Based on a comparison between the theories of N. Katherine Hayles (Chaos Bound: Orderly Disorder in Contemporary Literature and Science) and Vladimir Tasic (Mathematics and the Roots of Postmodern Thought), J. Linn Mackey asks the question "Is Chaos Theory Postmodern Science" and looks at parallels between the literature dubbed as "postmodern" and the scientific branch more properly termed "complex dynamics." His conclusions are surprising: both Hayles and Tasic miss important connective elements of both the science and the practice of postmodernism. Postmodern science does, in fact, exist, and literature just may be it.

Is Chaos Theory Postmodern Science?

J. Linn Mackey

<2> Hayles thus offers both a qualified "yes" and "no" to chaos theory as postmodern science. A more absolute "no" is given by Tasic in Mathematics and the Roots of Postmodern Thought.

Where does this leave us as to whether chaos theory should be considered postmodern science? I will argue in what follows that chaos theory does have parallels to literature and postmodernism and can be considered postmodern science in a qualified or limited way. I agree with Hayles on this point and disagree with Tasic, although the points he makes against the postmodern aspects of chaos theory are valid. I will also argue that there are problems with Hayles account of chaos theory and postmodernism.

<3> As much as I admire Hayles achievement in Chaos Bound, I believe her narrative of chaos theory, literature, and postmodernism is inadequate on several counts. I will argue that it is inadequate in accounting for the parallels between chaos theory and postmodernism. I will further argue that her analysis fails to adequately account for the distinct differences she herself identifies between chaos theory and postmodernism. I will propose that another accounting makes it possible to understand both the parallels between chaos theory and postmodernism and respond to Tasic's "no" to chaos theory as a postmodern science.

<4> Hayles seems well aware of the significance of the mythos of order/chaos in the Western tradition (16). Rather than dealing with the persisting privileging, until recently, of order over chaos, she chooses to pin her account to the changing nuances attributed to chaos through time in the West. A changed nuance in a positive meaning of chaos in information theory leads, in her account, to both chaos theory, postmodernism and certain contemporary forms of literature.

<5> Hall and Ames in Anticipating China give an illuminating account of the formation of the general features of Western cultural sensibility. They argue that this cultural sensibility was significantly determined by ideas invented or discovered in the ancient Greek world and culminating with the work of Augustine in the fifth century CE (xvi). This begins with cosmogonic myths of order conquering chaos. These cosmogonic myths were slowly transmuted into conceptual structures often expressed in theoretical traditions that privileged order, stability, and permanence over chaos (102).

<6> It is this account, rather than Hayles' changing nuances of chaos in the Western tradition, which allows us to understand the emergence of postmodernism and the continuing commitment to order over chaos that Hayles and Tasic find in chaos theory. From Hall's and Ames' perspective postmodernism emerges from a reversal of the tradition of order and permanence over chaos and change. This reversal gets underway by the end of the nineteenth century. Hall and Ames link this reversal to Kierkgaard and Nietzsche. This reversal gathers momentum through Heidegger and Derrida and culminates in postmodernism (106). This account of the origins of postmodernism is widely shared [2]. In contrast chaos theory maintains the traditional Western cultural sensibility of order over chaos.

<7> How then are we to explain the parallels between chaos theory, literature, and postmodernism? To account for the parallels, I believe it is necessary to provide a narrative very different from Hayles. Hayles explains the parallels as the result of all three emerging from the same cultural complex that was influenced by a positive nuisance given to chaos by information theory. In contrast, my explanation argues that chaos theory, literature, and postmodernism independently discover in their accounts the unusual, the unpredictable, and the uncertain. Unlike Hayles and the authors in Chaos and Order, which Hayles edited, I argue that the unusual, the unpredictable, and the uncertain were present in the origins of the novel in the eighteenth century [3].

