This is the third of my "Pythagorean" posts. (The other two are "The Pythagorean Dilemma" and "Fermat's Last Theorem - a dynamic qualitative approach").
The purpose of the post is to demonstrate using the Pythagorean Theorem how its interpretation subtly changes with each level (and sub-level) of the Spectrum. A detailed summary is given at the end.

The great shortcoming of conventional mathematical interpretation is its very reduced nature. Interpretation is given solely from the perspective of the rational paradigm. This represents just one level of the Spectrum of Consciousness (that I call L0).

Though I am illustrating this here with respect to the well-known Pythagorean Theorem, this can be generalised for all mathematical and scientific interpretation.

The famous theorem of course states that for any right-angled triangle the square on the hypotenuse (i.e. diagonal line) is equal to the sum of squares of the two other sides.

Conventional mathematical interpretation is based – as I have stated - on the rational paradigm of L0. This requires the separation of polar opposites in experience in horizontal, vertical and diagonal terms.

Thus horizontal poles (subjective and objective) are separated. Subjective interpretation is then either ignored or reduced in objective terms. Thus in terms of the rational paradigm of L0, the Pythagorean Theorem has an absolute objective validity (unaffected through subjective mental interaction).

The vertical poles (qualitative and quantitative) are also separated. Again qualitative (holistic) interpretation is either ignored or interpreted in reduced quantitative manner.

In terms of the rational paradigm of L0, the Pythagorean Theorem has an absolute quantitative validity (unaffected by qualitative considerations).

The diagonal poles (potential and actual) are also separated. Once more potential (infinite) interpretation is reduced in actual (finite) fashion.

So once again the Pythagorean Theorem has an absolute finite validity (unaffected through spiritual interaction).

Thus conventional mathematical interpretations emphasise merely the positive pole of experience (horizontally, vertically and diagonally). Truth is identified in terms of objective reality (horizontal). Truth is equally identified in terms of quantitative reality (vertical). Finally truth is identified in terms of finite reality (diagonal).

We can identify three sub-levels of interpretation at this level (L0). Again these are based on horizontal, vertical and diagonal type understanding (but in a very reduced manner).

The first sub-level is the concrete operational (conop).

The understanding of the Pythagorean Theorem is in concrete empirical terms. For any actual right-angled triangle the square on the hypotenuse is indeed equal to the square on the other two sides. For example if the three sides of a right-angled triangle are 3, 4 and 5 units respectively, it is easy to demonstrate that 3 (squared) + 4 (squared) = 5 (squared).
Through concrete investigation it would be possible to demonstrate that the proposition holds in each individual case.

Indeed this could eventually lead to an inductive kind of generalisation that – based on repeated concrete examples – the square on the hypotenuse (in any right-angled triangle) is equal to the sum of squares of the other two sides.

It seems clear that the validity of this proposition was known to the Babylonians (before Pythagoras). However they did not provide a formal proof.

The next sub-level of formal operational (formop) relates to the vertical (qualitative) aspect of experience which is interpreted in reduced quantitative terms. The purpose of this sub-level is to obtain an abstract general understanding of the nature of quantitative relationships.

Ken Wilber stresses the potential nature of this sub-level in that it opens up a wide range of possibilities in terms of understanding.This is certainly true. However Ken does not sufficiently stress the reduced nature of such understanding. Creative potential in understanding is directly provided by intuition (not reason). Thus when the rational aspect of understanding is emphasised it inevitably reduces an infinite set of possibilities to a limited number of actual cases.

With formal operational understanding, one can appreciate the abstract nature of mathematical proof. What has been proven to be true for the general case is now understood as true for all particular cases. Thus the general formal proof of the Pythagorean Theorem (which does not require concrete measurement) ensures its validity for all concrete examples. We therefore deduce the truth (in particular cases) from the general case.

However I must stress once again that the reduced mathematical interpretation of proof is very misleading. Indeed it involves a fundamental confusion regarding the relationship of finite to infinite notions.

