What is an Answer? An essay on the Theory of Functions

Historically the fields of numerical and symbolic methods for scientific applications have been widely separated, nearly isolated from one another. Symbolic (and algebraic) computation has used symbolic methods to obtain symbolic answers to problems. Numerical computation has used numerical methods to obtain numerical answers. The natures of the methods and answers obtained are seen by most to be completely distinct. One goal of this paper is to raise doubts that any such distinction actually exlsts. The focus is on problems whose answers are functions, e.g., temperature distributions or pressure fields, from phenomena modeled by differential or integral equations. A wide range of mathematical models exist for functions, both from mathematics and computer science. All these theories are candidates to be used to define answers, so we examine the nature of functions that are useful to obtain from either symbolic or numeric computation. We conclude that all these theories for functions fail to model the real world properly.