The notion of proof is very much at the heart of mathematics. It is rare for mathematical results to emerge or be discovered in the polished form in which they are ultimately presented in lectures and text books. Mathematicians often have an intuitive feeling about the truth or falsity of some statement. Nevertheless, a rigorous proof based on a logically sound argument is necessary in order to convince others of the result, and even for the originator to be certain that no error has been made; intuition can easily lead the unwary astray. ‘‘Results’’ which appear obvious, but are difficult to establish rigorously, have a nasty habit of turning out to be false.
It seems instinctively true that every surface in three-dimensional space (such as a sheet of paper) has two sides, namely a top and a bottom. However, if you take a narrow ribbon of paper, give it a twist through and join together the ends, then you have a surface known as a Möbius band, which has only one side. Try it, and trace your finger around its surface.
Having seen this example, we realize that the original statement is false, but without it one could perhaps be forgiven for accepting its truth.
When considering the notion of proof, one can identify two separate issues:
the philosophical question of whether there is an absolute notion of correct mathematical proof;
the practical issue of assessing whether a claimed proof is valid.
Mathematicians are usually more interested in (ii), which is a matter of interpreting published work. Most published items are generally accepted as being correct, and hence form a basis of prior knowledge from which to make progress, but there are sometimes controversies as to whether a claimed proof is valid. Sometimes published proofs contain minor errors that the reader can repair, and that do not make the proof invalid; in other cases, the proof can be obscurely written or hard to check.
A question which is more than 400 years old asks: given a large (infinite) supply of spherical balls of equal radius and a large (infinite) rectangular box, what is the best way of packing the balls into the box to fit in as many as possible?
Intuitively, the answer should be to arrange the bottom layer in a regular hexagonal array with each ball touching six neighbours. Then create a layer of balls on top in another regular hexagonal array, so that the balls fill in the gaps between the balls in the bottom layer. Then continue this construction and create further layers. The statement that this is indeed the optimal solution is known as Kepler’s sphere-packing conjecture (Kepler, 1611).
In the late 1990’s, Hales and Ferguson established the truth of this conjecture. Their proof, however, involved complex computer calculations which are very difficult to verify, and so its correctness was not universally accepted at the time. Hales and a large number of collaborators have subsequently carried out substantial work to overcome these issues.
The mathematician should be very wary of any assertion which has not firmly and thoroughly been established by a robust argument, or proof. Proofs can take a number of forms; in the following sections we shall consider the most important ones.
We use the symbol ‘‘’’ to mark the end of a proof. Other symbols may also be encountered in the literature, for instance ‘‘’’ or ‘‘QED’’. (The latter appears mainly in older texts; it is an abbreviation of the Latin phrase ‘‘quod erat demonstrandum’’ which means ‘‘which was to be demonstrated’’.)