The general proof of a proposition strictly speaking has only a potential validity (in applying to an infinite set of cases).
However particular instances of the proposition have an actual validity (in applying to a limited finite set of cases). We cannot move from potential (infinite) to actual (finite) notions in solely rational terms.

Intuition is also involved (which operates in a qualitatively different manner from reason). Therefore I consider it very important to restore this qualitative holistic dimension (that is directly intuitive in origin) to mathematics.

The formal operational sub-level strictly speaking relates to the vertical aspect of experience. It deals with the qualitative aspect of general form (as distinct from the quantitative aspect of concrete phenomena). However it interprets this experience in reduced rational fashion so that the qualitative aspect is reduced to the quantitative.

Thus the formal qualitative truth of the Pythagorean Theorem i.e. as holistically true for all (unspecified) cases is quickly given a (reduced) quantitative interpretation where it is understood as specifically true for any concrete situation.

The third sub-level of vision-logic properly relates to the diagonal aspect (which involves the interaction of both horizontal (concrete) and vertical (formal) sub-levels.

It certainly is the most dynamic and versatile of the rational sub-levels.

This involves both induction (moving from the particular to the general) and deduction (moving from the general to the particular).Because of the increased interaction between concrete and formal understanding a greater degree of intuition is implicitly generated. This in turn leads to a quasi-integrative capacity in understanding. However there are severe limits to the nature of this integration.

Ken Wilber in fact claims far too much for vision-logic.For example he says that it provides a means of synthesising opposites. This is true only in a limited manner (where opposite poles maintain a somewhat rigid identity).

Ken's own four-quadrant approach illustrates this point very well. Ken would of course accept that the in horizontal terms the right-hand should ultimately be integrated with the left-hand quadrants in experience; likewise the upper quadrants should be integrated with the lower. However Ken interprets these quadrants in a somewhat fixed and rigid manner.

For example he views interior experience in terms of left-hand quadrants and exterior experience in terms of right- hand quadrants. This exemplifies the one-directional nature of vision-logic.

However at the "higher" level of HL1, exterior and interior become entirely relative terms. Thus from this perspective exterior can equally apply to the left-hand and interior to the right-hand quadrants. Therefore it is only meaningful to fix quadrant locations (in a strictly relative sense).

So with vision-logic understanding keeps switching in flexible manner from concrete to formal (and formal to concrete). Though implicitly this process involves a greater degree of intuitive understanding, explicitly reality in rational is rationally translated in one-directional terms.

Vision-logic therefore give a more creative understanding of the Pythagorean Theorem providing greater insight into its (potential) theoretical and (actual) practical implications.

By extension the vision-logic of this level can lead to an understanding and application of conventional mathematical and scientific constructs that is both flexible and creative.
 

The transition from L0 to HL1 involves the unfolding of the important negative direction of understanding in what I call mirror understanding. This represents a substantial deepening of intuitive understanding.

There is in fact a simple dynamic relationship involving intuition and reason. We can only move to intuitive understanding through the negation of the positive (conscious) direction of reason.

Essentially this involves the undoing of conscious attachment to phenomenal symbols insofar as they relate to horizontal polarities (e.g. subjective and objective).

Existentially this is a very dark and painful process. In Christian ascetical terms it is known as "purgation", "learned ignorance", "the cloud of unknowing" etc.

Conscious understanding always involves undue attachment to one pole of experience. The opposite pole (which is in truth complementary) is thereby separated and repressed in the unconscious. So the initial step in moving towards genuine integration is the recovery of the "lost" negative direction of experience (i.e. mirror understanding).

The deeper development of intuition gradually leads to a switch from quantitative type understanding of phenomenal symbols to a holistic qualitative appreciation where they become rich archetypes of a universal order.

Thus the transition from L0 to HL1 involves a radical negation or undoing principally of the former concrete sub-level (i.e. conop). To a lesser extent it also entails the negation of formal conceptual understanding.

Concrete understanding of the Pythagorean Theorem (and mathematics and science in general) is severely eroded. As one's interest in empirical pursuits declines one gradually loses the ability to apply mathematical symbols in an analytical fashion. (This in fact is the preparation for a switch to a deeper holistic understanding).

