INTRODUCTION The attempt to mechanize mathematical reasoning belongs to the first experiments in artificial intelligence in the 1950 (Newell et al., 1957). However, the idea to automate or to support deduction turned out to be harder than originally expected. This can not at least be seen in the multitude of approaches that were pursued to model different aspects of mathematical reasoning. There are different dimension according to which these systems can be classified: input language (e.g., order-sorted first-order logic), calculus (e.g., resolution), interaction level (e.g., batch mode), proof output (e.g., refutation graph), and the purpose (e.g., automated theorem proving) as well as many more subtle points concerning the fine tuning of the proof search. In this contribution the proof planning approach will be presented. Since it is not the mainstream approach to mechanized reasoning, it seems to be worth to look at it in a more principled way and to contrast it to other appro

How can we understand reasoning in general and mathematical proofs in particular? It is argued that a high-level understanding of proofs is needed to complement the low-level understanding provided by Logic. A role for computation is proposed to provide this high-level understanding, namely by the association of proof plans with proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs We argue that Logic 1 is not enough to understand reasoning. It provides only a low-level, step by step understanding, whereas a high-level, strategic understanding is also required. Many commonly observed phenomena of reasoning cannot be explained without such a high-level understanding. Furthermore, automatic reasoning is impractical without a high-level understanding. We propose a science of reasoning which provides both a low- and a high-level understanding of reasoning. It combines Logic with the concept of proof plans, [Bundy, 1988]. We illustrate this with examples from mathematical reasoning, but it is intended that the science should eventually apply to all kinds of reasoning. 2 The Need for Higher-Level Explanations A proof in a logic is a partially ordered set of formulae where each formula in the set is either an axiom or is derived from earlier formulae in the set by a rule of inference. Each mathematical theory de nes what it means to be a formula, an axiom or a rule of inference. Thus Logic provides a low-level explanation of a mathematical proof. It explains the proof as a sequence of steps and shows how each step follows from previous ones by a set of rules. Its concerns are limited to the soundness of the proof, and to the truth of proposed conjectures in models of logical theories.

In proof planning, common patterns of mathematical reasoning are embodied in tactics. Each of these is associated with a method, specifying its preconditions and effects. Using methods, a plan for a proof can be constructed from tactics. This plan is then used to guide a theorem prover through the proof. We have implemented diagonalization, a common form of mathematical reasoning, in Clam, a proof planning system. The methods are based on a formalisation of diagonalization inspired by results of Cantor and Turing. Tested against several theorems these methods constructed plans for all of them. We have been successful in using these plans to generate informal, English proofs of the theorems. However, there is still scope to generalise the methods. Acknowledgements I would like to thank my supervisors Alan Bundy, Toby Walsh and Antony Maciocia for their invaluable guidance during this project. Thanks also to Manfred Kerber and Michael Kohlhase for their help. Thank you to my friends C...