The "LAPLACE demon", an intelligence capable of knowing the position and the velocity of every particle in the universe and so, using the laws of classical physics, knowing all their future states, seemed in the last century to pose a contradiction between the laws of physics and the freedom of the human will. In addressing this apparent paradox, MAXWELL, BOUSSINESQ and SAINT-VENANT were led to consider aspects of the theory of systems of differential equations. These aspects, though for different reasons, have come to assume a central place in the modern theory of dynamical systems.
1. The Laplace Demon
LAPLACE in his Essai philosophique sur les ProbabilitJs [10], in the section headed De la ProbabiBtO, includes a passage which is here given in full in a new translation.
"All events, even those which by dint of their pettiness seem not to participate in the great laws of nature, are consequences of these as surely as are the revolutions of the Sun. In our ignorance of the ties binding together the whole system of the universe, we have attributed them to final causes or to chance, depending upon whether they arise and succeed one another in a regular way or without apparent order; but these imaginary causes have been more and more displaced along with the limits of our knowledge, and vanish completely in the face of sound philosophical thought, which sees them merely as the expression of our ignorance as to the true causes. "Real events and their antecendents have a connection based on the manifest principle that nothing can come into being without a cause to produce it. This axiom, known as the principle of sufficient reason, applies even to occurrences that seem quite trivial. Thefreest will, in the absence of an efficient