|POLE PAPER SERIES||ISSN 1370-4508||Vol. 1, No. 1, January 1995|
War, Hypercomplexity, and Computer Simulationby Prof. Dr. Gustaaf Geeraerts
Despite numerous empirical and other studies on war, we do not as yet have much insight into why and how wars come about. This is especially true about how war as a certain and comparably rare form of conflict regulation is connected to conflict behaviour at lower levels of intensity as military disputes and international conflict behaviour in general. In other words, we still do not understand very well the principles or mechanisms underlying the origins of the war phenomenon, its persistence and evolution, as well as its complex interrelationships with serious disputes and lower-level international conflict. Without such understanding, however, it is hard to distinguish, explain, and predict different categories of (future) conflicts and their interrelated dynamics.
This situation, i.e. lack of insight into the deeper mechanisms governing international conflict and war, is problematic. The main reason for concern is that (world) society develops in ever faster and more complex ways because of continual technological innovation and change, a process that will continue to raise grave problems of behavioural and societal adaptation (1). As we know all too well from both historical and contemporary political experience, such adaptation is always a rather hazardous affair, liable to run out of hand. Moreover, this liability is even more probable in the extant international political and economic system. Lacking a world government or central authority vested with the capability to enforce rules, settle disputes, and maintain peace, the international system has but a low capacity for the effective and efficient (re)solution of a major crisis or challenge to the existing structure. Especially in times of rapid and drastic changes, this means that the predictability of social and political interaction shows a tendency to decrease rapidly. On the other hand, uncertainty, and with this the likelihood of injurious or violent behaviour, increases accordingly.
The question then is how our knowledge about war, and especially its dynamics, can be improved. Now, to say that the study of war needs improvement is one thing. It is quite another to specify the substance of such an effort. In this article I make an admittedly sketchy attempt to do just that by making a few programmatic remarks of which I suspect that they may be essential to theoretical progress in the study of war. The aim is to stimulate constructive criticism and further efforts at theoretical articulation.
When studying war we easily incline to treat it as an extraordinary form of behaviour. For one thing, statistically speaking war is a rare event (2). Moreover, it is an exceedingly costly and disruptive form of interaction between people and states. It is also a peculiar, not to say: absurd, form of dealing with each other, as it is so very often detrimental to all involved. Indeed, what could be more irrational than warfare or fighting among people or systems, a form of exchange, that is, in which they seriously harm or even destroy each other? All this seems to make war a rather special category of human interaction, quite unlike other forms such as trade and cooperation.
From a theoretical point of view, however, there are few if any valid reasons to place war in a special category or to regard it as an extraordinary form of interaction. Surely, from a rational point of view, people or states should try to avoid it as much as possible because of its disruptive and destructive effects. Yet, since people or systems do have the capacity to harm each other, it is perfectly understandable that under certain conditions they will be tempted to do so to further their own ends. And, it is equally understandable that interaction among systems or people sometimes creates situations in which some or all concerned come to see the use of violence as an attractive or necessary option. As we will argue below, social processes are governed by a definite dynamism of their own that is hardly under the participants' control. As a result, even among initially friendly, peaceful, and benevolent people or systems, there is always a risk that their behaviour towards each other will assume violent forms, up to and including actual warfare. As such, war represents a perfectly natural, but deplorable, outcome of ordinary social processes. Its explanation does not call for any special and especially bad or sinister forces, motives, intentions, or proclivities in man. Warfare is the outcome of a complex interplay of relationships between a set of actors, the nature of the system to which they belong, and the evolution of the interaction or process in which they participate (3).
War is thus a natural though relatively rare condition. As a social occurrence it belongs to those processes that are typically "caused" by "hypercomplex systems". (4) Such systems represent complex configurations of parts, factors, or forces that interact with each other, in particular as a consequence of diverse direct and indirect non-linear feedback relationships. Therefore, the condition or conduct of any single part is a function of the condition or conduct of the (in principle: all) others. By implication, the analysis of such hypercomplex systems can be anything but an easy matter. Moreover, it is intuitively clear that traditional notions of linear causal relationships and related "black box" models are hardly suited for a more than superficial analysis of such systems. These notions and models are simply not conceived to discern the internal dynamics of any system. In fact, they predict the system's response without explaining the properties of the system components and connections between them, that is the system's structure. (5) Therefore, their application only makes sense in those cases where one has to deal with (relatively) stable systems. In social systems, however, stability can hardly be taken for granted. Most certainly not if one tries to explain or predict social and political processes in the longer run. A black box approach, then, only seems plausible with regard to the study of statics (i.e., relationships at a point when change is not occurring). It predicts without explaining or gaining insight into the internal structure of the system, and its applications are therefore limited from an evolutionary perspective (6).
