Wednesday, March 26, 2008

III. Introduction to the Fractal Distribution of Words in Literary Texts

Strangely, my dissertation text is missing the following section -- one of the most important sections in the entire dissertation. Also, I could not for the life of me get the graphs on this site.

For Kundera, a novel investigates themes. “A theme is an existential inquiry” (AN, 84), the “examination of certain words, theme-words” (84). In other words, “A novel is based primarily on certain fundamental worlds,” which are “analyzed, studied, defined, redefined, and thus transformed into categories of existence” (84).

By the time I came around to rereading James Gleick’s book Chaos: The Making of a New Science, I had taken Kundera’s idea of theme-words in literature to heart, and was beginning to think about literature within the paradigm of chaos theory. Thus, I re-read Gleick’s book looking for metaphors that could describe literature. Mandelbrot’s observations regarding noise in a system, creating fractal time, made me realize that narrative, too, was an example of fractal time, the words acting as the “noise” in the “system” of the novel, similar to Cantor dust. This is best seen in a rewording of Gleick’s own words:

Mandelbrot saw the Cantor set as a model for the occurrence of errors in an electric transmission line. Engineers saw periods of error-free transmission, mixed with periods when errors would come in bursts. Looked at more closely, the bursts, too, contained error-free periods within them. And so on – it was an example of fractal time. (93c)

which I have reworded thus:

I see the Cantor set as a model for the occurrence of words (particular words) in a novel. There are periods where a given word does not appear, mixed with periods when the word does appear, mixed with periods when the word comes in bursts. Looked at more closely, the bursts, too contain periods without that word within them. And so on – it is an example of fractal time within the novel.

These dusts occur on smaller scales, each cluster giving clusters of spaces and clusters. In a novel one can only go down so far, but one can see the general principle holds. Fractals are repeated self-similar patterns. This is what made me think to graph words in a text to see if any patterns would make themselves apparent – and if they would show bifurcations and both steadiness in frequency and self-similar groupings of words. This is indeed happened with the word “friend” in Thomas Hardy’s Jude the Obscure at 1,000-word intervals, as we can see here.

This graph is what is called a “PoincarĂ© map,” which “removes a dimension from an attractor and turns a continuous line into a collection of points” (142). “Such pictures ... [begin] to reveal the fine fractal structure” (144) of the system – in this case, the novel. We see word distribution in a novel “as a Cantor set arranged in time” (Gleick, 92), where “the degree of irregularity remains constant over different scales. ... the world displays a regular regularity” (Gleick, 98). This suggested to me that meaning in a novel is both emergent and fractal – as one goes down, one sees ever-smaller elements of meaning – elements that finally stop at the level of words, or morphemes. There is also perhaps the level of multiple interpretations of words – especially in context of the emergent properties of the phrase, sentence, paragraph, etc. Going down helps us see the fractal repetitions while going up (looking at the patterns the words make, looking at how they are functioning in a particular sentence, paragraph, scene, etc.) helps us see the emergent levels of meaning. The mere repetition of a word is not enough – it has to repeat in a chaotic pattern to create the strongest levels of meaning. Each word “repeat[s] itself, displaying familiar patterns over time. ... But the repetitions [are] never quite exact. There [is] pattern, with disturbances. An orderly disorder” (15). If we look at the graph above, we can see that the pattern of word distribution between 33,000 and 59,000 resembles the pattern between 69,000 and 89,000, and the word distribution pattern between 94,000 and 176,000 words resembles both, though it is flatter than the first two. They all resemble, but are not identical, to each other, as one would expect in a fractal. This specifically happens when we get a bifurcation – in the frequency of the word, which would indicate a bifurcation in the word’s meaning within the text, since in bifurcation, the attractor the line was following turns into a repellor, dividing the line. Although the word “church” has a similar number of respetitions in Jude the Obscure, as we can see here:

the fact that it lacks this kind of periodic behavior (orderly disorder) – the word “church” does not have either steadiness in frequency nor a bifurcation – while the word “friend” does, tells you the word “friend” is a stronger theme-word, having been created through the tensions in the novel. An even stronger example of this is when we look at the distribution of the word “the”:

We would expect a random distribution of a word like “the,” as it is an article, the type of word one would not expect to receive thematic development in a work such as a novel (though it may have thematic importance in such works as Wallace Stevens’ “The Man on the Dump”, which ends with the line “Where was it one first heard of the truth? The the.” – but such works have different ways of emphasizing meaning). And that is what we see here. While repetition does create meaning in a text, it is not just any type of repetition (or else words like “the” would typically be the most important theme-words in a text), but patterned repetition. In another text, “church” may be a theme word. Here, it is not. But “friend” is. How does this happen? “Information is transmitted back from the small scales to the large... And the channel transmitting the information upward is the strange attractor, magnifying the initial randomness just as the Butterfly Effect magnifies small uncertainties into large-scale weather patterns” (261). So we would expect the peaks of the word “friend” to correspond to major plot points in the text.

Take another look at the first graph, of the distribution of the word “friend.” We find in the first peak of eight, at around 34,000-35,000 words entirely in chapter II-4, pg. 79-85, when Sue came to see Jude as he was working at his job as a stonemason on a Gothic church, and Jude takes Sue to meet Phillotson for the first time. The first “friend” occurs to say how Jude would not treat Sue, at first – to befriend her in hopes of it leading to romance (79). The second occurrence is of Sue saying to Jude in a latter she would have liked for her and Jude to have been friends while they had been in Christminster (81). She next speaks “with the freedom of a friend” (82), though this is their first real meeting as adults. As they begin to talk, Jude laments her leaving because she has “hardly any other friend. I have, indeed, one very old friend here somewhere,” speaking of Phillotson, whom he has yet to go see. They decide to go see him, and when they arrive, Jude says he came “to see him as an old friend” (82). Phillotson invites them in, after saying he had forgotten Jude, though the “old friendship was imperceptibly renewed” (83). When they leave Phillotson’s, Jude perceives that Sue’s “sentiments toward him were those of the frankest friendliness only” while “he loved her more than before becoming acquainted with her” (83-4). And when they part, Jude goes back to Phillotson to ask him for a teaching position for Sue, which Phillotson assures “Jude as a friend” (85) would be a waste of time for Sue unless she wished to pursue teaching.

What is going on here? Jude wants a lover, but gets instead a friend – thus inverting the way things “should” develop, from friend to lover. And the tragic situation is also set up in this chapter because of Jude’s friendship with Phillotson. It is because of this friendship that Sue and Phillotson meet, and that Sue works for Phillotson – all of which leads to Sue and Phillotson’s marriage. There is a complex of meaning and consequence created in this chapter by the strange attractor of friendship and the meaning of friendship.

The second peak occurs (at 51,000-52,000) after Sue’s expulsion of school, after her having gone to see Jude, the episode which leads into Jude telling Sue of his being married, Sue’s seeing Jude about her engagement to Phillotson, and the last two occurrences of “friend” in this peak in the second letter from Sue and the response from Jude regarding Sue’s marriage. Phillotson comes to see Jude about the scandal of Sue’s expulsion, and each felt their conversation “could not comfortably merge in a friendly discussion of their recent experiences after the manner of friends” (131). After Jude tells Sue of his being married, at that moment “She was his comrade, friend, unconscious sweetheart no longer” (132), though after further discussions they persuaded themselves that they could still be friends, and their “parting was in good friendship” (134). The next chapter (III-7) begins with Sue announcing in a letter her immanent marriage to Phillotson. In her second letter, Sue asks Jude to give her away in marriage, saying that her father is not “friendly enough to be willing” (136). Jude recommends in his letter that Sue marry from his house, not “from your school friend’s” (136). Here we go from not friendly, no longer friends, to a promise to remain friends, to not friendly and not another friend. We have a bifurcation in friendliness to unfriendliness, and a putting-off onto others the not-friendly and not-friend felt by Sue and Phillotson toward Jude. By negating others, by seeing Sue’s father as unfriendly, by rejecting Sue’s friend, Sue and Jude are able to remain friendly toward each other, despite Jude’s being married and Sue’s immanent marriage.

