Clausewitz, Nonlinearity and the Unpredictability of War

Originally presented at Clauswitz.com

Carl von Clausewitz, 1780–1831

By Alan D. Beyerchen, Ohio State University

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 (1780–1831). 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. (1) The book continues to draw the attention of both soldiers and theorists of war, although soldiers often find the ideas of Clausewitz too philosophical to appear practical, while analysts usually find his thoughts too empirical to seem elegant. 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." (2) Lofty but pragmatic, by a theorist who repudiated conventional meanings of theory, On War endures as a compelling and enigmatic classic.

Just what is the difficulty with Clausewitz that makes his work so significant yet so difficult to assimilate? On War's admirers have sensed that it grapples with war's complexity more realistically than perhaps any other work. 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. (3)

A second approach to the question is exemplified by Peter Paret's stress on the changing interpretation of any significant author over time. Clausewitz's writings have suffered more distortions than most, Paret has suggested, because abstracting this body of work from its times does violence to its insistence on unifying the universal with the historical particular. Thus for Paret the literature on Clausewitz has been "fragmented and contradictory in its findings" because of our lack of historical consciousness. (4)

A third route to explaining the difficulties encountered in coping with On War has been typified by Michael Handel, for whom the issue is not so much changes in our interpretations as changes in warfare itself. Those aspects of On War that deal with human nature, uncertainty, politics, and rational calculation "will remain eternally valid," he contended. "In all other respects technology has permeated and irreversibly changed every aspect of warfare." (5) For Handel, the essential problem in understanding Clausewitz lies in our confrontation with a reality qualitatively different from his. 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. I am not arguing that reference to a few of today's "nonlinear science" concepts would help us clarify confusion in Clausewitz's thinking. 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.

The correctness of Clausewitz's perception has both kept his work relevant and made it less accessible, for war's analytically unpredictable nature is extremely discomfiting to those searching for a predictive theory. An approach through nonlinearity does not make other reasons for difficulty in understanding On War evaporate. It does, however, provide new access to the realistic core of Clausewitz's insights and offers a correlation of the representations of chance and complexity that characterize his work. Furthermore, it may help us remove some unsettling blind spots that have prevented us from seeing crucial implications of his work.

What is "Nonlinearity"?

"Nonlinearity" refers to something that is "not linear." This is obvious, but since the implicit structure of our words often reveals hidden habits of mind, it is useful to reflect briefly on some tacit assumptions. 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).(6)

The result has been an authoritative guide for our Western intuition, but one that is idealized and liable to mislead us when the surrounding world and its messy realities do not fit this notion. 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.

"Linear" applies in mathematics to a system of equations whose variables can be plotted against each other as a straight line. 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. (7)

Nonlinear systems are those that disobey proportionality or additivity. 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. (8) If the behavior of a system can appropriately be broken into parts that can be compartmentalized, it may be classified as linear, even if it is described by a complicated equation with many terms. 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. (9) But often the equations needed to describe the behavior of nonlinear systems over time are very difficult or impossible to solve analytically. 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. (10)

In contrast, sophisticated analytical techniques for solving linear equations have been developed over the centuries, becoming the preferred tools in nearly all technical fields by the latter portion of the nineteenth century. 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." In commonly used terms, one thus goes from equations that "blow up" to those that are "well-behaved." In fact, mathematician Ian Stewart has noted:

Classical mathematics concentrated on linear equations for a sound pragmatic reason: it couldn't solve anything else.... 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. (11)

As is often the case, reality has been selectively addressed in order to manipulate it with the tools available. 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.(12)

The resort to idealized linearizations has been legitimated by the assumption, increasingly dubious, that reality is ultimately simple and stable. 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....it is a metaphysical doctrine that from the same antecedents follow the same consequents. 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." But here we have passed from sameness to likeness, from absolute accuracy to a more or less rough approximation. (13)

Thus Maxwell held that analytical mathematical rules are not always reliable guides to the real world. We must often rely on statistical probabilities or approximate solutions reached by numerical techniques.

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