3. Domain of the Rational

3. Domain of the Rational

Lower Middle Band

The Lower Middle spans the stages between the Lower and Higher Bands. From one perspective it represents both the ceiling of the Lower Levels and from another the floor of the Higher Levels.

We now move towards the specialisation of linear type understanding where phenomenal form is considerably abstracted (especially where cognitive understanding is concerned) from spiritual emptiness. So in holistic mathematical terms, 1 (as inherent in all form) is clearly differentiated from 0 (as inherent in emptiness).

Though the original meaning of 0 is of a deeply qualitative spiritual nature, through linear understanding it is given a reduced merely quantitative meaning (as the absence of form).

However such linear understanding is still of immense value when used with respect to its appropriate band of understanding. For example information technology depends on the binary digital system where - potentially - all information can be encoded in terms of the two digits 1 and 0.

L0 (LM1) - Fractions and Rational Understanding

As we have seen the previous level concluded with the ability to (implicitly) negate as well as (explicitly) posit phenomena. When properly understood in holistic mathematical terms, this then provides a remarkable insight into the true nature of rational understanding. 

L0 relates to localised understanding with respect to specific phenomena in what - in the well-known terminology of Piaget - is concrete operational understanding. For simplicity I refer to it as the concrete level.

We saw in the previous stage that linear understanding implies interpretation with respect to the (default) first dimension.

Initially in experience this dimension is merely posited in experience. However with growing stability with respect to phenomenal understanding, a child is now (implicitly) enabled to negate as well as posit this dimension.

Now this growing capacity has fascinating parallels in holistic number terms.

If we take the natural number 2, in linear terms this means that it implicitly is raised to the default dimension (i.e. power) of 1.

So here, 2 = 2¹.

However when this first dimension is negated we now have

2^-1 = ½.

In other words through raising the number to the negative (rather than positive) first dimension, we obtain the reciprocal (as a simple fraction) of the original number.

In experiential terms, this is deeply relevant to the dynamics through which the specialisation of analytic ability takes place.

Thus we have at this stage we see the growing tendency to operate on phenomenal perceptions (representing objects) and break them up into various parts (while maintaining their relationship to the original whole).

To illustrate let us take the - apparently - simple case where a cake is divided into two slices.

So in this context the original cake represents an identifiable whole (posited with respect to the conceptual - or dimensional - class to which it belongs). Each slice also represents (in terms of itself) another identifiable whole (again posited with respect to its own distinctive concept class).

However when (phenomenal) perceptions and concepts are related to each other the positing and negating of both aspects is required. So to switch from perception to concept, the perception must be negated; likewise to switch back from concept to perception, the concept must be negated. Also, again in this relational context, when one type of perception (and corresponding concept) is being negated the alternative type in each case is being posited. In this way one can keep relating the slices to the cake and the cake to the slices (with respect to both perceptual and conceptual recognition). What this in effect means is that each slice can be recognised as a whole (in its own localised terms) and also as a part with respect to the more general perception of the cake.

Likewise the cake can be recognised as a single whole in its own context while likewise comprising the sum of two wholes (now understood as parts) with respect to the individual slices.

The deeper implication of this is that actual experience necessarily takes place with respect to negative as well as positive dimensions (though of course we are confined with linear understanding to the 1^st dimension).

This can be accommodated to conventional understanding of 4 dimensions (3 of space, 1 of time) by recognising that the 3 dimensions of space essentially relate to the quantitative characteristics we associate with object perceptions. Thus an object perception is defined by 3 spatial dimensions (length, width and height).

So essentially with linear understanding we give a reduced quantitative interpretation to space.

Thus from a qualitative perspective, relating to time, linear experience is indeed of a one-dimensional nature.

Also we should not be surprised that this dimension can be negated in experience.

The idea that time moves positively forward in one-direction, is inconsistent with the spiritual awareness of a present moment in reality (that is continually renewed). So the very movement from phenomenal awareness of time moving forward to spiritual realisation of the present moment requires that the posited forward direction in experience be likewise negated. This in turn comes from a more subtle understanding of the complementary interaction of polar opposites so that the forward direction of time for (external) reality and the (internal) self - relatively speaking - takes place in opposite directions.  

