The approach of mathematics has subsequently developed in a highly reduced fashion. Many important types of mathematical quantities have been discovered which though they are not rational (i.e. irrational, imaginary, transfinite) are interpreted, in a highly reduced fashion, within a non-matching qualitative paradigm.
 
 

The rational paradigm is the product of the linear level. However just as the linear level is only one of several levels of understanding, the rational paradigm is only one of several possible paradigms.

In my "Transforming Voyage", I outlined in detail these various levels.

In general terms the circular level involves a translation as between intuitive and rational means of communication. This involves the attempt to express an inherently two-dimensional mode (the unity of complementary opposites) in reduced one dimensional format (the separation of opposites).

This inevitably leads to paradox. In other words when expressed in rational format, intuitive understanding seems deeply paradoxical. Indeed, from a rational point of view, intuition is irrational.
 
 

The whole process of mathematically extracting the square root of 2 reveals exactly complementary difficulties. Here there is the translation of a unified integer from a two dimensional mode to a reduced one dimensional format. Once more this results in the separation of poles, in obtaining both positive and negative roots, and the result from a rational perspective is deeply paradoxical (i.e. we get an irrational quantity).

The circular level is the correct psychological basis for the interpretation of irrational numbers. The circular level involves the mixing of intuitive (continuous) and rational (discrete) modes of understanding. An irrational number also involves mixing of discrete and continuous notions of understanding. The paradox of an irrational number is that despite being in one sense a finite quantity, its value extends indefinitely and is thus from this perspective infinite.
 
 
 
 

CONCRETE STAGES

(A) Irrational (Positive)
 
 

The supersensory structures (of the circular level) involves interpreting reality in the light of an illuminated spiritual vision. The local phenomena of the senses increasingly serve as archetypes of a global interconnected reality. In terms of the former (rational) understanding this latter (intuitive) understanding is deeply paradoxical. Absolute distinctions (appropriate to the separated poles of the rational format) break down to be replaced by a relative vision (appropriate to the dynamically complementary poles of intuition).

In other words, a higher form of subtle perception emerges, which in terms of the previous gross perception is irrational.
 
 

The positive concrete irrational number (i.e. Ö2) is the complementary mathematical quantity to this psychological quality (or paradigm).

It is highly interesting that Ö2 can be written as 2^1/2. In other words it represents a quantity raised to a (positive) fractional or rational dimension. By extension all irrational numbers (algebraic) represent numbers raised to fractional dimensions.

Thus we can in complementary terms describe the stage of supersensory understanding as experience of reality that takes place in fractional dimensions. In other words the conventional natural model (of 3 dimensions of space and 1 time) is at this level entirely inappropriate. Rather we have at this supersensory stage a unified (intuitive) space and (intuitive) time dimension with - in reduced rational format - twin component poles, as dimensions which are positive and negative respectively. Each dimension (in this reduced rational format) is therefore half of the unified intuitively understood dimension.
 
 
 

(B) Irrational (Negative)

The stage of supersensory mirror structures is relevant here. This simply puts a negative direction to the understanding of the previous stage.

The corresponding mathematical quantity is the negative irrational number,

(-Ö2 = -2^1/2). In other words we have here a negative quantity in a fractional dimension.

In complementary terms, phenomena during this stage, as well as taking place in fractional dimensions are experienced as moving - in relative terms - backwards in space and time. This is why subtle phenomena are literally negated, or erased from experience, as they can only be posited consciously in forward moving space and time.
 
 
 
 

FORMAL STAGES

(A) Irrational (Positive)
 
 

The suprarational stage is characterised by high level spiritual illumination at an intellectual level, deeply transforming fundamental conceptual experience. The attempt to translate this intuitive experience - at a reduced rational - level can only be achieved once more through polarisation and paradox. This time all concepts - in reduced rational format - are irrational.
 
 

The corresponding mathematical equivalent is the interpretation of a rational two dimensional quantity in reduced one dimensional format (this time considered from a conceptual basis).

Experience of numbers always involves the specific number perception in relation to the corresponding general number concept (dimension). To posit a specific concrete number, we hold the general number concept constant. Likewise to posit a general number concept we hold the specific number perception constant.

Mathematically, concrete specific definition of any number involves the horizontal format of expression of a quantity in a given (i.e. 1st) dimension.

The formal general definition of any number involves the vertical format of expression of a quality or dimension with respect to a given number.

Thus the concrete specific definition of the square root of two is (Ö2)¹. Here we are understanding the specific number perception (Ö2) while holding the concept to which it relates constant.

The formal general definition of the square root of two is 1^Ö2. Here we are positing the number concept (or dimension) to which Ö2 relates, while holding the number perception constant.

Therefore the complementary mathematical interpretation of the suprarational stage is that it involves experience in a (positive) irrational dimension.

Thus whereas experience at the concrete supersensory stages takes place in fractional dimensions (positive and negative), experience at the formal suprarational stages takes place in irrational dimensions. Thus an appropriate understanding of physical reality at this stage must involve a corresponding understanding of dimensions as irrational.

There is a mistaken view that many important mathematical ideas have an abstract rather than a practical physical validity. Thus we may entertain the notion of negative, fractional and now irrational quantities and powers (or dimensions) as mathematically valid notions we do not believe that these have practical relevance as a description of physical reality.

However - correctly understood - the correct notion of the nature of objects and dimensions is mathematical and all of these possible mathematical interpretations are replicated in reality.

Properly understood the understanding of number is always dynamic being as much psychological as well as mathematical. Thus there is a corresponding psychological and physical explanation - as I am demonstrating - for every mathematical interpretation of reality. However the appreciation of all this requires greatly transcending merely rational modes of understanding.
 
 
 
 

(B) Irrational (Negative)
 
 

Again, the suprarational mirror stage relates to experience of negative irrational dimensions. Psychologically the highly intuitive (irrational) concepts of understanding are greatly negated or eroded during this stage.
 
 

Thus there is a very definite progression through the circular level.

Firstly we have (relative) irrational understanding of objects in a positive direction, where intuitively based perceptions are posited in experience. Here experience is in fractional dimensions (supersensory stage).

The mathematical counterpart of this is (Ö2)¹.
 
 

Next we have (relative) irrational understanding of objects in a negative direction, where intuitively based perceptions are negated or eroded in experience. Again experience is in fractional dimensions (supersensory mirror stage).

The mathematical counterpart of this is -( Ö2)¹.
 
 

Then we have (relative) irrational understanding of dimensions in a positive direction, where intuitively based concepts are posited in experience. Perceptive development now is dormant (suprarational stage).

The mathematical counterpart of this is 1 ^Ö2.
 
 

Finally we have (relative) irrational understanding of dimensions in a negative direction, where intuitively based concepts are now negated or eroded. Perception is again dormant (suprarational mirror stage).

The mathematical counterpart of this is 1 ^{- Ö2}.