Integral

Originally published on Medium. In my previous essay, Sex, Pronouns, and Prepositions

, I made the case for expanding Ken Wilber’s integral perspectival mathematics system with a set of prepositional operators. I wanted to show how an enriched relational vocabulary can really open up new territory for Wilber’s perspectival notation. If you’re one of the very rare people for whom that actually sounds interesting or attractive, keep reading. Otherwise, just back away slowly and hopefully no one will get hurt. To illustrate how the expanded set of prepositional operators work, I took Otto Scharmer’s 3 Levels of Listening as a use case and offered some simple (or simple enough) notation to map certain relational dynamics involved at each stage. Here, I’d like to do something similar — this time to stretch the notational system a little more and demonstrate a more sophisticated use of it. Because I will mostly focus on mapping out a particular use case, a different (but similar) one this time, this paper will honestly be more of a slog to read than the last one. If you’re one of those people, like me, whose eyes glaze over when equations fill a page, I encourage you to relate to those sections mostly as aesthetic objects, and to focus on the descriptions that follow. When Wilber first introduced his “integral mathematics

of indigenous perspectives,” back in the early ‘00s, he commented that he would probably need to wait for some 20-something young gun to come along and really work the whole thing out. That person isn’t me. I’m 30 years too old for that shit. But I do have my little piece to add. And I do

feel it opens up something useful, and potentially illuminating. I just don’t know how far it can go. So, anyway, let’s give this a shot. Mathing the Triple-Loop Learning Process Related to the Levels of Listening model is the well-known systems thinking and organizational development model that looks at single-, double-, and triple-loop learning processes. Each level describes increasingly complex levels of learning and transformation, and because these levels build on each other in a fairly clear way, I thought they’d make an instructive use case for exercising our expanded perspectival notation system. For anyone not familiar, single-loop learning involves making corrections to actions without altering the underlying beliefs and values that govern those actions. It’s about solving problems within the existing system or framework. In single-loop learning, an individual modifies their actions or behaviors based on feedback from the environment or a task, without changing their underlying framework or assumptions or departing from the realm of their personal experience. They make an adjustment, in other words, within the existing parameters of their understanding or behavior. Double-loop learning goes deeper, questioning and potentially altering the underlying beliefs, values, and policies that lead to the actions in the first place. It’s about examining and modifying the framework itself. Double-loop learning reflects a deeper transformation where an individual not only modifies actions but also revises their underlying beliefs or framework, which influences their personal experience or understanding. The transition indicates a move from changing within the system to questioning and altering the system itself. Triple-loop learning involves transforming the way in which problems are framed and solutions are sought, often leading to a shift in the identity or self-concept of the learner. It’s about learning how to learn, questioning the very basis of the framework itself. Triple-loop learning involves a recursive reflection where the individual not only changes actions and underlying assumptions but also fundamentally transforms their perspective on learning and knowing itself. This is modeled by moving through an action change, to altering the belief system, and finally looping back to a broader, more encompassing understanding of the process of change itself, possibly incorporating a collective or universal perspective. This indicates a transformation in how the individual conceptualizes learning and their relationship to knowledge and self-world organization itself. In what follows, I’ll explore how integral perspectival math notation can illuminate the dynamics of each stage. As I discussed in my previous paper, I’ve developed an expanded set of prepositional operators, some of which I’ve already introduced. For this paper, I will be using just a handful of them: Overall, I’ve developed 21 operators that I’ve been experimenting with; time will tell how many are really needed to cover most perspectival situations. For our example here, 5 will be enough. Single-Loop Learning {[1p

