Two ways of looking at mathematics

It seems that in mathematics there are sometimes two or more ways of proving the same result. This is often mysterious, and seems to go against the grain, for we often have a deep-down feeling that if we choose the ‘right’ ideas or definitions, there must be only one ‘correct’ proof. This feeling that there should be just one way of looking at something is rather similar to Paul Erdos's idea of ‘The Book’ [1], a vast tome held by God, the SF, in which all the best, most revealing and perfect proofs are written.

Sometimes this mystery can be resolved by analysing the apparently different proofs into their fundamental ideas. It often turns out that, ‘underneath the bonnet’, there is actually just one key mathematical concept, and two seemingly different arguments are in some sense ‘the same’. But sometimes there really are two different approaches to a problem. This should not be disturbing, but should instead be seen as a great opportunity. After all, two approaches to the same idea indicates that there are some new mathematics to be investigated and some new connections to be found and exploited, which hopefully will uncover a wealth of new results.

I shall give a rather simple example of just the sort of situation I have in mind that will be familiar to many readers – one which will be typical of the kind of theorem we will be considering throughout this book.

Consider a binary tree. A tree is a diagram (often called a graph) with a special point or node called the root, and lines or edges leaving this node downwards to other nodes.