Is Finite Precision Arithmetic Useful For Physics?

Both empirical sciences and computations are fundamentally restricted to measurements/computations involving a nite amount of information. These activities deal with the FINITE { some nite precision numbers, coming out from measurements, or from calculations run for some nite amount of time. By way of contrast, as Leibniz expressed it, mathematics is the science of the INFINITE, which contains the concept of continuum. The related concepts of limit points, derivatives and Cantor sets also belong to mathematics, the realm of the in nite, and not to the world of the nite. One is then lead to wonder about the basis for the \unreasonable e ectiveness of mathematics in the natural sciences" (Wigner (1960)). This puzzling situation gave birth, over the centuries, to a very lively philosophical discussion between mathematicians and physicists. We intend to throw into the debate a few simple examples drawn from practice in numerical analysis as well as in nite precision computations. By means of these examples, we illustrate some aspects of the subtle interplay between the discrete and the continuous, which takes place in Scienti c Computing, when solving some equations of Physics. Is Nature better described by discrete or continuous models at its most intimate level, that is below the atomic level? With the theory of quantum physics, it seems that the question has received a signi cant push towards a discrete space. However one can argue equally that the time variable in Schrodinger's equation is continuous. We will not get involved in the scholarly dispute between the continuous and the discrete. Instead, we will show on simple examples taken from Scienti c Computing, the subtlety of the interplay between the continuous and the discrete, which can take place in computations, be it with nite precision or exact arithmetic. 1 Inexact versus exact arithmetic Almost all the mathematical real numbers require, to be exactly represented in a given basis, an in nite amount of digits. Therefore exact computations require an in nite amount of information. On the contrary, with the nite precision arithmetic of the computer, each machine number is represented by a nite number of digits, say p (usually p bits in base 2). Consequently computations in nite precision deal with a nite amount of information. It is reasonable to expect that the result of a calculation done with p digits would tend to the exact result if p would tend to in nity. This is indeed very often the case. For a presentation of 1 C. Calude (ed.). The Finite, the Unbounded and the In nite, Proceedings of the Summer School \Chaitin Complexity and Applications",Mangalia, Romania, 27 June { 6 July, 1995. Journal of Universal Computer Science, vol. 2, no. 5 (1996), 380-395 submitted: 13/5/96, accepted: 13/5/96, appeared: 28/5/96  Springer Pub. Co.