SOURCE: “Chaos, Rigor, and Hype,” in Mathematical Intelligencer, Vol. 11, No. 3, Summer, 1989, pp. 6–8.

[In the following essay, Hirsch objects to Gleick's misrepresentation of chaos theory in Chaos and his failure to focus on the contributions of mathematicians, particularly Stephen Smale, toward a scientific understanding of chaos.]

Gleick's book Chaos [reviewed by John Franks, Mathematical Intelligencer, vol. 11, no. 1, 1989] captures vividly and faithfully the personalities of the researchers, the atmosphere they worked in, and the spirit of the times (as far as I can tell—I don't know all the people in the book), as well as skillfully expounding many fascinating themes of current research.

But to my mind the book has one great defect: It doesn't do justice to the rigorous mathematics underlying a great deal of the research in nonlinear dynamical systems. In fact a nonmathematician or even a mathematician unfamiliar with the material would find it hard to tell that rigorous mathematical proof—as contrasted with conjecture, heuristic, experiment, and computer simulations—played a vital part in this research.

This book severely underrates the importance of rigorous mathematics and its influence on the understanding of chaotic dynamics. It is not easy to learn from it that many important chaotic systems, including the earliest and the most influential examples, were first identified and explored not by computer simulation, and not by physical experiment, but by mathematical proof (Poincaré, Birkhoff, Levinson, Smale, Anosov, Kolmogorov, Arnold, Moser …).

These and many other mathematicians achieved by rigorous mathematical analysis crucial insights into what is now called chaos. It is difficult to imagine that what they discovered could have been found through any kind of experimentation any more than the existence of irrational numbers—which was even more astonishing when it occurred—could have been discovered by computation. By ignoring this, Gleick missed an opportunity to attack the public's profound ignorance about the role and nature of mathematics; he has also misrepresented a chapter in the history of mathematics. Let me illustrate what I mean with Smale's work on horseshoes, which is far more important than the book suggests.

In discussing the horseshoe Gleick writes of Smale's “intuition,” claiming that he “turned his ideas about visualizing global behavior into a new kind of model”; he “put his horseshoe through an assortment of topological paces,” and “the horseshoe provided a neat visual analogue of the sensitive dependence on initial conditions that Lorenz would discover in the atmosphere a few years later”; “Smale's horseshoe stood as the first of many new geometrical shapes that gave mathematicians and physicists a new intuition about the possibilities of motion. In some ways, it was too artificial to be useful. …”

But the only mention of mathematics in this discussion is “mathematics aside”! Thus the reader learns that Smale had a new “model” (of what?), but it was merely an “analogue” of what Lorenz would actually “discover.” The horseshoe is “an enduring image,” but it was “artificial”; the only reason it is famous is because it caused a “paradigm shift.”

This is profoundly misleading. It conveys little of what Smale accomplished, what its importance is, why it is has been so influential. The horseshoe is important because Smale proved very interesting things about it, and these theorems tell us important facts about many other dynamical systems, facts that could not have been found by intuition, simulation, or experiment. In these theorems and facts lies the importance of Smale's work. The “paradigm shift” was due chiefly to them, not merely to a “new kind of model,” “a neat visual analogue,” or a “new geometrical shape” that was “too artificial to be useful.”

Smale proved the following things about the horseshoe map:

1. He proved it is chaotic. For example, arbitrarily near any point there are periodic points with arbitrarily high periods, and there is also a point whose orbit comes arbitrarily close to every other point—a precise and extreme form of sensitive dependence of long-term behavior on initial conditions.

2. He proved the horseshoe is structurally stable. This is not something that can be discovered or verified from calculations or simulations.

3. He proved that any system having a “transverse homoclinic orbit” (a concept due to Poincaré) must contain a horseshoe as a subsystem; such systems are thereby proved chaotic. This is a profound result because in many systems coming from physics, biology, etc., it is comparatively easy to demonstrate the existence of such orbits. This result has been used many times to prove that horseshoes are embedded in many particular systems, thus revealing their chaotic nature.

4. He proved that the chaotic dynamics of the horseshoe is isomorphic to the dynamics of the shift map in the space of bi-infinite sequences of zeroes and ones (so-called “symbolic dynamics”). Now, the dynamics of the shift map are quite transparent; for example, it is easy to see that periodic orbits are dense. Smale's isomorphism immediately opened up the possibility of similarly analyzing other chaotic systems. A great deal of research has since been done on this, greatly enlarging our understanding of chaotic dynamics.