The important implication of this transition is that every fact, hypothesis, theory is balanced in dynamic terms by a corresponding mirror fact, hypothesis, theory. Thus the Pythagorean Theorem has an important mirror explanation (or anti-theorem).

Theorem and anti-theorem must be understood in dynamic terms.

Thus if the positive direction (i.e. theorem) refers to the objective interpretation (in relation to the subjective mind), then the negative direction (i.e. anti-theorem) refers to the subjective mind (in relation to the objective interpretation). The positive direction is identified with rational (conscious) understanding; the negative direction is identified with intuitive (unconscious) understanding.

This simply highlights the fact that objective and subjective poles are equally important in the dynamics of understanding. Therefore it is misleading to identify truth statically with just one pole. Just as get an absolute number when we ignore the sign of the number in mathematics (positive or negative), equally we get an absolute interpretation of a theorem when we ignore the sign of the theorem. Thus the conventional static understanding of the Pythagorean Theorem (and by extension all mathematical theories) is of an absolute nature.

We would also have the negation or undoing of formal constructs in a mirror form of formop during this transition.

Finally we would have to a limited extent a passive stage relating to the negation of vision-logic. This would be more purely intuitive though in a largely secret interior manner.

Development in understanding then moves on to HL1. I refer to this as the circular level. It equates well with - in Ken's terminology - both the psychic and subtle realms.

A very different type of understanding emerges at this level which is characterised by a high degree of spiritual intuition. In rational terms it involves the attempt to combine the two horizontal poles of understanding (subjective and objective) as the complementarity of polar opposites.

This represents a very different logic from former understanding. Rational understanding is characterised by an either/or logic (where opposite poles are clearly separated). This new understanding is characterised by a both/and logic i.e. irrational logic (where these same poles are now complementary).

In terms of rational either/or logic a proposition is either true or false and has an absolute truth-value.In terms of this new irrational both/and logic a proposition is both true and false and has a relative truth-value.

So at HL1 the Pythagorean Theorem has a relative truth-value.

Understanding is now interpreted in terms of the dynamic interaction of both positive (objective) and negative (subjective) poles. Therefore the truth of the proposition depends on both the positive and the negative interpretation.

The clear implication of this as is that as each person's subjective interpretation of a proposition varies that mathematical truth really represents a special type of social consensus. Indeed it is quite possible that with mistaken social consensus a proposition could be interpreted as true that is in fact objectively false.

At this level understanding moves away from a quantitative to a more holistic qualitative perspective. Mathematical symbols increasingly become archetypes of a universal structural order.

In general the more important a proposition is from a quantitative perspective equally the more important it will be potentially in qualitative holistic terms.Therefore the Pythagorean Theorem has potentially a very important qualitative significance.I will now attempt to explain the precise nature of this significance.

Conventional understanding of mathematical symbols is based on the rational separation of polar opposites. Here the three sides of the right-angled triangle (adjacent, opposite and hypotenuse) have a specific quantitative interpretation in horizontal, vertical and diagonal terms.

Holistic understanding of these same symbols is based on the intuitive complementarity of polar opposites (which has an indirect rational formulation).
 

The three sides of the right-angled triangle now acquire a qualitative interpretation in horizontal, vertical and diagonal terms. In other words one now begins to see the importance of horizontal, vertical and diagonal polarities as the basic underlying symmetrical structure of reality at all levels. Two dimensions (squares) in quantitative terms have positive and negative polarities in dynamic qualitative terms.

As these three polarities are fundamental in Holistic Mathematics it is arguable that the qualitative interpretation of the Pythagorean Theorem is even more important that its quantitative counterpart.

We can use this qualitative interpretation to effectively explain the nature of the Pythagorean approach. The horizontal line represents numbers as quantities (which the Pythagoreans initially assumed were rational). The vertical line represents number as quality (the rational paradigm). They basically defined their scientific method in terms of this paradigm. The diagonal line represents number as the relationship between quantitative and qualitative aspects. Thus the Pythagoreans saw the mutual interaction these two aspects (quantitative and qualitative) as an appropriate means of attaining spiritual contemplation.

