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Reframing Relationships as a Mathematical Challenge

Jerusalem: a Parody of Current Inter-Faith Dialogue

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Prepared on the occasion of the European conference of the United Religions Initiative (Oxford, 1997)


It might be considered presumptuous and naive for anybody to seek windows of opportunity in the current tangled complex of Jerusalem. Many have devoted considerable attention to the matter. Nevertheless, as with a number of other faith-based territorial disputes, current approaches seem doomed to reinforce polarization, repression and violence. Jerusalem makes a mockery of present approaches to inter-faith dialogue -- mirroring too well the kinds of dialogue characteristic of the end of the 20th century. So maybe there is a case for fools stepping in where the wisest fear to tread -- for the initiatives of the wise are proving counter-productive. 

Stated simply, perhaps simplistically, Jerusalem exemplifies a condition in which different groups lay claim to some form of possession or sovereignty over the same territory. These claims are not going to disappear, however they are 'regulated' in the short or medium term. From this perspective, it could be considered futile to continue to seek solutions based on the assumption that one or other claim can be made to disappear, or can be effectively neglected as irrelevant in practice. Can 'a land for people without a land' be achieved for any length of time by making another people landless? 

The challenge then becomes to discover ways in which several claimants can maintain sovereignty over the same territory. This challenge needs to be framed in an exploratory, brainstorming mode to avoid premature closure of doors, and neglect of windows of opportunity that are as yet poorly recognized. The point to be strongly made is that such possibilities need to be on the table as a trigger to the imagination, whether to evoke richer possibilities or to discover ways of giving form to those which might be so articulated. It is useful to recall that many successful scientific developments are based on numerous (even hundreds or thousands) of trials. 

Excluded options

In what follows, options that have already been articulated and rejected are not considered. These include: 

Preamble to further exploration

It is absolutely vital to recognize that the 'land' under discussion is not to be understood in the same way as states normally recognize territory to which they lay claim. Territory in this sense is something which states acquire or lose in the course of history. It may then bebought or sold, carved up, or disposed of like any other asset -- a pawn in the game between states. This is a 'real estate' approach embodied in a legalistic language which limits non-conventional explorations. In the case of Jerusalem, Tibet, Kurdistan, or Northern Ireland, the symbolic significance of the land completely transforms the debate. A completely different way of perceiving is a feature of the claimants and is required of any resolution. This is characterized by cultural imponderables, associated with a sense of collective identity. Treating such land in the light of 'mechanical' and 'linear' approaches to its division is then completely meaningless and doomed to failure -- however successful such solutions may appear in the short term. 

This conventional approach, appropriate to other situations, needs to be labelled and positioned in a manner which can distinguish it from more relevant kinds of approach that need to be discovered. 

The argument of this note is that science has been obliged to discover ways of thinking which defy and transcend the constraints of conventional Euclidean geometry or Newtonian physics. These are now seen as special cases, or limit conditions, set within a more complex richer framework. In physics, for example, the Pauli exclusion principle specifically excludes the possibility of one electron occupying the same position as another. The question is under what conditions is this principle effectively violated in the manifestation of more complex and richer phenomena? How are such conditions to be understood? Why are they so challenging to the understanding? 

It is from such frameworks that conventional approaches to the search for 'a common ground' need to be evaluated. Where such ground is sought based on an implicit metaphor such as a 'town square' or 'village common', this objective needs to be challenged in terms of the inadequacy of its complexity. Jerusalem is characteritic of a situation in which claimants claim the totality of the common ground -- in contrast to the many simpler negoitating situations in which some sharing may be envisaged. It is precisely what makes the ground 'sacred' to a given faith that undermines simpler approaches to 'common ground' -- in a special sense it is an ethnic 'homeland'. These dimensions taken 'common ground' out of its metaphoric two-dimensions, so easily enshrined in legalese, and call for other levels of understanding. 

With respect to Jerusalem, given, its profound and fundamental symbolic significance, the key question is how the Jerusalem which is of such great symbolic importance is related to the land measureable by a surveyor. Is the symbolic Jerusalem to be solely understood through what are effectively Euclidean and Newtonian perspectives? Or does that Jerusalem call for a depth of understanding which is more closely analogous to concepts associated with complex geometries and fundamental physics -- which are a real challenge to the understanding? In this sense, does not the conventional Euclidean/Newtonian approach effectively demean what is essential to the symbolic and spiritual significance of Jerusalem and all that it represents? Can those who would defend the profundity of insight underlying the importance attached to Jerusalem truly believe that this insight can be validly expressed through conceptual frameworks that are many orders of magnitude less complex than those required for fundamental physics (in understanding atomic structure) or for astronomers (in understanding cosmic phenomena)? 

