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  • Print publication year: 2009
  • Online publication date: September 2009

6 - Interactive theorem proving

Summary

Our efforts so far have been aimed at making the computer prove theorems completely automatically. But the scope of fully automatic methods, subject to any remotely realistic limitations on computing power, covers only a very small part of present-day mathematics. Here we develop an alternative: an interactive proof assistant that can help to precisely state and formalize a proof, while still dealing with some boring details automatically. Moreover, to ensure its reliability, we design the proof assistant based on a very simple logical kernel.

Human-oriented methods

We've devoted quite a lot of energy to making computers prove statements completely automatically. The methods we've implemented are fairly powerful and can do some kinds of proofs better than (most) people. Still, the enormously complicated chains of logical reasoning in many fields of mathematics are seldom likely to be discovered in a reasonable amount of time by systematic algorithms like those we've presented. In practice, human mathematicians find these chains of reasoning using a mixture of intuition, experimentation with specific instances, analogy with or extrapolation from related results, dramatic generalization of the context (e.g. the use of complexanalytic methods in number theory) and of course pure luck – see Lakatos (1976), Polya (1954) and Schoenfeld (1985) for varied attempts to subject the process of mathematical discovery to methodological analysis. It's probably true to say that very few human mathematicians approach the task of proving theorems with methods like those we have developed.

One natural reaction to the limitations of systematic algorithmic methods is to try to design computer programs that reason in a more human-like style.