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Integral World: Exploring Theories of Everything
An independent forum for a critical discussion of the integral philosophy of Ken Wilber

Peter CollinsPeter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

An Integral Mathematical Stage Model of Perspectives

Part 2: Higher Levels

Peter Collins

The discussion in Part 2 continues into the higher levels with respect to both structures and perspectives. Integral mathematical interpretation cannot replace the need for authentic experience of the higher levels. However when appropriately based on such experience it can be of considerable scientific benefit in precisely clarifying the dynamic structural nature of these stages.

The Higher Levels

In the first part we dealt briefly with the lower and middle levels showing how the structure of each stage is inherently mathematical in a holistic binary digital sense. We then extended such mathematical notions into the discussion of perspectives indicating how - for each level - they are precisely defined by the dynamic integral interpretation of the major number types.

We now continue this discussion into the "higher" levels. 1 However because the proper interpretation of the "lower" depends on the mature understanding of the "higher", we will discuss the fundamental structural features of these levels in more depth.

Higher 1 (H1)

This stage - which I also refer to in holistic geometrical terms as the (mature) circular level - corresponds broadly with what Ken Wilber calls the psychic/subtle realm. 2

Entry to the stage is usually preceded by a profound existential crisis, the purpose of which is to gradually wean one from dualistic attachment. Though the direct basis of such attachment is volitional (in the misleading identification of the supreme spiritual good with limited phenomena) indirectly it is associated with the unbalanced intellectual appreciation of the nature of polar reference frames.

From the standpoint of the nondual, all dualistic notions of direction have a purely relative meaning. Thus to understood relationships in an asymmetrical fashion i.e. as dualistic, we must rigidly fix the polar reference frame in just one arbitrary manner (even though the opposite reference frame is equally valid). Heraclitus is perhaps the earliest Western thinker to have a clear grasp of the nature of the problem. Though I have repeated this example time and time again, I fear that its subtle but crucial implication for the study of all development processes is rarely grasped.

Imagine if we were stranded in a desert and encountered a straight road! Now by pointing in one direction we could arbitrarily fix it as "up", in which case the opposite direction would be unambiguously defined as "down". However we could initially point in the alternative direction along the road defining it as "up" in which case the opposite direction would be "down". So therefore what is defined as "up" in terms of the 1st reference frame is "down" in terms of the 2nd; equally what is defined as "down" in terms of the 1st is "up" in terms of the 2nd frame.

So when Heraclitus says that the way "up" is the way "down" and the way "down" is the way "up", he is pointing to the necessarily arbitrary nature of all polar reference frames (when considered in dynamic interactive terms).

From a nondual integral perspective polar opposites have a merely relative meaning. However to differentiate in dualistic fashion we must unambiguously fix the polar reference frame in just one arbitrary manner (though the opposite interpretation is equally valid). And as we have seen, for any pair of polar opposites, we have two equally valid ways of doing this. Though each of these differentiated interpretations is unambiguous when viewed in a linear independent manner, when both interpretations are viewed in circular terms as simultaneously interdependent, deep paradox results.

Therefore the consistent move from (dual) differentiated to (nondual) integral understanding requires that all polar relationships be viewed in a bi-directional linear manner (where two equally valid unambiguous interpretations are given for events that are opposite to each other in direction) Then when both interpretations are viewed in a simultaneous circular manner as interdependent, dualistic paradox inevitably arises. It is the implicit acceptance of such paradox that lessens dualistic attachment to the merely differentiated appreciation of phenomena, thereby facilitating transition to nondual spiritual awareness. Therefore though - in direct terms - integral awareness is always of a nondual spiritual nature, indirectly it is enabled - at least implicitly - through the keen recognition of paradox with respect to the dual world of (differentiated) form.

As vision-logic is still formally based on the one-directional interpretation of asymmetrical relationships, it thereby still affirms one-sided dualistic understanding (i.e. with reference to just one reference frame). This is then inevitably associated with consequent dualistic attachment from a cognitive perspective (generated through limited unambiguous interpretation). Therefore we cannot consistently move to nondual integral awareness while still maintaining the dualistic interpretation that vision-logic fosters.

The intellectual vision of H1 entails a much more subtle bi-directional appreciation of relationships - potentially applying to all development processes - than can occur at H0 (vision-logic).

Let me briefly illustrate the distinction. The understanding of H0 still entails looking at development from one direction (like looking up a road from a fixed vantage-point).

So here development appears to unfold in an unambiguous manner, with for example prepersonal followed by personal and then transpersonal stages.

However with the understanding of H1, one now characteristically looks at development from two opposite directions through continually switching reference frames. This is like being able to look up a road in two opposite ways through the ability to continually switch vantage-points. If for example the direction of the road is from North to South (and in the opposite direction South to North) at H0 one is only able to consistently deal with relationships by taking up one vantage-point i.e. by either facing North or South. However at H1 - because of the great increase in the dynamic interactive nature of phenomenal experience through growing interpenetration with Spirit - one is able to keep switching vantage-points, giving two equally valid opposite (unambiguous) interpretations. Then by simultaneously relating these two interpretations as interdependent, one can thereby keenly appreciate the paradox entailed.

Again continuing with this road analogy, if one heads North along the road, a left or right turn will have an unambiguous interpretation. Likewise if one heads South along the same road, again a left or right turn will have unambiguous meaning from this opposite vantage-point. However if we simultaneously attempt to relate both reference frames (i.e. vantage-points) again deep paradox results, for what is "left" in terms of the first is "right" in terms of the second; likewise what is "right" in terms of the first is "left" in terms of the second.

This is deeply pertinent to all development relationships (which are necessarily conditioned by polar reference frames). Clear understanding entails that every asymmetrical distinction we make (in terms of one fixed vantage-point) can be rendered invalid from the opposite reference frame.

Now this does not appear obvious at the level of H0 (vision-logic) because one is still accustomed to looking at relationships from just one vantage-point. So Ken Wilber for example fixes the forward spiritual (diagonal) direction of development with transcendence even though the immanent direction is equally valid. He fixes the vertical direction with holarchy (i.e. movement toward more collective wholes) even though the onarchical (i.e. movement towards more unique parts) is equally valid. Finally for example he would fix the horizontal (heterarchical) direction of physical relationships with his Right-Hand quadrants (even though physical interpretation in terms of the Left-Hand quadrants is equally valid).

However the bi-directional understanding of H1 is much more refined. One is now able to appreciate how an unambiguous asymmetrical interpretation of relationships can be given for each direction of development (treated as separate).

