Alan Beyerchen teaches German history in the Department of History at Ohio State University, where he is also affiliated with the Division for Comparative Studies and the Mershon Center. This paper is the first part of a larger project on the implications of nonlinear science for the liberal arts.
For their comments and encouragement, I am indebted to Kaushik Bagchi, Christopher Bassford, Lisa Barber, Leonard Jossem, Edward Merta, Raymond Muessig, Williamson Murray, Barbara J. Reeves, Oliver Schmidt, David Staley, Raymond Stokes, Ruud van Dijk, Paul Watkins, Barry Watts, Bostwick Wyman, David Young, Keith Zahniser; my students in Humanities 792 during Spring Quarter 1991; participants in colloquia held at the Institute for Contemporary History in Athens, Ohio, in November 1990 and November 1991; and members of the Implications of Nonlinear Studies Working Group at Ohio State University, especially Randolph Roth.
1. Carl von Clausewitz, On War, ed. and trans. Michael Howard and Peter Paret (Princeton: Princeton University Press, 1976). I use this edition for all quotations from On War in English unless otherwise indicated. For the German, see Vom Kriege, 18th ed. (complete edition of original text), ed. Werner Hahlweg (Bonn: Dümmlers, 1973). For other works in English, see von Clausewitz, Historical and Political Writings, ed. and trans. Peter Paret and Daniel Moran (Princeton: Princeton University Press, 1992).
2. Hans Rothfels, "Clausewitz," in Edward Mead Earle, ed., Makers of Modern Strategy (New York: Atheneum, 1969), p. 93. Christopher Bassford offers one impression of the reception of Clausewitz's work in his study of the Anglo-American reception of Clausewitz, 1815-1945 (Oxford: Oxford University Press, in press).
3. Raymond Aron, Clausewitz: Philosopher of War, trans. Christine Booker and Norman Stone (London: Routledge and Kegan Paul, 1983), p. 6. Original Penser la guerre, Clausewitz, 2 vols. (Paris: Gallimard, 1976). The suggestion has recently been made that the text was actually much more finished than has hitherto been thought: Azar Gat, "Clausewitz's Final Notes," Militärgeschichtliche Mitteilungen, Vol. 45, No. 1 (1989), pp. 45-50.
4. Peter Paret, Clausewitz and the State: The Man, His Theories and His Times (Princeton: Princeton University Press, 1985), pp. 8-9 (originally published by Oxford University Press, 1976). Azar Gat's argument, that Clausewitz's work is best understood as part of the Romantic backlash against the Enlightenment, also belongs to this approach. See Gat, The Origins of Military Thought: From the Enlightenment to Clausewitz (Oxford: Oxford University Press, 1989).
5. Michael I. Handel, War, Strategy and Intelligence (London: Frank Cass, 1989), p. 60.
6. Alan Beyerchen, "Nonlinear Science and the Unfolding of a New Intellectual Vision," Papers in Comparative Studies, Vol. 6 (1988-89), pp. 26-29.
7. The principle of proportionality means that if f is a function or an operator, a is a constant, and u is the system input (either a variable or itself a function), then f(au) = af(u). A more precise way of stating the principle of additivity is that the effect of adding the system inputs together first and then operating on their sum is equivalent to operating on two inputs separately and then adding the outputs together, so that f(u1 + u2) = f(u1) + f(u2). If f does not meet both of these conditions, it is nonlinear. In effect, if a system can be described adequately by the mathematical operations of addition, subtraction, multiplication by a constant, integration with respect to time or differentiation with respect to time, it can appropriately be thought of as linear. If it is necessary to multiply or divide variables by each other, raise to powers, extract roots, or integrate or differentiate with respect to dependent variables (that is, variables other than time), then the system is nonlinear.
8. The meaning of a "synergistic" interaction is indicated by the contrast between a common linear operation and a common nonlinear one. A linear operation such as multiplying by a constant obeys the principle of additivity: let f(u) = au, then f(u1 + u2) = a(u1 + u2) = au1 + au2, which is just f(u1) + f(u2) again. A nonlinear operation such as squaring, however, is different: let...