¹ Blake, William, “The Marriage of Heaven and Hell: Proverbs of Hell,” in Sampson, John, ed., The Poetical Works of William Blake (London: Oxford University Press, 1925) 251Google Scholar.

² This is the algorithm for the computer-generated image of a fern. To speak of an arithmetical algorithm as a soul is, of course, metaphorical, for a fern's nature cannot be reduced to its shape. However, to think that even the shape of a fern is mathematically describable is intriguing. This algorithm can be found in Stevens, Roger T., Fractal Programming in Turbo Pascal (Redwood City, CA: M & T, 1990) 430–31.Google Scholar Those with a mathematical eye will note that it is technically more correct in equations to distinguish the second value of x from its previous value by the use of subscripts—i.e., x ₂=x ₁² rather than x=x ₂. Computer programming has made us at home with the latter, simpler form, which I will use throughout the paper.

³ The name “chaos theory” unfortunately misconstrues the central point of the theory, which finds a deterministic order at the heart of disorder. Porush, David (“Making Chaos: Two Views of a New Science,” New England Review and Bread Loaf Quarterly 12 [1990] 439)Google Scholar observes that calling the new science “chaos theory” is comparable to calling Copernicus's theory “geocentrism.” Nevertheless, chaos theory has become the popular name for this constellation of ideas.

⁴ This analogy comes from Stanislaw Ulam, according to Gleick, James (Chaos: Making a New Science [New York: Viking Penguin, 1987] 68)Google Scholar, but Stewart, Ian (Does God Play Dice? The New Mathematics of Chaos [London: Penguin, 1990] 84)Google Scholar has the most fun with the analogy.

⁵ To offer predictions about the impact of chaos theory may be rash (and perhaps ironic). Conservative scientists argue that it is merely an extension of the tools of an increasingly more exact science. See Poole, Robert, “Chaos Theory: How Big an Advance?” Science 245 (1989) 26–28.CrossRefGoogle ScholarPubMed The conservative argument might be playfully rendered thus: “We have managed to build a better mouse trap within the framework of treating mice as very small, cheese-loving, trunkless elephants. Now we have the tools for looking at mice as mice, but that merely fine tunes what has been accomplished.” Because it weighs the significance of chaos theory only with respect to the advance of “hard” science, this conservative perspective is short-sighted. The impact of Newtonian science cannot be reduced to its scientific and technological achievements but must also be considered in terms of its social and cultural impact, for it reflected and confirmed a vision of reality that generated such important cultural perspectives as scientism, individualism, and romanticism. Chaos theory, I believe, will have a similar impact.

⁶ For those wanting a deeper understanding of the underlying concepts of chaos theory, I would recommend—for fellow mathematiphobes, at least—first Ian Stewart's Does God Play Dicel? and then two essays by Heinz-Otto Peitgen and Peter H. Richter, “Frontiers of Chaos” in idem, eds., The Beauty of Fractals (Berlin: Springer, 1986) 1-21 and “Magnetism and Complex Boundaries,” in ibid., 129-37. These provide an excellent introduction without assuming a background in calculus. Gleick's Chaos is entertaining but it avoids mathematics and philosophy altogether and thus does not fully elucidate the topic. Also, Gleick has been criticized (see Porush, , “Making Chaos,” 431–36)Google Scholar for marginalizing the work of Ilya Prigogine on the grounds that Prigogine rejects the Kuhnian framework in favor of an evolutionary view of science. Prigogine's, Order Out of Chaos: Man's New Dialogue with Nature (New York: Bantam, 1984)Google Scholar contributes to a well-rounded appreciation of the nature and scope of chaos theory.

⁷ One must distinguish here between linear and nonlinear determinism. Properly speaking, determinism applies where there is only one possible course of events. While chaos theory recognizes the centrality of bifurcations (that is, points at which a process can have one of a number of possible but unpredictable outcomes), this “indeterminacy” is deterministic in the sense that it is strictly determined by the process itself; indeterminacy arises from the fact that the fine details of the process cannot practically be known. Linear determinism (the Newtonian framework) implies a unilateral causality, where all effects can ultimately be reduced to atomic causes and where atomic forces are uninfluenced by more comprehensive (for example, biological) constraints and processes. Nonlinear determinism denies this unilateral causality.

⁸ Gleick, , Chaos, 8Google Scholar.

⁹ Peitgen, and Richter, , “Frontiers of Chaos,” 4Google Scholar.

