The founding father of computational mathematics is von Neumann, who first set the agenda for the development of programmable electronic computers with large memory, as well as programming languages, schemes of discretising continuum equations, and algorithms for solving the discretised equations. Everybody with the least familiarity with computing is aware of the continuing phenomenal improvement in the last fifty years of computer speed, memory capacity, software and graphics capabilities. Problems that at one time have strained computing capacities could, a few years later be done routinely, cheaply and fast. But most people are unaware of how much of this progress is due in equal measure to new mathematical ideas for tackling computational tasks. No doubt many more such mathematical shortcuts remain to be discovered.
Theorists can use computing in the manner of the experimentalists.
Perhaps the most astonishing discovery through numerical experimentation is Odlyzko's exploration of the connection between the distribution of the zeros of Riemann's zeta function and the distribution of the eigenvalues of random matrices. To prove rigorously what has been discovered by computation is a tremendous challenge of the future. Computing has also made its mark as an aid in proofs. The most spectacular example is Haken and Appel's demonstration of the four color theorem, where the computer is required to perform exact calculations. But computing is also an integral part of the logical structure of some analytical proofs, where the computations are only approximate.