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

Peter CollinsAs an economics’ student in Dublin in the late 1960’s, Peter Collins underwent a significant “scientific conversion”. Since then he has devoted considerable attention to the implications of a full spectrum developmental approach for radical new interpretations of mathematics and its related sciences. Though potentially of growing relevance for better understanding of our present problems, so far, he believes, these have been greatly overlooked by both the scientific and integral communities.

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Revisiting Perspectives

Part 6: The True Nature
of Integral Mathematics

Peter Collins


In Integral Spirituality, Ken Wilber describes “Integral Mathematics” in terms of inhabiting as many perspectives of others as one possibly can.

This in itself is an unsatisfactory definition. Whereas one might accept that a hidden mathematical structure is already deeply embedded within perspectives, one cannot confuse this structure with the direct manifestation of perspectives in experience.

Then to properly clarify the mathematical nature of perspectives, it is important that we can first understand them in an appropriate manner.

And as we have seen in earlier articles, this requires their appreciation in a dynamic interactive fashion that typifies everyday experience.

In fact, three levels of appreciation arise where perspectives are involved:

  • a distinct (differentiated) appreciation, where in a temporary freeze frame manner, the various primordial perspectives arise in a relative dualistic fashion.
  • a complementary (integral) appreciation, where these perspectives are also related in an interactive—ultimately nondual—manner with each other.
  • a comprehensive (radial) appreciation, where both dual and nondual aspects now suitably refined, can be coherently combined in an increasingly dynamic multi-varied composite fashion.

Though Ken Wilber is seeking to provide an integral approach to perspectives, because of the lack of coherent dynamic appreciation, he fails to ever make clear the crucial distinction as between their differentiated and integral nature. This then leads to a reduced interpretation, where he often identifies integration with the sum of his separate primordial perspectives.

This lack of dynamic appreciation also results—as we have seen—in the untenable distinction as between sentient and non-sentient holons (with their corresponding sentient and non-sentient perspectives). This then leads to a great lack of coherence in his overall treatment of perspectives.

His absolute interpretation of quadrant locations gives rise to an unduly rigid model where only four distinct quadrants are available. This leads initially to the identification of four perspectives, 1st person and 3rd person in singular and plural terms. Then by defining an inside and outside to each of these perspectives, he extends his model to 8 primordial perspectives. However, his inside and outside aspects entail the mixing of 1st person and 3rd person perspectives. So Wilber in his identification of inside and outside has moved from the primordial to a composite identification of perspectives.

However when quadrants are defined appropriately, in relative fashion, 16 distinct locations are available, which can then be identified directly with 16 primordial perspectives.

So to clarify, “he” for example can be objectively identified in 3rd person terms. However one could then recognise that “he” has also a subjective 1st person interior nature. So one can now link these two perspectives in an indirect composite manner.

However the direct recognition of this interior subjective dimension would take place in a 2nd person manner, where “he” now switches to “you”.

However, Wilber never gets to grips with the important distinction as between the primordial identification of perspectives (where one is directly involved) and the corresponding composite understanding (where two or more perspectives are indirectly linked with each other).

However, as we have seen in the last article, we can identify both an inside and outside, which face in opposite directions, to each primordial perspective. Furthermore this is vital for appreciation of the true integral nature of perspectives.

With its emphasis on 1st person and 3rd person, there is no natural place in Wilber's quadrant model for the 2nd person “you” perspective, which he attempts to bundle in an unsatisfactory manner with 1st person plural “we”.

Even less, there is no recognition of a distinctive 4th person perspective, which properly relates to the objective recognition of 1st person. [1]

So, just as one would identify, for example, in singular terms the objective recognition of 2nd person “you” in a distinctive 3rd person manner (“he, she or it), equally, one should identify the objective recognition of 1st person “I” in a distinctive 4th person manner.

Thus 1st person and 2nd person are natural counterparts of each other in personal terms; likewise 3rd person and 4th person are equally natural counterparts in a corresponding impersonal fashion. Therefore, recognition of this “missing” 4th person is again vital for a proper integral appreciation of both the quadrants and perspectives.

However, the normal use of pronoun language, with its emphasis on 1st 2nd and 3rd allows for no explicit recognition of a 4th person.

Then, Wilber in his “Integral Mathematics” moves from this initial approach to perspectives based on the four quadrants with its extension to 8 primordial perspectives (based on inside and outside definitions) to yet another approach based on the pronoun language of 1st, 2nd and 3rd persons, which with singular and plural usage, allows for 6 primordial perspectives.

But even Wilber recognises a big problem here.

“Also, “the inside and outside of the singular and collective” technically are not the same as 1st-, 2nd- or 3rd- person approaches or combinations thereof and some severe theoretical problems result if this equation is made.”

And Wilber makes it clear that he considers the former quadrant approach to be more important:

“the quadrants (inside/outside x singular/plural) are much more fundamental and prior differentiations in Kosmogenesis than are 123p (and in fact generate them)”. [2]

However, Wilber does not provide clarification of what these “severe theoretical problems” might entail (which are in fact rooted in his limited interpretation of the quadrants).

Therefore, he is never able to coherently explain the relationship between the four-quadrant and the 123p approach. So, he himself has unwittingly posed a major question mark regarding his subsequent treatment of “Integral Mathematics”

Wilber's Integral Mathematics

Though admittedly he has some valuable comments to make regarding its general nature—which should be interpreted in the context of an all-perspectives framework—because of his unsatisfactory interpretation of perspectives, he does not in fact accurately portray the existing nature of mathematics.

And in no meaningful sense can Wilber's approach be construed as an attempt to provide a proper “Integral Mathematics”.

Even by his own admission, this “Integral Mathematics” is designed as an ingenious notation to remind one to preserve an all-perspectives approach in the understanding of reality. And Wilber conceives the nature of this notation in a 3rd person abstract, or more accurately demi-abstract fashion, where abstract symbols are used to remind us of the need for perspectives, which themselves do not arise in an abstract manner.

The very nature of abstraction is that it attempts to differentiate meaning in an absolute dualistic fashion. However an integral approach by definition represents the other extreme in emphasising that what might initially appear as opposite perspectives, in a dualistic manner, are in fact intimately related with each other in complementary interactive terms.

Conventionally, mathematical symbols are given an absolute independent meaning, where they are interpreted in merely reduced quantitative terms.

The very point therefore of a true “Integral Mathematics” is to show how these symbols can be equally interpreted in a corresponding qualitative fashion, so that their holistic interdependence can now be appropriately recognised.

Because “integral” has already a well established analytic meaning in conventional mathematical terms, I customarily refer to “Integral Mathematics” as “Holistic Mathematics”. And I have been deeply engaged now for the past 50 years in formulating an integral appreciation, which has truly immense possibilities for a radical new form of mathematics that remains all but unrecognised in our present culture.

Therefore by no stretch of the imagination would I see Wilber's approach as representing “Integral Mathematics”, which again by his own admission is designed as an abstract notation system to conveniently represent perspectives. His “Integral Mathematics” again in effect represents but a form of short-hand notation with the laudatory intention of reminding one to avoid reductionism through providing an all-perspectives context for the interpretation of meaning.

However, even in terms of a satisfactory notation system there are many failings with his approach.

Insofar as his interpretation of perspectives is flawed then this consequent representation through mathematical notation will likewise be flawed. And as we have seen there are many problems indeed with Wilber's interpretation of perspectives.

