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Bill Torbert received both his BA in Politics and Economics and his PhD in Individual and Organizational Behavior from Yale. He served as Founder and Director of both the War on Poverty Yale Upward Bound Program and the Theatre of Inquiry. A professor of leadership and organizational transformation at SMU, Harvard, and Boston College between 1970 and 2008, Bill consulted widely, served on boards of directors, and published eleven books articulating and demonstrating the Collaborative Developmental Collaborative Inquiry meta-paradigm of social science. Since 2012, he and his associates at Action Inquiry Associates are devoted to creating coaching, consulting, and workshop activities that embody and further encourage the practice of collaborative, self-and-other-transforming leadership. See: http://www.williamrtorbert.com/

Six Dimensional
Space/Time

Mathematical Intuitions Underlying
the Integral Meta-Paradigm of Science Named
'Collaborative Developmental Action Inquiry'

Bill Torbert

Recently, I wrote a speech in honor of Chris Argyris (1923-2013), in thanks for the Lifetime Contribution award in his name given to me at the Academy of Management annual meeting (8/2/14). After acknowledging my gratitude for his immense influence on me, I sought to recognize once again the scientific approach that I first named “action science” (Torbert, 1976) and that was inspired by both Chris' body of work at that time (Argyris, 1971) and my own early work. Later, Chris would expound on the theme of action science quite differently (Argyris, Putnam & Smith, 1985).

As a result of writing that speech, I came to ponder the fact that I have never tried to articulate the most fundamental mathematical/ ontological/ action axioms that I have constructed (with the help of many people, books, and wisdom-traditions, including Ken Wilber's). These axioms serve as the basis for the new meta-paradigm of what I then named “action science,” but more recently have come to name Collaborative Developmental Action Inquiry (CDAI).[1]

My Activities and My Two Mathematical Intuitions During My Twenties

Bill Torbert

I call the two interconnected sets of mathematical/ontological axioms I came up with during my twenties:

1. “the four territories of experience”;
2. “the six dimensions of space and time.”

Increasingly contacting these four territories or six dimensions experientially in real time as one acts is what I understand to be the aim of integral development. (I sometimes call the four territories the 'deep four' in contrast to Wilber's AQAL 'flat four'.)

I begin with a very brief introduction to what I was doing in my early twenties to provide a little context for these rather abstract ruminations. In my 21st year, as I was graduating from college, I came into contact with three very different 'traditions' of inquiry—

1. the Yale PhD program in Individual and Organizational Behavior, a home of empirical, third-person, qualitative and quantitative inquiry, along with some experimenting in the direction of action research and second-person learning;
2. the almost brand-new world of systematic, second-person, action-emotional inquiry in Tavistock human relations conferences (based in the work of Freud, Bion, Klein, and Rice), at 'T-groups' associated with the US-based National Training Laboratories (based in the work of Lewin, Bradford, Bennis, and others), and Esalen-style encounter groups (based on the work of Perls and traditions of bodywork); and
3. a primarily first-person, spiritual inquiry into the transforming nature of one's attention in everyday life, intended to support one's own inner efforts to transition from living in a 'waking sleep' to more frequently 'awakening'; in my case, I began a 25-year participation in a form of inquiry called the Gurdjieff Work.

At that time, I was not yet able to articulate the broad distinction I have just made among first-, second-, and third-person approaches to inquiry-in-action (nor other central concepts of CDAI, such as the personal and organizational developmental action-logics [my version of integral theory, with its accompanying psychometric measure, the Global Leadership Profile] and their embrace of progressively more types of mutually-transforming power and feedback). Nevertheless, I earnestly engaged in all three types of inquiry during my leadership of the War on Poverty Yale Upward Bound program in the summers between graduate courses, and received more feedback than I could then name. A drastic one-phrase summary of the practical effect of the program was that, in spite of several crises and one or two catastrophes, it cut New Haven's dropout rate in half.

My own primary question was how I could become increasingly aware of when to exercise what kind of leadership (e.g. authoritative, collaborative, supportive, confronting, etc.) to help students, staff, the organization as a whole, and myself to transform developmentally. This inquiry into timely leadership action led to a second, more scholarly outcome of the Upward Bound program—my 1976 book about the whole experience. This book introduced both a comprehensive theory of organizational development (seeking to generalize about when in an organization's life different kinds of leadership are called for to deal with incipient crises before they become catastrophes), as well as a new way of doing social science, where the researcher and the research-subject/practitioner become the same person. At that time, I named this way of doing social science and social action at the same time “action science”; now, I call it CDAI (for gradually maturing versions of CDAI, see Torbert, 1987, 1991, 2004).

