algorithm (n.) A set of rules for solving a problem in a finite number of steps. Coined in the late nineteenth century as a variant of algorism, the art of computation using arabic numerals. The early computer language ALGOL is putatively an acronym for Algorithmic Language, but given the difficulty of some algorithms, it may derive from Greek algos, Pain.
Technobabble

If algorithms can be described as a recipe, or a set of instructions written in some kind of formal language, it follows that two such recipes are somewhat equivalent if they produce the same output from the same input. However, "students of programming are often asked to implement a particular algorithm (such as 'heap search' or 'binary search') in a language of their choice. This makes no sense unless there is some idea of 'algorithm' that is independent of any particular programming language" (Foster 1). Computer scientists use algorithms to solve problems in a way that is infinitely translatable for a number of different coding styles or languages. The application of mathematical algorithms in chaos theory is a similar given (with fractals as an illustration) but this is merely how the term is applied in the scientific community -- algorithms have made an important transition from mathematical computations to representation of spatial, scientific, and I argue, literary, relationships. For example, an algorithm for a dance step could state move from point A to point B. However, the ends are not the means -- the difference between a style of free dance or a classic en pointe maneuver are vast. But the algorithm still links the process -- it is the literary realization (the combination of input & output) of how to accomplish that process that has relevant significance.

I will first consider the history of the term algorithm from a typical computer science perspectives as well as cognitive science's view. Next, I will examine how algorithmic-meditated complexity relates to complexity theory as well as complex systems, which incorporates some chaos theory. Further explanation of algorithmic mediation and modeling in English Studies will utilize a textual model. Tom Stoppard's play Arcadia will serve as an attempt to analyze textual material utilizing algorithmic tools as well as noting certain relationships to chaos theory.

I. History of the term algorithm & its cognitive relation
Engineers use partial differential equations to model physical phenomena such as flow of fluids in pipes, pressure variations along airplane wings or how a structure deforms and distributes the loads that it carries. These models are created for the purpose of quantifying the critical characteristics of a design. Unfortunately, these partial differential equations cannot be solved explicitly and thus must be approximated using some sort of numerical technique. Before the advent of computers, usable approximations were invariably impossible and so actual prototypes had to be built and tested. With computers, however, the entire process can be done with mathematical algorithms. This is the typical characterization of the term algorithm prior to the advent of cognition studies. In computer science, an algorithm is the foundation for the coding of a particular program. Using the recipe analogy, the instructional style may differ (like Pascal, C, C++, Visual Basic) as long as the result (a sorted database) is the same. For example, you may use a set of directions to make lasagna, but it may differ from your father's way of making lasagna. However, the outcome is still recognizably baked lasagna. If a computer programmer has to repair a software program written by someone else, knowing the algorithm is a crucial step in attempting to fix the original code.

This metaphor is also appropriated by cognitivists, who view algorithms as analogous to computational models of cognition and complexity theory. Nevertheless, there is considerable debate, especially in cognitive science, about the lack of distinction using this definition. Cognitivists, in particular (as opposed to cognitive scientists), use levels of explanation to better articulate their interpretation of cognition. Since cognitivists espouse that intellectual cognition can be represented via such computational models, their use of the term algorithm is especially telling. David Marr originally suggested using three levels of explanation for complex systems. The top level is called the computational level and is generally regarded as the most abstract. The middle level is the algorithm level, and "at this level, there can be an explanation of how the system produces the appropriate output from the given input" (Foster 10). The third level is the "hardware implementation" level, which is the most concrete. This bottom level explains the physical properties or devices in which the algorithm is implemented. This can be a person or computer in Marr's view. The important aspect of this has to do with information processing as a top-down approach (similar to the term top-down design in computer science) where an algorithm is likely to be understood more readily by understanding the nature of the problem being solved than by examining the mechanism (and the hardware) in which it is embodied(Marr 27). Pylyshyn differs from Marr by referring to the levels using different terminology: the bottom is biological; the middle level is symbol processing or syntactic; and the top is semantic or representational. Although Foster takes issue with the difference of Pylyshyn (based on Newell) and Marr, such ongoing investigations of algorithmic-mediated complexity from a cognitive science perspective provides illumination for other uses of the term.

