Classical mathematics concentrated on linear equations for a sound pragmatic reason, Ian Stewart noted: it couldn’t solve anything else. 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…. 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.
In describing war, Clausewitz resorts to a striking metaphor of nonlinearity:
In the last section of Chapter 1, Book One, he claims that war is “a remarkable trinity” (eine wunderliche Dreifaltigkeit) composed of (a) the blind, natural force of violence, hatred, and enmity among the masses of people; (b) chance and probability, faced or generated by the commander and his army; and (c) war’s rational subordination to the policy of the government.(28) Clausewitz compares these three tendencies to three varying legal codes interacting with each other (the complexity of which would have been obvious to anyone who lived under the tangled web of superimposed legal systems in the German area before, during, and after the upheavals of the Napoleonic years). 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.” (29) What better image could he have conjured to convey his insight into the profoundly interactive nature of war than this emblem of contemporary nonlinear science? (30)
Although the passage is usually taken to mean only that we should not overemphasize any one element in the trinity, Clausewitz’s metaphor also implicitly confronts us with the chaos inherent in a nonlinear system sensitive to initial conditions. 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. Eventually, the energy dissipates under the influence of friction in the suspension mountings and the air, bringing the pendulum’s movement asymptotically to rest. 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.
My claim here is not that Clausewitz somehow anticipated today’s “chaos theory,” but that he perceived and articulated the nature of war as an energy-consuming phenomenon involving competing and interactive factors, attention to which reveals a messy mix of order and unpredictability. His final metaphor of Chapter 1, Book One captures this understanding perfectly. The pendulum and magnets system is orderly, because it is a deterministic system that obeys Newton’s laws of motion; in the “pure theory” (with an idealized frictionless pendulum), we only need to know the relevant quantities accurately enough to know its future. But in the real world, “a world like this” in Maxwell’s phrase, it is not possible to measure the relevant initial conditions (such as position) accurately enough to replicate them in order to get the same pattern a second time, because all physical measurements are approximations limited by the instrument and standard of measurement. 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. (31) Clausewitz’s message is not that there are three passive points, but three interactive points of attraction that are simultaneously pulling the object in different directions and forming complex interactions with each other. 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.” (32) Literally: “The task is therefore that the theory would maintain itself floating among these three tendencies as among three points of attraction.” The connotations of schweben involve lighter-than-air, sensitive motion; a balloon or a ballerina “schwebt.” The image is no more static than that of wrestlers. 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.
Secondly, Clausewitz’s employment of magnetism is a typical resort to “high-tech” imagery. 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. (33) His famous incorporation of the notion of “friction,” also a high-technology concept for his day, is another example of this characteristic of his thought.
Thirdly, and perhaps most importantly, the metaphor offers us insight into a mind realistically willing to abandon the search for simplicity and analytical certainty where they are not obtainable. The use of this image displays an intuitive grasp of dynamic processes that can be isolated neither from their context nor from chance, and are thus characterized by inherent complexities and probabilities. It encodes Clausewitz’s sense of war in a realistic dynamical system, not an idealized analytical abstraction.