Also, during my graduate school period, as my dissertation, I did a voluntary, non-manipulative laboratory experiment, with a great deal of ongoing feedback to participants, on what methods of intervention generated greater learning from experience. All this was to test a theory of what counted as learning from experience that I had just developed (Torbert, 1972).

One reader's responses on first reading this paper:
I notice a dryness, a tightness, a shortness of breath as I read the first paragraphs. Will I understand the paper? Am I clever enough?

The Four Territories of Experience

It was in this book that I first formulated the axiomatic first principle of all my ongoing inquiry and action since—namely, that a full accounting for and learning from all our experience involves the recognition of four distinguishable, yet also interweaving, “territories of experience.”

In order to render these four territories or qualities of experience initially effable, think of the three-dimensional “outside world” as one territory. Think of the realm of your own (anyone's own) sensation of your (their) embodied self (moving or still) as a second quality or territory of experience. We may be aware of the sensations of what we are doing (e.g. breathing), but often we are not (as is the case with each of the four territories). For example, I am concentrating so strongly on this writing right now that I am only intermittently aware of being on a train, or even of how I am being jounced around in my seat.

As just suggested, my lack of awareness of (a) the outside world and (b) my own embodied action occurs because I am so strongly focusing within a third territory or quality of experience—namely, the ongoing thinking of these thoughts I am writing. However, I am also experiencing frequent moments when I am aware of the seashore outside the train window, the sensation in my fingers as I type, and my thinking, all at once. I am experiencing these three territories all at once in the light (one might say) of yet a fourth quality of experience that we can label the “bare attention,” or “post-cognitive conscience and consciousness.” That I can recognize and appreciate the play of this fourth territory of experience is in part due, I believe, to my decades of intermittent guided practice in contacting this distinct quality of experience.

We can imagine the four territories as different dimensions, orthogonal to one another, like the three dimensions of space in the outside world territory. We will return to this idea in the next section. We can also imagine the outside world territory as the highest-density/lowest-frequency of the four territories, with each successive territory re-presenting and re-enacting a lower-density/higher-frequency than the previous one. Each higher frequency territory has some capacity for intervening in and organizing each higher density territory. Thus, our actions can “interpenetrate” and reorganize the outside world, as when I use a humanly-made tool to mow my lawn, cutting through parts of the outside world (grass and air) and avoiding really dense objects (stones and trees) as I do so. In this example, my thought in turn organizes the overall plan of my actions before I begin mowing, monitors the feedback from the environment as I work, and assesses when I've achieved satisfactory completion.

It is not clear that a post-cognitive attention is necessary for this mowing-the-lawn task, though we know that some of our most satisfying moments in life come from simply 'witnessing' ourselves at work or at play. Another possibility is that a tweak of conscience may interrupt my mowing, reminding me of some more profound duty. Attuning ourselves more frequently to this higher-frequency attention may be a critical process if we wish to accept the guidance of still higher-frequency qualities of intelligence and passion (if such exist).

(Same early reader)
I relax into my bed and the paper and start to enjoy the dance with the words. The description of the vibrational frequency / density of the four territories feels familiar and yet somehow newly cast—like a piece of fine jewelry. I both recognise the shape and marvel at the fine filigree inlaid upon the metalwork. A set of nested worlds within worlds—each one wielding a different set of laws.

The 'four-territory' construct or axiom gives us access to a multitude of fundamental questions, if we use the idea to do existential, living inquiry in the present moments of our lives. How can we cultivate the spacious awareness of all four territories of experience more often? How might we accept feedback from ourselves, others, and the sheer object-ing objectivity of the outside world that would show incongruities between our intent, plan, action, and effects on the outside world? How might we cultivate relationships with others that give them equal access to such spaciousness within themselves? Relationships, that is, not of uni-directional, unilateral power, but of mutually-respectful and potentially mutually-transforming power? Relationships that encourage a trustworthy testing of the validity and the transformational learning value of feedback among us? How can we gradually cultivate an ongoing balancing among our first-person passions, our second-person compassions, and our third-person dispassions? How can we learn to act in ways that are increasingly uniquely crafted to be situationally timely for this event, from the diverse perspectives of oneself, the others present, and the larger social/natural container(s) 'holding' us?