(1p) x 1p(3-p

) x 1p(3p

)] ↩ [1p(1p) x 1p(3-p) x 1p(3p) + 3p(3p)]} → [1p(1p) x 1p(3-p) x 1p(3p)x

] In this expression, we have three main components: Putting it all together, the amended notation for single-loop learning can be interpreted as follows: “The individual’s initial first-person perspective on their third-person view of their actions recursively feeds back into a process of incorporating external third-person feedback on the objective aspects of their actions. This recursive loop of integrating external input leads to a modified first-person perspective on their third-person view of their revised actions, without changing their underlying beliefs or assumptions.” This particular formulation highlights the role of external feedback and objective information in the single-loop learning process, showing how the individual’s perspective on their actions can be modified through the incorporation of additional input, even if their fundamental framework remains unchanged. I’m using the curly braces to enclose the recursive loop to help visually differentiate this first part of the expression, emphasizing its role as a distinct subprocess within the overall single-loop learning dynamic. However, as you’ll see in the next parts of this discussion, the layered bracketing quickly builds up and becomes hard to follow, so I’m using a coding-like convention of spreading the equation over multiple lines

to make the layers easier to track. Since I’m not (at all!) a coder, please let me know if this works and is helpful. Double-Loop Learning {{ {[1p(1p) x 1p(3-p) x 1p(3p)] ↩ [1p(1p) x 1p(3-p) x 1p(3p) + 3p(3p)]} → [1p(1p) x 1p(3-p) x 1p(3p)x

] } ↩ { } } → Initial Setup First Recursive Loop with External Feedback Action Modification Second Recursive Loop with Perspective Transformation Comprehensive Transformation This expression for double-loop learning attempts to layer the processes of action modification and perspective transformation, employing recursive loops to indicate feedback and reflection stages. The use of superscripts (^x

, ^y

) communicates changes at different levels: actions and perspectives. This structured approach aims to model the iterative, deepening process characteristic of double-loop learning, where individuals not only adjust their behaviors based on external feedback but also engage in deeper self-reflection that may lead to significant shifts in their beliefs, values, and ultimately, their self-concept. Triple-Loop Learning {{ {[1p(1p) x 1p(3-p) x 1p(3p)] ↩ [1p(1p) x 1p(3-p) x 1p(3p) + 3p(3p)]} → [1p(1p) x 1p(3-p) x 1p(3p)x

] } ↩ { } } → ↩ { } ⇒ Initial Reflection and Action Modification Deeper Reflective Process and Perspective Transformation Integration with Collective Understanding and Transformation of Self-Concept The shift to the plural mode (1pp

) in the triple-loop learning expression is meant to capture the idea that at this level of learning, the individual’s perspective on learning itself undergoes a fundamental transformation that incorporates a broader, more encompassing understanding of the learning process. This shift is not just a personal or individual one, but one that reflects a deeper awareness of the collective, intersubjective, and even universal dimensions of learning and knowledge. In triple-loop learning, the individual not only modifies their actions (single-loop) and their underlying assumptions and beliefs (double-loop), but also transforms their very understanding of what it means to learn, know, and change. This involves a recursive reflection on the learning process itself, which can lead to a profound shift in the individual’s identity, self-conception, and relationship to knowledge. By introducing the plural mode (1pp

) in the final recursive loop of the triple-loop learning expression, I’d like to suggest that this transformation involves (at least

) a recognition of the collective and intersubjective nature of learning and knowledge. The individual comes to see their own learning process as deeply embedded within and influenced by broader social, cultural, and historical contexts, and begins to incorporate this understanding into their perspective on learning itself. This shift to the plural mode suggests, in other words, a paradigmatic level of awareness, where the individual becomes more attuned to the collective conceptual frameworks, worldviews, and meaning-making structures that shape their own learning and knowing. It suggests a move beyond a purely individual or subjective understanding of learning, towards a more holistic and integrative perspective that acknowledges the complex interplay between personal experience, intersubjective dynamics, and broader collective structures of knowledge. However, there is an alternative, and deeper, way to understand the triple-loop learning phase, involving a shift to trans-conceptual awareness that the above notation doesn’t quite capture. So, for anyone interested in a subtler framing (or my best attempt at it), let’s proceed… Reconsidering Triple-Loop Awareness In an essay