<8> It was much later with Henri Poincare in the later nineteenth century that intimations of the unusual, the unpredictable, and the uncertain appeared in scientific representations. A more widespread awareness of these elements in science only begins to emerge with the work of Edward Lorenz in 1963. The self-conscious emergence of chaos theory that embodies the unusual, the unpredictable and the uncertain takes place in the 1970s as told by James Gleick in Chaos: Making A New Science. Chaos theory discovers these elements as the result of an anti-Cartesian move taken with iterative resequencing. It is this anti-Cartesian move that results in the appearance of the unusual, the unpredictable, and the uncertain in scientific representations and leads to the parallels between chaos theory and literature.

<9> The influence that Descartes had in shaping the modern worldview is well documented by Toulmin in Cosmopolis. Descartes was certainly heir to the Western cultural sensibility of the privileging of order, permanence, and rational objectivity which Hall and Ames so ably elucidate. According to Hall in Richard Rorty: Prophet and Poet of The New Pragmatism,

[Descartes] made the significant double move that defines the modern epoch ’Äì the internal move toward the grounding of the self in consciousness of itself, in securing repose of self-reflection and the outward move from the self to the material world armed with coordinates of analytic geometry. What is modern about the Cartesian stratagem is not merely the provision of an unsullied vantage point from whichreason could inventory the extended world [my emphasis]the modern impulse is found as well in Descartes use of the corpuscularian theory rediscovered by his contemporary, Pierre Gassendi. Atomic theory, which characterizes the cosmos in materialist, mechanistic terms, is a principal motor of modernity. (30, 31)

<10> With this mind/matter dichotomy, Descartes deprived material systems of any self-organizing capacity and left them bereft of any possibility of expressing the unusual, the unpredictable, and the uncertain. One of the key insights of postmodernism is that self-consciousness is a historically contingent concept (Hall and Ames, 70). Postmodernism drastically undermines Descartes first move. Rather than being a certain ground from which to think rationally about the world cogito ergo sum is, from the postmodern perspective, nothing but a historically contingent standpoint. The rationally constructed world from such a standpoint must contain historical contingencies. Another standpoint will lead to a different rationally constructed world, etc. I propose that with iterative resequencing chaos science repudiated the Cartesian vantage point. In so doing it aligns itself with postmodernism. Further, this move constructs representations of material systems that reveal the unusual, the unpredictable, and the uncertain which parallels what was present in the novel from its beginnings. What is surprising is that chaos theory, which as Tasic recognizes embraces Descartes' materialist, mechanist cosmos, discovers that material systems can self-organize when it explores the consequences of an anti-Cartesian, historically contingent standpoint through iterative resequencing. Material systems demonstrate the unusual, the unpredictable and the uncertain. What remains is to show that the unusual, the unpredictable, and the uncertain were incorporated in the novel at its origin and to explain and demonstrate the anti-Cartesian move of iterative resequencing which leads to representations that display the unusual, the unpredictable and the uncertain.

<11> Ian Watt in The Rise of the Novel: Studies in Defoe, Richardson, and Fielding, and J. Paul Hunter in Before Novel: The Cultural Contexts of Eighteenth English Fiction trace the origins of the novel to eighteenth century England. This is an interesting period to contrast the viewpoints of classical science and the humanities since the impact of Newton's work was felt throughout the culture of the time and the Enlightenment was flowering. Surveying the newly emerging form of the novel, Hunter makes clear that the 18th century novels had to take the new scientific order of Newton into account. Yet they went beyond this scientific order in conveying human experience. The novelist Henry Fielding illustrates this need to take into account the operation of scientific laws but the need to go beyond this to convey human experience. Hunter says of Fielding's novels:

Thus to create the feel of life experience, Fielding and the other eighteenth-century novelists had to pay homage to the law-like and deterministic world that emerged with seventeenth-century science, from the work of Galileo and Newton and their kin. But they had to do more. They had to represent a world that allowed for the unusual, the unpredictable and the uncertain, a representation of a world of strange and surprising events. The novelist's insistence on incorporating the unpredictable, the unexpected and the uncertain into their description of the world makes it antithetical to the classical Laplacian point of view of science striving for certainty and predictability.

<12> I will now examine the changed approach to representation in chaos science and fractals [4] and show that it embodies a viewpoint close to that of literature and an epistemology like that of postmodernism. Although the new approach to representation is widely used in chaos science and fractals, I believe the changed assumptions that they utilize and how these move toward the viewpoint of literature and postmodernism are little understood by scientists and others.