Mathematical Intuitions About the Four Territories

With this very condensed introduction to the “four territories of experience” idea, I proceed to the mathematical intuitions that have been helpful to me in establishing a relatively proper proportionality among the priorities in my everyday life. (I say “relatively proper proportionality” because my lived life to date feels like a relatively good life in many ways [while I remain aware that like Job's life, and like the lives of so many people in the world nowadays, especially in the broad band of the Middle East, events beyond my control may turn my life relatively much worse at any time].)

My first mathematical intuition is that the proper number to associate with each “territory of experience,” is 0 for 'bare attention,' 1 for thinking, 2 for inner sensation, and 3 for outside world.

0, or source, I associate with “bare attention” and all inclination and effort to turn our attention, from its gravitational flow down and out through thought and action into our impacts on, and perceptions of the outside world, toward its origin. This 0-territory is the noumenal realm, the non-dual realm, no-thing. 0 is the origin of the attention and of all numbering but is itself not just any number, but rather both real and imaginary (imaginary numbers are signified as 'i,' as in 'i = √-1' [an incalculable quantity]). Of utmost significance for CDAI, 0 is not an arbitrary, abstract, merely third-person origin for mathematics and other activity, but rather also a potential qualitative/ experiential origin for first- and second-person action inquiry and the ontological source of meaning.

1, or unity, or archetype, I associate with the “thinking” territory of experience, where each concept (e.g. 'tree') is a kind of archetype, where each rational whole number is a kind of archetype (though most are composites, leaving the prime numbers as the ur-archetypal numbers, a key link between the rational and the trans-rational, with no formula for how to calculate them (primes), because they are partly post-rational. (Based on this reasoning, the Riemann Hypothesis about prime numbers— the primary unresolved question in mathematics for the past century—is in principle unconfirmable [Derbyshire, 2003]). This archetypal territory is the realm of nominal numbering, distinctions, and statistics. The distinctions I am making here among 0, 1, 2, and 3, for example, all involve an emphasis on how they represent nominally distinct categories.

2, or duality, or temporal rhythm (breathing in and out, walking, the hands washing one another, first you speak then I, more than/less than, etc.), I associate with the territory of inner awareness of one's embodiment in interaction with others and the environment. This is the ordinal realm, the realm of first, second, third, fourth, the realm of 'some' where we use pre-rational emotional and sensual impressions as much or more than reason to calculate more-than/about-the-same/less-than distinctions, such as “she has more power than I do” or “I trust him.” Non-parametric statistics are the appropriate tools for quantitative analysis of phenomena in this territory. In terms of ordinal numbering, we can distinguish two different ordinal orders to these four territories of experience I am describing. As we grow up, we first gain some ability to organize them from the outside-in, from 3 to 0. First, we gain some control over aspects of the outside world—how to walk, ride a bike, or operate an i-phone (usually, about 2-10 years old); second, we gain some inward ability to manage our own behavior to conform to, or violate, socially normative patterns (usually, 8-16 years old); third, we may develop expertise in one or more areas of techno-culture (usually, 14-28 years old); and fourth, if at all, we come to inquire into our nouminal, ontological status via an exploration into post-cognitive conscience and consciousness. Thus, in durational terms, we first master the outside world (3); next we master our own behavior from the inside (2); then our thinking (1); and finally, if at all, we befriend our attention/ intention (0). However, once we engage in tasting all four territories and their interaction as a continuing practice, we tend to re-order them ordinally in reverse fashion, from the inside-out, from 0-3, wishing increasingly to have our source (0) and our principles (1) cause our actions from within (2), in response to (but not caused by) the external world (3).

3, or tri-alectic, or the many external, distinctly-boundaried things (apples, oranges, cars, dollar bills, etc.), I associate with the outer, 3-dimensional, object world. This is the interval realm of number—where things of any particular category can be counted, added, subtracted, multiplied, and divided; and where the assumptions of parametric statistics are generally valid.

It is interesting to recognize that the 0-1-2-3 range is the only case within the number system where four non-composite numbers (and two primes, 2 and 3) occur consecutively, dramatically highlighting quantity's origin in quality.