To arrive at a deeper understanding of the evolution of hypercomplex (social) systems it is necessary to reveal how their state, and eventually their structure, change over time. As historical analysis reveals time and time again, the actual state of any social system, as well as its structure, are to a large extent determined by their previous state or structure or, more generally, by their past history. Put differently, social systems are inherently historical (7).
The preceding has some fundamental heuristic implications for the study of the war phenomenon. Foremost it means that there is not so much sense in conceiving such an enterprise as consisting in the search for observable regularities, that is, of invariant relationships at the level of the empirical phenomena themselves. The reason for this is that due to the dynamic or historical nature of the systems and processes under study in social science, meaningful empirical generalizations are hard to obtain. This is certainly no less true regarding war-peace issues. Illustrative in this respect is the following remark by Singer (1986):
The hypercomplex nature of social life hardly allows any valid and non-trivial empirical generalization. As has been suggested earlier, what a system does or does not do, how it responds to any given stimulus or event, is to a significant extent determined by its current state as the product of its historical evolution. This implies that the development of a system will be governed by a definite dynamism of its own, and that the very same stimuli or events will usually have rather different effects on different systems.
In brief, the development and conduct of two systems, be it individuals or collectivities, will never be entirely the same, not even when the circumstances in which they find themselves appear to be identical. The same causes do not produce the same effects. This does surely not signify that scientific inquiry or theory formation about people or social systems is impossible. But it does imply that we should conceive such inquiry in a more abstract or fundamental fashion than has been the case so far. In other words, it is to provide us, not with a kind of summary description of empirical reality, a sort of generalization of experience, but with an insight into the mechanisms underlying the change or transformation of the many social phenomena. (8) To gain deeper insight into society we need to know the mechanisms by means of which people or systems generally influence each other's conduct. More specifically, how does the behaviour of any actor or system component vary with that of the (in principle: all) others?
The forgoing signifies that the explanation or prediction of the state of any social system, can only proceed by means of a dynamic analysis, as a function, that is, of the time path of change in the social process among the membership of the system. We must address our attention not so much to the behaviour of people or systems at some moment. Instead we need to ask how their behaviour with respect to each other changes, and how these changes, in turn, produce other changes. As such, the state of a system is but a transient stabilisation of ongoing processes of change.
The argument so far also sheds some light as to why research into the "causes" of war has been largely unsuccessful.The main reason is that the principle of causality is only suited for handling elementary relationships. As a heuristic device it renders no justice to the assumption that war is a hypercomplex system. More specifically, the assumption means that war is a social process made up of dynamic complexes of various elements influencing each other directly and indirectly. Consequently, how any single element behaves, is essentially determined by the behaviour of all the other elements. For instance, in any form of interaction between two or more actors, how any of them behaves is governed by expectations regarding the other parties' behaviour. This applies to all participants. Their interaction thus flows from an interrelated set of expectations.
One might then conjecture that a condition of conflict or hostility between actors generally would raise the chances of violence between them; and conversely, that the likelihood of their co-operating peacefully were somehow proportional to the degree of accord between them. Yet, at the same time such hostility or accord is itself also influenced by the very probability of violence or co-operation between them. On closer examination this probability turns out to be less a matter of their own wishes, than of the prevailing boundary conditions imposed on them by the system of which they are part (De Vree 1991:33). Individuals, for instance, obviously do rather different things in conditions of war than in peaceful and more secure social conditions; they behave differently in anarchic social surroundings than they do in highly ordered and regulated societies. (9)
All this also implies that we will in general not be able to predict what will happen in the future from the mere observation of what happened in the past. Whether an external threat will or will not stimulate internal unity or integration in a political system, or power shifts in a political system will or will not bring about violence or war, or oppression or deprivation will or will not lead to protest and revolution all depend on how the initial conditions are filtered through the current state and structure of the systems under consideration. It also follows that historical analysis of any specific case, system, period, or society, may not simply be projected on other cases, systems, periods, or societies, let alone be used as a basis for present policy-making. To give but one example, most analysts tend to agree that British and French policies of appeasement towards Nazi Germany have clearly contributed to the outbreak of the Second World War. However, can we take this to mean that giving in to a growing power will invariably lead to war? To do so would surely go much to far. Again, to deal with such questions in a more balanced and informative way we need deeper insight into the mechanisms through which people or systems generally influence each other's behaviour.