Thus the scientific interpretation of the forward direction of time (i.e. where it is merely posited) arises from the - ultimately mistaken - view that (external) reality can be viewed as somehow independent of the (internal) observer. And of course such interpretation is a direct product of linear understanding! 

We will now proceed briefly through the three sub-levels of the concrete level of L0 (LM1).

At SL1 the emphasis is more on perceptions that can be recognised as (fractional) parts than on the whole which - in reverse manner - combines theses parts.

This represents the starting point of specialised rational ability where understanding is largely confined to superficial local type phenomena.

At SL2, a gradual switch is in evidence to the conceptual basis underlining phenomenal perception. In this way the child is better able to combine the various parts into coherent wholes. Though experience is still of a largely concrete nature, the more general conceptual context leads to a certain deepening of experience.

Finally at SL3, both aspects can interact in a more dynamic manner where wholes can be broken into (lower) parts and in turn parts organised into (higher) wholes.

Then because this activity entails considerable progress in the ability to (implicitly) negate localised perceptions (and concepts), experience becomes more refined with one capable of becoming increasingly detached from specific phenomena. In other words one can now move from the consideration of the specific to the much more general nature of phenomena which characterises the next level. 

L0,H0 (LM2) - Rational Dimensions

We now move on to the very important stage i.e. L0 H0. This is the centre of the lower levels and can be looked on as both the ceiling of the lower levels and the turning point (if development unfolds successively) in terms of the higher levels.

In other words it represents the specialization of mere linear understanding suited to rational understanding, it also represents an extreme limitation with respect to circular intuitive type appreciation. Thus in search of greater balance, further development may later require corresponding development of the more unconscious intuitive element.

Again using Piaget’s terminology - which applies especially to the cognitive features of the stage - it represents the full unfolding of formal operational understanding. For simplicity I refer to it as the formal stage.

However my major purpose here is of course to demonstrate its holistic mathematical rationale in number terms.

During the previous stage we saw how growing interaction between both specific concrete perceptions (and corresponding concepts) takes place through the ability to (explicitly posit and (implicitly) negate both aspects in experience.

Then as understanding here is of a linear nature this implies that it is one dimensional where however one can negate as well as posit this default dimension.

And as we have seen when a number is raised to  - 1 we obtain the reciprocal (as corresponding fraction) of that number (e.g. 2 ^-1 = ½.) In like manner in holistic terms a child learns to break up original whole units into constituent parts and inverse manner organise lower parts into coherent wholes.

However the very growth in negating ability (though still of a largely unrecognized implicit kind) leads to a certain nullifying of posited phenomena. In this way the child is enabled to become increasingly detached from the specific features of phenomena and thereby abstract their more universal features in rational terms.

In other words we have a considerable growth with respect to more generalized conceptual understanding (which again is likewise associated with more general perceptions).

So the analytical ability to break wholes into parts now relates to more universal abstract notions relating to theoretical rather than practical investigation.

We can readily see how such understanding is so important with respect to the development of pure mathematics (where rational linear understanding of a most specialized kind reaches its zenith). However it also applies in various measures to all disciplines in a general overall grasp of their essential features.

Now one may wonder how we can break concepts (which represent dimensions) into parts as this process requires one dimensional understanding.

However as the very process of linear understanding entails the relationship of the qualitative to the quantitative aspect, this entails that concepts themselves now attain a quantitative meaning. So in this sense the reduced quantitative nature of concepts can also be expressed in a linear (one-dimensional) manner where both positing and negating take place.

We can see this for example in the manner that academic disciplines are broken down into various branches. So for example if we start with a general discipline indeed such as Mathematics, this can be divided and further sub-divided into further disciplines.

We now will give a brief summary of the three sub-levels.

SL1 This entails the breaking up of concepts (i.e. dimensions) into various sub-concepts as constituent parts. In this way, fractional dimensions now - literally - arise in experience.

SL2 Attention now focuses more on the reverse procedure whereby constituent part

concepts can be organised into more general whole concepts.

