BOX 7.1 COMPUTATION
The new terrain to be explored by computation is not a recent discovery, but rather comes from a recognition of the limits of calculus as the definitive language of science. Ever since the publication of Newton's Philosophiae naturalis principia mathematica in 1687, "calculus has been instrumental in the discovery of the laws of electromagnetism, gas and fluid dynamics, statistical mechanics, and general relativity. These classical laws of nature have been described," Smarr has written, by using calculus to solve partial differential equations (PDEs) for a continuum field (Smarr, 1985, p. 403). The power and also the limitation of these solutions, however, are the same: the approach is analytic. Where the equations are linear and separable, they can be reduced to the ordinary differential realm, and calculus and other techniques (like the Fourier transform), continued Smarr, "can give all solutions for the equations. However, progress in solving the nonlinear PDEs that govern a great portion of the phenomena of nature has been less rapid" (Smarr, 1985, p. 403).
Scientists refer to the results of their analysis as the solution space. Calculus approaches the solution of nonlinear or coupled differential equations analytically and also uses perturbation methods. These approaches have generally served the experimenter's purpose in focusing on a practical approximation to the full solution that—depending on the demands of the experiment—was usually satisfactory. But mathematicians down through history (Leonhard Euler, Joseph Lagrange, George Stokes, Georg Riemann, and Jules Henri Poincaré, to name only the most prominent) have not been as concerned with the practicalities of laboratory experiments as with the manifold complexity of nature. Smarr invoked mathematician Garrett Birkhoff's roll call of these men as predecessors to mathematician and computational pioneer John von Neumann, whose "main point was that mathematicians had nearly exhausted analytical methods, which apply mainly to linear differential equations and special geometries" (Smarr, 1985, p. 403). Smarr believes von Neumann "occupies a position similar to that of Newton" in the history of science because he realized the profound potential of computation to expand science's power to explore the full solution space; as Birkhoff put it, "to substitute numerical for analytical methods, tackling nonlinear problems in general geometries" (p. 403).
These distinctions are not mere mathematical niceties of scant concern to working scientists. The recognition that a vaster solution space may contain phenomena of crucial relevance to working physicists—and that computers are the only tools to explore it—has revitalized the science of dynamical systems and brought the word chaos into the scientific lexicon. Smarr quoted another mathematician, James Glimm, on their significance: "Computers will affect science and technology at least as profoundly as did the invention of calculus. The reasons are the same. As with calculus, computers have increased and will increase enormously the range of solvable problems'" (Smarr, 1985, p. 403).