¹⁰ This is done by rendering processes into simple numerical values. It is something like high school math: a single point was plotted on a graph to represent two variable values, x and y. If x and y are time and distance, a moment in the process of movement can be represented as a single point. Now extend this notion in two ways. Instead of a two-dimensional x-y grid, imagine a grid of three, four, five, or a thousand dimensions, each dimension representing a variable in a complex process. (If you can do this, your imagination is far more supple than mine. I just imagine a three-dimensional grid and think “etc.” Such grids are unimaginable, but they can be dealt with mathematically.) Second, imagine each possible position on the grid as having a numerical value. In two dimensions, imagine a checkerboard having its squares numbered from 1 to 64 so that any position, any x-y value, can be identified by a single number. Now extend this to multidimensional grids and recognize that real numbers (for example, 1.1 or 2.000001) may also be used. Each number on the grid conveys a certain state in a complex process.

¹¹ Stewart, , Does God Play Dice? 303Google Scholar.

¹² Gleick, , Chaos, 251Google Scholar.

¹³ Eilenberger, Gert, “Freedom, Science and Aesthetics,” in Peitgen, and Richter, , Beauty of Fractels, 175–80Google Scholar.

¹⁴ Barbour, Ian, Issues in Science and Religion (Englewood Cliffs, NJ: Prentice-Hall, 1966) 298–314Google Scholar.

¹⁵ Barbour, (Issues, 293–94)Google Scholar deals with this point at greater length in his treatment of the Complementarity Principle.

¹⁶ Stewart, , Does God Play Dice? 292–95.Google Scholar No one has found a way of extending chaos theory to quantum systems, and thus far chaos theory deals only with classical systems. The basic problem is that chaos theory deals with processes. Quantum systems reveal discrete and apparently discontinuous observables rather than continuous processes.

¹⁷ Barbour, , Issues, 308–9Google Scholar.

¹⁸ Gleick, , Chaos, 309.Google Scholar This balance of microscopic and macroscopic in snowflake formation accounts for the infinite variety that arises despite the small set of principles involved.

¹⁹ Skarda, Christine and Freeman, Walter I., “How Brains Make Chaos in Order to Make Sense of Worlds,” Behavior and Brain Sciences 10 (1987) 161–73CrossRefGoogle Scholar.

²⁰ Fischer, Roland, “The Time-like Nature of Mind: On Mind Functions as Temporal Patterns of the Neural Network,” Diogenes 147 (1989) 64.Google Scholar His comments are in reference to the work of Skarda and Freeman.

²¹ Ludwig von Bertalanffy presents the arguments against mechanistic philosophy together with the philosophical and scientific basis for general systems theory in Problems of Life: An Evaluation of Modern Biological Though (Published originally in German in 1949; London: Watts, 1952).Google Scholar The triumph of this perspective can be seen in Ilya Prigogine's Nobel Prize for modelling mathematically the emergence of self-organizing systems.

²² For those interested in the theoretical dimension of this resolution of determinacy and final causality, an excellent article is Rosen's, Robert “The Physics of Complexity,” in Trappl, Robert, ed., Power, Autonomy, Utopia: New Approaches toward Complex Systems (New York: Plenum, 1986) 35–42.CrossRefGoogle Scholar The context of this discussion is cybernetics, the study of feedback systems, which is how the mechanistic perspective has accomodated biological phenomena. It is particularly significant that this essay is the Ashby Memorial Lecture—Ashby represented the apex of mechanistic cybernetics—because the essay outlines the inadequacies of the simplistic mechanistic approach. Rosen's paper concludes by saying that the study of nonlinear or “complex” dynamics admits final causality back into the discussion of science “in a perfectly respectable, nonmystical way” (p. 41). Another significant article in this book is Francisco Varela's “Steps to a Cybernetics of Autonomy” (pp. 117-22).

²³ Dyke, Charles, “Strange Attraction, Curious Liaison: Clio Meets Chaos,” The Philosophical Forum 21 (1990) 369–92. A more theoretically oriented work of his isGoogle ScholarThe Evolutionary Dynamics of Complex Systems: A Study in Biosocial Complexity (New York: Oxford University Press, 1988)Google Scholar which is in part a response to the reductionism of sociobiology and a constructive theoretical and methodological effort in extending the insights of nonequilibrium thermodynamics to sociology and economics.

²⁴ Ibid., 383.

²⁵ Ibid., 383-84.