So the right-angled triangle perfectly symbolises in holistic terms their overall approach. The basic shortcoming was in reducing the vertical to the horizontal aspect. Thus the scientific paradigm of the Pythagoreans was basically confined to L0 (i.e. the rational linear level).

This relationship between quantitative (linear) and qualitative (circular) aspects can be expressed in another way. A right-angled triangle can be circumscribed by a circle with the hypotenuse or diagonal as line diameter. In like manner the qualitative interpretation (of horizontal, vertical and diagonal polarities) is enclosed within intuitive (circular) understanding.

We will look briefly at the sub-levels of HL1 (which now have both positive and negative directions).

The first corresponds to a new type of conop where concrete symbols become an archetypal expression of a universal order.

In like manner the concrete representation of a right-angled triangle begins to symbolise (in impressionistic manner) this deeper holistic level of reality comprising horizontal, vertical and diagonal polarities. This in holistic mathematical terms is the concrete expression of irrational number qualities (corresponding to irrational number quantities).

There are two horizontal directions to this concrete understanding. Thus we have positive and negative (concrete) irrational number qualities.

With the positive (external) direction, a degree of confusion is involved in that one tends to reduce the qualitative with the quantitative i.e. attempt to grasp holistic intuitive meaning of concrete symbols in rational terms.

With the negative (internal) direction, one tries to overcome this confusion through the negation (i.e. undoing) of rational understanding. This is a more purely intuitive stage (largely devoid of phenomenal translation).

The second corresponds to a new kind of formop where formal symbols become archetypes of a universal order.

There are two vertical directions to this formal understanding. Thus we have positive and negative (formal) irrational number qualities.

Here a deeper type of holistic understanding is possible (that is not dependent on specific concrete symbols). A more generalised philosophical understanding of the fundamental dynamic nature of polarities (horizontal, vertical and diagonal) becomes possible.

Again with the positive direction a degree of confusion is involved in that one tends to identify the truly intuitive dimension of holistic form with (reduced) rational understanding.

With the negative direction one attempts to undo this confusion in a more truly holistic understanding (largely devoid of symbols).

The third relates to a new kind of vision-logic (which represents the dynamic interaction of concrete and formal stages).

Because this stage is inherently more interactive it tends to be very intuitive and spiritual involving a high degree of transcendent intuition. It is largely devoid of symbols. Also positive and negative directions largely coincide.
 

We now move on to the very important transition from HL1 to HL2.

This leads to a new type of understanding (which I refer to as virtual understanding).

Though a considerable amount of intuition is evident at HL1, there is still the attempt to interpret this understanding which is qualitatively distinct in rational terms (thus keeping it under conscious control).

Thus in correct terms, HL1, represents in holistic mathematical terms a "real" form of understanding (i.e. interpreted from a conscious perspective).

Now the transition from HL1 leads to a new understanding of projected consciousness. Here the archetypal symbol (that is expressive of qualitative holistic meaning) is seen more correctly as an indirect expression of what is truly unconscious in meaning.

The spiritual unconscious is developed though the negative direction of understanding and involves two polarites (positive and negative). Virtual understanding is the indirect expression in (reduced) one-dimensional terms of this two-dimensional understanding. Thus it exactly complements - in qualitative terms - the meaning of imaginary numbers (in quantitative terms).

So projected virtual consciousness in holistic mathematical terms is "imaginary" rather than "real".

The transition from HL1 to HL2 involves the plentiful projection - in indirect conscious terms - of holistic archetypal meaning.

Understanding is no longer inspired in direct terms through the recognition of conscious symbols but rather indirectly through the projection of the unconscious (in "imaginary" form).

Symbolic representations now become a more spontaneous expression of the spirit and thereby acquire a far closer and meaningful association with this spirit (which generates them).

