We have see in the last contribution that when the quantitative to qualitative relationship is maintained that raising a rational number to a rational fraction results in an irrational quantity.
So for example in the best known case, the square root of 2 i.e. 2^(1/2) = 1 .4142... is an irrational number.
In corresponding fashion when rational perceptions are dynamically related to rational concepts that are fuelled by appropriate intuitive appreciation of a qualitative nature (algebraic) irrational perception (and later conceptualisation) results. Expressed in more common language this entails the more refined appreciation of phenomena that are inherently paradoxical in nature.
The very nature of an irrational number is that it necessarily combines both finite and infinite aspects. Thus an irrational number can be approximated in rational terms to any required degree of accuracy. However equally it possesses an elusive infinite aspect in that its exact value can never be known.
Likewise in corresponding qualitative terms, irrational understanding with respect to perceptions and concepts (in what is sometimes referred to as the psychic/subtle realm) combines both finite and infinite aspects. Thus phenomena in experience still possess a distinct identity (of a dynamic relative nature). However equally they possess a numinous spiritual quality that is infinite in nature.
As always we can identify - though in truth considerable overlap may be involved - three stages at this level.
Firstly we have the unfolding mainly of the more superficial refined phenomena that are qualitatively irrational in nature. Now once again the very basis of rational understanding is that interpretation appears - especially in a mathematical context - unambiguous in nature. Thus for example the positing therefore of what is true, implies the corresponding negation of its opposite as false.
However in the very dynamics of understanding at this level, phenomena that are consciously posited in experience are quickly negated (in an unconscious manner), Thus propositions take on a merely relative i.e. paradoxical truth value.
The second stage then entails deeper conceptual structures of understanding that are also irrational (in qualitative terms).
Finally at the most advanced stage we have the growing interaction of both irrational perceptions and irrational concepts paving the way for a remarkable transformation of experience to a new level.
When the mathematician Hilbert detailed 23 unproven propositions at his famous address in 1900, one related to finding a proof that any rational (or irrational) number when raised to an irrational dimension (power) would result in a transcendental number quantity!
A transcendental number resembles an even more subtle form of irrational number.
For example the the square root of 2 and the well known constant pi are both irrational numbers. However whereas the former - and indeed any number of this type - can be expressed as the solution to a polynomial equation, the latter cannot be expressed in this manner.
There is a fascinating corollary in qualitative terms. Paradoxical (2-dimensional) understanding based on the complementarity of opposites is clearly paradoxical in terms of conventional linear (1-dimensional) reason that is unambiguous in nature.
However if we define reason in terms of the former (2-dimensional) variety then from this perspective it is now rational. Indeed Hegel did precisely this in his writings defining reason in terms of his dialectic while treating conventional logic as a "lower" form that he termed "understanding".
The trouble is that Hegel then effectively reduced the nature of such dialectical reason through his failure to emphasise the corresponding need for the necessary supporting intuition provided through authentic contemplation.
This in turn is a regular failing at the irrational (psychic/subtle) level where secondary rigid attachments to the paradoxical symbols in experience emerge.
The resolution of this problem requires the profound negation of such attachment. In this way one gradually develops the ability to preserve an increasingly harmonious balance as between (conscious) reason and (unconscious) intuition.
Put another way this implies maintaining an appropriate relationship (that is quantitative as to qualitative) as between both the paradoxical perceptions and concepts that typify the level.
It was eventually proven in 1934 that when a rational (or irrational) number is raised to an irrational power that a transcendental number quantity results. And remember this was one of Hilbert's 23 propositions!
Remarkably we can provide the qualitative corollary to this proposition by saying that when rational (or irrational) perceptions are appropriately related to irrational concepts that a transformation in understanding takes place whereby experience of a transcendental nature emerges. And this is the important transformation that enables successful transition from the psychic/subtle to the causal level.
Now, we can understand the true nature of a transcendental number with respect to the nature of pi, which represents the pure relationship of the circular circumference to its line diameter. In like manner transcendental understanding (which typifies the causal realm) represents the pure relationship between circular appreciation (that is paradoxical) and rational understanding (of a linear nature). In other words it points directly to the common relationship as between both.
Now the center of a circle equally represents the midpoint of its line diameter. In like manner it is through the still point of being (representing the naked will through pure volitional intent) that both circular and linear type appreciation are reconciled. In this way the transcendental structures properly evolve. Because this entails approximating ever closer to this still point of being (in both physical and psychological terms) I have always referred to the causal level in holistic mathematical terms as the point level!
In terms of development of such structures the most refined possible are of an imaginary - rather than real - nature.
A real transcendental perception (using holistic mathematical language in a precise manner) relates to a consciousness of a specific phenomenon as representing the refined interaction of both conscious and unconscious aspects of experience (with both operating in close harmony).
A real transcendental concept then represents corresponding conscious experience of general universal categories as again representing the refined harmonious interaction of both conscious and unconscious aspects of experience.
However an imaginary transcendental perception is even more elusive as representing the indirect recognition of a projection emanating from the unconscious where again both conscious and unconscious aspects of recognition with respect to its temporary phenomenal identity are maintained in close harmony. And then finally an imaginary transcendental concept would entail the corresponding recognition with respect to indirectly projected universal categories of experience. In other words when conscious and unconscious aspects of recognition become so closely related in experience so as to approximate simultaneous identity, then - by definition - remaining involuntary attachment to phenomena largely ceases.
Some 20 years ago when I wrote the "Number Paradigms" I recognised in holistic mathematical terms that the most refined conceptual structures possible in experience are - in holistic mathematical terms - of an imaginary transcendental nature and that these typify the most advanced stage of the causal level (approaching pure spiritual union).
It was only later that I was able to properly make the connection as between such understanding and the Euler Identity and realise its deeper significance.