25 February 2009 | Draft
with Globality through Enacting a Klein bottle
Cognitive implication in a polysensorial "lens"
- / -
with Globality through Knowing Thyself
of a four-fold exploration. [Engaging with Globality
Dimension 1: Cognitive
; Dimension 2: Cognitive
; Dimension 3: Cognitive
; Dimension 4: Knowing
Overview of Engaging with Globality
Klein bottle: enacting a cognitive process
"Distracting" aids to comprehension
Psychodynamics of sexual experience
The following is one example of an exercise envisaged in Playfully
engaging with globality through re-categorization and re-classification (in Dimension
4). The context for this exploration of a metaphor with sexual connotations
is the subject of a series of comments provided separately as as Annex
with Globality through Playful Re-categorizing):
Humour | Playful
examples | Logical
vs Aesthetic correspondences | Meaningful
and orifices | Consumption
and consumerism | "Knowing" another | Courtship | Kama
The cognitive implications of the metaphor are initially developed in Annex
Governance via a Double-breasted Strange Attractor) and further developed
in Annex C (Engaging
with Globality through Dynamic Complexity) as an introduction to what
The exploration is premised on the assumption that sustainable governance
is necessarily sexy -- and if it is not then it is unlikely to be sustainable. By
designing sex out of governance, sex has become identified with the problematic shadow
If arguments regarding the consequence of sex (namely population
increase) are as well-founded as those which sustained the financial system
prior to its crash in 2008, then the argument of economists regarding population
stabilization should be viewed with the deepest suspicion. They obscure surprises
as unexpected as the crash of the financial system.
Perhaps the mystification
associated with "lost knowledge" could, in the case of governance,
be associated with a lost capacity to integrate the complex cognitive implications
of sexual dynamics into decision-making. More to the point, in the
absence of the cognitive dyanmic associated with sexuality, is there any
indication that the international community "engendered" anything
From the perspective of many cultures of governance, notably for males, a
"Klein bottle" is indeed considered an admirable approach to finding
the answer -- notably by enabling the question to be forgotten. However, if
the entheogenic associations to "triple crown cognition" (in Dimension
3) are valid, this approach may offer a fruitful approximation
in the absence of anything better.
Klein bottle: enacting a cognitive process
In contrast to the indications of Annex B (Global
Governance via a Double-breasted Strange Attractor) regarding
of engagement with globality -- to be caricatured as the "outer-game of
-- the focus here is on what might be termed the "inner-game of
sex" (a term seemingly first used by Robert Shea, The Inner
Game of Sex, Playboy, October 1978). The focus advocated here addresses
the cognitive insights associated with the processes of "making love" with
otherness, however radical, rather then "making war" -- a theme implict
in theology ("love thine enemy") and the politics of intercourse.
Globality is held, and intuited, to be closely associated with a sphere --
and how roundness and sphericity are understood and experienced. However the
paradoxes of experiencing globality in that case -- as with the necessary
diurnal rhythms of night and day -- call for a degree of cognitive engagement
that implies a more complex form of cognitive "convoluted involution". These
imply a dynamic beyond that associated with the "revolution" of the sphere.
there is a terminological confusion with revolution on an axis -- as in
per minute" -- otherwise termed "rotation".
"Revolution" is more correctly applied to orbital
movement, as with a planet around a sun. This
confusion may well be reflected in the psychosocial use of the term. The torus
may be understood as a topological development of the sphere -- suggested by
the manner in which it traces the path of (orbital) "revolution" of a sphere
around the axis of a central "hole". The cognitive challenge for greater engagement
with globality is a form of enantiodromia that "involves" a
process of cognitive "inside-outing" and "outside-inning" --
hence the potential insides from a Klein bottle -- as a further development
beyond the sphere and the torus.
Complexity of conventional explication: For the exploration
here to be fruitful, it is appropriate to expect that the nature of any proposed
construct -- and the understandings it might be used to interrelate -- may
readily invite an inhibiting degree of complexity in any explication. This
would be fundamentally counterproductive to its role in facilitating insight
into the engagement with globality that is the focus of this quest. In a sense,
its very form elicits the explicatory challenges for which it might be used
to provide clarification.
