In 1983, a mathematician named Benoit Mandelbrot published the second edition of a beautifully illustrated book, re-entitled The Fractal Geometry of Nature. The book is an extended discussion of mathematical objects with an exceptionally rich structure. The objects were called "fractals" by Mandelbrot, a contraction of "fractional dimensional measure."
The book is mathematically interesting, but the excitement it caused extended well beyond mathematics. Mandelbrot himself is an applied mathematician and has worked in many areas of applied mathematics, among them information theory, linguistics, biology, geography, economics, diffusion theory. In each, he found fractals. One of Mandelbrot's favorite trick questions is to ask, "How long is the coast of the eastern seaboard of the U. S.?" Most people will say something like 1,000-2,000 miles, and almost no one says 10 million miles. But both answers are equally plausible: it all depends on the scale of measurement. As you measure more and more finely, the coast "gets longer." Also, you will find that patterns found at coarser scales get repeated at finer scales. This phenomenon of scaling, or self-similarity, is characteristic of many fractals. When it is found, it suggests that a simple law is at work, being applied repeatedly to produce great complexity. The most familiar such patterns are snowflakes.
It has been known for a long time that order and symmetry underlie such crystalline structures. What is somewhat surprising is that the same sort of order appears in apparently disorderly systems. Such systems are called "chaotic," and the branch of mathematics that studies them is called "Chaos." (It used to be called "non-linear dynamics," but the new term is sexier.) It is beginning to seem that there are limits on Chaos, that there are certain types of disorder which can be classified and studied in theory and in the laboratory. A mathematician at University of California at Santa Cruz who has a small lab to study a dripping faucet has found that the faucet may begin dripping at one rate, but then the system changes to oscillate between two rates. When those rates split again, four unstable equilibrium points emerge, then eight, and so on, until the system becomes completely chaotic. This is the period-doubling route to Chaos, and it characterizes many chaotic processes. In 1976, a Los Alamos lab physicist, Mitchell Feigenbaum, was sitting at his desk playing with a desk calculator, fooling around with the equation for a parabola, and feeding the results of one iteration into the next. When he looked at the ratio of rates at which the equation approached certain values, he found a numerical constant showing up again and again. This was mildly surprising, but he at first thought that it was just a peculiarity of the particular equation he was studying. With a mathematician's instinct for generalization, however, he decided to see if the number showed up anywhere else. So he moved to a bigger computer and found very much to his and everyone else's surprise that the same number showed up in many cases of period doubling. Since period doubling as a route to chaos is common to many processes – heart attacks, stock market fluctuations, atomic phenomena, biological populations and epidemics, etc. – this is a significant result. It places a very stringent limit on how disorderly such processes can be. The numbers he discovered are now called "Feigenbaum numbers" (which are transcendental, like e and ).
[The systems I have been alluding to are usually modeled using differential equations describing continuous, as opposed to discrete processes. But it is possible to model fractal and chaotic systems as discrete processes. If our scale is fine enough, we won't be able to "see" the difference. This is what we do when we solve differential equations on a computer. We use difference equations to get approximate solutions to differential equations.]
I'll describe the amazingly simple rule that generates the most famous fractal: the Mandelbrot set, which has been called the most complex object ever discovered by human beings. It might even be compared in some respects to thinking things.
[(PRETTY SIMPLE) MATHEMATICAL FORMULAE AHEAD!]
First I need to tell you a bit about complex numbers. A complex number is simply an ordered pair of real numbers: z = (a,b). Call the first coordinate the "real part" of z and the second coordinate the "i-part" of z. We can think of such complex numbers as coordinates in the plane. The set of all complex numbers forms a mathematical set called a field if we define addition and multiplication as follows:
(a,b) + (c,d) = (a + c, b + d)
(a,b) * (c,d) = (ac - bd, ad + bc).
Note that (0,1)*(0,1) = (-1,0). We can define (distance and) magnitude as follows:
|z| = (a^2 + b^2).
We can have functions of complex variables, too. Consider the very simple one defined by
fc(z) = z^2 + c, for c some complex constant, possibly negative.
