Reframing Relationships as a Mathematical Challenge
Jerusalem: a Parody of Current Inter-Faith Dialogue
- / -
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
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
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.
In what follows, options that have already been articulated and rejected
are not considered. These include:
- some form of division of a land which each party sees as necessarily indivisible
- token or symbolic presence of one or other party
- formulae which effectively label one people as secondary to another
- formulae characteristic of Dayton-type peace agreements, of which the Oslo agreement may
prove to be another example.
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
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
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.
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:
- 1. The condominium case, as it has been made
- 2. Mathematical framework for reconciling N claimants to the same surface (where are
interested mathematicians to be found with the appropriate expertise?)
- 3. Visualization for comprehension (using computer graphics) of such higher dimensional
approaches to multiple claims (who would be interested in moving this forward, especially
in the light of the role of computer graphics in the Dayton agreement?)
- 4. Identification of metaphors (notably from the sacred texts of the claimants) to
justify a 'dematerialization' of the argument beyond two dimensions (this would
require involvement of mathematically oriented theologians)
- 5. Consideration of how imaginative alternatives can be communicated to wider audiences
to trigger imaginative involvement.
- 6. Creation of a context in which the above can be discussed (possibly on e-mail)
- 7. Reflection on procedures for grounding multidimensional realities in new kinds of
symbolic document that go beyond the constraints of legal texts dependent on agreement.
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
'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
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]