And When the Bombing Stops?
Territorial conflict as a challenge to mathematicians
- / -
An earlier abridged version of this paper was printed in Technological
Forecasting and Social Change, 61 (1999), pp. 297-301 under the title And After the Bombing Has Stopped?
Absence of new thinking on territorial conflicts
This communication is stimulated by reflections on responsibility for Kosovo,
Tibet, Kurdistan, Kashmir, Jerusalem, Sudan, East Timor, Taiwan, Gibraltar,
Malvinas, Quebec, Northern Ireland, Cyprus, Sri Lanka, Western Sahara, Scotland,
and many indigenous / ethnic land claims (notably Basque, American Indians and
Australian Aborigines). What happens when the bombing stops in Kosovo? Or when
access to disputed land is interdicted by vengeful use of anthrax?
As with Kosovo, it is always easy to identify and demonize immediate villains
(such as Milosevic or NATO) and to argue for immediate remedies in response
to the need of victims made widely visible by interested parties. The case for
yet more peaceful negotiation is also easily made, despite many years of essentially
futile exercises in developing the art of non-decision-making (Judge, 1997).
In this connection, it is useful to recall Albert Einstein's definition of insanity
as 'doing the same thing over and over and over and over again, but expecting
a different result'.
However, as the years of negotiation in a number of these cases have illustrated,
there is very little new thinking that can be put on the table as a basis for
viable longer-term solutions. Why is this? 'Urging peace', 'Doing
nothing' and 'Denouncing violence' cannot be considered appropriate
substitutes for creative thinking on these challenges.
Challenging the real specialists in complexity
This communication therefore explores the possibility of laying the underlying
and ultimate responsibility for such territorial conflicts at the door of the
following groups and the manner in which they interact: mathematicians,
legal theorists, lawyers and accountants,
international relations scholars, conflict
mediators, media specialists, 'psy-ops'
specialists, and theologians, together with systems
analysts, computer visualization specialists and
The basic argument here is that modern civilization has achieved amazing conceptual
advances in numerous areas, notably resulting in technical and organizational
advances capable of sustaining extremely complex systems. This has encouraged
elimination of many kinds of barriers in a spirit of global unification of one
kind or another, but it has increased the vulnerability of those previously
protected by those buffers against system instability. Beyond the orgy of triumphalism,
the nationalist tensions and conflicts resulting from the dissolution of the
USSR could have been anticipated. The disastrous Asian financial crisis is a
result of failure to understand the consequences of reducing barriers. Who is
responsible for failing to predict such consequences through appropriate simulation?
Or failing to listen to such predictions?
The point to be made is that these amazing conceptual advances have not yet
been focused on conflicting territorial claims and boundary disputes -- despite
the bloody conflicts and major massacres to which they continue to give rise.
Yet, despite widespread obsession with complexity and its management, simplicity
is all that is on offer in practical conflict situations. The opposing parties
in such territorial disputes, assembled around a negotiating table, are not
supplied with any framework based on insights more complex than those on offer
centuries ago. Why is this? It is therefore not surprising that agreements emerging
from such negotiations are simplistic, unsatisfactory and unsustainable.
Understanding territorial claims
Before discussing who is responsible for such failure, it is useful to recall
what underlies a territorial dispute. Most obviously it concerns conflicting
claims for ownership or possession of the same piece of territory. Underlying
this is the sense of identity which individuals and peoples have developed in
relation to such territory. They may experience it as absolutely fundamental
to their sense of who they are. Related to this is the importance of such territory
to community dynamics, whether in the form of pilgrimages, periodic assemblies
and ceremonies, or periodic movements (as with nomadic peoples, gypsies and
'travellers'). However much of Robert Ardrey's Territorial Imperative
remains relevant, humans remain to some degree compelled by instinct to possess
and define territory that they believe belongs exclusively to them -- even unto
death. Kosovo, for example, is imbued with such significance as the Serbian
heartland that it unleashes aggressive territorial instincts.
But what are the many distinctly meaningful ways in which people can identify
with land or territory in a more abstract sense? There is a burgeoning literature
on 'sense of place' (see Edward Casey, 1993). For those whose individual
and collective identity is intimately associated with particular shrines, who
lays claim to 'ownership' of St Peters in Rome, Lourdes, the Temple
on the Rock in Jerusalem, Mecca, or the many shrines that are holy to Hinduism,
Buddhism, and other faiths?
The conceptual responsibility for failure to respond to the subtleties of territorial
dispute needs to be laid at the door of the following groups -- and the manner
in which they fail to interact. They have a case to answer.
Mathematics purports to be the study of relationships, spaces, boundaries
and identity. It offers an incredibly rich panoply of approaches to configuring
relationships, whether statically or over time -- and through many dimensions.
Branches of this discipline focus on patterns, networks, topology and knots.
They offer special insights into the relationships between local and global.
It is however important at this time to ask how much thinking by mathematicians
is relevant to identification of windows of opportunity through which subtler
solutions to territorial claims can be identified. To what extent are territorial
claims a matter of tessellation (tiling) and packaging patterns, possibly in
more than two dimensions and possibly involving time? A symposium of the wise,
to celebrate the sesquicentennial of Boston University (Lance Morrow, Metaphors of The World, Unite!, Time, 16 Oct. 1989, p. 96) selected a tessellation
as the metaphor that best captured the spirit of the times. The European
Bioinformatics Institute is exploring use of tessellation's to manage its
bibliographic data. How can tiling patterns overlap, be subjected to transformation,
and offer alternatives patterns to perception through visual illusions?
Do the 17 types of tiling pattern on a surface suggest the range of territorial
patterns worth examining? Does the range of regular and semi-regular polyhedra
( with their respective tiling nets when unrolled onto a plane surface) suggest
other ways of organizing territory, especially in light of the dynamic transformations
between them, their duals, etc? What is the relevance of George W Hart's collection
of over 1,000 virtual reality polyhedra as a mean of organizing and visualizing
complex spatial relationships (http://www.georgehart.com/)?
Ironically, it has taken the real estate industry to invent 'time-sharing'
of the same territory. Sadly, ghettos and the zoning of Belfast and Palestine
are examples of tiling without the benefit of such insight. Do mathematicians
have something better to offer? Some thing other than the standard urban grid
There has been much application of mathematics to game theory and to global
economic modeling (whose merits are best exemplified by the recent, unpredicted,
Asian financial crisis resulting from 'globalization'). Complex decisions
by multinational are now being facilitated by 'analytic hierarchy process'
(AHP) software developed by mathematician Tom Saaty. These all focus on techniques
by which one party may "win" over one or more others -- as long explored by
military research establishments. None of these explores the possibilities of
new patterns of relationships between conflicting parties that need to attach
their identity to the "same" territory they can each claim as their own -- so
that they can all 'win' together, rather than set the stage for renewed
conflict. All that is on offer is a simplistic focus on resolving conflicts
over a two-dimensional surface that fails to honour cultural differences and
the importance attached to them. And what does the fashionable exploration of
'complexity' have to offer, other than assistance in speculation on
the financial markets?
Mathematicians -- having lent the full support of their discipline to the weapons
industry supplying the missile delivery systems -- would claim that their subtlest
thinking is way beyond the comprehension of those seated around a negotiating
table. They have however failed to tackle the challenge of the packing and unpacking
of complexity to render it comprehensible without loss of relationships vital
to more complex patterns. As with the protagonists in any conflict, they would
deny all responsibility for such failures and the manner in which these have
reinforced unsustainably simplistic solutions leading to further massacres.
This group is most closely associated with reinforcing definitions of sovereignty
in relation to territory and boundaries. It is they who are central to the process
of articulating constitutional options as an outcome for complex negotiations.
It is their theories which are most challenged by increasingly outmoded concepts
of sovereignty and the explosion of regional foci within nations. It is they
who are most terrified by the consequences of rethinking the significance of
territorial boundaries -- a consequence often labelled as a 'Pandora's
box'. It would be difficult to find a group less motivated, or capable,
of dialoguing effectively with mathematicians regarding new approaches to identifying
territories and to maintaining exclusive claims upon it.