Imagine that we pick some value for c and that we start iterating. Let z = (0,0) to begin with, plug it in to the equation and then feed the result back into the equation. That gives 0^2 + c = c. And with z = c, fc(z) = c^2 + c. At the next iteration we get (c^2 + c)^2 + c, and so it goes. When the magnitude of the numbers reaches a certain size, the result of iteration begins to grow very quickly, though just how fast it grows depends on the initial choice of c.
The Mandelbrot set is the set of all complex numbers c for which the magnitude of fc(z) = z^2 + c is finite even after indefinitely many iterations. We can represent this set with a black region in the complex plane, using colors from red to violet to represent different rates at which points outside the set "go off to infinity."
[DONE WITH FORMULAE.]
That's all there is to it. The result is amazing for its beauty and complexity. You owe it to yourself to take a look at some of the many books that feature pictures of this extraordinary object, and, if possible, to view some of the many videos made to exhibit its beauty and some of its infinite complexity. Only graphical descriptions of the Mandelbrot set will enable you to visually appreciate the fact that it is both deterministic and unpredictable: although each and every point in the graph has its position and color determined with absolute precision and rigidity, the global result is utterly unexpected.
This last feature – the paradoxical sounding combination of determinism and unpredictability – is a consequence of the key feature of chaotic systems: sensitive dependence on initial conditions. In nonchaotic systems, a very small change in the initial state will result in a correspondingly small change in successive final states. In chaotic systems, even the smallest change in how things start is magnified in ways that can't be anticipated exactly, and the resulting change in the system may be enormous. If a chaotic system changes fast enough and is complex enough, it will outrun even the capacity of the fastest physically possible computer to simulate its operation. Such a system would be unpredictable in a very profound sense: the laws of nature would make it impossible for us to know what it was going to do before it actually did it; as a matter of physical necessity, the most efficient way to find out about the system would be to watch it change.
Because of the prevalence of chaotic systems in nature and in the world human beings have created, the visual beauty of the graphs of chaos, and the tantalizing near-contradictoriness of such systems' basic natures, Chaos theory has received a great deal of attention in the popular media and has spawned a sizable cottage industry in fractal and chaotic products. James Gleick's Chaos: Making a New Science was on the New York Times best-seller list for many weeks, and many other popularizations have been published in its wake. An episode of NOVA was devoted to the "new paradigm of scientific explanation, Chaos theory."
Some scientists have seen the flurry of public attention as excessive, and there's been some discussion of just how big a change Chaos theory induces in previously well-established scientific perspectives on non-linear systems. Describing Chaos theory as bringing a "new paradigm" tends to heat up the discussion, sometimes to the point of chaos.
The notion of a paradigm was introduced into twentieth-century discussions of science by the historian of science Thomas Kuhn in his famous book, The Structure of Scientific Revolutions. Before Kuhn wrote the book, many scholars of science had shown an unfortunate tendency to view science as a steadily progressive enterprise with a largely stable stock of concepts, standards, and methods. Kuhn emphasized that scientific revolutions sometime occur in which conceptual and methodological upheaval results. One paradigm is discarded and replaced by another. Although Kuhn himself never endorsed the view, some even proposed that science itself is defined by the process of paradigm development and replacement: science and science alone is governed by paradigms. If this is correct, then it might yield a gatekeeper:
Chaos theory is scientific because it is governed by a paradigm
–and if, say, astrology, failed to be paradigm-governed it could be dismissed as pseudoscientific.
None of this is very helpful unless we have a very clear idea of what a paradigm is and are sure that the notion is coherent. Soon after The Structure of Scientific Revolutions was published, reviewers began to criticize it for failure on both scores. One of the earliest, most highly critical and influential reviews was by Dudley Shapere. Here is Shapere's selection of Kuhn's key remarks about paradigms:
Paradigms are "universally recognized scientific achievements that for a time provide model problems and solutions to a community of practitioners" (x).