In fact, as we shall see more clearly later, a number of other problems also arise.

Often the same symbol is used when in fact two or three distinct meanings are implied.

There is a lack of consistency in the use of his symbols, where a different notation can be used without clear explanation of the change.

Notation symbols are often used which are—strictly speaking—redundant, only serving to hinder a clearer understanding. In other cases, the representation of his perspectives through his notation does not accurately convey the original literal meaning.

Then the inferences that he attempts to reach from the use of this notation are often quite invalid.

And just as he has frequently used the inadequate “colour” terminology of Spiral Dynamics in a misguided attempt to stereotype other contributions, he now seeks to use this inadequate mathematical terminology to unfairly typecast the contributions of other integral theorists. [3]

The major problem however is that he provides no means, through his notation, of distinguishing the relative nature of distinct perspectives (as differentiated) from their corresponding interdependent nature (as integral). So a considerable amount of confusion characterises his overall approach.

However, these issues can perhaps be appreciated more clearly through looking in greater detail at his written contributions.

More on Wilber's Integral Mathematics

On P.47 of “Excerpt C, The Ways We are in This Together”,Wilber first introduces his Integral Math in this manner.

“If we call this “first event horizon” a first person experience of first person realities we could represent it as (1p x 1p) where 1p means “first person”.

However right away, 1p is here associated with 3 distinct meanings. Now “the first person experience” properly constitutes the 1st person perspective in this case. Then “the first person realities” relate to the various phenomena e.g. emotional feelings that arise (within this perspective). Also 1p refers to “1st person” (as the vantage point from which the perspective arises).

Then “you” as second person (2p) have your own first person experience of 1st person realities”, Wilber defines as 2p(1p x 1p).

Wilber translates “my perception of your first person” as 1p(1p) x 2p(1p). Then he states that if my perception of your first person matches your perception of your first person then we have mutual understanding:

1p(1p) x 2p(1p) = 2p(1p x 1p)

Note that there is even more unnecessary confusion created here as Wilber switches, without explanation from bold to normal type!

Wilber then states that this is the beginning of an integral mathematics not based on variables but on perspectives.

As I have stated before, he tends to use the word “perspectives” as if it has some magic quality that excludes all other terminology.

However perspectives are by their nature also variables. We can have variables as in mathematics that can take on absolute values, which represent one impersonal manifestation of perspectives. Also we can have variables such as feelings in actual experience that can undergo continual change (representing thereby a distinctive type of personal perspective). However the attempt to give perspectives a definition that excludes all related terminology, “not things, or events, or perceptions or processes” as Wilber states is quite mistaken. In this context, there is considerably irony in the fact that Wilber then uses the word “perceptions” frequently in his equations as synonymous with “perspectives”.

However when Wilber again returns to this theme in Appendix B An Integral Mathematics of Primordial Perspectives, he introduces new notation to refer to a perspective with 1-p, 2-p and 3-p now referring in turn to 1st person, 2nd person and 3rd person perspectives.

Personal Perspectives

If my first person view of you matches your 1st person view of yourself i.e.

1p(1p) x 1p(1-p) x 2p(1p) = 2p(1p) x 2p(1-p) x 2p(1p),

then according to Wilber, we have mutual understanding.

If we translate this equation literally, the Left Hand states that my 1st person, (which from my vantage point is also 1st person) has a 1st person perspective (again from a 1st person vantage point) of your 2nd person (which is also 1st person).

The Right Hand then states that your 2nd person (that is also 1st person) has a 1st person perspective (that from my vantage point is 2nd person) of your 1st person (that from my vantage point is 2nd person). Note that there is still confusion as 1p and 2p are used to represent both 1st and 2nd persons and 1st and 2nd person vantage points respectively!

If both sides of the equation match, again supposedly we have mutual understanding.

However, a number of problems exist with this formulation.

Firstly, it represents a very cumbersome notational interpretation of the original relationship, that if my view of you matches your view of yourself then we have mutual understanding. So, one could validly question whether the mathematical notation aids understanding of the original relationship in any meaningful sense.

Secondly, the mathematical equation suggested by Wilber does not in fact represent an accurate translation of the original relationship.

If my view of you is to match your view of yourself, then clearly I must adopt a 2nd person perspective.

However, Wilber misreads this crucial requirement on the Left Hand of his equation by substituting what is in fact a 1st person perspective (in place of 2nd person). Quite simply, I cannot have a 1st person perspective of your 2nd person. By definition I can only have a 1st person perspective (in relation to my own 1st person).

So, the whole point of adopting a 2nd person perspective is that I can thereby switch the focus of attention from my person to your person. I cannot therefore reduce your personal feelings to my own feelings, which means that I am still relating to myself rather than you.

And this again reveals a huge weakness with Wilber's overall approach in that it allows no proper place for the separate “you” perspective. And this lack of appreciation is illustrated through the manner in which he typically approaches critics, repeatedly seeking to impose his own “I” perspective on their distinctive contributions. [4]

Thirdly, though the equation can be validly interpreted in several different ways, no indication of this is given by Wilber.

As I have stated, there are three levels of interpretation.

The first is the differentiated interpretation where the relative separation of distinctive perspectives is maintained.

In this way, two people can have a personal relationship with no real communication taking place. And unfortunately this often typifies married couples, who after many years living together can predict their partner's respective emotional states with considerable accuracy, while sharing very little real intimacy with each other. And these emotional states become so predictable precisely because the transforming effect that genuine intimacy implies is no longer in evidence!

For example, I might recognise that you are feeling tired. So this represents a phenomenal event regarding my 2nd person perspective of you.

You might agree that you are feeling tired, which from my vantage point represents an event regarding your 1st person perspective.

So in this sense there is indeed mutual agreement. However it represents a very limited one-sided form relating to mere confirmation of an emotional state.

One could therefore in this context insist on mutual agreement from both sides. So, for example if I am feeling angry, then from my 1st person vantage point, I have a 1st person perspective, where the phenomenal event occurring is the feeling of anger.

Then, from your first person vantage point, you could have a 2nd person perspective, where you recognise that I am feeling angry. In this sense, there is again a mutual one-sided agreement between both of us regarding this feeling state.

However, though we are now in two-way agreement with each other regarding our separate emotional states, this does not necessarily imply any true degree of integral communication.

For this to occur, a recognition must exist—at least implicitly—that the two perspectives i.e. 1st person and 2nd person are indeed complementary with each other. This then leads to a shared form of mutual interdependence, where each person's feelings can be properly reciprocated by the other.

So an equality of understanding in this integral sense relates directly to a true holistic experience, ultimately of a mysterious nondual nature, that characterises the relationship.

Indeed just as there can be an undue degree of separation regarding emotional states, likewise in a sense there can be too much interdependence.

For example, a romantic couple in the first flush of their relationship may share a wonderful sense of intimacy, which still remains however confined to a relatively superficial set of experiences. This could then be used as a means of avoiding other important relationships lacking in such intimacy.

Both the differentiated and integral aspects of relationships are equally important. So ideally one should preserve the sense of oneself as a unique (separate) individual, while equally possessing the capacity to share intimate relationships with others (where separate individual barriers are to a degree eroded).

And this leads to yet a third sense of dynamic equality regarding perspectives i.e. where a proper balance can be maintained as between both the differentiated and integral aspects of perspectives.

Mutual agreement, in this context, implies an open-ended acknowledgement that both dual and nondual aspects should be maintained in appropriate harmony.