Re-cognizing the body: It would appear that it is in relation
to lacanian topology that most attention has been given to such possibilities;
topology is used to situate psychoanalytic concepts in structures that convey
interdependent relations. But Elizabeth Gorsz (Volatile
Bodies: towards a corporeal feminism, 1994) refigures the body so
that it moves to the center of discourses on subjectivity. She offers a theorization
of the female body based on the Möbius strip as a corrective to the male-biased
theories of Freud, Lacan, Merleau-Ponty, Foucault, Deleuze, Derrida, etc. on
the subject of the body -- notably challenging the visual bias of Merleau-Ponty.
In showing how images promote assumptions and conceal tensions in philosophical
works, how images persuade, and how they limit debate and exclude ideas, Marguerite
La Caze (The
Analytic Imaginary, 2002) suggests that a more open-ended and reflexive
approach can result in richer, more fruitful insight. She cites Grosz's understanding
of the Möbius strip, as the way the body is both imbued with subjectivity and
the way subjectivity is material:
The strip has the advantage of showing the inflection of mind into body
and body into mind, the ways in which, through a kind of twisting or inversion,
one side becomes another. This model also provides a way of problematizing
and rethinking the relations between the inside and the outside of the subject,
its psychical interior and its corporeal exterior, by showing not their fundamental
identity or reducibility but the torsion of one into the other, the passage,
vector, or uncontrollable drift of the inside into the outside and the outside
into the inside. (p. xii)
Intriguing however is the manner in which two such strips of different orientation
constitute a Klein bottle. This is a theme extensively explored by Melanie
Claire Purcell (Towards
a New E.R.A: epistemological resolution analysis in, from and through Klein
bottle wholeness and transdisciplinary education. 2006; Imperatives
for Unbiased Holistic Education: the Klein bottle, a universal structure: an
archetypal image, 1999).
Paradoxical recursion: The Klein
bottle is recognized as offering a means of giving visual expression
to the paradoxical cognitive challenges of self-reflexivity and to the psychodynamics
of "knowing thyself". It has been held to be the most unifying
topological structure by various authors, including David Bohm and Steven
is Radical Recursion?) -- as clarified by Melanie Claire Purcell
(2006). Its role in this respect is consistent with arguments originally
made by the psychoanalyst Jacques Lacan. Purcell, for example, sees it as
integrating the characteristics of wholeness, formative causation, and an
ontological bridge for comprehending consciousness through an epistemology
that invokes pantheism as the most universal spiritual; construct.
Visualization: In mathematics, the Klein bottle is a surface,
apparently taking the form of a "bottle" in three-dimensions, with
no distinct "inner" and "outer" sides, nor any boundary
(in contrast with the sphere). Being only a surface, it cannot act as a container.
It is related to the better known Möbius
strip which is a two dimensional surface, having only one side, but with
a single boundary. These structures need to be visualized in three dimensions
to be partially comprehended. Better still the visualization needs to be dynamic
to gain fuller comprehension -- especially in the case of the Klein bottle.
Of the many visualizations available on the web, the following were noted as
In fact, as is acknowledged, a three dimensional representation in the latter
case is itself completely inadequate (and misleading) -- implying an inappropriate
distortion that is nevertheless significant to what follows. The Klein bottle
is essentially a four dimensional construct -- beyond conventional representation.
Representation of globality: The argument here is that it
is precisely such paradoxical subtleties that address the widespread assumption
that the challenges of engaging with globality can be explicated with simpler
and more readily comprehensible forms. But, to the extent that such subtleties
are commonly experienced, the Klein bottle provides a surface with which they
may be experientially associated.
Globality is "universally" represented by the ideal form of a sphere
-- perhaps driven by traces of the original instinctual attraction of an egg
for a sperm, as required for fertilization and reproduction. This shape is
however not sufficiently complex to hold the psychodynamics of engaging with
the globality of an apparently "external" world whilst sustaining
an integrative ("global") sense of one's own personal "inner" identity.
Topological forms such as the Klein bottle, mathematically related as they
are as developments of the sphere, are therefore at the limit of the forms
of comprehensibility with which it is possible readily to identify. Hence the
interest of the explorations of lacanian psychoanalysis such as those of Rosine
Lefort and Robert Lefort (Birth
of the Other / Naissance de l' Autre, 1980 / 1994) citing Bernard
Morin et Jean-Pierre Petit (Le Retournement de la Sphère, Pour la
Science, janvier 1977).
