McNair Paper Number 52, Chapter 10, October 1996

All but one of the historical and conceptual elements necessary for this essay's fourth and final task-recasting general friction in modern terms-have now been introduced. The sole outstanding item is the concept of nonlinearity as it has come to be understood in fields like mathematics and physics since the early 1960s. By revealing how small differences in inputs can make large differences in outcomes, nonlinear dynamics will not only complete the task begun in chapter 6 of building indirect arguments for friction's undiminished persistence in future war, but furnish the last conceptual elements needed to update and extend Clausewitz's original concept.

Nonlinear science has been deferred to the end mostly to avoid burdening the exposition any earlier than necessary with a subject that various readers may find unfamiliar, difficult to grasp, or simply alien to the subject at hand. As mentioned in chapter 4, Clausewitz himself was not the least bit shy about appropriating concepts like friction and center of gravity from the physics of his day to illuminate the phenomena of war. Furthermore, in the winter 1992/93 issue of International Security Alan Beyerchen argued convincingly that Clausewitz himself not only perceived war "as a profoundly nonlinear phenomenon . . . consistent with our current understanding of nonlinear dynamics," but that his use of a linear approach to the analysis of war "has made it difficult to assimilate and appreciate the intent and contribution of On War."(Note 1) This author's experience has also confirmed that attempts to apply the ideas of nonlinear science to the study of war continue to be met with resistance, if not incomprehension, for precisely the reason Beyerchen cited: the widespread predominance of linear modes of thought. Hence, it seemed wise to defer nonlinearity until all the other evidence and arguments suggestive of its relevance had been deployed.

What is nonlinear science all about? The core ideas are not hard to describe. Nonlinear dynamics arise from repeated iteration or feedback. A system, whether physical or mathematical, starts in some initial state. That initial state provides the input to a feedback mechanism which determines the new state of the system. The new state then provides the input through which the feedback mechanism determines the system's next state, and so on. Each successive state is causally dependent on its predecessor, but what happens to the system over the course of many iterations can be more complex and less predictable than one might suppose. If the nonlinear system exhibits sensitive dependence on initial or later states, then at least three long-term outcomes are possible: (1) the system eventually settles down in some single state and remains there despite further iterations (long-term stability); (2) the system settles on a series of states which it thereafter cycles through endlessly (periodic behavior); or, (3) the system wanders aimlessly or unpredictably (so-called "chaotic" behavior). In the third case, detailed predictability of the actual state of the system can be lost over the course of a large enough number of iterations. (Note 2) Chaotic behavior, however, should not be confused with randomness. Successive tosses of a coin remain the exemplar of a random process; if the coin is not biased, then the probability of either "heads" or "tails" on one's next toss is 50 percent. The paradigmatic example of a chaotic process, by contrast, is a "flipperless" pinball machine of infinite length. Edward Lorentz has characterized its behavior as being sensitively dependent on a single "interior" initial condition, namely the speed imparted to the pinball by the plunger that players use to put each ball into play. (Note 3) On this view, chaos may be described as "behavior that is deterministic, or is nearly so if it occurs in a tangible system that possesses a slight amount of randomness, but does not look deterministic." (Note 4)

The "mathematics of chaos" that has been used since the early 1960s to explain the sort of nonlinear dynamics exemplified by Lorentz's infinite pinball machine can be easily demonstrated using a personal computer or a programmable calculator to explore a simple nonlinear equation such as the "logistic mapping," xn+1 = kxn(1 - xn) (where the variable x is a real number in the interval [0, 1], and the "tunable" constant k can be set between 1 and 4). (Note 5) Depending on the choice of k, the logistic mapping exhibits all three of the long-term outcomes just described: stable, periodic, and chaotic behavior. Unfortunately, those uncomfortable with mathematics and programming languages (however "user friendly") are easily deterred from such "experiments" even though, for the very simplest nonlinear functions, the requisite calculations do not require more than the arithmetic of real numbers. Yes, as a practical matter the amount of repetitive number crunching involved in any serious exploration demands machine assistance, and one does have to be meticulous about the number of places to the right of the decimal point to which calculations are carried. (Note 6) Still, the mathematics of elementary nonlinear functions like the logistic mapping are readily accessible to anyone willing to invest a modest amount of time and effort. (Note 7)