Thus once again, looking at the spiritual directions of development to illustrate, from the first reference frame (i.e. transcendent), it will appear that prepersonal are followed by personal and then transpersonal stages. Likewise from the second reference frame (immanent), it will again appear that prepersonal are followed by personal and then transpersonal stages. So we are now giving two asymmetrical differentiated interpretations of development (that in absolute terms seem identical). However when both of these interpretations - which are taken from opposite directions - are viewed as simultaneous, paradox inevitably results. So what is "prepersonal" in terms of the 1st is "transpersonal" in terms of the 2nd; likewise what is "transpersonal" in terms of the 1st is "prepersonal" in terms of the 2nd.

It is indeed quite true that from a linear (discrete) perspective - suited for differentiated meaning - prepersonal should be properly distinguished from transpersonal (and transpersonal from prepersonal). However at H1, we clearly realise that just like giving directions along a road can be given from two equally valid vantage-points e.g. facing North or South, likewise the interpretation of the direction of development from prepersonal to transpersonal can be taken in two equally valid (opposite) ways. (At H0 we clearly see only one direction!)

So from a circular (continuous) perspective - suited to integral meaning - where both frames are viewed as interdependent - "prepersonal" is complementary - and ultimately identical - with "transpersonal" (and "transpersonal" with "prepersonal").

Thus to move consistently to integral meaning we must first understand that differentiated interpretation can always be taken from two equally valid opposite directions. Though once again both of these interpretations are asymmetric and unambiguous (within their separate polar reference frames) they are rendered paradoxical (when both frames are viewed as interdependent).

Therefore from a formal intellectual perspective, a consistent integral approach is necessarily based on the bi-directional interpretation of all developmental relationships with respect to both linear and circular aspects.

Most of the problems that I find with Ken Wilber's integral positions (as for example his treatment of the pre/trans fallacy) springs from an inadequate mode of intellectual appreciation. Quite simply he predominantly uses the H0 (vision-logic) interpretation based on one-directional understanding, whereas the more subtle bi-directional interpretation of H1 (and "higher" levels) is required. Thus when we try to explain the process of integration as based on H0 understanding, it is inevitably reduced in intellectual terms to mere differentiated appreciation.

So to properly move from a H0 (one-directional) to H1 (bi-directional) understanding entails an enormous change in one's fundamental manner of approach (intimately affecting the treatment of all developmental concepts). It cannot therefore be included in a H0 vision-logic interpretation (constituting perhaps a Wilber 6 stage) as a further refinement of this approach.

So again for example the belief that the overall nature of development is holarchic strictly makes no sense from the bi-directional perspective of H1.

What is broadly required in this context is the ability to see development from its two opposite vertical perspectives as both holarchic (where every whole is also part of another whole) and onarchic (where every part is also whole in the context of other parts). So we can initially give development two linear (differentiated) interpretations from these two perspectives. Then from a circular (integral) perspective both of these are paradoxically understood in a bi-directional complementary manner thereby facilitating the movement to a purer nondual appreciation of the development process.

Thus the initial existential crisis that precedes H1, relates to the erosion of attachment to the dualistic appreciation of phenomena (based on rigidly fixed reference frames).

Therefore - provided that motivation is truly authentic - this erosion with respect to dualistic phenomena is associated with the incubation of spiritual energy at a deep unconscious level. Then given sufficient dualistic negation (of a rigid kind), the process is reversed when often in a dramatic "conversion" moment, spiritual energy is suddenly released consciously into understanding as pure light.

So the early phase of the H1 level is often characterised by considerable illumination where the phenomenal world - now perceived in a more refined manner - is bathed in a shimmering mysterious new radiance.

This would constitute the dream state aspect of the stage i.e. where a vision of great spiritual potency may be imparted.

However the quality of the spiritual state inevitably interacts with structures of form, which thereby undergo considerable dynamic transformation.

The initial transformation of experience largely takes place with respect to heterarchical polarities i.e. interior and exterior (and exterior and interior). Due to the erosion of dualistic attachment already taken place (leading to the consequent recognition of the bi-directional nature of polar relationships), dynamic switching as between opposite poles greatly increases. Because one is able to hold both directions in understanding simultaneously, this greatly facilitates the appreciation of the dynamic paradox associated with all dualistic connections. This enables the transition to nondual spiritual awareness, which in turn enhances the appreciation of paradox at the (reduced) dualistic level. So nondual insight (as an empty spiritual state) and the bi-directional appreciation of form (as dynamic structure) mutually serve each other in a complementary manner.

Whereas Ken Wilber certainly recognises the spiritual states (associated with H1) there is a major discontinuity in his approach in that the consequent interaction of states with fundamental structures (and structures with states) is not adequately interpreted. Therefore insofar as Ken actually deals with the basic structures of the "higher" levels (e.g. cognitive) it is largely through an extension of vision-logic understanding.

H1 is associated with development through several sub-stages alternating in varying degrees as between illumination (where new dynamic structures are posited) and purgation (whereby corresponding rigid attachment to such structures is negated). 3

Firstly after the initial illumination, bi-directional development (based on this spiritual intuitive understanding) takes place with respect to concrete structures. However an inevitable conflict gradually builds up as between the purity of such spiritual light and secondary dualistic attachment. So this eventually leads to the need for a prolonged period of purgation (dynamic negation) so as to erode such attachment.

Then when sufficiently complete, a new purer phase of illumination occurs leading to the formal development of more universally based bi-directional structures. 4 This in turn breeds secondary attachment requiring further dynamic negation in a deeper "dark night of the soul". 5 Then with secondary linear attachment with respect to both concrete and formal structures eroded a more purely contemplative state can be attained.

However a serious unresolved problem may remain. Because H1 is largely based on the attempt to reconcile interior and exterior (and exterior and interior) polarities at a conscious "real" level, it can create huge demands on the supporting unconscious (which is not yet sufficiently developed). Thus because of the dynamic link as between conscious and unconscious, one can be thrown back in development to the earlier "prepersonal" complementary stage i.e. L1, where unresolved shadow impulses surface in an intense fashion. Therefore though development initially proceeds with the attempt to reconcile conscious opposites (within a given level), insufficient reconciliation of vertical polarities can remain a problem.

