In these articles I am following a mapping of the "higher" transpersonal levels of the Spectrum which is directly suited for scientific translation.
We are presently dealing with H1 (Higher Level 1) which equates well with the Subtle Realm.
This is then divided in terms of three sub-levels.
Sub-level 1 relates to surface supersensory structures (concrete).
Sub-level 2 relates to deeper suprarational structures (formal).
Sub-level 3 relates to a more refined spiritual intuitive understanding, which arises from the growing interaction of the previous two sub-levels (vision). 1
As H1 is defined in terms of the specialization of horizontal polarities of experience, each sub-level here has two directions (exterior and interior). 2
The unfolding of each direction defines a stage, which incorporates distinctive development with respect to affective, cognitive and spiritual modes of understanding. 3
Finally the dynamics of each stage can be summarized - for convenience - in terms of five phases. 4
I have already dealt with the stages of sub-level 1 in previous articles ("Fascinating Connections" and "Quantum Mechanics and the Existential Decision").
So I am now moving on to the deeper formally based structures of sub-level 2, dealing in this article with the holophysical implications of the first stage (relating to its exterior direction).
So as before I will start with a brief outline of the main phases of development in relation to this stage. 5
As we have seen, the positive stages are marked by significant outpourings of spiritual illumination (i.e. pure intuition), which tend to dramatically transform one's vision of reality.
Generally, the quality and intensity of this illumination is related directly to the nature of the painful inpouring during the preceding purgative stage. Due to the severity of the previous stage it is now purer, more prolonged, and deeper than before conveying a far greater level of conviction and experiential truth.
Special peak experiences are likely at this time. These come in the form of mystical visions - passively communicated through the intellect - where one is given remarkable insight into the underlying structure of reality. In the intensity of this contemplative gaze, all boundaries between self and the material universe momentarily melt. Then, suddenly, in the wonder of an intimate embrace the secret of everything in creation is revealed as love.
Such mystical communications can occur for some time, where one, without any conscious effort, becomes enlightened and exalted with a special form of intuitive wisdom. Because of the experiential conviction and certainty of such insight, one can feel utterly transformed, with a burning desire to convey what one has truly seen to others.
After the initial peak experiences recede, there follows an attempt to translate or interpret reality through the new insight characteristic of the stage. This in what I refer to as suprarational understanding. It is based directly on holistic intuition (where mind and matter comprise a dynamic unity), though indirectly conveyed through the paradoxical language of the complementarity of (conceptual) opposites.
In terms of the psychological energy spectrum, it comprises radiation of extremely long wavelength and of very low frequency (i.e. holistic and passive in quality). This translation - intimately affecting the very dimensions of reality - goes to the heart of all formal systems of thought. Concepts - wherever used - are understood as strictly relative. There is now a more profound realization of the futility of trying to model any scientific system in absolute terms. The very concepts we must necessarily use to formulate this system are, in themselves, inherently relative.
This light can provide highly original insights into many diverse fields, springing from a strong innate sense of the fundamental interdependence of all reality. For someone with a natural intellectual bent, the phase can be most exciting and exhilarating.
After some time, inevitably, personality conflicts emerge. There is a tendency, where highly specialized development of structures takes place, for psychic energy to move from other structures to become (temporarily) excessively identified with the new stage. In other words, the ego becomes strongly identified now with suprarational understanding. One can then feel frustrated and rejected, when others show little appreciation for one's vision. Partly this is due to difficulties in communicating what is a non-linear form of knowledge. However, the sobering fact is that appreciation for really original insight is inherently very restricted. It can only be properly confirmed by people who have "seen" in like manner (and these are few in numbers).
The process of personal transformation is therefore likely to involve much isolation with two different kinds of suffering involved. The more obvious arises during the intense purgative periods of the interior stages. However, there is also a more subtle form involved during the positive stages of illumination. Though having a burning desire to communicate this authentic spiritual insight, one is frequently met with indifference and even hostility as reality is now experienced on a different wavelength from others. Also, because one is still in a process of rapid change without yet reaching stable personality equilibrium, it creates continual difficulties in terms of maintaining meaningful personal relationships.
These problems in large measure are the inevitable consequence of an authentic commitment to transcendence.
Though less a feature of the stage, there is likely also to be some high level affective development. Due to commitment to transcendence, natural emotional expression may be repressed. Thus affective response becomes somewhat spiritualized involving deeply holistic archetypal patterns.
Whereas cognitive development represents the unfolding of "masculine" ideas, affective represents the (complementary) unfolding of "feminine" images. Both of these in turn represent an understanding of God (Supreme Being) essentially as a projection of one's own divine personality.
Because of the considerable transfer of psychic energy to the emerging suprarational structures, little may be left for customary responsibilities (from which one derives little satisfaction).
A familiar pattern emerges, whereby one sees the need for reconciling the structures of the stage with earlier development, but finds this in practice very difficult to achieve.
A crisis now starts to loom. Undue identification with the secondary suprarational structures creates increasing problems in terms of pure spiritual desire. As the clash intensifies one is driven more and more inwards in an attempt to get to the roots of ego need. Gradually this mood of despondency starts to predominate in experience. One slowly realizes that the only solution is to surrender entirely this subtle attachment to reason. For someone, with a natural intellectual gift, this constitutes an enormous sacrifice. Earlier in the stage one felt that one's true vocation in life was at last about to be realized. Now, instead, one slowly prepares to leave everything behind to journey on in faith into a very uncertain future.
The cognitive aspect of Phase 2 is especially relevant for scientific understanding. 6
Once again this differs from the cognitive experience of the first sub-level in that it is based on far richer intuitive insight which penetrates right to the depths of formal intellectual systems. One can see clearly into the inadequacies of any rational system based on absolute conceptual notions.7
It is important to clarify the true dynamic nature of this experience.
