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In his book Chaos Making a New Science, James Gleick examines the birth and foundations of chaos theory and fractal geometry. As expressed by Gleick, chaos theory is defined by a close examination of randomness and a method of identifying patterns in phenomena that behave in a manner that is by definition random. For example, one wonders whether there must be some sort of a regular pattern in things such as economics or the weather, both of which move and change within certain limits. In his book, Gleick considers how Benoit Mandelbrot examined the nature of this movement and in so doing, offers, through Mandelbrot’s findings, a unique way of looking at the idea of patterns.
According to Gleick, what Mandelbrot discovered was that a series of events, like prices of cotton for example, are indeed random and hold no apparent pattern when considered relative to each other on a single scale; that is to say, if one looks at today’s price compared to yesterday’s price and that of the day before and so forth, no pattern can be identified. However, if one were to chart the prices from day to day and the separately chart yearly prices, the curves formed by the two charts would match, such that the curve of the yearly or monthly prices would, in closer detail, consist of smaller versions of the same curve. Thus, Mandelbrot found, in certain cases, a pattern or symmetry is to be found not between parallel events or points of information, but between large and small scales. What came out of these findings was a method of understanding irregular shapes and phenomena by examining their very irregularity, based on the idea that the degree of irregularity remains constant across scale. A sense of this is perhaps most easily understood as evident something like a rough piece of land, which has a certain roughness. When looked at from a great distance, the land appears just as rough as from very close up. That is to say, if one were to take a photograph of this piece of land (disregarding the limits of photography due to resolution and such), the degree of roughness would appear the same in the original photo as it would if the image were greatly magnified. Thus, it is not that the phenomenon in question is not random, but that the degree of this randomness is not random. This idea of a constant degree of irregularity is essentially based on the tendency of the large scale to imitate the small scale and vice versa. This tendency and the infinite character of its occurrence within itself are referred to as fractal.
If we extend this idea that a phenomenon maintains a certain level of complexity no matter how much it is magnified, then we must conclude that this principle allows for an infinite amount of detail and thus implies that the phenomenon continues infinitely. For example, a coastline can appear to constitute a particular length. However, the more detail with which we measure this coastline, the longer it becomes; measured with a yardstick, an estimate of the length overlooks certain twists and turns, nooks and crannies, that can be measured with a ruler, such that measurement with a ruler will yield a greater estimate of the length, and even greater when measured by the inch, the centimeter, the millimeter, and so forth ad infinitum. Therefore, using this principle of the fractal, the length of a coastline is infinite. In Gleick’s words, “in the mind’s eye, a fractal is a way of seeing infinity.” However, while we can say whatever we like about this coastline, it cannot be denied that the piece of land outlined by it is of a finite area; regardless of its coastline, Britain clearly does not extend infinitely into the rest of Europe or into the Atlantic Ocean. The mind-boggling but undoubtedly accurate consequence of this analysis is that it is possible for a finite amount of space to be circumscribed within an infinitely long line.
It thus seems that all we are left with after examining this system of fractal analysis is an unsettling bouquet of paradoxes. Patterns of randomness. Infinite lines surrounding finite spaces. What can we possibly make of these absurd combinations of words? From such descriptions, it would seem that any coherent understanding must be hopelessly complex. Although this conclusion is admittedly tempting, we must perhaps consider instead that it is not the above described tendencies of the universe that are complex, but rather our language for understanding them. Gleick recognizes that Mandelbrot’s observations run somewhat contrary to the common language of scientific understanding “How did nature manage to evolve such complicated architecture? Mandelbrot’s point is that the complications exist only in the context of traditional Euclidean geometry.” In considering the reasons why the system of blood vessels, which is certainly fractal, has not been widely analyzed as such by members of the field, Gleick remarks that “the language of anatomy tends to obscure the unity across scale.” The language of science is not intuitively conducive to a fractal understanding.