In holistic mathematical terms, H1 is defined by the development of mature circular understanding in terms of the horizontal polarities with somewhat rigid linear understanding however remaining with respect to both vertical and diagonal polarities. In this way it can be clearly seen how H1 (psychic/subtle) is directly complementary with L1 (mythic) with the mature circular understanding - with respect to horizontal polarities - of H1 interpreting the corresponding confused circular understanding of L1. 6

Perspectives of Stage

The continued dynamic negation of rigid dualistic phenomena that takes place at H1 is associated with considerable corresponding erosion of higher order distinct perspectives. The problem with such perspectives is that they largely represent conventional notions of relationships that can prove somewhat inauthentic when experienced from an overall integral nondual perspective.

Though the erosion of higher order structures can create significant temporary problems of adjustment to a world (defined by such perspectives), it also enables clear spiritual insight into the dynamic nature through which primary perspectives interact.

I mentioned before that implicit within every revealed perspective (as posited in experience) is an opposite hidden perspective, which requires negation of the former before it can be revealed.

Therefore because H1 is associated with a great increase in the ability to dynamically negate rigid perspectives (where the unambiguous revealed aspect is solely recognised), it thereby facilitates ready switching as between complementary opposite perspectives (which at H1 are mainly confined to the heterarchical aspects of experience). Put another way the flexible interchange as between both personal and impersonal perspectives (of a primary nature) is greatly facilitated. In other words the emphasis now mainly switches to the integration of perspectives (within an increasingly nondual framework) rather than the further differentiation of distinct perspectives.

Once again as with all of the previous levels, its structure is defined by the holistic appreciation of one of the key number types and at H1 the holistic philosophical interpretation of (algebraic) irrational numbers is especially relevant. 7

Though this might all seem unusually far fetched and removed from customary understanding, we cannot really appreciate the development of Western mathematics (over the last 2500 years) without dealing with this matter.

I mentioned in the last article (in Note 26) how the Pythagorean School attempted to maintain a balance as between the analytic quantitative understanding of mathematical operations and overall holistic qualitative appreciation (of a philosophical kind). Now for a while the perfect marriage seemed to exist where the assumption that all quantities were rational (i.e. with an unambiguous representation as fractions) corresponded with a philosophical belief in the rational paradigm (where the laws of nature could be revealed through scientific investigation).

However the discovery of irrational numbers - ironically through the use of the famed right-angled triangle forever associated with the School - dealt a mortal blow to this belief system. For in the simplest possible case where opposite and adjacent lines of the triangle are 1, the hypotenuse (or diagonal line) has an irrational value (as the square root of 2).

Subsequently in the West, mathematics has largely developed through concentration on the merely reduced quantitative aspects of the system (without corresponding emphasis on philosophical interpretation of its results). Therefore with respect to irrational numbers, it is sufficient in such mathematics to demonstrate that a number is irrational without dealing with the philosophical significance of what an irrational number means.

However in the more comprehensive mathematical approach that I am pursuing, this is unacceptable. The basic premise from which I start is that in principle, for every mathematical symbol or relationship of an analytic quantitative nature, a fully coherent holistic qualitative explanation exists. Therefore in scientific terms because we have irrational number quantities, we must therefore explain the philosophical nature of an "irrational" qualitative paradigm (or rather metaparadigm) which can properly incorporate such behaviour.

Looking at the square root of 2 firstly in quantitative analytic terms, it is important to recognise that it has two equally valid answers, which are positive and negative with respect to each other. Thus approximating the value correct to 4 decimal places, the square root of 2 is either + 1.4142 or - 1.4142.

Secondly the very nature of an irrational number is that finite (discrete) and infinite (continuous) elements are necessarily intertwined.

The finite aspect relates to the fact that the value of the number can be approximated to any degree of accuracy. However this accuracy is necessarily of a relative rather than absolute nature. So once again correct to 4 decimal places the value is 1.4142. However correct to eight decimal places (for example) the value can be expressed more accurately as 1.41421356.

The infinite aspect relates to the fact that the decimal sequence of its value continues on indefinitely (in no predictable sequence).

There are remarkable holistic connections here which ultimately explains the philosophical significance of (algebraic) irrational numbers.

Corresponding to 2 in holistic terms is the notion of duality. Now we can take a linear (one-dimensional) notion of duality that is absolutely unambiguous. This relates to the situation where opposite poles are clearly separated (with only one pole posited in understanding). However we can equally take a "higher" circular (two-dimensional) interpretation of duality. Here opposite poles - which are separated at the linear level - are now considered as complementary (i.e. both positive and negative). So reason at H0, is still linear and unambiguous (based on the separation of polar opposites); reason however at H1 is circular and paradoxical (based on the complementarity of these same polar opposites).

However to express this circular two-dimensional understanding of the nature of duality in reduced linear terms we must take what is the holistic equivalent of taking a square root. And when we do this we find that there are always two equally valid (unambiguous) polar interpretations that are positive and negative with respect to each other (which I have been illustrating earlier in the article).

Secondly, interpretation at H1 is strictly relative (rather than absolute) with both finite and infinite meanings interpenetrating in any context. In other words phenomenal symbols become increasingly imbued with spiritual (archetypal) meaning. Therefore, though from the phenomenal standpoint they can still be identified in finite terms (as form), from the spiritual perspective they increasingly resonate with mystery (reflecting the emptiness from which they originate).

There is one other interesting point! Just as x (raised to the power of 2) = 2 is rational (when considered from a two-dimensional perspective) equally the interpretation of duality (in terms of the complementarity of opposites) is also rational (from this "higher" perspective). 8 So what is rational from the "higher" perspective is "irrational" when interpreted from a reduced linear standpoint. In other words paradox is only "irrational" when we look at reality from a linear (one-dimensional) rational perspective.

The deeper implication of all this is that we can solve the Pythagorean dilemma and provide a qualitatively consistent "irrational" paradigm which equates with the behaviour of irrational numbers. However this requires the "higher" understanding of H1 to be achieved satisfactorily.

Therefore in scientific terms we now have at H1 a whole new vision of integral science that is based directly on the "higher" reason of the complementarity of polar opposites (in conscious "real" terms). The true purpose of such science is to increasingly show how the fundamental structures of reality are (ultimately) identical with respect to their physical and psychological aspects. Therefore the realisation of such complementarity greatly facilitates (at all levels) true integral experience. 9

Higher 2 (H2)

I also refer to this in a holistic geometrical manner as the point level (the centre of emptiness). It would broadly equate with what Ken refers to as the causal realm.