In this way, two people can share a degree of mutual intimacy, while equally maintaining a realisation of their separate identities as unique individuals.

However, Wilber never properly clarifies the distinction as between these three types of agreement as represented by his equality sign.

In any case, his notation system is not geared for the acknowledgement of such important distinctions.

And even in his terms, while failing to distinguish the different meanings associated with perspectives, his attempt to convey their relationship through his notation is far too cumbersome.

So if my view of you agrees with your view of yourself, then this could be more simply expressed as:

1p(2-p) = 1p(1-p) i.e. my 2nd person perspective (from a 1st person vantage point) agrees with your 1st person perspective (from a 1st person vantage point).

In fact, one could simplify further by just using 1 (for 1st person vantage point), 1p (for 1st person perspective) and 2p (for 2nd person perspective). Therefore,

1(2p) = 1(1p)

Note that in this notation, both sides of the equation are considered from 1st person vantage points. If we in turn the consider both sides from 2nd person vantage points, we obtain,

2(1p) = 2(2p)

And this now starts to reveal the true complementary nature of the relationship, where both vantage points and perspectives keep switching in experience, thereby revealing, through their mutual interdependence, the integral aspect.

However, once again such notation, either with Wilber's or my more simplified version, strictly only allows one to identify the relationship between perspectives (without reference to the phenomenal events included).

Impersonal Perspectives

Wilber then conveys “if I were a scientist trying to study you in 3rd person mode” as:

1p(1p) x 1p(3-p) x 2p(3p)

So again, fully translated this states that my 1st person (from a 1st person vantage point) has a 3rd person perspective (from my 1st person vantage point) of your 2nd person (that is also a 3rd person).

However there are a number of problems with this apparently straightforward translation.

Though we have now moved to an objective stance with respect to the “I”, Wilber is still using the notation that applies to the subjective “I” (as 1st person).

So, just as we distinguish the subjective other (as 2nd person “you”) from the objective other (as 3rd person “he, she or it”), likewise we need to distinguish the subjective one (as 1st person “I”) from the objective one (as 4th person “I”).

It would be helpful if ordinary language distinguished these two uses. Unfortunately however, “I” is used in both cases. So 1p in Wilber's notation should here be replaced by 4p.

Then, there is also a confusing conjunction of 2nd and 3rd person perspectives. Strictly speaking, I cannot have a 3rd person perspective of “you” (which entails personal recognition).

So, a 3rd person perspective relates to the objective form of “he, she or it”.

Of course, in dynamic interactive terms, one keeps switching in experience as between both personal and impersonal perspectives. So at one moment, I relate directly to “you” in subjective 2nd person terms. Then at the next, I indirectly relate to “him, her or it” in an objective 3rd person manner. And with the composite—as opposed to primordial—appreciation of perspectives—these two can then be coherently linked with each other so that I can see a “you” as also a “him, her or it” and a “him, her or it” also as a “you”. [5]

Thus to properly include both 2p and 3p in our notation we would need likewise to include both 2nd and 3rd person perspectives.

Wilber then allows plural forms for his notation through his asterisk notation. So 1p*pl for example represents 1st person (plural) “we” and 1-p*pl represents the 1st person plural “we” perspective.

However again there is a decided limitation in such notation in that it does not enable one to distinguish between the quantitative differentiated notion of “we” (as made up of a number of distinct individuals) and the qualitative integral notion (where “we” shares a common group characteristic). And of course this also implies that it cannot distinguish the true dynamic situation, where the “we” involves the interaction of both quantitative and qualitative characteristics.

Wilber then provides his notation for the statement “I think we agree that George is a fine person”, i.e. 1p x 1p x 1p*pl(1-p*pl)x 3p(1p) that raises several further questions. He interprets his own notation here as “my first person has a perception of our (first person plural) perception of George's first-person”. However this is not an accurate meaning of this notation, which literally states that my first person (from a 1st person vantage point) has a first person plural perspective (from a first person plural vantage point) of his 3rd person (which is also 1st person). In any case, neither version accurately relates to the original statement “I think we agree that George is a fine person”.

In his earlier statements, agreement (or mutual agreement) entailed the use of an equation (with an equality sign) connecting two distinct perspectives. So this should likewise be the case here.

Also, it might be construed that “I think we agree” entails a level of doubt, which would then be very difficult to express through Wilber's notation.

However if we overlook this possible doubt, then the original statement regarding George could be expressed—now assuming that we are viewing George in an impersonal manner—as “my view of him (i.e. George) agrees with their view of him. And in terms of Wilberian type notation, this would be rendered something like:

1p(1p) x 1p(3-p) x 3p(1p) = 3p*pl(3*pl) x 3p*pl(3-p*pl) x 3p(1p)

So, even this apparently simple statement leads to a very unwieldy form of notation, which would be much more difficult to express if we tried to incorporate both 2nd and 3rd person views in the statement.

And even then this notation does not really provide an accurate translation of the original statement, as we should include not only the perspectives involved but also a means of representing the information revealed by these perspectives (i.e. that George is a fine person).

Thus, it could be validly said that this attempt to provide suitable mathematical notation for perspectives—beyond the most basic type of statement—quickly becomes far too unwieldy and complicated to be of any real benefit.

One could perhaps argue, that the very difficulty in providing comprehensive notation, alerts one to the inherent intricacies contained in perspectives, which remain concealed through the normal use of language. However the real requirement is the proper understanding of such intricacies, rather than an attempt to provide a convoluted form of mathematical notation for poorly understood perspectives, that quickly becomes unintelligible.

Still, Wilber's attempt to provide a mathematical notation system represents an interesting experiment, which at the very least serves to remind one that normal experience entails many distinctive perspectives, which cannot be successfully reduced in terms of just one interpretation.

Also, it provides perhaps an ingenious way of attempting to reflect on these issues in a new manner.

However it certainly—for the reason that I have outlined—does not constitute a valid “Integral Mathematics” and by being proposed as such could in fact significantly weaken the quest to find a more authentic version.

Mathematics and Perspectives

Though I have deep reservations regarding Wilber's particular notation system to represent perspectives, in a general way I agree with much of what he claims.

I would for example accept that the phenomenal world is indeed built on endless iterations of perspectives, which have evolved through evolution into all the complex forms of self-reflective understanding that we can now freely exercise with respect to this world.

Also I would accept that mathematics itself is deeply embedded within the very structure of such perspectives and thereby contained within the 1st person, 2nd person and 3rd person language that we commonly use. However, as I have repeatedly stated, the anomalies of such language usage, to a significant extent, conceal the true nature of this mathematics.

Therefore to uncover its inherent nature, it is vital to understand perspectives in a proper dynamic interactive manner, where once again the differentiated nature of the primordial perspectives (as relatively separate from each other) can be coherently related with their corresponding integral nature (entailing appreciation of their complementary links in horizontal, vertical and diagonal terms).

And vital to this appreciation is the recognition of the “missing” 4th person (which normal language usage fails to explicitly identify).

So once again I identify three distinct levels of appreciation of perspectives:

1) their differentiated nature, where at a minimum 16 distinctive primordial perspectives arise directly from a relative appreciation of quadrant locations involved. Insofar as mathematics can aid such understanding (as for example through the application of basic notions of group theory) the conventional approach applies.

2) their integral nature where the two-way complementary links between these perspectives are investigated.

