The field of computational science is concerned with generating feasible algorithms to solve mathematical problems, usually those that are important in scientific applications. An important difference between such feasible algorithms and traditional algorithms considered in computability theory is that, in general, they involve various forms of approximation. We will see that there is a common strategy in computational science that can take a problem that is not feasibly computable, and then generate a (more) feasible algorithm to a slightly modified problem. We will see how one, more general, version of this strategy underlies numerical computing, which uses approximations, and how a more restricted version underlies symbolic computing, which is exact. The nature of this feasible computation strategy in these two branches of computational science has some consequences for computability theory. We will also consider its roots in the history of science. It emerges, therefore, that feasible computing is a fundamental part of a great deal of scientific inference, as well as at the core of advanced algorithms for solving problems in computational science.