I find both Eastern and Western treatments of this stage unsatisfactory in key respects. Though the contemplative aspects of the level have been delineated to an extraordinary degree in Eastern traditions, unfortunately the emphasis has been predominantly on states (rather than structures). However because form and emptiness (and emptiness and form) are interdependent the strong emphasis on (mere) states is unbalanced. In the Western traditions a failure to properly distinguish the conscious from the unconscious aspects of mystical experience typically has led to the corresponding failure to properly distinguish the subtle and causal realms. 10

So perhaps surprisingly, the holistic mathematical approach can offer unexpected new insights with respect to this important level.

The previous stage culminates in an advanced contemplative type awareness. However paradoxically this can be associated with the extremely severe psychological crisis of the "dark night". Though physical and emotional difficulties are perhaps inevitable given the prolonged purgation entailed, a pathological depressive element is often involved due to the continued neglect of basic instinctive desires. Because the emphasis is primarily on reconciliation of conscious polarities at H0, deep purgative experience (leading to considerable dynamic negation of phenomena) can put an undue weight on the less developed unconscious considerably repressing its basic expression.

So often the "dark night" crisis represents an extreme in terms of the merely transcendent direction of development (in an attempt to overcome the limitations of the body). Therefore this crisis is only relieved through switching towards the immanent direction of experience. This then enables the unconscious to gradually project itself into experience in an "imaginary" refined manner. This happens mainly in two different ways.

The spiritual (potential) self as ultimate Spirit is made manifest through "high" level archetypal spiritual projections that passive arise. By its very nature - as the unconscious inherently relates to the complementarity of opposite polarities - it can only be expressed in pure fashion in a very short-lived manner through virtual phenomena (which tend to disappear soon after they arise). Therefore the closer we come to unveiling the unconscious in a truly spiritual manner, the more refined and short-lived are its secondary phenomenal manifestations as form. 11

However because of the complementarity of Spirit and matter, physical instincts (e.g. of an increasingly intimate erotic nature) expressive of the involuntary unconscious, are also projected in a "lower" level unstable manner. So during H2, though a permanent spiritual centre has been already stabilised, considerable dynamic interaction with respect to short-lived virtual phenomena (that are directly expressive of the unconscious) takes place. Thus it is through the harmonisation of pure spiritual with instinctive bodily projections that the hierarchical integration of "higher" and "lower" levels mainly takes place.

The overall structures of H2 are - literally - more "complex" than at H1. As we have seen at H1, both bi-directional linear and circular appreciation unfolds with respect to "real" conscious relationships. However now at H2 we now have bi-directional linear and circular appreciation with respect to both "real" and "imaginary" relationships.

I gave the analogy of H1 understanding, as like a person who is able to look up and down a road from two vantage-points, North and South (which keep switching direction). However the analogy at H2, understanding is akin to a person who is not only able to look up and down the road from North and South (which keep switching) but also up and down an intersecting road from East and West (which also keep switching direction). Indeed it is even more complicated than this as now two of the directions are "real" and two "imaginary". So as well as establishing the bi-directional relationship as between "real" and "imaginary" opposites separately (using Type 1 Complementarity), we must also establish the very subtle relationship as between "real" and "imaginary" opposites (using Type 2 Complementarity).

Indeed one fruitful way of appreciating such "real" and "imaginary" interaction is through the relationship as between states and structures (and structures and states).

Therefore structures and states are "real" and "imaginary" with respect to each other. Thus if we designate structures as "real" then - in relative terms - states are "imaginary". So - in this context - the bi-directional hierarchical interaction as between structures will be in "real" terms whereas the corresponding bi-directional hierarchical interaction as between states will be "imaginary"(both based on Type 1 Complementarity). However the bi-directional interaction as between states and structures will be relatively "real" and "imaginary" with respect to each other (based on Type 2 Complementarity).

Once again we can delineate major sub-levels in concrete, formal and vision (contemplative) terms. So we move from a more generalised concrete experience of virtual interactive phenomena to the more abstract formal experience (closer to emptiness) before finally reaching pure contemplation (that is largely absent of phenomena of either a "real" conscious or "imaginary" unconscious kind).

In holistic mathematical terms, H2 is defined by mature circular (bi-directional) interaction with respect to both horizontal and vertical polarities with however linear differentiation (of a rigid kind) remaining with respect to diagonal polarities.

(Now these definitions are given in terms of structures. However because structures and states are "real" and "imaginary" with respect to each other at this level, we can incorporate the dynamic interaction of both aspects in the mathematical configuration of the level by allowing for switching as between "real" and "imaginary" polarities).

Finally because H2 is complementary with L2, the mature understanding of H2 (causal) is needed to properly unravel the confused understanding of L2 (magic).

Perspectives of Stage

In an earlier article, I distinguished as between actual (relating directly to conscious) and potential perspectives (relating directly to unconscious appreciation) respectively. In the dynamics of experience actual and potential aspects continually interact (though typically in a confused and somewhat reduced manner). Indeed the potential aspect is vitally required to enable change with respect to actual perspectives to take place. Therefore with advanced mystical development - where the potential for radical transformation reaches its zenith - the potential aspect becomes especially important where it is understood in an increasingly refined manner (as the pure archetype of Spirit). This also entails that little rest with respect to actual perspectives can take place (as this would lead to phenomenal rigidity). Therefore due to the increasing influence of the "imaginary" potential aspect (during H2) dynamic interactivity as between opposite (conscious) polarities is greatly accentuated. Thus ultimately when this process reaches a peak, phenomena arise and disappear so quickly in experience that they do not seem to be even present. Thus the continuous stable attainment of spiritual emptiness (ultimately without perspective) is thereby associated with the increasingly refined dynamic interaction of short-lived temporary distinct perspectives (of both a "real" and "imaginary" kind).

So the movement here with respect to the intimate experience of the inherently dynamic nature of primary (distinct) perspectives, is associated with the purer attainment of an overall spiritual perspective (largely without phenomenal distinction).

Once again the holistic interpretation of number types can throw remarkable light on the nature of this level.

We now move on to a second type of irrational number, which is referred to as transcendental. So the holistic interpretation of transcendental numbers precisely defines the scientific nature of H2.

Though this may seem very rarefied, in fact one of the best known of all numbers is transcendental i.e. pi.

Now pi in quantitative terms measures the ratio of the circumference of a circle to its line diameter and is constant (irrespective of the size of the circle) with a value correct to 4 decimal places of 3.1416.

So the very clue as to the holistic meaning of pi (which in turn defines all transcendental numbers) lies in the notions of line and circle.

Thus in holistic terms, H2 is intimately concerned with establishing the (ultimate) relationship as between circular (based on the complementarity of opposites) and linear understanding (based on the separation of opposites).