Mathematics aside, nonlinearity has a crucial contribution to make toward completing the case for the view that general friction will persist more or less undiminished in future war regardless of technological developments. Specifically, the role nonlinearity plays is to close the door once and for all to the sort of fully predictable (at least in principle), "clockwork" universe advocated most persuasively during Clausewitz's lifetime by the mathematical physicist Pierre Simon de Laplace (1749-1827).

The idea that the subsequent motions and effects of physical phenomena could be completely predicted on the basis of their earlier states was first argued at length in the 1750s by the Jesuit priest Roger Boscovich (1711-1787). (Note 8) However, it was Laplace who, more than anyone else, seemed to make good on this heady promise. At an early age, he set himself the task of tying up the loose ends of the Newtonian enterprise. Using the improved calculus developed by various colleagues, particularly Joseph-Louis Lagrange, Laplace was widely perceived to have "removed all the known errors from, and explained all known anomalies in, the Newtonian cosmology and physics." (Note 9) Whereas Isaac Newton (1642-1727) had never been fully convinced of the stability of the solar system, suggesting that it might require some divine correction now and again, (Note 10) Laplace eventually claimed to have proven "that every known secular variation, such as the changing speeds of Saturn and Jupiter, was cyclic and that the system was indeed entirely stable and required no divine maintenance." (Note 11) Laplace also completed the theory of tides and solved another of Newton's famous problems, the deduction from first principles of the velocity of sound in air. (Note 12) This unbroken string of triumphs in removing all the known anomalies in Newtonian mechanics led Laplace to conclude that the universe was rigidly deterministic in the spirit of Boscovich.

Laplace's clearest expression of this wholesale mechanization of the world picture can be found in his 1814 Essai philosophique sur les probabilitJs [Philosophical Essay on Probabilities], which is a lucid nontechnical introduction to his principal work on the laws of chance, the Th(orie analytique des probabilitiJs. (Note 13) "Given for one instant," he wrote, an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it, an intelligence sufficiently vast to submit these data to analysis, it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. . . . The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that due to our ignorance. (Note 14)

On Laplace's understanding of reality, the operation of the universe, down to the most minute details and the smallest particles, is strictly determined by quantitative, predictive, mathematical laws. The world is quite literally a giant clockwork. To Laplace's so-called "demon", meaning a sufficiently vast intelligence with sufficiently accurate and complete data about the universe at any point in time, all past and future states are calculable. Regardless of human ignorance or shortcomings in such matters, the mathematical laws of nature leave nothing to chance, not even combat outcomes or the emergence and evolution of life on Earth. (Note 15) The difficulty with this Laplacian outlook is not, of course, its plausibility or enduring appeal. By the beginning of the twentieth century the vast majority of working physicists accepted "Laplacian determinism", meaning causality plus long-term predictability, as a well-established scientific fact, and many people still do so today. The problem is that the universe we happen to inhabit is not broadly deterministic in the full sense Laplace meant, not even quantitative domains like physics and pure mathematics. There are processes like the Earth's tides and solar eclipses that can, barring unforeseen perturbations, be highly periodic and, hence, precisely predictable and precisely retrodictable across the majority of their dynamic range. However, there are also processes such as the evolution of the precise weather conditions at a given location on the Earth (temperature, winds, humidity, cloud conditions, the presence or absence of precipitation, etc.) that are so sensitive to the slightest disturbances or differences in initial conditions that detailed predictability is generally lost over time spans as short as two weeks. (Note 16) While strongly nonlinear processes like long-term weather prediction are "deterministic" in the restricted or narrow sense of being causally determined, they predominantly exhibit long-term unpredictability that is inconsistent with full-blown "Laplacian determinism." At best, these sorts of highly nonlinear processes harbor occasional islands of predictable behavior within a sea of unpredictability. (Note 17)