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On this point, it is necessary to acknowledge that scientific understanding does indeed have a specific language, which is just as much shaped by subjective experience and culture as any national tongue. This point is illuminated in Brian Rotman’s book Signifying Nothing The Semiotics of Zero. Rotman describes the process by which the use of zero and of the number system that we commonly use today came into being. He illustrates how a common system of expressing and communicating numerical quantities had to be consciously developed. Upon considering numbers and quantification this way, one is forced to look at science and mathematics in a decidedly different light than the one we are used to. In light of Rotman’s description of this evolution of the way numbers are expressed, it comes to mind that numerical discourse can be considered as a language, and when considered as such, we end up with a much different view of all its implications. Just as a verbal language affects (perhaps even creates?) the way we conceive the thoughts we express through it, so numerical language must influence the thoughts we are capable of.
For the most part, we like to think of numbers and mathematics as uniquely objective and universal, as a raw truth that self-selects based on what works and what does not, what makes sense and what does not. We consider that in this manner, it self-generates. This view, however, becomes more ambiguous when we consider that we choose the way it expresses itself. Its language then, like the language of any thought, is not necessarily contained within it, but rather is assigned (perhaps somewhat arbitrarily?). If we then consider that the language of mathematics is not objective (simply by virtue of being a language) and also that the language in which ideas are expressed often defines the structure and substance of those ideas, we must consider that mathematics is perhaps just as subjective as any other field of thought, and we end up with a very different picture of scientific and mathematical discourse.
The truth is that this fractal picture of the universe simply does not fit into the language by which we understand our lives. If we can consider space as fractal, then surely we must also consider a fractal interpretation of time. If we can take a finite amount of time and understand it as an endless structure of smaller and smaller patterns within themselves, then we must conclude that this finite amount of time is at once infinite. Given this, what do we make of the space of a lifetime? Surely we cannot consider it as infinite. Five minutes, an hour, a day, or a year goes by, and that space of time is over; there are tangible things that we did or didn’t do within it. If infinity is defined as the time in which all things that can happen do happen, surely infinity has not passed within a single second. But perhaps this is exactly what Mandelbrot meant; a second is indeed infinite because everything that will ever happen has happened within it and within the moments within the second, and will continue to happen in the same fractal pattern for hours and years and centuries ad infinitum. Indeed this would imply that thought also is fractal, that all human experience occurs in some chaotic pattern. Perhaps the tides of history and human thought behave in an ever-repeating self similar process. We can then extend the example of the curve of cotton prices to suggest that the same curve, the same essential model of thought and experience, is formed in a single second, any second, of a single individual’s life as is achieved by the whole of humanity after thousands of years. A fractal consideration of existence would ultimately lead to a very dismal outlook; indeed it would appear to render our conception of the self, the individual life, and all of this elusive idea of historical progress meaningless.
If we are doomed to forever repeat ourselves in an endlessly self-similar process, if the curve of any individual life is already ordained by its place in a larger but identical curve and by its composition of smaller identical curves, then why bother? Fractals rob us of identity. Inherent in this dismal way of thinking is an assumption that infinity renders us meaningless. Identity is able to exist only because it is finite. This idea of infinity simply does not fit into the structure of our minds. And yet we are ever inclined to consider it. What then, can this concept of infinity possibly mean to us? It is somewhat of a paradox and a lost effort for us to conceive of an existence that continues infinitely, because we cannot readily identify this characteristic in any tangible aspect of our lives or identities. Why then do we think about infinity? Did we invent it? Why would we invent a concept that we cannot possibly understand? Examined in the light above rendered, it comes to mind that perhaps we invented this idea simply because we cannot understand it; this seems to be the only feasible reason, as it would appear that this idea of infinity is definitively contrary to the way we understand ourselves and the world around us. Indeed perhaps this is the primary nature scientific inquiry. It reveals an inherent desire to reach beyond ourselves and our own distinctly human understanding and to achieve some sort of objectivity, separate from the subjective human experience. It is perhaps this effort that foremost characterizes scientific discourse. But this of course presents an inherent dilemma; while we are constantly fascinated with the question of what exists beyond our own experience and independent of it, we have nothing but the subjective tools of human understanding with which to engage in this effort. This is the central paradox that sustains rational inquiry. It is most eloquently articulated by Immanuel Kant in the preface to his Critique of Pure Reason
The human reason has this peculiar fate that in one species of its knowledge it is burdened by questions which, as prescribed by the very nature of reason itself, it is not able to ignore, but which, as transcending all its powers, it is also not able to answer.
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