²⁶ Peitgen, Heinz-Otto and Saupe, Dietmar, eds., The Science of Fractal Images (Berlin: Springer, 1988) 39–42Google Scholar.

²⁷ Barnsley, Michael, Fractals Everywhere (Boston: Academic Press, 1988).Google Scholar This textbook aims precisely at merging art and fractal geometry. See particularly the color plates 9.8.1 through 9.8.14. See also plates 16a-16d in Peitgen, and Saupe, , The Science of Fractal ImagesGoogle Scholar.

²⁸ Stewart, , Does God Play Dice? 229Google Scholar.

²⁹ In the words of the German mathematician, Hermann Weyl: “My work has always tried t o unite the true with the beautiful and when I had to choose one or the other I usually chose the beautiful” (quoted in Peitgen and Richter, “Frontiers of Chaos,” 4).

³⁰ Porush, , “Making Chaos,” 441Google Scholar.

³¹ The theme of control runs throughout Bernard Lonergan's works. Strikingly, he contrasts his affinity for science with his inability to identify with mystic revelation: “Praise of the scientific spirit that inquires, that masters, that controls, is not without an echo, a deep resonance within me, for, in my more modest way, I too inquire and catch on, see the thing to do and see that it is properly done” (Insight: A Study of Human Understanding [New York: Harper & Row, 1978] 324).Google Scholar Elsewhere Lonergan deals with confession (as a type of rite related to Eliade's “myth of the eternal return”) in terms of control: “Eliade criticizes these rites as a flight from history. But I think one can also think of them as a primitive means, on the symbolic level, to deal with and dominate history” (“The Philosophy of Education,” lectures given at Xavier University, Cincinnati, OH, 3-14 August, 1959 transcribed and edited by James Quinn and John Quinn, 1979] 82).

³² Idislike the implication of “postmod[ernity” that modernity can be reduced to the Enlightenment with its consolidation of society according to the dictates of universal reason. The Enlightenment was but one expression of a modernity that has its roots in a new attitude to human, mundane existence, a modernity that found its more vital, youthful expression in a “modern devotion” and a novum organum.

³³ Hayles, Katherine, “Chaos as Orderly Disorder: Shifting Ground in Contemporary Literature and Science,” New Literary History 20 (1989) 305–22CrossRefGoogle Scholar.

³⁴ Gleick, , Chaos, 163, 197Google Scholar.

³⁵ Augustine himself did not assert this mutual unintelligibility but it is apparent as a tension in his works. He lived within the classical Greek perspective which affirmed the priority of knowing in loving (see De trinitate 10.1). Yet he held this in tension with the notion that knowledge depends on love (see In Joh. Ev. trac. 46) The fundamental tension is most apparent in his treatment of the Trinitarian procession in De trinitate 9. The Holy Spirit is treated as love and Christ as the Word: the Word emerges through a movement of desire to know, to represent knowledge. Although in De trinitate 9.12 Augustine does not want to call this desire love, he does so in 9.8: the inner word (the intelligibility that grounds language) is conceived in and born of love, a love which is either lust or charity. At the heart of this historically novel approach is the recognition that understanding relies upon either wholesome or unwholesome desire, which finds concrete expression in terms of attention. Thus knowledge depends on will, and will on knowledge. This tension in Augustine does not lead us to assert the priority of love over intellect, as some “Augustinians” hold; it would be truer to say that Augustine held in tension the “priority” of both intellect and will. He recognized the mutual unintelligibility of intellect and will; one can not make sense of one without the other.

The final point to be made is that Platonic tradition sees moral and aesthetic experience as intertwined. For Augustine, the inner word conceived by charity (a love for the truly good) is born when it comes to full fruition, when it is known in its fullness, when in and of itself it is a source of delight. One only knows justice truly when it is beautiful (De trinitate 9.8).

³⁶ Hayles, , “Chaos as Orderly Disorder,” 314Google Scholar.

³⁷ Lonergan, , Insight, 678.Google Scholar Each of the five arguments addresses a way in which the world is incompletely intelligible and needs something beyond itself to be fully intelligible.

³⁸ This is how I would summarize the debate on the relevance of metaphysics that appears in Alfred J. Ayer and Frederick C. Copleston on the relevance of metaphysics. See “Logical Positivism—A Debate,” in Edwards, Paul and Pap, Arthur, eds., A Modern Introduction to Philosophy (2d ed.; New York: Free Press, 1957) 726–56Google Scholar.