5. He proved horseshoes exist in all dimensions greater than or equal to 2. Therefore, structurally stable chaotic systems exist in great abundance in those dimensions (for flows one more dimension is needed). This was new and striking information. Earlier examples of chaotic systems, such as Birkhoff's, were strictly 3-dimensional.

These discoveries and the subsequent work they inspired gave strong impetus to the now prevalent belief that chaos is a common phenomenon. They are different in kind from the simulations of Lorenz and others, have had a different kind of influence, and have led to different insights. Today finding horseshoes is one of the chief ways of analyzing the dynamics of a chaotic system and one of the very few ways of rigorously demonstrating chaos. Far from being merely an artificial image, the horseshoe is a natural source of chaos and a basic tool for investigating it in all fields of applied dynamics.

Many of the new insights into nonlinear dynamics arising from the work of many mathematicians would probably not have been discovered (or appreciated, if they had been discovered) without their rigorous mathematical foundations. Moreover, these discoveries are not isolated facts; many of them are parts of extensive theories that have been shown to apply to many fields of science. In fact, they have also been used in reverse to obtain new theorems about nonchaotic systems. In contrast, the insight provided by a simulation such as Lorenz's, no matter how striking, does not extend very far to other systems.

I do not mean to denigrate the importance and influence of simulations and intuitions such as that of Lorenz and many others. The chaotic phenomena they have discovered are important, seemingly quite different from horseshoes, and apparently less tractable to rigorous analysis.

Curiously, the horseshoe has sharpened appreciation of the Lorenz system, because the latter seems highly resistant to a rigorous analysis. I believe that only recently has it been proved that it has periodic orbits of arbitrarily high period. The horseshoe is a tame kind of chaos; compared to it the Lorenz system seems quite wild.

I've discussed Smale's work because I know it well and it's extremely important. But there are many other examples of rigorous mathematics leading to insights about chaos. For example, Birkhoff proved what Poincaré had conjectured: the existence in the restricted planar 3-body problem of infinitely many periodic orbits. Gleick mentions many names, but again he fails to distinguish between computer simulation (e.g., by Ulam) and rigorous mathematics (e.g., by Kolmogorov).

Both Poincaré and Birkhoff deserve much more space than Gleick gives them. And why isn't poor Levinsion's name revealed on page 48? After all, he sent Smale the letter with the “robust and strange” example, with “stability and chaos together”; following which, Smale's disbelief “slowly melted away.” The discoverer of such an important example deserves credit. He's mentioned without explanation on page 182, but he is not indexed.

Another difficulty with this influential book, as Franks emphasized, is that Gleick hyperbolizes chaos into “a way of doing science,” “a method, a canon of beliefs” (p. 38), something with “universal laws” (p. 304). What does this mean? What are these laws?

A remark of René Thom is pertinent here: “It is to be expected that after the present initial period of wordplay, people will realize that the term ‘chaos’ has in itself very little explanatory power, as the invariants associated with the present theory—Lyapunov exponents, Hausdorff dimension, Kolmogorov-Sinai entropy—show little robustness in the presence of noise” (Behavioral and Brain Sciences, 10 (2) (June 1987), 182).

Thom was criticizing the hypothesis of chaotic dynamics in a biological model. Indeed, I have observed many biologists looking to chaos as a magic key to unlock nature's secrets. Inquiry revealed that in fact they had a superficial knowledge of nonlinear dynamics and a respect for its difficulties. They had the impression, however, that there is a “theory of chaos” that would somehow solve their mathematical problems. I told them that a chaotic system is really bad news—if your system seems chaotic, perhaps it is, but you probably can neither prove nor disprove this, nor can you accurately simulate its long-term behavior. In any case, to call a system chaotic is not to say much more than that it is nonlinear and complicated.

This was not a welcome message. After all, they had heard wonderful things about the wonderful world of chaos. Now Gleick tells them that chaos is indeed a “method,” with “universal laws.” Such hype prejudiced a generation of biologists against the truly original ideas in Thom's catastrophe theory. Nonlinear dynamics is in no danger of a similar fate; but exaggerated claims may send some scientists on a wild goose chase, while discouraging others from seeking the valuable insights that dynamics can offer.

The foregoing complaints are of interest only to mathematicians. The defects I ascribe to the book will not interfere with anyone else's pleasure in reading it. Let me emphasize that, despite my criticisms, I think Gleick's book is really first rate. Anyone who has tried to explain mathematical concepts to nonscientists will appreciate how remarkable is its exposition of difficult ideas. But I wish he had given less publicity to the nonexistent science of chaos and more to rigorous mathematics.