This untidy situation reared its head early in the development of Newtonian physics. During the drafting of the first edition of Newton's Principia Mathematica, which appeared in 1687, Newton ran into difficulties moving from the problem of two bodies mutually attracted to one another by gravitation, which he easily solved, to the problem of describing the dynamics of many such bodies (the many-body problem). (Note 18) In the summer of 1694 he returned to this problem in the form of the dynamics of the moon moving about the Earth, which was in turn orbiting the sun (the three-body problem). (Note 19) Once again, though, Newton's achievements fell short of his aspirations. In retrospect, Newton's difficulties with the irregularities of lunar motion are wholly understandable. As we now know, the three-body problem "does not admit a general analytic solution." (Note 20) So Newton's renewed efforts in 1694-1695 to find such a solution to the "inequalities" in the moon's orbit were doomed to failure, just as they had been in 1685-1686 when he was laboring to complete the first edition of Principia Mathematica. The problem he labored to solve was literally an impossible one in the sense in which Newton aspired to solve it. (Note 21)

Again, Laplace thought he had proved that the observed perturbations of the planets were periodic rather than cumulative: they would "repeat themselves at regular intervals, and never exceed a certain moderate amount," thus substituting dynamic stability for divine intervention. (Note 22) Unfortunately, physicists no longer consider Laplace's proof of the stability of the solar system rigorous, and "all attempts to make it so have failed." (Note 23) Indeed, in recent years evidence has been accumulating to show that the orbit of Pluto, and the solar system as a whole, appear to be unstable or chaotic on time scales of 4-20 million years. (Note 24)

The first individual to recognize that the three-body problem included unstable or nonperiodic behavior was Poincar(. In an 1890 essay he showed that a gravitational system involving only three bodies would not always give rise to predictable or periodic motion. Specifically, in the case of an idealized form of the three-body problem in which the third body is vastly smaller than the other two, PoincarJ discovered motion so complex and irregular, "homoclinic tangles" to use the technical term, that he did not even attempt to draw the corresponding figure. (Note 25) This "chaotic" behavior is "fundamental" or built in; neither "gathering more information," nor processing it better, will eliminate the unpredictability. (Note 26) As Poincar( wrote in 1908 of chance in the sense of causes too small to be discernible giving rise to large, noticeable effects:

A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. (Note 27)

The seminal experiment that first showed chaos had physical as well as mathematical reality was carried out on superfluid helium by Albert Libchaber and Jean Maurer at the Ecol( Normale in the mid-1970s. As Mitchell Feigenbaum declared when he saw Libchaber's jagged graph of a period-doubling cascade in the waves traveling up and down the vortices that heating created in Libchaber and Maurer's tiny experimental device: "The moment Albert did his experiment, and chaos showed up in a real thing, not to mention in a fluid, it completely changed the reaction of the [physics] world [to chaos]." (Note 28) Since then, scientists have confirmed the kind of chaotic(Note 29) or unpredictable behavior that is the essence of contemporary nonlinear dynamics in a wide range of physical phenomena, including electronic circuits, mechanical and electromechanical systems, hydrodynamics, acoustics, optics, solid-state physics, biology, and even ecology. (Note 30) Chaotic behavior, which lies between order and disorder, is quite widespread in the real world, if not fairly ubiquitous. Granted, stable and periodic dynamics, as exemplified by the Earth's tides, are also widespread. Even a "nonlinear" system like the logistic mapping exhibits stable or periodic behavior over most values of the tunable parameter k; only for k greater than 3.57 or so(Note 31) (depending on the computer and program being used) does truly unpredictable, chaotic behavior ensue. That said, pockets or areas of chaos infect a wide range of physical phenomena, and within those "chaotic regions" detailed predictability is lost. In this sense, chance abounds there.