Essentially it involves the interaction of pure spiritual intuition with a very refined form of bi-directional reason. 8
Intuition is qualitatively very different from conventional reason (working according to an alternative logical system). Therefore the essential requirement in moving from reason to intuition is to create paradox in terms of all dualistic understanding. This then prepares the mind for that qualitative transformation in intuition (where opposites are reconciled). 9
Bi-directional understanding is the means by which this rational paradox is achieved.
Growth in intuition in turn facilitates bi-directional understanding (in the ready recognition of paradox).
So in dynamic terms intuition and (bi-directional) reason are inseparable.
The implication of this stage is that no formal theory can be absolute (even in mathematical terms).
Let us take the famous Pythagorean Theorem as an illustration.
It states that in any right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Now conventionally this theorem is true in absolute terms.
However from the perspective of this stage it has - like all theorems - a merely relative truth-value.
In conventional science no distinction is made as between the exterior and interior aspects of understanding.
However from a dynamic perspective these directions are complementary opposites (in horizontal terms).
In mathematics, the absolute value of a number requires that the sign be ignored. Thus +2 and –2 have an absolute value of 2.
In holistic mathematical terms it is precisely similar. When we ignore the direction of a concept, absolute understanding results.
Thus the absolute understanding of the rational linear level (H0) - on which conventional mathematical interpretation is based – is very reduced (with opposite directions confused).
However in terms of this stage, any theory necessarily has a relative interpretation.
So if the Pythagorean Theorem is defined in relation to its (objective) exterior, then the corresponding mirror theorem is defined with respect to its (subjective) interior aspect.
In dynamic terms, understanding arises from the interaction of these complementary opposite theorems giving a relative truth-value. 10
This has very practical implications where truth only has a relative meaning.
Each person’s understanding of a theorem – such as the Pythagorean - necessarily varies. In experiential terms its "truth" is different for every person.
So from a dynamic perspective, mathematical truth represents no more than a special form of social consensus.
This raises a very deep issue, which is ignored in the presentation of (conventional) mathematical truth.
Though this understanding necessarily involves a dynamic interaction of both reason and intuition, formally it is interpreted in merely rational terms and is thus very reduced.
Reflection will show that intuition is extremely important for mathematical and scientific understanding and especially necessary for truly creative work in the fields. 11
Even in conventional terms, understanding is not possible without intuition for it is essential for one to literally "see" the truth implied by logical reasoning.
Thus though not explicitly recognized, intuition is always implicitly required.
In fact most people have a very limited capacity to appreciate abstract thought and respond more readily to intuitive rather than rational explanations. 12
A comprehensive appreciation of mathematics necessarily entails that intuition be incorporated with reason in its formal interpretation.
So what we commonly understand as Mathematics in fact represents a special limiting case where dynamic interaction (of reason and intuition) is ignored.
This criticism is also very true of scientific understanding in general (including Physics).
It is still formally based on the (conventional) rational paradigm. This gives rise to reduced static interpretations which ignore the essential dynamics of experience.
Many of the findings of modern Physics (e.g. quantum mechanics) point to the inherent dynamic interaction of natural processes. However because these are still interpreted through a static (non-interactive) paradigm they appear counter intuitive.
With a more comprehensive paradigm, all findings in Physics would be fully intuitive (when interpreted from the perspective of this more refined understanding).
The integral translations represent a very significant step in the development of this more comprehensive worldview.
The Pythagorean Dilemma
The main problem with the modern interpretation of mathematics is that the quantitative has become largely divorced from its corresponding qualitative aspect. This has led to a great loss in appreciation of the deeper dynamic meaning inherent in all its symbols. Also, it has resulted in a corresponding failure to recognize the enormous potential of the qualitative mathematical interpretation for a truly integral appreciation of science.
It has not always been this way. The Pythagoreans for example, sought to maintain an essential balance as between quantitative and qualitative aspects.
Indeed it is ironical that the famed triangle (named after them) led to a problem which threatened to undermine the very basis of their approach.
This was the discovery that the square root of 2 is irrational.
Because of its great importance it is worth exploring the precise nature of this problem at greater length.
Once again the Pythagoreans sought to maintain harmony as between the quantitative and qualitative aspects of mathematical understanding. Indeed through maintaining the relationship between these two aspects, mathematics was seen as a highly valuable meditative tool ultimately leading to pure spiritual contemplation.
Before the discovery of the square root of 2 this essential harmony appeared to exist.
The Pythagoreans scientifically interpreted reality in terms of the rational paradigm (which provided the qualitative aspect).
Likewise they believed that all numbers were rational (i.e. could be expressed as fractions).
So the quantitative number system was in direct correspondence with the qualitative interpretation (of this system).
However the simplest of all right angled triangles (where horizontal and vertical lines = 1) raised an insoluble dilemma.
For - in accordance with their theorem - the diagonal line of this triangle (i.e. the hypotenuse) has a length of the square root of 2. As the Pythagoreans could not express this as a fraction, it was therefore an irrational number. 13
So the essential balance as between quantitative and qualitative aspects was broken.
The Pythagoreans were not able to explain how an irrational number could arise within the context of a paradigm, which was rational. Putting it another way they lacked a deep philosophical appreciation of the meaning of an irrational number.
Subsequently in Western society a merely reduced interpretation of mathematics has been developed (where quantitative interpretation is largely separated from philosophical appreciation).
Since the Pythagoreans, many other number types - besides the rational - have been discovered.