Regarding these mechanisms, De Vree (1988, 1990) has introduced some interesting general assumptions. More specifically he presumes that at all societal levels, from the family up to and including world politics, interaction springs from the fact that the actors involved are interdependent. This means that, quite aloof of their wishes or intentions, actors will be inclined to interact to the degree that they have the capacity to affect, positively as well as negatively, each other's power. The latter concept is generally understood as an actor's endowment to survive or maintain himself (10). As such, an actor's power implies the capacity to affect another actor's power, both to support, aid, or strengthen, and to harm, weaken, or injure him. Starting from these assumptions, De Vree (1988:12) derives the following conclusion:
All this means that actors in a system must find ways of adjusting to shifts in their mutual power relationships. In such context appeasing may often turn out to be the only sensible thing to do, namely bow to the inevitable, and avoid a costly and useless violent test of strength. Of course, to be able to decide in this matter, the actors involved need the information that permits them to assess what is inevitable, possible, or beneficial - precisely the kind of information which in practice is all too often lacking, especially in a system as intricate as the international system (Geeraerts 1991).
Apparently, if history is to teach us any lessons, we must look for them at a more fundamental level than that of directly observable relationships. This level, which De Vree (1991) depicts as that of the general structure of things, is a much more complex one altogether, in the sense that the structure of a system is defined in terms of a (matrix of a) larger or smaller number of different relationships or functions. As such it determines the state transitions of a number of observable magnitudes, and produces quite different (observable) results depending upon the initial state of the system concerned.
In all then, it seems not very fruitful to ask for the causes of violent conflict and war. Instead, the basic query about phenomena of war and peace should read: What are the mechanisms that make for social processes at times to destabilize and escalate to certain levels of injurious interaction (low level conflict, serious disputes, war)? And in the same vein: What mechanisms make a social process stabilize or reach equilibrium, so that more supportive forms of interaction (trade, co-operation) become possible? The underlying idea then is a vision of war as one of the many possible conditions in the international political process, in other words, a state that is connected in various complex ways with other injurious forms of interaction as serious disputes and low level conflict behaviour, but also supportive behaviour like trade and co-operation.
From the point of view of scientific understanding, what precedes also comes down to the necessity of adopting a 'holistic' or 'systems approach' (11) to the study of war. Indeed, as already mentioned, in hypercomplex systems, the state and development of any one component is, in principle at any rate, a function of the state and development of every other component. By the same token, the behaviour and evolution of such component is determined to a large extent by its position in, and by the nature or evolution of the whole system of which it is a part. As pointed out previously, actors behave rather differently in conditions of war than in a peaceful and secure social setting. They also have different expectations regarding the behaviour of their fellow actors in anarchic social systems than they do in highly ordered and regulated societies. (12) And what the system as a whole does, its state and evolution, is not simply a function of how its components behave, but also of the way these components are arranged, of the system's internal structure or organization. When studying the dynamics of war and peace we therefore badly need to think in terms of entire systems. It is imperative to regard specific occurrences of war and peace as being produced by some evolving system as a whole, that is, by a moving complex of a greater or smaller number of interacting actors and forces.
All this is quite demanding. It calls for a continual awareness of many different relationships at the same time. It conflicts with rather deeply ingrained mental habits such as prevailing anthropomorphic conceptions of things human and social.Last not least, it requires a comparatively high level of methodological sophistication. In this connection, Kirkpatrick and Widmaier (1985) point to the necessity of formalisation. Indeed, as a consequence of the complex nature of the subject under consideration the thrust of an argument is more often than not dependent on the accurate specification of the intricate interrelationships involved. They also argue that formalization means much more than simply writing regression equations or statements in one of the many available computer languages. It involves the mathematical formalization of whole systems of equations, an undertaking that necessarily leads to questions of logical or internal consistency and equilibrium. If we are to analyze and understand (hypercomplex) social systems beyond the black box level, clearly new approaches and techniques are imperative. Examples of inspiring and theoretically progressive attempts in this regard are Cusack and Stoll (1990), De Vree (1990) and Wolfson, Puri and Martelli (1992).