As we can see from looking at just the first two peaks, Hardy creates a development of Sue and Jude’s friendship that coincides with major plot points, even when we do not realize in our first reading that what is happening is a major plot point or crisis, as we saw with the first peak with Jude introducing Sue to Phillotson. The peaks correspond with major plot points and with the development of the theme-words in the novel. And we also see the first peak of the second pattern (pg. 192-4, at about peak 71,000) gives a re-affirmation of the ideas of friendship developed around the first peak of the first pattern, even as they occur right when Sue leaves Phillotson for Jude. But of course this reaffirmation is one done in light of the previous development of the meaning of the word “friend,” which means it becomes affirmed in a different light, and is thus given a different meaning, than was the first peak the reader encounters.

The existence of fractal word patterns shows us that a novel is a particular kind of fractal It is self-similar at lower intervals, but as one goes up, new forms are made, self-similar to what came before, but having emergent properties (meanings). What we see in deconstruction is a concern only with the “infinite coastline” of the novel, at the expense of the emergent meaning of that coastline in delineating the complete form of the novel. Since the novel is now seen to be both regular and irregular, to be, in essence, fractaline, one could perhaps see Gleick’s observation that “The degree of irregularity corresponded to the efficiency of the object in taking up space” (100) could be taken as a literary judgement. Is there perhaps a correlation between a novel’s degree of irregularity as a fractaline object and our finding that novel beautiful (and giving it long-term survival)? This complexity that a fractal view of the novel illuminates is also another way of judging a novel (or understanding how novels have perhaps been judged in the past) since, as Gleick says, “Simple shapes are inhuman. They fail to resonate with the way nature organizes itself or with the way human perception sees the world” (116-7). We have to be careful when we say the word “simple,” since “simple systems can do complicated things” (167), as anyone who has read Hemingway knows. Further, “as [a] system becomes chaotic ..., strictly by virtue of its unpredictability, it generates a steady stream of information” (260). This is undoubtedly why we consider both predictable stories and stories that are not retrodictable to be bad stories. A chaotic story would be one that is not predictable, but is certainly retrodictable.

Gleick also says irregular patterns and infinitely complex shapes have “a quality of self-similarity. Above all, fractal meant self-similar” (103). Further, “self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern” (103). The presence of meaning in a novel is also not determined by its scale. Its words have meaning, and so do its plots, and every level in between. But, in the same way eddies of air are the same as a hurricane, only at different scales, the effect of the higher levels of meaning is as different from word to plot as the effects of an eddy of air are to that of a hurricane. All the same, an eddy of air can, building on other eddies of air, build into a hurricane over space and time in the same way as words, building on other words, build into a novel over space and time. This is because “each change of scale [brings] new phenomena and new kinds of behavior” (115). The existence of meaning applies “without regard to scale” (108) in a novel. And greater meaning emerges as we go up in scale, since these scales are hierarchical (116). “Fractal scaling [is] not just common but universal in morphogenesis” (Gleick,110). Since fractal geometry is “nature’s own” (114), and a novel is a part of nature inasmuch as it is a creation by a living organism, we should not be surprised to find that novels have fractal geometry. Further, “A geometrical shape has a scale, a characteristic size. To Mandelbrot, art that satisfies lacks scale, in the sense that it contains important elements at all sizes” (117). A good novel (that satisfies the reader) lacks scale, in the sense that it contains important elements at all scales, from words to plot.

