Definition

Noun: A rigorous, logical demonstration that a mathematical statement (theorem, proposition, lemma, corollary) is necessarily true, based on a set of axioms and previously established theorems. It is a sequence of deductive arguments that establishes the truth of the conclusion beyond any doubt, given the accepted premises.

Usage

A mathematical proof is the definitive verification within mathematics. It is not merely evidence or a persuasive argument but a complete logical justification. - The validity of a conjecture is not accepted by mathematicians until a formal mathematical proof is provided. - Constructing a mathematical proof requires a clear understanding of definitions, axioms, and logical rules.

Examples

• The teacher asked the students to write a for the Pythagorean theorem.
• After decades of work, the mathematician finally published the of the famous conjecture.
• A single error in a can invalidate the entire argument.

Advanced Usage

• Proof by contradiction (reductio ad absurdum): A method where one assumes the statement to be proven is false and shows this assumption leads to a logical contradiction, thereby proving the original statement must be true.
• Proof by induction: A technique used to prove statements about all natural numbers by showing it holds for a base case (e.g., n=1) and that if it holds for an arbitrary number , it must also hold for .
• Constructive proof: A proof that not only demonstrates the existence of a mathematical object but also provides a method for creating or finding it.
• "The proof is left as an exercise for the reader": A common, sometimes humorous, phrase in textbooks indicating that the author expects the student to complete the proof themselves.

Variants and Related Words

• Prove (verb): To establish the truth of a statement by a mathematical proof. (e.g., )
• Proof theory (noun): A branch of mathematical logic that studies proofs as formal mathematical objects.
• Proof sketch (noun): An outline of the main ideas of a proof without the detailed, rigorous steps.

Synonyms

• Demonstration (in a mathematical context)
• Derivation (especially for formulas)
• Logical verification

Related Phrases and Idioms

• "Q.E.D." or "∎" (Quod Erat Demonstrandum): An abbreviation meaning "which was to be demonstrated," traditionally placed at the end of a mathematical proof to signify its completion.
• "Elegant proof": A proof that is particularly concise, insightful, or aesthetically pleasing.
• "The burden of proof": While a legal term, in mathematical discourse it emphasizes that the one making a claim (e.g., a conjecture) is responsible for providing the proof.

Noun

1. proof of a mathematical theorem