<Table> <Tr> <Td> </Td> <Td> This article has an unclear citation style . The references used may be made clearer with a different or consistent style of citation and footnoting . (May 2018) (Learn how and when to remove this template message) </Td> </Tr> </Table> <Tr> <Td> </Td> <Td> This article has an unclear citation style . The references used may be made clearer with a different or consistent style of citation and footnoting . (May 2018) (Learn how and when to remove this template message) </Td> </Tr> <P> In mathematics, a proof is an inferential argument for a mathematical statement . 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 are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning (or "reasonable expectation"). 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 unproved proposition that is believed to be true is known as a conjecture . </P> <P> 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 . </P>

What do we call statements that we assume to be true without proofs