Even though at H1, the clear recognition of circular (bi-directional) appreciation unfolds, one is not able to properly reconcile it with linear (one-directional) understanding. So one tends to alternate as between circular understanding (that remains to a degree rigid) and linear understanding (that also retains rigidity). Therefore complete spiritual freedom - eliminating such rigidity - requires that circular and linear appreciation be properly reconciled.

Now we have already seen a remarkable demonstration in a previous article as to how this takes place. Ultimately - in analytic terms - what is common to both circle and line diameter is the point, which lies at the centre of both.

Likewise - in holistic terms - what is common to both circular and linear understanding is the spiritual point centre of being (that unites both). So this point or apex of being ultimately defines H2 (causal realm) as the centre of emptiness. In other words with the reconciliation of both linear (unambiguous) and circular (paradoxical) form, no distinct phenomena (as rigid) remain.

Another important connection can be made in this context. Though circular and linear interpretations are qualitatively distinct, they can be converted into each other's format through a remarkable translation.

Thus "real" circular understanding can be indirectly expressed in an "imaginary" linear fashion. Likewise "real" linear understanding can be indirectly expressed in an "imaginary" circular fashion. So from a psychological perspective, the interaction of (linear) conscious and (circular) unconscious is "real" and "imaginary" with respect to each other.

Thus when we obtain the four roots of unity (as the structure of H2) we generate the four quadrants which thereby mathematically represents this structure. So here we have the relationship of "real" polarities of form (represented by horizontal lines) with "imaginary" polarities of form (represented by vertical lines).

However we can only experientially express the holistic transcendental notion (as the relationship between line and circle) either through "real" (conscious) or "imaginary" (unconscious) phenomena. So just as pi can be given a "real" or an "imaginary" value, likewise the dimensions of H2 (causal) level can be represented mathematically through either "real" or "imaginary" transcendental notions. In each case this understanding is associated with the very refined appreciation that such phenomena do not represent either the conscious or unconscious (in isolation) but rather the dynamic interaction as between both aspects. In this way what appears directly as conscious phenomena (already incorporates to a degree the unconscious through interaction). Equally what appears as of unconscious origin likewise incorporates to a degree the conscious. Therefore because we are moving here so close to the true intersection of both conscious and unconscious (as pure Spirit) phenomenal representation is thereby of an increasingly transient nature. 12

Finally just as the integral science of H1 is defined by an "irrational" metaparadigm (based on the complementarity of "real" opposites), the integral science of H2 is defined by a more complex "transcendental" metaparadigm based on complementarity of both "real" and "imaginary" polar opposites (the precise relationship between which, are provided by the holistic interpretation of the four-quadrant diagram).

Higher 3 (H3)

I also refer to this in holistic geometrical terms as the null level. It would correspond in Ken Wilber's approach to nondual reality. 13

Though considerable spiritual progress is made at each of the "higher" levels H1 and H2, they end with unresolved issues. At H1 the problem is due to repression of the unconscious aspect (due to greater emphasis being placed on - merely - conscious interactive dynamics). At H2 the other extreme can arise. Due to considerable emphasis now on the specialisation of the unconscious aspect (through short-lived "imaginary" projections), imbalance with respect to the conscious may occur. This can lead to practical problems whereby - due to the presence of virtual type activity of an extremely short-lived nature - insufficient stability of normal phenomena may remain (with which to carry out customary responsibilities).

Put another way an unbalanced emphasis on the "imaginary" (unconscious) aspect leads to a consequent imbalance in the relationship as between form and emptiness (with undue attention to the empty aspect). This is associated with a very subtle attachment - not strictly to emptiness as such - but rather to the process of continually attaining emptiness in experience. So in the end, the goal of the causal level to inwardly reach the centre of being (as the infinitesimal point of pure emptiness) is somewhat unbalanced, as the centre of the self equally has an outward expression as the centre of every phenomenon in reality.

Thus H1 tends to emphasise the integration of the conscious aspect (over the unconscious) H2 then tends to emphasise the integration of the unconscious (over the conscious). So finally at H3 the emphasis is on preserving the equal balance as between both the ("real") conscious and the ("imaginary") unconscious which simultaneously arise in experience. This in turn requires the very pure spiritual exercise of the will.

Because the unconscious is now largely purified, even "imaginary" projections of a short-lived nature tend to die out. However a more profound level of the unconscious is experienced. From one perspective this relates to a deep existential fear relating to death of the physical body. Indeed overcoming this fear may require a dangerous crisis (e.g. serious physical illness) where one's life is directly threatened. Associated with this physical fear is the counterpart deep fear - even at this advanced stage - of losing total commitment to Spirit through (selfishly) embracing the material world. 14 Also the deepest psychosexual instincts surface at this time. Indeed from this perspective, I would see this stage as in many ways a return to the womb where the most intimate notions of sexual identity are faced. So we have the two complementary sides. From the transcendent perspective this final "higher" stage represents an "entombment". Here one learns to most intimately die to self (in psychological terms) so that pure emptiness can be attained. Equally from the immanent perspective it represents an "enwombment" where one is again united with the mother (representing the phenomenal world) free of self centred attachment. This is likewise necessary so that world of form can be correspondingly made spiritually immanent with Mother Nature now freely embraced in all creation.

In this way through the balanced reconciliation of both perspectives, the bodymind (as form) can be fully integrated with Spirit (as emptiness).

In many ways the actual experience of H3 can seem very mundane as one learns to go about one's normal dualistic activities with a minimum of conscious effort. Indeed one can operate very much as if sleepwalking (or perhaps even more accurately) sleepwaking through one's daily activities. In other words, the conscious mind (that is rigidly identified with phenomena) is now largely dormant enabling the purer life of Spirit attain continual permanency. Therefore the memory becomes as an empty storehouse with only very limited use made of it on a short-lived temporary basis. Equally this stage can resemble - in reverse fashion - a period of waking sleep which I would liken to the sleep mode on a computer. In other words one can attain a deep state of relaxation (as if asleep) yet retain the capacity to respond to conscious stimuli as they arise.

In structural terms (which now through indirect "real" to "imaginary" translation also includes states) we now reach the most "complex" of the bi-directional interpretations.

So using our earlier analogy it is like having a road running North and South (depending on which reference point is taken) intersecting with another road running East and West with two diagonal roads cutting through this intersection point. So now one is able to look at development relationships bi-directionally in both "real" and "imaginary" directions (considered separately) and also in both "real" and "imaginary" directions (considered simultaneously).