But this group does have a particular approach to the use of number and sets
that merits the attention of mathematicians. Most legal documents, from constitutions
to contracts and including every variety of agreement, declaration, and manifesto,
is subdivided into a series of articles, possibly nested into sub-sections.
Society is governed through the provisions and procedures organized in this
way (see Anthony Judge, Representation, comprehension and communication of
sets: the role of number, 1979). From a mathemetical perspective,
it would be difficult to be more simplistic than this shopping list approach.
What does it imply when such thinking is applied to Kosovo? What alternatives
do mathematicians have for the organization of such sets?
It is also interesting that contracts and agreements between two parties require
'confirmation' by a 'third party', namely one or more 'witnesses'.
In this way the law imposes a process which surveryors would understand as triangulation
from a baseline to map out territory. The need for such triangulation is obvious
to mathematicians. As with surveyors, society may then be understood as a collection
of inter-triangulated contracts.
Again, the real estate industry has managed to provide a legal framework for
"time sharing" despite the rigidity of the legal theoreticians. There are many
formula for sharing facilities. And many people have more than one passport
or declared residence -- even though in many cases this is "illegal". From 1906
the New Hebrides was operated for some 50 years as a condominium under the UK
and France (suggesting equivalents for Northern
Indeed the founding father of Zionism, Theodore Herzl stated: "We'll simply
extra-territorialize Jerusalem, which will then belong to nobody and yet to
everybody, the holy place common to the adherents of all faiths. The great condominium
of culture and morality." A search on the web under 'shared sovereignty'
indicates further options.
Andorra provides another example, as does St Martin in the Caribbean. Monaco,
Liechtenstein, Jersey, Isle of Man, and the Cayman Islands, offer others. How
many degrees and variants of 'autonomy' and 'independent protectorate'
are on offer -- without including 'wholly-owned' client-state formulae?
What about 'international zones' such as Tangiers (1925-1956), the
'international compound' of Shanghai, or the many 'export
processing zones'? What identity and ownership issues are involved
in UNESCO's 582 'World
Heritage Sites', in IUCN's 50,000 'protected
areas', or in 'safe havens' for refugees? It is curious that
the diplomatic community has been able to enshrine in international treaties
the notion of an embassy as being effectively a part of the sovereign territory
of the country represented there. Diplomatic 'missions' enjoy similar
creative protection, as do ships on the high seas. What about the legal status
of governments 'in exile'? Who 'owns' Antarctica?
Finally, what about property on the Moon (see http://www.asi.org/adb/03/05/lunar-property.html)
and its current sale (http://www.moonshop.com/)?
What about sale and ownership of real estate on a range of other extraterrestrial
each with its own webserver?
Is there a typology of all such examples which would suggest opportunities
for as yet unexplored possibilities? If not, why not? These examples have been
treated as curiosities by legal theorists who have significantly failed to explore
any new formulae that might be placed on the table during such crucial negotiations.
Why is this?
Lawyers and accountants -- handling territory in practice
It is these people who have been most "creative" in assisting multinational
corporations to design innovative ways to function in a global community --
despite antiquated regulations and mindsets. It is they who render practical
and operable any agreements in principle.
The most successful have been able to design complex networks of holding companies
to manage extremely complex relationships amongst interwoven corporations of
different cultures. They invented "offshore" operations to circumvent territorial
restrictions. Conventional boundaries do not constrain their clients. But their
skills and insights do not seem to have been devoted in any systematic way to
the challenges of territorial disputes. Why is this?
International relations scholars
It is this group, with peace researchers, which is purportedly most closely
associated with innovations in policy-making and political thinking at the international
level. It is they who are most concerned with power politics and the manner
in it is articulated and configured within global society. They are closely
associated with the non-economic dimensions of global modeling and simulation
of alternative policy opportunities. Many of these scholars have mathematical
skills or access to those who do. Many have legal backgrounds. It is therefore
this group which would be expected to develop alternative formulae for consideration
at the negotiation table.
It is ironic that it is this group that is so attentive to the 'points'
made from opposing 'positions', often in a 'line' of argument,
possibly from a particular 'angle' or 'orientation', and
regarding a subject 'area'. As with the legal theorists, this does
indeed imply a crude sense of the fundamentals of geometry. Unfortunately there
is little ability to develop such geometrical insights to enable the construction
of new 'spaces', which would require configuring the different lines
and areas according to a three-dimensional design. As with the legal theorists,
this group focuses almost exclusively on proving that one 'point'
or 'line' is correct. There is an interesting parallel between the
valued ability to make stronger 'points' in an argument and the ability
to fire better bullets in a territorial conflict -- in both cases the quality
of the 'ammunition' is valued. Such point making tends to trap dialogue
in linearity -- although again both would aspire to being able to cow or extinguish
any opposition by using more powerful explosives. Consequently, with such primitive
elements, nothing ever gets built that provides a sustainable environment for
groups aligned to that space in different ways (see also: https://www.laetusinpraesens.org/docs/conftran/xvgeom.php).
Many of them would however claim that their main function was to provide post-facto
explanations for territorial conflicts rather than furnish thinking to assist
in preventing or remedying such situations. Nevertheless they are happy to be
consulted on such occasions.
It is from this group, and their network of think
tanks, that statesmen and negotiators would expect clear articulation of
the most comprehensive range of frameworks which opposing parties might explore
at the negotiating table. How many such options are normally on the table during
such negotiations? Who excluded the other options and according to what criteria?
Have such scholars really contributed so little to new options for territorial
and boundary disputes beyond those articulated over past centuries? Why is this?
It is the expertise of this group that is involved in assisting the negotiation
process. Such mediators may be specialists in their own right or represent skills
developed by policy specialists of governments assisting in the mediation process.
It is they who are responsible for directing the attention of the opposing parties
to the opportunities for more fruitful interaction and to the frameworks that
might be acceptable to the different parties. Given their responsibility, and
in the light of recent scandals of croneyism at the highest levels, through
what processes are they selected and with what transparency are their skills
assessed, even after the event? What frameworks do they choose to present and
through what processes? Do they, like commercial representatives, have a catalogue
of frameworks for discussion -- so that the advantages and disadvantages of
each may be discussed prior to narrowing the focus onto what is meaningful to
the particular situation? What do they exclude from that catalogue and why?
What is their responsibility for ensuring premature closure around simplistic
options -- "Getting to Yes"?
Given their people-oriented skills, do they themselves only have limited capacity
to comprehend options of sufficient complexity to contain the complexities of
the situation (cf Ashby's Law in cybernetics)? Do they really understand the
meaninglessness of signed 'agreements' within some cultures -- especially
for people whose relatives have had their throats cut by the opposing group
over several generations, and have responded in kind? Are they so committed
to "resolving the conflict" and to "agreement" -- at all costs -- that they
ignore options needed to hold centuries-old intractable differences within new
kinds of framework? Why might this be? Have the psychic needs of such negotiators
been assesed in the light of this obsession?
Why the need to 'reach an agreement' when viable situations might
be achieved by new insights into 'reaching a disagreement'? Such a
viable 'disagreement' is perhaps best illustrated by the mathematical
principles underlying a geodesic dome -- surely an example of a 'standing
disagreement', reminiscent of standing wave phenomena. But of similar nature
is the possibility of patterns of relationship based on resonance
hybrids -- alternating over time between very different structural formula,
and achieving stability in a manner reminiscent of crop
Increasingly territorial conflicts, and any negotiation, are described and
maintained within a media context. Blow-by-blow accounts are provided by CNN-type
news services. Initiatives of any kind have to be credible to the media, or
else conducted in secret (as with the Oslo negotiations). The media has a basic
need to describe incidents and options in comprehensible terms. One approach
is to focus on whatever is simple. This undermines options which are necessarily
more complex. However the media are capable of responding creatively to very
complex initiatives, as shown by coverage of a space exploration, provided that
resources are devoted to rendering such initiatives comprehensible (as NASA
has discovered and respected).