(The early reader:)
The sequencing of numbers feels right. true. good. I really connect with the description of 0—I feel a mystical quality, the secret of the real / imaginary somehow being contained within the curves.
I'm not so clear on the 1—the archetypes of concepts. I feel foggy and then see into the distance, before the fog closes in again. I see an infinite matrix of uniqueness stretched out before me and am quickly overwhelmed. My mind creates concepts that cluster, concepts to shield me from the awe-fulness of the whole. I become subtly aware that the 1 is the 'right' number. The One in the Many and the Many in the One.
The 2 eludes me. I am split—my mind is working hard to make sense and my body is lying here, subtly excited but somehow deaf, mute, numb. My sensitivity is turned down too low to register the frequencies being transmitted.
I have a crude sense of the 3—the multiplicity of all objects. And strangely this also feels beyond my sensitive range—as if I am looking through infrared glasses at the world. I see shapes, hotspots, but no detail.

The Universe and Our Experience as Six Dimensional

If we can momentarily and provisionally accept this first formulation of the four territories of experience, including the notion of the outer world territory of experience as three-dimensional space… A question that arises is: what kind of dimensions do the other three more inward territories of experience re-present?

My second mathematical/ ontological/ epistemological intuition is that the universe we can potentially experience and at least partially know is organized in six dimensions (not eleven, as current string theories hold).[2] These six dimensions can be considered orthogonal to one another, three spatial and three temporal. We can think of the three dimensions of time as analogous to the three dimensions of space, with a horizontal x axis, a vertical y axis, and a depth z axis that generates volume. In the case of time, we can easily imagine the horizontal x axis as a line of durational time from past to present to future, but few have any ready image of the other two dimensions of time.

In our world virtually all humans come, by the end of childhood, to share a three-dimensional appreciation of the outside world territory of experience. In the famous book Flatland, by contrast, the two-dimensional squares, triangles, and circles have some trouble living with their differences, but they are really disturbed when one member begins to be able to experience the third or volume-dimension of space. The others end up isolating him and persecuting him for his weird 'belief'…

Not only do we humans tend to master the three dimensions of space, but by adulthood a very high proportion of us come,—through stories, relationships, and more formal types of education—to gain a rudimentary competence in dealing with the fourth dimension—durational time. Virtually all of us gain a sense of what an hour is or a week and how to get to a meeting on time, as well how to name and negotiate general behavioral patterns over time (e.g. the beginning, middle, and end of a project). But I say that we tend to come to a “rudimentary competence” in dealing with one another and the outside world in terms of durational time because fifty years of closely examining my own and others' behavior patterns in family and organizational settings has made me distinctly aware of how relatively rare it is for adults to actively study their own or larger groups' patterns of behavior across durational time. Even rarer are those who study these temporal patterns in such a way as to be in a position, not so much to comment authoritatively on the patterns enacted, as to themselves act in a truly timely manner in gesture, tone, and word (perhaps breaking their own typical behavior patterns in order to do so). Timely action and interaction becomes especially crucial at those inflection points when relatively unique situational acts and patterns of interaction are called for that can lead to a transformational change in the prevailing pattern.

Even those of us, such as myself, who make deep personal, professional and scholarly commitments to such self-and-other study of temporal patterns in the midst of action, in order to act in a more timely fashion, discover again and again the degree to which we are exhibiting shadow patterns, not yet penetrated by that “bare attention” at the moment of action, and the degree to which we need our friends' best challenges and support to act in a more timely fashion. Thus, while all moving animals and humans have a rudimentary sense of the fourth dimension of temporal duration that is specifically associated with the second territory of experience—our own inner sensation of ourselves, still or in movement—this realm remains largely “terra incognita” for all of us entering adulthood and remains a major frontier for anyone considering engaging in CDAI.

If our appreciation of the fourth dimension, or the first dimension of time, is in general rudimentary, we are at least called back to the dilemmas of how to use our time continually. Some of us feel bored a lot. Time is hanging from our hands. We don't know what to do with it. Our desired fantasies are completely disconnected from our everyday reality. Contrarily, many of us often feel as though there is not enough time. We are constantly rushing and squeezing things in. How to decide which priority to follow for how long? Why does such rushing seem to drive deep intimacy and the direct experience of love from our lives? How can we ever find enough time, given our common limit of 24 hours per day?