<13> Let us compare the representations of the novel with those of science. The novelists' representations are in words; they use language. Even when novelists use the omniscient voice, the representations are about particular persons and event chains and plots in the medium of written words. The representations of science and mathematics are sometimes expressed in written words, but in line with science's goal of generality, they are often expressed as equations such as Galileo's equation for an object moving under the influence of gravity:

where S stands for distance an object has moved and t stands for the time it has moved. The a is a constant related to the gravitational attraction.

<14> How can terms like unusual, unpredictable, or uncertain be attributed to equations like the one above? One needs to recognize that equations like Galileo's are never adequate representations of the real world, even the world of scientific experiment and experimental validation. To make an equation like Galileo's applicable to a real situation or to explore its full implications, scientists must take additional steps. To make the equation applicable to a real world situation, detailed particulars must be applied to the equation. Thus in Galileo's case, for example, a brass ball is held at the top of a plane that is say four feet long and inclined at an angle of 45 degrees from horizontal. With these particulars specified, Galileo's equation will predict how long it takes for the ball to roll from the top of the board to the bottom. It is worth noting that this is very much akin to what a novelist does in fleshing out a plot as Fielding does for his protagonist Tom Jones.

<15> Scientists engage in another process of representation in order to understand equations and their implications. This is the process of plotting or I prefer resequencing. Let us examine the processes of resequencing that lead to scientific representations. I will argue that it is the assumptions embedded in the anti-Cartesian process of resequencing used by chaos science and fractals that brings them closer to the viewpoint of literature and toward a postmodern perspective.

<16> By and large until the 1970s scientists followed a Cartesian resequencing process for representing the implications of equations. This is the way this resequencing process proceeds. The variables are displayed on a Cartesian grid. One selects data for the independent variable and puts that into the equation to calculate the dependent variable. Selecting zero as the first value for the independent variable and then selecting values of regular increasing magnitude for the independent variables followed by calculating the dependent variables is usually how this is done. The corresponding values of independent and dependent variables are then plotted on the Cartesian grid. The pattern obtained from a small set of data can be extrapolated to see the overall pattern.

<17> Let us examine the assumptions embedded in this Cartesian resequencing process. We might expect that this way of approaching the equation embodies the objective standpoint of a self-conscious observer independent of the world. The approach is omniscient, God like. There is no sense of time or process. We are not constrained in our choices. We can arbitrarily choose any value of independent variable and find the corresponding dependent variable. We can start with the independent variable small and work up or with it large and work down. We can instead choose to jump back and forth in independent variable values. The choices we have made in the past have no bearing on our future choices. We stand outside any contingency of choice. Given a modest set of corresponding independent and dependent values plotted, we can extrapolate the resulting pattern. It is surprise-free.

<18> Contrast this with human experience, the experience the novelist attempts to capture. Our human experience is temporal. We are trapped in time. Our present issues from a specific, inescapable past, and we sense we are caught up in a process toward an indeterminate future that contains surprises, the unexpected. Note how different this is from the omniscient, Cartesian resequencing approach above. There is little vestige of the nature of contingent human experience in the Cartesian approach.

<19> I now examine a new resequencing process that came to prominance in the 1970s with chaos science and fractals. It differs from the Cartesian resequencing process in two important ways: it uses a process of iteration, which is taking the calculated dependent variable and using that as the next independent variable inputted into the equation. It focuses on the outcome or limit of the iteration process not on the particular outputs of the equation. In this new resequencing process, one picks a starting value for the independent variable, then puts this in the equation to calculate the value of the corresponding dependent variable. One then uses the just calculated dependent value as the next value of the independent variable to put into the equation. The new dependent value becomes the next independent value and so on. This process continues and one sees if the iterated series of values approaches a limit. On a coordinate grid, one labels or colors the initial starting value of the independent variable with the limit label or color. One picks a second starting value for the independent variable and repeats the iteration process ending with the starting value labeled with color of its limiting value. Another starting value is picked and so on. Each new starting point my lead to different outcomes, One quickly grasps the necessity of a computer to map out the results of such a resequencing process. An Appendix is included for the more mathematically inclined which gives examples of both Cartesian and iterative resequencing. The later example demonstrates the unpredictable and uncertain that results from this anti-Cartesian approach.