What bearing has the discovery of nonlinearity's physical reality on friction in future war? Clausewitz wrote of war that no other human activity "is so continuously or universally bound up with chance" and suggested that war most resembles a game of cards in its sensitivity to chance. (Note 32) He and Scharnhorst believed that chance (Zufall(Note 33)) could not be eliminated from military affairs. Clausewitz identified chance events as an explicit source of general friction, although he did not (and could not) explain how small differences from what is expected or predicted could potentially turn success into failure and vice versa. What the empirical fact of nonlinear dynamics does is to explain how such small differences or "chance" occurrences of "the kind you can never really foresee" can give rise to long-term unpredictability. (Note 34) Laplace believed that human judgments about chance and probability were simply the result of ignorance. In many cases, including games of chance like cards, he felt that we simply do not know enough, or have adequate calculation time, to predict the outcomes. Expressed in the language of nonlinear dynamics, Laplace's presumption is that human ignorance prevents us from completely eliminating tiny differences between our representations of phenomena and their actuality. If, however, these small differences cannot be eliminated, then nonlinear dynamics explain how global or macroscopic unpredictability can arise from the structural dynamics of iterated feedback when the feedback function exhibits, in at least some part of its domain, extreme sensitivity to initial or later conditions. Since there is increasingly persuasive evidence from a number of fields, especially mathematical logic and physics, that any coherent or formal system we develop "to represent or deal with large portions of reality will at best represent or deal with that reality incompletely or imperfectly," it appears that these differences and mismatches cannot be eliminated. (Note 35) Consequently, the existence of nonlinear systems confirms some of the deepest insights Clausewitz and Scharnhorst had into the nature of combat processes and the fundamental role of chance in those processes. It also suggests that unforeseen and unforeseeable differences in initial or later conditions, which, on present evidence, cannot be wholly eliminated even by Laplace's demon, allow us to subsume chance within the framework of nonlinearity.

How might these insights into the connections between nonlinearity and at least two components of general friction help us recast the concept in more modern terms? Perhaps the most thought-provoking piece of research bearing on this question was carried out by James Dewar, James Gillogly, and Mario Juncosa of the RAND Corporation. Their aim was to see if nonlinear effects arising from "mathematical chaos" could be demonstrated in a simple computer model of land combat. (Note 36) Their point of departure was the fact that computer models of combat often produce "nonintuitive" results, by which they meant "nonmonotonicities in which a capability added to the side of one combatant leads to a less-favorable result for that side." (Note 37) Since such unruly behavior in computer models can arise from other numerous sources than nonlinearityCincluding round-off error, the wrong step size (time-step granularity), or delayed feedbackCthey chose a class of very elementary "Lanchester square law," attrition models that were designed to facilitate the elimination of such alternative sources of nonlinear behavior. (Note38) As the exemplar of this class of combat models in table 1 indicates, they also included reinforcement criteria based on the state of the battle at the end of a given step. This feature was a mathematical surrogate for human decision making (or intervention) in the battle in response to the amount of attrition the two sides had suffered to that point. What would constitute intuitive and unintuitive, or monotonic and nonmonotonic, behavior in such a model? Dewar, Gillogly, and Juncosa's 1991 paper offers the following characterization. As an example of monotonic behavior, fix Blue's initial strength at 830 troops and allow Red's initial strength to vary from 1,500 to 3,500 troops. Each value of Ro represents a separate "battle" from which a winner is determined when one side or the other withdraws at the end of some step. In this particular case, the behavior is exactly what one would intuitively expect and desire. Blue wins from Ro = 1,500 until Ro = 2,696; thereafter, Red wins all the battles; the situation appears to be entirely linear. However, with Bo = 500 and Ro varying from 700 to 1,800, seriously non-monotonic behavior ensues: ARed wins when starting with as few as 884 troops, loses when starting with as many as 1623 troops, and suffers a surprising number of reversals of fortune in between." (Note 39) This unpredictable flip-flopping of which side wins as Ro varies from 884 to 1,623 troops is what constitutes unintuitive or nonmonotonic behavior.