For example there are transcendental numbers (which represent another type of irrational quantity). Then there are imaginary, complex and transfinite numbers. However all of these are understood in a merely reduced quantitative fashion. Their deeper qualitative interpretation - with immense significance for philosophy and science - has been largely lost.
The rational translation of this stage can however solve the Pythagorean dilemma and throw considerable light on the nature of Holistic Mathematics.
When one obtains the square root of 2, the answer is expressed in terms of opposite polarities. In other words the answer can be given as either + 1.4142 or - 1.4142.
Equally the understanding of H1, i.e. the complementarily of horizontal directions, is expressed (in reduced linear format) in terms of opposite polarities. In other words it has both positive and negative directions.
The structure of the "higher" (psychological) qualitative exactly complements the "lower" (mathematical) quantitative translation.
Thus when we mathematically reduce in quantitative terms a number from two to one dimensions (by obtaining the square root of 2), the interpretation always splits into two opposite aspects.
Likewise when we reduce in a psychological qualitative sense by expressing two-dimensional understanding in reduced one-dimensional terms, the interpretation again splits into two opposite aspects.
The similarities do not end here.
An irrational number (such as the square root of 2) has both finite (quantitative) and infinite (qualitative) aspects.
Its quantitative value can be given in (reduced) terms as a rational approximation.
Thus correct to four decimal places the square root of 2 is 1.4142.
Thus every irrational number has a (reduced) rational interpretation which can be expressed to any required degree of accuracy.
Because mathematics operates out of the rational paradigm it essentially interprets irrational numbers in this reduced fashion.
However every irrational number also has a qualitative aspect in that its decimal sequence continues indefinitely. Its value in other words can never be absolutely expressed in rational terms.
Again this exactly complements the nature of psychological understanding at H1.
Symbols still carry a quantitative (finite) meaning. Thus one can distinguish object phenomena.
However they now equally carry a qualitative (infinite) meaning.
Thus when I look at an object such as a rose it has a deeper qualitative significance pointing to a universal spiritual order.
Thus the objects no longer have an absolute interpretation at this level, but carry more profound meaning where quantitative interpretation is increasingly imbued with a richer more mysterious qualitative significance.
Thus "lower" irrational number behavior (with its finite and infinite characteristics) once again complements "higher" psychological understanding.
I will just point to one more connection.
Though the square root of 2 is irrational in one-dimensional terms, when expressed in two-dimensional format it becomes rational (with a value of 2).
Likewise bi-directional understanding is irrational (i.e. paradoxical) from a one-dimensional (i.e. linear) perspective. However when interpreted from the appropriate "higher" level, it becomes fully rational (in terms of the understanding of this level). 14
The Pythagorean Dilemma cannot be explained in terms of H0 (the rational linear level).
However it can be explained in terms of the understanding of H1. Here, "higher" psychological understanding precisely complements the "lower" number behavior of the square root of 2.
When properly appreciated, the (quantitative) finding of the Pythagoreans that the square root of 2 is irrational has a precise complementary qualitative interpretation with major philosophical implications.
This could be expressed by saying that duality - when interpreted from the perspective of the two-dimensional understanding of H1 - is irrational (i.e. paradoxical). 15
By extension the understanding of the "higher" levels is always paradoxical when expressed in (reduced) linear (i.e. one-dimensional) terms. 16
Holistic Mathematics provides the qualitative interpretation of this important implication of the Pythagorean Theorem. Potentially it can provide dynamic qualitative interpretations for all mathematical symbols and relationships.
These in turn provide the most precise basis for true integral understanding where science and philosophy merge.17
Rational and Irrational Paradigms
Our very use of the term "rational paradigm" indicates a qualitative mathematical interpretation.
A rational number can be subdivided into parts (fractions); equally the whole number is defined as the sum of its parts.
Likewise the very basis of the rational paradigm in science is the belief that reality can be subdivided and analyzed with respect to its parts. Equally "whole" understanding is then defined in terms the sum total of its constituent parts.
Thus the rational paradigm is inherently very limited as it creates - in any context - a marked tendency to reduce qualitative to mere quantitative understanding.
The rational paradigm - as employed in science - simply represents the understanding of H0.
However just as the science of H0 can be given a holistic mathematical interpretation, equally the science of all other levels can be defined in like manner (using the appropriate number types).
We have seen that the understanding appropriate to H1 has a complementary qualitative interpretation to that of an irrational number (the square root of 2).
Thus if the scientific approach of H0 is characterized as the rational paradigm, then the paradoxical scientific approach of H1 can be accurately characterized as the irrational paradigm.
The implication - by extension - is that every number type (when qualitatively defined) is associated with a distinctive scientific paradigm. 18
So just as we can have rational and irrational numbers (in quantitative terms), we can also have rational and irrational paradigms (in a qualitative fashion).
Thus the scientific understanding of H1 corresponds precisely - in holistic mathematical terms - to the irrational paradigm.
Every mathematical symbol, operation, theorem etc. can be given a richer qualitative interpretation (with precise complementary characteristics).
As we know number is the most appropriate means for ordering quantitative data.
However - what is not yet appreciated - is that number is equally the most appropriate means for ordering qualitative data (when given a holistic interpretation).
This means for example that the various stages of the Spectrum of Consciousness can be classified most precisely through using the qualitative interpretation of the number system. 19
Let me illustrate this once again with respect to the key numbers 0,1 and 2 and the operations of addition and subtraction.
In quantitative terms 0 and 1 are extremely important. All numbers can be expressed in terms of the binary system using these two digits. The digital revolution offering limitless possibilities for the encoding of information is based on the binary numbers.
In deeper qualitative terms, the binary digits are equally important. Ultimate awareness is intimately linked to the interpretation of these two numbers where they are inseparable.