<20> It is the representations of this resequencing process used in chaos science and fractals that show the unusual, the unpredictable and the uncertain. The best way to experience these representations is to see them and hear them [6]. Mandelbrot, the founder of fractal mathematics writes of the computer-generated displays of fractal representations in Leonardo

John Briggs in Fractals: The Patterns of Chaos writes about the representations of chaos science and fractals. He says that with the development of chaos science and fractal math we no longer inhabit a world shaped by lifeless mechanically interacting fragments driven by mechanical laws, but a world that is alive, creative, and diversified, born of an unpredictability ultimately beyond our control (180). What the non-scientist and even some scientists my not grasp is that all this is a result of the new, anti-Cartesian resequencing process just outlined. It has become a clichˆ© used by scientists and non-scientists alike to attribute the unusual results of chaos science and fractals to "non linearity" or non linear equations (Pigliucci 63). Yet, nonlinear equations have been around for three centuries, since the birth of modern science, and no one produced the unusual representations of chaos science and fractals. Yes, chaos science and fractals use non linear equations, but the unusual, the uncertain and the unpredictability they produce result from the new anti-Cartesian resequencing process they use.I argue that it is the changed assumptions embedded in that resequencing process, assumptions akin to those of literature and postmodernism, that should be credited with emergence of the unusual, the unpredictable and the uncertain in these representations of chaos science and fractals.

<21> We now examine those assumptions embedded in the resequencing process of chaos science and fractals. When we compare the iteration resequencing approach to the Cartesian resequencing approach and to human experience, we see the striking similarity of iteration resequencing to human experience. In the anti-Cartesian resequencing process of chaos science and fractals, once we make a choice for a starting value of the independent variable to put in an equation, we are caught up in an inescapable process. This starting point is consequential. The past, in terms of our initial choices and their subsequent development, is significant. The process, once set in motions, moves to future states which are unknown until they are achieved (Mackey 58). Given that the process of iteration resequencing mimics that of human life experience, the life experience that novelists attempt to convey in their literary representations, it is perhaps not surprising that the representations that appear in chaos science and fractals show the unusual, the unpredictable and the uncertain.

<22> We have argued that chaos science stays within traditional Western cultural sensibility by privileging order over chaos unlike postmodernism which embraces the reversal of chaos over order. As a result, as Hayles and Tasic indicate, chaos science is modern rather than postmodern. But contrary to Tasic and like Hayles, chaos science does have parallels to literature and postmodernism in that its representations show the unusual, the unpredictable, and the uncertain. This is a result of the anti-Cartesian move to iterative resequencing. In making this anti-Cartesian move chaos science restores self-organizing properties to material systems.

Appendix: Examples of Cartesian and Iterative Resequencing

The Cartesian resequencing process for equations: Lay out the variables on a Cartesian grid. Select data for the independent variable and calculate the dependent variable using the equation. This is usually done by starting at zero as the first value for the independent variable and then selecting values of regular increasing magnitude for the independent variable. Plot the corresponding points of independent and dependent variables on the Cartesian grid. The pattern obtained from a small set of data can be extrapolated to see the overall pattern. We take as an example Galileo's equation for a falling body S = at². For simplicity let a = 1.

Values for t and S are listed in the following table, and a Cartesian plot of these values is shown.

The plot is a parabola. Until the 1970s it was assumed that this was the only significant representation that could be obtained from the general equation. We now know that other resequencing schemes exist. Other resequencing allows us to see some of the assumptions hidden in the Cartesian scheme.

The iterative resequencing process for equations: Pick any starting value for the independent variable. Substitute that value in the equation and carry out the operations to calculate the dependent variable. Use that output dependent variable as the next input variable and so on. Continue this process and see if the iterated series values approach a limit. On a coordinate grid, label or color the initial starting value with the limit label or color. As an example take an equation only slightly more complex than Galileo's used earlier,

S = 1 ’Äì a t².

