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July 2000

And When the Bombing Stops?

Territorial conflict as a challenge to mathematicians

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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 architects.

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 (

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 pattern?

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.

Legal theorists

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 Ireland, Cyprus, Jerusalem). 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 and its current sale ( What about sale and ownership of real estate on a range of other extraterrestrial bodies ( 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:

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?

Conflict mediators

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 rotation.

Media specialists

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 is this?

'Psy-ops' specialists

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 with them.

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.

Systems analysts

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 . 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 sense!

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 this?


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:

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:

Once again, these different ways of understanding relationships include:

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 up.

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 whole.

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.


Christopher Alexander:

Robert Ardrey and Philip Turner (Editor). The Territorial Imperative : A Personal Inquiry into the Animal Origins of Property and Nations.Kodansha International, 1997

Ron Atkin:

Stafford Beer. Beyond Dispute: the invention of team syntegrity.  Wiley, 1994 [commentary]

Steven J. Brams and Alan D. Taylor:

Stewart Brand. How Buildings Learn : what happens after they're built. Penguin, 1995

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 Verlag, 1987

Keith Critchlow. Order in Space: a design source book.  Thames and Hudson, 1969

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:

Bruce Edmonds. Hypertext Bibliography of Measures of Complexity by, 1997 [text]

R Buckminster Fuller. Synergetics: explorations in the geometry of thinking. Macmillan, 1979

George W. Hart. Encyclopedia of Polyhedra. (including 1,000 virtual reality polyhedra) [access]

Patrick A. Heelan:

Will Hively. Dividing the Spoils. Discover Magazine, March 1995 [text]

Douglas Hofstadter:

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]

Anthony Judge:

Peter Kruse and Michael Stadler. Ambiguity in Mind and Nature : Multistable cognitive phenomena. Springer Verlag, 1995 (Springer Series in Synergetics, Vol 64)

George Lakoff and R E Núñez. The Metaphorical Structure of Mathematics. In: L. English (Ed.), Mathematical Reasoning: Analogies, Metaphors and Images. Lawrence Erlbaum.(1996)

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 and outline]

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.

Matthias Raith:

Matthias Raith and Anne Chwolka. Group Preference Aggregation with the Analytic Hierarchy Process. University of Bielefeld, Department of Economics, DP No. 412, Feb. 1999 [text]

Matthias Raith and John Reinhard. Optimizing Multi-Stage Negotiations. University of Bielefeld, Department of Economics, Discussion Paper No. 351, Aug. 1997 / Jan. 1999 [text]

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 controllable "Virtual Tables.") [text]

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., 1998.

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 [ see]

Ron Surratt. The Mediator: Mediation, Negotiation and Conflict Resolution Software - a program which allows you to easily create the best win-win agreements possible. [text]

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 [text]

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]

Tuan Yi-Fu:

Jeff Zaleski. The Soul of Cyberspace. HarperEdge, 1997

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