We have already seen that each theory has a corresponding mirror theory. Thus the Pythagorean Theorem is complemented (in horizontal terms) by its own anti-theory.

In like fashion we now see that each theory has also a corresponding virtual theory. In other words the Pythagorean Theory has both a "real" and an "imaginary" explanation in vertical terms.

The "real" theory in direct terms is based on conscious understanding; the "imaginary" theory is based directly on unconscious understanding, which then obtains an indirect conscious expression through spontaneous projection from this unconscious source.

Just as with virtual particles in physics, horizontal polarities are very closely associated (as particle and anti-particle),likewise with virtual understanding horizontal polarities are also very closely associated (positive and negative).

What this means in practical terms is that virtual understanding provides a very pure representation of archetypes, where the phenomenal representation of truth is emitted as a spontaneous expression of what it signifies. Also because of the close fusion of horizontal polarities, these phenomena are very transparent and short-lived returning quickly to the source from which they emerge. So with virtual understanding, the pure intuitive nature of archetypal forms (and not indirect conscious representations) are chiefly affirmed.

Thus the transition from HL1 to HL2 leads to a purely qualitative understanding of the holistic aspect of symbols.

The Pythagorean theorem now gives a clearer insight into the underlying spiritual nature of reality, of which (holistic) complementary polarities (horizontal, vertical and diagonal) are a phenomenal qualitative expression.

Again we can take this "imaginary" understanding in terms of the three sub-levels.

With "imaginary" concrete understanding, symbols are projected from the intuitive unconscious in holistic archetypal form. Thus here the appearance of the image of a right-angled triangle becomes a concrete holistic expression of the underlying spirit expressed by dynamic horizontal vertical and diagonal symmetries.

With "imaginary" formal understanding symbols are projected in a more generalised manner. This understanding represents the same truth in deeper holistic form.

With "imaginary" vision-logic, greater interaction as between concrete and formal understanding takes place. Phenomena are so short-lived in experience that their underlying intuitive origin is continually present in consciousness.
 

We now move on to HL2 which I call the point level. (It equates well with the causal realm).

This involves an even more subtle form of understanding.Again L0 is concerned with the specialisation of linear (analytic) understanding. HL1 relates to the specialisation of circular (holistic) understanding.

The key problem during HL1 is the attempt to grasp what is qualitative and holistic in reduced rational terms (which inevitably is linear in nature).

As HL1 overemphasises holistic understanding it inevitably leads to confusion with linear understanding (which is complementary). For example to maintain that all things are relative is itself an absolute statement.

Indeed the key problem during HL1 is the attempt to keep intuitive development under "higher" conscious control. One thus strives to interpret reality in "real" terms.However the birth of "imaginary" understanding - where understanding becomes directly intuitive – paves the way for a more balanced and subtle understanding.

Understanding is now seen neither in linear (L0) nor circular terms (HL1) solely but rather in terms of the interaction of both. This is exemplified well by the number pi, which is the best example of a transcendental quantity and one of the most important numbers in mathematics.

The same point is both the centre of the straight line diameter and its circumscribed circle. Thus this point is central to both. Likewise it is this concentrated spiritual focus (of the point level) that is central to the reconciliation of both linear (analytical) and circular (holistic) paradigms.

Just as the sub-levels of HL1 have two distinct horizontal directions (positive and negative), the sub-levels of HL2 have two distinct vertical directions (real and imaginary).

Understanding keeps switching from real to imaginary format (and from imaginary to real). At the "higher" level of transcendent understanding, one attempts to translate reality in real terms. In holistic mathematical terms this is the understanding of transcendental numbers (where reality is seen in terms of the interaction of linear and circular logic).

At the "lower" level of immanent understanding, where one attempts to translate reality in imaginary terms. This again is transcendental understanding of a more directly intuitive kind. In the former case spiritual meaning is seen as lying beyond phenomenal symbols i.e. literally without symbol (transcendence). In the latter case spiritual meaning is seen as inherent within symbol (immanence).

Again with transcendence we move beyond (unique) parts to a (collective) whole.

