Computing and Proving

Abstract

Computer mathematics is the enterprise to represent substantial parts of mathematics on a computer. This is possible also for arbitrary structures (with non-computable predicates and functions), as long as one also represents proofs of known properties of these. In this way one can construct a ‘Mathematical Assistant’ that verifies the well-formedness of definitions and statements, helps the human user to develop theories and proofs.

An essential part of the enterprise consists of a reliable representation of computations f(a) = b, say for a, b in some concrete set A. We will discuss why this is so and present two reliable ways to do this. One consists of following the trace of the computation in the formal system used to represent the mathematics ‘from the outside’. The other way consist of doing this ‘from the inside’, building the assistant around a term rewrite system. The two ways will be compared.

Other choices in the design of a Mathematical Assistant are concerned with the following qualities of the system

• 1. reliability;

• 2. choice of ontology;

• 3. choice of quantification strength;

• 4. constructive or classical logic;

• 5. aspects of the user interface.

These topics have been addressed by a number of ‘competing’ projects, each in a different way. From many of these systems one can learn, but a system that is satisfactory on all points has not yet been built. Enough experience through case studies has been obtained to assert that now time is ripe for building a satisfactory system.