The Fifth Dimension of Space/Time, or the Second Dimension of Time—The Eternal Now

By contrast, the very fact that virtually no one speaks about the other two dimensions of time indicates just how unknown they are in most people's daily experience. The border between the first two dimensions of time is, we might say, a general border for human experience historically speaking. Nevertheless, once we start naming and exploring this second dimension of time, we will find that almost all of us have very occasional, fleeting experiences of this kind of time, experiences that are somehow deeply meaningful even if we cannot explain them.

We have taken durational time as the x axis, a horizontal line on a three-dimensional graph of time. Now we draw the second dimension of time as the y axis, a vertical line that can pass through the durational time line at any point. What can this represent or mean? This dimension of time is not durational. In this dimension of time, it is always now, the eternal now. The experience of this moment—of, say, mutual, loving intimacy with this person—is all we want and have ever wanted. 'Time' feels full. We are in no rush whatsoever to get anywhere else. The longer this unending moment lasts the better. Or: our deepest conscience is evoked; we see clearly and feel shriven by the archetypal incongruity that we are currently enacting and that we have re-enacted again and again throughout our life. To turn our glance away this time would be the ultimate cowardice. We accept the depth of our rootedness in this incongruity (and thereby gain a small new measure of choice for the future). For another account of the experience of this kind of time, see the story below, from a young woman who read this passage:

After I read this (writes a second reader), I had a very strong experience of the eternal now. It happened on a conference call initiated by my Jewish mentor with a few of us from Muslim and Jewish backgrounds to try and talk about what's happening in the middle east (during the Israel-Hamas shelling, 7-8/14, ed.). Instead of talking about it, which all of us were finding very difficult to do, we decided to just take the time to grieve together. So we sat in four corners of the world and wept for half an hour, saying a little here and there about what was coming to our mind. (There was a particularly powerful exchange between my very dear friend Itzak, who sobbed like I have never seen a man sob and told me that he felt he was losing not only the one place in the world that feels like home (Israel) but also a person who feels like a sister (me). I told him it was true I have been avoiding him and my other Jewish friends, and all I really wanted to do was go over to his house and smoke pot with him and stay up all night not talking about any of this stuff!) The whole thing turned out to be a most humanizing and intimate experience. There was such a feeling of timelessness. The moment felt detached from the chain of events that were coming before or after it, but felt connected with all the other moments of true intimacy in my life, as if there were beads on a string hanging down from the heavens…

In Latin this Eternal Now experience (that typically dissipates within moments) is called the Nunc Stans—the Standing Now. Such moments can either arise entirely spontaneously, or be experienced as a moment of grace in the midst of a spiritually-oriented ritual (e.g. Communion)… or as an in-and-out-of-body experience of elation and witnessing-presence amidst the danger of participating in a non-violent political action or a staged psychodrama. Such moments seem to appear more frequently as a by-product of efforts to turn our attention toward its source, while witnessing our patterns of thought, action, and impact dispassionately, in prayer, meditation or action. R.M. Bucke in Cosmic Consciousness (1902), a book I found during my twenties, recorded hundreds of experiences of this 'elevated' state. This five-dimensional experience of time and space seems to touch and integrate all four territories of experience at once—the noumenal (0), the archetypal (1), one's present embodiedness (2), and the outside world (3).[3] It also seems to embrace the fourth dimension of duration without disconnecting from the present, sometimes revealing our life's journey as if in a single lightning flash.

Just as the experience of the first dimension of time is especially associated with the embodied awareness territory, so the experience of the second dimension of time is especially associated with the archetypal thinking territory. The special challenge here is how to develop archetypal thought patterns that we actually exercise in action and that, rather than entrapping us in the thinking territory alone, generate an active listening into all four territories at once.[4]

The Sixth Dimension, or Third Dimension of Time, or Volume of All Possibilities

Now we can briefly address the least-known or written-about dimension of space-time in this 6-dimensional conception. What can it mean to speak of the third dimension or z-axis of time? What might we mean and what might we experience as the 'volume of time'? Many of us have gained some awe-ful impressions of the volume of space from dizzying planetarium films about the unfathomable distances between galaxies and the trillions of stars. These films and current books on astrophysics also tell us that we are living 13 billion years after the big bang inaugurated our version of the universe. But, however overwhelming the notion of 13 billion years may be, all that time is measured along the x-axis of durational time and doesn't give us a hint about the other two dimensions of time—y-time and z-time. What happens if, standing, relaxed, you explore entering all four territories of experience at once, focusing lightly after a minute or two on the vertical sense of levity/gravity that runs through you, following it down to planet core, then sun core, and on 'back' to physical origin?

