SOURCE: A review of Chaos, in Mathematical Intelligencer, Vol. 11, No. 1, Winter, 1989, pp. 65–69.

[In the following unfavorable review of Chaos, Franks disparages the notion of a “chaos revolution” and objects to Gleick's misrepresentation of chaos theory, fractal geometry, and mathematical methodology.]

[Chaos] is a book about new ways in which mathematics is used to model phenomena in the real world. It is intended for a general audience. The author is James Gleick, formerly a science reporter for the New York Times. He does a good job explaining what constitutes a mathematical model (by which he means a differential equation or a difference equation) and what it does. The theme of the book is that even rather simple non-linear models can, and typically do, exhibit extremely sensitive dependence on initial conditions. What this means is that two solutions of a non-linear ordinary differential equation can start with very close initial conditions but then diverge rapidly from one another while remaining in a bounded region. The effect is that after a moderate amount of time the position of a solution can appear to be a random function of the initial condition. This seems quite paradoxical because the differential equation certainly generates a deterministic system.

Only recently have scientists begun to realize the extent to which simple dynamical systems produce this random-seeming, complex, “chaotic” behavior. The story that this book recounts is the sometimes painful process of acquiring this understanding and the histories of the scientists involved. The result, in Gleick's view, has been a revolution in the way science views nature—a “paradigm shift.” The book is subtitled “Making a New Science.” We are told, “Where chaos begins classical science stops” and “Chaos poses problems that defy accepted ways of working in science.” This revolution, Gleick says, was carried out in the face of numerous obstacles by “a few free-thinkers, working alone, unable to explain where they are heading.” These stories of individual achievement, in diverse disciplines but with a common theme, are the heart of his book.

We are presented with an exciting and romantic story, but I suspect that Gleick has greatly overestimated the achievements of chaos theorists. Only time will tell. Also I think he missed the obvious explanation of the sudden interest in chaos in many branches of science. Thirty years ago there was very little a scientist could do with a typical non-linear model. It is generally impossible to solve non-linear differential equations. And it is not even clear what a scientist could do with a complicated, explicit, closed-form solution if one could be found. It would probably be no more enlightening than the original differential equation. As a result researchers have emphasized linear systems, which are much more tractable but greatly limited in the range of phenomena they can model. This may even have reached the point, as Gleick suggests, of scientists training themselves not to see non-linearity in nature.

The necessity of this oversimplification changed with the advent of inexpensive, easy-to-use digital computers. Now one can numerically approximate the solutions of non-linear differential or difference equations and, equally important, graphically display the results. The computer is a viewing instrument for mathematical models that will, in the long run, be more significant than the microscope to a biologist or the telescope to an astronomer. Nearly all of the scientists whose work is discussed in this book made heavy use of computers and were among the first to do so in their discipline. It is no more surprising that numerous types of complex dynamical phenomena have been discovered in the last twenty years than would be the discovery of numerous kinds of bacteria if thousands of biologists were, for the first time in history, given microscopes.

Most researchers now understand that seemingly random behavior is an inherent element of non-linear dynamical systems and not just the result of experimental error or “noise.” This is an important insight. I would suggest that its widespread acceptance has more to do with readily accessible computers than the brilliance or courage of Gleick's protagonists. What is surprising and fascinating is the resistance to accepting it that is documented in this book. Indeed, it was well known to Henri Poincaré and George David Birkhoff, but only when computers made it impossible to ignore did it gain acceptance.

We are probably nowhere near the end of the exciting new insights into nature that can be gained by computer viewing of mathematical models. All of the models discussed in Chaos: Making a New Science are relatively simple ordinary differential equations or difference equations. As computers become more powerful and supercomputers become widely accessible and easy to use, we can expect similar insights into non-linear partial differential equations. If non-linear ordinary differential equations are intractable without computers the typical non-linear partial differential equation is hopeless. The breakthroughs came first in simple ordinary differential equations because a personal computer (in some cases even a hand calculator, as we learn in this book) is adequate to experiment with them. Much greater computational power is necessary for partial differential equations, because their numerical solution requires many more arithmetical steps.

Despite these advances, I would speculate that the real evolution in scientific modeling and the greatest impact of computers on science is probably yet to come. Computers have made it easy to view the solutions of non-linear differential equations. But now that we have computers it is no longer clear that differential equations are the best models to use, especially in areas like biology or the social sciences. One can envisage a wholly new mathematical construct, better suited to digital computers, which could provide a new type of model. The emergence of such a construct capable of accurately modeling a variety of phenomena would indeed be a true paradigm shift. Nothing this dramatic appears to have happened yet.