The limitations of the spherical globe in this respect are
helpfully illustrated by the distinction between night and day as experienced
on the surface of the planet. Definitive local assertions
are brought into question by its rotation. This results in
symbolically significant encounters with "shadow" and the celebration
(externally lit) -- a cognitive dynamic essentially excluded from static metaphoric
respresentations of global and globalization. The Klein bottle offers a richer
means of comprehending the paradoxical continuity between seemingly definitive
Identification rather than representation: The stress in
what follows is precisely with respect to experiential identification rather
than with description or representation. Assumptions regarding the conventional
objectivity of description are in fact called into question (Creative
Cognitive Engagement: beyond the limitations of descriptive patterning,
2006). In a sense, in order to carry and focus the significance of "know
thyself", the challenge is to enable and give form to the cognitive process
through the paradoxical complexity of the Klein bottle.
In effect that form must be cognitively embodied or "felt" as a
dynamic process, rather than visualized statically and "looked at".
It is sensed through familiarity with a process with which one is identified
rather than through detached observation. This is consistent with arguments
made elsewhere (Stepping
into, or through, the Mirror: embodying alternative scenario patterns,
Challenge of Polysensorial Knowledge: bringing the "elephant" into "focus",
"Distracting" aids to comprehension
The challenge is to locate clues to enable comprehension of such embodiment
of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics,
2007). These may perhaps be understood as mnemotechnics or Ars
A. Yates, The Art of Memory, 1966).
Whilst many of these are associated
with extensive and insightful commentary (as with that of Melanie Claire Purcell),
the suggestion here is that it is the more radically experiential "aid" (of
the subsequent section) on the Psychodynamics
of sexual experience that is more fruitful than what immediately follows.
These clues are then best used for possible clarification and as a source of
complementary insight. Arguably they may even be understood as various symptoms
of the problem of the challenge of communication of insight. They might also
be considered, following the arguments of Dimension
2, as forming a circlet that implies the subsequent cognitive focus.
Text: The challenge of conventional discursive text explanations
may be caricatured by the classic example of any effort to provide a text description
of a spiral staircase (or sexual intercourse) for those who have never had
the experience. Carolyn G. Guertin (Quantum
Feminist Mnemotechnics: the archival text, digital narrative and the limits
of memory, 1996) provides a helpful review of many of the issues --
but in a text that is necessarily subject to the reservations that she highlights
(as with this one, or that of Melanie Claire Purcell).
The problematic assumptions conventionally made regarding text are discussed
in detail in Beyond
the plane: form and medium in terms of the calculus of indications (part
of Requisite Variety for Sustainable Psychosocial Dynamics: Transforming a
matrix classification onto intertwined tori, 2006). The arguments
made there with regard to the torus are also relevant to consideration of the
Specifically, in summary, in contrast to the plane surface of a simple matrix,
a torus holds an interesting position in the discussion of the relationship
between form and medium as fundamental to advanced theories of communication.
This notably featured in the work of Niklas Luhmann (Die Gesellschaft der
1997) and discussed by Michael Schiltz (Form
and Medium: a mathematical reconstruction, Image [&] Narrative,
6, 2003) in relation to the calculus of indications of George
Spencer-Brown (Laws of Form, 1969/1994). Schiltz notes that form/medium
is "the image for systemic connectivity and concatenation", as described by Humberto
Maturana and Francesco
Varela. Schiltz notes, that the notion of "space" is the key to reflexivity
appropriate to any discussion of form and medium, citing Spencer-Brown as follows:
In all mathematics it becomes apparent, at some stage, that we have for
some time been following a rule without being aware of it. This might be
described as the use of a covert convention. [… Its] use
can be considered as the presence of an arrangement in the absence of an
agreement. For example, in the statement and theorem.... it is arranged (although
not agreed) that we shall write on a plane surface. If we write on
the surface of a torus the theorem is not true […] The fact
that men have for centuries used a plane surface for writing means that,
at this point in the text, both author and reader are ready to be conned
into the assumption of a plane writing surface without question. But, like
any other assumption, it is not unquestionable, and the fact that we can
question it here means that we can question it elsewhere […]
To what extent are the various approaches to sustainable development, and
the search for alternative paradigms, to be considered as efforts to achieve
new -- and more encompassing -- forms of closure?