The following explorations assume that conventional approaches, which lend themselves to ready verbal articulations, effectively conflate simple three-dimensional understanding with the multidimensional understanding which is fundamental to the real significance of Jerusalem. The multidimensional Jerusalem tends to be seen, and discussed, through the framework of the two- or three-dimensional Jerusalem -- although existentially people relateto the multidimensional reality which is beyond ready verbal articulation. This in no way denies the significance of the three-dimensional Jerusalem, but it is effectively a constrained projection of the larger symbolic complex with which people identify. The three-dimensional Jerusalem contains mnemonic markers to features of the larger symbolic complex. 

The question is whether there are not multidimensional understandings of Jerusalem which totally legitimate a number of seeming incompatible or incommensurable understandings of Jerusalem. Such richer multidimensional understandings would justify particular interpretations, each apparently incompatible with the other. In the case of physics, an example is the 'wave vs. particle' understandings of light. Both are correct at one level, but their reconciliation is a fundamental challenge to comprehension -- even for those with long training in physics and mathematics. Given the symbolic importance of Jerusalem, is it to be expected that comprehending its psycho-cultural significance should be less or more complex than that required to understand the nature of light? 

Lines of exploration

The question is whether windows of opportunity can be discovered beyond the three-dimensional discussion which demeans the understanding that all parties have, in principle, of Jerusalem and its significance. Fundamental to this exploration is the identification of metaphorical frameworks which clarify the distinction between particular understandings of Jerusalem. Some possibilities include: 

1. Mapping the globe: The spherical nature of the globe makes it impossible to represent appropriately and simultaneously, the areas on the Earth's surface and the distances between locations. Geographers use many different 'projections' in compromise representations which are accurate in one respect, but necessarily not in another. Can the challenge of Jerusalem be considered in this light? Are the ways in which the identity of different claimants is attached, perhaps unnecessarily, to one projection rather than another? Is there a challenge of misplaced concreteness? 

2. Horizon effects: A person on one side of the Earth telephoning to a person on the other can claim that the noon Sun is shining, and be faced with the riposte from the other party that it is in fact totally dark. This can give rise to lengthy argument and discussion of fact and misunderstanding. What justifies both perspectives simultaneously is their respective positions on a particular kind of surface -- of which neither need be aware. The question is whether the disputing claimants to territory such as Jerusalem are, or could be, fruitfully understood as being positioned differently on a surface -- of which neither may be aware. Mathematics, and especially topology, has made incredible discoveries concerning the nature of complex surfaces. These could well reconcile perspectives and claims which would appear ridiculous if expressed within a conventional three-dimensional framework. 

3. Comprehension constraints: A particular branch of mathematics, known as Q-analysis, is used to clarify comprehension boundaries in complex psycho-social systems. It helps to establish how people are constrained and divided by the geometry of their communication patterns and limits on their comprehension. Typically it clarifies how people able to understand an N-dimensional structure, but not an N+1 dimensional structure, may then be constrained in their communications within the geometry of the N-dimensional structure. In the case of Jerusalem, the question is whether current discussions are about an N-dimensional Jerusalem, when any possibility of reconciling claimants can only be made through an N+3 dimensional discussion. To illustrate the non-trivial potential of this branch of mathematics, the originator was taken under contract to a government defence department after developing this approach. 

4. Resonance hybrids: In chemistry certain molecules, notably some fundamental to organic life, do not have a static structure as readily understood. Many molecules can be easily represented by balls (atoms) linked by sticks (molecules). Some molecules however, such as those based on the benzene ring, can only exist because of the way in which the sticks effectively alternate in between several neighbouring atoms. The resultant resonance structure is effectively a dynamic 'average' between several inherently unstable structures between which the bonds are alternating. Given the dynamic complexity of this structure and its fundamental significance for biological life, is it not possible that Jerusalem (as a complex of equivalent significance to the psycho-cultural identity of several peoples) may depend upon a similar degree of dynamic complexity? 

5. Architecture: Superficial discussion of Jerusalem focuses on visible aspects of its architecture, notably religious buildings -- most tragically the Holy Sepulchre of Christendom. These are however understood to be effectively traces of symbolic 'buildings' whose architecture juxtapositions elements and energies fundamental to the respective claimants and their identity. The design of the symbolic Jerusalem is of profound significance to many. This design may be understood as a configuration of elements that protect and give expression to that which cannot be effectively expressed through material construction --the holy place, necessarily 'hidden' or 'invisible' in some special way. Approaching a complex structure of this kind, different perspectives are possible -- there may be many planes of symmetry, each a form of 'common ground' for more limited understandings. As when passing an orchard of aligned trees, impenetrable confusion may suddenly give way to a dramatic sense of pattern, which may then be lost again. The question is whether there is an architecture to the symbolic Jerusalem which would legitimate the different perspectives of the claimants. 