Or, what happens if you follow that vertical sense 'up' into the 'future'—into the volume of all possibilities—a notion more or less aligned with cosmologists' current speculations about the possibility that we are experiencing but one of many currently developing universes. Perhaps, once we have learned something about how to open ourselves to increasingly prolonged experiences of four-territory awareness in the present (y-time), we will encounter an increasing number of moments when we can focus lightly on a future intended act, and discover one, or many, that are not just in line with our personal archetypal patterns, but also responsive to the other, the group, the societal, and environmental needs (or the scientific data in front of us). Perhaps, we can learn to remain quiet and observant as the first more-creative/responsive possibilities occur to us, until a virtual waterfall of possibilities pass before our inner eye (z-time).

Over a period of decades, I discovered first that, when planning a future meeting agenda, or class, or consulting intervention, if I 'forced' strategizing by creating agendas from outlines of decisions to be made, the subsequent meetings tended to be relatively lifeless events. On the other hand, if I could nurture an alert, relaxed state within myself, then a kind of meditative, creative imagination suffused with color and feeling would develop and produce a sense of the energetic shift necessary at the core of whatever work was in the offing. In a kind of waking-dreaming, I would find myself spontaneously immersed in imagining myself and others interacting in the forthcoming event. Taking notes of these 'process-thoughts' would generate an agenda consisting of exercises to help us think and decide things together. These agendas were both an exploration and a joy to enact, challenging me to generate 5th and 6th dimensional awareness during the event, and seeming to encourage others' creativity and the production of inspiring results to which we were deeply committed.

Only after some time did I develop the patience, not to accept the first creative offering, leave the state, and craft a result, but rather to remain in the open, listening state and experience the “rain of possibilities.” My experience of this 6th dimension is, I am quite sure, minimal and inadequate to its full timely and timeless spaciousness. I include this short discussion of the 6th dimensional rain of potentially timely possibilities only as a small effort to help others' future action inquiry.

I feel more confident about the overall implication of a three-dimensional theory of time—namely, that the past, present, and future cannot properly be visualized or understood on a one-dimensional line. The past line of actual events and our ordinary sense of sequential movement lies in one dimension of time (x-time). The more fully presencing work of opening oneself to four-territory awareness occurs in a second dimension of time (y-time). And the play of waking/dreaming a future worthy of enacting, in the midst of presencing awareness and embodied action (including intentional in-action), is found in a third dimension of time—in the volume of all possibilities (z-time). I trust that one conclusion of this way of thinking-experience and experiencing-thinking—namely, that our source lies more in our future and in our presencing than in our past—proves provocative and encouraging for some of you.

(First reader's response:)
I feel awakened and enlivened by the 6 dimensions of space and time. I recognise the Eternal Now and have come to befriend this fine sliver of existence that endures for an exquisite moment. A look into my wife's eyes, a quickening of my heart as my children laugh, a moment of the purest silence as a client works out an answer, a period of patience as the present tips over into the future and the future unfolds in the now. The more I recognise / remember the existence of this dimension, the more able I am to simply relax and enter it's potent portal.
The 6th dimension feels like something on the other side of a shroud, a curtain, a door of perception perhaps? And yet I have had one or two experiences in recent months where I have felt 'detached' from my body, where durational time has slowed down and I have entered the Eternal Now. And from this 'place', I have had a view of a myriad of choices, paths of possibilities making themselves visible to me as I have slowly 'turned around' to resolve the needs of the present.
I notice how limited my temporal labels are—how challenging it is to think outside a space and durational time framework. And how I feel these other time dimensions deep within and beyond me.

As physicists ever since Einstein's Theory of Relativity (which would more accurately be named the Theory of Relationality) have dimly known, there will be no adequate cosmological theory (just as there will be no adequate social theory), that does not include the evolutionary dynamics of consciousness, love, and time, along with more elegant explanations of astro-bio-evolution. (It can hardly be co-incidental that Riemann discovered a deeper intimacy with how the primes work than anyone before him by entering via the realms of the imaginary numbers (an archetypal, 5th dimensional, mathematical mode) and the calculation of where the zeroes (or sea level points) lie on a ley line (found at 'West ½' in a zeta landscape) (du Sautoy, 2003). Such is the modern mathematical analogue for the 6-D experience or z-time.