Poetry: One clue is to be found in the pattern of associations
held succinctly in poetic form, as with that of T
S Eliot (in Little
Gidding, 1942), echoing the life-cycle experience of a salmon:
|We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.
Haiku may be understood as
specifically serving such a purpose (Ensuring
Strategic Resilience through Haiku Patterns: reframing the scope of the "martial
arts" in response to strategic threats, 2006), although the more
general relevance of poetry remains to be explored (Poetry-making
and Policy-making: arranging a marriage between Beauty and the Beast,
Plato's Cave: Another clue might be to suggest that the much
cited cognitive challenge of Plato's Allegory
of the Cave might be more appropriately understood if it took the
form of a Klein bottle. However, whilst the traditional symbol of the Ouroboros is
consistent with the cognitive challenge posed by the Klein bottle, its meaning
as a symbol has to be derived elsewhere.
Music: For the musical, Robert W. Peck (Klein-Bottle Tonnetze, Music
Theory Online, 9, 3, August 2003) departs from the toroidal Tonnetz of
neo-Riemannian theory, to construct a generalized Klein-bottle Tonnetz.
He then examines associated transformational graphs and analytical contexts,
using various cyclic group operators. He ends by considering relationships
among entire Klein-bottle Tonnetze, and places them into recursive
supernetworks. Ernest G. McLain (The Myth of Invariance: the origins
of the gods, mathematics and music from the Rg Veda to Plato, Shambhala,
Reflexivity: The exploration of Douglas
Hofstadter beyond the reflexivity of Gödel,
Escher, Bach: an Eternal Golden Braid (1979) is exemplified by the
title of his more recent work, I
Am a Strange Loop (2007), which is indicative of the cognitive challenge.
Psychoanalysis: The pioneering work of Jacques
Lacan and his followers, informed by topology and knot theory, is clearly
vital in offering indications, especially with respect to the significance
of the manner in which the surface of the Klein bottle appears to be intersected
by a portion of itself. ****
This diagram [the Möbius strip can be considered the basis of a sort
of essential inscription at the origin, in the knot which constitutes the
subject. This goes much further than you may think at first, because you
can search for the sort of surface able to receive such inscriptions. You
can perhaps see that the sphere, that old symbol for totality, is unsuitable.
A torus, a Klein bottle, a cross-cut surface, are able to receive such
a cut. And this diversity is very important as it explains many things
about the structure of mental disease. If one can symbolize the subject
by this fundamental cut, in the same way one can show that a cut on a torus
corresponds to the neurotic subject, and on a cross-cut surface to another
sort of mental disease. [Lacan (1970), pp. 192-193]
In this space of jouissance, to take something bounded, closed,
is a location, and to speak about it is a topology. ...What does the most
recent development of topology allow us to put forward concerning the location
of the Other, of this sex as Other, as absolute Other? I will put forward
the notion of compactness. Nothing is more compact than a fracture; clearly,
the intersection of everything that closes being admitted as existing on
an infinite number of sets, it follows that the intersection implies this
infinite number. It is the very definition of compactness [Lacan (1975)]
Such issues are the subject of extensive relevant comment by subsequent psychoanalysts
of lacanian persuasion:
- Charles Shepherdson, The
Intimate Alterity of the Real: A Response to Reader Commentary on "History
and the Real" (Postmodern
Culture, 6, 3, May, 1996)
- Jacques-Alain Miller. The symptom:
universalism versus globalization
- Gérard Wajcman, translated
by Ron Estes, Jr. (Intimate
Extorted, Intimate Exposed, Journal for the Jan van Eyck
for Lacanian Ideology Critique, 1, 2008, pp. 58-77)
Misrepresentation: However it is precisely these paragraphs
of Lacan that are cited as an example of pseudoscience by the mathematician
Kuroki Gen (Lacan,
30 August 1999). It was such argumentation by Lacan that was incorporated into
an article by mathematician Alan
Sokal, submitted for publication in a respected journal as a deliberate
hoax -- resulting in the notorious Sokal
Affair (Alan Sokal, Transgressing
the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity, Social
Text 46/47, Spring/Summer 1996, pp. 217-252).