The question is whether the media have been sufficiently demanding in relation
to territorial conflicts. Given the easy footage of conflict, have they ignored
their responsibility in challenging the opponents for the oversimplicity of
their thinking in the face of complex issues? How much of the coverage of Kosov
reports on possibile formulae through which the situation might be stabililized
sustainably? Have they ignored their responsibility in endeavouring to identify
more complex options and render them comprehensible -- or challenging others
to do so? Why is this?
Metaphor specialists and spin-doctors
The media are increasingly dependent upon sound-bites and photo opportunities.
It is the job of the spin doctor to compress complexity into such small windows
of comprehensibility -- to signal new thinking and opportunities under exploration.
A prime tool in this respect is metaphor, much used by the advertising industry,
politicians and legislators. The question is what metaphors have been designed
to render comprehensible more complex options in relation to territorial conflict?
There is no catalogue of metaphors exemplifying such options -- and adapted
to communication across cultures, as required by many of these conflicts. Why
This is the term used by the military and intelligence agencies for their "psychological
operations" to ensure strategic advantage. And it is the military intelligence
people who specialize in destroying threatening patterns of relationship and
understanding rather than creating ones that reflect new and more appropriate
insight. In a wiser world, seeking to respond creatively to territorial conflicts,
these specialists would include psychotherapists and political psychologists
of various persuasions. They would provide insights into identity across cultures
and the windows of opportunity for reframing the "surface of attachment" currently
articulated as being "land". Regrettably, those with most insights into such
opportunities tend to be focused on the therapy of isolated individuals and
are uninterested in the group challenges of wider society. Worse, when they
are concerned, they fail to recognize the additional complexities of those contexts.
Significantly, they have never effectively addressed their own intense territorial
conflicts as opposing schools of practice. Why is this?
Can emerging global society be based on traditional patterns of identity dependency
on space-place and time-space, or do substitutes have to be found that are sufficient
to serve the same fundamental need in the human psyche? The question for the
(immediate) future, in a global situation wherein a rapid increase of the number
of bodies in the space available is still inevitable, is what the criteria are
to be for anyone occupying the available space in this world. Control of entry
into and exit from this existence will become critical, regardless of who comes
to occupy territorial space in between these events. Many disciplines and schools
of thought, whether arts or sciences, have led the way in defining, laying claim
to,and quarreling over, 'territory' in 'knowledge space'.
How is the occupation of this space, and 'intellectual property',
to be handled in a global information society? If, as the 'property'
notion implies, these challenges are to be dealt with in the same manner as
'territorial property', then society is faced with a sorry repetition
of the centuries of bloody quarrels over physical territory.
Why is new thinking on the relation of a person, or a group, to 'territory'
so limited and simplistic at a time when such thinking is so desperately needed?
After all, it is possible that cultures and belief systems with other understandings
of their relationship to 'territory' may have insights from which
many might benefit -- without detracting from truths which appear immutable
and eternal. Is there something to be learnt from the manner in which people
of different cultures attach erotic significance to quite different parts of
the body -- whilst being relatively indifferent to other parts? Without questioning
the value of such pschological identification (even fixation) with erogenous
'zones', does it suggest contrasting ways in which a person or a culture
might attach deep, behaviour determining significance to territory?
Mathematics has spaces beyond ready comprehension -- commensurate with the
insights of those who have explored other modes of consciousness and relationships
to reality. Who is afraid of exploring them?
Architects and planners
Perhaps the groups with most concrete experience of designing spaces for people
are architects and planners. In doing so they are obliged to reconcile and balance
many conflicting needs. They also have to provide for both common space and
privacy, notably in the light of desires for conflicting use of space (eg parties,
reflection, ceremonial, convalescence, etc). However, as noted by Stewart Brand
(How Buildings Learn, 1995), architects have tended to distinguish themselves
by lack of attention to how real people need to use spaces, especially as their
needs evolve within a building effectively designed to be incapable of evolving
This is in contrast to the remarkably insights of spatial planners such as
Christopher Alexander (Timeless Way of Building, 1979), and especially
his Pattern Language (1977) -- effectively marginalized by the architectural
profession. Through their focus on fashionable high rises and serial construction,
it is this group which has designed the urban environments that are increasingly
detrimental to quality of life and non-standard family structures. The unaddressed
challenge for this group is most evident in the case of multicultural environments
where spaces have to be designed to allow for habitation by people with different
lifestyles (Gregory Penoyre and Sunand Prasad, 1999). It could be argued that
they have designed the spaces to encourage rather than alleviate social problems.
Their current focus on 'gated suburbs' echoes the simplistic recommendations
for 'safe havens' for minorities as in Kosovo.
It is surprising how little has been learnt from the spatial explorations of
R Buckminster Fuller (Synergetics, 1979), perhaps because of his failure
to focus on social relations rather than the organization of material resources
(as continues to be the case with his enthusiastic followers). It is only recently
that the subtle Chinese geomantic discipline of feng shui has become
acceptable to modern architects (under pressure from Chinese real estate developers),
but again there has been no effort to apply such insights to the challenges
of conflicting approaches to territory typical of Kosovo.
In principle it is this group that is most sensitive to the possibility of
using mathematical insights to understand and respond to complexity in new ways.
Their success is evident in the many highly complex systems which are basic
to the infrastructure of modern society -- petrochemical complexes, organizational
information systems, transportation and delivery systems, etc. It would however
be difficult to identify non-technical areas to which this expertise has been
applied, or been considered relevant, either by systems analysts or those concerned.
Systems analysis is basic to weapon delivery and battlefield intelligence systems,
as exemplifed by Yugoslavia. Its insights and expertise are significantly absent
in the associated humanitarian crisis response, in which even the supply of
basic rations, tents and identity procedures have usually been chaotic -- although
systems analysts are sensitive to the potential of chaos theory. Complex modelling
exercises have never been able to effectively design in the determining effects
of corruption at every systemic level -- as increasingly recognized as characteristic
of systems in Eastern Europe and at the highest decision-making levels in most
countries in the light of new scandals. Is there a built-in inadequacy to mainstream
systems thinking, focused on use of large sets of complex equations, that precludes
development and use of insights based on spatial and topological insights (a
point argued elsewhere)
Of necessity this group is concerned with system dynamics -- besides relationships,
the behaviour of complex systems is a function of system structure and the dynamics
of those relationships. They argue that the underlying principles of systems
thinking embedded in feedback theory are equally valid in the analyses of natural,
social, physical, technical and other systems. From this perspective, treatment
of Kosovo as a territorial problem itself constitutes very linear thinking --
there are clearly two or three more dimensions to the Kosovo problem. If the
problem is viewed as a state space domain, this might involve territorial dispute,
power struggle, history and outside environment. As a four-dimensional problem,
this is quite complex to address, so there is no attempt to do so. Unfortunately
the only systems that systems 'analysts' are able to design and build
are based on tangibles. They have been unable to (and uninterested in) responding
to the territorial dynamics of communities and individuals -- or even the feuding
between the different factions of their own discipline.
This is not for lack of trying as indicated by the theme of the gathering of
the Society for General Systems Research (London, 1979) on 'Improving the
Human Condition: Quality and Stability in Social Systems' (London, 1979).
Rather that there is a crucial gap -- the most fundamental 'north-south
gap' -- between the effort to articulate insights within the discipline
about such issues and the relevance of what is so articulated to enabling new
patterns of relationship between those in conflict. Why is this?