To seek to live on a daily basis in all six dimensions of space-time seems to me to generate a truly rich good life of deepening existential inquiry and relatively timely interaction, lived with other humans in both ongoing non-violent struggle and deep periods of harmony, and with the cosmic capacity for real-izing heretofore uncontemplated possibilities.

(Elliot Benjamin comments: I think I basically understand what you mean, if I take the z-axis of time to be all those profound mysterious elements of time that are "possibilities"—such as pre-cognition, past-life communications, etc.—for which I take the perspective of what I refer to as "experiential skepticism." But I must also say that I find your whole mathematical scheme basically poetical, as the analogy of 3-dimensional time with 3-dimensional space is not there, regarding plotting points in any kind of meaningful way, forming graphs and equations, etc. However, I also find your whole scheme much more digestable than I find Collins' "qualitative mathematics" for exactly this reason. Collins tries to define a whole formal new realm of mathematics, which I strongly think should not be categorized as mathematics, as I have written about on Integral World. I find your ideas much less pretentious, and if I take it as an interesting context that does not intend to have mathematical properties per se, then I actually like it.)

With gratitude to Elliot Benjamin, Aftab Erfan, Ian Harcus, Jason Harrison, and Grady McGonagill for their constructive and critical commentaries on earlier drafts of this essay.

NOTES

[1] The closest I have come to articulating these 'axioms' is in my chapter on Pythagorean mathematics in Sources of Excellence (1993), in my "27 Flavors of Action Research" article (w/ D.Chandler, 2003), and in my 2013 article "Listening into the Dark." (Only since first drafting this paper have I more closely read and come to appreciate Peter Collins' (1999-2014) Integral World essays into marrying a qualitative, experiential mathematics with quantitative mathematics, along with Elliot Benjamin's informative attendant critique of Collins.)

[2] Since the 11-dimensional theory suggests that the 'extra' dimensions are enfolded within the three known spatial dimensions, we may be able to save that aspect of that theory by saying that there may be other infinitely 'smaller' space/time worlds enfolded within ours.

[3] Thus, the 5-dimensional experience of time and space opens us, not just to single-loop corrective feedback (between the action and outside world territories, correcting our hammering so the nail enters straight), but also to double-loop feedback (among our 'theory,' our action, and its impact, making it possible to correct our whole strategy [e.g. substituting a screw, a small drill, and a screwdriver for the hammer and nail])… as triple-loop feedback (among intent, strategy, performance, and outcome) will become meaningful in the context of 6-D experience.

[4] While it is far beyond the scope of this short paper to investigate such archetypal thought patterns, I will name three in increasing order of complexity:

1. so-called polarity thinking that helps people recognize their habit of polarizing opposites and treating them as implacable enemies, and helps them to experience the transforming creativity of treating opposites as mutually necessary and finding solutions that include both;
2. trialectical thinking that recognizes that any resolution will be improved if it passes through a thesis-antithesis-and-synthesis process (as, note, the polarity coaching/consulting process does); and
3. the so-called enneagram figure. The enneagram figure, unfamiliar to most people, but eminently google-able, is a much more complex archetypal symbol that shows a circle, with a triangle (representing the trialectic) inscribed within it, along with another six-sided figure. The triangle touches the circumference at points 0, 3, 6, and the lines of the six-sided figure successively touch the circumference at points 1, 4, 2, 8, 5, 7, and back to 1 (note that this generates the recurring fraction for the sevenths: 1/7= .1428571 etc., 2/7 = .2857142etc., etc.). The result is an octave of development around the circle where the angles of the inscribed figures touch the circumference. Dancers may swirl sufi-like around the paths within the circle, offering an embodied sense of cosmic laws in action. At the center, an apparently crazed figure jitters and trembles about. In our own practices, can we situate ourselves in the center of the enneagram and experience the archetypal patterns of the dance of the personal, the immediate-group, the societal, and the planetary-environmental layers of our own action situation in 6-D?

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* These citations can be accessed by entering "William Torbert" at:
http://dcollections.bc.edu/R/?func=search&local_base=gen01-bcd03