17 Proof
Mathematicians attempt to justify their claims by proofs. The quest for cast iron rational
arguments is the driving force of pure mathematics. Chains of correct deduction from
what is known or assumed, lead the mathematician to a conclusion which then enters
the established mathematical storehouse.
Proofs are not arrived at easily – they often come at the end of a great deal of
exploration and false trails. The struggle to provide them occupies the centre
ground of the mathematician’s life. A successful proof carries the mathematician’s
stamp of authenticity, separating the established theorem from the conjecture,
bright idea or first guess.
Qualities looked for in a proof are rigour, transparency and, not least,
elegance. To this add insight. A good proof is ‘one that makes us wiser’ – but it
is also better to have some proof than no proof at all. Progression on the basis of
unproven facts carries the danger that theories may be built on the mathematical
equivalent of sand.
Not that a proof lasts forever, for it may have to be revised in the light of
developments in the concepts it relates to.
What is a proof?
When you read or hear about a mathematical result do you believe it? What
would make you believe it? One answer would be a logically sound argument
that progresses from ideas you accept to the statement you are wondering about.
That would be what mathematicians call a proof, in its usual form a mixture of
everyday language and strict logic. Depending on the quality of the proof you are
either convinced or remain sceptical.
The main kinds of proof employed in mathematics are: the method of the
counterexample; the direct method; the indirect method; and the method of
mathematical induction.