Beyond binary logic: But whatever the merits of Sokal's perspective,
the many comments on his Affair offer a remarkable illustration of the
range of binary perspectives (open/closed, right/wrong, agreement/disagreement,
knowledge/ignorance, reality/illusion, objectivity/subjectivity,
identification/detachment) that any richer understanding of globality
must necessarily embody. They offer a metaphor of many "two
culture" challenges of "clashing civilizations" and the
arrogant discrimination these evoke as a reaction to perceived naivety
and gullibility. Such perspectives might therefore be fruitfully mapped
onto the Klein bottle to hold the continuity between such apparently
A particular justification for this 4-dimensional transcendence of binary
logic is in relation to the polarity subject/object, especially as it is highlighted
by feminist scholars and lacanian psychoanalysts with respect to women framed
as a sexual "object". Hence the value of the sexual metaphors explored here
in order to reframe this simplistic distinction. Of related interest is the
study by Max Deutscher (Subjecting and Objecting : an
essay in objectivity,
Curiously Sokal would seem to have had little insight to offer into these
matters (outside his discipline), despite the fundamental cognitive challenges
posed by paradoxes to physicists (Etienne Klein, Conversations with the
Sphinx: paradoxes in physics, 1996) . The understanding he brings to bear
on the cognitive challenges are not of requisite complexity to encompass their
diversity and scope -- and the gullibility of those who expected more of him.
Psychodynamics of sexual experience
Associations of the Klein bottle with sexuality -- especially its pathology
-- have been a theme of the lacanian approach to topology. The more radical
argument here is that, rather than explore such "explanations" as
discursive text and its comprehension, more fruitful is the use of the form
itself as a psychoactive "inplanation". At issue is not how to communicate
or transmit insights to others but the very nature of mutually meaningful intercourse
with another ("Human
Intercourse": "Intercourse with Nature" and "Intercourse
with the Other", 2007). This may be understood in terms of:
These dimensions relate in various ways to arguments regarding embodiment
and the understandings of enactivism (George
Lakoff and Mark Johnson, Philosophy In The Flesh: the embodied mind and
its challenge to western thought, 1999; Francisco
Varela et al. The Embodied Mind, 1991. En-minding
the Extended Body Enactive engagement in conceptual shapeshifting and deep
ecology, 2003). The form is then not to be "looked at" as
signifying something "elsewhere", rather it is to be experienced
in the flow (in the sense of process
philosophy or flow
psychology) as the embodiment of an essentially sexual dynamic, however
generically this is understood. ****recursion -- image of itself -- mirroring
Dynamics (4D): Whilst any concept of a 4th dimension is notmally
associated with time as understood by physicists, this tends to obscure other
understandings of time, notably its circularity -- as exemplified by the poem
of T. S. Eliot (above) or perhpas the symbol of the Ouroboros. Thje argument
here is that there is an intimate relation to time that offers an experiential
understanding of a pattern of dynamics, notably as in any sensational process
such as intercourse -- of which the Klain bottle offers a formal representation
-- but challenged by some instances of greater/vaster scope (enantiodromia)
of that pattern.
In this sense, in its 4-dimensionality, the Klein bottle might be understood
as a standing wave in a flow process -- cycling through an attractor that transforms
(enantiodromia). Failiure to encompass the cyclic process results in a sense
of discontinuity -- as intimated by the broken circlets of Dimension
challenge to time-bound learning is one of entering into that wave in ways
reminiscent of the engagement of surfers -- learning to traverse the cycle
as is called for in the literature on the resilience required in engaging in
an adaptive cycle..
Enantiodromia: The possible implications of such a process
have been discussed elsewhere (Psychosocial
Energy from Polarization within a Cyclic Pattern of Enantiodromia,
2007; Patterns of alternation: toward an enantiomorphic
Polarization: This cognitive tendency is widely recognized
as a major challenge. Associated experiences are suggestively addressed
by lacanian psychanalysts as noted above, notably with respect to "image" vs.