published in Integral Review in 2005, Anne Starr and Bill Torbert attempt to explain and illustrate a transconceptual mode of experience which they call triple-loop awareness. This is related to triple-loop learning, of course, but the discussion focuses more on its connection with or expression through a mode of awareness that involves both a post-reflective experience of the Now and creative participation in the future (akin to Scharmer’s account of Generative Listening), and so is especially relevant to our concerns here. To get a better grasp of what Starr and Torbert mean by triple-loop awareness, I recommend taking the time to read their whole article. I will offer just a brief (re)definition for now. Triple-loop awareness or learning is conceived as a developmentally sophisticated, transconceptual

mode of action and inquiry, incorporating and building on simpler modes of awareness (and temporal consciousness). As first described by Gregory Bateson and later elaborated upon by Peter Senge, and as we’ve reviewed above, single-loop learning involves incremental learning and behavioral adjustment, whereby new skills and capabilities are gradually acquired over time. Double-loop awareness and learning involve apprehension of and change in underlying patterns of organization, not just behaviors. We perceive and work with process in addition to content. And triple-loop awareness and learning, then, involve a transrational apprehension of all patterning or schemata – where conceptuality itself, in both its weaknesses and strengths, becomes an object of awareness. As Starr and Torbert put it, This is clearly different from pre-reflective consciousness, where in one’s embeddedness in the present, one cannot step outside of one’s context to take in all of these rich territories of experience. The temporal implications of this mode of awareness and learning are clear, as Starr and Torbert also recognize. In discussing these issues, Starr and Torbert propose a multi-dimensional model of temporality. Normally, they argue, humans operate with either a zero- or one-dimensional time consciousness: either we are oblivious to time, simply caught up unreflectively in our activities, or we are aware of the linear pressure of time – say, while waiting for an appointment or trying to make a deadline. One-dimensional time consciousness, which apprehends time sequentially, allows for single-loop learning, in which one is able to “identify a gap between act and intended outcome, then adjust one’s action, and [possibly] achieve one’s goal.” As they point out, sophisticated versions of this mode of temporal consciousness inform certain historical models of evolution and development. Two-dimensional time contrasts with the narrow moving point-instant of one-dimensional time, involving an expanded sense of open presence – the timeless Now or nunc stans

of the mystics, within which functions such as memory or anticipation arise as ornaments or expressions of the overall field. Starr and Torbert suggest visualizing it as a “line” which intersects linear, sequential time, creating a plane – an open state which does not foreclose and may include zero- or one-dimensional activities (sensory engagement, sequential reflection and anticipation). And three-dimensional time, then, “can again be imagined as orthogonal (the Z axis) to the plane defined by chronological time (X axis) and eternity (Y axis). The three-dimensional ‘volume’ of time can be imagined as holding all possibilities, all the potentialities of the future and the still-hidden meanings of the past, some of which emerge into the present (become act-ualized) and then pass into linear, historical time… [This mode of time involves a] different quality of awareness that goes beyond a deepened sense of presence in the present to sensing oneself as a creative subject actively participating in midwifing an emerging future.” With all of this in mind, we can offer a different interpretation of the Triple-Loop Learning phase, while keeping the basic notation intact. In this interpretation, the triple-loop learning section [1p(1pp) x 1p(3-pp) x 1p(3pp)]

represents the individual’s (1p

) adoption of a collective, multi-dimensional perspective (1pp)

on the learning process itself (3-pp)

and its outcomes (3pp

). This collective perspective encompasses an awareness of all territories of experience, including the outside world, one’s own behavior, thoughts, and feelings, and a witnessing consciousness. The recursive loop ↩

