Proofs

In mathematics, a proof is a logically rigorous argument that establishes the truth of a proposition, assuming the underlying axioms and rules of inference. A proof shows that, starting from these axioms and from previously established results, every step follows necessarily and the conclusion cannot be false if the premises are true. Proofs aim for certainty within the chosen mathematical framework and distinguish themselves from conjectures or informal explanations.

Proofs are structured around statements such as theorems, lemmas, and corollaries. A proof consists of a sequence of justified steps, each derived from definitions, axioms, or earlier results. The goal is to make the reasoning transparent and free of gaps, so that a counterexample cannot arise. In practice, proofs are often presented for readers or researchers to verify, and they may be scrutinized for logical precision and completeness.

Common proof techniques include direct proofs, where the conclusion follows from a chain of implications; indirect proofs such as contrapositive or proof by contradiction; and induction, including demonstrations over natural numbers. Constructive proofs provide explicit objects or algorithms, while non-constructive proofs establish existence without construction. Some proofs are formal, written as symbolic derivations in a specified logical system; others are informal but still aim to communicate a valid argument clearly.

Proofs depend on a chosen set of axioms or a formal framework. They illustrate the universality of mathematical truth within that framework, though Gödel’s incompleteness theorem shows that no sufficiently powerful system can prove all true statements. Proofs are central to mathematics and differ from empirical demonstrations used in the sciences, which rely on evidence and models rather than universal deduction.