In this example, let the constant a = 1.5. The equation is now,

S = 1 ’Äì 1.5 t².

We examine this equation in the interval 0 to 1.

Cartesian resequencing:

Iterative resequencing: We use the same equation but with a changed notation which emphasizes the iterative procedure.

Tn+1 = 1 - 1.5 Tn².

The points on the plot never settle down to a limit or regular behavior. They appear to jump around randomly, but are produced by a regular equation and are quite deterministic. This deterministic yet stochastic behavior is called chaos. It was not observed until 1971 and was quite surprising and strange to scientists. (Gleick 70).

Works Cited

Briggs, John. Fractals: The Patterns of Chaos. New York: Simon and Schuster, 1992.

Pigliucci, Messimo. "Chaos and Complexity: Should We Be Skeptical?" Skeptic 8.3 (2000): 62 ’Äì70.

Gleick, James. Chaos: Making A New Science. New York: Viking, 1987.

Hall, David L. Richard Rorty: Prophet and Poet of the New Pragmatism. Albany, NY: StateUniversity of New York Press, 1994.

--- and Ames, Roger T. Anticipating China: Thinking Through the Narrative ofChinese and Western Culture. . Albany, NY: State University of New York Press, 1995.

Hayles, N. Katherine. Chaos Bound: Orderly Disorder in Contemporary Literature and Science. Ithaca, NY: Cornell University Press, 1990.

---, ed. Chaos and Order: Complex Dynamics in Literature and Science. Chicago: The University of Chicago Press, 1991.

Hunter, J. Paul. Before Novels: The Cultural Contexts of Eighteenth Century English Fiction. New York: W. W. Norton and Co., 1990.

Mandelbrot, Benoit. The Fractal Geometry of Nature. New York: W. H. Freeman, 1977.

---."Fractals and an Art for the Sake of Science." Leonardo 22, Supplemental Issue, (1989). 21-24.

Mackey, J. Linn. "Narrative and the Physical Sciences." Issues In Integrative Studies 11, (1993). 45-62.

Stewart, Ian. Does God Play Dice? Oxford, Eng.: Blackwell, 1989.

Tasic, Vladimir. Mathematics and the Roots of Postmodern Thought. Oxford: Oxford University Press, 2001.

Toulmin, Stephen. Cosmopolis: The Hidden Agenda of Modernity. Gllencoe, IL: The Free Press, 1991.

Watt, Ian. The Rise of the Novel: Studies in Defoe, Richardson, and Fielding. Berkely: U. of California P, 1957.

Notes

[1] Some examples would be Ceclie Brennan, "Beyond Theory and Practice: A Postmodern Perspective," Counseling& Values 39:2 (1995), 99-108, Harland Bloland, "Postmodernism and Higher Education," Journal of Higher Education 66:5 (1995), 521-57, Mario Markus, "A Scientists Adventures In Postmodernism," Leonardo 33:3 (2000), 179-186. Particularly significant would be Katherine Hayles, Chaos Bound (Ithaca: Cornell UniversityPress, 1990), and the authors in Chaos and Order: Complex Dynamics in Literature and Science, ed. Katherine Hayles (Chicago: The University of Chicago Press, 1991). [^]

[2] Examples include From Modernism to Postmodernism, ed. Lawrence Cahoone (Cambridge, MA: Blackwell Publishers Inc. 1996), Vladimir Tasic, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford University Press, 2001), and Stuart Sim in "Postmodernism and Philosophy" in The Routledge Companion To Postmodernism, ed. Stuart Sim (London: Routledge, 2001). These sources do not include Kierkgaard. [^]

[3] Although Shelia Emerson, Linda Hughes and Michael Llund in Chaos and Order, ed. Hayles find parallels to postmodernity in earlier forms of literature, none of them trace the parallels back to the origins of the novel. [^]

[4] Chaos theory and fractals have been shown to be related. Ian Stewart in Does God Play Dice? (Oxford: Blackwell, 1989) says "Fractals present us with a new language in which to describe the shape of chaos." (216) Both utilize iterative resequencing. [^]