Some methods of Mathematical Proof

Abstract

The word "proof" comes from the Latin probare meaning "to test". In mathematics, a proof is an inferential argument for a mathematical statement . Proofs remain important in mathematics because they are our bellwether for what we can believe in, and what we can depend on. They are timeless and rigid and dependable. They are what hold the subject together, and what make it one of the glories of human thought. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs make use of logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice in mathematics. The concept of a proof is formalized in the field of mathematical logic. In this study we first discuss, The role of some methods of Mathematical Proof of theorems of mathematics. For this two main methods are formal proof and other one is direct proof. A formal proof is written in a formal language instead of a natural language. In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. Also aim of this study is to see impact of methods, namely: Proof by mathematical induction, Proof by contraposition, Proof by contradiction, Proof by construction, Proof by exhaustion, Probabilistic proof, combinatorial proof, Nonconstructive proof, Undecidable statements, Elementary proof of some mathematical results.