This is well expressed in the most famous of Buddhist sutras.
"Form is not other than Void
Void is not other than Form."
This could be equally expressed as
Oneness is not other than Nothingness
Nothingness is not other than Oneness.
So we can now use our customary number digits in a holistic fashion,
1 = 0
And 0 = 1,
where the numbers here obey an alternative logical system.
The quantitative (analytic) interpretation is based on clear differentiation and separation of digits (logic of form).
However the qualitative interpretation is based on the complementarily and ultimate identity of these same digits (logic of emptiness).
Just as the two binary digits - with a quantitative (analytic) interpretation - can be used to encode all information, equally the same two digits can be used to encode all transformation when given a corresponding qualitative (holistic) interpretation.
Thus all transformation processes can be precisely encoded through using the logic of form and the logic of emptiness. 20
The two most basic operations in mathematics are addition and subtraction (positing and negating).
These two operations have a fundamental dynamic significance.
Conscious understanding is literally based on the positing of phenomena.
Unconscious experience requires their corresponding negation.
Thus psychologically - in direct terms - we make conscious through (dynamic) positing of phenomena; we make unconscious - in direct terms - through (dynamic) negation.
Thus experience involves a continual process of positing and negating phenomena.
These two operations are inherently combined in dynamic terms with the binary numbers.
Purely intuitive experience (spiritual emptiness) entails the total negation of (posited) phenomena.
So phenomena are then literally erased from consciousness the very instant they arise.
So just as 1 - 1 = 0 (in quantitative terms), equally 1 - 1 = 0 (in qualitative terms).
In other words (spiritual) emptiness results from the total negation of (conscious) form.
Now when one tries to interpret reality in (solely) conscious terms then both horizontal aspects of experience are given a positive interpretation.
So just as 1 + 1 = 2 (quantitatively), equally 1 + 1 = 2 (qualitatively).
In other words duality results from the merely conscious interpretation of phenomena (in relation to both poles).
I have briefly illustrated here with reference to the fundamental numbers and operations.
However the potential scope of Holistic Mathematics may be better appreciated when one grasps that every mathematical symbol and relationship, with a quantitative interpretation, can be equally given a deeper dynamic appreciation which is holistic.
The crucial implication of this is that all mathematical concepts have a dynamic experiential interpretation (with complementary applications in both physical and psychological terms).
The reason why we fail for example to see that an imaginary number relates directly to experience is due to the inadequate means by which we attempt to scientifically interpret this experience (within the confines of the rational paradigm). However when correctly understood the holistic interpretation of imaginary numbers has an extremely important experiential application. 21
I suggested a holophysical interpretation for Einstein's Special Theory of Relativity in "Fascinating Connections" The corresponding interpretation for some of the key insights of his General Theory would be properly associated with this stage.
There are several interesting parallels.
Einstein showed how light - assumed to travel in a straight line - could be bent by gravity in the presence of matter. Ultimately with sufficient gravity - as in a black hole - light is completely bent in a circular fashion (and unable to escape the event horizon).
Psychological reality can be understood in reverse complementary manner.
At the rational stages, matter (i.e. phenomenal reality) is understood in a straight-line fashion (i.e. the linear level). Then at the circular level, due to the presence of spiritual light (i.e. intuition), matter begins to bend with phenomena understood in increasingly relative terms. When this spiritual light is sufficiently intense - as in the "dark night" stage, reality becomes completely circular with all phenomena in understanding becoming internalized in the unconscious.
Einstein simply explained gravity as due to a distortion in the fabric of space-time.
So in the presence of matter the force of gravity would exercise itself through curving space-time.
Now the concept of gravity has complementary psychological interpretations (which are rarely recognized).
In psychological terms the dimensional framework of our experience is provided - directly - through concepts. So all perceptions (as phenomena) must be placed for interpretation within a framework of concepts (i.e. space and time).
In dynamic terms perceptions and concepts ceaselessly interact. So concepts are continually changed through interaction with perceptions.
Conventional science operates very much in terms of flat space-time (in psychological terms). Indeed Einstein was a prime proponent of this position.
Flat space-time in this context simply represents the linear view that conceptual formations are unaffected through interaction with perceptions.
In other words the truth of a theory is unaffected through relationship to particular
However in dynamic terms this is untenable. Concepts are bent (become relative) through interaction with perceptions. Thus the strictly linear viewpoint breaks down in terms of the natural dynamics of experience.
Space-time - psychologically - is always curved to some extent and becomes especially pronounced with a high level of interactivity.
Gravity in physical terms is necessary to hold matter together.
It is similar in psychological terms where a force is needed to hold our phenomenal experience in place.
This binding force comes through the interaction of our conceptual structures with perceptual phenomena. Through this we can place - in a psychological sense - matter in a meaningful framework of space and time.
When the psychological dimensional framework is insufficiently developed, stable structures are not possible. In other words psychological gravity cannot properly exercise its force.
It is somewhat puzzling that we readily recognize psychological equivalents to physical notions of light and energy. However this does not equally apply to denseness and gravity.
And yet if we reflect on it there are clues in our language. We may refer to a person as having a grave or serious manner. Indeed in stereotypical terms this is often associated with someone with a decidedly intellectual bent.
Now the very essence of intellectual activity is the formation of general conceptual structures with which to organize perceptual phenomena. Not surprisingly this can lead to a greater development in psychological gravity.
By contrast someone with a lighter personality, generally relates better to immediate perceptual phenomena (without the same attempt to organize them conceptually). This then leads to a more superficial personality (and a corresponding absence of gravity).
Gravity in (interior) psychological terms (i.e. grief) can be accurately defined in terms of this curving of space-time which involves the increasing internalization of experience through the dynamic negation of psychological dimensions.
