Preface

This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first draft of the notes was written during a terms sabbatical leave in 1980. Later they were used for a similar course at undergraduate level.

While there are now several textbooks on Artificial Intelligence techniques and, more particularly, on Problem Solving and Theorem Proving, I felt the need for a book concentrating on applications of these techniques to Mathematics. There was certainly enough material, but it was scattered in research journals, conference proceedings and theses. If it were collected together I hoped it might prove of interest to a wider audience than the usual artificial intelligentsia; I hoped that mathematicians and educationalists might find it a eye opener to how computational ideas could shed light on the process of doing Mathematics.

1.1 Why read this book?

This book is aimed at people interested in the question

How do you do Mathematics?

i.e. at professional mathematicians, students of Mathematics, teachers of Mathematics, psychologists studying mathematical reasoning and anyone else who is curious about the apparently mysterious processes by which mathematicians: conjecture theorems, formulate definitions, construct proofs and build mathematical models. My theme is that light can be shed on these `mysterious' processes with the aid of a wonderful tool { the digital computer. By building computer programs which `do' mathematics we can explore how it is possible to do mathematics; what the vital are talents that separate success from failure; and how we all can learn to be better mathematicians.

The building of computer programs for doing mathematics is part of the new science of Artificial Intelligence. The aim of Artificial Intelligence is to study all aspects of intelligence by `computational modelling', mathematical reasoning being just one such aspect of intelligence. Other aspects which are studied include: the ability to coordinate hand and eye to manipulate objects; the ability to hold a conversation in a so-called `natural' language like English (as opposed to an artificial programming language, like ALGOL or FORTRAN) or the ability to diagnose an illness and prescribe a cure.

Whatever aspect of intelligence you attempt to model in a computer program - the stacking of bricks, a cocktail party conversation or the proving of the compactness theorem - the same needs arise over and over again.

* The need to have knowledge about the domain
* The need to reason with that knowledge.
* The need for knowledge about how to direct or guide that reasoning.

Mathematical reasoning is a particularly convenient domain for studying intelligence, because the knowledge required is neatly circumscribed and the goals clear and unambiguous.