"reality". The latter polarity is intriguingly addressed by mathematicians
in their understanding of the complex
plane and the real and imaginary axes
in terms of which it may be represented. The Klein bottle, as with the Möbius
strip, reframes possible understandings of polarization which otherwise constrain
responses to psychosocial dynamics.
In addition to its implications for enantiodromia, the issues are further
considered elsewhere (Polarities
as Pluckable Tensed Strings: hypercomprehension through harmonics of value-based
choice-making, 2006; Discovering
Richer Patterns of Comprehension to Reframe Polarization, 1998; Responding
to Conceptual and Value Polarities: learnings from sexuality, 1998; Antagonistic
Dualities: Polarization and Paradox, 1983).
"Triple crown": Given the arguments in Dimension
3, it is intruiging to look at the Klein bottle as a form of triple crown
-- possibly partially projected into the form of the pharaohnic triple crown.
Are three possible cognitive stages to be distinguished in the process of
knowing represented by the Klein bottle? Are these to be associated with
stages in some drug-enahnced psychedelic experience as implied there? Is
the crown, to be understood as evolving through geometric transformations:
sphere, torus, etc?
What might such insights apply for engaging with globality in the development
of strategy by those so crowned?
Mirroring: The implications of mirroring are a theme of lacanian
insights. The relationship between the Möbius strip and the Klein bottle
are of interest in that respect. They have been discussed elsewhere in terms
of the implications of the mirror
recognition test of self-awareness (Self-reflective
Embodiment of Transdisciplinary Integration (SETI) the universal criterion
of species maturity? 2008).
Does the Klein bottle imply the possibility of a 3-fold mirroring -- the requisite
complexity to encompass experiential paradox?
Attractor: If the Klein bottle form is effectively to be
cognitively enacted, how is any attractor to be perceived so as to ensure the
cyclicity of the process of enantiodromia? This implies issues of mystery and
"perspective" potentially well-illustrated by sexual courtship and intercourse
-- perhaps best understood generically, through "grokking", to encompass all
the forms of engagement through the senses.
This implies cognitive processes of:
- encompassing that by which one is encompassed
- engulfing that by which one is engulfed
- enfolding that by which one is enfolded
- enveloping that by which one is enveloped
- enthralling that by which one is enthralled
- embodying that by which one is embodied
- embedding that in which one is embedded
Of potentially greater interest is the manner in which values functions as
Values as Strange Attractors: coevolution of classes of governance principles,
Pathology: Again it is the lacanian psychoanalysts that have
been very interested in the sense in which the Klain bottle is "cut" by the
intersection in 3-dimensions (which is not apparent in 4). Also of interest
is the possibility of "stretching" in different ways as potentially indicative
-- through the degrees of asymmetry -- of different cognitive pathologies.
Symbolism: As noted above, Melanie
Claire Purcell (2006; 1999) has commented extensively on the relation between
the Klain bottle and the well-known taoist Taijitu
symbol -- the one being potentially a 2D projection of the other. This
relationship is also discussed in Snoring
of The Other: a politically relevant psycho-spiritual metaphor? (2006).
Some quotes from the Tao Te Ching are suggestive of experiential
understandings associated with cognitive "exploration" of the Klein
bottle as a process experience.
|Tao: The Great Mother
as related to the cognitive experience of a Klein bottle process?
The Valley Spirit never dies
It is named the Mysterious Female.
And the doorway of the Mysterious Female
Is the base from which Heaven and
It is there within us all the while;
Draw upon it as you will, it never
runs dry. (chap. 6)
The Tao doesn't take sides;
it gives birth to both good and evil....
The Tao is like a bellows:
it is empty yet infinitely capable.
The more you use it, the more it produces;
the more you talk of it, the less you understand. (ch. 5)
The Tao is called the Great Mother:
empty yet inexhaustible,
it gives birth to infinite worlds.
It is always present within you.
You can use it any way you want. (ch. 6)
The Tao is infinite, eternal.
Why is it eternal?
It was never born;
thus it can never die.
Why is it infinite?
It has no desires for itself;
thus it is present for all beings.
|How do I know this is true?
By looking inside myself. (ch. 54)