Formal Rational: External
There is a fundamental distinction as between perceptions and concepts which is rarely appreciated.
Experience necessarily entails a dynamic interaction of (specific) perceptions with (holistic) concepts. This is true of all scientific and mathematical data. For example we cannot meaningfully understand a specific number perception (e.g. 2) without relating it to the corresponding concept of number.
In conventional rational understanding concepts are reduced to perceptions with their fundamental qualitative difference thereby overlooked.
Let us illustrate this point carefully with respect to the understanding of a number.
The perception of a number represents a specific number datum or content (e.g. 2).
The concept of number by contrast represents the general class or form of number.
The meaningful understanding of a number therefore necessarily entails this dynamic interaction of content and form (i.e. of quantity and quality).
However a merely reduced quantitative appreciation of number exists in conventional mathematics.
To recognise a specific finite number phenomenon (e.g. 2) it must be rationally posited in consciousness (and thereby differentiated from other finite number phenomena).
To move to the general concept of number the specific number phenomenon must be dynamically negated in consciousness (enabling integration with other numbers). It is this fusion of positive and negative poles which causes a crucial switch from the actual finite understanding of number as a specific quantity to the potential infinite understanding of number as a general quality. Thus correctly understood the number concept - though related potentially to all number phenomena - initially does not relate actually to any specific phenomenon.
However in conventional rational understanding this potential qualitative notion of concept (which is inherently spiritual and intuitive) quickly collapses as it were to a reduced actual quantitative notion. In other words the number concept now is simply interpreted as relating to all actual number phenomena.
In this way what is a dynamic interactive process involving both actual conscious and potential unconscious understanding is interpreted in reduced static terms as being merely conscious. Alternatively what inherently involves the subtle interaction of reason and intuition is explained as merely rational.
Thus there is both horizontal and vertical aspects to all understanding. Whenever one switches from perception to concept (from content to form), understanding necessarily switches as between the extensive rational horizontal and the intensive intuitive vertical aspects of understanding. However once again this crucial vertical aspect of understanding is entirely missing in conventional rational understanding.
When one interprets reality from the flatland of the reduced (solely) horizontal perspective, understanding becomes literally linear (one dimensional).
This is well illustrated for example in mathematics by the quantitative number system. If we use numbers such as 2 or 2,000 it is implied that these number are one dimensional (i.e. raised to the power of 1). Thus both numbers - strictly speaking - should be written 21 and 20001. Whenever a number is initially expressed with respect to another dimension e.g. 22, its value ultimately will be given in (reduced) one dimensional terms (i.e. 41 or more simply 4).
However, this equally applies to all concepts used in (conventional) science.
If for example in physics we can have many different particle phenomena. However it is assumed that all these relate to the one concept of particle.
Thus - in this view - though we can have many partial phenomena (as quantities), we have only one holistic phenomenon (as quality) which is the concept of particle.
In other words the concept of particle - and by extension all concepts in physics - are understood in strictly one dimensional terms.
Just as number quantities in mathematics are defined with respect to the 1st dimension, likewise all phenomenal quantities in science are defined with respect to an invariant 1st dimension (i.e. the general class to which these phenomena belong).
This is fundamental. The objects of experience are (directly) given by perceptions. The dimensions in experience are (directly) given by concepts.
When we interpret concepts in one dimensional terms physical reality not surprisingly appears as one dimensional. The three space dimensions are essentially measurements we apply to objects (i.e. quantities). Time is the remaining qualitative aspect of experience and universally interpreted as one dimensional.
Our physical view of reality therefore mirrors the limited and reduced manner in which we rationally interpret reality.
Revolutionary New World-view
When one gives equal consideration to the qualitative - as well as quantitative - aspects of experience, the possibilities for a remarkable new world-view in terms of both physical and psychological reality becomes possible.
This is where the qualitative vertical number system is of immense value. While going greatly beyond mathematics - as conventionally understood - paradoxically it greatly enhances the applicability of mathematics to reality. When correctly appreciated, one realises that whatever is possible mathematically is actually replicated - at a certain level of experience - in the real world (physical and psychological). In this "new" world both physical and psychological aspects are dynamically complementary with an underlying structure which is mathematical.
We have seen that corresponding to the quantitative number system
11, 21,31, 41…… is a complementary qualitative number system
11, 12, 13, 14……
I now wish to give this second qualitative system its dynamic interpretation in the context of experience.
Conventional experience freezes interaction by keeping the concept fixed. All perceptions of a certain class are interpreted absolutely within that (same) conceptual class. Thus psychologically - as we have seen - all particles for example are interpreted within the same concept of "particle". Likewise physically all objects are interpreted within the same dimension of time.
However in dynamic terms perception and concept are equal partners in dynamic interaction.
Thus the perception of the number 1, has a dynamic interaction with the number concept (to which 1 is related).
The relationship works both ways.
We have the (quantitative) perception of the number 1 (in relation to the concept of number). This is 11.
We have also the (qualitative) concept of number (to which the number "1" is related. This is also 11.
Next we have the (quantitative) perception of the number 2 (in relation to the concept of number). This is 21.
In reverse fashion we have the (qualitative) concept of number (to which the number "2" is related). This is 12.
In like manner we can generate complementary numbers for 3, 4 etc.
What this really entails is that each number perception - in dynamic terms - has a unique relationship with the corresponding number concept (and the concept in turn with the perception). For each number perception (quantitative) there is a corresponding unique number concept (qualitative).
By extension all quantitative phenomena in mathematics and physics have unique relationships with corresponding qualitative conceptual classes. Thus each particle phenomenon - in dynamic terms - is related to a unique particle class.
Amazingly, when correctly appreciated we live in a multidimensional reality where there are as many (qualitative) dimensions as (quantitative) phenomena.
This viewpoint - though not usually articulated in this fashion - is the illumined mystical vision where heightened intuitive awareness is maintained in harmonious balance with rational understanding.
Dynamics of Development
Whereas the concrete rational stages are correctly represented in terms of the quantitative number system, the formal rational stages - relating to conceptual understanding are more correctly represented by the corresponding qualitative system.
We studied in detail - at the concrete stage - the breaking of a cake into 4 (quantitative) slices is represented by ¼, i.e. (¼)1.
The corresponding breaking of the concept of cake into four (qualitative) sub concepts is represented by 11/4 .
What is remarkable about this is that even this simple conceptual analysis involves operating in fractional dimensions.
However because of reductionism this is not appreciated. Conventionally, all understanding - both physical and psychological - is interpreted within one qualitative dimension of time.
However as we have already seen, objective and subjective experience involve - what in dynamic relative terms - are positive and negative dimensions.
Now we see that conceptual analysis involves operating in what literally are fractional dimensions. As we will see all the other options permitted by the number system in terms of irrational, imaginary, complex, transfinite numbers are actually replicated at appropriate levels of experience.
As before this stage of development has both affective and cognitive aspects.
Normally with conscious development, affective sense development precedes cognitive rational development.
The affective phase here would relate to a more holistic capacity in terms of general sense involvement with reality. One is now more able to screen out immediate attachment to objects to obtain an overall impression of the environment. However as the thrust of development becomes increasingly rational, conscious affective experience is subject to much repression becoming less spontaneous.
This often leads to the release of the primitive unconscious through fantasy in an attempt to compensate.
The cognitive phase relates of course to a more holistic mental capacity, whereby one can increasingly reason using abstract mental objects. Pure logic and mathematics best exemplify this analytical ability.