something that has been hard won over a period of many hundreds, indeed thousands, of years. In mathematics, the only way that you know something is absolutely true is by coming up with a proof. In science, it seems to me the only way you know something is true is by the standard experimental method. You first become educated. You develop hypotheses. You very carefully design an experiment. You perform the experiment, and you analyze it and you try to draw some conclusions."
"Now, I believe absolutely in approximation, but successful approximation should have an error estimate and should lead to some conclusion. One does not typically see this in fractal geometry. Mandelbrot himself says that typically what one does is one draws pictures."
Indeed, Devaney remarked, "many of the lectures one hears are just a bunch of pretty pictures without the mathematical content. In my view, that is tremendously unfortunate because there is a tremendous amount of beautiful mathematics. Beyond that there is a tremendous amount of accessible mathematics."
Yet these concerns seem relatively minor compared to the insights that have been achieved. It may well distress mathematicians that claims for chaos have been exaggerated, that the field has inspired some less than rigorous work, and that the public may marvel at the superficial beauty of the pictures while ignoring the more profound, intrinsic beauty of their mathematical meaning, or the beauty of other, less easily illustrated, mathematics.
But this does not detract from the accomplishments of the field. The fundamental realization that nonlinear processes are sensitive to initial conditions—that one uses linear approximations at one's peril—has enormous ramifications for all of science. In providing that insight, and the tools to explore it further, dynamical systems is also helping to return mathematics to its historical position, which mathematician Morris Kline has described as "man's finest creation for the investigation of nature" (Kline, 1959, p. vii).
Historically, there was no clear division between physics and mathematics. "In every department of physical science there is only so much science . . . as there is mathematics," wrote Immanuel Kant (Kline, 1959, p. vii), and the great mathematicians of the past, such as Newton, Gauss, and Euler, frequently worked on real problems, celestial mechanics being a prime example. But early in this century the subjects diverged. At one extreme, some mathematicians affected a fastidious aversion to getting their hands dirty with real problems. Mathematicians "may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from