Proof is the essence of mathematics. It's what distinguishes mathematics from all other disciplines. The process is one in which all rules are adhered to very strictly, there is no grey area. Unfortunately there is an everyday use of the word which is far from strict and this often causes difficulties for mathematicians.

In other disciplines there are established principles of proof: a Lawyer's interpretation, the Historian's use, the Scientist's use and the Mathematician's proof.

Both proof and refutation are important pieces of kit in the mathematician's tool-box.

SECONDARY: what is proof? Difference between proof in everyday language and in mathematics.

Theorems in geometry and algebra. Counter-examples and paradoxes.

SIXTH FORM: plus proofs by exhaustion, deduction, contradiction and induction. Examples

of historically important proofs, some unresolved conjectures and the Millennium problems.