**Mathematical Proof**

In mathematics, a **proof** is a demonstration that if some fundamental statements (axioms) are assumed to be true, then some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing *all* possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproven proposition that is believed to be true is known as a conjecture.

Proofs employ 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, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Read more about Mathematical Proof: History and Etymology, Nature and Purpose, Undecidable Statements, Heuristic Mathematics and Experimental Mathematics, Ending A Proof

### Famous quotes containing the words mathematical and/or proof:

“An accurate charting of the American woman’s progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of time—but like a *mathematical* curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.”

—Susan Faludi (20th century)

“O, popular applause! what heart of man

Is *proof* against thy sweet, seducing charms?”

—William Cowper (1731–1800)