Mathematical proof
The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms,[2][3][4] along with the accepted rules of inference.
A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity.
The word proof derives from the Latin probare 'to test'; related words include English probe, probation, and probability, as well as Spanish probar 'to taste' (sometimes 'to touch' or 'to test'),[5] Italian provare 'to try', and German probieren 'to try'.
The legal term probity means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status.
[6] Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.
[7] It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, which originated in practical problems of land measurement.
Mathematical proof was revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today.
In the 10th century, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers.
[11] An inductive proof for arithmetic progressions was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.
As practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement.
To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected.
The mathematician Paul Erdős was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem.
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.
A variant of mathematical induction is proof by infinite descent, which can be used, for example, to prove the irrationality of the square root of two.
Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example.
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately.
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory.
In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates.
In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it.
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.
In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs.
One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry.
Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.
Some illusory visual proofs, such as the missing square puzzle, can be constructed in a way which appear to prove a supposed mathematical fact but only do so by neglecting tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated.
More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis.
A particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States.
Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new evidence or information is acquired.
Mathematician philosophers, such as Leibniz, Frege, and Carnap have variously criticized this view and attempted to develop a semantics for what they considered to be the language of thought, whereby standards of mathematical proof might be applied to empirical science.
