Mathematical proof

In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true.

Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In the context of proof theory, where purely formal proofs are considered, such not entirely formal demonstrations are called "social proofs". The distinction 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: where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false.
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.

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 tell you how such an X will be obtained. A constructive proof, conversely, will do so.

A statement which is thought to be true but hasn't 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.