[1p(1p)

y

x 1p(3-p)

y

x 1p(3p)

xy

• 1p(3pp)]

indicates the integration of the individual’s transformed perspective and actions from the double-loop learning process with this multi-dimensional, transconceptual understanding of learning. The emergence (⇒) of the final expression [1p(1p)

yz

x 1p(3-p)

yz

x 1p(3p)

xyz

]

represents the individual’s attainment of triple-loop awareness, a developmentally sophisticated mode of consciousness that transcends and includes single-loop and double-loop learning, and involves a simultaneous, presential awareness of multiple layers of patterning and schemata. The xyz

notations can do double-service here, also standing for the temporal axes they associate with single-, double-, and triple-loop awareness. Here, the notation itself doesn’t clearly communicate the subtleties Torbert and Star discuss, but they can be “read in” without too much distortion. However, we could also try an alternate approach (but really, this last part is only for people who do not yet have a splitting headache). Intersection with Generative Listening Processes Torbert and Starr’s emphasis on midwifing an emergent future is reminiscent, of course, of my discussion in Sex, Pronouns, and Prepositions

of the Generative Listening phase in Otto Scharmer’s work. As a reminder, here is that discussion and the perspectival notation I offered: Generative Listening is the deepest level of listening, where we tune into the emerging future potential. We listen not just to the other person, but to the larger field or system that we’re both part of. This requires a letting-go of our old identities and assumptions, and a willingness to let something new emerge through the interaction. [1p(1p)

⊘ 1pp(3-pp) x 3p]

⊘ [1pp(1pp) → 1pp(3-pp) x 3/pp)] There is co-presence (⊘) between my (an individual’s) first-person experience [1p(1p)

] and our (the collective’s) vision [1pp(3-pp)

] of the future [(3p

)], and this movement itself is co-present or intimately entangled with our (the collective’s) experiential [1pp(1pp)

] movement towards (→

) our third-person apprehension [1pp(3-pp)

] of shared or emergent potential [(3/pp)

]. The use of the third-person perspective in the first phrase [(3p

)] suggests that the future we’re sensing into has a certain objectivity or reality to it – it’s not just a subjective fantasy, but something that we’re discovering or uncovering together. The second part, [1pp(1pp) → 1pp(3-pp) x 3/pp)

], shows the movement (→

) of our collective first-person experience (1pp(1pp)

) towards a third-person apprehension of our shared potential (1pp(3-pp) x 3/pp

). This captures the generative, forward-looking nature of this stage – we’re not just dwelling in the present moment, but actively leaning into and bringing forth the future that we sense is possible. In this stage, the boundaries between individual and collective start to become more fluid and permeable. The individual attunement to the future is not a separate process, but a participatory aspect of the collective movement (which the double use of the co-presence operator [⊘] is meant to convey). Can we take inspiration from this framing to modify the notation for Triple-Loop Learning that better reflects the nuances Torbert and Starr emphasize? Let’s see. Triple-Loop Learning as Transconceptual Midwifery {{ {[1p(1p) x 1p(3-p) x 1p(3p)] ↩ [1p(1p) x 1p(3-p) x 1p(3p) + 3p(3p)]} → [1p(1p) x 1p(3-p) x 1p(3p)x

] } ↩ { } } → ↩ { [1p(1p)y

⊘ 1p(1pp) x 1p(3-pp) x 1p(3pp)] ↩ [1p(1p)y

x 1p(3-p)y

x 1p(3p)xy

• 1p(3pp)] } ⇒ {[1p(1p)yz

⊘ 1p(1pp)z

x 1p(3-pp)z

x 1p(3p)xyz

] ⊘ [1pp(1pp)z

→ 1pp(3-pp)z

x 1pp(3/pp)z

]} In this revised notation, you’ll notice that most of it is the same as in the previous version; where the changes come in are in the final two expressions. The key changes are in the triple-loop learning section and the final emergent state. Step 1: { [1p(1p)

y

⊘ 1p(1pp) x 1p(3-pp) x 1p(3pp)]

↩

[1p(1p)

y

x 1p(3-p)

y

x 1p(3p)

xy

• 1p(3pp)] } This step represents the recursive loop within the triple-loop learning process. The first part, [1p(1p)

y

⊘ 1p(1pp) x 1p(3-pp) x 1p(3pp)]

