Mathematical proof

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In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true.[1][2] Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to be true is known as a conjecture.

The statement that is proved is often called a theorem.[1] Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.

Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, 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

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.[3] The development of mathematical proof is primarily the product of early Greek civilization. Thales (624–546 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. Euclid (300 BCE) began 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. 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).

The word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscis”, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),[4] 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.[5]

Nature and purpose

There are two different conceptions of mathematical proof.[6] The first is an informal proof, a rigorous natural-language expression that is intended to convince the audience of the truth of a theorem. Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof. In order to be considered a proof, however, the argument must be rigorous enough; a vague or incomplete argument is not a proof. Informal proofs are the type of proof typically encountered in published mathematics. They are sometimes called "formal proofs" because of their rigor, but logicians use the term "formal proof" to refer to a different type of proof entirely.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet. This allows the definition of a formal proof to be precisely specified without any ambiguity. The field of proof theory studies formal proofs and their properties. Although each informal proof can, in theory, be converted into a formal proof, this is rarely done in practice. The study of formal proofs is used to determine properties of provability in general, and to show that certain undecidable statements are not provable.

A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic-synthetic distinction, believed mathematical proofs are synthetic.

Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs he found particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.

Methods of proof

Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.[7] For example, direct proof can be used to establish that the sum of two even integers is always even:

Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x+y has 2 as a factor and therefore is even, so the sum of any two even integers is even.

This proof uses definition of even integers, as well as distribution law.

Proof by mathematical induction

In proof by mathematical induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases.[8] Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is infinite descent. Infinite descent can be used to prove the irrationality of the square root of two.

The principle of mathematical induction states that: Let N = { 1, 2, 3, 4, ... } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that

• (i) P(1) is true, i.e., P(n) is true for n = 1
• (ii) P(n + 1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n + 1) is true.

Then P(n) is true for all natural numbers n.

Mathematicians often use the term "proof by induction" as shorthand for a proof by mathematical induction.[9] However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning.

Proof by transposition

Proof by transposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".

Proof by contradiction

In proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction toward the absurd"), it is shown that if some statement were so, a logical contradiction occurs, hence the statement must be not so. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that is an Irrational number:

Suppose that is a rational number, so where a and b are non-zero integers with no common factor (definition of a rational number). Thus, . Squaring both sides yields 2b² = a². Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a² is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b² = (2c)² = 4c². Dividing both sides by 2 yields b² = 2c². But then, by the same argument as before, 2 divides b², so b must be even. However, if a and b are both even, they share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that is an irrational number.

Proof by construction

Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example.

Proof by exhaustion

In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.

Probabilistic proof

A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This is not to be confused with an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument' and is not a proof; in the case of the Collatz conjecture it is clear how far that is from a genuine proof.[10] Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.

Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.

Nonconstructive proof