So the perspectives are initially defined regarding both personal and impersonal categories relating in turn to the affective mode (of feeling) and the cognitive mode (of thinking).

Horizontal integration then entails appreciation of the two-way complementary links in interior and exterior (and exterior and interior) terms, regarding both types of perspectives.

Vertical integration then entails corresponding appreciation of the two-way complementary links in individual and collective (and collective and individual) terms again regarding these perspectives.

Finally diagonal integration relates to the simultaneous appreciation of both horizontal and vertical integration, which becomes the very means through which the combined integration of both types of perspectives (personal and impersonal) can then take place.

A true integral mathematics—based on a qualitative rather than quantitative interpretation of symbols—can, as we shall see, play a vital role in clarifying the nature of these complementary relationships.

3) their true dynamic nature, representing the most comprehensive appreciation of perspectives, where the continual interaction of both the differentiated and integral aspects can take place.

And though it is beyond the scope of this article, an even richer mathematics i.e. radial, in turn can play an equally important role in the clarification of this multi-varied appreciation.

So, embedded in the appreciation of perspectives at each of the three levels of understanding, differentiated, integral and comprehensive are three corresponding forms of mathematics, which I refer to as conventional, holistic and radial respectively.

And it is the purpose of this article to focus primarily on the true integral dimension (which I have long referred to as Holistic Mathematics).

Reduced Nature of Mathematics

Though Wilber has some valid comments to make regarding the reduced nature of mathematics, in an important manner, he mischaracterises the nature of such reductionism.

Indeed throughout his career, he has swung between two opposite positions. For example in Excerpt C, he characterises conventional mathematics (as with his equation x = 3y) as relating to 3rd person abstractions in an exterior topographical space.

However, in other places, as for example in “The Marriage of Sense and Soul” he points to the deeply interior nature of mathematics:

“thus a direct interior mental experience (or empiricism in the broad sense) has guided our every move through the mathematical domain and these inwardly experimental moves can be confirmed or rejected by those who have performed the same interior experiment (run the proof through their minds).”

Wilber would characterise interior mental experience as 1st person.

Thus in effect, he is arguing in different writings that mathematics is 1st person and 3rd person, without showing how both are reconciled in practice. Of course, one important problem is that the interior mental “I” needs to be properly distinguished (as 4th person) from the corresponding feeling “I” (as 1st person) just as the personal “you” (as 2nd person) needs to be properly distinguished from the objective “he, she or it” (as 3rd person).

So, in true dynamic terms, mathematical experience principally entails the interaction of 3rd person and 4th person perspectives.

However in conventional fashion, the crucial assumption is made that the 4th person corresponds directly with 3rd person interpretation.

For example, we could represent the famed Pythagorean Theorem in 4th person terms as a statement regarding interior mental constructs.

However, equally we could represent it in a 3rd person manner as relating to exterior objects in an abstract mathematical space.

In formal terms, both 4th person and 3rd person perspectives are assumed to correspond with each other in an absolute manner. Therefore, the qualitative nature of understanding (relating to the mutual interdependence of both perspectives) is thereby completely ignored and in effect fully reduced to the quantitative aspect.

Thus, the most important statement that can be made regarding conventional mathematics is that all its relationships are thereby explicitly interpreted in a reduced quantitative fashion, through ignoring the dynamic interdependence of related perspectives.

However, implicitly in understanding, such dynamic interdependence as between interior and exterior aspects must take place. And indeed such interdependence becomes especially important where truly original mathematical work is involved.

So, for example, a gifted mathematician may seek an important new proof that fits (interior) mental constructs to (exterior) objective relationships, in a creative manner. However in true experiential terms, the struggle to find a solution can entail considerable recognition of the mutual interdependence of both interior and exterior aspects.

This holistic qualitative dimension of experience occurs at a hidden unconscious level. Then, when sufficient development has taken place with respect to the finding of a proof, the mathematician may clearly see, in a sudden flash of illumination, the answer to the problem.

Now this illumination, which may well constitute the peak moment of a mathematician's career, occurs at a holistic intuitive level of understanding, where all aspects of the proof are seen as simultaneously related to each other in a wonderfully coherent manner.

However the formal presentation of the proof is then provided in a merely reduced sequential rational manner.

Thus though implicitly the unconscious qualitative aspect of mathematics is so important in the generation of appropriate creative insights, in explicit terms it is allowed no role in the formal presentation of mathematical truth.

And the key issue that has yet to be even addressed is that this intuitive level of appreciation, which is necessary for all mathematics—and especially truly original work—is of a qualitatively distinct nature from rational understanding.

Thus, when one attempts to reduce the qualitative nature of intuitive appreciation in a merely quantitative rational manner, a gross distortion of the true nature of mathematical experience takes place.

And it needs to be repeatedly stated that despite the admitted brilliance of many mathematicians regarding their use of ever more abstract thought processes, an enormous blindness still characterises the overall profession in its refusal to properly examine the true nature of mathematical understanding, which is very different indeed from what is commonly represented.

Thus in terms of perspectives, the holistic qualitative nature of mathematics arises, when one recognises, in relative terms, the mutual interdependence of the 4th person (interior) and 3rd person (exterior) perspectives, just as the hermeneutic circle regarding emotional relationships arises when one recognises the mutual interdependence of both 1st person (interior) and 2nd person (exterior) perspectives.

Therefore, this integral appreciation in the first instance occurs, when one accepts the two-way complementary nature of 3rd and 4th person perspectives in a horizontal manner, regarding exterior and interior aspects.

And such appreciation relates directly to intuitive rather than rational understanding.

So, conventional mathematics, in explicit terms, is rational (though implicitly intuitive understanding is also necessarily involved). However, by contrast, holistic mathematics, in explicit terms, is intuitive (though implicitly, rational understanding is likewise involved).

Thus, conventional mathematics relates to the conscious interpretation of symbols (in quantitative terms); holistic mathematics by contrast makes explicit their unconscious nature (in a qualitative manner).

Then radial mathematics, which is the most comprehensive type of appreciation, makes explicit both the quantitative and qualitative nature of symbols in coherent dynamic interaction with each other.

Nature of Mathematical Proof

However there is equally an important vertical dimension relating to the individual and collective. Again this is given a merely reduced quantitative interpretation in formally accepted mathematics .

This then leads for example to the notion of mathematical proof as in some sense possessing an unambiguous validity.

So again, using the example of the Pythagorean Theorem, one could maintain that the accepted proof has an absolute truth value.

Underlying this belief is a merely quantitative notion of the collective as comprising a number of separate independent individuals.

Thus for a commonly accepted proof in this manner, one takes the view that it is possible for each individual to separately verify the proposition in the same absolute manner as every other individual.

Therefore in verifying the truth of the Pythagorean Theorem as an individual, I am confident that the mathematical community (comprising in this context other separate individuals) will likewise accept the same truth through a similar process of verification.

However, this represents a somewhat impoverished notion of the collective group as lacking any shared quality and in fact misrepresents once more the true experiential situation, where mathematical proof represents but a special form of social consensus that can only claim a relative truth value.

Indeed the limits of such a social consensus have been adequately demonstrated in recent years, with proof becoming increasingly difficult to firmly establish.

For example due to the rapid advances of the discipline, only a very limited number of individuals possess the appropriate technical expertise to verify an important proof. And because of the length of such a proof, with the associated difficulties involved, the possibility that an important error might be overlooked, significantly increases.