, indicates the co-presence (⊘) between the individual’s transformed perspective from double-loop learning [1p(1p)

y

] and their sensing into the collective perspective and potential [1p(1pp) x 1p(3-pp) x 1p(3pp)]

. This highlights the fluid boundaries between individual and collective awareness in triple-loop learning. The second part, [1p(1p)

y

x 1p(3-p)

y

x 1p(3p)

xy

• 1p(3pp)]

, represents the individual’s transformed perspective and actions from double-loop learning [1p(1p)

y

x 1p(3-p)

y

x 1p(3p)

xy

]

being integrated with the collective understanding of the learning process [1p(3pp)]

. This integration is a crucial aspect of triple-loop learning, as it allows for the individual’s learning to inform and be informed by collective knowledge and understanding. Step 2: ⇒ {[1p(1p)

yz

⊘ 1p(1pp)

z

x 1p(3-pp)

z

x 1p(3p)

xyz

]

⊘ [1pp(1pp)

z

→ 1pp(3-pp)

z

x 1pp(3/pp)

z

]} This step represents the emergent state resulting from the triple-loop learning process. It consists of two expressions linked by the co-presence operator (⊘). The first expression, [1p(1p)

yz

⊘ 1p(1pp)

z

x 1p(3-pp)

z

x 1p(3p)

xyz

]

, represents the individual’s transformed perspective that is co-present with the field of intersubjective/collective perspective and potential. The superscript ^yz

indicates the transformation brought about by the integration of double-loop learning (^y

) and the sensing into the collective perspective (^z

). The superscript ^xyz

on 1p(3p)

suggests that the individual’s actions have been transformed through the integration of single-loop (^x

), double-loop (^y

), and triple-loop (^z

) learning processes. The second expression, [1pp(1pp)

z

→ 1pp(3-pp)

z

x 1pp(3/pp)

z

]

, switches to the plural 1pp(1pp)

framing to emphasize the collective’s first-person plural experience (1pp(1pp)

z

) moving towards (→

) a collective third-person apprehension of shared potential (1pp(3-pp)

z

x 1pp(3/pp)

z

). This framing highlights the collective’s agency

and active participation in the triple-loop learning process, as it senses into and works to bring forth the future potential. The use of the superscript ^z

throughout the second expression indicates the transformation in the collective perspective and potential brought about by the triple-loop learning process. By using the 1pp(1pp)

framing in the second expression, we give a nod to the previous notation for Generative Listening – and, I think, better capture the dynamic, participatory nature of the collective’s

movement in triple-loop learning, as opposed to just the individual’s attunement to a collective understanding. This refinement aligns the expression more closely with the Generative Listening notation and emphasizes the active role of the collective in shaping the future. The co-presence operator (⊘) linking the two expressions continues to highlight the intimate entanglement and fluid boundaries between the individual and collective in triple-loop learning and awareness – indicating that the movement here is properly transjective

, in John Vervaeke’s sense. As with my last paper, I have to close with the caveat that these experiments in perspectival notation are provisional and incomplete. Human relational and developmental fields are, of course, incredibly complex, and the idea of ‘mathing’ subjectivity at all is, shall we say, fraught

. But with all of that acknowledged, I’m hopeful this little exercise has nudged expectations (and questions) about what we can do with a system like this up another notch. My use of the ‘spread out’ notations was experimental, just to see if it helps to better visualize and follow the nesting relationships common to perspectival interplay. Other conventions could be used, like color coding, or images to accompany the notations. If this were a formal paper, submitted to a professional journal, I’d wrap up with a consideration of critical perspectives, a discussion of findings, etc. But it’s too early for all of that, so instead I’m just offering this into the dialogical

ring. I do see lots of potential applications for an expanded notation system like this, once it is well developed – in, say, the fields of educational theory and pedagogy, developmental psychology and psychological practice, metatheory or interdisciplinary studies, social analysis, human-AI interface or possibly AI empathy training, and so on – so that’s why I’ve just sacrificed another of my weekends to write this nerdy treatise up. Let me know what you think.