Thus grief arises from the erosion of the conceptual framework we use to organize reality. When this erosion is great the experience of grief is profound.
So the death of a loved one can lead one to question the fundamental assumptions on which life is built. This serves to erode deep-rooted concepts, which distorts the very fabric of psychological space-time and results in the feeling we know as grief.
The key insight that led to Einstein's formulation of the General Theory was the realization that gravity and acceleration are equivalent.
Once again this - by definition - has a holophysical interpretation.
Thus if we were in an elevator that was allowed to fall freely to earth we would not experience gravity.
Now the corresponding acceleration in psychological terms entails the authentic exercise of faith.
This tends to loosen the attachment to conceptual poles thus reducing grief.
The problem of grief arises due to attachment to our concepts and assumptions. When this attachment is removed we no longer experience loss. So in this way we literally become free of grief.
One of the great needs in modern physics is to reconcile gravity and light (the gravitational and electromagnetic forces).
I hope to eventually show through this integral approach that they are intimately connected in a more fundamental symmetrical fashion. Also in this integral approach, physical is fully complementary with psychological understanding.
Our conventional interpretation of space and time is asymmetric.
So we operate on the principle of three space dimensions and one time dimension.
This understanding of space-time is certainly suited to the world of causality and phenomenal analysis of phenomena (reflecting the differential approach).
However it is inappropriate for true integral understanding which is based on a symmetrical approach to relationships.
Now in mathematical terms as we have seen when we express a two-dimensional relationship in (reduced one-dimensional) terms that we get two real opposite poles.
Thus if we take the very simple equation
x is here defined mathematically in two-dimensional terms (i.e. is raised to the power of 2) and has a value of 1.
Now when we express the value of x in reduced one-dimensional terms (i.e. raised to the power of 1),
x = + 1 or x = - 1.
Now what is fascinating is that the integral (symmetric) understanding of dimensions corresponds perfectly to the mathematical notion.
So there is in fact a deeper qualitative mathematical structure underlying the most fundamental interpretations of space-time. Somehow however these have been largely missed in scientific understanding.
So, when from the perspective of H1 (which is inherently two-dimensional) 23 the nature of physical dimensions are expressed in standard linear terms, then both space and time are symmetric with positive and negative polarities.
So from this symmetrical perspective, space is two dimensional with positive and negative (real) directions;
Likewise time is two-dimensional equally with positive and negative (real) directions.
This symmetric interpretation is fully consistent with the nondual view of reality.
Reality here is essentially spiritual and continually exists in the absolute present moment.
Space and time are then understood as an expression of the secondary phenomenal characteristics of reality, which have a merely relative interpretation.
So all relative interpretations are grounded in the absolute present moment.
They arise from - and return to - this continual present and extend relatively in opposite directions in both space and time.
So we could say that in absolute terms, all holons are grounded in the present moment.
However in relative phenomenal terms, movements in space always take place in positive and negative directions;
Equally – and very importantly – movements in time also always take place in both positive and negative directions.
This symmetric understanding of space and time has very important practical applications, which I will illustrate, in the next section.
Once again nondual reality exists in the continual present. All relative phenomenal expressions of reality are grounded in this absolute moment.
However the conventional way of looking at time is clearly inconsistent with nondual understanding.
In this view the arrow of time moves irreversibly forward. This then serves to greatly cut off phenomenal reality from the present moment.
So from a linear perspective past events have already happened and are irreversible.
Future events have not yet happened and are unattainable (in the present moment).
This simply reflects a linear view of space and time (where both move in one direction only).
However the integral perspective of this level is very different. Here both past and future events are connected through the present moment.
Let us look at this more closely.
Whenever I reflect on some "past" event, say that happened one year ago, I am by definition starting from experience of the present moment.
Now there are in fact two ways of interpreting this "past" event.
From the reference point of the self, then the event is relatively in the past. Thus from this perspective it happened one year backward in time (-1 with respect to the present).
However from the reference point of the actual event, then the self is relatively future (in the present). In other words the reflecting self is now one year forward in time (+1 with respect to the event).
So in dynamic interactive terms, time is merely relative where the self and the event are both positive and negative with respect to each other.
It is somewhat similar in terms of "future" events. If I anticipate an event that is to happen in a year’s time, in dynamic terms two equally valid relative interpretations are involved.
Relative to the self, the event is one year forward in time.
However from the perspective of the event, the self is – relatively – one year backwards in time.
So the dynamic interaction once again involves positive and negative directions.
It is the very appreciation of this necessary two-way relativity of all "past" and "future" events that leads one to the realization that the present moment alone absolutely exists.
Now the problem with conventional interpretations is that it continually confuses these two directions of time (which are merely relative). Thus by ignoring the sign of time (i.e. as positive or negative) one then literally obtains an absolute interpretation (where time moves only in one direction).
So it is not possible to reconcile a rigid absolute interpretation of time (which is one-directional) with the nondual present moment (which alone is truly absolute).
Thus the misleading interpretation of time (as one-dimensional), that is so rigidly maintained in science, is one of the greatest barriers to its reconciliation with authentic spirituality.
From the correct integral perspective space and time are merely relative expressions of the present moment (each having opposite directions).
This leads naturally to a more dynamic view of experience where both the "past" and the "future" are continually recreated out of the present moment.
So from this perspective the meaning of a "past" event is continually changed through being related to the present.
Likewise the meaning of a "future" event is continually altered with reference to the present.
This is extremely liberating for it gives us power over both our past and future. We need no longer be hapless victims of the "past" or passive slaves to the "future".
This symmetric notion of time is extremely important in psychological terms.
