Artificial-life techniques--specifically, agent-based models and evolutionary learning algorithms--provide a potentially powerful new approach to understanding some of the fundamental processes of combat. These models are designed to illustrate how certain aspects of land combat can be viewed as emergent phenomena resulting from the collective, nonlinear, decentralized interactions among notional combatants. Background In , F. Lanchester introduced a set of coupled ordinary differential equations--now commonly called the Lanchester Equations 1 LEs --as models of attrition in modern warfare. This essay discusses the relationship of postpositivist research and nonlinear science, particularly in terms of their importance to the educational researcher. Postpositivist researchers have been among those attempting to explore alternatives to this paradigm in the hopes of creating a better world, even when questioning the very idea of progress.

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Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating. Clausewitz, On War, Book One, Chapter 1 Despite the frequent invocations of his name in recent years, especially during the Gulf War, there is something deeply perplexing about the work of Carl von Clausewitz In particular, his unfinished magnum opus On War seems to offer a theory of war, at the same time that it perversely denies many of the fundamental preconditions of theory as such—simplification, generalization and prediction, among others.

Members of both groups sense that there is too much truth in what he writes to ignore him. Yet, as the German historian Hans Rothfels has bluntly put it, Clausewitz is an author "more quoted that actually read. Just what is the difficulty with Clausewitz that makes his work so significant yet so difficult to assimilate?

Its difficulty, however, has prompted different explanations even among Clausewitz partisans. Raymond Aron has spoken for those who believe that the incomplete and unpolished nature of On War is the primary source of misunderstanding: as Clausewitz repeatedly revises his treatise, he comes to a deeper understanding of his own ideas, but before his untimely death he brings his fully developed insights to bear only upon the final revision of Chapter 1 of Book One.

Thus for Paret the literature on Clausewitz has been "fragmented and contradictory in its findings" because of our lack of historical consciousness. Those aspects of On War that deal with human nature, uncertainty, politics, and rational calculation "will remain eternally valid," he contended.

Each of these approaches has merit, yet none satisfies completely. I offer a revision of our perception of Clausewitz and his work by suggesting that Clausewitz displays an intuition concerning war that we can better comprehend with terms and concepts newly available to us: On War is suffused with the understanding that every war is inherently a nonlinear phenomenon, the conduct of which changes its character in ways that cannot be analytically predicted.

My suggestion is more radical: in a profoundly unconfused way, he understands that seeking exact analytical solutions does not fit the nonlinear reality of the problems posed by war, and hence that our ability to predict the course and outcome of any given conflict is severely limited.

An approach through nonlinearity does not make other reasons for difficulty in understanding On War evaporate.

Furthermore, it may help us remove some unsettling blind spots that have prevented us from seeing crucial implications of his work.

What is "Nonlinearity"? Like other members of a large class of terms, "nonlinear" indicates that the norm is what it negates. Words such as periodic or asymmetrical, disequilibrium or nonequilibrium are deeply rooted in a cultural heritage that stems from the classical Greeks.

The underlying notion is that "truth" resides in the simple and thus the stable, regular, and consistent rather than in the complex and therefore the unstable, irregular, and inconsistent.

An important basis for confusion is association of the norm not only with simplicity, but with obedience to rules and thus with expected behavior—which places blinders on our ability to see the world around us. Nonlinear phenomena are thus usually regarded as recalcitrant misfits in our catalog of norms, although they are actually more prevalent than phenomena that conform to the rules of linearity. This can seriously distort perceptions of what is central and what is marginal—a distortion that Clausewitz as a realist understands in On War.

For a system to be linear it must meet two simple conditions. The first is proportionality, indicating that changes in system output are proportional to changes in system input. Such systems display what in economics is called "constant returns to scale," implying that small causes produce small effects, and that large causes generate large effects. The second condition of linearity, called additivity or superposition, underlies the process of analysis. The central concept is that the whole is equal to the sum of its parts.

This allows the problem to be broken up into smaller pieces that, once solved, can be added back together to obtain the solution to the original problem. They may exhibit erratic behavior through disproportionately large or disproportionately small outputs, or they may involve "synergistic" interactions in which the whole is not equal to the sum of the parts.

If interactions are irreducible features of the system, however, it is nonlinear even if described by relatively simple equations. Nonlinear phenomena have always abounded in the real world. Systems with feedback loops, delays, "trigger effects," and qualitative changes over time produce surprises, often abruptly crossing a threshold into a qualitatively different regime of behavior. The weather, fluid turbulence, combustion, breaking or cracking, damping, biological evolution, biochemical reactions in living organisms, and hysteresis in electronic systems offer examples of nonlinear phenomena.

Although some analytical techniques have been generated over the centuries to cope with systems characterized by nonlinearity, until the advent of numerical techniques offered by computers its study has been relatively limited.

Due to the structural storability of a linear system, once we know a little about it we can calculate and predict a great deal. The normal procedure has thus been to find mathematical techniques or physical justification for an idealized "linearization" of a natural or technological system.

