STEVE SMALE Finding a I lorseshoe on the Beaches of Rio

What is Chaos? A mathematician discussing chaos is featured in the movie Jurassic Park. James Gleick's book Chaos remains on the best-seller list for many months. The characters of Tom Stoppard's celebrated p l ayArcad i a discourse on the meaning of chaos. What is the fuss about? Chaos is a new science that establishes the omnipresence of unpredictability as a fundamental feature of common experience. A belief in determinism, that the present state of the world determines the future precisely, dominated scientific thinking for two centuries. This credo was based on mechanics, where Newton's equations of motion describe the trajectories in time of states of nature. These equations have the mathematical property that the initial condition determines the solution for all time. This was taken as proving the validity of deterministic philosophy. Some went so far as to see in determinism a refutation of free will and hence even of human responsibility. At the beginning of this century, with the advent of quantum mechanics, the untenability of determinism was exposed. At least on the level of electrons, protons, and atoms, it was discovered that uncertainty prevailed. The equations of motion of quantum mechanics produce solutions that are probabilities evolving in time. In spite of quantum mechanics, Newton's equations govern the motion of a pendulum, the behavior of the solar system, the evolution of the weather, many macroscopic situations. Therefore the quantum revolution left intact many deterministic habits of thought. For example, well after the Second World War, scientists held the belief that long-range weather prediction would be successful when computer resources grew large enough. In the 1970s the scientific commtmity recognized another revolution, the theory of chaos, which seems to me to deal a death blow to the Newtonian picture of determinism. The world now knows that one must deal with unpredictability in understanding common experience. The coin-flipping syndrome is pervasive. "Sensitive dependence on initial conditions" has become a catchword of modern science. Chaos contributes much more than extending the domain of indeterminacy, just as quantum mechanics did more than half a century earlier. The deeper understanding of dynamics underlying the theory of chaos has shed light on every branch of science. Its accomplishments range from analysis of electrocardiograms to aiding the construction of computational devices. Chaos developed not from newly discovered physical laws, but by a deeper analysis of the equations underlying Newtonian physics. Chaos is a scientific revolution based on mathematics----deduction rather than induction. Chaos takes the equations of Newton, and uses mathematical analysis to establish the widespread unpredictability in the phenomena described by those equations. Via mathematics, one establishes the failure of Newtonian determinism by using Newton's own laws!