For example when Andrew Wiles initially first announced his proof of Fermat's Last Theorem in 1993, as he had been working in secret, he was the only person that had yet accepted it as “absolutely true”. So it was really an exercise in faith at that stage by the mathematical community that the proposition had indeed been proven.

Then later, when one of the referees (representing just a handful of mathematicians capable of following his solution) discovered an error, this led to a new attempt to fix the proof. which was eventually achieved in an unexpected manner.

However, though one might pragmatically now accept that with no more errors detected in the intervening years, that Fermat's Last Theorem has indeed been proven, this expresses in probability terms a high level of confidence, rather than absolute acceptance.

There are also new dificulties with proof in that in some cases, as with the 4-colour and the sphere packing problems, extensive computer analysis, which would be impossible to humanly check, has been necessary.

Then another interesting problem has emerged in that the very nature of obtaining proofs has moved considerably away from the notion of one individual providing a complete contribution, that can be independently verified by others, to co-operation amongst differing mathematicians—and even different disciplines—as with the classification of finite simple groups.

And as the classification of such groups runs to 10,000 journal pages, it would be impossible for any one individual to verify the proof!

In any case, it is widely accepted that there may be many remaining problems with the classification, which are assumed however to be too small to derail the overall proof.

So in this environment, it is no longer possible to view mathematical contributions in a truly independent manner. However, once again, the qualitative notion of interdependence finds no place in understanding the formal nature of proof.

Poincare Conjecture

Then the recent proof of the famed Poincare Conjecture—one of the seven outstanding unsolved mathematical problems posed by the Clay Institute at the turn of the millennium—raises many other issues.

This conjecture relating to topological surfaces in three dimensions, goes back more than 100 years to the French mathematician Henry Poincare and had proven notoriously difficult to solve.

Then a gifted Russian mathematician Grigori Perelman, in the early 2000's, started posting excerpts on the Internet claiming that he had found a proof.

And Perelman seemed determined to break many existing mathematical conventions.

One relates to an acceptance that proposed proofs should appear in a recognised mathematical journal (where the appropriate peer review can take place).

However, Perelman deliberately ignored this convention by posting directly to a site on the Internet, in the confident expectation that the few mathematicians capable of appreciating his efforts, would take due note of his contribution.

Another major convention requires that all steps in the proof should be clearly developed, without leaving the reader to fill in the gaps. However Perelman once again bypassed this route by offering a short and somewhat sketchy proof.

However over time, it became largely accepted by the recognised experts that the conjecture had been proven. Perelman then broke another convention in refusing any prize for his proof on the grounds that he believed an earlier contributor Richard Hamilton—who had significantly influenced his own approach—should also be included. So, Perelman was again supporting the notion that the proof was in fact a collaborative effort, where the distinct contributions of the two principle people involved could not be clearly separated.

However, it is not universally accepted that the Poincare Conjecture (though recognised as proven by the Clay Institute) has indeed been solved. At least one prominent mathematician, who himself has been awarded the Fields Medal, is of the opinion that the issues involved in the proof are so complex that no one can yet properly appreciate their precise implications.

Still, there exists a majority consensus within the very small group of qualified experts in the field that the problem has been solved. However, for the wider mathematical community, acceptance as to whether the conjecture has indeed been proven, represents an act of faith, or lack of faith, as the case may be.

Indeed the position of proof is beginning to resemble that of political parties, where a candidate may receive substantial backing from one section of the mathematical community, while failing to convince other sections. And national pride can play a huge role. In fact, shortly after Perelman had posted his contribution on the Internet, two prominent Chinese physicists announced before an international gathering in Beijing what they claimed as a valid proof of the conjecture (which left no unfilled gaps).

And this again reveals the true nature of mathematical proof that strictly has always represented but a special form of social consensus (with a merely relative truth value).

Properly understood, there is a qualitative as well as quantitative dimension to every mathematical proof. However, once again, this qualitative dimension has throughout its history been reduced in mere quantitative terms.

So the increasing difficulties in the recognition of “purported proofs”, indirectly is calling for incorporation of this unrecognised qualitative aspect.

Mathematics and the Personal Perspectives

Once again mathematics must be understood in the context of all perspectives. So, we have been concentrating here on bringing the integral qualitative dimension to the understanding of mathematics, directly in terms of the cognitive (impersonal) 3rd and 4th person perspectives.

However, indirectly the affective (personal) perspectives, 1st and 2nd person, are also involved in mathematical understanding.

For example, mathematics must necessarily use symbols which ultimately are sensibly verified. This is true even at the most abstract levels of pure mathematical reasoning. In particular, childhood understanding of mathematical relationships is initially heavily rooted in concrete understanding based on affective type experience.

And the appreciation of beauty can form a very important part of a mathematician's mindset as for example in the understanding of an elegant proof. The mathematician G.H. Hardy famously stated:

“The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

Unfortunately, as recent developments e.g. in relation to the already mentioned four colour and sphere packing problems demonstrate, some important mathematical proofs can be very ugly indeed, though it is possible that in time more elegant solutions will emerge.

This notion of beauty arises in a special way in connection with the Riemann Hypothesis, which is generally accepted as the outstanding unsolved problem in mathematics.

This problem, which intimately expresses the two-way manner in which the primes and natural numbers are related to each other, gives rise to a new set of numbers known as the zeta zeros (or Riemann's zeros) which arouse deep feelings of wonder and appreciation of remarkable beauty among those familiar with the area. Indeed in Jungian terms it can be said that the study of these numbers often leads to intimate expressions of the feminine unconscious (i.e. anima) among the prominent male mathematicians working in the field. [6]

However, as presently constituted, mathematics provides no explicit means of giving expression to such beauty (as it is based on mere conscious rational principles).

However with true integral appreciation that explicitly incorporates the unconscious dimension of understanding, such intimate feelings of beauty in the study of mathematical objects can be directly incorporated in what now truly becomes an all-perspectives approach (where both personal and impersonal meaning can interact with each other).

So the first stage of integral mathematical appreciation is based on incorporation of the holistic unconscious dimension of understanding regarding the impersonal perspectives (3rd and 4th person).

However the second more advanced stage now is additionally based on extending such appreciation equally to the personal perspectives (1st and 2nd person).

Then the radial approach to mathematics truly constitutes an all-perspectives approach, where overall experience is directly understood as encoded in mathematical terms. In other words, human experience is then understood as but the decoding of its inherent mathematical nature, as deeply embedded within all perspectives.

The Integral Mathematical Approach.

I now wish to address in a little more detail how a proper integral mathematical approach can be brought to the appreciation of perspectives. And I believe I can do this with a degree of authority, as I have become deeply involved in the development of this holistic dimension of mathematics over the past 50 years.

Indeed my first real breakthrough came earlier at the age of about 10, when measuring the areas of fields during the arithmetic class in primary school.

Using the old imperial measurements of the time, 1 acre = 4840 square yards, I remember then thinking to myself that a field with length 80 yards and width 60 yards would have an area of 4800 square yards (and be very close therefore to an acre). So in multiplying here the length by the width (80 x 60) we move from linear (1-dimensional) to square (2-dimensional) units.

Then it struck me that in the customary treatment of multiplication, this change in the dimensional nature of units is simply ignored, with all resulting numbers expressed in 1-dimensional terms (as lying on the same number line).

And even at that tender age, I realised that this was a key problem that needed addressing. Put another way, though I would not have been able to describe it properly at the time, I clearly saw that a merely reduced interpretation of multiplication was being adopted in mathematics. And in the coming years, I became aware of many more important examples of such reduced interpretation at work e.g. in the treatment of the infinite notion.

