McNair Paper Number 52, Chapter 10, Notes, October 1996

1. Alan Beyerchen, "Clausewitz, Nonlinearity, and the Unpredictability of War," International Security (Winter 1992/93): 87-88.

2. While this description of nonlinear dynamics and chaos is adequate for purposes of reconstructing general friction, it overlooks definitional controversies and details that are far from trivial, particularly to mathematicians. For example, the "absence of periodicity has sometimes been used instead of sensitive dependence [on initial conditions] as a definition of chaos" [Edward Lorentz, The Essence of Chaos (Seattle: University of Washington, 1993), 20]. However, if a system is not compact, meaning that "arbitrarily close repetitions" need never occur, "lack of periodicity does not guarantee that sensitive dependence is present" (ibid., 17 and 20). Readers interested in exploring various definitions of chaos should consult: Lorentz, 3-24 and 161-179, and James A. Dewar, James J. Gillogly, and Mario L. Juncosa, "Non-Monotonicity, Chaos, and Combat Models," RAND Corporation, R-3995-RC, 1991, 14-16.

3. Lorentz, Essence, 10 and 23-24.

5. Heinz-Otto Peitgen, Harmut Jhrgens, and Dietmar Saupe, Chaos and Fractals: New Frontiers of Science (New York: Springer-Verlag, 1992), 585-587; also, Ian Stewart, Does God Play Dice? The Mathematics of Chaos (Oxford: Basil Blackwell, 1989), 155-164. For a rigorous treatment of the logistic mapping (alias the "quadratic iterator"), see chapter 11 in Peitgen, Jhrgens, and Saupe; as they note, the final-state or "Feigenbaum diagram" (after the physicist Mitchell Feigenbaum) for the logistic mapping "has become the most important icon of chaos theory" (587). Mathematica's built-in function NestList renders the research mathematics needed for a basic understanding of the logistic mapping almost trivial. However, the same calculations can be easily carried out on a calculator like the Hewlett Packard HP-48SX.

6. Edward Lorentz's discovery of chaos was precipitated by an attempt to take a research shortcut. At the time he was trying to develop a weather model and making successive runs on a computer. The output of the model was calculated weather patterns over a period of "months." At a certain stage in the research he made some adjustments to the model and then, rather than completely running it from the beginning as he had done previously, Lorentz tried to save time in generating a new run by entering the weather data from midway through the last run to three decimal places rather than six. As a result of this minor change, Lorentz saw his weather "diverging so rapidly from the pattern of the last run that, within just a few months, all resemblances had disappeared" (James Gleick, Chaos: Making a New Science (New York: Viking, 1987), 15-16).

7. Readers interested in nonmathematical introductions to nonlinear dynamics may wish to consider John Gleick, Chaos: Making a New Science, John Briggs and F. David Peat, Turbulent Mirror, or Edward Lorentz, The Essence of Chaos.

8. For the crucial excerpt from Boscovich"s most famous and widely read work, his 1758 Philosophiae Naturalis Theoria Reducta ad Unicam Legem Virium in Natura Existentium [A Theory of Natural Philosophy Reduced to a Law of Actions Existing in Nature], see John D. Barrow, Theories of Everything: The Quest for Ultimate Explanation (New York: Fawcett Columbine, 1991), 54. While it is not certain how far the Theoria influenced the subsequent development of atomic theory, this work was widely studied, especially in Britain where Michael Faraday, Sir William Hamilton, James Clerk Maxwell, and Lord Kelvin stressed the theoretical advantages of the Boscovichian atom over rigid atoms [Lancelot L. Whyte, Boscovich, Roger Joseph, in The Encyclopedia of Philosophy, ed. Paul Edwards, vol. 1 (New York: Macmillan and the Free Press, 1967), 351].

9. R. HarrJ, "Laplace, Pierre Simon de,"The Encyclopedia of Philosophy, ed. Paul Edwards, vol. 4 (New York: Macmillan and the Free Press, 1967), 392.

10. Isaac Newton, Optics, in Great Books of the Western World, vol. 34, 542; also, Richard S. Westfall, Never at Rest: A Biography of Isaac Newton (New York: Cambridge University Press, 1980), 777-778.

13. James R. Newman, "Commentary on Pierre Simon de Laplace," The World of Mathematics: A Small Library of the Literature of Mathematics from A=h-mosJ the Scribe to Albert, vol. 2 (Redmond, WA: Tempus,1988; reprint of 1956 edition), 1296. Laplace=s Th(orie analytique appeared in 1812 and was dedicated to Napoleon.

