Proof — the essence of mathematics. Part I

Abstract

A case is made for restoring the idea of proof to a fundamental position in school mathematics, thereby giving a truer picture of what mathematics is really about, and closing the gap which exists between sixth form and early undergraduate versions of the subject. I attempt to achieve three objectives: to show that proofs of apparently obvious propositions are necessary and useful, and not just an indication of mathematicians' perversity; to make the ideas involved in proofs rather more natural by showing some apparently isolated ideas at work in a wide variety of contexts, so that ‘tricks’ become ‘standard techniques'; to emphasize the aesthetic element of mathematics by presenting a few examples of proofs which may incidentally be ‘useful’ or ‘important, but whose main value is that they are beautiful. Part I is devoted mainly to the first of these aims.