There is indeed a revolution in progress, but it is not, as this book suggests, the “Chaos revolution.” Instead it is the computer revolution, and the current popularity of chaos is only a corollary. This revolution may still be in its infancy, but computers have already taught us one remarkable fact. There is mounting evidence that sensitive dependence on initial conditions may well be as close to the rule as to the exception for non-linear dynamical systems. This is a surprising fact, well documented in this book. It surprises us because it was invisible before the computer, but with computers it is easy to see, even hard to avoid.

I would have liked to see more information on the mathematics behind chaos. Much of this mathematics can be readily described at the same technical level as the rest of the book. For example, a discrete-time dynamical system discovered by Stephen Smale and commonly called the “Smale horseshoe” is described, but its significance is never discussed.¹ It was one of the first examples of sensitive dependence on initial conditions to be completely and rigorously understood. But more important, a theorem of Smale shows that the horseshoe or its analogue for continuous time systems occurs as a subset of almost any system that displays chaotic behavior.

An even more serious omission is the absence of any discussion of the close connection of the Smale horseshoe with the doubling map on the interval. The doubling map is the function on the unit interval whose value at x is defined to be the fractional part of 2x. An understanding of this simple dynamical system goes a long way toward demystifying the paradox of deterministic randomness. If we think of x as written as a binary number with a leading “decimal point” then this function simply shifts the decimal point to the right one place and deletes the first digit. Clearly, if an initial x is chosen whose value is significant to 16 binary places, then after 16 iterates its value will appear to be completely random and independent of the starting value. This is essentially the same mechanism that underlies the complex behavior of most chaotic dynamical systems. The opportunity Gleick missed here was to give the non-specialist reader a real insight into the nature of chaos. The doubling map is readily understandable and closely related to the dynamics of the Smale horseshoe. The Smale horseshoe, in turn, is an essential ingredient of almost all chaotic dynamics!

I was also surprised to note that, despite a lengthy discussion of the Lorenz attractor, there was no mention of the work of John Guckenheimer and Robert Williams concerning this remarkable example. It is a great achievement to be the first to observe an important phenomenon in nature, but it is equally important to be the first to understand and explain it. Edward Lorenz wrote a system of three first-order ordinary differential equations as a simplified model of atmospheric convection. When he studied the system with a computer he observed all the hallmarks of chaos: very complicated trajectories of solutions, sensitive dependence on initial conditions, seemingly random behavior in a deterministic system, etc. His paper² on this example was published in the Journal of Atmospheric Science in 1963. Gleick tells of the extraordinary influence that the work of Lorenz, and this example in particular, has had in the development of non-linear dynamics. However, there is no mention of the work of mathematicians who explained the remarkable things Lorenz had discovered and catalogued. This work by Guckenheimer and Williams³ represents one of the outstanding achievements of mathematics in the understanding of so-called strange attractors. It reduces the dynamics of a system of three non-linear differential equations to the study of a class of functions from the interval to itself, in fact, one not unlike the doubling map described above. This omission is especially surprising because one of Gleick's main points is that nature seems to have penchant for one-dimensional dynamics, but the one-dimensionality is often well disguised. Guckenheimer and Williams certainly showed this to be the case for the Lorenz attractor.

The sections on fractal geometry and its inventor, Benoit Mandelbrot, are the only parts of the book that have little to do with chaotic dynamics. Fractal geometry is a static theory only marginally related to any kind of dynamics. Mathematicians may find it disconcerting to learn that the term fractal geometry does not refer to a body of mathematics, as, for example, the terms projective geometry or algebraic geometry do. In fact, there appears to be no formal mathematical definition of the word fractal. This book mentions no theorems in fractal geometry. A highly regarded mathematical text⁴ with the title The Geometry of Fractal Sets deals primarily with mathematics developed prior to 1950, which is certainly not what Gleick is describing. There may be recent mathematical results in fractal geometry, but, if so, they clearly play a secondary role. Instead, fractal geometry is viewed by its proponents as a new framework and set of tools for describing nature. Gleick tells us, “In the end, the word fractal came to stand for a way of describing, calculating, and thinking about shapes that are irregular and fragmented, jagged and broken-up.”

The principal tenet of this descriptive framework is that nature is extremely irregular and the degree of irregularity remains constant when viewed on different scales. That is, natural phenomena exhibit a self-similarity of complexity across scales. “Above all fractal meant self-similar,” according to Gleick. The most important tool for fractal geometry is Hausdorff dimension. This is a numerical invariant of metric spaces that is defined as a limit of numbers obtained from covers of the space by smaller and smaller balls. For manifolds it equals the topological dimension, but in general it is not an integer.