14. Pierre Simon de Laplace, "Concerning Probability," The World of Mathematics, vol. 2, 1301-02; the cited selection is an excerpt from Laplace's 1814 Essai philosophique sur les probabiliti(s.

15. For Laplace, the fundamental objects of probability theory were not "chance events, but degrees of belief" necessitated by the imperfections of human knowledge [Theodore M. Porter, The Rise of Statistical Thinking: 1820-1900 (Princeton, NJ: Princeton University Press, 1986), 72].

16. David Ruelle, Chance and Chaos (Princeton, NJ: Princeton University Press, 1991), 45-47; Lorentz, 77-110.

17. While the contrast between the gross behavior of tides and detailed perturbations in the weather at a given location is legitimate, it is not the entire story. The earth's weather exhibits predictable regularities such as higher average temperatures in the summer than during winter, and even something as "regular" the times of future sunrises can be delayed or advanced a millisecond or so as a result of measurable decreases or increases in the Earth's speed of rotation. Lorentz, therefore, has a fair point in noting that, "when we compare tidal forecasting and weather forecasting, we are comparing prediction of predictable regularities and some lesser irregularities with prediction of irregularities alone" (79). In this sense, nonlinear science is not so much an alternative to the classic physics of Laplace as an expansion that puts irregular processes on equal footing with regular ones. In this regard, Lorentz is on record as objecting to the presumption that regular behavior is more fundamental or "normal" than chaotic behavior (69).

18. Westfall, 430. A second edition of Principia Mathematica appeared in 1713 and a third in 1726, the year before Newton's death.

21. The history of mathematics is littered with impossible problems. A classic example from antiquity is the problem posed by the Greeks of "squaring the circle, that is, constructing a square with an area equal to that of a given circle" with the aid of a straight-edge and compass [Howard DeLong, A Profile of Mathematical Logic (New York: Addison-Wesley Publishing, 1970), 29]. While some of the Greeks suspected that this problem was impossible under the stated condition, it was not until the 19th century that a proof was finally given which showed, once and for all, that such a construction is logically impossible (ibid., 69). The impossibility in this case was tied to the specified means, and the calculus of Newton and Gottfried Wilhelm Leibniz provided an alternative method that allowed the circle to be squared, if not exactly, at least to whatever degree of precision might be practically required. By comparison, G(del's incompleteness theorems present a more severe type of mathematical impossibility because there do not appear to be, even in principle, alternative means to the deductive methods of inference to which these theorems apply (ibid., 193). The "impossibility" associated with the three-body problem is somewhat different. The impossibility is not that there are no solutions at all, or even no stable ones, but that in certain regimes the dynamics become so unstable that future states of the system cannot be predicted even approximately.

22. Laplace quoted in Newman, 1293.

23. Tony Rothman, "God Takes a Nap: A Computer Finds that Pluto's Orbit is Chaotic," Scientific American, October 1988, 20.

24. Gerald Jay Sussman and Jack Wisdom, ANumerical Evidence That the Motion of Pluto Is Chaotic,@ Science, 22 July 1988, 433; also Sussman and Wisdom, AChaotic Evolution of the Solar System,@ Science, 3 July 1992, 56.

25. Stewart, Does God Play Dice? The Mathematics of Chaos, 70-72. For Lorentz's discussion of George William Hill's "reduced" version of the three-body problem, see Lorentz, 114-120 and 192-193.

26. Crutchfield, Farmer, Packard, and Shaw, "Chaos," Scientific American, December 1986, 46.

27. Henri Poincar(, Science and Method, trans. Francis Mailand (New York: Dover, 1952), 67-68. This passage has been often quoted by nonlinear dynamicists See, for example, Crutchfield, Farmer, Packard, and Shaw, 48. Insofar as it recognizes other sources of "chance" than human ignorance, the passage constitutes an explicit rejection by Poincar( of Laplace's "demon" or vast intelligence. Does it also recognize "chaos"? As Lorentz has noted, Poincar('s work on the three-body problem was not begun in search of chaos. Instead, Poincar( sought "to understand the orbits of the heavenly bodies" and found chaos in the process (Lorentz, 121). And while we cannot be certain, Awe are left with the feeling that he must have recognized the chaos that was inherent in the equations with which he worked so intimately" (ibid., 120).