If we consider literary theoretical practice using this same kind of model, you can substitute relevant terms in order to construct a similar methodology. The bottom level is the "text" or the source of the information, i.e., the physical manifestation. The algorithm could refer to the particular theory that mediates the textual examination, and the top level is the product or the abstraction of the original material produced through application of the algorithm. In a similar cognitive model, the top level is at a higher level of complexity, and the same may be said of our textual model. This is where issues of complexity are clearly interconnected with algorithmic mediation. When the algorithm shifts, the complexity level increases accordingly. For example, consider a current "algorithm" in literary theory -- feminist theory. Processing a text through feminist methodologies/philosophies will increase the level of complexity, especially if utilizing particular aspects of the theory and if the text is modern [1] in origin. As interesting as these parallelisms may be, the cognitivists' theoretical practice differs from other cognitive scientists because they determine definitions, like "algorithm" in an entirely different way.

Current cognitive scientists like Barbara Rogoff rely, in some part, on the theories of Ludwik Fleck, a German scientist who wrote Genesis and Development of a Scientific Fact almost 55 years ago. In the translated version, the preface is by Thomas Kuhn who mentioned Fleck's ideas in his text The Structure of Scientific Revolutions. Kuhn reinforces the importance of Fleck's theories in this preface, as well as acknowledges that his reference to Fleck in his book may not have been completely fair, due to Kuhn's poor translation of the German's original text. What Fleck posits is the idea of thought styles and thought collectives (158). Basically, his view of cognition means that it is more than a dual relationship between the knowing subject and the object to be known. The existing fund of knowledge must also be added as the third member (prior paradigms, so to speak). Cognition is also not an individual process, but a result of social activity:

for if we define "thought collective" as a community of persons mutually exchanging ideas or maintaining intellectual interaction, we will find by implication that is also provides the special "carrier" for the historical development of any field of thought, as well as for the given stock of knowledge and level of culture. This we have designated thought style. The thought collective thus supplies the missing component. (Fleck 39)

This concept is used by cognitive scientists who argue that definitions are transformed by the community of practice, and are not specified according to some dictionary definition. This community usually has specialized sense of the word, and its meaning is entirely dependent upon how the community utilizes it in their work. Therefore, the term "algorithm" is part of a collective consciousness in a particular community that is not translatable to other communities (because the meaning will shift accordingly). Rather than try to argue one side or the other, I would rather utilize both interpretations as a necessary step in analyzing the meaning of the term algorithm.

Since I have already addressed the former, consider the latter's notion of a community of practice, though often exemplified by science-oriented collectives, is just as useful in considerations of the academic (in the literary sense) collective. If we agree the algorithms of the profession are differing literary theories, then definitional meaning is decided by the academic collective, which is no longer relegated (although still controlled in large part) by the academy's canon. They become the arbiters of meaning, like determining panels for the MLA. A cultural studies panel is just that -- the academy's notion of what constitutes cultural studies in literature. However, the reality is not so simple. I believe that complexity is a major part of the problem. As the number of algorithms have proliferated, i.e., theoretical practices like feminist studies, postmodernity, and cultural studies, the complexity level has increased as well. The notion of algorithmic complexity is an important one to consider, first through a careful examination of complexity and complex systems, and then in terms of the implications for the academy.