With immanence we move within each unique part to find the (collective) whole.

Thus the all-important relationship as between whole and part (and part to whole) should be translated holistically in "complex" mathematical terms. If the whole is "real" then the part - in relative terms - is "imaginary". If the part is "real" then the "whole" is imaginary.

It is during HL2, that mature holistic mathematical understanding develops. It is only here that one fully begins to realise the two-fold meaning of all mathematical symbols in quantitative and qualitative terms. One moves from (refined) reason to (refined) intuition. This in turn enables one to move from quantitative understanding of mathematical symbols as "real" to qualitative understanding as "imaginary" and vice versa. ("Real" and "imaginary" of course are purely relative terms).

We will now look briefly at the various sub-levels of HL2.

We again have a new highly subtle version of conop where concrete symbols alternate sequentially as between their linear (analytic) and circular (holistic) interpretations. One would now view the Pythagorean Theorem from a dual perspective alternating as between a quantitative and qualitative appreciation of its symbols.The extension of this understanding is very important in scientific terms.

One would now look on reality dynamically in psycho-physical terms (where every physical relationship is complemented by a psychological counterpart). Interpretation here is at a highly globalised transcendent level (where the parts are included in the whole). (This is the qualitative concrete understanding of "real" transcendental numbers).The "imaginary" counterpart of this stage involves the switch to a very localised immanent type of understanding (where the whole is included in each part).

We then have the corresponding version of formop where more formalised symbols now alternate sequentially as between linear and circular understanding.

Thus in terms of the Pythagorean Theorem we have a deeper formalised understanding of how its symbolic representations (i.e. the squares of horizontal, vertical and diagonal lines) have equally important quantitative and qualitative manifestations.

The extension of this understanding to all mathematical symbols leads directly to a formalised interpretation of Holistic Mathematics. Every mathematical symbol as number, operation, function or relationship is now understood as having holistic (as well as analytic) interpretations.

So just as Conventional Mathematics - in its fullest sense - is a cognitive expression of the formal operational stage of L0, Holistic Mathematics - in its developed sense - is a cognitive expression of the corresponding formal operational stage of HL2. (This is the qualitative formalised understanding of the "real" transcendental paradigm).

Again the "imaginary" counterpart of this stage involves the switch to an extremely localised immanent type of understanding (where the whole is included in each part).

The vision-logic counterpart of HL2, is extremely intuitive (where concrete and formal understanding interact).

Real and imaginary aspects are harmonised to a greater extent as experience switches between transcendence and immanence.
 

The next transition is that from HL2 to HL3.

This involves the entry into a continual experience of the spiritual present. Again it has a dual rational and intuitive interpretation which can be remarkably demonstrated by the Pythagorean Theorem itself.

With the conclusion of HL2, one achieves freedom from confusion of "real" (quantitative) and "imaginary" (qualitative) understanding. Both are considered equally important.

In quantitative terms we can represent this by a right-angled triangle (where horizontal and vertical sides are equal in length. If we give the horizontal side a "real" interpretation, then the vertical side will have an "imaginary" interpretation.Thus the diagonal side (i.e. the hypotenuse) will be the sum of squares of these two sides and equal to zero.

In complementary holistic terms when we attempt to integrate horizontal and vertical polarities of experience (each having been equally differentiated), we attain to the experience of a (transfinite) spiritual void.

Thus this transition involves an extremely subtle form of understanding which is simultaneously both "real" and "imaginary" (i.e. "complex"). Paradoxically this leads to a purely simple underlying appreciation of reality as transfinite (i.e. infinite).

The understanding of the Pythagorean Theorem would be extremely subtle during this transition.
The quantitative interpretation of its symbols would lead (almost instantaneously) to corresponding qualitative interpretation. Equally qualitative would lead quickly to quantitative interpretation. Symbols would therefore become extremely transparent and spiritual during this time.

HL3 of course eventually culminates in nondual reality.