What did Dewar, Gillogly, and Juncosa conclude about mathematical chaos in this type of model? After eliminating computational and input sources of nonintuitive behavior such as roundoff error and time-step granularity, they were able to demonstrate non-monotonicity in a version of the underlying model they termed the "force-ratio-only mapping":

Specifically, in a simple model with unlimited reinforcements, we have shown for a specific decision (when to call in battle reinforcements) based on the state of the battle (specifically, on the ratio of the opposing forces numerical strengths) that the underlying dynamics of the model satisfy four mathematical conditions characteristic of chaotic systems. . . . The "misbehavior" of this model is structural rather than computational, it is in the nature of the phenomenon being modeled, decisions based on the state of the battle. (Note 40)

They further noted that if the reinforcement decisions are "scripted" so that they are no longer a function of the state of the battle, then "the nonlinearities, the chaos, and the nonmonotonicities generally disappear." (Note 41)

What implications, if any, can we draw from this research concerning friction in future war? Dewar, Gillogly, and Juncosa were reluctant to make any inferences about real war as opposed to war on paper based on the mathematical features of the force-ratio-only mapping: historic battles have been known to hinge on very subtle effects of decisionmaking and have been described as "chaotic." This research holds out the promise that mathematical chaos and the chaos of war might be related. As a strong caveat here, it is too easy to presume that they are necessarily connected. Whether or not the behavior in our simple model is akin to behavior in a real battle is an interesting question but one that requires serious thought and research. (Note 42)

This essay has sought to furnish precisely the serious thinking and historical research that Dewar, Gillogly, and Juncosa recommended prior to linking mathematical chaos in their simple combat model to something like the general friction manifested by real war. Given the complexities of actual combat, this sort of linkage need not insist that decision making in the force-ratio-only mapping faithfully models any concrete instances of decision making in actual combat. Nevertheless, nonlinear behavior has been confirmed in a wide range of physical processes. Thus, the demonstration of non-monotonicity in a mathematical model of combat, however simplified, is certainly suggestive as to how effectiveness, results, and overall outcomes in war might be unpredictable.

One can go further. As first noted in chapter 4, Clausewitz insisted in On War's second book that the very nature of two-sided interaction between opposing sides was bound to make interaction unpredictable. (Note 43) Nonlinear dynamics in general, and Dewar, Gillogly, and Juncosa's results with their force-ratio-only mapping in particular, reveal how inspired Clausewitz's observation was. It is no longer a mystery to explain how unpredictability in war can arise from human purposes and decisions without any suspension of causality. In nonlinear systems that are sensitively dependent on initial or later conditions, the interaction of iterative feedback can so magnify the smallest of differences, including those stemming from human decisions, as to render combat outcomes structurally unpredictable. In other words, while technological advances might temporarily mitigate general friction, they could neither eliminate it nor substantially reduce its potential magnitude.

With these insights added to those of earlier sections, general friction can now be reconstructed in modern terms. The presumption underlying this specific reconstruction is that general friction ultimately arises from three elementary sources:

! Human beings and their purposes

! The spatial-temporal inaccessibility of key information in military affairs

! The unpredictability of chaotic processes.

This hypothesis suggests the following list of general friction's sources as a late-20th-century alternative to the eight "Clausewitzian" sources listed at the end of chapter 4:

1. Constraints imposed by human physical and cognitive limits, whose magnitude or impact are inevitably magnified by the intense stresses, pressures, and responsibilities of actual combat

2. Informational uncertainties and unforeseeable differences stemming, ultimately, from the spatial-temporal dispersion of information in the external environment, in military organizations, and in the mental constructs of individual participants

3. The structural nonlinearity of combat processes which can give rise to the long-term unpredictability of results and outcomes by magnifying the effects of unknowable small differences and unforeseen events (or, conversely, producing negligible results from large inputs).