Software specialists in visualization of complexity
Most people, and especially children, are aware of the truly amazing innovations
in graphics software, special effects, and the like. Artificial worlds, even
based on different physical laws and higher dimensionality, have been designed
(see the popular Active Worlds site at http://www.activeworlds.com/)
. Much use is made of these in the major industries associated with computer
games, movies and advertising. The question is to what extent are these skills
used in articulating options for new frameworks relating to real territorial
conflicts? Some thoughtful computer games move in this direction. Some significant
use was made of such technology in the Dayton process to visualize and clarify
boundary issues. But no serious effort has been made to use these techniques
to explore windows of opportunity in conflict situations. Why is this?
Theologians and philosophers
In response to an early draft of this text, one response was why not mention
theologians. There is a strong argument for this since it is widely recognized
that the most bitter territorial conflicts have been associated with religion,
and with views that theologians continue to be responsible for reinforcing.
Obvious examples include: Jerusalem, Sudan, Northern Ireland, Yugoslavia, and
Kashmir. Milosevic ensured full TV coverage (on 20th April 1999) of his wholehearted
support from the Russian and Serbian orthodox hierarchy. Little needs to be
said about the conflicts arising from differences between the monotheistic religions
(Christianity, Islam, Judaism). Their disparagement of other religions, notably
those of indigenous cultures has been repeatedly documented. The state of the
Holy Sepulchre in Jerusalem is an extreme metaphor, with six Christian religions
in extremely hostile relationship over use of the floor space of that edifice.
And what conflicts, including Kosovo, have not been sanctioned on each side
by the 'universal' god of a particular religion?
Theologians of most religions are adept at discussing that which cannot be
comprehended by the uninitiated. Most present skilled arguments for the merits
of the intangible over the tangible, for the eternal over the temporal, and
for detachment over attachment. Ironically, in a world of increasing land shortage,
some religions continue to emphasize the need for defined burial space following
the pharaohonic tradition -- and remains are brought back from distant parts
to fill it. And what of the implications for fruitful understanding of territoriality
in the light of : "Foxes have holes and the birds of the air have nests, but
the Son of Man hath no place to rest His head" (Mathew 8:20)? In no way
do such insights seem to translate into creative thinking concerning conflicting
claims over territory. Few theologians deal with the issue of how two sincerely
held, but apparently totally incompatible, perspectives can be equally true
in some transcendent sense. The fact that there are mathematical, geometric
and visual models in support of this does not seem to evoke any alternative
behaviour. Ironically, Gnostics made adopted an unusual procedure that anticiaptes
the thinking that is required. According to the person who declared them heretics,
namely Ireanaeus of Lyon (ca.180 A.D) in his Libros Quinque Adversus Haereses
(1.13.4), they drew lots before every religious service to determine who should
be bishop, priest or deacon -- a remarkable response to possession of territory.
Ironically it is Islam, as a historical torchbearer of mathematics, which through
its architectural motifs, is most sensitive to complex tiling patterns over
three-dimensional surfaces. There is a traditional antipathy of Christianity
to mathematics, dating back to St Augustine: 'The good Christian should
beware of mathematicians, and all those who make empty prophecies. The danger
already exists that the mathematicians have made a covenant with the devil to
darken the spirit and to confine man in the bonds of Hell.' [St. Augustine
(354-430), DeGenesi ad Litteram, Book II, xviii, 37]. This despite,
or because of, contemporary works such as those of Iamblichus on The Theology
of Arithmetic. But, ironically again, the much quoted Biblical injunction
to 'go forth and multiply' or 'Be fruitful and multiply'
(Genesis 1:28) -- on which policies with disastrous and far reaching
implications for the planet are based -- is not subjected to dialogue between
theologians and mathematicians.
At the time this injunction was written, multiplication was not understood
as a mathematical operation. Only addition and subtraction were possible. It
might therefore be better translated as 'increase', as featured in
some Papal writings that also deal with the remainder of that same injunction,
namely to 'fill the earth and subdue it" (see Papal Encyclical Mater
et Magistra, 15 May 1961). From a mathematical perspective, 'increase'
may occur in many ways, some of which might be better associated in a theological
perspective with 'increase in comprehension', some 'expansion'
of the person, or 'growth in wisdom'. It makes all the difference
whether such increase is based on concentric circles radiating 'out'
from the person (to greater understanding of external reality), moving 'within'
the person (as progressively increasing understanding of internal reality),
or rather as some form of serial replication. Similarly 'subdue',
and associated notions of 'domination', would be well understood by
mathematicians as achieving some form of cognitive control or comprehension
of a complex phenomenon. It would be ironic if theologians have bought into
a notion of increase in a material sense when the intent was increase in a non-material
But it would seem that no theologian has explored the manner in which the theological
arguments of the major religions are enriched and exemplified by the complexities
of modern geometry, such as to open up windows through which alternative perspectives
could be simultaneously valid. Although a recent paper by James B Miller (1998),
aims 'to demonstrate how the use of mathematics as metaphor can be of heuristic
value in an effort to understand a theological concept which often seems paradoxical
when described in ordinary or traditional language'. Both theologians and
mathematicians have their respective difficulties with the three-body problem.
But why is it left to mathematicians to appreciate the transcendent implications
of their work? As with P Gordon, on being exposed to Hilbert's work in variant
theory: 'This is not mathematics, it is theology' [Quoted in P. Davis
and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.]
No theologian deals with the issue of how people can usefully identify with
particular surfaces and spaces, and the consequent constraints and new freedoms
for identity shown to be possible in the light of modern mathematics. Why is
Following the failed peace talks between Arafat and Barak (Camp David, July
2000), Bill Clinton claimed that in searching for possibile ways forward over
Jerusalem 'no stone had been left unturned'. Who is to believe that?
The issue is which 'stones' were turned and which ones were not --
and why not?
In further mulling over this matter, there would seem to be a need for some
sort of complexity measure relating to each of the proposals presented at a
negotiating table and to the situation addressed. We need to be aware when a
"Level 3" proposal is being presented to deal with a "Level 5" situation. Or
that opposing parties are only prepared to think of proposals at complexity
"Level 2". Or that the identity concerns of one party is a "Level 6" matter
for them, and the sensitivity of another is only to "Level 4" issues. When the
bombing stops, what new thinking will be on offer? Or will it be yet more of
the same -- setting the stage for further conflicts and massacres? It is the
above groups who hold the conceptual skills of humanity to reframe such conflicts.
Our civilization has invested heavily in their acquisition of that competence.
They have proven themselves significantly incompetent to date in respect of
the challenges faced by humanity. Why is this? Is it not time that those with
such skills should be 'forced' into interaction (if only electronic)
on the real opportunities for more complex solutions worthy of experiment? Is
it also not time that such interactions should be carefully monitored to exclude
those with a conceptual investment in "more of the same" or 'doing nothing'?
What is this nature of this psychic space into which these people force us in
order to watch humanitarian disasters and do nothing -- accepting their sophisticated
justifications for doing so?
Is it this mindset that is responsible for failure to respond to other problems
like unemployment and the continuing destruction of the environment by 'development'?
Who exactly are these people who are ultimately responsible for the Kosovo situation?
How much human sacrifice to their gods do they require before they are prepared
to explore alternatives?
A. Practical leads for those concerned about next steps
Moving beyond the mindsets that keep creative practical thinking off the negotiating
table, what might be some of the insights that could be brought together
in a context that encourages fruitful interaction:
- Common ground: Rethink the much bandied notion of 'common ground',
considered fundamental to 'agreement',in the light of more complex
mathematical perspectives that might allow for non-simplist occupation of
it dependent on the exclusion principle (see: https://www.laetusinpraesens.org/docs/common97.php)
- Unity: In the light of mathematical insights, produce a hyperlinked
web catalog of ways in which 'unity', 'universal', 'global',
'unified', 'holistic', 'integrated', 'diversity',
etc can be understood, in order to avoid the simplistic assumptions implicit
in many 'global' and 'universal' initiatives that ignore
the challenge of articulating diversity and variety in any unified framework.
