Mathematical proof

P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.[1]

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. 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 accepted rules of inference. Proofs are examples of exhaustive deductive reasoning or exhaustive inductive reasoning which establish logical certainty, and are distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Enumerating many confirmatory cases is not enough for a proof, which must demonstrate that the statement is always true (occasionally by listing all possible cases and showing that it holds in each). An unproven proposition that is believed to be true is known as a conjecture.

Proofs employ logic expressed in mathematical symbols, along with some amount of natural language which usually admits some ambiguity. 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.

History and etymology

The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", and "probability", the Spanish probar (to smell or taste, or (lesser use) touch or test),[5] Italian provare (to try), and the German probieren (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.[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 originally meant the same as "land measurement".[8] The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) proved some theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true from the Greek "axios" meaning "something worthy"), and used these to prove theorems using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century.[9] In addition to theorems of geometry, such as the Pythagorean theorem, the Elements also covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers.

Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for "lines." He used this method to provide a proof of the existence of irrational numbers.[10] An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.[11]

Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see Axiomatic set theory and Non-Euclidean geometry for examples).