Proof

Mathematics depends on proof for its development, usefulness and veracity.

Proof is the foundation of mathematics, so proving is a foundation skill in school mathematics.

Sometimes, only one example is needed to prove something: a visitor to Australia from England only needs to see one black swan as proof that swans can be black as well as white. Students sometimes transfer this logic to mathematics, arguing that because something holds true for one example (or a number of examples), then it has been 'proved' to be true.

In mathematical proof, a mathematical axiom, which is a basic proposition, is assumed to be true. The axiom is used to show that a consequent mathematical statement will always be true. Proof is deductive in nature and 'if–then' statements are used in deductive reasoning.

An unproven proposition that is believed to be true is known as a conjecture. Proof adds the authority of clear, deductive, sequential argument that applies to all cases.