It is a mistake to identify this pure (non-phenomenal) awareness with intuition (rather than reason). Rather it is the state where polarities as between reason and intuition lose their meaning.We have here the ultimate paradox that rational activity is so dynamic that its generated phenomena elude detection. So the passivity of this state is the dynamic result of balancing active poles.

Clearly the Pythagorean Theorem has no phenomenal interpretation here. Its symbols are inseparable from what they ultimately represent (i.e. spirit).
 

We now move into Radial Reality which involves the most comprehensive understanding.

The understanding of all the previous levels is restored.

Each level of the Spectrum now has a partial differentiated validity it its own right and yet is dynamically integrated with all levels.

Thus we can have the freedom to understand the Pythagorean Theorem from a linear quantitative perspective (typical of L0).

Likewise we have the freedom to understand from a circular qualitative perspective (HL1).

We can also understand from the point perspective of HL2 (where quantitative and qualitative aspects are complementary).

Finally we can understand from a purely nondual perspective (HL3) where no phenomenal symbols are involved.

However each of these partial interpretations is progressively integrated in an ever-changing dynamic understanding.

Radial Reality can for convenience be subdivided into two main stages.

In the earlier finite (phenomenal) and transfinite (spiritual) understanding remain somewhat separated.In the second stage they are progressively integrated. Understanding now reaches its highest creative expression (quantitatively and qualitatively).

Fascinating insights and applications could now spring from this comprehensive understanding of the Pythagorean Theorem (and by extension all theories).

I have not mentioned explicitly the "lower" levels LL3, LL2 and LL1 in terms of understanding the Pythagorean Theorem.

As appropriate interpretation of the Theorem requires the understanding of L0, they would not be directly involved in their confused state. However as the development of the "higher" levels requires returning to the "lower" levels (to unravel confusions), they are ultimately involved.

Indeed in dynamic terms integrated intuitive understanding always involves corresponding confused instinctive understanding.
Though this instinctive behaviour can be progressively refined it can never be entirely eliminated and remains dynamically necessary for the generation of intuition.

So ultimately all levels of the Spectrum are involved in a comprehensive understanding of the Pythagorean Theorem (and indeed any mathematical or scientific theory).
 

I will conclude with a summary of this multilevel interpretation of the Pythagorean Theorem.
 

L0.
 

This operates on the either/or logic of separate poles and is the basis of the rational (linear) paradigm.

Sub-Level 1 (Horizontal)

Concrete Operational (Conop). This is a quantitative understanding (with specific actual triangles) of the Pythagorean Theorem (i.e. that the square on the hypotenuse of a right-angled triangle is equal to the squares of the other two sides).

Sub-Level 2 (Vertical)

Formal Operational (Formop).

This is also a quantitative generalised understanding of the Pythagorean Triangle.

Sub-Level 3 (Diagonal)

Vision-Logic. This implicitly is a more intuitive quantitative understanding (where concrete and formal understanding increasingly interacts).

Transition from L0 to HL1.

This involves the negative (intuitive) direction of understanding. It leads to the undoing of rigid (one-directional) rational understanding of phenomena.

Sub-Level 1 (Horizontal)

This involves the negation (undoing) of concrete understanding of the Pythagorean Theorem in quantitative terms (with the corresponding development of deeper holistic understanding).

Sub-Level 2 (Vertical)

This involves the negation (undoing) of formal understanding of the Pythagorean Theorem in quantitative terms (again with the corresponding development of deeper holistic understanding).

Sub-Level 3 (Diagonal)

This involves the negation (undoing) of both concrete and formal understanding of the Pythagorean Theorem in quantitative terms. This is a very intuitive in character.
 

HL1.

This operates on the both/and logic of complementary poles. In direct terms it is understood intuitively. Indirectly it can be given a (reduced) rational translation (i.e. the irrational paradigm).

Sub-Levels at HL1 have both positive and negative directions

Sub-Level 1 (Horizontal) - Positive Direction

This is a concrete qualitative understanding of the Pythagorean Theorem (as symbols of horizontal, vertical and diagonal polarities which underlies reality at all levels).