(For a preliminary attempt, without the benefit of mathematical formalisms,
see Integrative knowledge and transdisciplinarity, plus associated database
In July 1999 the Taiwan-China crisis reached new intensity because of interpretations
of 'one nation, two countries' (the new Taiwan position) versus
'one country, two systems' (the Chinese position on Hong Kong).
The statement was made in English because the distinction between 'nation'
(in the sense of people or race), 'country' (separate from, or against
-- from the Latin contra), and 'state' as a unit of government,
cannot be effectively made in Chinese. The subtleties of these distinctions,
which may well be cause for military intervention. The challenge is to find
ways to articulate what is separate but part of a common framework -- and
what weight to attach to each notion. Surely mathematicians could present
an array of possibilities on the underlying principles given their importance
in some many territorial situations -- and the need to avoid simplistic interpretations
of unity that do not meet understanding of necessary distinctions in practice.
- Global: Distinguish the cognitive and integrative dimension of 'global'
from the predominant geographical usage that is currently obscuring possibilities
vital to the integration of variety within 'global society' (https://www.laetusinpraesens.org/docs/converse.php)
- Shared sovereignty and property: Produce a web catalog identifying
the range of formulae for 'shared sovereignty', 'co-ownership',
condominiums, etc., without prejudging their value in unforeseen situations.
Important learnings could be derived from the work of the International Association
for the Study of Common Property (IASCP), founded in 1989 and devoted to understanding
and improving institutions for the management of environmental resources that
are (or could be) held or used collectively by communities in developing or
developed countries (see http://www.indiana.edu/~iascp/).
Related to this is the evolution of 'land trusts' to conserve natural
- Comprehension: Explore the significance of Ron Atkin's work (1981)
on q-analysis as a way of articulating different levels of understanding of
complexity in complex societies. See overview in: (Comprehension: social organization
determined by incommunicability of insights )
- Epistemological frameworks: Explore the insights of Magoroh Maruyama
into poly-ocular vision in the light of alternative epistemological frameworks,
especially those deriving from different cultural perspectives. (See overview
in: Systems of categories distinguishing cultural biases https://www.laetusinpraesens.org/docs/cultcat.php
- Changing classificatory frameworks: Review the work of the Jesuit
mathematician Patrick Heelan, notably with respect to the logic of changing
classificatory frameworks http://www.math.hmc.edu/~su/papers.html
- Fair division algorithms: The theory of fair division is concerned
with finding ways to divide an object among n people fairly (http://www.math.hmc.edu/~su/fairdivision/).
According to Francis Edward Su (http://www.math.hmc.edu/~su/papers.html),
there are many possible notions of fairness. The notion that he has explored
is finding envy-free divisions, i.e., one in which each person feels she received
the best piece, in her own estimation. Existence of such divisions has been
known for some time. Of greater interest are algorithms for achieving such
divisions. For instance, in dividing a desirable object, such as cakes, a
2-person algorithm has been known since antiquity, a 3-person algorithm since
at least 1960, but an exact algorithm for achieving n-person envy-free cake
divisions was not produced until Brams and Taylor's solution in 1995 (features
of which have been used in a software package developed by Ron Surratt (http://www.mcn.org/c/rsurratt/conflict.html).
Su and Elisha Peterson have recently extended these techniques to provide
an exact envy-free chore-division algorithm for n people. However, such methods
tend to be quite complicated as n grows large. Su has been involved in development
of web-based Fair Division Calculator (http://www.math.hmc.edu/~su/fairdivision/calc/).
Matthias Raith has also developed ARTUS, a Web-based Adaptable Round Table
with a User-specific Surface (http://artus.wiwi.uni-bielefeld.de).
Su and Raith are in process of integrating their two approaches to create
a fair division platform for any number of parties that is accessible from
any remote location 24 hours a day.
- Tiling and quasicrystals: Review the new relationship between tiling
patterns (Roger Penrose) and the emerging studies of quasicrystal ordering
(Petra Gummelt, Paul Steinhardt, Hyeong-Chai Jeong). It has now been demonstrated
that in certain crystals decagons could be the fundamental unit of quasiperiodic
arrangement -- if they are allowed to overlap each other. Understanding the
radical possibilities of 'overlapping' could prove fundamental to
new thinking about occupation of territory. The classical mathematical constraints
on tiling can be replaced with a more physical requirement whereby quasicrystals
form low-energy clusters -- the greater the overlap, the lower the energy
-- leading to a large universe of quasiperiodic structures. Highly charged
territorial challenges call for equivalent 'low-energy' opportunities
of organization. (see David Voss. Smash the system. New Scientist,
27 Feburary 1999, pp. 44-46)
- Topological options: In 1999, the new prime minister of Israel has
expressed renewed support for a plan to link Palestinian territory in the
West Bank with Gaza using a 30 kilometer overpass across Israel. Although
simplistic, this illustrates new possibilities in relating to territory through
what is essentially a very simple property in topology. The question is whether
topology does not offer many other such possibilities suggestive of unexplored
ways of relating to territory.
- Voronoi diagrams: Given a pattern of objects in continuous space,
Voronoi diagrams provide a means of naturally partitioning the space into
subregions. This is a rapidly expanding topic as these diagrams find application
in such areas as spatial data manipulation, modelling spatial structures and
spatial processes, pattern analysis and locational optimization. These areas
can be found within many different scientific fields, and consequently this
increases not only the use of Voronoi diagrams but also the demand for knowledge
about them. (see Atsuyuki Okabe, et al. Spatial
Tessellations: Concepts and Applications of Voronoi Diagrams, 2000
- Flexagons: These origami-like flexible structures were discovered
in 1939 by Arthur H. Stone, and developed by him with colleagues. They have
been publicized by Martin Gardner in his "Mathematical Games" columns in Scientific
American. Kathryn Huxtable notes that there are many kinds of flexagons, with
varying numbers of faces and edges, although she finds the symmetric hexaflexagons
the most interesting. They are characterized by three-fold symmetry at the
"center" with two-fold symmetry in the "outer" layers. They can be made with
3 times any power of two sides, that is, 3, 6, 12, 24, etc., though the higher
order flexagons are difficult to make, due to the mechanical limitations of
paper. She has a Java program on her site to illustrate their flexibility.
It would be intriguing to explore ways in which the different faces could
be used to represent territories that could be juxtaposed in various configurations.
In this sense they hoild insights into different kinds of 'unity'
and 'agreement' amongst the surfaces -- based on dynamics rater
- Temporal options: The extremely acrimonious, and much publicized,
debate ovdr many months in 1998-9 concerning selection of the new director-general
of the World Trade Organization has finally converged on a solution involving
a shared leadership with one nominee to occupy the post for the first half
of the term and the other for the second. This can hardly be described as
an earth-shaking breakthrough in mathematical terms but it does suggest how
some very simple alternative formula could usefully be available as a resource
under such circumstances.