SL3 Finally, the relationship between perceptions (as wholes or fractions) and concepts (also as wholes and fractions) starts to develop in an interactive manner.

H0 (LM3) - Bridging the Rational and Irrational Numbers

The understanding of this level is often referred to as vision-logic. From one perspective it represents the most developed expression of (linear) reason where vast networks of concepts and perceptions (representing varied fields of experience) can dynamically interact with each other. In this sense it brings together deduction (moving from the general to the particular) and induction (moving from the particular to the general) in an enhanced manner.  This implies the ability to continually apply theoretical constructs in the interpretation of relevant data and likewise intuit from facts general conclusions.

So expressed in another way we have the two-way interplay here of both the concrete and formal levels.

However there is an inevitable paradox at the centre of such understanding for the very flexibility required for such understanding (which implicitly is of a direct spiritually intuitive nature) tends to undermine the very assumptions on which linear understanding is based.

Put simply linear understanding implies the dualistic capacity to maintain the separation of opposite poles (such as internal and external).

However the creative flexibility required for vision-logic to operate, entails continual negation of the perceptions and concepts initially posited in experience. Thus when both poles (positive and negative) are continually related in this manner, the strong relational aspect involved tends to undermine linear understanding (based on clear separation of opposite poles).

So this stage can represent a decisive turning point. When experience is somewhat extensive ranging over a wide rage of interests and pursuits, spiritual intuition operates as a considerable catalyst for the creative and flexible use of linear understanding. Development will tend to plateau without further dramatic changes taking place.

However when of a more intensive nature, considerable erosion of posited phenomena is likely to take place. This in turn can then lead to a profound existential crisis which decisively sets the stage for entry to the stages of the Higher Band.

Now a remarkable holistic mathematical transformation is involved.

As is well-known for example from the Pythagorean Theorem when one attempts to obtain the square root of 2, an irrational number results.

Alternatively this can be expressed by saying that when we raise 2 to the fractional dimension (i.e. power) of ½ an irrational number results.

And this point can be generalised. Whenever a rational number (either whole or fractional) is raised to a fractional power, an irrational number results. This is universally apart from some trivial exceptions.

For example 4^1/2  = 2 (which is rational).

However 4 can itself be expressed as a number to a higher dimension (than 1). So 4 = 2². Therefore when this expression is raised to ½, a whole (rather than fractional) dimension results.

Now we will deal in more detail with the precise psychological transformation associated with this result at the next level.

However for the extensive type of experience we can say that the irrational experience, which ultimately relates to the generation of spiritual intuition, remains of an implicit nature that does not seriously undermine the assumptions of linear understanding.

Though the horizontal (quantitative) and vertical (qualitative) aspects essentially relate to distinctive types of understanding (that are linear and circular with respect to each other) when experience is strongly linear, the qualitative aspect is significantly reduced in quantitative terms.

Therefore a reduced merely quantitative interpretation is given with respect to the irrational number transformation involved.

Though the square root of 2 is irrational, its value can be approximated to any degree of accuracy (in merely quantitative terms). And by operating in this reduced manner, irrational numbers can thereby be incorporated within the conventional rational paradigm.  Likewise psychological understanding that is properly - in corresponding holistic mathematical fashion - irrational can be incorporated within linear understanding.

Thus once again, for many the conflicts inherent within vision-logic understanding remain merely implicit and - through acceptance of this inevitable quantitative reductionism - never seriously threaten the predominance of the rational paradigm. 

We will go briefly now through the three sub-levels.

At SL1 The emphasis is mainly on the interplay as between a wide range of concepts and perceptions from the more concrete inductive perspective (where generalisations arise from the appropriate arrangement of facts). Holistically, this corresponds to number (as quantity) raised to an irrational power (or dimension).

At SL2, the emphasis is more on the reverse interplay from a more theoretical deductive perspective (where the facts are interpreted through general hypotheses).

Holistically this corresponds to numbers as dimension (expressed in reduced quantitative terms) raised to an irrational power.

Finally at SL3 we get the two-way interplay of both approaches where facts and hypotheses and hypotheses and facts are continually related in dynamic manner.

This is the highest expression of rational type understanding that however is already discovering its own inherent limitations.