26 1. Differential Equations and Their Solutions considering) that model the time-evolution of various real-world pro- cesses. So it should not be surprising that the sort of issues that we have just been discussing have important practical and philosophical ramifications when it comes to evaluating and interpreting the pre- dictive power of such laws, and indeed some of the above theorems were developed for just such reasons. At first glance, it might appear that theory supports Laplace’s ringing deterministic manifesto quoted above. But if we examine matters with more care, it becomes evident that, while making de- pendable predictions might be possible for a god who could calculate with infinite precision and who knew the laws with perfect accuracy, for any lesser beings there are severe problems not only in practice but even in principle. First let us look at the positive side of things. In order to make reliable predictions based on a differential equation dx dt = V (x), at least the following two conditions must be satisfied: 1) There should be a unique solution for each initial condition, and it should be defined for all t R. 2) This solution should depend continuously on the initial condition and also on the vector field V . Initial value problems that satisfy these two conditions are often re- ferred to as “well-posed” problems. The importance of the first condition is obvious, and we will not say more about it. The second is perhaps less obvious, but neverthe- less equally important. The point is that even if we know the initial conditions with perfect accuracy (which we usually do not), the finite precision of machine representation of numbers as well as round-off and truncation errors in computer algorithms would introduce small errors. So if arbitrarily small differences in initial conditions resulted in wildly different solutions, then prediction would be impossible. Similarly we do not in practice ever know the vector field V per- fectly. For example, in the problem of predicting the motions of the planets, it is not just their mutual positions that determine the force law V , but also the positions of all their moons and of the great mul- titude of asteroids and comets that inhabit the solar system. If the
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