- Structural auto-generation: Review the visualization possibilities
of the auto-generating structures produced using Struck software (developed
by Gerald de Jong), including
the possibilities for transformation between such structures developed by
the Struck user community (http://www.critpath.org/idioverse/struck/web.html)
and related explorations in the computerization of of Buckminster Fuller's
- Polyhedral representation of complexes: Develop techniques of representing
complexes of issues, strategies, organizations and/or values using the full
spectrum of semi-regular polyhedra (as an extension of current virtual reality
web experiments with their profile data at the Union of International Associations,
in the light of
Configuring strategic dilemmas in intersectoral dialogue)
- Poetry and policy-making: Identify mathematical poets capable of
embodying complex insights in resonant metaphors (https://www.laetusinpraesens.org/docs/poetry/poetpolg.php),
following the initiative of MIT in setting up a centre under Gyorgy Kepes
to cross-fertilize insights between scentists and visual artists
- Metaphor and policy: Identify those working at the interface between
metaphor and policy-making (cf George Lakoff, David Cooperrider, Donald Schon)
- Configuring domains of dialogue: Review possibilities for configuring
complex dialogues in three dimensions. See overview in: Spherical configuration
of interlocking roundtables: electronic enhancement of global self-organization
through dialogue patterns (https://www.laetusinpraesens.org/docs/interlok.php)
- Configuring electronic dialogue: Explore electronic protocols to
configure and pattern electronic dialogue so as to facilitate the emergence
of higher orders of consensus (and avoid the disappointments of current
notably in the light of the syntegration experiments of cybernetician Stafford
Beer (see http://www.oise.on.ca/~gfilewod/ts.html)
- Methodological clarification: There is widespread interest, outside
the academic mainstream, in issues touching on the interface between alternative
modes of understanding, religion, geometric structure, number, visualization,
and the like. What is lacking is a means of mapping such initiatives in relation
to one another and to the mainstream, and 'sorting the wheat from the
chaff', in order to highlight fruitful avenues of investigation in relation
to the territorial challenges. Regretably few of these issues seem to focus
on concrete implications and, ironically, most have strong views on other
initiatives in this territory. (For one useful point of departure into this
rich, but confusing array, checkout links from http://www.new-universe.com/pythagoras/)
- Visualization approaches: Just as the bioinformatics community catalogues
visualisation sites, applications, techniques and papers that may be of interest
to the biological community (see http://industry.ebi.ac.uk/~alan/VisSupp/VisAware/VisAware.html)
there is a strong case for developing an analogous web catalogue for sites
of relevance to territorial conflicts.
- Reframing agreements and declarations: In the light of hypertext,
visualization, geometric interfaces (including virtual reality) and the need
to configure text to reflect complexity that exceeds the capacity of legal
points on a two-dimensional surface, how might the future of legal agreements
be framed in new ways? How might the text of a (dis)agreement be embodied
in a visualized 'information structure' in which it is the structure-plus-text
which is the (dis)agreement (as with a technical patent) rather than the text
alone? And how might this reflect the emerging challenges of reflexivity?
(see for example: https://www.laetusinpraesens.org/docs/conftr0.php#patt
- Sense of space: Relating to the concerns of architects and planners,
much could be gained by exploring the increasing sensitivity to 'sense
of place' (many web documents) and 'spirit of place', exemplified
by Yi-Fu Tuan (Space and Place, 1987) and Edward Casey (Getting
Back into Place, 1993). It is only though developing respect and understanding
of such ideas in a multicultural environment that over-simplistic and unsustainable
approaches to territorial conflicts can be avoided. Such explorations can
usefully be extended into non-physical spaces, exemplified by the emerging
challenge of the organization of territory in cyberspace (see for example:
but especially in personal relationships. There is an illustrative anecdote
concering such relationships as established between orthodox Jews (the world
wide Chabad-Lubavitch Movement, http://www.chabad.org/)
and Zen Buddhists that had constructed neighbouring temples in the Active
Worlds virtual environment. The use and occupation of cyuberspace by religions
has been discussed by Jeff Zaleski (The Soul of Cyberspace, 1997).
- Configuration of mediators: To avoid the trap of cosy dependence
on (self-)selected mediators in a negotiation, there is a case for finding
ways of using simultaneously a set of quite distinct teams of mediators characterized
by very different styles and (incompatible) mindsets. Rather than design out
perspectives from the process, the challenge is to design a diversity of perspectives
into the process. There will always be a case for the conservative, establishment
(more-of-the-same) perspective which arrogantly dominates diplomatic thinking.
However the procedure of interrogators using a 'nice-guy, bad-guy'
technique could be extended in the light of the approach developed by Edward
de Bono in Six Thinking Hats (1987) and Six Action Shoes (1991)
for use in corporations. These books deal with what he has called "operacy".
This is the skill of action, of getting things done and making things happen
-- which, as with literacy and numeracy, is not well developed in everyone.
As de Bono points out: "The six hat method has been widely accepted because
it is simple, it is practical, and it works. It actually changes how thinking
takes place in meetings and elsewhere: instead of the usual to and fro arguments
it makes it possible for people to have constructive discussions.". The six
pairs of action "shoes develop the action dimension of the thinking
associated with the six 'hats'. De Bono's 'hats' involve
participants in a discussion in a type of mental role playing:
These categories may themselves be too compatible to respond effectively to
the real challenges of negotiation. As with many vital skills, they are also
'proprietary' metaphors whose use is controlled by de Bono (see
Judge, 1992). The more essential question is whether apparently zany perspectives
may not have something to contribute, especially to negotiations amongst cultural
groups that have not bought into the western mindset that many mediators would
like to believe is 'universal' (as clarified by Maruyama). The question
to be asked regarding any negotiation is which approaches to negotiation have
been prematurely designed out, by whom, and why? Any responsibility for failure
of the negtoiation can then be more appropriately discussed.
- White hat: An objective look at data and information.
- Red hat: Legitimizes feelings, hunches, and intuition.
- Black hat: Logical negative, judgement, and caution.
- Yellow hat: Logical positive, feasibility, and benefits.
- Green hat: New ideas and creative thinking.
- Blue hat: Control of the thinking process. (1987)
- TO BE CONTINUED.....
B. Territorial conflicts as a mirror of conceptual disarray
There is widespread recognition of a crisis of governance. This is typified
by the policy disarray in the aftermath of the Asian financial crisis and the
challenges of managing the global financial system. The emerging crises, such
as unemùployment, social security and environment, challenge old patterns
of thinking. Territorial conflicts might usefully be considered as the end result
of conceptual failure of a very high degree.
The bloody territorial conflicts like Kosovo seem conveniently distant to many.
But conceptually the pattern which underlies them is very close to home. Most
competitive business is faced with conflicts of territory. What academic common
room lacks dynamics isomorphic with the Balkans? Consider the traditional feuding
between natural and social scientists, for example. Is it not common for one
discipline or school of thought to exhibit a form of righteous 'nationalism',
accompanied by a disparagement of some other conceptual persuasion? Are the
values of some disciplines not subject to harassment and violence at the hands
of another -- even including wholesale 'massacre' and its cover-up?
Are there not 'refugees' from such conflicts? Like peasants everywhere,
are not most academics desperate for 'tenure' and vigilant about their
'intellectual cproperty'? Would any discipline have difficulty in
detecting phenomena in academia analogous to 'ethnic cleansing' --
ensuring the removal of interlopers and representatives of negatively labelled
perspectives? Is there not a sense in which mainstream academia can be perceived
as engaged in 'bombing raids' on alternative perspectives? Are there
not schismatic challenges of 'separatism' that have to be resisted
to preserve the unity of a discipline, because of some equivalent to 'sovereignty'?
Can 'interdisciplinarity' be considered as anything more meaningful
or effective than the United Nations?
In this light there is merit to exploring the parties accused in this paper
from an even more immediately challenging perspective. Each of them might usefully
be considered as ways of knowing about relationships between territories
and the identities attached to them. They are disciplines about relationships.
For each of the groups (above, and reviewed below), therefore, it is useful
to distinguish (and recognize parallels) between insights they might apply to:
- the conflicts within the group, and the mindset of the discipline
that reinforces their exemplification in territorial conflict on the ground;
- the challenges of configuring more fruitfully the relation between these
different ways of knowing about relationships;
- the challenges of applying these insights to territorial conflict
on the ground.
Once again, these different ways of understanding relationships include:
- Mathematics: This is the formal study of relationships. Mathematicians
pride themselves on the capacity to abstract from visible phenomena, but have
proven themselves incompetent in applying those insights in ways capable of
offering windows of opportunity in territorial conflicts. But this is notably
true in the relationships between different schools of mathematics and in
the relationship of mathematics to other disciplines. Has mathematics been
used to address its own schismatic tendencies or even to map out the multitude
of domains of mathematics -- of which most specialized mathematicians are
happy to profess both ignorance and disinterest.