The connection of Hausdorff dimension with the ordinary concept of dimension is somewhat tenuous, but to the uninitiated the idea that an object can have a dimension of 1.2618 has a kind of science-fiction-like appeal. It would have been better if the word “dimension” had never been attached to this number. Often in mathematics or physics a common word will assume a special or technical meaning sometimes quite different from its usual sense. The use of the word dimension here is one example, and I am told that “universal” is used by physicists in a different sense than its common use. Any exposition of mathematics or physics for non-specialists should take pains to explain which words are being used in an unusual way. Unfortunately, this has not been done by Gleick in this book nor in general by the adherents of fractal geometry. On the contrary, I feel that the fact that Hausdorff dimension assumes fractional values may have been emphasized to glamorize the concept. This sort of mystification should be avoided in science, rather than catered to. Incidentally, Gleick suggests that mathematicians prefer the term Hausdorff dimension to fractal dimension or fractional dimension because they spitefully want to deny credit to Mandelbrot. This is both false and insulting.

One of the principles of fractal geometry holds that Hausdorff dimension is an important measure for physical objects.

As Mandelbrot himself acknowledged, his program described better than it explained. He could list elements of nature along with their fractal dimensions—seacoasts, river networks, tree bark, galaxies—and scientists could use those numbers to make predictions.

The mathematical definition of Hausdorff dimension, involving limits as size goes to zero, may not make sense for a physical object, but one can make similar philosophical objections to measuring length. In any case, Gleick tells us,

Mandelbrot specified ways of calculating the fractional dimension of real objects, given some technique of constructing a shape or given some data, and he allowed his geometry to make a claim about the irregular patterns he had studied in nature. The claim was that the degree of irregularity remains constant over different scales. Surprisingly often, the claim turns out to be true. Over and over again, the world displays a regular irregularity.

Like fractal geometry this book describes better than it explains. The reader may be entertained, but is unlikely to gain new insights into nature. My greatest disappointment was the way in which mathematics is portrayed. One could read this book and come away with the view that mathematical proofs are an obstacle to the pursuit of truth—a sort of self-imposed mental straitjacket worn by stodgy old pedants. This probably overstates Gleick's view somewhat, but he definitely feels that mathematics has greatly suffered from an interest in rigor. A reader for whom the concepts of proof and rigor are vague at best could easily conclude they are better avoided. I had hoped for a more sympathetic view of the discipline.

Mathematics has a methodology unique among all the sciences. It is the only discipline in which deductive logic is the sole arbiter of truth. As a result mathematical truths established at the time of Euclid are still held valid today and are still taught. No other science can make this claim. The phrase “mathematical certainty” is commonly used in general discourse to represent the highest standard of truth. A theorem proven today may be forgotten in the future because it is uninteresting but it will never cease to be true (barring drastic changes in our standards of rigor). I would contend that an important criterion for judging a scientific discipline is the half-life of its truths. Mathematics does extremely well by this measure and mathematicians are justifiably proud that their standards of truth are higher than those of other sciences.

The use of deductive logic rather than empiricism is taken for granted by mathematicians to such an extent that they rarely contemplate alternatives. Scientists in other disciplines seldom appreciate this methodology and in some cases even disdain it. With few exceptions, Gleick's treatment of mathematics echoes this attitude. Rigor is blamed for increased narrow specialization in mathematics and for decreased contact between mathematicians and other disciplines. This seems to me to be implausible; in particular I note no lack of narrow specialization in empirical disciplines and no greater tendency toward interdisciplinary research.

In fairness, these views may only be reflections of the attitudes of some of the scientists interviewed by Gleick. We are told, for example, that Mandelbrot felt it necessary to emigrate from France because of the stifling influence of Bourbaki. Curiously, this book contains a fair amount of Bourbaki bashing—not because of Bourbaki's pedagogical style, where, in my view, it might be deserved, but for their rigor.

The methodology of mathematics does raise valid philosophical questions that need more attention. There is an apocryphal story of a meeting between a youthful Albert Einstein and an aging Henri Poincaré at which Einstein said, “I considered taking up mathematics, but I decided against it because there is often no connection with the real world in mathematics and it is impossible to tell what is important.” To which Poincaré replied, “Well, in my youth I considered becoming a physicist, but I decided against it because in physics it is impossible to tell what is true.” Clearly there is a tradeoff between high standards of truth and applicability. Sadly, this issue attracts little attention from either side.