II. Complexity Theory and Complex Systems
Like the levels described by cognitivists, complexity "at any given level is a consequence of the operation of relatively simple rules one level lower down"[2] (Cohen & Stewart 219). A common derivation of this thought assumes that complexity is just a result of many simple interactions. But Cohen and Stewart argue that consideration of where simplicity comes from is just as important (222). [3] Complex systems incorporate a network of independent agents that act and interact in parallel to one another, simultaneously constructing and reconstructing their environments. John Holland points out that

A complex adaptive system has no single governing equation, or rule, that controls the system. Instead, it has many distributed, interacting parts, with little or nothing in the way of central control. Each of the parts is governed by its own rules. Each of these rules may participate in influencing an outcome, and each may influence the actions of other parts. (21)

Mitchell Waldrop argues the richness of such interactions allows the systems to spontaneously generate self-organization. Self-organization within chaotic fields notwithstanding, Waldrop also claims that "all these complex systems have somehow acquired the ability to bring order and chaos into a special kind of balance. This balance point-often called the edge of chaos-is where the components of a system never quite lock into place, and yet never quite dissolve into turbulence, either. The edge of chaos is where life has enough stability to sustain itself and enough creativity to deserve the name of life" (12). It would not be a stretch to include the literary community as a complex system in this sense either, and consideration of such implications has much to offer. An especially compelling case has to do with the Silvio Funtowicz and Jerome Ravetz's idea of emergent complex systems.

Funtowicz and Ravetz distinguish between ordinary complexity (which has a simple teleology) and emergent complexity that can be used to influence related systems for hegemonic benefit, enabling a new conception of scientific practice, "involving its epistemology, methodology and power relationships" (Sardar & Ravetz 565). This is similar to the relationship between prior literary algorithms and the advent of postmodernity. Recognizance of emergent complex systems implicate complexity itself as well as promote a new understanding of science. Funtowicz and Ravetz claim that "formalisms and computations are no longer taken to represent the core of immutable truth and certainty in a world of flux; but they are used with respect for the variability and uncertainty of the world of experience" (568).[4] Emergent systems oscillate between hegemony and fragmentation, and an extension of that can "resolve the contradiction between hegemonic reductionism and fragmented relativism, which characterizes the post-modern condition" (569). Another indicator of emergent complex systems is novelty among the elements. Apparently, Funtowicz and Ravetz consider novelty an expression of individuality that is not present in ordinary complex systems, and another marker of their notion of postmodernity. This border zone of difference, like that the chaotic tightrope Waldrop claims complex systems walk, may well factor in the technology of cyberspace, where notions of "ordinarily complex systems" might be questioned (571). This border zone is different from "the edge of chaos" where "post-normal science" can use places of turmoil (contradictions appearing as differences of perception and value) to effect containment and occasion for mutual respect and learning.

This is a positive advance in complexity theory, at least from a scientific standpoint, but before we spend more time engaging algorithms and their complex manifestations, we also need to note other implications of complexity and chaos theory for science. Ziauddin Sardar makes an eloquent case against ethnocentricism in science and complexity in his essay "Conquests, chaos and complexity: The Other in modern and postmodern science." Sardar thinks complexity is nothing new, and in fact many of its ideas, like order out of chaos and interconnection can be found in many non-Western cultures. Even more telling, Sardar claims that the Western idea of conquest has been central to Western science and its appropriation and consumption of non-Western sciences and "in its evolution and development as modern science which has been intrinsically connected with colonialism and empires; and now, in its postmodern form, as chaos and complexity, its seizure of world views and outlooks on nature of Third World cultures" (Sardar 567, 666). By placing certain practices of Other cultures outside the realm of sanctioned Western science, their knowledge could be utilized and recycled "under its (Western) rubric" without recognizing its true origins. Scientists like Donna Haraway have noted such marginalizations as well, especially in the left science tradition where "important writers and thinkers are regularly not cited when science studies is professionalized and located in the academy alone" (Darnovsky 75). But how does this relate to our algorithms? If we incorporate a scientific definition of algorithms and algorithmic complexity, we must also acknowledge inherent problems in those institutions. Western science, as Sardar presents it, is monolithic, something the literary academy is still dealing with in terms of cultural practices and theory. Is there a postmodern algorithm that incorporates this, or does it replicate marginalizations while claiming to encompass the complexity and multiplicity of cultural systems? [5]