Concrete proposal: the condominium approach

The above examples point to the possibility of windows of opportunity at levels of complexity which would draw upon the conceptual riches of the most challenging disciplines -- which are particularly well-represented in Israel. There is however a concrete proposal which is merely one step beyond the conventional pathetically polarized dialogue based on outmoded linear thinking. 

The condominium approach to Jerusalem is exemplified by 50 years of co-sovereignty of the New Hebrides by the governments of the UK and France. The same territory had two governments simultaneously, with two parallel administrative structures, and corresponding citizenship arrangements. It has been articulated in detail by John Whitbeck since 1988 in relation to Jerusalem (and Northern Ireland) in appropriate Middle East journals (Middle East Policy (December 1994, pp. 110-118) and Jerusalem Times (15 March 1996, pp. 8-9)), as well as being the subject of a detailed report the International Herald Tribune (17 November 1994), and presented unsuccessfully to US Vice-President Al Gore. It is ironic, in a period when many in international capitals possess two or more passports -- surely 'irrational' in terms of purported understandings of soveriegnty -- that consideration should not be given to some such formula. 

Next steps

Given the unsuccessful presentation of the condominium solution, and the apparent lack of success with Al Gore, it is increasingly intriguing why such arguments get brushed aside when what is offered as a substitute is so intellectually primitive. Is the conventional search for 'common ground' to be likened to dialogue enthusiasts equipped solely with a 'hamme'r who then need to define every problem as a 'nail'? There may be other conceptual tools. 

The condominium formula is as an example of a formula which is one step beyond the current unworkable formula. But what are the steps beyond the condominium, as a class of solutions of increasing complexity, capable for that reason of reconciling increasingly intransigent political (or religious) perspectives. 

The issue is how two (or more) parties can lay claim to the same territory. With respect to the condominium option, Whitbeck has done a great service in clarifying the issue from a political and administrative perspective. This needs to be supplemented and reinforced by arguments from certain spatially oriented branches of mathematics (toplogy, etc). It is precisely in mathematics (the study of relationships) that 'higher dimensionality' is legitimately explored, and it is through such higher dimensionality that claims that are irreconcilable in two dimensions can in fact be reconciled, or at least shown to be simultaneously valid. 

Further steps might be envisaged as follows, all of which (and almost certainly others) are required to get anything moving: 

It is an irony that the country with the greatest number of mathematicians per capita is Israel. And the country with the greatest number of lawyers and graphics people is the USA. What is inhibiting their imaginative creativity? 


However clear mathematically, it may be argued that this ignores the sense of both 'true belief' and political psychology. The question is whether mathematics can offer useful insights into alternative perspectives. Experience with the systems dynamics tools publicized by the Club of Rome is an illustration of the limitations. It may however be argued that these used particular bracnhes of mathematics, neglecting those which might have communicated insights into more complex spaces appropriate to the sacredness of 'common ground'. 

It cannot be established whether the debate is trivial or psychologically dubious until it is undertaken by those with appropriate insight. The argument here is that given the apparent dearth of insight of any complexity currently applied to such territorial disputes, it is at least worth exploring. The Pentagon explored much crazier options in its heyday. 

The meta question is really why there so little interest in doing so? Why is there so much tolerance of a level of debate which leads naturally to the violence appearing daily in the news? 

It may be argued that mathematical language, as other additional languages, provide additional perspectives and thereby: provide better understanding to some; put issues into new formats; and sometimes may indicate new solutions. 

However their chance of impacting successfully on large-group or mass phenomena is zero (uses by Pentagon elite planners are quite a different matter) until publics are much more enlightened, which is a matter for the future. It is also important to be warned of the error that better 'understanding' always (or even usually) produces agreement. If the problem is better understood, those with that understanding may then 'want it all' or regard the other as 'an enemy'; better understanding will be then used for more effective violence. 

Whilst new perspectives on Jerusalem should indeed be made available, and some groups working on that issue may indeed benefit from such views, little 'political' and 'popular' impact will necessarily result. 

The 'rational' nature of some languages, such as mathematics, topology etc., may complete misrepresent/ignore the real nature of a problem, which may be related to 'extra-rational' and even 'irrational' dimensions of human identity and existence, which may be much deeper and more meaningful. 

But much of mathematics and physics now deals daily with multidimensionality that can well be considered 'extra-rational' because of its challenge to limited understanding. The question is whether such rational languages can be used metaphorically to suggest other levels of understanding -- rather then becoming trapped in their own impenetrable logics, as is so often the case.


The basic argument here is that people need to be challenged by more imaginative insights into the challenge of Jerusalem -- whether or not such proposals are doomed to failure. The chances of success can be no smaller than those proposals currently on the table. 


Anthony Judge. And When the Bombing Stops? Territorial conflict as a challenge to mathematicians.  Technological Forecasting and Social Change, 61, 1999, pp. 297-301 [text]

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