All of us have been wounded in various ways by "past" events. However we have the continual power to recreate this "past" through the present and achieve healing.
Likewise we are all anxious and fearful to some degree regarding the future.
However this ultimately reflects a lack of sufficient faith in our power to recreate this "future" through the present. Ultimately it is only the present moment continually renewed that truly exists.
People who have a healing gift generally are very intuitive and implicitly operate on a symmetric view of time (that is grounded in the present).
This symmetric view is inherent in all natural processes and has not been incorporated into mainstream scientific understanding.
This in turn is due to the fact that science is still heavily rooted in a limited and inadequate paradigm.
The symmetric view of dimensions (space and time) is fully consistent with paranormal findings.
Since all events are bi-directionally rooted in the present moment, then one potentially has the power to influence any event (through the present).
Thus from this perspective there is a coincidence or synchronicity of all events in the universe (connected through the present moment).
Thus through spiritually connecting events (through this present) one potentially has the power to influence all events.
So this synchronous acausal principle of connectivity is simply an expression of an integral symmetric view of reality (and is very much consistent with this stage of understanding).
It is important to stress once again the complementarity of all physical and psychological relationships (in integral terms).
Thus the notion that space and time have both positive and negative directions applies equally to physical and psychological reality.
The recognition of this complementarity in turn leads to a strong growth in spiritual intuitive understanding.
When this is sufficiently developed, a remarkable synchronous interplay can take place between the psychological self and natural world. Essentially this involves looking on all reality as a living organism so that one treats nature as an intimate friend. And when nature is treated in this fashion it tends to respond in like manner. 24
Once again – despite its great achievements - conventional scientific interpretation is fundamentally unbalanced and itself breeds considerable alienation from the environment.
Once one accepts the complementarity of all physical and psychological processes, then "inanimate" nature no longer has meaning.
One important implication of this perspective is that a physical law is not - strictly speaking - absolute.
Such a law represents a dynamic interaction pattern which maintains remarkable stability.
However potentially, every law can be changed – however slightly – through psychological interaction.
The belief in fixed laws is itself an expression of a worldview where objective reality is given an independent existence. However once we accept that all laws involve a two-way mind-matter interaction, then the potential for change becomes apparent.
In future ages physical "laws" will become subject – to an increasing extent – to psychological interference.
We have looked more deeply at the underlying nature of the Integral 1 Method to discover –remarkably - that it has a purely mathematical structure (when interpreted in a qualitative sense).
I term this complementary qualitative interpretation of mathematical symbols as "Holistic Mathematics" (Holomatics).
Though it provides a precise basis for a true integral approach to reality it has been greatly ignored in the development of science. 25
1. This would correspond to the vision-logic stage (Centaur) of the previous level.
Basically it arises from the interaction of the concrete and formal understanding of the earlier sub-levels, which greatly facilitates intuitive insight.
Just as the three sets of polarities (horizontal, vertical and diagonal) can be used to define the various levels, equally they can be used to define the sub-levels (within each level). In relative terms if sub-level 1 (concrete) involves horizontal, then sub- level 2 (formal) involves vertical and sub-level 3 (vision) involves diagonal understanding respectively.
2. In qualitative mathematical terms, horizontal polarities can be defined more precisely as the positive and negative directions (in real terms).
Geometrically, they correspond to the two roots of unity, + 1 and – 1 (with a qualitative interpretation). The extremities of these lines represent two points on the circle (with radius one unit) and can be connected by a horizontal straight line (representing the x axis stretching in both directions from the center to the circumference).
3. Once again the development of affective, cognitive and spiritual development within a stage can be precisely defined - relatively – in horizontal, vertical and diagonal terms.
Thus switching from affective to cognitive understanding involves a decisive transformation from a horizontal to vertical mode of understanding. The precise qualitative mathematical significance of this will be clarified at H2.
Likewise the shift from either affective or cognitive to spiritual understanding involves a more complex switch to a diagonal mode of understanding. This has an even more fascinating qualitative mathematical interpretation (which will be clarified at H3).
4. The use of five phases is very convenient in terms of highlighting the cyclical wave characteristics of the stages of H1 (concrete and formal).
5. There are definite similarities in the manner of development of H1 (the circular level), with the rational stages of the linear level. At the linear level, we have firstly the more concrete stages (based on the senses), which develop in an exterior and later in an interior direction. It is this very change in direction, which lessens exclusive identification with the senses, paving the way for the formal stages which are far more conceptually based, again first in an exterior, and then in an interior direction.
Likewise at the circular level, we start with the "higher" intuitively based development of the senses in an exterior positive direction. Over-involvement, with this form of knowledge, necessitates withdrawal (through mirror structure development), leading to the subsequent development of the intuitively based senses in an interior direction, finally stabilizing - in terms of direction - in a more central and neutralized spiritual equilibrium.
Through detachment from the intuitive senses thus exercised, one is now free to develop the deeper more conceptualized regions of the personality, again in this explicitly intuitive fashion. I am now dealing with the formal sub-level (in its exterior direction).
6. Perhaps the most developed Western expression of this understanding is to be found in Hegelian philosophy. However strictly speaking Hegel gives a reduced view of the understanding of this level.
In dynamic terms its very essence consists in the maintenance of a balance as between spiritual intuition and bi-directional understanding.
Whereas Hegel’s philosophy undoubtedly is derived from authentic spiritual insight, its expression is largely collapsed in terms of a bi-directional form of understanding that is somewhat rigid. Thus it does not lead on to purer form of contemplative insight. Rather it tends to reduce this truly spiritual understanding to philosophical speculation (of a bi-directional rational kind).