And what is truly remarkable is that the diagonal lines have interpretations as both form and emptiness. In other words when we combine both horizontal and vertical aspects equally, we get the interpretation of form (as conscious and unconscious interacting in a purely balanced manner). However when we take the diagonal lines directly, we get the interpretation as spiritual emptiness (where the magnitude of the lines = 0).

So therefore the pure nondual experience as spiritual emptiness, in dynamic interactive terms is associated with the truly balanced interaction of both conscious and unconscious (with respect to phenomenal form).

The full holistic mathematical interpretation of these relationships (entailing Type 1, Type 2 and Type 3 Complementarity) is given by the eight-sector diagram which is literally the geometrical expression of a reduced form of oneness.

So whereas I refer to the integral approaches associated with H1 and H2 as Integral 1 and Integral 2 respectively, this most advanced appreciation - associated with H3 - is Integral 3. 15 So therefore to sum up in holistic mathematical terms, with H3 the mature circular integration of all polarities horizontal, vertical and diagonal is achieved. Though this requires appropriate recognition of the potential importance of restoring stable form at a later stage, such form still remains largely hidden so that the overall experience represents the specialisation of (mere) contemplative spiritual awareness.

Therefore it requires the further development of the radial levels to properly combine the mature expression of differentiated form with the equally mature experience of contemplative emptiness.

Perspectives of Stage

We now approach closest to the pure experience of overall nondual integral perspective (that is without distinct perspectives). Indeed at the peak of this stage, experience becomes so dynamically interactive that even primary perspectives lose their stability. So we come here most closely to appreciating the very basis of all primary perspectives through direct experience of the interface as between dual and nondual realms (which makes primary perspectives possible). As I stated in another article, the pure holistic binary understanding arising here serves as the very basis for deep appreciation of how all reality is digitally encoded in a binary digital manner. It is this most fundamental holistic mathematical appreciation that therefore underpins all other interpretation of development.

Once more we find that the holistic interpretation of another important number type provides wonderful insights as to the scientific nature of this most refined spiritual stage.

As I have stated before, conventional mathematics is based on very reduced linear notions of understanding. Nowhere is this more obvious that in the conventional treatment of the infinite. Mathematics has traditionally attempted to deal with the infinite as a linear extension of the finite notion. So we tend to think of the infinite in quantitative terms as a number that is larger than any finite number that can be imagined. 16

This also leads to the notion that there can only be one infinite quantity (as when any finite number is added the quantity remains infinite). Actually there are close parallels here with the transcendent notion of infinite wholeness (which is assumed to contain all finite parts). 17

However late in the 19th century, the pioneering work of George Cantor proved very disturbing for this traditional view of the infinite. For Cantor proved by his ingenious methods that there is more than one infinite quantitative notion. Indeed ultimately we can generate an "infinite" set of such infinite notions.

The most telling demonstration of this finding was the proof that the set of all real numbers (including both rational and irrational numbers) referred to by Cantor as C denoting the continuum, is greater than the set of all rational numbers. It has long been accepted - using conventional axioms - that the set of all rational numbers is infinite. So Cantor was showing that he had found another infinite quantity that was "larger" than the first.

Now there is no satisfactory way of accommodating Cantor's findings in conventional terms. Here they remain as a disturbing set of paradoxes threatening the very logical foundations of the subject.

However when we look at the matter from a proper philosophical perspective (where the rational and intuitive aspects of mathematical understanding are combined) it makes much greater sense.

So as we have seen in our holistic study of irrational numbers (especially transcendental) that we have in fact the interplay of two logical systems that are qualitatively distinct i.e. the linear and circular. In other words we have the interplay of the holistic binary digits. So just as ultimately we can create any number of finite quantities from the binary digits (as analytic) equally we can create any number of infinite qualities through the interplay of the holistic binary digits (as two qualitatively distinct forms of logic).

However when we attempt to express such findings through just one logical system (i.e. linear) we are left with our unresolved paradoxes.

So the real meaning of an infinite number of infinite quantities must be appreciated in a proper dynamic manner (where the two logical systems interact).

What this means in effect is that there are two ways of looking at the relationship of the finite and infinite. We can look at the finite as contained in the infinite (as in the conventional transcendent interpretation). However we can equally look at the infinite as being contained in the finite (as in the immanent phenomenal interpretation).

In other words though we can refer to Spirit as an infinite "whole", it can reflect itself in an infinite number of partial ways in the finite realm (as every finite phenomenon is a unique revelation of infinite Spirit).

And indeed this very understanding matures most clearly at H3 where the transcendent and immanent directions of Spirit are successfully united.

So the clue to the transfinite paradoxes discovered by Cantor is not really analytic at all but rather holistic.

And here the message is now clear. Just as the number system starts with the quantitative recognition of the binary digits (1 and 0), the overall philosophical understanding of this system ends with the clear recognition of how qualitatively, the binary systems of logic (linear and circular) are necessary to provide an unreduced understanding of the nature of number and indeed all mathematical activity. So we have travelled full circle. We started at the "lowest" with the confused interaction of the binary system and now end with the mature philosophical appreciation of this dynamic binary system at the "highest" level.

Therefore to obtain a more comprehensive understanding (that can enable both the differentiated and integral appreciation of reality) we are required to understand mathematical activity itself in binary terms as representing the interaction of both reason and intuition though linear and circular type understanding respectively.

With analytic understanding the rational aspect is direct (and the intuitive indirect). However with holistic understanding the intuitive aspect is now direct (and the rational indirect). Finally with the most comprehensive radial understanding both of these aspects interpenetrate in a balanced and creative manner.


1. All stages have both linear interpretations (as discrete) and circular interpretations (as continuous with other stages). When I am referring to the discrete interpretation I use for example higher and lower. However with continuous interpretation, I use "higher" and "lower". So the use of "higher" here is designed to suggest its complementary nature (with the "lower" level).

2. I would see my treatment of this level (and indeed all the higher levels) as differing substantially however from Ken Wilber. Ken clearly broadly approaches these stages from an Eastern mystical perspective (where there is considerable emphasis on spiritual states). However I would see my own approach as much more typically Western. Thus I have always emphasised the importance of the structural characteristics of these stages e.g. as a precondition for development of an integral scientific approach. Indeed I would see Ken as offering a mystical type system to Western readers that is ultimately misguided. In other words - whatever his actual intention - he tends to sell the idea that "higher" contemplative states can be married successfully to the advanced structures of the middle level (without the need for radical "higher-level" transformation in the understanding of such structures). In this way even advanced mystical development is thereby misunderstood as an enhancement of commitment to the world (defined in structural terms by mere vision-logic understanding). And all the research-based activity of the Integral Institute is consistent with this vision-logic perspective which frankly I find very reduced and somewhat superficial. If the Institute stressed that it was offering one distinctive type of vision-logic integration (i.e. in tune with Ken Wilber's approach) I would have no objections. However even here, other distinctive vision-logic interpretations of what is integral can be offered. More importantly much more advanced integral systems - based on the many stages of development beyond vision-logic - can be developed. Indeed from the perspective of these stages it is highly questionable whether vision-logic interpretation would even properly qualify as "integral". Therefore I think it is very important that other voices should be heard pointing to notions of the integral that may initially prove less popular, but perhaps ultimately more rewarding.

