Cauchy's Defence of the ‘Principle of Continuity’

The method of proofs and refutations is a very general heuristic pattern of mathematical discovery. However, it seems that it was discovered only in the 1840s and even today seems paradoxical to many people; and certainly it is nowhere properly acknowledged. In this appendix I shall try to sketch the story of a proof-analysis in mathematical analysis and to trace the sources of resistance to the understanding and recognition of it. I first repeat the skeleton of the method of proofs and refutations, a method which I have already illustrated by my case-study of the Cauchy proof of the Descartes–Euler conjecture.

There is a simple pattern of mathematical discovery – or of the growth of informal mathematical theories. It consists of the following stages:

(1) Primitive conjecture.

(2) Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures or lemmas).

(3) ‘Global’ counterexamples (counterexamples to the primitive conjecture) emerge.

(4) Proof re-examined: the ‘guilty lemma’ to which the global counterexample is a ‘local’ counterexample is spotted. This guilty lemma may have previously remained ‘hidden’ or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem – the improved conjecture – supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.