- Theology: This is the study of the relationship between the human
being and whatever is understood to be universal and transcendent. It is especially
attentive to the challenges of understanding that which does not lend itself
to ready comprehension. Theologians have however proven to be almost totally
incompetent in addressing concerns raised by the set of religions -- other
than in a purely comparative mode. This is reflected in their treatment of
each other when they are of different persuasion. Whilst addressing 'unity'
in its highest manifestation, they are completely unable to reconcile this
with the diversity of understandings of that unity -- other than by demonizing
those who fail to agree with a favoured perspective.
- Media: The media are in the business of providing communication between
perspectives through information, commentaries, documentaries, annecdotes,
news, and the like. Its development has reached the point that governance
is now conducted through the media. As many have remarked, media thrives on
bad news and is unchallenged by it. In this process it amplifies selected
significance at the expense of insights which merit articulation. The savage
competition within the media, fighting for territory and coverage, reflects
mindsets common in any bloody territorial conflict that it covers. The media
has not found a way of systematically catalyzing the imaginative development
of society -- although government controlled media have often tried a dangerously
distorted, heavy-handed approach to this. Is it any wonder that the young
generation is more nourished by Neighbours and Star Trek than
by real territorial conflicts? Such conflicts are not used to facilitate new
- International relations scholars: This group, charged with understanding
the international system, has failed to articulate a way forward. They have
either become locked into a sycophantic relationship to government, in the
hopes that their services will be required, or locked into a pattern of systematic
criticism of government that is not accompanied by any viable alternatives.
Again, amongst themselves there is much infighting. It is not surprising that
studies of international relations have long failed to focus on levels of
croneyism, nepotism and other dubious practices in intergovernmental relations.
These practices have long been part of the pattern though which foreign policy
advice was selected. Locked into a mainstream thought pattern, this group
has been systematically surprised by events, notably the emergence of nongovernmental
organizations. It has been reluctant to address the question of the necessary
diversity of insights relevant to any adequate response to the complexities
of the global system.
- Conflict mediators: Conflict mediators each tend to have well-formed
views on the merits of their own particular (if not patented) process and
the inadequacies of those favoured by colleagues. In a policy-crisis, like
any discipline, each mediator has a vested interest in stressing the absolute
value of their particular approach. How then is a team of mediators to be
chosen for a negotiation process? How would mediators address the challenge
of their own territorial conflicts? How might mediators mediate their own
conflicts? By failing to develop means of reconciling their own processes,
they fail to respond both to the challenges of developing a fruitful relationship
between the different groups identified here, or to the needs of those faced
with territorial conflict on the ground.
- Visualization software specialists: Many have remarked on the limitations
of linear text in articulating complexity for more adequate comprehension.
There is now a heavy investment in visualization of information through patented
software packages. These are primarily designed for corporations concerned
at improving their comparative advantage in the emerging knowledge society.
Those with such skills are therefore in intense territorial conflict for rights
to intellectual property. Not only do such people have little interest in
clarifying options for more fruitful relationship between the groups identified
here, they have no interest in applying their skills to clarifying options
in territorial conflicts on the ground. In fact, a major market for their
skills lies in increasing the realism with which territorial conflict can
be experienced vicariously through video games.
These are some of the ways of knowing about relationships in which our global
civilization has invested. As such they are failing society abysmally -- if
it is their processes which are failing to offer new ways of framing options
for alterantives to the bloody regional territorial conflicts and the social
disasters perpretrated in the name of globalization.
Rather than continue to identify how these patterns play out within other groups
and disciplines that may be vital to a more creative approach to territorial
and relationship conflicts (even of an interpersonal nature), it is perhaps
more useful to consider how these insights might be explored further.
It would appear that we are dealing with a paradoxical pattern of self-reflexiveness
of the kind long relished by Douglas Hofstadter and his colleagues. It is unfortunate
that their insights too are not translated into more fruitful responses to territorial
conflict. However, using this self-reflexive approach, what could usefully be
explored is a visual configuration of the various 'spheres of influence',
identified above (and subsequently) concerned with knowing relationships and
identity. This configuration must necessarily seek to articulate the relationship
tensions both within each sphere and between each sphere. The
challenge is to render this visual configuration comprehensible, possibly using
web-based virtual reality techniques (supplemented by text, etc), as well as
suggestive of windows of opportunity. The suggestion assumes a need to shift
from a surface-oriented approach to a topological approach, if the degrees of
freedom required for the resolution of territorial conflicts are to be opened
This display must be susceptible to packing (folding) and unpacking in response
to different needs for simplicity and articulation. Those familiar with current
web-based experiments in displaying polyhedral structures in virtual reality
are aware of the numerous opportunities. Many of these have developed from early
work by Keith Critchlow (Order in Space) or Buckminster Fuller (Synergetics)
which again have not yet been effectively used by their followers to address
the kinds of territorial conflict challenges identified here -- despite their
rich insights into design and spatial relationships. Critchlow in particular
has explored close packing relationships between spheres as a way of holding
information on the range of distinct regular and semi-regular polyhedra. Each
of these is a pattern of spaces reflecting different ways of articulating a
The question is whether such fundamental solids can be used to articulate and
underpin the cognitive and symbolic understandings of unity that are associated
with religious concerns about territory -- concerns that are the domain of yet
other 'spheres of influence' that should be held in that visual configuration.
This goes to the heart of how people relate to, and identify with, territory
and the 'spirit of place' -- yet another 'sphere of influence'.
Critchlow himself has explored the use of such patterns in Islamic religious
motifs (Islamic Patterns). Related concerns, from a Jewish perspective,
have been explored by Stan Tenen. Such patterning approaches can be used as
a means to respond to the challenges of articulating the relationships between
sets of strategic dilemmas of the kind highlighted during the Rio Earth Summit
(see Configuring dilemmas in intersectoral dialogue)
The argument here is that it is frameworks of this kind that might hold the
necessary richer patterning through which windows of opportunity for higher
orders of consensus could emerge in response to territorial conflicts. Kosovo
is proving that it is both expensive and disastrous to invest in simplistic
thinking. Civilization has invested in disciplines that are not being applied
to the issues which will contiue to torture our planet.
Some are recognizing a complementarity between the school massacre in Littleton
(Co, USA) and the violence in Yugoslavia. In commenting on the perpetrators
of the former, an editorial in the International Herald Tribune (23 April
1999) argues 'The cultural fragments out of which they invented themselves,
and their deaths, are now ubiquitous in every community, urban, suburban, rural.
What matters is not the fragments but how they were combined.' Is it possible
that it is only such massacres that will eventually focus attention on how the
insights (disciplines and groups) noted above are 'combined', rather
than on their various jealously-guarded, fragmented realities. Is any more attention
given to how such fragmented perspectives are to be constructively combined
than in the manner in which they are destructively combined by perpetrators
of violence? As Gregory Bateson argued long ago, it is a question of the pattern
that connects, but especially the nature and quality of that pattern. As any
cook knows, the ingredients are one thing but combining them is another.
It is time to explore, and make meaningful, the strategic complementarity of
such ways of knowing about relationships -- as well as the manner in which failure
to rise to this level of complexity reinforces the inability to offer meaningful
options to those who need them. It is they who then become the victims of territorial
conflict. People are being massacred because of intellectual irresponsibility.
Seeking to identity others to blame is consistent with the forms of denial associated
with this irresponsibility.
- The Timeless Way of Building. Oxford University Press,
- A Pattern Language: towns, building, construction. Oxford
University Press, 1977
Robert Ardrey and Philip Turner (Editor). The Territorial Imperative : A
Personal Inquiry into the Animal Origins of Property and Nations.Kodansha International,
- Combinatorial Connectivities in Social Systems: an application
of simplicial complex structures to the study of large organizations. Birkhäuser, 1977
- Multidimensional Man: Can man live in three dimensional space?
