In mathematics, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. In contrast, an indirect proof begins with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. Mathematics is the study of quantity, structure, space and change. ... See the disimbiguation page for the term, true which points out that the derivative term, truth can mean, in simplest terms, A statement that is in accord with the actual state of affairs in any particular case. ... In mathematics, a lemma is a proven statement, typically named as such to distinguish it as a truth used as a stepping stone to a larger result rather than an important statement in and of itself. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language[?].
Proof by contradiction: where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false.
A probabilistic proof should mean a proof in which an example is shown to exist by methods of probability theory - not an argument that a theorem is 'probably' true.
Proof is the establishment of a disputed or controverted matter by lawful means or arguments.
Proof is perfect, or complete, when it produces full conviction, and enables the judge without further investigation to pronounce sentence: imperfect, or incomplete, if it begets probability only.
Two imperfect proofs cannot constitute perfect proof in criminal cases, in which proof must be clearer than the noonday sun; in matrimonial cases, when there is question of the validity of a marriage already contracted; or in civil actions of a grave character.
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