Philosophical Significance of Computational Science

This article focuses not on computer science but on computational science. First we will elucidate some properties of computer software which are the tool of computing. What are the scientific foundations of "computer science"? ACM curriculum '68 stresses mathematics as the foundation of computer science. But ACM curriculum '91 stresses three paradigms, theory, abstraction, and design. The first paradigm of theory actually refers to mathematics. This shift from '68 to '91 means that the mathematical logic (especially Turing Machine model) is not adequate to the foundations of "computer science". Supplemental grounds for this shift can be found in "software engineering" and the program verification debate. Secondly we will deal with some problems of numerical analysis. There are two critical features. First, our electric computer is digital. Indeed ordinary set theory represents both natural number as well as rational and real number. But by reason of rounding error, ordinary set theory does not strictly represent digital numbers on our computers. Second, computational complexity is a real constraint when computers are used in our scientific studies. We generally cannot have any analytic solution of partial differential equations. Therefore we must solve these equations by numerical analysis of partial differential equations which sometimes result in ill-posed problems. All this means that calculation cannot be understood simply in terms of logical reasoning. Under these constraints, numerical simulations are carried out. We examine the history of "the weather prediction by numerical simulation". Some simplifications or modifications of numerical models are necessary to gain solutions. And computational complexity demands the modifications of the numerical models. It is difficult for even the numerical simulations to deal with nonlinear interactions of many factors. So modeling and predictions in simulations demands more than logical operations. Especially predictions that are not any sort of deductions in a strict meaning. Therefore we cannot have any simple criteria of evaluations for the numerical simulations. Usually philosophy of science is interested in the truth or verification of scientific knowledge. But computational science focuses on solutions and predictions. Because of our bounded rationality, it is difficult to reach these solutions and predictions. But it is philosophically interesting and necessary to make the difficulties clear