The Holistic Binary System

However the first truly holistic understanding started to emerge as a college student when I began to deeply reflect on the nature of 1 and 0.

In a sense, these two numbers are the most important of all and sufficient to build a satisfactory number system i.e. the binary, which provides the basis for the digital information revolution.

So, all information can be potentially encoded through the use of the two numbers 1 and 0, where both are considered in analytic terms as absolutely independent of each other.

Then I realised that the same numbers can be said to play a very distinctive role in the mystical religious traditions, with ultimate spiritual attainment expressed as a plenum-void, where form and emptiness become inseparable from one another. Form, in this context, strictly expresses the pure interdependence of all form (without separate phenomenal content) and this equates with the ultimate holistic notion of one (as oneness).

Emptiness in turn equates with the holistic notion of nothingness as the potential for the creation of all form.

So we have two contrasting interpretations of 1 and 0.

First, we have the standard quantitative notion (dual), where 1 and 0 are absolutely separated in an independent manner from each other.

Secondly, we have the largely unrecognised qualitative notion (nondual), where 1 and 0 are understood as ultimately completely interdependent with each other in a holistic manner.

So in analytic terms, the binary digits, 1 and 0, in quantitative fashion, represent one and zero (i.e. nothing) respectively. In holistic terms, the binary digits, 1 and 0, in a corresponding qualitative manner, represent oneness and nothingness respectively.

And just as all information can be potentially encoded through the quantitative interpretation of the binary digits 1 and 0, likewise all transformation processes can potentially be likewise encoded through the qualitative interpretation of the same digits.

Human development for example represents an especially important transformation process.

Therefore, all stages of such development can be encoded through the holistic aspect of the binary system, where 1 is associated with linear (asymmetrical) understanding, directly related to the differentiated aspect of development and 0 associated with circular (paradoxical) understanding, directly related to the corresponding integral aspect. [7]

And I have persistently argued over the years that when viewed from this binary perspective, deeply rooted in integral mathematical notions, that Ken Wilber's approach is exposed as a reduced intellectual interpretation of development (where integration is consistently confused with multi-differentiation).

And I have yet to see any coherent attempt within the integral community to address this fundamental issue.

So my point here is to show that starting with the most basic holistic number notions, that an extremely powerful approach applicable to all development is thereby made possible.

This also helps to explain my deep pessimism regarding our present ability as a species to properly address the wide range of problems that we currently face on the planet.

Increasingly these problems such as the impact of climate change will require a deep holistic appreciation, if they are to be properly addressed.

However, the very nature of information technology is that it reduces in every context the inherent qualitative nature of relationships in a quantitative manner.

So, what is now urgently needed in our world is a truly deep transformation in personal and social terms, rather than fragmented activity based on the generation of ever more superficial information.

However, without proper recognition of the holistic aspect of understanding as embedded in all scientific appreciation—and most importantly in the very nature of mathematics itself—we will never attain the depth of qualitative understanding that is required to make sense of our growing problems.

I do believe that such understanding will eventually emerge. However it would be much better for everyone's sake if we could start accepting this message now, before many catastrophic developments eventually force a radical reappraisal of our unquestioned assumptions.

So with this early reflection on the binary digits 1 and 0, I began to appreciate how all mathematical symbols contain both a qualitative as well as quantitative meaning.

In fact this relationship between 1 and 0 gives rise to a holistic mathematical appreciation of the operations of addition and subtraction respectively.

Again in conventional mathematical terms 1—1 = 0; so + 1 and—1 are treated in analytic terms as independent.

Likewise, in holistic mathematical terms 1—1 = 0; however + 1 and—1 are now treated as fully interdependent with each other. So the relationship here is akin to that between a matter and anti-matter particle in physics, where the fusion of both leads to the generation of physical energy.

Likewise the fusion of + 1 and—1 as representing opposite poles in understanding, leads to the generation of a pure psycho-spiritual energy, i.e. intuition.

Also in this context, + 1 relates to the act of (conscious) positing with respect to a given phenomenon.

Then—1 relates to the corresponding (unconscious) act of negating with respect to the same phenomenon. In this holistic manner, addition (positing) and subtraction (negating) are seen as intimately related to the conscious and unconscious aspects of understanding respectively.

The Circular Number System

The next major advance occurred indirectly as a result of reading “One Dimensional Man” by Herbert Marcuse, which was very popular at the time (the late 60's) in student political circles.

I began to realise that conventional science (and most especially mathematics) can be accurately characterised, in this new holistic mathematical manner that I was adopting, as 1-dimensional.

For example, this intimately applies to the problem regarding multiplication, earlier mentioned. So when we multiply two numbers, though the nature of the units changes from 1 to 2 dimensions, the result is expressed in a reduced 1-dimensional manner (as a number on the real line).

Then at a deeper level, 1-dimensional applies to all dualistic understanding, where truth is understood in an unambiguous manner (with reference to just one pole of interpretation). So the conventional view of science, for example, is that we can understand reality in an objective manner (without consideration of our subjective interaction with the world).

So I began to ponder as to what, for example, 2-dimensional appreciation—especially as applied to science and mathematics—might imply.

And largely as a result of studying Hegelian philosophy at the time, I began to get a better insight into the nature of such understanding.

This philosophy is often explained in terms of the dialectic where a thesis is opposed by its antithesis leading then to transformation in a new synthesis.

This can also be expressed in line with the mystical nature of Taoism as the complementarity of opposite poles (such as the yin and the yang).

An earlier version in Western philosophy is provided by Heraclitus, which is often summarised in the paradoxical statement,

“the way up is the way down and the way down is the way up.”

This statement in fact point to the purely relative interpretation of opposite poles, which keep switching in the very dynamics of understanding.

And this intimately applies to the manner in which perspectives change in experience. So, one initially posits for example in an exterior conscious manner—say—a 3rd person perspective (relating to an objective “it” phenomenon).

However this strictly has no meaning in the absence of its corresponding mental perception, which now relatively is of an interior 4th person nature.

In this way, a continual dynamic switching therefore takes place in experience as between 3rd person and 4th person perspectives, which moves consciously from exterior to interior and in reverse manner from interior to exterior recognition.

However for this switching to occur, one must to a degree (unconsciously) negate what has been (consciously) posited. And the very ability to achieve this relates to a holistic recognition that exterior and interior are interdependent with each other (as complementary poles of understanding).

And once more, just like matter and anti-matter particles in physics, this holistic recognition of exterior and interior aspects as interdependent, creates a psycho-spiritual fusion of energy that we recognise as intuition.

And whereas the horizontal differentiation of perspectives involves the separation of exterior and interior aspects (in dualistic terms), their corresponding integration entails complementary appreciation of their relative interdependence (in a nondual unconscious manner).

And this relationship is intimately related to the integral mathematical appreciation of the qualitative notion of “2” (as twoness).

Two Dimensional Understanding

This two-dimensional understanding is expressed through a circular rather than linear notion of number.

So to (consciously) posit one polar aspect in experience, the opposite aspect must be (unconsciously) negated. So for example, one can only posit the exterior recognition of a perspective by negating the interior, where it is thereby given a relatively independent existence. Likewise one can only posit the interior recognition (as the self) through negating the exterior aspect (relating to the outside world).