3. Evelyn Underhill provides the classic Christian description of these alternating illuminative and purgative stages in her "Mysticism". I would not consider however that Ken Wilber is temperamentally in tune with this tradition. See "INTEGRAL APPROACH: A COMPARISON OF UNDERHILL AND WILBER". However there is a significant weakness evident in that no clear distinction as between subtle and causal stages is made (which seems to me typical of the majority of Western accounts). Though St John of the Cross - which I would see as the most profound of Christian writers - does implicitly deal with development of the causal level in a poetic imaginative manner, even here gaps remain in intellectual terms (with respect to his overall synthesis). I would see this ultimately as a failure to properly distinguish the conscious and unconscious aspects of experience.

4. The great philosopher Hegel is one who I would see as using the bi-directional formal structures of H1 in his writings. However with respect to Hegel's writings I would maintain that we are given but a reduced interpretation, due to lack of sufficient balance being maintained with the spiritual intuitive state (associated with H1). In other words though there certainly was a genuine contemplative dimension to Hegel (especially with respect to earlier writings), the - merely - structural philosophic aspect was subsequently over-emphasised (thereby effectively reducing in some measure contemplative experience to philosophical form). Indeed there is a danger for any writer or artist whose work is genuinely inspired by spiritual intuition, that the subsequent demands of regular phenomenal output may lead to the gradual loss of the initial creative energy giving it birth.

5. The "dark night" as used in Christian literature is a somewhat vague term. In this context it symbolises the spiritual unconscious. More specifically it relates to the intense activity of "dynamic negation" which is also referred to as purgation. During such times no conscious indication of spiritual development is given which incubates in a very hidden manner in the unconscious. St. John of the Cross gives the most detailed (and precise) accounts of the "dark night". He distinguishes as between active and passive "nights". The active "nights" would relate to dynamic negation with respect to linear type understanding (which would seriously commence before the onset of H1). The passive "nights" would relate to dynamic negation with respect to the indirect phenomenal representations of circular (archetypal) type understanding.

He also distinguishes as between "nights of sense" (which can be identified with concrete understanding) and "nights of spirit" (which can be identified with both the formal and vision sub-stages). The most prolonged "dark night" referred to by Underhill as "the dark night of the soul" would in St. John's account refer to the "passive night of spirit" relating to dynamic negation with respect to the indirect phenomenal representations associated with the two later sub-levels of H1. However it is clear from St. John that this "dark night" really overlaps with what would be referred to in my terms as H2 and H3. In his later writings he gives a good account of the causal aspects without however satisfactorily relating it to his earlier treatment.

6. Strictly speaking in dynamic terms it is the mature interaction of H1 and L1 that interprets the corresponding confused interaction of H1 and L1. In other words we must treat these levels in a vertically complementary manner.

7. An algebraic irrational number can be expressed as a solution to a polynomial equation. (A polynomial equation in x consists of one or more terms raised to integral powers of x!)

For example x (raised to the power of 2) - 2 = 0 is a simple example of a polynomial equation (which gives the square root of 2).

(Algebraic) irrational are distinguished from another class of irrational numbers called transcendental. This latter type cannot be expressed as the solution to equations with rational coefficients.

Incidentally in holistic terms, I would maintain that there is complementarity as between H1 and L1 i.e. between the holistic interpretation of (algebraic) irrational and natural nos.

Remarkably evidence of this complementarity also exists in analytic terms, as all (algebraic) irrational numbers can ultimately be expressed as the solution to polynomial equations (using natural number coefficients).

8. Hegel for example tended to identify his dialectical interpretations as "reason". In other words he identified "reason" with the dynamic form of circular (i.e. two-dimensional) appreciation. He then referred to conventional reason - in its linear (one-dimensional) form - as "understanding.

9. The interpretation of space and time fundamentally changes from an integral perspective with dimensions literally corresponding - in holistic terms - to the mathematical interpretation of dimensions. One of the implications of this is that space and time move relatively forward and backward with respect to each other. In other words we have negative as well as positive dimensions (which are dynamically complementary). So from a linear perspective at H0, we would still believe that the arrow of time moves unambiguously forward with respect to all development processes. However - in the spirit of Heraclitus - we can see that at H1, it is somewhat subtler. So now the understanding dawns - because the (interior) self and (exterior) world constitute two distinct reference frames - that if time moves forward with respect to the world (as exterior) then it moves - relatively - backwards with respect to the self (as interior). Likewise if time for the (interior) self moves forward then time for the (exterior) world moves - relatively - backward. In other words when we fully understand the necessary dynamic relativity of time (and space), positive and negative directions cancel out and we experience in the continual (spiritual) present. So implicitly this is how all truly spiritual people remain in a continual present moment while dealing with relative arrangements in space and time.

10. I think it is very important to distinguish the scholarly approach - based on extensive research - from more personal accounts that are based directly on experience. Though research is certainly very necessary in an appropriate setting, I would contend that it is of much less value in a spiritual context where authentic experience of what is talking about is of the utmost importance. So the very activity of research - leading to a distance as between the self and others - can create a certain barrier to the very ability to experience in an authentic 1st hand manner. In other words research necessarily represents 2nd or higher order experience, which is of progressively less value in a truly integral approach.

There is also another problem. The accepted manner of scholarly recognition based on peer-group review, though of great value in weeding out poor quality material, can in some respects be too conservative (especially in transitional stages of thinking where new possibilities open up quickly). For example if one is offering genuinely original ideas, by definition no accepted framework of recognition will yet exist for such thinking. As no scholarly "peer" group exists, the accepted review process is therefore inappropriate. Initial recognition in such cases is more likely to come from fellow pioneers who perhaps are beginning to appreciate along the same lines (though again without recognition from established academic circles). Therefore there is a considerable potential benefit to forums such as this where original integral ideas may be launched (ahead of the willingness of the accepted scholarly community to recognise them). Of course there is also a risk as inevitably inferior material will also enter the process. And initially it may well be difficult to clearly distinguish both categories. However I for one would believe - in the interests of widening the integral debate - that this risk is well worth taking.