The preceding assumptions have considerable theoretical and methodological implications for the study of war and peace. An emphasis on social dynamics and complexity calls for different accents in explanation, modelling and testing (Hanneman 1988: 324). Foremost, it means that the states of war and peace come to be seen as transient stabilisations (outcomes) of the interactions between a number of actors within (usually multiple) interdependent dynamic processes. Their variability is understood to be the result of different over-time conjunctures of common underlying behavioral mechanisms and ensuing processes of interaction (i.e., unit level of analysis) occurring within and, at the same time, being influenced by the system's structure (i.e., structural level of analysis). (13) As a result the testing of similar dynamic systems theories acquires a clearly 'historical═ or 'irreversible' flavour under both controlled conditions (as in computer simulations and experiments where the same dynamic process is replicated across variables and parameters) and uncontrolled conditions (as in the study of the rise and fall of empires (e.g. Kennedy 1987), or again in the analysis of processes of state formation (e.g. Tilly 1975) and integration (e.g. Jansen and De Vree 1985). It is the capacity of the hypothesized system or process(es) to produce, predict, and postdict sequences of related classes of events that becomes central to the evaluation and critical testing of a theory.
Greater attention to the complex dynamics of war and peace demands both a kind of theory and methodology that I conjecture to be quite different from most of the current work in the field. To begin with, when studying war and peace from a dynamic systems perspective, we will usually be at a loss to say what any given observation in and by itself means as regards the theory involved in the first place. The crucial point is that the state of hypercomplex systems, how they behave or what happens within them, results from the interplay of various factors and forces, and may typically come about in many different ways (degrees of freedom problem). (14) As a result, actual developments may differ vastly even though governed by the same principles and mechanisms. As hinted earlier, such diversity is to be expected: working on different initial conditions, the same principles or mechanisms will produce different results, and create different circumstances as starting-points for different subsequent developments, and so on. In a way, the same causes do not produce the same effects. For this reason historical regularities may be expected to be the exception rather than the rule. To give an example: bipolarity may breed war, as it did on the eve of World War I, but it may also generate relatively stable and peaceful international political relations, as was the case in the post-World War II era. In the same vein, international wars, revolutions, and the historical evolution of national societies, occurring in different regions or periods, differ vastly from each other.
All this implies that our ability to predict the future course of events will also be rather sharply limited in most practical cases, and the more so, the longer the time-span involved becomes. Surely, knowing the mechanisms governing real events or developments allows us to predict what is more or less likely to happen from given initial conditions. That is, it allows us to predict the system's state at one moment, t+1, givent the previous moment, t. When the system is not subject to outside disturbances, or when, as in experiments, we are able to insulate it against them, one may repeat the operation and predict the system's state at t+2 from that at t+1, and so on (see figure 1).
Figure 1: Iteration
Human behavioral and social systems, however, are continually subject to outside disturbances from which they cannot be artificially insulated. As mentioned before, the phenomena under consideration here are typically "hypercomplex", in the sense that they represent, and in their turn belong themselves to multifarious configurations of a varying number of different elements or variables that influence each other continuously in many direct and indirect ways. Accordingly, deterministic models of predictability as the ones we encounter in classical mechanics for example, can hardly be a realistic ideal when studying similar phenomena. For this reason, it is difficult to say anything generally valid, definite and concrete about social phenomena as war and peace. Accordingly, we must often do with approximative conclusions regarding the direction which their development will take, implying, however, the possibility of larger or smaller divergences in concrete instances.
As a consequence, when evaluating theories dealing with hypercomplex phenomena such as war and peace, observed facts have meaning only in the context of the entire theory involved in conjunction with the whole complex of variables describing the system's state. Such complete interrelationships can hardly be described by the kind of hypotheses that are commonly being proposed and tested in studies on war, while, conversely, the actual outcome of testing such hypotheses will usually not tell us anything meaningful and decisive at all.
Moreover, if we wish to avoid premature and therefore non-cumulative empirical research, we must first of all have a much more accurate insight into the actual behaviour of our models or theories than mere "paper-and-pencil" work and "ad hoc testing" can provide. For instance, it is imperative that we know whether the model contains any hidden inconsistencies, whether it does indeed behave as intended, whether it does develop an equilibrium or lead to a definite condition, and whether it is indeed stable and does not "explode" or produce nonsense. This is the more necessary as highly complex and dynamic systems often do confront us with theoretically quite unpleasant surprises. The basic tool for doing this rather essential research, a kind of pre-empirical inquiry, is the technique of computer-aided modelling and simulation (Whicker and Sigelman 1991; Neelamkavil 1987; Spriet and Vansteenkiste 1982). Using this technique, we let the computer work upon simplified data concerning social systems that are modelled in more restricted and therefore more manageable ways.