In this way, one can keep switching as between exterior and interior (and interior and exterior) perspectives in a differentiated manner.

However the experience of separate perspectives, in this context, involves the implicit recognition that both exterior and interior are mutually interdependent.

So + 1 =—1 and—1 = + 1. This in fact is a perfect holistic mathematical characterisation of the paradoxical statement of Heraclitus, “The way up is the way down; the way down is the way up”, which provides a purely relative notion of direction.

Thus if, as a direction, the way up is designated as + 1, then the way down in this context is—1; and if in reverse the way down is designated as + 1, then the way up is now—1. And this is the situation at a crossroads, where, potentially, what is up and down can initially be taken in either direction.

Then in reduced linear terms, such 2-dimensional appreciation is given by the two roots of 1 i.e. + 1 and—1 (where both are considered as absolutely independent of each other). And this is the actual situation that arises when up and down are now arbitrarily fixed with just one direction.

And these two poles can be illustrated on the circle of unit radius as the end points of the horizontal diameter drawn through the circle.

The development of such circular paradoxical type understanding is directly related to the first of the contemplative stages of development (which I refer to as Band 3, Level 1).

This would equate well with the psychic/subtle realm in Wilber's approach. However, I have always preferred the more Western type appreciation, where continuity with previous stage type structures is more easily maintained. So my own treatment would be broadly similar to that provided by Underhill in “Mysticism”.

Entry to this level typically follows an existential crisis, where one begins to suffer considerable disillusionment with conventional expectations.

For a while, one remains suspended between worldly and spiritual aspirations, before eventually identifying with ones “higher self”. This can then lead to the sudden release of an intuitive energy—that had been incubating in the unconscious during the previous period of struggle—through a peak form of illumination.

This in fact marks the first period of the 2-dimensional nature of nondual understanding, where exterior and interior aspects of experience become ever more closely related.

Initially, this applies at a more superficial concrete level in the form of a heightened supersensory awareness, then spreads to the cognitive mental faculties and to a lesser degree at this stage the inner self.

Then the growing clash as between this fledgling new form of spiritual awareness and the deep rooted dualistic habits of one's former existence can lead to a further immersion in the unconscious (in what St. John of the Cross refers to as the passive night of the senses). Then given sufficient development, a more intense period of illumination can follow, which now informs one's deeper formal structures both of reason and emotion.

And this level of intuition is properly required to truly appreciate the holistic nature of mathematical symbols in 2-dimensional terms.

However, there are limitations to such understanding, which is not yet sufficiently developed to enable one appreciate the true qualitative links in vertical terms as between individual and collective notions.

So an even further prolonged immersion in the unconscious, which St. John refers to as the passive night of spirit, may be necessary to fully erode the roots of one's rigid attachment to dualistic type understanding.

Therefore, this first level of contemplative development leads to appreciation of the nature of mutual interdependence in a truly qualitative manner, where the exterior and interior aspects of perspectives are concerned. And the fundamental nature of such interdependence is then perfectly described in integral mathematical terms, through the holistic (circular) dimensional understanding of 2.

So, with conventional dualistic understanding (of 2), opposite poles (such as exterior and interior) are understood as independent; however with corresponding holistic appreciation (of 2) these poles can now be understood as deeply interdependent in a qualitative integral manner.

Four Dimensional Understanding

However it is only at the next level (Band 3, Level 2) that a true 4-dimensional appreciation (entailing both horizontal and vertical aspects) is obtained.

And such appreciation is required for a proper integral mathematical understanding of the 4 quadrants.

I have not the space here to go into the kind of detail that is required to fully explain the highly refined intuitive nature of this new understanding (which would equate to the causal realm or to what St. John in the Christian mystical tradition refers to as “Spiritual Betrothal”), where holistic phenomenal symbols tend to operate in a pure archetypal manner that become increasingly transparent in experience.

Suffice it to say that it provides the means of properly distinguishing the true qualitative nature of wholes from the quantitative interpretation that defines conventional mathematics.

In standard terms, the whole is simply treated in reduced terms as the sum of its individual parts.

However, true qualitative appreciation gives the whole a distinctive meaning that cannot be confused with the quantitative. And there are here two complementary definitions.

The first is the notion of wholeness that transcends all of the constituent parts. So we recognise here that the whole nature of an organism such as a cell for example cannot be reduced to its constituent parts.

The second is the notion of wholeness that is made immanent in each specific part. So here, each part in any system achieves a qualitative whole nature through being related to all other parts (as interdependent) in the system.

And often these two notions of wholeness are combined. For example, a great painting will possess an overall quality that cannot be identified with any specific part aspect. Likewise each part aspect in turn obtains a special quality of wholeness through being related to the overall painting. So in the first case each individual aspect makes its contribution to the overall quality of the painting. In the second, each aspect in turn gains a unique quality through its relationship to the painting.

And in a precise integral mathematical fashion, we can denote the quantitative and qualitative aspects of wholes and parts as real and imaginary with respect to each other.

Just as we can have positive (+) and negative (–) in real terms, equally this applies in an imaginary manner, relating to the two types of qualitative wholeness identified.

In the past few centuries, the acceptance of imaginary as well as real notions has revolutionised the quantitative appreciation of number.

Therefore, in like manner, the acceptance of imaginary notions can revolutionise the qualitative appreciation of number, as for example intimately related to all perspectives (regarding their individual and collective aspects).

So the integral mathematical appreciation of four dimensions provides wonderful clarity on the qualitative nature of perspectives.

And this equates with the circular representation of the four roots of 1, i.e. + 1,—1, + i and—i (in a quantitative manner).

Thus we have two real aspects (exterior and interior) that are positive and negative with respect to each other relating to the quantitative nature of phenomena.

Then we have likewise two imaginary aspects that are likewise positive and negative with respect to each other, relating to their corresponding qualitative nature.

In fact this enables us to precisely model the dynamic relationship as between holons and onhols.

So basically, holons and onhols must be conceived in a complex manner (in integral mathematical terms) where real and imaginary aspects keep dynamically switching between each other (depending on context).

Eight Dimensional Understanding

Briefly the “highest” of the contemplative bands (Band 3, Level 3) is directly related to diagonal integration where both horizontal (exterior/interior) and vertical (individual/collective) appreciation with relation to the qualitative nature of perspectives simultaneously arises.

Eventually, phenomena become so refined that they no longer even appear to arise in experience, as one approaches pure nondual reality.

The additional 4 roots here, as the extremities of the diagonal lines, contain both real and imaginary components (that are of equal magnitude).

What this implies, in a holistic qualitative manner, is that experience now becomes so refined that one is enabled to switch equally as between the conscious (real) and unconscious (imaginary) appreciation of perspectives, without undue attachment to the phenomenal representation of either aspect. This means that phenomena can thereby become so spiritually transparent that they no longer even appear to arise in experience.

In fact this is beautifully expressed in the diagram. If one looks at either triangle in the UR quadrant, using the Pythagorean Theorem— because both horizontal (real) and vertical (imaginary) sides of the triangle are equal— implies that the diagonal line (as the hypotenuse of each triangle) = 0.

So on the one hand we can represent these diagonal lines in terms of form (with real and imaginary coordinates of equal magnitude). Alternatively, they can be represented in terms of emptiness as null lines i.e. with a magnitude of 0. These null lines play an important role in the Theory of Relativity, where physical light can be understood to “travel” in the present moment (with no time passing).