This book contains considerable discussion of the dynamics of complex functions. I believe that, as far as we know today, this is pure mathematics that models nothing in nature, yet has great appeal for even the most down-to-earth experimentalist. It is difficult to view pictures of Julia sets or Mandelbrot sets without becoming something of a Platonist. These sets seem to have a reality of their own even if they don't reflect some part of what we glibly refer to as the real world.

What constitutes a good mathematical model? This question needs to be paid much more attention by all scientists. How do we judge if a given piece of mathematics accurately reflects some aspect of nature? This is not a mathematical question; at least, it is not a question whose answers are subject to proof. Historically the ability to make non-trivial quantitative predictions has been an important test for models. My own view is that a good model must explain, at least to some extent. For example, one could videotape some phenomenon, digitize the result, and use that to produce a “mathematical model” capable of reliably describing the process. But such a model would add nothing new to our understanding. Similarly, it is not enough to find a mathematical system that exhibits similar behavior to a physical experiment, but has no apparent connection with it.

I fear that many of the models of chaos may be faulted for this deficiency. In reading this book I was repeatedly struck by the parallels between chaos and catastrophe theory. Gleick quotes extravagant claims of chaos proponents, such as “twentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos.” Fifteen years ago similar claims were made for catastrophe theory. I must admit that I was one who at least entertained the possibility that those claims were correct and that catastrophe theory represented a new “paradigm shift.” Now such a view seems extremely improbable.

Gleick himself yields to the temptation to engage in hyperbole on behalf of chaos. After a brief account of the inadequacy of pre-chaos science in dealing with complex natural phenomena, he says,

Now all that has changed. In the intervening twenty years, physicists, mathematicians, biologists, and astronomers have created an alternative set of ideas. Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. And most important, the laws of complexity hold universally, caring not at all for the details of a system's constituent atoms.

Despite having carefully read this book, I do not know to what he refers when he speaks of universal laws of complexity.

Like chaos, catastrophe theory was highly interdisciplinary, claimed breakthroughs in numerous areas, was warmly received by researchers in several disciplines, and received a great deal of media attention. Both chaos and catastrophe theory are based on some interesting and beautiful mathematics, dynamical systems and singularity theory, respectively. But both have been rather phenomenological in their modeling, finding a similarity in a mathematical model and a physical phenomenon to be an adequate basis for adopting that model.

In discussions with colleagues I have encountered considerable objection to the comparison of chaos to catastrophe theory. Some feel it is unfair to chaos while others think it is unfair to catastrophe theory. Probably both groups are right. Comparing catastrophe theory to fractal geometry may be unfair to catastrophe theory since it has a great deal more substantial mathematics behind it than fractal geometry. On the other hand a comparison between catastrophe theory and non-linear dynamical systems may under-rate the impact of non-linear dynamics. The insight that seemingly random, complex behavior is inherent in non-linear dynamics and not a result of error or noise is an important one. For me it is difficult to think of it as a recent breakthrough or a revolutionary idea since it was taught to me in a routine fashion over twenty years ago when I was a graduate student. But the full impact of this idea on the conduct of scientific research is only now being felt. Catastrophe theory has not had a comparable impact.

Researchers in numerous disciplines are now obliged to study non-linear dynamics if they hope to provide good mathematical models. They have to know about disorder if they are going to deal with it. This new attitude is a very good thing—I believe that non-linear dynamics is the place most researchers should be looking for models. Unfortunately, our knowledge of chaotic systems, beyond the fact that they exist in profusion, is still extremely limited. This is true in both a theoretical and a practical sense. I am concerned that extravagant claims like those quoted above raise unrealistic expectations that have no chance of being met in the near future. Despite its portrayal in this book, chaos is not a new tool that can solve the problems of every discipline. It is difficult for professional scientists, much less the general public, to distinguish excessive hype from solid scientific advances. Chaos has given us both.

Whether in the long run chaos enjoys a greater success than catastrophe theory in providing useful mathematical models remains to be seen. This judgment will not be made by mathematicians or by popular science writers. It will be made by the practitioners of the various disciplines to which the techniques of chaos are being applied. And the criterion by which chaos will be judged is the quality of the models it provides for physical phenomena.

Notes

1. S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (S. S. Cairns, ed.), Princeton: Princeton University Press (1963), 63–80.

2. E. N. Lorenz, Deterministic non-periodic flow, Jour. Atmos. Sci. 20 (1963) 130–141.

3. J. Guckenheimer and R. F. Williams, Structural Stability of Lorenz Attractors, Publ. Math. IHES, 50 (1979) 59–72.

4. K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. #85, Cambridge University Press (1985).