The importance of looking at fractal distributions of words can be understood in Prigogine’s explanation of what occurs in a dissipative system: “One of the most interesting aspects of dissipative structures is their coherence. The system behaves as a whole, as if it were the site of long-range sources. ...the system is structured as though each molecule were “informed” about the overall state of the system” (171), a dissipative system being one that has both structure and disorder in it (143). One can see a sentence as having this very structure (Turner, The Culture of Hope). One can, in a sense, see how each “molecule” of the word “friend” is “informed” about the “overall state of the system” of the novel, helping it to cohere and have meaning. This is also how beauty is created: literature achieves beauty through linguistic density. Charles Kahn defines linguistic density as “the phenomenon by which a multiplicity of ideas are expressed in a single word or phrase” (89). This occurs in the relationship between the sign and the signified

if by sign we mean the individual occurrence of a word or phrase in a particular text, and by signified we mean an idea, image, or verbal theme that may appear in different texts. Then density is a one-many relation between sign and signified; while resonance [among signs] is a many-one relation between different texts and a single image or theme. (89)
One form of resonance “is a repetition of the very same word,” while others include “occurrences of the same theme in cognate words” (90), “the recurrence of a single image or theme which may or may not be expressed by the very same words: sleeping and waking” and “between words of similar or related meanings” (90). We can see this when we compare the graphs for “life” and for “live”. One is a noun, the other a verb, and yet the graphs independently give fractal distributions of words, though those fractals have a different geometry than did the distribution for the word “friend”. We would, of course, expect there to be differences in the fractal patterns for different words. What would be unusual, it seems to me, would be if one did come across words similarly patterned within a word. It would certainly suggest a strong correlation between the meanings of the two words. But let us look at the words live and life. First, Live:

And now let us look at Life:

The first thing to note is that they appear to be almost mirror images of each other in the way the major peaks are distributed. But it is life, with the most occurrences of the word, which is the stronger theme-word. And it has the stronger fractal geometry too, with the flatter single- followed by double-peaks pattern. We do see a similar pattern between “live” and “friend” to the extent that the single-double-peak pattern is there, but the pattern is less obvious because of the high number of words spread between them. However, we also notice here that there are very large peaks in each of these, particularly, again, in the graph for “live.” For the peak at 78,000, we have the repetition of the word “live” starting on pg. 176, where Sue is asking her husband, Phillotson, to allow her to move out. To which he replies: ‘And do you mean, by living away from me, living by yourself?’ And she responds with: ‘Well, if you insisted, yes. But I meant living with Jude.’ He asks her to consider his reputation, and she agrees instead to live in his house, but separate from him. What we have in this passage is the irony of them talking about her living with Jude, while "living" is hardly what is being proposed in her staying with Phillotson. Thus we have a bifurcation of the meaning of “living” in this peak. We know she will not really live if she continues to live with Phillotson, and this is confirmed in the next chapter when she jumps out of the window of her bedroom to escape her husband when he accidentally walks into her bedroom to go to bed. She says she was asleep when she did it – and perhaps she was half-asleep – but what this then suggests is such a deep disconnect between living and living with Phillotson versus her desire to live with Jude, that it became expressed in a leap from a second story bedroom window rather than having to sleep in the same bed with her husband.

These graphs and the analysis they suggest is highly suggestive of further work in this direction. More works of literature should be analyzed using this method, to see what sorts of patterns develop in other works, to see how theme-words are developed. It seems likely that it would be in long prose works, particularly in novels, where fractal patterns of word distributions would be seen. There is a limit to the level of complexity that is comfortably taken in by the human brain. Thus we would expect simpler (though likely still fractal) patterns of word distribution in more poetic works such as Shakespeare’s plays, since there is already a great deal of complexity in the rhythms and the rhymes and other poetic techniques. But the apparent simplicity of prose allows for the development of more complex fractal word patterns, allowing for a different kind of meaning development of the theme-words. And it might also be interesting to look at both the retention of fractal patterns and development of the meanings of theme-words for a particular novelist, as well as the history of such patterns and theme-words through the history of literature. Can we find a genealogy of the development of the word “friend” that is traceable through the use of the kind of fractal pattern Hardy uses? And is the kinds of fractal patterns of word distribution limited? And what about issues of translation? Can a translation be judged better than another if it maintains these patterns? These and many other questions can be raised and investigated using this technique I have introduced here.

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