The problem with using a postmodern algorithm to mediate between texts (at the lowest level) and practical theory (at the highest, most complex level) is the nature of postmodernity as open to disorder and order within the texts themselves. In fact, the algorithmic mediation process drops out of the analysis entirely. There are no "recipes" for filtering postmodernity so that the input and output are weakly equivalent for different applications, or even insuring a higher order of complexity results. Complexity is now located within the text. The texts are transformed into verbal algorithms, meaning these emergent complex systems contain their own examples of self-organization amid chaotic states. There are now too many variables to chart texts using simplistic algorithms. In calling postmodern texts emergent complex systems, they oscillate between hegemony and fragmentation, trying to "resolve the contradiction between hegemonic reductionism and fragmented relativism, which characterizes the post-modern condition" (Funtowicz and Ravetz 568). Even more importantly, N. Katherine Hayles argues that "the disappearance of a stable, universal context is the context for postmodern culture" (272). [6]

III. The algorithmic shift
We may acknowledge that the term or idea of algorithm must still be established, but its inherent problems of deterministic properties (when is a recipe no longer just a recipe? Perhaps it's the difference between brownies and molotov cocktails) are no longer a factor when two levels (text/algorithm) are combined. Similarly, constructions of multiple meaning in texts involve complexity of an algorithmic order. Certain theoretical positions, as well as practices, are constructed using verbal algorithms to model or codify the continuation of the theory. In this example, algorithms are not unlike Thomas Kuhn's paradigms, for they exemplify a movement that seeks to replicate a particular subject position as well as expand upon it (Like using it in another coding language). Hayles posits chaos theory as a potential site for a paradigm shift, but it is still too new to be sure (145).[7] One important aspect of chaos theory has to do with the initial conditions of prechaotic states. This is where her definition of algorithm is played out in her book Chaos Bound. She defines algorithm as "a set of procedures a computer uses to solve a problem" (161), which is a common computer science definition. Algorithmic complexity theory is "concerned with defining and calculating the amount of information an algorithm contains," which basically translates as naming the size or complexity of the name of the shortest computer program that can print whatever sequence. But, "almost all real numbers are random in the sense that they cannot be generated by a finite computer algorithm" (162). Ford implies the randomness of real numbers that lack accuracy means there is documented unpredictability in deterministic systems. Algorithmic complexity fails to mediate the "unknowable" of these numbers (162). This failure implies the meaning and the nature of "algorithm" must shift again. To incorporate the infinite and the "unknowable" as well as the transient nature of knowledge, the algorithm must be relocated temporally and spatially. It can still function as a set of instructions, but these are embedded within the physical properties of the texts themselves. In other words, postmodern texts are transformed into verbal algorithms that are modeled in terms of complex systems and creative writing is an especially useful example. Hayles argues that it (creative writing) is located

within complex fields of intertextual resonances that affect signification not only in the narrow sense of the way words are understood but also in the broader sense of the ways plots are structured, characters conceived, actions represented. Combined, these factor make the literary texts more concerned than either chaos theory or deconstruction with the aura of cultural meanings that surround chaos [or postmodernism]. (19)

IV. Text study
Stoppard has always encouraged flux in his playwriting. He does not regard his plays as "finished products," but rather ongoing, chaotic processes that organize themselves in various ways. For example, he has had multiple versions of the same play in print, not favoring one revision over another but regarding them as almost pastiche, utilizing postmodern sensibilities.[8]

Using chaos theory to mediate Tom Stoppard's play, Arcadia, helps illuminate his appropriation of current scientific concepts to present a critique of literary theorists. Arcadia is a realization of a textual algorithm. Stoppard uses scientific theories of chaos and mathematical complexity in his verbal algorithm to subvert traditional scholarship (postmodernity) as well as illustrate it (through the fragmentation of narrative and temporality). It also functions as a representative of an emergent complex system by oscillating between hegemony and fragmentation that is partially realized through novelty and individuation of his characters and their temporal replications.