28. Madhusree Mukerjee, "Profile: Albert Libchaber's Seeing the World in a Snowflake," Scientific American, March 1996, 42. Libchaber and Maurer first published their discovery of chaotic behavior in superfluid hydrogen in 1978 [Hao Bai-Lin, Chaos (Singapore: World Scientific Publishing Company, 1984) 559].

29. Tien-Yien Li and James A. Yorke first introduced the term "chaos" in their 1975 paper, "Period Three Implies Chaos," to denote the unpredictability observed in certain "deterministic" but nonlinear feedback systems (Bai-Lin, 3 and 244). Their choice of this term remains, at best, mischievous because it tends to blur the notion of randomness with that of local unpredictability within predictable global bounds. For example, the well-known "chaotic" attractor named after Edward Lorentz is unpredictable in that, "even when observed for long periods of time," it does not ever appear to repeat its past history exactly; yet the beautiful "owl's mask" pattern it generates in state space is by no means wholly random (Lorentz, ADeterministic Nonperiodic Flow," in Bai-Lin, 282 and 289). In their 1975 paper, however, Li and Yorke insisted on using the term "chaotic" to describe the nonperiodic dynamics of certain equations despite advice from colleagues that they "choose something more sober," and the term has stuck (Bai-Lin, 245; Gleick, 69).

30. Bai-Lin, 67-71; for a more recent example of chaotic in physical systems, see "Chaotic Chaos in Linked Electrical Circuit," Science News, 14 January 1995, 21-22.

31. Peitgen, J(rgens, and Saupe give an estimated value of 3.5699456. . . for the onset of chaos in the quadratic iterator (Chaos and Fractals, 612).

32. Carl von Clausewitz, On War, trans. Peter Paret and Micael Howard (Princeton, NJ: Princeton University Press, 1976), 85 and 86.

33. Carl von Clausewitz, Vom Kriege, ed. Werner Hahlweg (Bonn: Dhmmler, 1980 and 1991), 207 and 237.

35. John R. Boyd, "Conceptual Spiral," unpublished briefing, July-August 1992, slides 14 and 31. Boyd lists nine features of the various theories, systems, and processes we use to make sense of the world that, unavoidably, generate mismatches or differences, whether large or small, in initial or later conditions. These features include: the numerical imprecision inherent in using the rational and irrational numbers in calculations and measurement; mutations arising from replication errors or other unknown influences in molecular and evolutionary biology; and, the ambiguities of meaning built into the use of natural languages like English or German as well as the interactions between them through translations (ibid., slide 32).

36. Dewar, Gillogly, and Juncosa, "Non-Monotonicity, Chaos, and Combat Models," iii.

38. Ibid., v and 4-5. Note that because this model is :piecewise continuous, not continuous,: Dewar, Gillogly, and Juncosa's :definition of chaos was similar to, but not the same as, definitions 39>39. Ibid., 5.

44. As Beyerchen ("Clausewitz, Nonlinearity, and the Unpredictability of War," 77) noted, "nowhere" does Clausewitz provide "a succinct definition of chance" in war.

45. Poincar( Science and Method, 67-70 and 72-76. Beyerchen has argued that PoincarJ left the door open to yet another form of chance: the unpredictability of interaction between the "slice" of the universe we can apprehend and some other part that we, as finite beings, do not or cannot (Alan Beyerchen, e-mail message to Barry Watts, 12 June 1996). "Is this a third way of conceiving of chance?" PoincarJ asked. In reply he wrote: "Not always; in fact, in the majority of cases, we come back to the first or second." (ibid., 76). Given that PoincarJ said "Not always" rather than "Never," there appear to be, on this understanding, three aspects of chance that transcend human ignorance as well as Laplace's demon: imperceptible microcauses that, through, amplification, have noticeable effects; the stochastic effects of causes too multitudinous or complex to be unraveled; and, the interaction of causes arising from different "slices" of universe which inevitably surprise us because we cannot take in the universe as a whole. Beyerchen speculates that PoincarJ was less concerned about this third guise of chance than was Clausewitz (e-mail message, 12 June 1996). For Lorentz's discussion of Poincar('s treatment of chance, see The Essence of Chaos, 118-120.

46. Andrew Marshall reiterated this concern as recently as November 1995, after reviewing the first complete draft of this essay.