John R. Rice

Function Theory: Mathematical and Real World

Two frameworks for function theory are discussed, the traditional mathematical framework and one more suited for real world functions. The shortcomings of the mathematical framework can be summarized as
(1) its dependence on polynomials conflicts with intuitive ideas of complexity and
(2) it is founded on set theory rather that computation.
Splines provide an alternative basis for function theory which is compatible with intuition and computation. In 1970 Collatz and Rice made an informal conjecture: The work to solve a PDE numerically is no more than the work to tabulate a closed form solution. This conjecture could not be stated precisely in the traditional mathematics framework and was never publicized. We examine this conjecture in more detail using a "real world" framework for function theory based on splines and computation. Considerable evidence has been developed to support this conjecture even though it still does not have a "clean" statement.