Sub-Level 1 (Horizontal) - Negative Direction

This is a more intuitive understanding of the previous sub-level (where rigidity in secondary rational translation is eroded).

Sub-Level 2 (Vertical) - Positive Direction

This is a formal qualitative understanding of the Pythagorean Theorem (as symbols of horizontal, vertical and diagonal polarities which underlies reality at all levels).

Sub-Level 2 (Vertical) - Negative Direction

This again is a more intuitive understanding of the previous sub-level (where rigidity in secondary rational translation is eroded).

Sub-Level 3 (Diagonal) - Positive and Negative Directions

This involves increasing interaction of concrete and formal understanding leading to a highly intuitive qualitative understanding (in transcendent terms).The dynamic relationship of "higher" HL1 with "lower" LL1 is established. The integration of the "higher" level depends on dissolving the confusion of the "lower" level.
 

Transition from HL1 to HL2.

This involves virtual understanding which relates to the spiritual unconscious that is indirectly projected in conscious terms. In holistic mathematical terms this more purely archetypal understanding is "imaginary" rather than "real". Every theory has a virtual image counterpart i.e. every theory has both a "real" and "imaginary" interpretation.
 

HL2.

This is characterised as the relationship between the line and the circle i.e. as between (linear) quantitative and (circular) qualitative understanding. In holistic mathematical terms it represents the transcendental paradigm and is extremely subtle representing the dynamic intersection of two logical systems. It requires the continual presence of a strong spiritual centre (point) to successsfully integrate both. The Pythagorean Theorem is now understood in terms of the relationship between its quantitative (analytic) and qualitative (holistic) interpretations.

Sub-levels of HL2 have both "real" and "imaginary" directions.

Sub-Level 1 (Horizontal) – Real Direction

The Pythagorean Theorem is understood in concrete "real" terms as the relationship between quantitative and qualitative understanding (transcendence).

Sub-Level 1 (Horizontal) – Imaginary Direction

The Pythagorean Theorem is understood in concrete "imaginary" terms as the relationship between quantitative and qualitative understanding (immanence).

Sub-Level 2 (Vertical) – Real Direction

The Pythagorean Theorem is understood in formal "real" terms as the relationship between quantitative and qualitative understanding (transcendence).

Sub-Level 2 (Vertical) – Imaginary Direction

The Pythagorean Theorem is understood in concrete "imaginary" terms as the relationship between quantitative and qualitative understanding (immanence).

Sub-Level 3 (Diagonal) – Real and Imaginary Directions

The Pythagorean Theorem is understood in both concrete and formal terms as the relationship between quantitative and qualitative understanding . This combines both "real" and "imaginary" directions (transcendence and immanence). Again a dynamic relationship of "higher" HL2 with "lower" LL2 is established. The integration of the "higher" level depends on dissolving the confusion of the "lower" level.
 

Transition from HL2 to HL3

This is the interaction of both concrete and formal understanding in a highly intuitive experience. Phenomena quickly dissolve in the continual awareness of the spirit.

Sub-level (Diagonal)

The boundaries as between "real" and "imaginary" understanding disappear. Likewise concrete and formal understanding interpenetrate. The Pythagorean Theorem in both immanent and transcendent terms is a now transparent mirror of spiritual truth.

HL3 Null Level (Nondual Reality)

The Pythagorean Theorem has no phenomenal meaning. This is a purely simple intuitive state. Equally it is a purely "complex" rational state (where the balance as between polarities is perfectly maintained). Once more the dynamic relationship of "higher" HL3 with "lower" LL3 is established. The integration of the "higher" level again depends on resolving the confusion of the "lower" level.

Radial Reality This involves the continuing relative differentiation of all levels and corresponding integration. The understanding of the Pythagorean theorem at all the previous levels and sub-levels is restored (and progressively integrated with each other).

During the earlier stage of Radial Reality though linear and non-linear levels are properly differentiated some difficulty in their full integration still remains.

During the later stage dynamic differentiation and integration of all levels is closely approximated.