At least three observations are in order to help explain and motivate this reconstruction of general friction. First, this revised list is shorter than any of those offered in chapter 4. The reason is that the reconstruction attempts to focus on the most fundamental or elementary sources of friction. Derivative sources, such as poor intelligence or human reactions to the imminent threat of death or mutilation inherent in combat, are clearly implied, but need not be called out separately. Given the emphasis that Scharnhorst and Clausewitz both placed on chance, (Note 44) it may be surprising to see chance, too, apparently reduced to a derivative source of friction. In this instance, however, a more accurate description would be that chance has been distributed across all three of general friction=s sources. The participation of finite human beings, the distribution of information in war, and unpredictabilities of nonlinear processes give rise to surprising and unforeseen discrepancies that cannot be eliminated, and we are right to gather them, in all their diverse guises, under the notion of chance. Poincar('s view was that even if human ignorance could be set aside, chance would still manifest itself in several guises. These unavoidable manifestations of chance include imperceptible small causes that have large and noticeable effects, the reverse in which great causes yield small differences, and causes either too complex or too numerous for us to grasp. (Note 45) When nonlinear processes amplify such structural differences and uncertainties, they render large-scale results unpredictable and give rise to ever larger differences between what we expect and what happens that, in turn, feed back into nonlinear processes. Chance in its various guises, therefore, is rendered pervasive but, once again, without letting go of causality.

Second, the reconstructed list of friction's sources suggests a way of dealing with a recurring objection to the entire concept of general friction. The objection, which has been consciously ignored to this point, is that the unified concept of a general friction (Gesamtbegriff einer allgemeinen Friktion) embraces so much of war that it does not provide a very precise instrument for analyzing the phenomena at issue. (Note 46) If we return to the notion of general friction as the entire panoply of factors that distinguish real war from war on paper, Clausewitz's reason for pulling so diverse a collection of things together under a single concept is clear: general friction was the bridge between war in the abstract and war in reality. Still, the objection suggests that some parsing of his unified concept into separable-but-fundamental components could prove fruitful. Whether the three proposed in this section will do so remains to be seen. However, constraints on military operations stemming from the physical and cognitive limits of human participants, uncertainties rooted in the spatial-temporal distribution of the information on which action in war is unavoidably based, and the unpredictabilities inherent in nonlinear dynamics seem more precise and, potentially, more promising as conceptual tools than any of the decompositions of general friction in chapter 4.

Third, the principal merit of the late 20th-century recasting of Clausewitzian friction proposed in this section is the transparency it gives to general friction's place in military affairs. Human limitations, informational uncertainties, and nonlinearity are not pesky difficulties better technology and engineering can eliminate, but built-in or structural features of the violent interaction between opposing groups we call war.

As a consequence, general friction's potential to dominate outcomes, as Proposition I in chapter 8 implied, seems likely to persist regardless of what changes technological advances bring to pass in the means of combat. Why? Because at least one of the root sources of Clausewitzian friction lies, when all is said and done, not in the weapons we wield but in ourselves. The presence of humans in the loop, with all the diverse frailties, physical and cognitive limits, purposes, and decisions which their presence and participation entail, alone seems sufficient to render Clausewitzian friction impossible to eliminate entirely and, in all likelihood, extraordinarily difficult to reduce greatly in any permanent sense. At the same time, human participation cannot be isolated from the spatial-temporal distribution of information on which combatants act, and those actions, in turn, involve processes that can be highly nonlinear. On this reconstruction of Clausewitz's concept, therefore, general friction arises, to paraphrase Roberta Wohlstetter, from fundamental aspects of the human condition and unavoidable unpredictabilities that lie at the very core of combat processes.