Overview
A mathematical proof is a sequence of reasoning that establishes the truth of a mathematical statement from accepted premises. A proof shows that a claim holds in every case asserted, given a specified framework. Proofs connect assumptions, such as definitions and axioms, with previously proved results to justify new conclusions. The target claim is often called a theorem, proposition, or lemma; the objects and relations involved are part of a mathematical object or theory.
Structure and typical elements
Formal proofs consist of premises, intermediate steps, and a conclusion. Mathematicians use stated definitions to avoid ambiguity and may introduce lemmas or corollaries to organize long arguments. A rigorous proof must make clear how each step follows: by a logical rule, an axiom, or a previously established result. Informal proofs written in textbooks aim to convey the main ideas while omitting routine formalities.
Common methods
- Direct proof: deducing the conclusion from the premises by a chain of logical implications.
- Proof by contrapositive: proving an equivalent statement of the form "if not B then not A."
- Proof by contradiction: assuming the negation and deriving an impossibility.
- Mathematical induction: establishing a base case and an inductive step for statements about integers.
- Case analysis, constructive proofs, combinatorial and probabilistic methods, and geometric constructions.
History and development
Proof has been central to mathematics since ancient times: Euclid's Elements gave early systematic axiomatic proofs. Over centuries the axiomatic method and standards of rigor evolved, with modern formal logic and set theory providing frameworks for precise formulations. In the 20th century work by mathematicians and logicians clarified what can be proved within a given system and exposed limits, for example through results about undecidability and independence.
Uses, examples, and importance
Proofs serve to verify results, build coherent theories, and transfer reliable knowledge to applications in science and engineering. Famous examples include elementary proofs in number theory, geometric constructions, and large-scale results such as the classification of finite simple groups. Some proofs use computers to check many cases or to verify intricate calculations.
Distinctions and notable facts
Proof differs from empirical evidence: a proof guarantees truth within its assumptions, while experiments provide supporting data. There is also a distinction between formal proofs, which can be written in a symbolic logical system, and informal proofs that rely on human-readable explanations. Some mathematical statements are independent of chosen axioms and thus neither provable nor disprovable within those axioms. Computer-assisted proofs and proof verification systems have become increasingly important tools in modern practice.