Mathematical Proof

A mathematical proof explains why a statement must be true, rather than merely showing that it works in some examples. It is a logically valid argument that derives the statement from accepted axioms, definitions and previously established results. Empirical confirmation through selected examples supports a conjecture but does not, by itself, establish a claim about every possible case.

Why proof establishes a general result

The decisive distinction is deductive reasoning: each step follows logically from the assumptions or earlier steps. A proof of a universal statement must cover every case allowed by its conditions, not just those examined individually.

• Axioms are accepted starting points, while theorems are statements established by proof within the mathematical framework.
• For example, any two even integers can be written as 2a and 2b, where a and b are integers. Their sum is 2(a + b), so it is even.
• This argument proves the result for all even integers because a and b are arbitrary, rather than particular numbers selected for testing.

What examples can and cannot establish

Examples help reveal patterns, suggest conjectures and expose mistakes. However, agreement across selected cases does not exclude the possibility of failure in an untested case.

• There is an important asymmetry: many supporting examples cannot prove an unrestricted universal claim, but one counterexample can disprove it.
• For example, the statement that every prime number is odd is disproved by 2, which is prime and even.
• Checking every member of a genuinely finite domain can establish a claim about that domain, provided the check is correct and demonstrably exhaustive. This is different from selecting representative examples.

Different routes to proof

Proof need not always take the same form, but every valid method must justify the conclusion rather than merely make it plausible.

• A direct proof derives the conclusion from the given conditions, as in the even-integer example.
• A proof by contradiction assumes the proposed conclusion is false and shows that this assumption leads to a logical contradiction.
• Numerical testing, whether performed by hand or by computer, remains example-based evidence unless it forms part of an argument establishing complete coverage of the claim.