A proof explains why a mathematical conclusion follows rather than merely showing that it works in selected examples. Its validity depends on the assumptions, definitions and logical steps used. A formal proof expresses those steps in a language suitable for precise checking.
A valid proof of one statement does not automatically establish a stronger or differently formulated claim. Its significance and the suitability of its assumptions remain separate questions. This distinction matters when evaluating model-generated mathematical results.
ELI5
Trying many examples can suggest that a rule is true. A proof explains why the rule must hold under its stated conditions, including cases nobody has individually tried.
For example, an even number of objects can be arranged in pairs. Combining two collections of pairs still leaves everything paired, explaining why adding two even numbers gives another even number.
Is checking many examples a proof?
Not generally. Examples can support a conjecture, but a proof must establish the stated claim under its assumptions.
Does a checked proof establish every related claim?
No. It establishes the exact statement proved, which may be narrower than another claim people associate with it.