L Penguin, 1981 [summary]
Stafford Beer. Beyond Dispute: the invention of team syntegrity. Wiley, 1994 [commentary]
Steven J. Brams and Alan D. Taylor:
- Fair Division: From cake-cutting to dispute
resolution. Cambridge University Press, 1996
- The Win-Win Solution: Guaranteeing Fair
Shares to Everybody. W.W. Norton, 1999 (analyzes several international territorial
Stewart Brand. How Buildings Learn : what happens after they're built. Penguin,
Edward S. Casey. Getting Back into Place: toward a renewed understanding
of the place-world. Indiana University Press, 1993
Zong Chuanming. Sphere Packings. Springer Verlag, 1999
Paul Chilton and George Lakoff. Foreign policy by metaphor. CRL Newsletter,
June 1989, 3, 5 [text]
J C Conway and N J Sloane. Sphere Packings, Lattices and Groups. Springer
Keith Critchlow. Order in Space: a design source book. Thames and
Ubiratan D'Ambrosio. Mathematics and peace: Our resposibilities. Zentralblatt
für Didaktik der Mathematik/ZDM, Jahrgang 30, Juni 1998, Heft 3, pp. 67-73.
Edward de Bono:
- Six Thinking Hats. Penguin, 1987
- Six Action Shoes. Fontana, 1991.
Bruce Edmonds. Hypertext Bibliography of Measures of Complexity by, 1997
R Buckminster Fuller. Synergetics: explorations in the geometry of thinking.
George W. Hart. Encyclopedia of Polyhedra. (including 1,000 virtual reality
Patrick A. Heelan:
- The Logic of Changing Classificatory Frameworks. In: J
A Wojciechowski (Ed) Conceptual Basis for the Classification of Knowledge.
K G Saur Verlag, 1974, pp. 260-274
- Space-Perception and the Philosophy of Science. University
of California Press, 1983 [text]
Will Hively. Dividing the Spoils. Discover Magazine, March 1995 [text]
- Fluid Concepts and Creative Analogies: Computer models
of the fundamental mechanisms of thought. Basic Books, 1996
- Gödel Escher Bach: an Eternal Golden Braid. Basic Books,
Nick Huggett (Ed). Space from Zeno to Einstein: classic readings with a
contemporary commentary. MIT Press, 1999
Margaret James, Phillip Kent and Richard Noss. Making sense of mathematical
meaning-making: the poetic function of language (Paper was presented at "Psychology
of Mathematics Education-21", Lahti, Finland, July 1997) [text]
- "Be Fruitful and Multiply": the most tragic translation error?
- Challenges in applying mathematical insights to comprehension
of world problems and communication amongst international organizations about
strategic responses 1999 [text]
- Human Values as Strange Attractors: Coevolution of classes
of governance principles (Paper prepared for the 13th World Conference of
the World Futures Studies Federation,Finland, August 1993). [text]
- Learnings for the Future of Inter-Faith Dialogue:Insights
evoked by intractable international differences, 1993 [text]
- In Quest of Uncommon Ground: Beyond impoverished metaphor
and the impotence of words of power, 1997 [text]
- Sustaining the coherence of dialogue through apartness: patterns
of systematic configuration of entities through hypertext 1997 [text]
- Poetry-making and Policy-making: arranging a marriage between
Beauty and the Beast, 1993 [text]
- A Middle Eastern Identity: Experimental articulation through
a dynamic system of metaphors [text]
- Globalization of Knowledge and Insight: Envisaging a paradigm-shifting
software package, 1997 [text]
- Anthony. Judge. From Information Highways to Songlines of the Noosphere:
Global configuration of hypertext pathways as a prerequisite for meaningful
collective transformation, 1996 [text]
- Representation, comprehension and communication of sets:
the role of number, 1979 [text]
- Geometry of Organizations, Policies and Programmes, 1992
- Reframing Relationships as a Mathematical Challenge: Jerusalem
as a Parody of Current Interfaith Dialogue, 1997 [text]
- The Art of Non-Decision-Making and the manipulation of categories, 1997
- Future Coping Strategies: Beyond the constraints of proprietary
metaphors, 1992 [text]
- Guiding metaphors and configuring choices, 1991 [text]
Peter Kruse and Michael Stadler. Ambiguity in Mind and Nature : Multistable
cognitive phenomena. Springer Verlag, 1995 (Springer Series in Synergetics,
George Lakoff and R E Núñez. The Metaphorical Structure of Mathematics. In:
L. English (Ed.), Mathematical Reasoning: Analogies, Metaphors and Images.
Magoroh Maruyama (Ed) Context and Complexity : Cultivating Contextual Understanding.
Springer Verlag, 1992
Magoroh Maruyama, et al. Mindscapes: The Epistemology of Magoroh Maruyama.
Gordon and Breach, 1994
James B. Miller. From the Garden to Gauss: Mathematics as theological metaphor
(Paper presented at the 7th European Conference on Science and Theology, March
31-April 4, 1998, Durham, England) [text]
Atsuyuki Okabe, Barry Boots, Kokichi Sugihara and Sung Nok Chiu. Spatial
Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, 2000 [details
Gregory Penoyre and Sunand Prasad. Accommodating Diversity: housing design
in a multicultural society. 1999
Clifford A. Pickover. The Loom of God: Mathematical tapestries at the edge of
time, Plenum, 1997.
- The Structure of Fair-Division Problems and the Design of
Fair-Negotiation Procedures. In: I. Garcia-Jurado, F. Patrone, S. Tijs (Eds.):
Game Practice, Kluwer, forthcoming 1999
- Supporting Cooperative Multi-Issue Negotiations. University
of Bielefeld, Institute of Mathematical Economics, Working Paper No. 299,
June 1998 [text]
Matthias Raith and Anne Chwolka. Group Preference Aggregation with the Analytic
Hierarchy Process. University of Bielefeld, Department of Economics, DP No. 412, Feb.
Matthias Raith and John Reinhard. Optimizing Multi-Stage Negotiations. University
of Bielefeld, Department of Economics, Discussion Paper No. 351, Aug. 1997
/ Jan. 1999
Matthias Raith and Helge Wilker. ARTUS: The Adaptable Round Table with a
User-specific Surface. University of Bielefeld, Institute of Mathematical Economics,
Working Paper No. 298, May 1998 (ARTUS is a process control system that provides
an infrastructure for group decision making over the Internet in the form of
Matthias Raith with Andreas Welzel. Adjusted Winner: An Algorithm for Implementing
Bargaining Solutions in Multi-Issue Negotiations. University of Bielefeld, Institute
of Mathematical Economics, Working Paper No. 295, February 1998 [text]
Rodney Riegle and D Michael Risen. Mastering Metaphor: policy analysis
for visionary leaders. Normal, IL: PIP, 1993.
Jack Robertson and William Webb. Cake-cutting Algorithms. AK Peters Ltd.,
Simon Schama. Landscape and Memory. 1995
Doris Schattschneider. Introductory References about Tessellations (tilings)
and Sources for Illustrations [text | software]
Donald Schon. Configuring: Using disagreements for superordinate frame configuration
Ron Surratt. The Mediator: Mediation, Negotiation and Conflict Resolution
Software - a program which allows you to easily create the best win-win agreements
Stan Tenen. The God of Abraham: A Mathematician's View. Torus: Journal
of the Meru Foundation, Vol 2, 3, July 1993 [text]
Kirby Urner. The Invention Behind the Inventions: Synergetics in the 1990s
Robin Waterfield (Tr). The Theology of Arithmetic: On the mystical, mathematical,
and cosmological symbolism of the first ten numbers. Phanes Press (attributed to Iamblichus,
4th century AD)
Kapila Vatsyayan (Ed). Concepts of Space: Ancient and Modern. New Delhi,
Vedams Books, 1991, 680 p., [text]
- Space and Place: the perspective of experience. University of
Minneapolis Press, 1987
- Passing Strange and Wonderful: aesthetics, nature and culture.
Island Press, 1993
Jeff Zaleski. The Soul of Cyberspace. HarperEdge, 1997