The above does certainly not imply that the validity of theories dealing with the dynamics of war and peace could not be assessed at all. But it definitely means that such assessment or testing will be a rather more involved, indirect, and slow affair than most actual scholars apparently deem it to be. In particular, it will necessitate the development and adoption of a rather different research methodology in which formal techniques combined with computer-aided modelling and simulation will have to play a central role. For it is, practically the only way that we can hope to successfully manage, that is, study and test, compute and manipulate, the dynamic complexities which form the subject of our trade. (15) As Meadows and Robinson (1985) point out, this has everything to do with the following potential advantages of computer models. (16) A first such advantage relates to rigor. The assumptions in computer models must be specified explicitly, completely, and precisely so that no ambiguities are possible. Every variable must be defined, and assumptions must be mutually consistent.
A second advantage pertains to the comprehensiveness of computer models. With the aid of a computer one can manipulate more information than is possible with the human mind and keep track of many more interrelationships at one time.
A third kind of advantage has to do with logic. If programmed correctly, the computer can process very complicated sets of assumptions to draw logical, error-free conclusions.
A fourth advantage resides in the accessibility of computer models. Because all the assumptions must be explicit, precise, and unambiguous in order to communicate them to the computer, critics can examine, assess, and alter computer models at will.
Finally, computer models are highly flexible. They can be made to test a wide variety of different conditions and policies, providing a form of social experimentation that is much less costly and time consuming than tests within real social systems.
Although the usefulness and advantages of computer simulation are apparent, some qualifications seem in order. For one thing, sceptics might rightfully point out that since they were first introduced in the early 1970s computer models have not been tremendously successful. In this regard the several models that were developed with support of the Club of Rome may serve as a well documented example.
Still, one is entitled to ask whether the deficiencies of these models were inherent to the technique of computer simulation, or whether they were due to the fact that these models were build upon an insufficient or reductionist theoretical base. With Deutsch (1987) I believe the latter to be the case. The crucial problem with the above models was that they did not take into account the different underlying (cultural, political, social, and economic) mechanisms that make for the evolution of (world)society. In a way the creators of these models can hardly be blamed for these omissions as such basic theoretical insight was not - and still is not - available (De Vree 1990; Wolfson, Puri and Martelli 1992). At the same time, the above implies that a more successful application of computer modelling and simulation in social science implies the need for a more explicitly theoretical (and mathematical) approach of the several hypercomplex social phenomena - war and peace being among them. After all, computers are relatively stupid things, meaning that what you get out of them cannot be better than the theoretical models you feed into them. It is precisely the extant paucity of theory which has so far stood in the way of a convincingly successful use of computer simulation in our field. As De Vree (1991, 47) has written recently:
A further qualification pertains to the fact that computer models are essentially "procedural" as opposed to "symbolic" models (Bremer 1987:22). Symbolic models are basically mathematical models that allow for the direct deduction of the behaviour of a referent system through application of a calculus to abstract symbols and functional relationships. Models of this type are very powerful deductive devices due to the generality of their conclusions and parsimony of their structures. However, their use becomes problematic when confronted with representing highly complex systems containing non-linear relationships and many interdependent variables, as is the case with (hypercomplex) social phenomena. In such cases procedural models are practically the only solution. The problem with the latter is that they do not have general solutions. As a consequence, the deductions they generate are valid only for a specific set of initial conditions (Kleijnen 1987:133). So, when changing these conditions, one is never quite sure whether rerunning the model might not produce radically different results.
The picture even gets more bleak if one wants to take into account that real social systems are not only very complex, but also face turbulent environments whose dynamics introduce a highly stochastic element into social processes. Indeed, as Bremer (1987, 24-25) remarks, working with highly stochastic models requires the use of Monte Carlo techniques, whereby each run of the model must be repeated many times before the result of an experiment can be assessed. (17) This procedures has serious practical drawbacks if one is dealing with large and highly complex models. For example, the introduction of random factors in the central equations of a complex model, would imply that several hundreds of runs might be required before the central behavioral tendencies of such a model could be evaluated. Nevertheless, if computer power and speed continue to increase as they have done so far (and it is generally expected that they will), it is very likely that these obstacles will be surmounted.