7. Any formal notions of an absolute "objective" reality break down fully at this level. We are always participators in the reality we attempt to interpret and this necessarily changes that reality. So all theories (including mathematical) have here a merely relative truth-value.
8. Bi-directional understanding is very subtle and is difficult to communicate in linear language.
It is not enough to accept that relationships involve opposite poles. (This would occur at the vision-logic stage of H0).
It is also necessary to appreciate that the understanding of these poles is purely relative.
Thus what is exterior and what is interior in any context depends on a fixing a frame of reference (with respect to one pole). However from the alternative frame of reference (fixed with respect to the other pole), the opposite interpretation is equally valid. So what is exterior and what is interior themselves keep switching in understanding until ultimately with pure intuition they no longer have a separate meaning.
The failure to appreciate the relativity of frames of reference leads – from an integral perspective – to an unbalanced and inconsistent approach where truth tends to be identified with just one pole.
This of course entails that the vision-logic of H0 is not suitable as a true integral approach.
9. Throughout development, vertical transformation to a new stage is then followed by horizontal translation through the structures appropriate to that (new) stage. Whereas the vertical upward transformation of structures intensifies experience, the horizontal translation within a given structure tends to make it more extensive. However transcendent development can be likened to a building, shaped like a pyramid (or a cone) with several stories. On the ground floor, which involves little or no vertical elevation, there is the greatest extension of space available. However as we proceed upward to higher floors, progressively less space is available, until finally we reach an apex or point at the top. In like manner, in psychological terms, it is easier to translate experience horizontally (i.e. rationally) when vertical experience is minimized. However as the intensity of vertical (i.e. intuitive) experience increases, it becomes more difficult to carry out this translation (which at best can only be done increasingly through indirect means). At the highest level of pure mystical contemplation, the attempt to translate horizontally is abandoned altogether.
By phase 2 of the exterior stage the purity and intensity of the initial illumination will have receded somewhat with the intellect more actively involved. However, as we have already seen, this translation - of what is essentially intuitive - can only be done through the indirect medium of paradox.
10. The very interpretation here of a theory is bi-directional with positive and negative aspects (in real terms).
In other words in dynamic terms "truth" relates to the interaction of each theory with its mirror theory.
11. The decisive break through, which enabled Andrew Wiles to solve " Fermat's Last Theorem", is a good example of a rare intuitive revelation which may come just once in a lifetime.
In Physics, Albert Einstein is a prime example of one who had a particularly strong reliance on simple but profound intuitions for the development of his theories.
12. The avoidance of mathematical equations in the "popular" presentation of Physics is sometimes carried to extremes as with Stephen Hawking's "A Brief History of Time". However it certainly indicates the reluctance of editors to trust in the abstract rational capacities of their readers.
13. Strictly defined the square root of 2 is an algebraic irrational number (i.e. can be expressed as a solution to a polynomial equation with rational coefficients).
So the square root of 2 is the solution for x to the simple equation x2 = 2.
The other type of irrational number - which cannot be expressed in this manner as a solution to a polynomial equation - is transcendental.
p (pi) is the best known transcendental number.
The acceptance of Holistic Mathematics would suggest a very different kind of qualitative mathematical proof (which is based on establishing satisfactory complementarity as between mathematical and psychological behavior.
In all I would suggest three main types.A comprehensive proof belongs to what I call Radial Mathematics. Again it requires that satisfactory correspondence be established as between both quantitative and qualitative "proofs".
Analytic Proof. This is the standard notion of (quantitative) mathematical proof.
For example the Greeks proved 2,500 years ago that the square root of 2 is an irrational number.
Synthetic Proof. This is the qualitative notion of proof (based on dynamic complementary connections).
I have suggested such a qualitative proof for the square root of 2, which provides a deeper integral explanation.
(The Pythagoreans sought unsuccessfully for such a proof).
Comprehensive Proof. This combines both analytic and synthetic approaches with parallel connections between both approaches.
Thus a radial mathematical proof that the square root of 2 is irrational would require comparing both analytic and synthetic proofs so as to establish a satisfactory correspondence as between both approaches.
14. This is well illustrated by Hegelian philosophy. When Hegel employed Reason, he referred to the dynamic bi-directional (i.e. two-dimensional) interpretation characteristic of H1 (i.e. dialectical reason).
Hegel referred to the "lower" conventional use of reason – based on one-directional interpretation – as "understanding".
15. Dimensions and directions are in fact synonymous terms, and have - as we have seen - a precise mathematical interpretation. Thus the two-directional understanding of H1 is - in qualitative mathematical terms - two-dimensional. (H0 - the previous level - is one-directional and therefore literally one-dimensional i.e. linear).
16. In mathematics when any higher dimensional quantity is reduced in one-dimensional terms, at least one irrational solution will result. Likewise if the understanding of any "higher" level is reduced in linear rational terms, paradox will arise.
Thus if x10 = 3, then x has an irrational solution.
However there is a decisive difference as between two dimensions and those greater (i.e. higher) than two. Whereas reduced interpretations in respect of two dimensions will always be real, this will never be so in the case of three or more dimensions where complex results arise. (Interestingly, this is the simple basis for a qualitative interpretation of Fermat's Last Theorem).
Likewise, in psychological terms there is a decisive difference as between the two-dimensional understanding of H1 and the greater dimensional understanding of the other higher levels (which will be clarified in due course).
17. The great physicists of this century e.g. Einstein, Heisenberg, Schrodinger, Pauli were invariably deeply interested in the epistemological issues underlying their scientific work. At a fundamental level of discovery, Physics tends to merge with Philosophy.
In an important sense, at the "higher" levels of understanding, Holophysics becomes inseparable from Philosophy with an underlying structure that is precisely mathematical (in a qualitative sense).