If we regard Stoppard's play as a complex system, remember that the system has no single governing equation or rule that controls it. Instead it has many distributed, interacting parts or elements. Each of these elements acts as if governed by its own rules, and each of these rules may influence an outcome or the actions of other elements (Holland 21). These are the algorithms that construct themselves, sometimes into spontaneous self-organization. The rules, or algorithms I wish to consider in Stoppard's play include: the nature of proof, models and temporality. The elements will vary, but so do the implications.

In Arcadia one of his central characters Thomasina Coverly, attempts to develop a mathematical proof for Fermat's Last Theorem. In 1637 French lawyer Pierre Fermat had written in the margin of his copy of the works of Appollonius that he had "discovered a truly remarkable proof which this margin is too small to contain" (Cipra 32). The theorem asserts xn + yn = zn has no solution when the exponent n is greater than 2. Thomasina attempts various methods of representation to prove the theorem, using geometric models to help illustrate her points. In Act I, Scene 4, in a later time frame, Valentine Coverly uses computer modeling to iterate Thomasina's attempt at solving the problem. He discovers that she uses algorithmic iterative techniques, a feedback loop of sorts, to generate her results. Surprised, he points it out to Hannah (an author friend) by telling her "It's the technique I'm using with my grouse numbers, and it hasn't been around for much longer than, well, call it twenty years." The irony, of course, is that Thomasina was working on it in 1809.

Although proof has a mathematical meaning in this text, the nature of evidence or "proof" in a literary sense is implicated as well. The nature of proof comes up in the current time frame, where Bernard Nightingale, an instructor and writer, is trying to prove that Lord Byron fought a duel on the Derbyshire estate and had to flee the country as a result. He is also trying to link Byron's authorship to a couple of reviews about the poet Byron supposedly killed in the duel. Temporality is also a factor in terms of proof, for the audience views both time periods and "knows" the truth about the incident. However, such assumptions are dangerous in Stoppard's complex system.

Hannah: I wouldn't worry about Chloe, she's old enough to vote on her back. 'Byron Fought Fatal Duel, Says Don.' Or rather - (sceptically) 'Says Don!' Valentine: It may all prove to be true. Hannah: It can't prove to be true, it can only not prove to be false yet. Valentine: (Pleased) Just like science. (2.7)

By layering literary proofs with mathematical proofs, Stoppard gives his audience an opportunity to question both. Not even "textual" evidence can be proved in terms of authorship. Bernard tells Valentine that

Well, by comparing sentence structures and so forth, this chap showed that there was a ninety percent chance that the story had been written by the same person as Women in Love. To my inexpressible joy, one of your maths mob was able to show that on the same statistical basis there was a ninety per cent chance that Lawrence also wrote the Just William books and much of the previous day's Brighton and Hove Argus. (1.2)

