereas at the earlier linear stage, intuition remained purely implicit, at the later stage it is made explicit. Thus, expressed in reduced one dimensional terms, all concepts of the suprarational stage are profoundly paradoxical, involving the complementarity of opposite polar directions. Now as we have already seen, psychologically, dimensional experience is created through concept formation. Thus whereas earlier (in the linear stage), rational dimensions are created through rational concept formation, in this suprarational stage irrational dimensions are created through irrational concept formation.

In other words, expressed in reduced one dimensional linear terms, the intuitively based concepts of the two dimensional circular level are paradoxical and thereby irrational.
 
 

Concepts are essentially the vertical counterparts of horizontal perceptions.

Higher level perceptions develop with initially lower level concepts remaining intact. Thus at the linear level, natural perception takes place out of prime dimensions. In other words natural perception (horizontal) precedes corresponding concept formation (vertical) which remains primitive. Natural concepts are developed to dynamically interact with these perceptions. Then, rational perception (i.e. concrete operational thinking) precedes corresponding rational concept development. Later with formal operational thinking, rational concepts are now developed enabling greater dynamic interaction with earlier perceptions.
 
 

This pattern is again repeated at the circular level. Irrational perception (i.e. the supersensory stages) precedes irrational concept development. Now at these suprarational stages irrational concept development takes place.

Putting it in mathematical terms, we first have irrational movement - the supersensory stages - with respect to rational dimensions. (For example Einstein’s Theory of Relativity is a perfect example of this kind of understanding).

However we now have the corresponding irrational dimensions - the suprarational stages - with respect to rational movement. In other words movement of phenomena now takes place in irrational dimensions.
 
 
 

Irrational Dimensions
 

We have already dealt Ö2 as the appropriate example of an irrational number quantity. Now this represents the horizontal number perception. Indeed writing it fully we have
(Ö2)¹.

The corresponding irrational number quality or dimension represents the vertical number concept. This could be written as 1^Ö2 which is the positive direction of the (irrational) concept to which Ö2 relates.

Once again, we can see, how actual understanding is so inadequately expressed in terms of the conventional scientific worldview. The essence of this suprarational stage is that one actually experiences reality in irrational dimensions.

What this entails is that the there is a dynamic two-way relative understanding of all concepts. There is therefore an interaction involved of external concept (to internal mind) which is the positive direction, and internal mind (to external concept) which is the negative direction.

Thus the number concept expressed in reduced one dimensional linear terms (in direction) is both positive and negative simultaneously, which by the logic of this level, is irrational. Of course in terms of unified two-way intuitive understanding, this understanding is highly rational.
 
 
 

Negative
 

Mirror Development
 

This corresponds to the mirror suprarational structures, where previous positive structures are negated.

In this context it would mean, therefore the understanding in negative irrational structures. This could be represented as - 1^Ö2, which is the negative direction of the (irrational) concept to which Ö2 relates.

Thus we can see that understanding goes from positive irrational movement in rational dimensions to negative irrational movement (the supersensory stages), and then to positive irrational dimensions followed by negative irrational dimensions (the suprarational stages).

During the circular level, specialised understanding of each stage is developed separately as it were. However the irrational (i.e. relative) nature of movement and of dimensions is not yet combined simultaneously. This form of interaction, leads to a further transformation in understanding which leads into the point level.