Back to "Real" Analysis is a Degenerate Case of Discrete Analysis

From wilf at math dot upenn dot edu Sat Mar 10 21:25:35 2007 Z., I was reading something of yours about how analysis is a degenerate case of discrete mathematics, and I was reminded of one of the most beautiful pieces of science that I have ever seen, due to the late, great Indian astrophysicist Subrahmanyan Chandrasekhar. In his elegant book "Radiative Transfer", there is one chapter in which the following sequence of events takes place: 1. He formulates some astrophysical problem as the solution to an integro-differential equation. 2. Despairing of solving that by any standard method, he replaces the integral sign by a numerical approximation, maybe trapezoidal rule with mesh size h, or something of that kind. 3. The resulting system of approximate equations can be solved analytically, by standard linear algebra methods. 4. Now the fun part. In the solution, which is the exact solution of the approximate equations, there is the mesh size h, and he takes the limit as h->0. Successfully. The result is that he has found the *exact* solution of his original integro-differential equation. So what he did was this. Confronted with a tough problem in continuous mathematics, he approximated it by a problem in discrete mathematics, found the exact solution of the approximate problem, and took the limit to make the approximate solution exact. Continuous mathematics was too feeble to solve the problem, so he stood on the shoulders of discrete folks to do it. W.

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