18. The word "paradigm" can be confusing. I am using the term here in the broadest sense to represent a distinctive logical framework of interpretation for science.
In terms of this interpretation, basically all the developments in conventional science have taken place within the same paradigm.
So from this perspective, Einstein's Theory of Relativity employs the same paradigm as Newton's Theory of Gravitation.
In the sense I use it a paradigm is more strictly a "meta-paradigm" and would be synonymous with my definition of a "Method of Translation".
Thus rational (analytic) science broadly constitutes one meta-paradigm (Differential I Method).
The meta-paradigm appropriate to H1 is the Integral 1 Method.
19. What is truly fascinating is that the number system (with its wide range of different number types) provides the most appropriate way for ordering the entire Spectrum of Consciousness.
Every level - indeed every sub-level and stage can be precisely defined in terms of the qualitative interpretation of the mathematical number system.
Indeed from this perspective, the Spectrum of Consciousness is the qualitative number system.
There is therefore a two way correspondence as between numbers as quantitatively defined and numbers as qualitatively defined (where they relate to different levels of development).
Thus if a major new number type was to be discovered in quantitative terms, then that would imply that a corresponding psychological level could be identified (in qualitative terms).
Equally if a "new" level of psychological development were to be discovered, then this would imply the corresponding existence of a "new" quantitative number type.
I will outline briefly here the levels of development and their corresponding qualitative number mapping.
L3 - Binary Nos.
Transition from L3 to L2 - the number 2
L2 - Prime Nos.
Transition from L2 to L1 - number as dimension
L1 - Natural Nos.
Transition from L1 to L0, H0 - Integers
L0, H0 - Rational Nos.
Transition from H0 to H1 - Negative Numbers
H1 - (Algebraic) Irrational Nos.
Transition from H1 to H2 - Imaginary Numbers
H2 - Transcendental Nos.
Transition from H2 to H3 - Complex Numbers
H3 - Transfinite Nos.
R1 - Growing differentiation of each number type separately combined with integration of all types simultaneously (in both quantitative and qualitative terms)
R2 - Dynamic interpenetration of quantitative and qualitative interpretations in simple understanding.
So in the articles on Holophysics, I have addressed the unfolding negative numbers in qualitative terms (i.e. the negative direction of mirror understanding) which relates to the transition from H0 to H1.
I have also addressed in this article the qualitative counterpart to (algebraic) irrational numbers though outlining the deeper roots of paradoxical understanding.
20. This is a simple idea with enormous implications. It means that all transformation processes - physical and psychological - can be encoded through the qualitative binary system (using the two logics of form and emptiness).
I demonstrated briefly in "Spectrum of Methods" how this encoding could be done for the Spectrum of Consciousness.
21. Perhaps the most fundamental of all relationships is that between whole and part (part and whole).
Quite simple when we try to translate this relationship in "real" terms - as in (conventional) science the qualitative aspect is inevitably reduced to mere quantitative interpretation.
However, mathematics provides the precise (unrecognized) basis for a correct translation.
The comprehensive number system comprises imaginary as well as real members.
To translate the relationship between whole and part (and part and whole) involves moving to a complex interpretation of reality (with real and imaginary aspects).
Thus correctly interpreted, insofar as we order reality in a qualitative rather than a quantitative manner we "see" imaginary numbers.
However this is a topic properly reserved for H2 understanding.
22. The historical evolution of scientific understanding itself contradicts the flat space-time view.
The dynamics of continually testing theories by observations leads ultimately to the emergence of new theories.
23. Again there is full complementarity as between physical and psychological interpretations of space and time.
If for example space or time have a positive direction in physical terms, then they has a negative direction – relatively - from a psychological perspective.
"Higher" dimensions give rise new to fascinating interpretations of the qualitative nature of space and time (in these dimensions).
24. Holons - at whatever level in nature - have exterior and interior aspects. However, Physics still only operates in terms of the exterior aspect. So the dynamic interaction of both aspects - exterior and interior - implies that all holons in nature is "animate" in a fundamental sense.
25. I have naturally been interested in finding historical precedents for Holistic Mathematics in Western culture. However I have found very little that is both coherent and systematically developed.
Even the Pythagoreans had no genuinely integral conception of Mathematics.
They did indeed recognize the importance of the qualitative as well as the quantitative dimension of mathematical symbols, but their interpretation was in terms of a reduced rational framework. Thus they were unable to find a satisfactory explanation for the square root of 2.
Leibniz was probably the greatest Western thinker with a holistic mathematical approach to understanding.
His philosophical conception of the "monads" was an especially interesting attempt to grapple with the fundamental relationship of whole and part.
He invented the quantitative binary system (based on 1 and 0). He was also deeply aware of the metaphysical significance of 1 and 0 (So implicitly he also hit on the notion of a complementary qualitative binary system).
Nicholas of Cusa combined mystical insight and reason in a holistic mathematical fashion.
Jung is the best known modern thinker who adopted an approach that – at least implicitly – is mathematical in the holistic sense. Not surprisingly many of his concepts (e.g. Personality Types) lend themselves readily to a more explicit holistic mathematical formulation.
Holistic Mathematics requires a particular combination of authentic spiritual insight allied to rigorous rational interpretations of a qualitative nature, (which are not usually combined). Quite simply it cannot be understood from a (reduced) mathematical perspective.
In Enneagram terms it would be consistent with a mystical personality that combines characteristics of both the 4 and 5 types.
So the map of development I have outlined, would represent just one possible profile of mystical development that would fit the 4-5 Enneagram type.
However this map is especially suited for the purposes in hand which is to demonstrate how an integral scientific approach – rooted deeply in Mathematics – can be formulated.