11. There is a very close parallel here with the nature of virtual particles. Virtual particles lie very close to the fundamental ground of matter and arise in pairs where complementary opposite particles are very closely associated. So the momentary short-lived existence of such particles requires that these opposite poles briefly separate. In this sense virtual particles represent the "imaginary" aspect of matter. Indeed properly understood, they represent the process whereby underlying "real" dimensions are indirectly expressed in "imaginary" phenomenal form.

12. Though transcendental numbers are extremely common (more so than any other number) remarkably few are really well known. After pi the most important transcendental number is e = 2.71828... which is the base of the natural logarithm system. Indeed in this context we can make a fascinating connection. Remember that in holistic terms both H2 and L2 are complementary. This implies that the holistic interpretation of transcendental (H2) and prime (L2) numbers is also complementary. Well there is fascinating connection evident her even in analytic terms! Thus the famous prime number theorem which is concerned with the general frequency of prime numbers in the natural number system is related to the simple formula n/log n (to the base e). Thus for the 1st 100 numbers this formula would predict 100/4.60517.. prime numbers = 22 (approx). (There are in fact 25 prime numbers from 1 - 100. However the approximation improves as n increases).

One of the remarkable properties of e relates to differentiation and integration in calculus. If y = e (raised to the power of x) then the rate of change of y with respect to x, i.e. dy/dx is the same = e (raised to the power of x). Then in reverse when we integrate, the same unchanged value arises. Now holistically in development, differentiation is associated with linear and integration with circular understanding respectively. Thus the situation where integration and differentiation are both identified, corresponds to exactly the same situation with respect to our discussion on pi where it occurs at the centre of the circle (which is also the centre of the line). So at the centre of emptiness we reach the (spiritual) point that is common to both differentiation and integration (through which both are identified). And once again this quest experientially defines H2 (the causal level).

Indeed the reason why a transcendental number cannot be a solution to a polynomial equation (with rational coefficients) is due to the fact that by its very nature it inherently represents the interaction of two logical systems (which are qualitatively distinct from each other).

Though an (algebraic) irrational number is irrational when considered from the perspective of a "lower" dimension from its own dimensional setting (or grouping of dimensions) it is rational. (In this sense it is defined with respect to one logical system). Thus if x (raised to the power of 2) = 2, then 2 is rational with respect to the 2nd dimension (defined in terms of two-dimensional understanding). It is only irrational when expressed in terms of the 1st i.e. as the square root of 2 (corresponding to linear one-dimensional understanding).

13. Ken tends to refer to nondual reality both as the culmination of contemplation (as the pure experience of emptiness) and also with respect to the more advanced stages where emptiness and the world of form are fully integrated. However this makes little sense. Even from the more restricted contemplative perspective emptiness is always associated with form (though of an exceedingly dynamic short-lived nature). So strictly speaking it is inaccurate (though perhaps loosely permissible) to refer to such experience as "nondual". However with respect to the full integration of form and emptiness (and emptiness and form) there is even less reason to refer to this as nondual. The very recognition of form requires dualistic distinction. Therefore the final stages - that I refer to as radial - relate to the very refined interpenetration (in a balanced manner) of both dual and nondual aspects.

14. Christ's temptations in the desert would symbolise this problem (for a Christian). Mystical development represents a great gift. Therefore the temptation to abuse such a gift for self-gain always remains a possibility. Indeed I would see evidence of this abuse for example with many leaders of religious cults who undoubtedly have achieved considerable mystical development but who also operate in some respects in an unacceptable egotistical fashion (e.g. pursuing power, sexual gratification or material riches). I would see this behaviour as resulting from the reverse counterpart to the lucid childhood mystical experience that subsequently acts as a considerable catalyst for authentic growth. Here we have (for someone who has reached the higher spiritual stages) an especially important "illucid" experience or temptation that subsequently acts as a considerable catalyst for the corruption of powers already attained. So the great "sin" of such religious cult leaders is in the identification of purely selfish "prepersonal" behaviour with authentic spiritual desire.

15. Of course beyond the integral are the radial approaches (Radial 1, Radial 2 and Radial 3) where integral and differentiated appreciated are mutually combined.

I sometimes also refer to the (reduced) integral approach of H0 (vision-logic) as Integral 0.

16. The same problem relates to the notion of the "infinitesimal" where a value approaches 0 (and is less than any arbitrary finite value that can be assigned). However this is very reduced thinking. A value that is smaller than any assigned finite value is still finite! In other words the infinite is qualitatively distinct from the finite and cannot be approached - except in a very reduced manner - from the finite rational perspective.

17. Mathematical proof is based on a very reduced notion of the infinite which ultimately is untenable (in its accepted form). If for example if the truth of a general mathematical proposition e.g. the Pythagorean Theorem (that the square of the hypotenuse in a right-angled triangle is equal to the sum of the squares of the other two sides) is accepted, this is then believed to apply to any actual case that may arise.

However the general result applies potentially to "all" cases and not strictly to any actual case.

Thus with mathematical proof there is a fundamental reduced assumption that the truth of the "actual" case (in any given instance) follows from the truth of the general case which again strictly applies potentially to "all" cases (which cannot be finitely demonstrated). In other words the infinite notion is reduced in mathematical proof to finite interpretation. The deeper reason for this problem relates to the fact that even though in dynamic terms mathematical understanding entails both rational (finite) and intuitive (infinite) understanding, the intuitive is reduced in formal interpretation to the rational.

Strictly speaking the proof of any mathematical proposition is relative (relating to a prevailing social consensus).

For example initially when it was thought that Andrew Wiles had provided the solution to Fermat's Last Theorem, a problem was subsequently found with the proof. This was later remedied and now it is generally accepted that the theorem has indeed been satisfactorily proven. However this truth is of a merely relative nature representing a social consensus among the small group of mathematicians capable of appraising his efforts. So this consensus cannot be absolutely guaranteed to remain.


Jones, Dr. R, 1982 Physics as Metaphor; Abacus

Davis, P.J & Hersh, R. 1980 The Mathematical Experience; Penguin

Collins, P. 1994 The Number Paradigms

Collins, P. 2003 Development: The Radial Approach

Mc Dermott, M. (1998) Knowledge and the Knower; Complexity and the Self. This monograph is available for download at the following web address.

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