Formal methods in mathematics

Abstract

Formal methods in computer science seems to have begun with the observation of people like John McCarthy that programming languages usually lack a precise mathematical semantics and that the latter was necessary in order to even pose important questions like compiler correctness. The fact that people appear to perform meaningfully computations using these languages does not guarantee their well definedness. An episode from the history of mathematics provides a useful comparison. The first modern advance in European mathematics over the ancients and the work of Islamic scholars was the discovery of a formula for the solution of the general cubic (degree 3) and quartic (degree 4) algebraic equations in the sixteenth century. This appears to be the first new discovery in mathematics to be found in Europe in over a millenia! The general solution of the quadratic equation (degree 2) which we all learn in high school goes back to the Babylonians several thousands of