So, if proof is indeterminate, then indeterminate models will not be able to replicate what cannot be fixed. This is an important algorithm of Stoppard's play. The models are only representations of representations [9] and they are continually in flux. For example, the sketch book of Richard Noakes, a landscape artist, models the garden in terms of past and future (another temporal intersection). His pages are cut so that the after picture is superimposed over the before image. Stoppard uses the element of landscape as a model for the "natural," which is no more ordered than the household on which it borders. English gardens in particular, were modeled on idealizations of what epitomized the natural, which of course, is constructed. The attempt to generate order from the natural "disorder" reflects the chaotic nature of the text. Lady Croom tells Thomasina that "The rill is a serpentine ribbon unwound from the lake peaceably contained by meadows on wich the right amount of sheep are tastefully arranged--in short, it is nature as God intended, and I can say with the painter, 'Et in Arcadia ego!' 'Here I am in Arcadia," Thomasina" (1.2). But her model of Arcadia is disrupted by the disorder from the house, the landscape and her own interpretation of the "natural." Stoppard'sArcadia is truly a model of a complex system possessing chaotic elements. However, the natural is not the only modeling present, but Valentine's grouse modeling as well as Bernard's attempts to model events of an earlier period are all problematic. None of the models are accurate, because the initial conditions cannot be reconstructed and so what appears at first to be right (Bernard's assertions regarding Byron for example), gradually diverges into a different path altogether. As time is factored in, the divergence grows exponentially. Septimus tells Thomasina that God "has mastery of equations which lead into infinities where we cannot follow"(1.3) and later, Thomasina comments about the "lost plays of the Athenians! Two hundred at least by Asechylus, Sophocles, Euripides--thousands of poems--Aristotle's own library . . . How can we sleep for grief" (1.3)?

But time, according to Septimus, has its own feedback loop. In fact, most of the play is a realization of that feedback loop regarding intersections of temporality and knowledge. He comforts Thomasina by answering her:

You should no more grieve for the rest than for a buckle lost from your first shoe, or for your lesson book which will be lost when you are old. We shed as we pick up, like travellers who must carry everything in their arms, and what we let fall will be picked up bythose behind. The procession is very long and life is very short. We die on the march. But there is nothing outside the march so nothing can be lost to it. The missing plays of Sophocles will turn up piece by piece, or be written again in another language. (1.3)

These algorithms (proof, models and time) influence the interaction of character elements and artifacts to form a chaotic state where order and disorder oscillate between hegemony and fragmentation. If Arcadia is an emergent complex system, then as Funtowicz and Ravetz claim, "formalisms and computations [or old algorithmic methodologies] are no longer taken to represent the core of immutable truth and certainty in a world of flux; but they are used with respect for the variability and uncertainty [and disorder] of the world of experience" (568). Hopefully, such readings can generate similar questions about the shared nature of literary and scientific disciplines -- we already know their language can.

Works Cited

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Cipra, Barry. "Fermat's Last Theorem Finally Yields." Science 2 July 1993: 32-33.

--. "Math Attendees Find There's Life After Fermat Proof." Science 4 February 1994: 614-615.

Cohen, Jack and Ian Stewart. The Collapse of Chaos: Simplicity in a Complex World. New York, Viking Press: 1994.

--. "Why Are There Simple Rules in a Complicated Universe?". Futures: July- August 1994: 648-664.

Darnovsk, Marcy. "Overhauling the Meaning Machines: An interview with Donna Haraway," Socialist Review 21:2. April-June, 1991. 75-76.

Fleck, Ludwik. Genesis and Development of a Scientific Fact. Trans. Fred Bradley, Thaddeus J. Trenn. Chicago: University of Chicago Press, 1979.

Foster, C. L. Algorithms, Abstraction and Implementation: Levels of Detail in Cognitive Science. London: Harcourt Brace Johanovich, Academic Press Ltd: 1992.

Holland, John H. "Complex Adaptive Systems." Daedalus 121.1 (1992): 17-30.

Kuhn, Thomas S. The Structure of Scientific Revolutions. 2nd ed. Chicago: University of Chicago Press, 1970.

Pylyshyn, Zenon. Computation and Cognition: Toward a Foundation for Cognitive Science. Cambridge, Mass. MIT Press, 1984.

Sardar, Ziauddin. "Conquests, chaos and complexity: The Other in Modern and Postmodern Science." Futures: July-August 1994: 665-682.

Stoppard, Tom. Arcadia. Dir. Trevor Nunn. Lyttelton Theatre, Royal National Theatre. 13 April 1993.

Waldrop, M. Mitchell. Complexity: The Emerging Science at the Edge of Order and Chaos. New York: Simon & Schuster, 1992.