Because a paradigm is "at the start largely a promise of success discoverable in selected and still incomplete examples" (23-24) it is "an object for further articulation and specification under new or more stringent conditions" (23); hence from paradigms "spring particular coherent traditions of scientific research" called "normal science." Normal science consists largely of "mopping-up operations" (24) devoted to actualizing the initial promise of the paradigm "by extending the knowledge of those facts that the paradigm displays as particularly revealing, by increasing the extent of the match between those facts and the paradigm's predictions, and by further articulation of the paradigm itself" (24).
A paradigm provides "a criterion for choosing problems that, while the paradigm is taken for granted, can be assumed to have solutions" (37).
"Scientific revolutions are . . . those non-cumulative developmental episodes in which an older paradigm is replaced in whole or in part by an incompatible new one" (91).
A paradigm is [simply?] "a set of recurrent and quasi-standard illustrations of various theories," and these are "revealed in its textbooks, lectures, and laboratory exercises" (43).
Paradigms, as accepted examples of prevailing scientific practice "include law, theory, application, and instrumentation together" (10).
A paradigm consists of a "strong network of commitments - conceptual, theoretical, instrumental, and methodological" (42) among which are "quasi-metaphysical" ones (41).
A paradigm is, or includes, "some implicit body of intertwined theoretical and methodological belief that permits selection, evaluation, and criticism" (16-17). Such a body of beliefs is never implied by the facts, and so "it must be externally supplied, perhaps by a current metaphysic, by another science, or by personal or historical accident" (17).
"From [paradigms] as models spring particular coherent traditions."
So far, astrology, phrenology and Scientific Creationism seem to be doing very well. But things get murkier.
Most fundamentally, they are not rules, theories or the like, or a mere sum thereof, but something more "global" (43) from which rules, theories, etc., are abstracted, but to which no mere statement of rules or theories, etc., can do justice. So anything that allows science to accomplish anything can be a part of, or somehow involved in, a paradigm.
"Once a first paradigm through which to view nature has been found, there is no such thing as research in the absence of any paradigm" (79).
"There can be no scientifically or empirically neutral system of language or concepts, [so] the proposed construction of alternate tests and theories must proceed from within one or another paradigm-based tradition" (145).
Paradigms are open to "direct inspection" although they cannot in general be formulated adequately (44).
"An apparently arbitrary element . . . is always a formative ingredient" (4) of a paradigm.
A paradigm is "the source of the methods, problem-field, and standards of solution accepted by any mature scientific community at any given time, . . . the reception of a new paradigm often necessitates a redefinition of the corresponding science . . .. And as the problems change, so, often, does the standard that distinguishes a real scientific solution from a mere metaphysical speculation, word game, or mathematical play. The normal-scientific tradition that emerges from a scientific revolution is not only incompatible but often incommensurable with that which has gone before" (102). Thus the paradigm change entails "changes in the standards governing permissible problems, concepts, and explanations" (105). "The differences between successive paradigms are both necessary and irreconcilable" (102).
"The competition between paradigms is not the sort of battle that can be resolved by proofs" (147), but is more like a "conversion experience" (150). In a scientific revolution (paradigm-shift), "What occurred was neither a decline nor a raising of standards, but simply a change demanded by the adoption of a new paradigm" (107). "In these matters, neither truth nor error is at issue" (150). "We may . . . have to relinquish the notion . . . that changes of paradigm carry scientists . . . closer and closer to the truth" (169).
With the notion of paradigm characterized in this way, Kuhn's use of it seems to suggest not just that standards can change in science, but that there really are no standards in science and that any hope of distinguishing between science and pseudoscience had best be abandoned. In the second edition of The Structure of Scientific Revolutions, Kuhn disowns this highly provocative first edition view that almost all his critics took him to be proposing. As modified in the second edition, his views seem to come to no more than rather bland remarks about the social context of science, for example, "scientists often work in groups, sometimes disagree with members of their own research groups, but may sometimes be able to resolve such intra-group disagreements more easily than inter-group disagreements."
To return to Chaos theory, we can fairly say that it offers some surprises and some hope for gaining a better understanding of a turbulent world, but there seems no point in saying that it involves a new paradigm.
2000 David F. AustinGo to next section: Lessons on Gatekeeping
