What distinguishes mathematics from the other sciences is the concept of proof. The mathematician can prove with 100% certainty that the primes will never run out. Whilst in other sciences old ideas are thrown away to be replaced by new theories, mathematical ideas remain true for all time. In no other scientific subject are we still teaching the discoveries of the Ancient Greeks. The Greeks believed that matter was made from earth, wind rain and fire. That view of chemistry seems absurd today. But their discoveries about primes are still as true today as they ever were.

A proof is like a mathematical story or symphony. As we read or listen to a proof, the ideas and themes gradual change, starting from a place where we feel familiar, until we find ourselves transported at the end of the piece into a new and exciting world.

Some people describe a proof like a journey. In the distance is some far distant mountain. Somewhere we would like to reach. A conjecture like the Riemann Hypothesis about prime numbers is like some huge Everest in the mathematical landscape. A proof is a path from the gentle pastures where we are happy and understand the twists and turns of the mathematical world into the unknown jungle. Our task is to find a way through to the top of the far distant mountain.

Some have compared constructing a proof to a game of chess. Each step in the proof is like a move of a chess piece. There are very strict rules about how each piece is allowed to move. In mathematics too there are strict rules about what you are allowed to do at each step. Each move in the mathematical game must be logical. Some say the constraints of logic limit the creativity of mathematics. But this is not true. It is a constraining as composing music within a twelve note scale or writing a sonnet. Quite often the constraints of the poetic form force the poet into unexpected directions that the freedom of prose might not have revealed. So too the mathematician finds the constraint of logic push him or her into surprising places.

Proof is so valued by the mathematician because evidence in mathematics can be very misleading. You might have checked a conjecture for a million numbers. But that does not mean that it won’t break-down for the next number you choose. Without proof, don’t believe anything in the mathematical world. Nature can be quite mischievous when it comes to the universe of numbers.