Such an idealized version of a system is often constructed by throwing out the nonlinear "approximation. So docile are linear equations that the classical mathematicians were willing to compromise their physics to get them.

So the classical theory deals with shallow waves, low-amplitude vibrations, small temperature gradients. Clausewitz pointedly contrasted his own approach with the implicit dependence upon such selectivity among military theorists of his era, such as Heinrich von Bulow or Antoine-Henri de Jomini. This assumption works well for linear systems, and even relatively well for those nonlinear systems that are stable enough to be treated using the techniques of linear analysis or control theory.

But it turns out to be misleading when applied to the many more systems that are unstable under even small perturbations. As Stewart implied, this was understood by the more thoughtful of the classical mathematicians and physicists. James Clerk Maxwell, one of the greatest scientists of the nineteenth century, displayed a keen awareness of the limitations of assuming that systems in the real world are structurally stable: When the state of things is such that an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable; but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the condition of the system is said to be unstable.

It is manifest that the existence of unstable conditions renders impossible the prediction of future events, if our knowledge of the present state is only approximate, and not accurate No one can gainsay this. But it is not of much use in a world like this, in which the same antecedents never again concur, and nothing ever happens twice The physical axiom which has a somewhat similar aspect is "That from like antecedents follow like consequents.

We must often rely on statistical probabilities or approximate solutions reached by numerical techniques. What is new is that computers have allowed us to attack nonlinear problems numerically, in the process highlighting patterns of instability that have captured scientific and popular imaginations alike. The various fields of "nonlinear science"—such as those that deal with solitons, fractals, cellular automata, and self-organization systems far from thermodynamic equilibrium—have been stimulated and enhanced by powerful computer graphics techniques for scientific visualization or "mathematical experiments.

Furthermore, the very nature or definition of the system can change, and can do so rather abruptly, with transitions that usually depend on the parameters of the system more than on the variables within the system. In effect, parameters set the context, and the idealized boundaries they represent often contrast starkly with the indistinctness of boundaries in the real world. Chaotic systems have raised some fundamental questions about relationships among order, randomness, and predictability, especially since the equations that represent them can be surprisingly simple.

One of the first contemporary examples of chaos was encountered in meteorology in the early s when the applied mathematician Edward Lorenz set up three linked first-order differential equations in a computer model of weather development.

With certain parameters, the system proved so sensitive to the initial conditions that it was estimated that quite literally a butterfly flapping its wings in one part of the world would be sufficient to cause a major storm to emerge somewhere else.

An arbitrarily small change could generate an entirely different history for the system. Obviously, acquisition and management of the precision and the amount of input data necessary for exact prediction pose an impractical problem, but the large scale of the atmospheric system is actually not the issue.

The difficulty arises merely from multiplying pairs of the variables in two of the three coupled equations. The question is whether, according to Clausewitz, wars are also nonlinear systems. Is War Nonlinear for Clausewitz? In Chapter I of Book One, Clausewitz engages the reader with three increasingly sophisticated definitions of war, each one of which is prominently marked by nonlinearity.

The first definition is that war "is nothing but a duel [Zweikampf] on a larger scale The course of a given war becomes thereby not the mere sequence of intentions and actions of each opponent, but the pattern or shape generated by mutually hostile intentions and simultaneously consequential actions.

The contest is not the presence or actions of each opponent added together. It is the dynamic set of patterns made in the space between and around the contestants. This may not be immediately evident if we think of a duel with swords or with pistols. But it is obvious in a match between two wrestlers, which is how Clausewitz himself suggests we imagine the Zweikampf literally "two-struggle" between opponents in war: the bodily positions and contortions that emerge in wrestling are often impossible to achieve without the counterforce and counterweight of an opponent.

Only if war were some hermetically sealed phenomenon could its fundamental nature rage on unchecked. This would require that war a be an isolated and sudden act without prelude, b consist of a single decisive act or set of simultaneous ones, and c achieve a result perfectly complete in itself. But Clausewitz contends that an actual war never occurs without a context; that it always takes time to conduct, in a series of interactive steps; and that its results are never absolutely final—all of which impose restrictions on the analytically simple "pure theory" of war.

Any specific war is subject to historical contingencies; thus he concludes that the theoretical basis for prediction of the course of the war dissolves from any analytical certainties into numerical possibilities. The unique political situation is the context that bounds the system constituted by a given war. It must be considered carefully, Clausewitz argues, for the same political object can elicit differing reactions from different peoples, and even from the same people at different times Between two peoples and two states there can be such tensions, such a mass of inflammable material, that the slightest quarrel can produce a wholly disproportionate effect—a real explosion.

Yet the relationship is not static; it implies neither that the instrument is unchanging nor that the political goal or policy itself is immune to feedback effects.

Using another image of explosion, he argues: War is a pulsation of violence, variable in strength and therefore variable in the speed with which it explodes and discharges its energy.