Although our knowledge of war and international conflict has definitely increased, we do not as yet have substantial insight into why and how wars come about. Especially, we are still very much in the dark as to how war as a certain and comparably rare form of conflict regulation is connected to conflict behaviour at lower levels of intensity as military disputes and international conflict behaviour in general. Theoretical progress in the study of war demands the adoption of a dynamic systems perspective. This means that the war issue is placed within an evolutionary and unified or systems-theoretic point of view. The latter specifically implies a vision of war as a certain and one of the many possible phases in the international political process, that is linked in various ways to other injurious forms of interaction as serious disputes and low level conflict behaviour, but also supportive behaviour like trade and co-operation. Although war may be rare and tragic, it nevertheless constitutes a "normal" event. There is no need to regard it as an extraordinary form of interaction, the explanation of which demands distinct explanatory principles. War, no less than other social phenomena such as peace and co-operation, is the result of different overtime conjunctures of common underlying behavioral mechanisms and resulting processes of interaction occurring within the higher level system's structure. In view of the highly complex nature of the concerned dynamics, I believe that the development of computer (simulation) models will become one of the essential research methods in the study of war and peace, as it already is, and for the same reasons, in the physical sciences and engineering, as well as more recently in the social sciences.
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See, for example, Rosenau (1990). Although almost no historical period is without some war, most nations are at peace (or at least not at war) most of the time. Statistically speaking, war constitutes a rare event. Only 67 inter-state and 51 other wars (involving at least one nation-state against an unrecognised state) were fought between 1816 and 1980. See Vasquez (1987:118). Compare with Singer (1981:3). See above all De Vree ( 1990), especially chapter. 29. This simply means that they the systems involved are both complex and dynamic in nature. In a more technical sense, I use the term 'hypercomplex═ here as it is introduced by Vree (1990:100). It serves to express that in mathematical terms social phenomena are usually located in multi-dimensional (hyper)space, and that they essentially involve both feedback and non-linear relationships. Here and in the rest of this article the term "structure" refers to the entire set of relationships (as determined or defined by a particular theory) which determines how a system"s state at some moment, t, is being transformed into another such state at the next moment, t+1, either under the influence of certain environmental stimuli or disturbances, or under that of its own internal dynamism. A such, one can say that structure describes the internal make-up of a system, usually in the form of more or less complex transformation matrix. See e.g. Neelamkavil (1987). See, among others, Benjamin (1982:69-98). To avoid misunderstandings, it should be mentioned that the term "mechanism═ merely refers to some set or system of relationships or functions governing the transition of one system"s state into another. It does not imply anything 'mechanic or 'mechanistic═. The current events in Bosnia-Hercegovina may serve as a dramatic reminder of this very principle. For a far more elaborated, and also mathematical treatment of the concept of power, and especially its possible role as a general explanatory mechanism or principle, see De Vree (1990, 1993). In order to thwart any misunderstandings, it should be mentioned that 'systems approach═ has no substantial meaning here. It merely serves to emphasize a methodological issue, viz. the explicit recognition of the fact that in social life phenomena are interdependent, are made up of still other phenomena that thus are interdependent, and should therefore be studies as such, or, conversely that there is little sense in studying social phenomena in isolation. This point has also been stressed by Waltz (1979). Of course, the system═s structure too can be changed as a result of the interaction between the units making up the system. An example of such dynamics would occur when the density and reliability of interaction between a set of actors becomes sufficiently high to override the deep structural effects of anarchy. More in particular such development could override the tendency for similarity of units to avoid relationships of interdependence, as already is the case among the advanced industrialized states. It would incite some units to survive and prosper through specialization. This position is very close to 'structural realism═ as presented in Buzan, Jones & Little (1993). See also Little (1994: 19). See especially G. Kampis (1991:206-208). Even the study of a very simplified system of interaction (viz. between only two systems, each having but two basic options: to support or to injure each other) must already take place in a 8-dimensional hyperspace. As such, it involves 8x8 transformation matrices, consisting of 64 fairly complex functional relationships. As the study of the evolution of such a system over any period in time involves the iteration of the involved calculations for a number of times, it is hardly conceivable that this could reliably be done by means of pencil and paper, that is without the aid of a computer. Compare with Cusack & Stoll (1990: 10-15). See also Bouton & Mai (1990).