Mathematical proof

In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. The assumed axioms are ZFC unless indicated otherwise.

Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. Purely formal proofs 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.

Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone. Once a theorem is proved, it can be used as the basis to prove further statements. The so-called foundations of mathematics are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques.

Some common proof techniques are:
 * Direct proof: where the conclusion is established by logically combining the axioms, definitions and earlier theorems
 * Proof by induction: where a base case is proved, and an induction rule used to prove an (often infinite) series of other cases
 * Proof by contradiction (also known as reductio ad absurdum): where it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true.
 * Proof by construction: constructing a concrete example with a property to show that something having that property exists.
 * Proof by exhaustion: where the conclusion is established by dividing it into a finite number of cases and proving each one separately

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. The latter type of reasoning can be called a 'plausibility argument'; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. Probabilistic proof is one of many ways to show existence theorems, other than proof by construction.

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Usually a one-to-one correspondence is used to show that the two interpretations give the same result.

If we are trying to prove, for example, "Some X satisfies f(X)", an existence or nonconstructive proof will prove that there is a X that satisfies f(X), but does not explain how such an X will be obtained. A constructive proof, conversely, will do so.

A statement which is thought to be true but has not been proven yet is known as a conjecture.

Sometimes it is possible to prove that a certain statement cannot possibly be proven from a given set of axioms; see for instance the continuum hypothesis. In most axiom systems, there are statements which can neither be proven nor disproven; see Gödel's incompleteness theorem.