War moves on its goal with varying speeds; but it always lasts long enough for the influence to be exerted on the goal and for its own course to be changed in one way or another That, however, does not imply that the political aim is a tyrant.

It must adapt itself to its chosen means, a process that can radically change it; yet the political aim remains the first consideration. The constant interplay is an interactive, feedback process that constitutes an intrinsic feature of war. War is, he says, a "true chameleon" that exhibits a different nature in every concrete instance. Then he concludes with a visual metaphor: "Our task therefore is to develop a theory that maintains a balance between these three tendencies, like an object suspended between three magnets.

The demonstration usually starts with a magnet pendulum hanging over one magnet; when the pendulum is pulled aside and let go, it comes to rest quickly. Positioned over two equally powerful magnets, the pendulum swings toward first one, then the other, and still settles into a rest position as it is captured by one of the points of attraction. But when a pendulum is released over three equidistant and equally powerful magnets, it moves irresolutely to and fro as it darts among the competing points of attraction, sometimes kicking out high to acquire added momentum that allows it to keep gyrating in a startlingly long and intricate pattern.

The probability is vanishingly small that an attempt to repeat the process would produce exactly the same pattern. Even such a simple system is complex enough for the details of the trajectory of any actual "run" to be, effectively, irreproducible. His final metaphor of Chapter 1, Book One captures this understanding perfectly. And what is needed is infinitely fine precision, for an immeasurably small change in the initial conditions can produce a significantly different pattern.

Nor is it possible to isolate the system from all possible influences around it, and that environment will have changed since the measurements were taken. Anticipation of the overall kind of pattern is possible, but quantitative predictability of the actual trajectory is lost. There are a number of interconnected reasons for the pendulum and magnets picture to be emblematic for Clausewitz, and all of them go to the heart of the problem of understanding what he meant by a "theory" of war.

First of all, the image is not that of any kind of Euclidean triangle or triad, despite its understanding as such by many readers.

Given his attacks on the formulation of rigidly "geometric" principles of war by some of his contemporaries, such an image would have been highly inapt. In fact, even the standard translation given above is too static, for the German original conveys a sense of on-going motion: "Die Aufgabe ist also, dass sich die Theorie zwischen diesen drei Tendenzen wie zwischen drei Anziehungspunkten schwebend erhalte.

The nature of war should not be conceived as a stationary point among the members of the trinity, but as a complex trajectory traced among them. The relationship of magnetism to electricity was just beginning to be clarified in a way that made it a cutting-edge concept for its time.

It is quite possible that he actually observed a demonstration of a pendulum and three magnets as envisioned in the metaphor, for he was a man of considerable scientific literacy.

EO 13526 PDF

## CLAUSEWITZ NONLINEARITY AND THE UNPREDICTABILITY OF WAR PDF

Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating. Clausewitz, On War, Book One, Chapter 1 Despite the frequent invocations of his name in recent years, especially during the Gulf War, there is something deeply perplexing about the work of Carl von Clausewitz In particular, his unfinished magnum opus On War seems to offer a theory of war, at the same time that it perversely denies many of the fundamental preconditions of theory as such—simplification, generalization and prediction, among others. Members of both groups sense that there is too much truth in what he writes to ignore him. Yet, as the German historian Hans Rothfels has bluntly put it, Clausewitz is an author "more quoted that actually read.

DICTIONNAIRE INFERNAL DE COLLIN DE PLANCY PDF

Modern chaos theorists like to emphasize this point. James Clerk Maxwell noted another chaotic concept over a century ago: When the state of things is such that an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable; but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the condition of the system is said to be unstable. It is manifest that the existence of unstable conditions renders impossible the prediction of future events, if our knowledge of the present state is only approximate, and not accurate…. No one can gainsay this.

DECONSTRUCTIVIST ARCHITECTURE PHILIP JOHNSON PDF

WritingContest Earlier this year, The Strategy Bridge asked university and professional military education students to participate in our first annual writing contest by sending us their thoughts on strategy. Now, we are pleased to present one of the essays selected for honorable mention, from Julie Anna Glascott of the U. In the legal trinity, the Charter of the United Nations holds the position of the government, the Geneva Conventions represents the people, and the Rules of Engagement cover the military. The legal discipline recognizes this potential energy, foresees the damage they can inflict if left untended, and seeks to tame these war horses through the application of the law of war. In On War, Clausewitz emphasizes the central role politics plays in war. The position that war should be entered into as a last resort to solve legitimate political disputes is the same position the United Nations took when it met at the end of World War II.

CANTOS NEOCATECUMENALES PDF

Technical Report Blair, J. This article deals with military genius from an historical and a classical theory perspective. The author modifies an approach developed by Carl von Clausewitz that makes use of theory as a framework for the study of history. Clausewitz used theory to study campaigns of Napoleon. Using seven qualities of military genius that Clausewitz lists in his ON WAR, a study was made to ascertain commonalities of behavior displayed by great battlefield generals.

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