However, equally they can be given an important psychological explanation in terms of spiritual light that likewise continually remains in the present moment.

Later Development

Initially in contemplative development, both personal and impersonal perspectives tend to be separated to a degree from each other, with a predominant emphasis on one side (largely dependent on personality characteristics).

So, the second major journey regarding the spiritual descent is likewise necessary before personal and impersonal perspectives can themselves be properly integrated with each other.

This in turn is associated with a much richer appreciation, where both personal and impersonal perspectives can now also be directly understood in an integral mathematical manner.

With this development, the remarkable realisation can dawn that all experience is in fact intimately encoded in a mathematical manner.

Thus, the language of perspectives, in this deepest sense is indeed mathematical. And we will, I believe, eventually learn to understand all perspectives that comprise our experience as but the decoded nature of their inherent mathematical nature. 8


1. In his “Integral Spirituality” Wilber provides the following notation for meditation i.e. 1-p x 1-p x 1p, which he defines as the inside view of the interior awareness of my 1st person. However, literally this means the 1st person perspective of the 1st person perspective of my 1st person. Now the very point of meditation is to move from the initial differentiated appreciation of perspectives (where they are understood in relative separation from each other) to a more holistic integral view, where they can be seen as interdependent with each other in an ultimately nondual fashion. However as Wilber has no clear means of distinguishing these two notions of perspectives, he is led to a duplication of his terminology in a misleading attempt to convey this distinction.

In any case, it is very mistaken to identify meditation solely with the 1st person. In fact, meditation can arise initially with any of the four persons, 1st person, 2nd person, 3rd person and 4th person respectively. However the intention in each case would be to eventually integrate this starting perspective with all others in a nondual fashion. So there is a heavy overemphasis throughout in Wilber's treatment of the 1st person perspective. As we have seen, he identifies the distinctive 4th person as 1st person and also misleadingly attempts to equate the 2nd person with 1st person (plural) leading to a distinct imbalance regarding his treatment of perspectives.

2. Both quotes here come from a footnote to “Integral Spirituality” P.59.

3. I will refer here as an illustration to Wilber's response in “Integral Spirituality” to Edgar Morin's “Homeland Earth: A Manifesto for the New Millennium” where both stereotypes are used. “Morin is a wonderful writer in so many ways, but misses the essential integral message; he attempts a teal/turquoise “unitas multiplex” in methodology, but he is basically a modernist attempting green inclusivity.”

Then Wilber follows by stating “that a genuine understanding of intersubjectivity is almost entirely missing in his work” when in fact this comment would be much better directed at the nature of his own criticism. He then says: “He also fails to grasp the injunctive nature of 1st-, 2nd-, and 3rd- person knowledge, and so his “integral” thought is really just a 3-p x 1-p x 3p, at most.” Not satisfied with this, Wilber later elaborates by stating that it is basically a 3-p(1-p + 2-p + 3-p) and equates this to “a horrifically imperialistic subtle reductionism”.

Now this sadly shows up Wilber as a man who has become increasingly trapped within his own categories of thinking. Basically, he seems to be claiming that because Morin—and indeed each one of those other writers that he briefly reviews in ”Integral Spirituality”—do not interpret reality exactly in the manner that he sees fit, therefore they cannot be seen to be offering a comprehensive integral approach, which by definition can be solely equated with his own perspective.

4. In fact nowhere in his examples in his Appendix does Wilber use a 2nd person perspective, which is always conveyed misleadingly in 1st person terms. This again highlights his difficulty with the “you” perspective, which can find no natural home in his interpretation of quadrants. At a deeper level, it reflects his failure to appreciate the dynamic interaction as between sentience and non-sentience that characterises all holons (and related onhols).

This problem is brought out in another way in his discussion of quarks on P.141. He attributes sentience to the individual quark. However the individual notion of a quark is meaningless in the absence of its relationship with other quarks. So for the 2nd person to arise there must a relationship to another quark. Thus two quarks are necessary for the relationship between 1st and 2nd persons to arise. But this collection of quarks is simply a heap in Wilberian terms (lacking sentience).

So put another way, the untenable notion of individual holons alone possessing sentience, eliminates the very possibility of 2nd person perspectives arising. And Wilber's entire treatment demonstrates a marked inability to properly recognise 2nd person notions. So again, he never once adopts a 2nd person i.e. 2-p perspective in the Appendix, but rather continually reduces this 2nd person to the 1st person perspective.

5. As regards these composite perspectives, Wilber only seems to recognise one side of the relationship. Therefore, though he can readily recognise that 2nd person “you” can switch in objective terms to 3rd person “he, she or it” he does not appear to recognise that equally 3rd person “he, she or it” can switch to 2nd person “you”. Thus he recognises the inside of the 3rd person “he or she” in 1st rather than 2nd person terms. Then he seems to fail completely to recognise that the 3rd person “it” can equally switch to 2nd person “you” so that a rock for example which is viewed initially in objective 3rd person terms can undergo a change in its identity to 2nd person “you” (where it is viewed in an aesthetic rather than a scientific manner).

6. For example the French mathematician Alain Connes, who at one stage was considered as the most likely to prove the Riemann Hypothesis, is quoted as saying in Karl Sabbagh's “Dr. Riemann's Zeros”: “I believe I have found a very nice framework (to prove the Riemann Hypothesis) but this framework is still awaiting the main actor. So there is the stage—it is perfectly well arranged and so on—but we are still expecting the heroine to come and complete it”.

7. I find it fascinating how the very symbols we use to represent one and zero i.e. 1 and 0 are themselves closely related to the line and circle respectively. In fact, generally speaking the symbols that have become widely accepted to represent key mathematical notions are rooted in their deeper qualitative meanings.

8. In the above article, I have dealt especially with the even-number dimensions i.e. 2, 4 and 8, which have a special importance with respect to the three types of integration, horizontal, vertical and diagonal that I have identified. However all number dimensions can be given an important holistic mathematical significance. The odd-numbered dimensions i.e. 1, 3, 5, 7, … operate in a very distinctive manner from the even, though clearly a fuller treatment of this issue goes beyond the scope of this article.

Basically both the even and odd numbered dimensions tend to be associated with two distinctive expressions of development.

The odd numbered dimensions (especially the earliest i.e. 1 and 3) relate to the more active expression of mysticism. The even numbered dimensions (as in this article) are by contrast associated with the more passive contemplative experience.

Also, the higher-numbered dimensions both even and odd, greater than 8 are directly related to the more advanced expressions of radial development, where activity and contemplation become associated with each other in a largely spontaneous manner.


Ken Wilber; Integral Spirituality: A Startling New Role for Religion in the Modern and Postmodern World; Shambhala Publications, US Reprint Edition, 28th December, 2007

Ken Wilber; Excerpt C: The Ways We are in This Together;, 2006

Ken Wilber; Rnase Enzyme Deficiency Disease; Integral World, 2002

Ken Wilber; The Marriage of Sense and Soul: Integrating Science and Religion; Broadway Books, Paperback ed. 1st May 2000.

G.H. Hardy; A Mathematician's Apology (Canto Classics 1st Ed; Cambridge University Press (March 26th 2012)

Sabbagh, Karl: Dr. Riemann's Zeros: Atlantic Books, September 2003

Sylvia Nasar and David Gruber; Manifold Destiny, New Yorker, August 28th 2006

Peter Collins; Original Vision: personal web-site with 1000 blog articles, relating directly or indirectly to Holistic Mathematics

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